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iii

Table of Contents

Preface - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - viVersion 15.0 New Features - - - - - - - - - - - - - - - - - - viii

1. Introduction - - - - - - - - - - - - - - - - - - - - - - - - - - 1.12. Theoretical Overview - - - - - - - - - - - - - - - - - - - - 2.1

2.1 Problem Definition - - - - - - - - - - - - - - - - - - - 2.12.1.1 Static Response - - - - - - - - - - - - - - - - - - - - - 2.12.1.2 Dynamic Response - - - - - - - - - - - - - - - - - - - 2.2

2.2 Cost Function - - - - - - - - - - - - - - - - - - - - - - - 2.42.3 Constraint Functions - - - - - - - - - - - - - - - - - - 2.5

2.3.1 Displacement Constraints - - - - - - - - - - - - - - 2.52.3.2 Stress Constraints - - - - - - - - - - - - - - - - - - - - 2.52.3.3 Natural Frequency Constraints - - - - - - - - - - - 2.62.3.4 Buckling Load Factor Constraints - - - - - - - - - 2.62.3.5 Amplitude Constraints Under Harmonic

Excitation - - - - - - - - - - - - - - - - - - - - - - - - - 2.72.4 Design Variables - - - - - - - - - - - - - - - - - - - - - 2.8

2.4.1 Definition of Design Variables - - - - - - - - - - - 2.82.4.2 Local and Global Design Variables - - - - - - - - 2.82.4.3 Design Variable Linking - - - - - - - - - - - - - - - 2.9

2.5 Design Sensitivity Analysis - - - - - - - - - - - - 2.122.5.1 Formulation of Design Sensitivity Analysis - -2.122.5.2 Mathematical Representation of Sensitivity

Coefficients - - - - - - - - - - - - - - - - - - - - - - - -2.143. Optimization Algorithms - - - - - - - - - - - - - - - - - 3.1

3.1 Generalized Reduced Gradient Method (GRG) 3.23.2 Recursive Quadratic Programming Method (RQP) 3.23.3 Optimum Cost Bounding Method (OCB) - - - - 3.3

4. Capabilities and Features of STROPT - - - - - - - 4.14.1 Finite Element Library - - - - - - - - - - - - - - - - - 4.1

iv

4.2 Section Library - - - - - - - - - - - - - - - - - - - - - - 4.34.3 Stress Library - - - - - - - - - - - - - - - - - - - - - - 4.104.4 Program Architecture - - - - - - - - - - - - - - - - 4.244.5 Forms in STROPT - - - - - - - - - - - - - - - - - - 4.254.6 Output Features and Postprocessing - - - - - - - 4.26

4.6.1 General Description of the Output - - - - - - - - -4.264.6.2 Post Processing of the Results - - - - - - - - - - -4.28

5. Input Description for STROPT - - - - - - - - - - - - 5.15.1 Overview of Input Data - - - - - - - - - - - - - - - - 5.15.2 Supplement to Modified Executive Commands in

NISA - - - - - - - - - - - - - - - - - - - - - - - - - - - - 5.55.2.1 Executive Commands - - - - - - - - - - - - - - - - - 5.5

5.3 Description of Optimization Input Data - - - - - 5.95.3.1 *OPTPAR Data Group - - - - - - - - - - - - - - - - 5.95.3.2 *DVGROUP Data Group - - - - - - - - - - - - - -5.135.3.3 *DVGELMT Data Group - - - - - - - - - - - - - -5.175.3.4 *DVLOCAL Data Group - - - - - - - - - - - - - -5.21

5.4 Supplement to Modified Data Block In NISA 5.335.4.1 PRINTCNTL Data Group - Selective Printout

Control for Sensitivity Coefficients - - - - - - - - -5.33A. Example Problems - - - - - - - - - - - - - - - - - - - - - A.1

A.1 Optimum Design and Design Sensitivity Analysis of a Deep Beam Optimum Design of a Deep Beam - - - A.1A.2 Optimum Design of a Simply Supported Beam A.16A.3 Optimum Design of a Four-Bar Truss - - - - - A.24A.4 Optimum Design of a Cantilever Plate - - - - - A.33A.5 Optimum Design of a Transmission Tower - A.50A.6 Optimum Design of a Geodesic Dome - - - - A.64A.7 Optimum Design of a Piston - - - - - - - - - - - A.80A.8 Optimum Design of a Composite Plate - - - - A.89A.9 Optimum Design of a Sandwich Plate - - - - A.105

v

A.10 Optimum Design of a Circular Plate - - - A.116A.11 Optimum Design of a Hemispherical Dome A.123A.12 Optimum Design of a Water Tank - - - - - A.132A.13 Optimum Design and Design Sensitivity

Analysis of a Swept Wing - - - - - - - - - - A.140A.14 Optimum Design of a Control Arm - - - - A.163A.15 Optimum Design of a Pressure Vessel - - - A.171A.16 Optimum Design of a Simply Supported Beam

with a Generalized Thin Walled Section - A.180A.17 Optimum Design of a Two-Bay, Two-Story

Frame - - - - - - - - - - - - - - - - - - - - - - - - A.191A.18 Dynamic Response Optimization of a Simply

Supported Beam - - - - - - - - - - - - - - - - - A.202A.19 Dynamic Response Optimization of a Simply

Supported Beam Under a Sinusoidal Pressure Load - - - - - - - - - - - - - - - - - - - - - - - - - A.211

A.20 Optimum Design of a Passenger-Compartment-Structure - - - - - - - - - - - - - - - - - - - - - - A.220

A.21 Optimum Design of a Cantilever Plate with Buckling Constraint - - - - - - - - - - - - - - A.241

A.22 Optimum Design of a Four-Bar Truss with Buckling, Stress, and Displacement Constraints - - - - - - - - - - - - - - - - - - - - A.250

B. Comparison of Design Sensitivity Coefficients - B.1B.1 Comparison of Design Sensitivity Coefficients

Between Exact Solution and Numerical Solution using STROPT - - - - - - - - - - - - - - - - - - - - - B.1

C. Output Format for Example Problem A.1 Using Optimum Cost Bounding Algorithm - - - - - - - - C.1

vi

D. Important Hints for Using STROPT - - - - - - - - D.1E. PCG Iterative Solver in STROPT - - - - - - - - - - E.1

vii

Preface

You have with you the latest version of NISA - A standard in Finite ElementAnalysis. NISA is one of the few, commercially available, proven and robustFinite Element Analysis software that has enjoyed a long-standing presence inthe arena of engineering analysis and design. Today it is the result of more thanthree decades of innovation and dedication of highly skilled scientists, technol-ogy architects and software engineers. As a result, generations of scientists,engineers and researchers have come to depend on NISA to solve their mostcomplex engineering problems.

NISA which has a heritage of more than 30 years, changed hands from EMRC,to Cranes Software, Inc. in July of 2005. Cranes Software, Inc is a whollyowned subsidiary of Cranes Software International Limited - a global softwareproducts and solutions provider. With this change comes an induction of freshtalent and resources which is poised to take NISA to a new level in the world ofFEA. NISA Version 15.0 is the achievement of a great development teamwhich has worked rigorously for the past year in accordance to the best in breedsoftware development life cycle management practices. As manuals are veryimportant to us, a lot of thought has gone into the design and content of themanuals. These have been totally revamped as per the new features and look ofthe product. New additions to the various modules have been consolidated andpresented in an integrated manner.

The NISA Shell is now more appropriately called Application Launcher. Thenew application launcher makes selecting modules, input/ output files, analysistype and various CAD/FEA translators a simple task. Significant updates anderror correction have been made to individual modules of the NISA suite ofprograms.

Version 15.0 New Features

NISA II: Improved iterative and sparse matrix equation solvers; end release forpipe and elbow element; General spring element (NKTP=38) upgraded to ageneral spring and damper element; facility to input user element stiffness,mass, and damping matrices; rigid link forces output for linear transientdynamic analysis; ability to post process larger problems involving multipleload cases.

ADVANCED DYNAMICS: Multiple Support Excitation, for shock spectrumanalysis to handle non-uniform support excitations. This feature also includesseven modal summation rules namely ABS, SRSS, CQC, Grouping Method,Ten percent Method, Double Sum Method, and with & without Missing Mass;Centroidal Stresses, Stress Resultants at Element Centroid and Base Shearcomputation for Shock Spectrum Analysis.

DYSPAN: Spectrum compatible Power Spectral Density generation.

ENDURE: Improved functionality of features in Shell; expansion of crackpropagation configurations; Automatic identification of EDI path; Automaticmesh generation for plate, pipe, and elbow with different types of cracks; andfatigue initiation theory based on the MANSON approach.

DISPLAY III: Pre & post processing support new features of NISA modules.In addition to this, general features are: Viewing the input function such as timeamplitude, spectra etc. as a graph; Realistic plot for 3-D General Beam, 3-DStraight pipe, 3-D Elbow elements (NKTP=48); Automatic selection of Masterand Slave node for Rigid links; Post processing of Non-linear Beam element(NKTP 39) stresses; Post processing of 3-D beam results such as - filtering ofthe results and report generation, reporting maximum stresses and ASME ratiosat critical points across section; Option to plot the XY points in the graph with-

viii

out the connecting lines; Area under the curve with respect to X-Axis; Historyplot for nonlinear results for external results; Reading/ Writing of MultipleExternal results for linear and nonlinear static analyses; Viewing results forcomplete model using Symmetry for external results; Variation of stress triaxi-ality along a line of nodes; Crack mouth opening displacement graph; Crackopening area calculation; facility to integrate fluid quantities such as pressure,temperature on surface.

DISPLAY IV: Pre & post processing support new features of NISA modules.Key additional features in DISPLAY IV are: Enhancement of Dialog boxes,accelerator keys. XP style file open dialog; enhancement of Entity-Status viewin the workspace to facilitate the deletion of a single entity or the entire groupusing mouse right click or by using ‘Delete’ key on the keyboard; a wizard thathelps navigate analysis data.

Up-gradation of translators such as SAT2NEU translator to support ACIS R16geometry kernel; Solid Works™ translator S/W 2007; geometric import fromIGES 5.3 version.

EMAG: 3D-Magnetodynamic analysis (harmonics) and 3D Transient Mag-netic analysis capabilities using magnetic vector potential and electric scalarpotential (with and without massive conductors).

NISA/HEAT3 and 3DFLUID: Heat Flux computation and Printing of HeatBalance Sheet, Sparse Matrix Equation Solver has been implemented; printingof Local Reynolds and Peclet numbers.

NISA/CIVIL: Revised code of practice conforming to ACI 318 -2005, BS8110-1997, BS 5950-1:2000 and LRFD 2002 for concrete and steel designs,Module to design pipes conforming to ASME-NB, NC & ND codes is now partof NISA/CIVIL. More emphasis is given to produce good structural designdrawings of RCC slabs, Beams, Columns, Footings, & structural steel draw-ings, Inclusion of concrete and steel quantities in structural drawings, custom-ized design report generation in ASCII, MS-Word and MS-Excel formats,stress resultants, contours for shell elements, reinforcement contouring for shellelements, combined isolated footings for expansion joint columns, improvedrealistic plots, standard animation feature for all Stress Resultants and EigenModes, animation of Color Contours.

ix

Chapter

1
Introduction
STROPT is a general purpose structural optimization program integrated with thefinite element analysis package NISA II. It has been developed by us withemphases on ease of data preparation, comprehensive data checking facilities,powerful finite element library, state-of-the-art optimization techniques withefficient design sensitivity computation.

The optimization process involves appropriately re-sizing the design variablesthat describe the structural system, while simultaneously satisfying the prescribedlimits on structural response, in order to find an optimum design with theminimum possible material volume/mass/weight. Prescribed limits on theresponse may refer to the nodal displacements and element stresses under staticloading, eigenvalues for free vibration and stability, as well as the maximumresponse (amplitude) under harmonic loading. In addition, lower and upperbounds on design variables can also be placed. STROPT offers the followingoutstanding features.

Efficient Design Sensitivity and Optimization Techniques

The capability to calculate design derivatives of structural response has beenimplemented by making use of existing features of NISA II with efficient designsensitivity analysis techniques and database management schemes. STROPToffers various optimization algorithms which employ recent advances in nonlinearprogramming techniques and are modified for use in structural optimization.

1-1

Extensive and Powerful Finite Element Library

NISA II features an extensive and powerful library of finite elements. As a result ofthis, various types of finite elements have been implemented, which gives adistinctive versatility to the program STROPT.

Design Variable Linking and Fixing

The design variables can be grouped and linked in many ways. It is also possible tofix them in certain regions of the structural system. This facility is especially usefulfor uniformity of design in certain regions of the structure, or for creating asymmetrical design under unsymmetrical loading conditions, and satisfying certaindesign specifications.

Multiple Load Cases and Boundary Conditions

Optimum design for multiple load cases and associated boundary condition isimportant in practical design environment. This capability can be handled in asingle optimization run.

A Variety of Loading Types

The loads may consist of concentrated nodal forces, displacements, pressure,temperature, centrifugal forces and accelerations.

Easy to Use Free Format Input

Input specification is designed to ease problem definition by the user. All the inputdata are in free format and preceded by NISA II input data.

Automatic Data Generation Features

To reduce the bulk of data, automatic data generation is available in geometrydescription, loading and boundary conditions, design variable groups, andconstraints.

Extensive Data Checking Prior To and During Optimization

A comprehensive set of data checks, ranging from the most trivial format errors tothe more complicated optimization input errors can be performed prior to

1-2

Introduction

optimization. During optimization, the magnitudes of the round-off error and thepivot are automatically checked to detect any inadequacy and/or instability of thestructural model.

Comprehensive Output Options

Various output options are offered to meet individual requirements. Results at theinitial and optimum (final) design can be saved for postprocessing. In addition, allthe sorted input data are printed out.

Availability Of STROPT

STROPT has been designed to be machine independent and can be implemented onmost computers with minimum effort. STROPT has already been successfully andefficiently implemented on various computers including PC, micro, supermini,mainframe and supercomputers.

STROPT is being expanded continuously and updated to reflect the latest advancesin structural optimization techniques.

1-3

Chapter

2
Theoretical Overview
2.1 Problem Definition

2.1.1 Static Response

A mathematical model for optimal design of linear elastic structural systemsunder static loads is defined as follows:

Find a design variable vector that minimizes the cost function

F(X) = Volume or mass or weight of the structure, (2.1-1)

and satisfies the state equations as well as the constraints stated as follows:

(2.1-2)

(2.1-3)

(2.1-4)

(2.1-5)

X ε Rn

K X( )U P X( )=

K X( )Y λM X( )Y K X( )Y λKG X( )Y=,=

Gi X U λ, ,( ) 0≥ i = 1,2,......m

XiU Xi Xi

L≥ ≥ ) i = 1,2,......n

2-1

Problem Definition

Where

2.1.2 Dynamic Response

A mathematical model for optimal design of linear elastic structural systems underdynamic loads is defined as follows:

Find a design variable vector that minimizes the cost function given inEquation (2.1-1) and satisfies the dynamic equation of motion as

(2.1-6)

K(X) = An LxL symmetric nonsingular linear structural stiffness matrix.

P(X) = An L-vector of equivalent nodal loads for the finite element model.

U = An L-vector of nodal displacements for the finite element model.

L = Number of degrees of freedom for the finite element model.

M(X) = An LxL structural mass matrix.

KG (X) = An LxL geometric stiffness matrix.

= An eigenvalue related to the natural frequency or the buckling load factor.

Y = An eigenvector related to the natural frequency or the buckling load factor.

= Constraints on displacements, and/or stresses, and/or natural frequencies and/or buckling load factors.

= Upper and lower bounds on design variables.

λ

Gi X U λ, ,( )

XiU and Xi

L

XεRn

t)P(X,K(X)UUC(X)UM(X) =++•••

2-2


with initial conditions

(2.1-7)

and subject to the constraints on structural response of point wise and functionalform

Additional notations defined here are as follows:

i = 1,2,....., m , and t [0,T] (2.1-8)

j = 1,2,.......,p (2.1-9)

i = 1,2,.......,n (2.1-10)

= An L-vector of nodal accelerations for the finite element model

= An L-vector of nodal velocities for the finite element model

C = An L x L structural damping matrix

t = Time

=Constraint functionals which may be used to represent a cumulative measure of performance over the time interval [0,T]

00 U)0(U,UU(0)••

==

Gi X U t, ,( ) 0≥ ε

Hj X U t, ,( ) td 0≥0T

∫

XiU Xi Xi

L≥ ≥

••U

•U

Hj X U t, ,( )0T

∫

2-3

Cost Function

2.2 Cost Function

In the present formulation of STROPT, the material volume or mass or weight ofthe structure is minimized. The cost function is given as

(2.2-1)

Where Vi, Mi, and Wi are the volume, the mass, and the weight, respectively, of theith element in the structure and NEL is the total number of elements in thestructure.

X( ) Vii 1=

NEL∑ or= Mi

i 1=

NEL∑ or Wi

i 1=

NEL∑

2-4


2.3 Constraint Functions

2.3.1 Displacement Constraints

Allowable limits on nodal displacements may be imposed as follows:

(2.3-1)

where Ui is the displacement (translation or rotation) of the ith degree of freedomand Ui* (> 0) is its allowable limit. Equation (2.3-1) may be normalized as:

(2.3-2)

2.3.2 Stress Constraints

STROPT offers wide range of options in imposing stress constraints. A glance atStress Library given in Section 4.3 shows them. Additional options can beincorporated easily due to modularity of the program. All stress constraints arenormalized and are of the form

(2.3-3)

where:

All stress constraints are imposed on element stress values. This means that theuser may choose Gauss points, node points or centroid of an element for imposing

= Absolute value of stress from structural analysis, and

= Allowable stress value (always positive) specified by the user.

ii UU* ≥

0U

U1 *

i

i ≥−

01 ≥σ

σ−

∗

σ

σ∗

2-5

Constraint Functions

stress constraints. However, stress constraints for 3D general beam and 3D sparelements are defined differently and are discussed in Stress Library.

2.3.3 Natural Frequency Constraints

Natural frequencies for the structure are obtained by solving eigenvalue problem ofEquation (2.1-3). Eigenvalues of the structure are related to the natural frequenciesas follows:

(2.3-4)

where and fi are the ith eigenvalue and the ith natural frequency for the

structure. If and are the lower and upper bounds on fi then the two naturalfrequency constraints can be written as:

(2.3-5)

The normalized frequency constraints are expressed as follows:

(2.3-6)

The lower and upper bounds on natural frequencies are specified in cycles/sec(Hz).

2.3.4 Buckling Load Factor Constraints

Buckling load factor for the structure is obtained by solving the eigenvalueproblem of Equation (2.1-3). Since, generally, the buckling load factorcorresponding to the first mode shape is of practical importance, STROPT allowsusers to impose a constraint on the lowest buckling load factor for only one loadcase. The normalized constraint on buckling load factor can be expressed as:

λi 2πfi( )2=

λi

fiL fi

U

fi fiL and fi

U fi≥≥

1fi

L

fi----- 0≥– and 1

fi

fiU

----- 0≥–,

2-6


(2.3-7)

where is the lowest buckling load factor for the ith load case and is thecorresponding lower bound.

2.3.5 Amplitude Constraints Under Harmonic Excitation

Amplitudes and phases of the response for the structure are obtained by recoveringthe physical response from generalized modal response. Presently, STROPTsupports amplitude constraints under harmonic excitation. Loadings can beconcentrated nodal forces and pressure loading which may be specified asfunctions of the exciting frequencies. The steady state response quantities are all inthe frequency domain and the maximum amplitude in this domain is constrained.The normalized constraint on maximum amplitude over the frequency domain canbe expressed as:

(2.3-8)

where Pi is the maximum amplitude of the ith degree of freedom on the domain anPi* (>0) is its allowable limit.

01or0,1 *

*

i

i

i

i ≥−λ

λ≥

λλ

−

λi*iλ

0PP

1 *i

i ≥−

2-7

Design Variables

2.4 Design Variables

2.4.1 Definition of Design Variables

The concept of design variables is central in the topics related to optimization. Thisconcept is described in this section. The entities that can be considered as designvariables have a bearing on the type of optimization problem. The discussion hereis in the context of the optimization problem defined in Section 2.1.

A design variable is defined as that entity which describes the system. The designvariables are to be chosen by the designer and serve as to assure the specificationsfor fabrication. Examples of design variables are: thicknesses of plates and/orshells; widths, depths, thicknesses, radii, moments of inertia of beams and/orcolumns; cross-sectional areas of truss members, layer orientations in composites,etc.

A design variable may be fixed or free. Optimization algorithms change the valuesof design variables iteratively to obtain optimum solution. Sometimes designermay want to assign fixed (constant) values to some design variables fromconsiderations like aesthetics, code requirements, or functional restraints. This ispossible in STROPT by treating such design variables as fixed design variables.The initial (or starting) input values of these design variables are not changed in theprocess of optimization. Obviously the values of upper and lower bounds for such adesign variable should be the same as the initial value. Other design variableswhose values are changed during optimization are termed as free design variables.

2.4.2 Local and Global Design Variables

STROPT uses finite element method in ascertaining behavior of a structure. Designvariables associated with an element or a section type are referred to as localdesign variables. Information on local design variable numbering and the types ofelements or sections related to them is given in the Section Library ( Section 4.2).The library is continually updated as new elements or section types areincorporated in the program.

Local design variables of all the elements of a structure constitute global designvariables. To bring out the meaning of local and global design variables, consider a

2-8


10-bar truss. Each bar of the truss has one local design variable which is its area ofcross-section. However, there are ten global design variables.

2.4.3 Design Variable Linking

Design variable linking is employed to impose equality of design variables in someregions of the optimized structure. Many design variables can be linked and thentreated as only one design variable in the process of optimization.

Significant savings in terms of array storage and computational time can beachieved by judiciously reducing the number of global design variables. STROPTprovides two ways of reducing the number of global design variables. One way isto group the local design variables.

Consider the two-story building frame shown in Figure 2.4-1. There are 10 beamelements (NKTP = 12). Each element is rectangular in section and has two localdesign variables viz. the width b and the depth d. From the Section Library, allelements have ISECT = 5. The total number of global design variables in this caseis therefore 20 (= 10 * 2). If optimum solution is sought with these 20 designvariables then the widths and depths of all the members may, in general, havedifferent values at optimum. However, if a designer wants to have: (1) same crosssectional dimensions (b and d) for the 3 first level columns (i.e. elements 1, 2 and3) and (2) Same cross-sectional dimensions for the 3 second level columns (i.e.elements 4, 5 and 6), then he may do so by grouping the local design variables asfollows:

Design Variable Group Number

Element Numbers

No. of Local Design Variables

No. of Global Design Variables

1 1, 2, 3 3 * 2 = 6 2

2 4, 5, 6 3 * 2 = 6 2

3 7 1 * 2 = 2 2

4 8 1 * 2 = 2 2

5 9 1 * 2 = 2 2

2-9

Design Variables

It can be seen that the number of global design variables is now reduced to 12 from20.

Another way of reducing the number of global design variables is by linking localdesign variables of two design variable groups. Continuing with the building frameexample, let it be assumed that the designer also wants the depths of elements 7and 8 to be the same and so also of elements 9 and 10. This is possible by linkingthe local design variables of groups 3 and 4 and also of groups 5 and 6. This linkingwould mean that the program would treat depths of elements 7 and 8 as one globaldesign variable (instead of 2) and similarly the depths of elements 9 and 10 wouldbe treated as one global design variable (instead of 2). It may be noted that the totalnumber of global design variables would further reduce to 10 from 12.

One would therefore use grouping of local design variables (first approach) if allthe local design variables of an element or section are to be linked with one or moreof identical elements or sections. However, if only a few local design variables ofan element or a section are to be linked with another element or a section (notnecessarily identical) then the second approach can be used. Depending on thedesigners requirements, one may use the two approaches separately orsimultaneously. User is strongly recommended to employ these approacheswherever deemed necessary

6 10 1 * 2 = 2 2

20 12

2-10


.

Figure 2.4-1 Building frame

2-11

Design Sensitivity Analysis

2.5 Design Sensitivity Analysis

2.5.1 Formulation of Design Sensitivity Analysis

The structural optimization techniques require design sensitivity coefficients, i.e.,derivatives of structural response such as stress, displacement, eigenvalue, andamplitude with respect to design variables. Consider a constraint Gi, which mayrepresent stress, displacement, eigenvalue, or amplitude of the structure. Using thechain rule of differential calculus, the total derivative of the constraint can bewritten as:

(2.5-1)

In structural optimization, such derivatives are not available in a closed form. Theyare obtained from differentiating the equilibrium equation for the structural systemgiven in Eqs. 2.1-2 and 2.1-3.

For brevity, K(X), P(X), M(X) are written as K, P, and M respectively in thefollowing equations.

By differentiating Eqs. 2.1-2 and 2.1-3, we have

(2.5-2)

and

(2.5-3)

where M(X)Y = 1 and the superscript ~ denotes that the variable is to be heldconstant during partial differentiation.

dGidX---------

∂Gi∂X---------

∂Gi∂U--------- dU

dX-------

∂Gi∂λ--------- dλ

dX-------++=

KdUdX------- ∂

∂X------- KU( ) ∂P

∂X-------=+

dλdX------- Y ∂

∂X------- KY( ),

λ Y ∂∂X------- MY( ),

–=

YT

2-12


Eq. 2.5-2 can be rewritten as:

(2.5-4)

To carry out design sensitivity analysis, right hand side of Eq. 2.5-4 should beknown, which is a matrix of the so called pseudo-force vectors. In most cases it isdifficult to express the stiffness and the mass matrices as explicit functions ofdesign variables. Thus, differentiation of such matrices is inconvenient and it isdesirable to transform the differential form into the incremental form as:

(2.5-5)

and

(2.5-6)

where the subscripts “o” and “m” represent original and modified quantities,respectively and represents a design perturbation. Eq. 2.5-5 can be solved for(dU/dX), which is called direct differentiation approach. The efficiency of thisapproach depends on the number of design variables (NDV) and number of loadcases (NLC) for a problem. For design problems with the number of activeconstraints (NCON) less than the number (NDV*NLC), the direct application ofEq. 2.5-5 can still place a severe computational burden. In this case, an adjointvariable vector is introduced as the solution of the following equation.

(2.5-7)

where the superscript T denotes the transposition. Substituting from Eqs. 2.5-4and 2.5-7 into Eq. 2.5-1, we obtain

KdUdX------- ∂P

∂X------- ∂

∂X-------– KU( )=

KdUdX-------

Pm Po–∆X

------------------ Km Ko–( )U∆X

-------------------------------–=

( ) ( )

∆−

λ−

∆−

=λ

XY~MM,Y

XY~KK,Y

dXd omom

.

∆X

Kηi∂Gi∂U---------

T

=

2-13


(2.5-8)

Thus, Eq. 2.5-8 involves the solution of Eq. 2.5-7. If the design derivatives areneeded only for active constraints, Eq. 2.5-8 can be more efficient. This is so calledadjoint variable method.

In STROPT, there are three basic procedures for design sensitivity analysis: finitedifference, direct differentiation, and adjoint variable methods. While they areintended to provide the same information, their computational procedures vary insome fundamental ways. A semi-analytic finite difference method is implementedusing the concept of direct differentiation approach. That is, Eq. 2.5-5 is alsosolved for (dU/dX) to avoid decomposition of the modified stiffness matrix. Itdiffers from the direct differentiation method in a way the total derivatives of theconstraints are computed. In addition, hybrid method is also available to select anefficient sensitivity method automatically during the optimization process bycomparing the number of active constraints (NAC) to the number (NDV*NLC).

2.5.2 Mathematical Representation of Sensitivity Coefficients

Before we can use the gradients in the design process, we need to understand whatvarious gradient components or sensitivity coefficients mean. These coefficientsare the first derivatives of the response function with respect to a design variable asgiven in Eq. 2.5-1. For example. if Gi is the ith stress constraint for the structureand xj is the jth component of design variable vector of dimension K, then Gi, j =first derivative of Gi with respect to jth design variable. To see how these gradientsshould be used in the design process, we need to write first order Taylor seriesexpansion of the response function. For example, for Gi

(2.5-9)

where H.O.T. stands for Higher Order Terms.

dGidX---------

∂Gi∂X--------- ηi

∂P∂X------- ∂ KU( )

∂X-----------------–

, ∂Gi

∂λ--------- dλ

dX-------⋅+ +=

Ginew Gi

current Gi j, xjnew xj

current–( ) +j 1=

k∑+= H.O.T.

2-14


Suppose is positive and we would like to change the design such that newdesign makes new smaller than current value, viz. . How can we doit? By looking at the second term on the right hand side of Eq. 2.5-9, i.e.

one can make it negative such that becomes smaller than .Actually, this term gives a criterion for changing the current design. That is, if Gi,j

is <0, an increase in the jth design variable would cause a decrease in

the , however, if Gij is >0, an increase in would amount to an

increase in .

Thus, sign of sensitivity coefficient tells the designer whether to increase ordecrease a design variable. However, it is not desirable to change all of xjs tocorrect . There are two reasons for this restriction. The first reason is thatsome of the sensitivity coefficients are quite small (i.e. Gi,j) as compared to others.So a large change in corresponding xj will be required to see significantcontribution in the correction of . Therefore, relative magnitude ofsensitivity coefficients play a key role in selecting design variable changes in thegiven design to correct particular responses. The second reason for not changing allthe design variables at the same time is due to consideration of other coefficients ofresponses under considerations associated with the same design variable must havethe same sign. We demonstrate the use of the above mentioned criteria in someexample problems in Appendix A and Appendix B.

Gicurrent

Ginew Gi

current

Gi j, xjnew xj

current–( )1=

k∑

Ginew Gi

current

xjcurrent( )

Gicurrent xj

current

Gicurrent

Gicurrent

Gicurrent

2-15


2-16

Chapter

3
Optimization Algorithms
The use of nonlinear programming techniques is becoming a standard approach instructural optimization due to their general applicability. They use the designphilosophy of improving the values of design variables iteratively using functionsand their gradients. In each design iteration, two basic decisions have to be madeto find a design change; they are direction search and line search. Direction searchinvolves design sensitivity analysis and line search involves structural analysis,respectively. Since a major computational effort is spent on structural and designsensitivity analyses, it is essential to achieve a rapid rate of convergence whileeconomizing on the number of structural analyses during line search.

The optimization algorithms which are available in STROPT are: GRG(Generalized Reduced Gradient), RQP (Recursive Quadratic Programming) andOCB (Optimum Cost Bounding). It is necessary to provide several differentalgorithms, because there is no unanimous choice of an optimization algorithmsuitable to every type of problem. The capabilities of each optimization algorithmdiffer from one another, so that they cannot be applied to the same physicalproblems with the same efficiency. Therefore, the user needs some options tochoose one of them depending on his particular interest and experience. For thisreason, there will be many more to account for a wide variety of structuraloptimization.

The use of an active set strategy is of utmost importance in structural optimizationproblems, owing to the size of the problem. That is, only active constraints areconsidered at every iteration to determine a design change.

3-1

Generalized Reduced Gradient Method (GRG)

3.1 Generalized Reduced Gradient Method (GRG)

This algorithm is based on the variable elimination technique which solves theoriginal problem in the reduced space. The inequality constraints are converted toequality constraints by adding slack (or state) variables. This implies that allconstraints are active and hence have to be differentiated. However, the reducedgradient can be computed by using an active set strategy which defines thebounded equality constraints as the active constraints. This enables us to avoidcalculating and inverting the design derivatives of all constraints. The step size isestimated by using the first-order Taylor series expansion for the most criticalconstraint. To maintain feasibility, Newton’s method is used to adjust slack (orstate) variables. If the initial design is not feasible, then STROPT scales the designappropriately to obtain a feasible design. The main feature of this algorithm is thatit reduces the objective function monotonically and gives a number of feasibledesigns. It may be found to suffer from failure of line search precluding theoptimum solution.

3.2 Recursive Quadratic Programming Method (RQP)

This algorithm is based on the solution of a QP subproblem which is obtained bylinearizing the original problem with a quadratic step size constraint. The solutionof this QP subproblem yields a search direction. The step size along this directionis then estimated and a new design is obtained. This operation is repeated until anoptimum solution is found. This method is also referred to as Variable Metric orModified-Newton method for constrained optimization.

There are many ways of expressing the QP subproblems. Here, the objectivefunction of the QP subproblem involves a Hessian approximation to theLagrangian function as

minimize

subject to

where W = A positive definite Hessian matrix

= Gradient of objective function

∇F S,( ) 0.5 S W S,( )+

∇Gi S,( ) Gi 0 i I∈,≥+

∇F

3-2

Optimization Algorithms

The matrix W is chosen to be a positive definite approximation to the Hessian ofthe Lagrangian function:

It is usual to estimate the matrix W by Quasi-Newton updates that require onlyfirst-order gradient information. In fact, the modified BFGS formula of Powell isused to maintain positive definiteness of Hessian. The main feature of thisalgorithm is that it converges super-linearly near the optimum solution. However, Itdoes not reduce the objective function monotonically and may give infeasibledesign before reaching an optimum solution.

3.3 Optimum Cost Bounding Method (OCB)

The basic idea of this algorithm is to first establish lower and upper bounds on theoptimum value of the objective function for the problem. The design spacebetween the established bounds is systematically searched to compute betterbounds until optimality conditions are satisfied or no design improvement ispossible. The search is conducted by executing one of the four design stepsdepending on the design environment at the current design. The four steps are costreduction, constraint correction, constraint correction without cost change, andconstraint correction with a limit on cost increment. Step size is determined bysolving a dual problem resulting in efficient computation of Lagrange multipliersfor the primal. The main feature of this algorithm is that it does not rely on linesearch to compute a step size at any design iteration. This feature makes OCB very

= Gradient of ith constraint function

I =

S = A vector of search direction

(a,b) = Inner product of two vectors a and b

where = The Lagrange multipliers of the problem.

∇Gi

i:Gi+ε 0 i,≥ 1 m,={ }

L X µ,( ) F X( ) µiGi X( ) i I∈;i 1=

m

∑+=

µi

3-3

Optimum Cost Bounding Method (OCB)

efficient in terms of CPU time per design iteration. However, it may fail toestablish a correct lower bound on optimum cost, which may result in moreiterations to converge. It may be noted that the algorithm is based on the gradient ofthe cost function. In case, if the cost is not a function of some design variables thenthe cost-gradient will have zero-components corresponding to those designvariables. In such a situation, the problem should be understood as the one withreduced number of design variables. Refer to problems on composite and sandwichplate elements where the cost is not a function of orientation angles.

3-4

Chapter

4
Capabilities and Features of STROPT
4.1 Finite Element Library

An extensive library of finite elements gives STROPT a distinctive versatility.Currently, the structures formed out of one or more of the types of finite elementsgiven in Table 4.1-1 can be optimized. If the other types of elements which arenot supported by STROPT are used in the structure, then the design variables forthose elements are assumed to be fixed during the optimization process. Forexample, spring or point mass element can be included in the structure. Inaddition, the use of rigid elements is supported to model spot welds, bolted joints,load transfer, etc.

Table 4.1-1 Finite element library

NKTP Element Description Design Variable

1 2D Plane Stress Element thickness

7 3D Composite Solida Layer thickness and angle

11 3D Tapered Beam* Cross-sectional dimensions

12 3D General Beam Cross-sectional dimensions

14 3D Spar Cross-sectional area

20 3D General Shell Element thickness

32 3D Composite Shell Layer thickness and angle

4-1

Finite Element Library

33 3D Sandwich Shell Layer thickness and angle

36 Axisymmetric Shell Element thickness

40 3D Thin Shell Element thickness

a. under development

NKTP Element Description Design Variable

4-2


4.2 Section Library

STROPT offers an option of selecting any of the following cross-sectional shapesfor the finite elements. The section library defining the local design variablesassociated with the different section types is given in Table 4.2-1 .

Table 4.2-1 Section library

ISECT PossibleElement

Types

Section/Element Sketch

Total Num-ber of Local

Design-Variables

Numbering of Local Design-Variables

1 1,20, 36, 40

1 1: Uniform thick-nessover the ele-ment, t

2 1,20, 36, 40

Number of Nodes (N)

1:

2:

i:

N:

Thickness at node 1, t1Thickness at node2, t2 Thickness at nodei, ti Thickness at nodeN, tN

4-3

Section Library

3 12, 14 1 1: Radius, R

4 12, 14 2 1:2:

Radius, RThickness, t

5 12, 14 2 1:2:

Width, b Depth, d

Table 4.2-1 Section library (Continued)


Types



Design-Variables


4-4


6 12, 14 4 1:2:3:4:

Width, bDepth, dThickness, t1Thickness, t2

7 14 Any Arbitrary Shape

1 1: Area of crosssection, A

8 32, 33 2*NLAY (NLAY:

Number of layers (layer 1 is always

the top layer))

1: t1

2: }Layer 1

3: t2

4: }Layer 2

2i-1: ti

2i: }Layer i

ti: layer thickness : layer orientation

(in degrees)



Types



Design-Variables


θ1

θ2

θi

θi

4-5

Section Library

9 12, 14 4 1:2:3:4:

bdtftw

10 12, 14 6 1:2:3:4:5:6:

btbbdtttbtw

11 12, 14 4 1:2:3:4:

bdtftw



Types



Design-Variables


4-6


12 12, 14 4 1:2:3:4:

bdtftw

13 12, 14 4 1:2:3:4:

wf ht

14a 12, 14 5 1:2:3:4:5:

wfht1t2



Types



Design-Variables


4-7

Section Library

21* 12, 14 Number of Sides (N)

1:2:...N:

l1l2

lN

22 12, 14 Number of Sides (N)

1:2:...N:

t1t2

tN



Types



Design-Variables


4-8


23 12, 14 1 1: t

a. Under development



Types



Design-Variables


4-9

Stress Library

4.3 Stress Library

Various options for imposing stress constraints are available in STROPT. A stressoption number (INNATE) represents the stress constraint type which signifies thenumber of stress constraints depending on the locations where stress constraints areimposed. Stress options must be chosen to be compatible with the element types.The stress library of STROPT is given in Table 4.3-1

Table 4.3-1 Stress Library

INNATE NKTP Location Description

1 1, 20, 36, 40 Gauss Points One or more of the six stress components (viz. NGST11, NGST22, NGST33, NGST12, NGST23, NGST13) at all Gauss points in global coord.

2 1, 20, 36, 40

Gauss points Same as for INNATE = 1 but in local coord

3 1, 20, 36, 40 Gauss points One or more of the six stress components averaged over all Gauss points of an element in local coord.

4 1, 20, 36, 40 Gauss points One or more of the principal stresses (viz. PRIN1, PRIN2, PRIN3) computed from averaged Gauss point stress components

5 1, 20, 36, 40 Gauss points Von Mises stress computed from the averaged stress components over all Gauss points

6 1, 20, 36, 40 Gauss points Maximum shear stress computed from the averaged stress components over all Gauss points

7 1, 20, 36, 40

Gauss points Octahedral shear stress computed from the averaged stress components over all Gauss points

8 1, 20, 36, 40 Gauss points One or more of the principal stresses at all Gauss points (viz. PRIN1, PRIN2, PRIN3)

9 1, 20, 36, 40 Gauss points Von Mises stresses at all Gauss points

10 1, 20, 36, 40 Gauss points Maximum shear stresses at all Gauss points

11 1, 20, 36, 40 Gauss points Octahedral shear stresses at all Gauss points

4-10


12 32, 33 Gauss points Failure stress function at all layers and Gauss points. Stress functions for different element types (NKTP) are as follows:

• for NKTP=32: Hill-Mises• for NKTP=33:

• for face sheets: ratio of 3 actual inplane stresses (i.e., σxx, σyy, σxy) to their respective allowable stresses and,

• for core material: ratio of 2 transverse shear stresses (i.e. σxz and σyz) to their respective allowable stresses

13 32, 33 Gauss points Failure stress function at all Gauss points (maximum over all layers). Stress functions for different element types (NKTP) are as follows:




14 32, 33 Gauss points Failure stress function for all layers (maximum over all Gauss points of each layer). Stress functions for differ-ent element types (NKTP) are as follows:




Table 4.3-1 Stress Library (Continued)


4-11

Stress Library

15 32, 33 Gauss points Maximum failure stress function over all layers and Gauss points. Stress functions for different element types (NKTP) are as follows:




16 32 Gauss points Failure stress function at all layers and Gauss points. Stress functions for different element types (NKTP) are as follows:

• for NKTP=32: Tsai-Wu

17 32 Gauss points Failure stress function at all Gauss points (maximum over all layers). Stress functions for different element types (NKTP) are as follows:


18 32 Gauss points Failure stress function for all layers (maximum over all Gauss points of each layer). Stress functions for differ-ent element types (NKTP) are as follows:


19 32 Gauss points Maximum failure stress function over all layers and Gauss points. Stress functions for different element types (NKTP) are as follows:


20-30 unused

31 1, 20, 36, 40 Node points One or more of the six stress components (viz. NDST11, NDST22, NDST33, NDST12, NDST23, NDST13) at all node points in global coord.

32 1, 20, 36, 40 Node points Same as for INNATE = 31 but in local coord.

33 1, 20, 36, 40 Node points One or more of the principal stresses at all node points (viz. PRIN1, PRIN2, PRIN3)



4-12


34 1, 20, 36, 40 Node points Von Mises stresses at all node points

35 1, 20, 36, 40 Node points Maximum shear stresses at all node points

36 1, 20, 36, 40 Node points Octahedral shear stresses at all node points

37 32, 33 Node points Failure stress function at all layers and node points. Stress functions for different element types (NKTP) are as follows:


• for face sheets: ratio of 3 actual inplane stresses (i.e. σxx, σyy, and σxy) to their respective allowable stresses and,


38 32, 33 Node points Failure stress function at all node points (maximum over all layers). Stress functions for different element types (NKTP) are as follows:


• for face sheets: ratio of 3 actual inplane stresses (i.e., σxx, σyy, and σxy) to their respective allowable stresses and,


39 32, 33 Node points Failure stress function for all layers (maximum over all nodes of each layer). Stress functions for different ele-ment types (NKTP) are as follows:



• for core material: ratio of 2 transverse shear stresses (i.e., σxz and σyz) to their respective allowable stresses



4-13

Stress Library

40 32, 33 Node points Maximum failure stress function over all layers and node points. Stress functions for different element types (NKTP) are as follows:




41 32 Node points Failure stress function at all layers and node points. Stress functions for different element types (NKTP) are as follows:


42 32 Node points Failure stress function at all node points (maximum over all layers). Stress functions for different element types (NKTP) are as follows:


43 32 Node points Failure stress function for all layers (maximum over all nodes of each layer). Stress functions for different ele-ment types (NKTP) are as follows:


44 32 Node points Maximum failure stress function over all layers and node points. Stress functions for different element types (NKTP) are as follows:


45-60 Unused

61 1, 20, 36, 40 Centroid One or more of the six stress components in global coord. (viz. NGST11, NGST22, NGST33, NGST12, NGST23, NGST13)

62 1, 20, 36, 40 Centroid Same as for INNATE = 61 but in local coord.

63 1, 20, 36, 40 Centroid One or more of the principal stresses (viz. PRIN1, PRIN2, PRIN3)



4-14


64 1, 20, 36, 40 Centroid Von Mises stress

65 1, 20, 36, 40 Centroid Maximum shear stress

66 1, 20, 36, 40 Centroid Octahedral shear stress

67 32, 33 Centroid Failure stress function for all layers. Stress functions for different element types (NKTP) are as follows:


• for face sheets: ratio of 3 actual inplane stresses (i.e. σxx, σyy, and σxy) to their respective allowable stresses and,


68 32, 33 Centroid Maximum failure stress function over all layers. Stress functions for different element types (NKTP) are as fol-lows:


• for face sheets: ratio of 3 actual inplane stresses (i.e. σxx, σyy, σxy) to their respective allowable stresses and,


69 32, 33 Centroid Failure stress function for all layers. Stress functions for different element types (NKTP) are as follows:


70 32, 33 Centroid Maximum failure stress function over all layers. Stress functions for different element types are as follows:


71-90 Unused

91 12, 14 Critical points

Stresses due to axial force, bending moment, and shear force



4-15

Stress Library

Notes:

1. The value of INNATE signifies the number of stress constraints. It may benoted that the number of Gauss points and/or nodes depend on the type(NKTP) and order (NORDR) of the element (NISA User Manual). The follow-ing examples illustrate the parameters involved and the associated number ofstress constraints for an element.

Example 1.

In this case the number of Gauss points is 4 (see the NISA II User Manual).Thus,

number of stress constraints = number of Gauss points * number of stress types= 4 * 2 = 8.


One or more of the principal stresses(viz. PRIN1, PRIN3)


Von Mises stress


Maximum shear stress


Octahedral shear stress

96-100 Unused

A: Element type : 2D Plane stress (NKTP = 1)

B: Element order : Linear (NORDR = 1)

C: Stress option : INNATE = 1

D: Stress types : NGST11 and NGST12



4-16


Example 2.

In this case, the number of nodes = 9 (see the NISA II User Manual). Thus,number of stress constraints = number of nodes * number of stress types = 9 *6 = 54.

2. NISA input pertaining to stress calculations should be consistent with thestress constraint input (*STRCON and *STRELM groupids) data. As anexample, user cannot expect to impose stress constraints at Gauss points ifstress calculations at Gauss points are not asked for in the NISA input(*LDCASE groupid) data. Similarly one cannot expect to impose constraintson global stress components if only local stress components are asked for inNISA input (*LDCASE groupid) data.

3. NISA computes stresses at middle, bottom and top surfaces for 3D GeneralShell (NKTP = 20), 3D Thin Shell (NKTP = 40), and Axisymmetric Shell(NKTP = 36). STROPT imposes constraints on appropriate (nodes and/orGauss points and/or centroid) maximum absolute values from among the threesurfaces. As an example, consider 3D General Shell element with NORDR =1.

A: Element type :

3D General Shell (NKTP = 20)

B: Element order :

NORDR = 6

C: Stress option :

INNATE = 32

D: Stress types :

NDST11, NDST12, NDST33, NDST22, NDST23 and NDST13 (all six components)

4-17

Stress Library

Let the nodal stress values at node1 be as follows:

Constraints can be imposed on the maximum absolute values viz. 20, 25, 12,28, 11 and 35 only. Similarly for other nodes (2, 3, and 4) maximum absolutestress values are computed by STROPT.

4. In case of 3D General Shell (NKTP = 20), 3D Thin Shell (NKTP = 40), andAxisymmetric Shell (NKTP = 36) elements, average values (averaged overnodes or Gauss points) of stress components are computed for each surfacefirst, and then the maximum absolute values are computed for the pertinentaveraged stress components.

5. PRIN1, PRIN2 and PRIN3 are minimum, intermediate, and maximum princi-pal stresses, respectively. In case of a 2D element constraints on PRIN1 andPRIN3 only, can be imposed.

6. STROPT computes von Mises, Maximum shear, and Octahedral shear stressesfrom the expressions given below.

A. Von Mises Stress:

(4.3-1)

B. Maximum Shear Stress:

NDST11

NDST22

NDST33

NDST12

NDST23

NDST31

Top Surface -10 -15 12 -8 11 -30

Middle Surface -5 -10 8 18 11 -35

Bottom Surface 20 25 -5 28 11 35

Max. Abs. Val-ues

20 25 12 28 11 -35

σvon12

------- σ1 σ2–( )2 σ2 σ3–( )

2 σ3 σ1–( )2

+ +1 2⁄

=

4-18


(4.3-2)

C. Octahedral Shearing Stress:

(4.3-3)

where σ1, σ2 and σ3 are maximum, intermediate and minimum principalstresses, respectively.

7. NISA computes values of Hill Mises failure stress function at all nodes and forall layers of 3D composite shell element (NKTP = 32). The failure stress func-tion is defined as follows:

(4.3-4)

where

σxa = Failure stress under unidirectional loading in X direction

σya = Failure stress under unidirectional loading in Y direction

σsa = Failure stress in pure shear.

Failure of a layer is implied if the failure stress function attains a value of 1.STROPT computes stress constraint in this case from the expression:

(4.3-5)

and the constraint is defined as:

(4.3-6)

τmax Maximum of 12--- σ1 σ2–( ) 12

--- σ2 σ3–( ) 12--- σ3 σ1–( ), ,=

τoct13--- σ1 σ2–( )

2 σ2 σ3–( )2 σ3 σ1–( )

2+ +

1 2⁄

=

F σxx σyy σxy, ,( )σxxσxa--------

2 σxxσxa--------

σyyσxa--------

σyyσya--------

2 σxyσsa--------

2+ +–

1 2⁄

=

1 F σxx σyy σxy, ,( )–

1 F σxx σyy σxy, ,( )– 0≥

4-19

Stress Library

8. The Tsai-Wu tensor failure theory defines the following failure surface:

(4.3-7)

Except for the linear terms, which allow the difference between tensile andcompressive failure stress to be modeled, this looks much the same as the Hill-Mises equation. However, the definition of the coefficients F1, F2,... F12 ismore general. We have:

(4.3-8)

where

Note that compressive failure stresses are also input as positive values.

The above coefficients are all computed from the user-input engineering fail-ure stresses. The calculation of F12 is left to the user, since it depends upon theexperimental method chosen to perform a biaxial failure test. Hence, the valueof F12 is input directly as a material property in *MATERIAL data group.

σxt, σxc = x-direction tensile and compressive failure stress, respectively

σyt, σyc = y-direction tensile and compressive failure stress, respectively

σs = Inplane (xy) shear failure stresses

F F1σxx F2σyy F11σxx2 F22σyy

2 F33σxy2 F12σxxσyy+ + + + +=

F1 1σxt------- 1

σxc--------–=

F2 1σyt------- 1

σyc--------–=

F11 1σxtσxc---------------+=

F22 1σytσyc---------------+=

F33 1σs

2-----+=

4-20


9. For 3D laminated sandwich general shell element (NKTP=33), NISA com-putes the ratio of actual stress to allowable stress where stresses are computed.If the ratio exceeds 1.0, layer failure is presumed to have occurred. The ratio isdefined as:

where

STROPT computes stress constraint in this case from the expression:

(4.3-9)

and the constraint is defined as:

(4.3-10)

10. Stress options 91 to 95 are for 3D General Beam (NKTP = 12) and 3D spar(NKTP = 14) elements. Constraints on stresses due to axial force and/ormoment about y axis and/or moment about z axis and/or shear force on yzplane (i.e. INNATE = 91) are defined as follows:

(4.3-11)

(4.3-12)

σ = Actual stress. For face sheets actual stresses are inplane stresses (i.e., σxx, σyy, and σxy) and for core material actual stresses are transverse shear stresses (i.e., σxz and σyz)

σ* = Allowable stress associated with each stress component

F σ( ) σσ∗------=

1 F σ( )–

1 F σ( )– 0≥

1σa

σa*

--------

Cmyσby

σby*

----------- Cmzσbz

σbz*

-----------+ +– 0.0≥

1σs

σs*

--------

– 0≥

4-21

Stress Library

where

Note that y and z are assumed to be the principal axes passing through centroidof a cross section.

For 3D spar element, the stress constraint is defined as:

(4.3-13)

Constraint defined by Appendix 4 is calculated at a point on the boundary ofthe cross section where the term in the bracket viz.

(4.3-14)

σa = Normal stress due to axial force

σ*a = Allowable normal stress under axial force

σby, σbz = Maximum normal stresses due to bending moments about y and z axes, respectively. These stresses are experienced by fibers farthest from the y and z axes, respectively.

σ*by, σ*bz

= Allowable normal stresses due to bending moments about y and z axes, respectively.

σs = Maximum shear stress due to shear and twisting moment. The location, where this maximum stress is experienced, depends on the shape of cross section and the relative magni-tudes of Vy and Vz where Vy and Vz are shear forces in y and z directions).

σ*s = Allowable shear stress.

Cmy, Cmz

= Magnification factors for stresses due to bending moments. Current version of STROPT takes Cmy = Cmz = 1.

1σa

σa*

--------

– 0≥

σa

σa*

--------σby

σby*

-----------σbz

σbz*

-----------+ +

4-22


has a maximum value. Similarly constraint Appendix 4 should be calculated ata point on the boundary of the cross section where is maximum. Tolocate these points is a difficult task. STROPT adopts a strategy based on soundengineering judgement to circumvent this difficulty. It computes the values ofstresses at some critical points on the boundary of the cross-section. STROPTcomputes values of expression no. (4.3-14) at critical points and uses the maxi-mum value for constraint evaluation given in Appendix 4. Similarly, a maxi-mum value for is used for constraint evaluation given in Appendix4.

Stress options 93 to 95 are computed based on Equations (4.3-1) to (4.3-3).These stresses are calculated at a point on the boundary of the cross sectionwhere equivalent stress has a maximum value.

11. Refer to sections on 2.3.2 and 5.3 (Groupid = STRCON and STRELM) formore information.

σs σs*⁄

σs σs*⁄

4-23

Program Architecture

4.4 Program Architecture

The program architecture of STROPT is shown in Figure 4.4-1. It is designed toprovide design engineers with an integrated CAD/CAE system. It has a modularset-up with defined interfaces. While the module DISPLAY III is used for pre- andpost-processing, the real optimization calculation takes place in STROPT.

Figure 4.4-1 Architecture of program STROPT

4-24


4.5 Forms in STROPT

STROPT uses form design. Forms are a user friendly interface added to DISPLAYIII command structure. They enable the user to input information without having toremember the command structure and with as little keyboard usage as possible.The forms are almost wholly mouse driven, thereby providing a fast and efficientsubstitute to typing commands.

In general, for any STROPT run two types of data are required, analysis data anddesign related data. Manual generation of the data is quite tedious and cumbersomebecause substantial cross referencing and repetitions are required. Forms caneliminate unnecessary data repetitions by grouping several data groups (data deck)in a set graphically within DISPLAY III.

In STROPT, a total of 13 data groups are required to identify a structuraloptimization problem consists of displacement, stress, natural frequency, bucklingload factor and amplitude constraints. Users need to specify all 13 data groupsseparately to define the problem. As an example, to define stress constraint,STROPT needs two data groups namely STRCON and STRELM. The two datagroups can be generated using one form within DISPLAY III. With DISPLAY III,13 data groups can be identified with seven forms for:

1. Key parameters and convergence criteria

2. Design variable group information

3. Displacement constraint information

4. Stress constraint information

5. Frequency constraint information

6. Buckling load factor information

7. Amplitude constraint information

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Output Features and Postprocessing

4.6 Output Features and Postprocessing

4.6.1 General Description of the Output

This section discusses the information provided in the STROPT printout. The inputdata for STROPT consists of four sets followed by an input data termination. Thesefour sets are: executive commands, model data block, analysis data set, andstructural optimization data block. The first data set is made up of executivecommands related to NISA program and additional commands strictly relevant tocomputations of response structural optimization. The second and third data setsare NISA input data which is called NISA related input data and the 4th data set isstructural optimization data block which is called STROPT related input data. Thissection explains outputs related to STROPT only, i.e. structural optimization anddesign sensitivity analysis.

Table 4.6-1 gives a general description of the printout for structural optimizationand references the pertinent sections in this manual for detailed information abouteach output item. A detailed output description for a sample structural optimizationis given in Appendix C

Hard copy printouts associated with STROPT only are printed in two phases. Thefirst phase is during the reading and processing of structural optimization whichstarts with *OPTPAR data group and terminates with *ENDDATA group data. Inthis stage, the program prints echoes of input and sorted values which are strictlycontrolled by the program. In the second phase the following are printed: value ofcost function, values of design variables, constraint function values, cost functiongradient, number of active (critical) constraints, sensitivity coefficients, and detaildiagnostic information associated with selected optimization algorithm which arestrictly controlled by the program. Program prints these information and user hasfull control over the volume of the printout.

Sensitivity analysis is a major task in STROPT. Before printout of sensitivitycoefficients, STROPT prints a table of global design variables numbering scheme.This table relates global design variable numbers to design variable group numbersgiven in *DVGROUP data group (detailed definitions of local and global designvariables are given in Section 2.4). The user should be cautioned that sensitivitycoefficients are printed with respect to all global design variables which follows a

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sequence of printouts depending on the command option of SOSEnsitivity given inexecutive commands (see Section 5.2).

For SOSEnsitivity = ASCENDING, printout response quantities are: first globaldesign variable number and then sensitivity coefficient for the first design, secondglobal design variable number and then sensitivity coefficient for the seconddesign, and so on.

For SOSEnsitivity = ASCENDING, printout response quantities are in ascendingorder according to increasing absolute values. In this case, the sequence of printoutis: global design variable number corresponding to the smallest absolute value ofsensitivity coefficients followed by the sensitivity coefficient itself, global designvariable number corresponding to the second smallest absolute value of sensitivitycoefficients followed by sensitivity coefficient itself, and so on.

For SOSEnsitivity = DESCENDING, printout response quantities are indescending order according to decreasing absolute values. In this case the sequenceof printout is: global design variable number corresponding to the largest absolutevalue of sensitivity coefficients followed by sensitivity coefficient itself, globaldesign variable number corresponding to the second largest absolute value ofsensitivity coefficients followed by sensitivity coefficient itself, and so on. Also,for all the printed responses, each case prints the most critical (i.e. the mostimportant global design variable number corresponding to the largest absolutevalue of design sensitivity coefficients followed by sensitivity coefficient.

For the printout of sensitivity coefficients, most of the output items may besuppressed, printed in its entirety or selectively printed for a subset(s) of itsmembers (nodes, elements, natural frequency number, buckling load case number,or frequency response number). See the *SETS and *PRINTCNTL data groups(Sections 6.6.1 and 7.5.3 in the NISA User Manual and Section 5.4 in the STROPTUser Manual) for details. Users are cautioned to review the default options given in*PRINTCNTL data group. These default options are chosen such that the printoutof lengthy output items (e.g. sensitivity coefficients related to element stress) issuppressed unless explicitly requested by the user.

STROPT prints out cost function histories at the end of optimum (final) designiteration which can be utilized to predict best manufacturable designs. It also prints

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responses at final design iteration. Thus the user can compare initial and optimumdesign to find trend of design improvements.

4.6.2 Post Processing of the Results

The following graphs can be plotted using DISPLAY III:

1. Displacement and stress contours.

2. Sensitivity coefficients vs. global design variables.

3. Cost function histories.

Graphical representation of the results may be obtained interactively through thepostprocessing module of the DISPLAY III program which is a 3-D color graphicsprogram with extensive features for displaying results.

All analysis types are interfaced with the DISPLAY III through two binary files:the basic data file (file 26) and the post data file (file 27).

The postprocessing procedure for a typical analysis type is as follows:

1. Run STROPT and save files 26 and 27 through the executive commands‘FILE NAME’ and ‘SAVE FILE’ (Section 5.3.1 in NISA User Manual).

2. Use the DISPLAY III program for interactive processing of the results.For further details see DISPLAY III User Manual.

Table 4.6-1 Output Description for STROPT

OUTPUT ITEM REQUESTED BY+

General output

*Total number of global design variables (fixed and free) Automatic

*Total number of global design variables (free) Automatic

*Global design variable numbering Automatic

Design variable related output

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*Design variable group information containing design variable group number, section type, and number of local design vari-ables

*DVGROUP(5.2.2) (Automatic)

*Design variable group numbers and associated element num-bers

*DVGELMT (5.2.3) (Automatic)

*Local design variable information containing design variable group number, local design variable number, status of design variable, starting value, lower bound, and upper bound on design

*DVLOCAL (5.2.4) (Automatic)

*Linking of local design variables containing cross linking between different design variable groups

*DVLINIC (5.2.5)

Displacement constraint information

*Displacement table containing displacement ID number, nature of the constraint, displacement type code number, and displace-ment limit

*DISCON (5.2.6)

*Node number and displacement ID number *DISNOD (5.2.7)

*Values of displacement constraint Automatic

*Sensitivity coefficients of displacements with respect to global design variables

*PRINTCNTL (5.3.1)

*Sensitivity coefficient of displacement with respect to global design variable number corresponding to the largest absolute value of displacement sensitivity coefficients, i.e. critical design variable number

*PRINTCNTL (5.3.1)

Stress constraint information

*Stress table containing stress ID number, nature of the con-straint, stress type code number, and stress limit

*STRCON (5.2.8)

*Element number and stress ID number *STRELM (5.2.9)

*Value of stress constraint Automatic

Table 4.6-1 Output Description for STROPT (Continued)


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*Sensitivity coefficients of stresses with respect to global design variables

*PRINTCNTL (5.3.1)

*Sensitivity coefficient of stress with respect to global design variable number corresponding to the largest absolute value of stress sensitivity coefficients, i.e. critical design variable number

*PRINTCNTL (5.3.1)

Frequency constraint information

*Frequency information containing natural frequency number, type of frequency, lower and upper bound values on natural fre-quency

*FRQCON (5.2.12)

*Value of natural frequency constraint Automatic

*Sensitivity coefficients of natural frequency with respect to glo-bal design variables

*PRINTCNTL (5.3.1)

*Sensitivity coefficient of frequency with respect to global design variable number corresponding to the largest absolute value of frequency sensitivity coefficients, i.e. critical design variable number

*PRINTCNTL (5.3.1)

Buckling load factor constraint information

*Buckling load factor information containing buckling load fac-tor information and load factor limit

*BUKCON (5.2.11)

*Value of buckling load factor constraint Automatic

*Sensitivity coefficients of buckling load factors with respect to global design variables

*PRINTCNTL (5.3.1)

*Sensitivity coefficient of buckling load factor with respect to global design variable number corresponding to the largest abso-lute value of the sensitivity coefficients, i.e. critical design vari-able number

*PRINTCNTL (5.3.1)

Amplitude constraint information



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+: refer to section shown between parentheses for details

*Frequency response table containing amplitude ID number, nature of the sensitivity, amplitude type code number, and ampli-tude limit

*FRSCON (5.2.12)

*Frequency number and frequency response ID number contain-ing node number and amplitude ID number

*FRSNOD (5.2.13)

*Value of amplitude constraint Automatic

*Sensitivity coefficients of amplitude with respect to global design variables

*PRINTCNTL (5.3.1)

*Sensitivity coefficient of amplitude with respect to global design variable number corresponding to the largest absolute value of amplitude sensitivity coefficients, i.e. critical design variable number

*PRINTCNTL (5.3.1)



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4-32

Chapter

5
Input Description for STROPT
5.1 Overview of Input Data

The input data for STROPT consists of five (5) sets followed by an input dataterminator. The input data must be arranged in the sequence shown in Figure 5.1-1. The first four sets are exactly the same as for NISA analysis. All the input datafor design optimization are contained in the last set. Currently, a total of thirteen(13) data groups are used to provide the optimization input data. If the data are notgiven in proper sequence, the program will stop while processing each data group.Input format follows the same rules and guidelines as applicable for NISA. A briefdescription of optimization data block is as follows:

Optimization Data Block:

Last data block before data terminator in STROPT is optimization data block. Thisdata block describes data pertinent to information required for structural designoptimization. There are two sets of data groups. The first set is always required.Data groups associated with this set are: *OPTPAR, *DVGROUP, *DVGELMT,and *DVLOCAL. The second set is optional. This set consists of two subsets. Thefirst subset is *DVLINK which links local design variables. The second subset isnot always completely required and depends on the type of constraints imposed.The following data groups are needed for each constraint type: *DISCON and*DISNOD for displacement, *STRCON and *STRELM for stress, *FRQCON forfrequency, *BUKCON for buckling, and, *FRSCON and *FRSNOD foramplitude. Table 5.1-1 shows a list of data groups and their constraint types. Adetailed description of executive commands and required input data foroptimization data groups is given in the following section.

5-1

Overview of Input Data

Modifications on *PRINTCNTL Data Group:

To control the bulk of printout, new LABELS in *PRINTCNTL data group areintroduced. This data group is part of NISA analysis data block (see NISA UserManual Section 7.6.3). Detailed descriptions for this data group are given in“Section 5.4.

Figure 5.1-1 STROPT input data sequence

5-2


Table 5.1-1 Valid group identification names and associated constraint types for the optimization data block

Section Number and Group ID (1) Description

Applicable Analysis(2)

Is It Always Require

?

Applicable Constraint

Type

5.2.1 *OPTPAR Key parameters and conver-gence criteria ALL Yes —

5.2.2 *DVGROUP Design variable group table ALL Yes —

5.2.3 *DVGELMT Design variable group and asso-ciated element numbers ALL Yes —

5.2.4 *DVLOCAL Local design variable informa-tion ALL Yes —

5.2.5 *DVLINK Linking of local design vari-ables ALL No —

5.2.6 *DISCON Displacement constraint table ALL No Displace-ment

5.2.7 *DISNOD Node numbers associated with *DSICON ALL No Displace-

ment

5.2.8 *STRCON Stress constraint table ALL No Stress

5.2.9 *STRELM Element numbers associated with *STRCON ALL No Stress

5.2.10 *FRQCON Frequency constraint informa-tion EV No Frequency

5.2.11 *BUKCON Buckling load factor constraint information BU No Buckling

5.2.12 *FRSCON Amplitude constraint table FR No Amplitude

5.2.13 *FRSNOD Node numbers associated with *FRSCON FR No Amplitude

5.3 *PRINTCNTL (Also see Section 7.5.3 in NISA User’s Manual).

Selective printout control for sensitivity coefficients ALL No —

(1) Minimum abbreviations are in bold face(2) ST : Linear static

5-3

Overview of Input Data

EV : Eigenvalue FR : Frequency response (amplitude) BU : Buckling ALL : ALL: (ST, EV, FR, BU)

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5.2 Supplement to Modified Executive Commands in NISA

5.2.1 Executive Commands

As described in NISA user manual, the executive commands constitute the firstdata block in a typical NISA input deck, and they define general control parametersfor the program execution (chapter on Input Setup and Executive Commands). TheSTROPT program includes additional executive commands, as shown in Table 5.2-1 , which follow the same rules and guidelines as applicable for NISA. Thefollowing general notes apply to all executive commands:

Embedded blanks in executive commands are ignored. Command names may be abbreviated. Minimum abbreviations are shown inbold face letters. Various command options are shown between square bracket. Default valuesof options are shown between braces.

The following commands are optional when user runs STROPT.

5-5

Supplement to Modified Executive Commands in NISA

PERFORM DESIGN OPTIMIZATION OR DESIGN SENSITIVITY ANALYSIS ONLY

DESIGN =

where

OPTImization : Perform both structural optimization and design sensitivity analysis together

SENSitivity : Perform design sensitivity analysis only

SAVE SENSITIVITY COEFFICIENTS FOR POSTPROCESSING

SASEnsitiviy =

where

OFF : Do not save sensitivity coefficients on file 27.

PRINTED : Save sensitivity coefficients on file 27 only for the specified responses given in *PRINTCNTL data group and computed in design sensitivity analysis data block. *PRINTCNTL data group (Section 5.4) is used to control the bulk of sensitivity coefficients printout for the output quantities such as displace-ments, stresses, and so forth.

ALL : Save all sensitivity coefficients on file 27 for the computed responses in design sensitivity analysis data block. Note that this option also saves the bulk of sensitivity coefficients print-out controlled by *PRINTCNTL data group (Section 5.4).

OPTImization{ }SENSitivity

OFF{ }PRINTEDALL

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SORT SENSITIVITY COEFFICIENTS FOR PRINTOUT

SOSEnsitivity =

where

OFF : Printout sensitivity coefficients of response quantities in order of appearance.

ASCENDING : Printout sensitivity coefficients of response quantities in ascending order according to increasing absolute values.

DESCEND-ING

: Printout sensitivity coefficients of response quantities in descending order according to decreasing absolute values. This option is more desirable because response quantities are sorted in degree of importance.

OFF{ }ASCENDINGDESCENDING

5-7

Supplement to Modified Executive Commands in NISA

Table 5.2-1 Alphabetical list of executive commands exclusively for STROPT

Section No. and

Executive command Name(1)

Description When is It Needed?

Is It Always

Required?

Applicable Analysis (2)

DESIGN “Section 5.2.1

Perform structural optimization or design sensitivity analysis only

Design optimiza-tion or design sensi-tivity analysis — ALL

SASEnsitivity “Section 5.2.1

Save sensitivity coefficients for postprocessing on file 27

Plots of sensitivity coefficients vs. glo-bal design vari-ables with DISPLAY III pro-gram

No ALL

SOSEnsitivity “Section 5.2.1

Sort sensitivity coefficients for out-put printouts

Printouts of sensi-tivity coefficients of response quantities in ascending or descending order

No ALL

(1) Minimum abbreviations are in bold face(2) ST : Linear static

EV : Eigenvalue

FR : Frequency response (amplitude)

BU : Buckling ALL : (ST, EV, FR, BU)

5-8


5.3 Description of Optimization Input Data

5.3.1 *OPTPAR Data Group

1. Program Controlling Parameters:

GROUPID = *OPTPAR Program controlling parameters and convergence criteria for the optimization algorithm are pro-vided by this data group. The group contains two cards. User may omit input for both the cards or for the second card. Default values are automati-cally assigned by the program for the omitted entries.

Entry No : 1 2 3 4 5 6 7

Variable : ICOST IPRINT ITRS METH MDS NEQL NIQL

Max char : 10 10 10 10 10 10 10

entry variable description

1 ICOST Cost function type (default = 1)

= 1 - Volume

= 2 - Mass =

= 3

- Weight

2 IPRINT Printing code (default = 1)

= 0 - Initial and final values only

= 1 - Summary of every design iteration

= -1 - Detailed diagnostic output

5-9

Description of Optimization Input Data

2. Convergence Criteria And Limiting Values

3 ITRS Maximum number of iterations

= 0 - Initial analysis only

> 0 - Design optimization

4 METH Parameter to select optimization method (default = 2)

= 1 - Generalized Reduced Gradient (GRG)

= 2 - Recursive Quadratic Programming (RQP)

= 3 - Optimum Cost Bounding (OCB)

5 MDS Sensitivity method (default = 2)

= 1 - Finite difference (semi-analytic)


= 2 - Direct differentiation

= 3 - Adjoint variable

= 4 - Hybrid (recommended)

6 NEQL Number of user-supplied equality constraints (pres-ently 0)

7 NIQL Number of user-supplied inequality constraints (presently 0)

Entry No : 1 2 3 4 5 6 7

Variable : ICOST IPRINT ITRS METH MDS NEQL NIQL

MAX char : 10 10 10 10 10 10 10

5-10


Notes:

1. The parameter ITRS limits the maximum number of design iterations beforeterminating optimization process. It is impossible for even an expert in optimi-zation to predict exact number of design iterations in which a given problemwould converge to an optimal solution. Number of design iterations requiredfor convergence depend on the values of initial design variables, specifiedacceptable tolerances on convergence criteria and type of optimization algo-rithm employed. A good practice to start with, would be to allow the programto go through reasonable (10 to 20) number of design iterations in the first runand then study the output. If the program fails to find optimal solution in thespecified number of maximum iterations (ITRS) then one would try to assessthe best solution obtained. Every design iteration provides a solution to theproblem. So 10 design iterations mean 10 different solutions for comparison toassess the best solution (or design). The best design is one which satisfies thefollowing three conditions simultaneously

Maximum constraints violation should be minimum,Convergence criterion should have minimum value, andThe cost should be minimum.


1 ACT Parameter to define active constraint set (default = 0.01)

2 ACV Acceptable constraint violation at optimum (default = 0.005)

3 ACS Acceptable tolerance on convergence criterion for opti-mum solution (default = 0.01)

4 DEL Design perturbation for gradient calculation of functions by Finite Difference Method (default = 0.01)

5 ALS Tolerance on line search convergence criterion (default = 0.005)

6 GRAVTY Gravitational acceleration (default = 9,810 mm/sec2)

5-11


If one finds the best solution that is very close to the optimum solution thenprobably it is not necessary to rerun the program with new initial values ofdesign variables. However, if the best solution is not satisfactory then onewould rerun the program with only one change in the input data. The change isin the initial (or starting) values of the design variables. These values should besame as those obtained for the best design of the previous run.

2. If the input for both of the cards is not given, then the program assigns adefault value of 10 to the maximum number of iterations (ITRS).

3. STROPT carries out initial structural analysis only and saves the post-process-ing files for the initial design if ITRS is set to zero.

4. User-supplied equality and inequality constraints should be incorporated inspecially written subroutines. Current version of STROPT does not supportthis capability. Input 0 for NEQL as well as NIQL for the present.

5. Value of the parameter ACT decides what constraints are to be consideredactive during a design iteration. An active constraint is that which is either vio-lated or is on the brink of violation. Larger the value of ACT greater will be thenumber of active constraints. Usually a value of 0.01 is reasonably good. How-ever, for the optimum cost bounding algorithm a value as high as 0.4 is recom-mended.

6. ACV is the value of acceptable constraint violation at optimum. Ideally, theoptimum solution should be a feasible design that is to say that the maximumconstraint violation should be zero. A value of 0.005 for ACV may be consid-ered satisfactory. Maximum constraint violation of 0.004 means 0.4 percentviolation. If the violation is of a stress constraint with a limiting value of 30 ksithen a 0.4 percent violation would mean a stress of 30 + 30 *.004 = 30.12 ksi.Larger the specified value of ACV, smaller would be the optimum cost andalso the number of design iterations to converge to the optimum.

7. ACS is the acceptable value of convergence criterion for optimum solution.Ideally, the convergence criterion should have a value of zero at optimum.However, in numerical algorithms it is difficult to attain zero value in reason-able number of iterations. Sometimes, for highly nonlinear cost and constraintfunctions, convergence criterion needs to be relaxed to the extent of 0.05 or so.A value of 0.01 for ACS may be considered reasonable for most of the optimi-zation problems.

5-12


8. ALS is the acceptable tolerance on the value of stopping criterion for linesearch. This is similar to ACS and is required only for the first algorithm, i.e.Generalized Reduced Gradient Method.

5.3.2 *DVGROUP Data Group

.

GROUPID = *DVGROUP Design variable group table. This data group is always required. It defines all the design variable groups for the structure. Design variable groups may be generated or defined separately.

Entry No : 1 2 3 4 5

Variable : IDVG1 IDVG2 INCR ISECT NDV

Max char : 10 10 10 10 10


1 IDVG1 Starting value of design variable group number

2 IDVG2 Last value of design variable group number (default = IDVG1)

3 INCR Increment for generating design variable groups between IDVG1 and IDVG2. (default = 0)4 ISECT

4 ISECT Parameter defining totality of local-designvariable-type, sec-tion-type and element-type

= 0 - Unsupported element types

= 1 - Constant thickness over the element

= 2 - Variable thickness at each node

= 3 - Solid circular section

= 4 - Tubular circular section

5-13


= 5 - Solid rectangular section

= 6 - Tubular rectangular section

= 7 - Cross-sectional area

= 8 - Laminated composite shell

= 9 - Tee section

= 10

- I section

= 11

- Channel section

= 12

- Angle Section

= 13

- Open hat section

= 14

- Closed hat section

= 21

- Generalized solid section

= 22

- Generalized thin-walled section with non-uniform wall thickness

= 23

- Generalized thin-walled section with uni-form wall thickness

5 NDV Number of local design variables in IDVG1. NDV is always positive and cannot be greater than 70 in the current version of STROPT. Default values for different ISECT values are:

ISECT

= 0 - NDV

= 0

= 1 - = 1

= 2 - = No default value


5-14


= 3 - = 1

= 4 - = 2

= 5 - = 2

= 6 - = 4

= 7 - = 1


= 9 - = 4

= 10 - = 6

= 11 - = 4

= 12 - = 4

= 13 - = 4

= 14 - = 5



= 23 = 1


5-15


Notes:

1. Input data about design variables is collected from the groupids *DVGROUP,*DVGELMT, *DVLOCAL and *DVLINK. All the information about designvariables is correlated through design variable group number (IDVG). EachIDVG has one element or section type (ISECT) associated with it. Once avalue for ISECT is assigned, the program can identify the types and numberingof local design variables. Refer to Section Library given in Table 4.2.1 formore information. Each IDVG may have one or more elements associated withit. This information is collected from the input of groupid *DVGELMT. Onealso needs to input the status (free of fixed), initial, upper and lower boundinformation for each of the local design variable. This information is given ingroupid *DVLOCAL. Information about linking of local design variables isthen collected from the groupid *DVLINK.

2. Design variable group number must be positive.

3. If the second and third entries are zeroed or blanked then design variable groupnumbers are not generated and only one design variable group number -IDVG1 - gets specified.

4. If a design variable group number is input more than once then the data per-taining to the last input of the design variable group number is used by the pro-gram.

5. For more information on ISECT and NDV refer to Section Library given inTable 4.2-1 .

6. NDV includes fixed and free design variables.

7. Two or more design variable groups can have same ISECT value.

8. IDVG and ISECT are set to zero automatically by the program for the elementtypes (NKTP) which exist in the structure but are not supported in STROPT. Insuch a case, these element types relevant to the given design variable groupsare not considered in the design optimization process. However, sectionalproperties should be provided in *RCTABLE data group to analyze the struc-tural system.

9. Design variable group numbers must start from 1 with an increment of 1.

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5.3.3 *DVGELMT Data Group

First Alternative

GROUPID =

*DVGELMT Design variable group number and associatedelement numbers. Element numbers must bethe same as defined in *ELEMENT or *E1data card. This data group is always required.It offers two alternatives to associate elementswith design variable groups. The first alterna-tive is for element generation and the secondone for assigning randomly numbered ele-ments to the design variable groups.

Entry No : 1 2 3 4 5 6 7

Variable : IDVG1 IELM1 IELM2 INCR1 IDVG2 INCR2 INCRDV

Max char : 10 10 10 10 10 10 10


1 IDVG1 Base (starting) design variable group number

2 IELM1 Base (starting) element number for IDVG1

3 IELM2 Last element number for IDVG1 (default = IELM1)

4 INCR1 Increment for element generation within one design variable group (default = 0)

5 IDVG2 Last design variable group number (default = IDVG1)

6 INCR2 Increment over the last element number of the previous design variable group number (default = INCR1)

7 INCRDV Increment for generating design variable group numbers (default = 0)

5-17


Notes:

1. All design variable groups generated or specified here must be defined in*DVGROUP data group.

2. This alternative offers three options in assigning elements to the design vari-able groups.

(i) If only first two entries are given and the rest are zeroed or blanked thenonly one element IELM1 is associated with one design variable groupIDVG1.

(ii) When first four entries are given and the remaining zeroed or blankedthen several elements between IELM1 and IELM2 with an increment ofINCR1 are associated with the design variable group IDVG1.

(iii) When first five entries are given and the remaining zeroed or blankedthen design variable groups with increment of one can be associatedwith the group of elements.

(iv) When all the entries are given then elements as well as design variablegroups can be generated and their association can be defined. Designvariable groups between IDVG1 and IDVG2 with an increment ofINCRDV are generated. Elements are assigned to the first design vari-able group as per option (ii) and for the subsequent design variablegroup, equal number of elements as in the previous design variablegroup starting with element number (IELM2+INCR2) are assigned. Asan example consider a plate discretized in 24 elements as shown inFigure (a). Element numbering is as shown in the following. Let there

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be 6 design variable groups and the associated element numbers asshown in Figure (b) be as follows:

The input in this case can be given as: 10,1,4,1,60,7,10. Note that 7 here represents 7 = (11-4), 7 = (21-14), 7 = (31-24), 7 = (41-34), and 7 = (51-44)

Second Alternative

1. Design Variable Group Number and Number of Elements

Figure (a)Figure (b)

Entry No : 1 2 3

Variable : IDVG NGEN NELM

Max char : 10 10 10

entry variable description1 IDVG Design variable group number. Element numbers specified

in the second card are linked with this group number

5-19


2. Element Numbers

Notes:

1. A maximum of 8 entries can be given in the second card. If NELM in the firstcard is greater than 8 then use as many second cards as are necessary (e.g. IfNELM = 20, one would need 3 second cards).

Input for this groupid may contain,

(i) All the cards from the FIRST ALTERNATIVE, or

(ii) All the cards from the SECOND ALTERNATIVE, or

2 NGEN Identifier for no generation of elements = a non-positive inte-ger

3 NELM Number of elements associated with IDVG. NELM cannot be more than 300 (default = 1)

Entry No : 1 2 3 4 5 6 7 8

Variable : IELM1

IELM2

IELM3

IELM4

IELM5

IELM6

IELM7

IELM8

Max char : 10 10 10 10 10 10 10 10

entry variable description1 IELM1 First element number associated with IDVG2

8

Subsequent element numbers associated with IDVG

IELM2

IELM8

5-20


(iii) Some cards from the FIRST and some cards from the SECOND ALTER-NATIVE

5.3.4 *DVLOCAL Data Group

1. Design Variable Groups

All design variable groups defined in this card should have the same local designvariables as defined in *DVGROUP data group.

GROUPID = *DVLOCAL Local design variable information. This data group is always required. Information about all local design variables associated with different design variable groups is specified here by giv-ing two data cards as defined below. Additional input cards 3-7 are needed for generalized beam sections.

Entry No : 1 2 3

Variable : IDVG1 IDVG2 INCR

Max char : 10 10 10


1 IDVG1 Base (starting) design variable group number

2 IDVG2 Last design variable group number (default = IDVG1)

3 INCR Increment for generating design variable group numbers (default = 1)

5-21


2. Local Design Variable Information.

Entry No : 1 2 3 4 5 6 7

Variable : ISTAT

START

BOT-TOM

UPPER

ISTAT

ISTART

BOT-TOM

Max char : 10 10 10 10 10 10 10

Entry No : 8

Variable : UPPER

Max char : 10

5-22


Notes:

1. If the second and the third entries are zeroed or blanked in the first cardthen design variable group numbers are not generated and only onedesign variable group number - IDVG1 - gets specified.

2. Each first card input must be followed by the second card input.

3. Design variable group numbers defined in the first card have the localdesign variable information as input in the *DVGROUP data group.

entry vari-able

description

1 ISTAT Design variable status

= 0 - fixed

= 1 - free

2 START Initial value of design vari-able

For local design vari-able number 1

3 BOT-TOM

Lower bound on design variable

4 UPPER Upper bound on design variable

5 -

Same as for entries 1, 2, 3 and 4 but for local design variable num-ber 2

6 -

7 -

8 -

5-23


4. If a design variable group number is input more than once then the datapertaining to the last input of the design variable group number are usedby the program.

5. Group of four entries in second card should be specified for each of thelocal design variable and in the same sequence as defined in the SectionLibrary given in Table 4.2-1 . Fixed design variable refers to no changein the value of the design variable during optimization.

6. If there are more than two local design variables then use as many sec-ond cards as are necessary. (e.g. one would need 3 second cards if num-ber of local design variables is 5 or 6).

7. If a design variable is fixed (ISTAT = 0) then the program assigns initialvalue (START) to the lower (BOTTOM) and upper (UPPER) bounds.

8. Values of the entries in the second card may be changed while linkingdesign variables in *DVLINK data group to provide consistency in theoverall input data.

9. In case of generalized beam sections, each second card input must be fol-lowed by additional card(s) input as follows:

- Card 3 only for generalized thin walled section with uniform wall thick-ness.

- Cards 4, 5, and 6 only for generalized thin walled section with non-uni-form wall thickness.

- Card 7 only for generalized solid section.

Generalized Thin Walled Section With Uniform Wall Thickness

3. Section Property Constants

Input from this card is required only if the design variable group IDVG1 defined infirst card pertains to thin walled section with uniform wall thickness (i.e. ISECT =23). Refer to Section Library given in Table 4.2-1 .

Entry No : 1 2 3 4 5 6 7 8

5-24


Notes:

1. It is assumed that thickness of the cross section is much smaller than theoutside dimension. In which case section properties are nearly linearwith respect to thickness.

2. y and z coordinates are similar to entries of 19 and 20 given in Table4.12.3 of NISA II User Manual. This point is for stress calculation only.

3. Computation of entries 1, 2, 3, 4, and 5 should be based on the thin-walled section assumption. In such a case starting design value given insecond card of *DVLOCAL can be different than the thickness used incalculation of 1, 2, 3, 4, and 5 entries.

Variable : XA XIYY XIZZ XJ XIYZ BRED

TYCOR

DZCOR

D

Max char :


1 XA Ratio of cross-sectional area to wall thickness

2 XIYY Ratio of moment of inertia about y-axis to wall thickness

3 XIZZ Ratio of moment of inertia about z-axis to wall thickness

4 XJ Ratio of torsional constant to wall thickness. For open thin walled sections ignore this entry.

5 XIYZ Ratio of product of inertia with respect to y-z axes to wall thickness

6 BREDT Bredt’s area which is the area enclosed by the wall center-line of the cross section.Should be set to any non-positive number for open thin-walled section.

7 YCORD y-coordinate of the stress point calculation

8 ZCORD z-coordinate of the stress point calculation

5-25


Generalized Thin Walled Section With Non-uniform Wall Thickness

4. Control Parameters for the Generalized Thin Walled Section

Input from cards 4, 5 and 6 is required only if the design variable group IDVG1defined in the first card pertains to the Generalized thin walled section (ISECT =22). Refer to Section Library given in Table 4.2-1 .

Notes:

1. For the purpose of input for cards 4, 5, and 6, centerlines of straightboundaries or sides of a thin walled section are termed as ’members’ andthe points where two or more members meet as well as the free ends ofthe members are called ’nodes.’

Entry No : 1 2 3 4 5

Variable : NODES IZSM IYSM NTCL NICL

Max char : 10 10 10 10 10

entry variable description1 NODES Total number of nodes used for defining the thin walled sec-

tion2 IZSM Indictor for symmetry of section about centroidal Z axis.

= 0 - Section is not symmetric = 1 - Section is symmetric

3 IYSM Indicator for symmetry of section about centroidal Y axis =

0 - Section is not symmetric

= 1 - Section is symmetric4 NTCL Total number of cells5 NICL Number of interior cells

5-26


2. Node numbers should be in sequence, starting with 1 and with an incre-ment of 1.

3. For cellular section, one additional node for each cell should be intro-duced on one of the members forming the cell. The additional nodes mayoverlap the other nodes.

5. Nodal Coordinates for the Generalized Thin Walled Section.

Entry No : 1 2 3 4 5 6

Variable : NODES ZCORD YCORD NODE ZCORD YCORD

Max char : 10 10 10 10 10 10

5-27


Notes:

1. Z and Y coordinates of the nodes are with respect to any convenientlychosen origin in the YZ plane of the element local coordinate system(Refer NISA II User Manual, Section 3.2).

entry variable description 1

2

3

4

5

6

NODE

ZCORD

YCORD

-

-

-

Node number

Z coordinate of the node

Y coordinate of the node

Same as for entries 1, 2 and 3 but for another node number

5-28


2. Node numbers need not necessarily be specified serially for the entries 1,4, 7, etc. However, all nodes should be specified in this card.

3. If the number of nodes (NODES) is even then there will be NODES/2lines of the fifth card input else the number of lines will be (NODES/2) +1.

As shown in the illustration, there are 1 - 13 regular nodes required todefine the geometry of the section. However, there are four cells andtherefore four additional nodes are introduced. These are 14-17. Note thatnodes 14 and 1, 15 and 2, 16 and 7, and 17 and 5 are overlapping nodesalthough they are shown distinct in the illustration for the sake of clarity.Additional nodes indicate that the cells are cut open at those nodes tofacilitate calculation of the shear flows.

4. If the section is cellular then the cells should be numbered serially start-ing from 1 and with an increment of 1.

5. Total number of cells includes number of exterior and interior cells. Asan example, for the section shown in the illustration under note 3, NTCLand NICL should be 4 and 1 respectively. Note that there are 3 exteriorand 1 interior cell in this case.

6. Member Connectivity and cell Information for Generalized Thin Walled sec-tion

Entry No : 1 2 3 4 5 6 7

Variable : MEMBER NNEAR NFAR NTC1 NTC2 NTC3 NTC4

Max char : 10 10 10 10 10 10 10

5-29


Notes:

1. Members should be numbered serially starting from 1 with an incrementof 1.

2. Members are given directional property by assigning near end and farend node numbers. Member connectivity gets defined by specifyingNNEAR and NFAR for all members.

3. The sixth card input for the illustrative section (shown under notes ofcard 5) is given to help user understand all the entries of this card.


1 MEMBER Member number

2 NNEAR Near end node number for the member

3 NFAR Far end node number for the member

4 NTC1 Exterior cell number to the left of the member as one goes from the near to the far end of the member

5 NTC2 Exterior cell number to the right of the member as one goes from the near to the far end of the member

6 NTC3 Interior cell number to the left of the member as one goes from the near to the far end of the member.

7 NTC4 Interior cell number to the right of the member as one goes from the near to the far end of the member.

1, 1, 2, 0, 1, 0, 02, 2, 3, 4, 1, 0, 03, 3, 4, 0, 1, 0, 04, 4, 14, 0, 1, 0, 05, 3, 5, 4, 0, 0, 0

5-30


4. If the section has no cells (open thin walled) then entries 4-6 may be leftblank or zeroed.

5. There will be as many lines of the sixth card input as there are number ofmembers.

7. Nodal Coordinates for Generalized Solid Section.

Input from this card is required only if the design variable group IDVG1 defined inthe first card pertains to the Generalized Solid Section (ISECT = 21). Refer toSection Library given in Table 4.2-1 .

6, 5, 6, 4, 3, 0, 07, 6, 7, 4, 0, 0, 08, 7, 8, 4, 0, 2, 09, 8, 15, 4, 0, 0, 0

10, 8, 9, 2, 4, 0, 011, 9, 10, 2, 4, 0, 012, 10, 16, 2, 4, 0, 013, 6, 11, 0, 3, 0, 014, 11, 12, 0, 3, 0, 015, 12, 17, 0, 3, 0, 016, 11, 13, 0, 0, 0, 0

Entry No : 1 2 3 4 5 6

Variable : NODE ZCORD YCORD NODE ZCORD YCORD

Max char : 10 10 10 10 10 10

1, 1, 2, 0, 1, 0, 0

5-31


Notes:

1. Nodes are the points where two sides of a solid section meet. Nodesshould be numbered serially starting from 1 with an increment of 1 in aclockwise or counterclockwise sense. Number of nodes equals the num-ber of side lengths or the number of local design variables.

2. Z and Y coordinates of the nodes are with respect to any convenientlychosen origin in the YZ plane of the element local coordinate system(Refer NISA II User Manual, Section 3.2).

3. Node numbers need not necessarily be specified serially for the entries 1,4, 7, etc. However, all nodes defining the geometry of the solid sectionshould be specified. As an example, consider a quadrilateral solid sec-tion. The section must have four nodes. User may specify 2, 3, 1, 4 as theinput against entries 1, 4, 7, 10 respectively.


1 NODE Node number

2 ZCORD Z coordinate of the node

3 YCORD Y coordinate of the node

4

5

6

-

-

-

Same as for entries 1, 2 and 3 but for another node number

5-32


5.4 Supplement to Modified Data Block In NISA

5.4.1 PRINTCNTL Data Group - Selective Printout Control for Sensitivity Coefficients

Applicable analysis types: STATIC, EIGENVALUE, BUCKLING, FREQUENCY

This data group may be used to control the bulk of printout for data groups relatedto NISA and STROPT. It must be placed among NISA data groups, i.e.before*OPTPAR data group. It can be used to control the bulk of sensitivity coefficientsprintout for the output quantities such as displacements, stresses, naturalfrequencies, buckling load factor, and amplitudes. The printout of a typicalsensitivity coefficients output may be provided in its entirety, totally suppressed orprovided for a subset(s) of nodes or elements.

In some cases, only the group-ID card may be needed, see note 2.

GROUP ID card: *PRINTCNTL

Entry No : 1 2 3 4 5 6 7

Variable : LABEL I1 I2 I3 I4 I5 I3

Max char : 12 6 6 6 6 6 6

Entry No : 8 9 10

Variable : I4 I5 9

Max char : 6 6 6

5-33

Supplement to Modified Data Block In NISA

Printout control card set:


1 LABEL Output label (up to 12 characters, only first 4 charactersare used). Allowable characters are:

The following labels are applicable to Static, Eigenvalue,Buckling, and Frequency Response Analysis

Default Option

SEAMPLITUDES Sensitivitycoefficients foramplitudes

-1

SEBUCKLING Sensitivitycoefficients forbuckling load factor

-1

SEDISPLACE-MENTS

Sensitivitycoefficients fordisplacements

-1

SEFREQUEN-CIES

Sensitivitycoefficients fornatural frequencies

-1

SESTRESSES Sensitivitycoefficients forstresses

-1

2 I1 Output option or set identification number

<0 Output designated by LABEL will be suppressed

=0 Output designated by LABEL will be provided in its entirety (e.g., stress sensitivity coefficients for all ele-ments)

5-34


Notes:

1. If more than 9 sets are required to define the subset for a typical output quan-tity, continue on additional cards. Additional cards must start with a tab($)character. Zero set ID numbers are ignored.

2. In some cases, the group ID card may be all what is needed to fully define theprintout options as follows:

Supplying just the group ID card has the effect of duplicating the printoutoptions defined in the preceding load case in static analysis (eigenvalue outputblock in eigenvalue analysis or event in nonlinear static analysis).The group ID card may take the form: *PRINTCNTL,IREF = n, where n is aninteger 0, referring to a previous load case ID is static analysis (eigenvalue out-put block in eigenvalue analysis or event in nonlinear static analysis). This hasthe effect of duplicating the printout options defined in the referenced loadcase.

3. This data group controls the printout of a typical output quantity provided thatthe computation of that quantity has been requested in the *LDCASE datagroup (for static analysis), in the *EIGOUT data group (for eigenvalue analy-sis), the *NLOUT data group (for nonlinear static analysis), or in the*LDCOMB data group (for load combination in static analysis).

4. For Static, Eigenvalue, Buckling, and Nonlinear Static Analysis, outputrequests made in alternate data groups (*I5, *N5, *REGIONS) supersedethose specified in this data group.

5. Sensitivity coefficients of buckling load factor can be printed for only one loadcase as given in *LDCASE data group. Consequently, label ‘SEBUCKLING’should be specified for the same load case only.

>0 Output designated by LABEL will be provided for members of the set with an ID of I1, which has been defined in the *SETS data group.

3 - 10 I2 to I9 Additional set ID numbers to complete the definition of the subset for which the output is requested, valid if I1>0 only.

5-35

Supplement to Modified Data Block In NISA

6. Sensitivity coefficients for all types of responses are with respect to globaldesign variables. Before printout of sensitivity coefficients, STROPT prints atable of global design variables numbering scheme. This table identifies therelationship between global design variable numbers and design variablegroup number given in *DVGROUP data group. Detailed definitions of localand global design variables are given in the STROPT User Manual Section 2.4.

7. Labels SEAM, SEBU, SEDI, SEFR, SEST also print out the most criticaldesign variable numbers (i.e. the most important) and their sensitivity coeffi-cients. Here, critical design variable number is defined as the global designvariable number corresponding to the largest absolute value of response sensi-tivity coefficients. For example if sensitivity coefficients for displacement atnode 5 are 0.4 for design variable number 1, -1.2 for design variable number 2,and 0.9 for design variable number 3, then the most critical design variablenumber is 2 having sensitivity coefficient of -1.2 (largest absolute value).

5-36

Appendix

A
Example Problems
A.1 Optimum Design and Design Sensitivity Analysis of a Deep Beam Opti-mum Design of a Deep Beam

Title

Optimum Design and Design Sensitivity Analysis of a Deep Beam

Problem

A deep beam with dimensions shown in Figure A.1-1 is subjected to a uniformpressure of 1 lb/in. The pressure load is substituted by equiplollent nodal forces inthe finite element model. Due to symmetry, only half of the structure is modeledas shown in Figure A.1-2.

The following objectives are desired:

1. To find element thicknesses that minimize the beam volume while satisfyingthe constraints on von Mises stresses at Gauss points.

2. To compute sensitivity coefficients of von Mises stresses at all Gauss points(i.e. INNATE=9).

Element Type

2D Plane Stress (NKTP = 1, NORDR = 2)

A-1

Optimum Design and Design Sensitivity Analysis of a Deep Beam Optimum Design of a Deep Beam

Section Type

Uniform Thickness (ISECT = 1)

Material Properties

Design Data

Elastic Modu-lus = 30,000 ksi

Poisson’s Ratio = 0.3

No. of Design Variables = 16 (uniform shell thickness for each element)

No. of Load Cases = 1

No. of Elements = 16

No. of Constraints = 64 (one at each Gauss point of every element)

Starting Thickness of all Shell Elements (Design Variables) = 1.0 in.

Upper Bound on Design Variables = 3.00 in.

Lower Bound on Design Variables = 0.01 in

Stress Limit (von Mises) = 30.0 ksi

A-2

Example Problems

Remarks

Detailed output format for this Example problem using Optimum Cost Bounding isgiven in Appendix C.

Figure A.1-1 Simply supported deep beam

A-3


Figure A.1-2 Finite element model

Structrual Optimization Input Data for Example Problem A.1 ANALYSIS = STATICLOAD=1AUTO=ONMAXWAVE=750WARN=GO** TOTAL NO.OF ELEMENTS = 16** TOTAL NO.OF NODES = 81*TITLE Problem TitleOPTIMUM DESIGN OF A DEEP BEAM*ELTYPE Element Type Selection (2-D Plane Stress)1,1,2*RCTABLE Real Constant Table1, 81.0///////*E1 Element Connectivities

A-4

Example Problems

$$,-4,18,41,1,2,3,12,21,20,19,10,1//4,2,1*NODES Nodal Coordinates-9,0,9,9,0.0,1.251,0//,0.0/9,0,1,0,10.0,0.0*MATERIAL Material Property DataEX,1,0,30000NUXY,1,0,0.3*LDCASE Static Load Case Control Card0, 0, 3, 0, 0, 0, 1, .0000E+00, .0000E+00*SPDISP Specified Displacements9,UX,0.0,81,91,UY,0.0*CFORCE Concentrated Nodal Forces73,FY,-0.625,7374,FY,-1.25,80,181,FY,-0.625,81*PRINTELST,0*OPTPAR Control Parameters for Optimization1,1,20,2,40.1,0.01,0.01,0.01*DVGROUP Design Variable Groups and Section Type1,16,1,1,1*DVGELMT Design Variable Groups and Element Numbers1,1,0,0,16,1,1*DVLOCAL Local Design Variable Information1,16,11,1.0,.01,3.0*STRCON Stress Constraint IDs1,VON,30.0

A-5


Input Data for Example Problem A.1 (continued) *STRELM Element Numbers For Stress Constraint IDs1,1,91,16,1*ENDDATA Data Deck Terminator

Design Sensitivity Analysis Input Data for Example Problem A.1PROBLEM=EXAM01ANALYSIS=STATIC***************** Perform Design Sensitivity Analysis ******design=sensitivity************************************************************LOAD = 1AUTO = ONMAXWAVE = 750WARN = GO** TOTAL NO.OF ELEMENTS=16** TOTAL NO.OF NODES =81** STROPT Example Problem #01*TITLEOPTIMUM DESIGN OF DEEP BEAM*ELTYPE1,1,2*RCTABLE 1, 81.0///////*E1$$,-4,18,4 1,1,2,3,12,21,20,19,10,1//4,2,1*NODES-9,0,9,9,0.0,1.251,0//,0.0/9,0,1,0,10.0,0.0*MATERIALEX,1,0,30000.NUXY,1,0,0.3*LDCASE0, 0, 3, 0, 0, 0, 1, .0000E+00, .0000E+00*SPDISP

A-6

Example Problems

Results for Structural Design Optimization

Table A.1-1 on page 8 shows optimum results obtained by three optimizationalgorithms (GRG, RQP and OCB). Computational effort required in this problem isgiven in Table A.1-2 on page 9 with the total numbers of structural and sensitivityanalyses. The convergence parameters ACV, ACS and ALS used in this exampleare 0.01, 0.01, and 0.001, respectively. Parameter ACT is assigned a value of 0.01for GRG and 0.1 for both RQP and OCB. Design iteration histories are given intables Table A.1-3 - Table A.1-5. It is observed that initial design is feasible and areduction of 90 % in volume is achieved. Figure A.1-3 shows graphically theobjective function histories.

9,UX,0.0,81,91,UY,0.0*CFORCE73,FY,-0.625,7374,FY,-1.25,80,181,FY,-0.625,81*PRINTELST,0sest,0*OPTPAR1,-1,20,3,20.1,0.01,0.01,0.01,0.001*DVGROUP1,16,1,1,1*DVGELMT1,1,0,0,16,1,11,16,1*DVLOCAL1,1.0,.01,3.0********** For Stress sensitivity Stress limit is 1.0 ******STRCON1,VON,1.0 ************************************************************STRELM1,1,91,16,1*ENDDATA

Design Sensitivity Analysis Input Data for Example Problem A.1 (Contin-

A-7


Table A.1-1 Optimum results of deep beam Design Group

Number

User Element Numbers

Initial Design (in.)

Optimum (Final) Design

GRG RQP OCB

1 1 .10000E+01 .44815E+00 .44844E+00 .44859E+00

2 2 .10000E+01 .97430E-01 .98796E-01 .10081E+00

3 3 .10000E+01 .10633E+00 .10461E+00 .10493E+00

4 4 .10000E+01 .11337E+00 .11294E+00 .11291E+00

5 5 .10000E+01 .20174E+00 .19651E+00 .19621E+00

6 6 .10000E+01 .69499E-01 .72272E-01 .73134E-01

7 7 .10000E+01 .50303E-01 .18129E-01 .20188E-01

8 8 .10000E+01 .69577E-01 .10000E-01 .10000E-01

9 9 .10000E+01 .88104E-01 .83972E-01 .84194E-01

10 10 .10000E+01 .10558E+00 .10910E+00 .10851E+00

11 11 .10000E+01 .79604E-01 .59818E-01 .59113E-01

12 12 .10000E+01 .66839E-01 .20097E-01 .22488E-01

13 13 .10000E+01 .59086E-01 .42021E-01 .42174E-01

14 14 .10000E+01 .52116E-01 .58720E-01 .57340E-01

15 15 .10000E+01 .59672E-01 .70212E-01 .70086E-01

16 16 .10000E+01 .71199E-01 .95142E-01 .94565E-01

Total Volume (in3) .20000E+03 .21733E+02 .19999E+02 .20065E+02

A-8

Example Problems

Table A.1-2 Comparison of optimization algorithms for deep beam

Table A.1-3 Design iteration history for deep beam with GRG

Algorithm NITa

a. Total number of design iterations

NSAb

b. Total number of structural analyses

NDSc

c. Total number of sensitivity analyses

CPU timed (sec)

d. ELXSI 6400 system

CPU time/Design

Iteration (sec)GRG 12 79 11 256.26 21.36RQP 11 16 9 77.7 7.1OCB 16 16 13 98.5 6.2

Iteration Number

Objectivea

Function

a. Material volume of one half of the deep beam

Convergenceb Parameter

b. Squared norm of projected reduced gradient (ACS)

Convergencec Parameter

c. Estimated stepsize (ALS)

1 .10000E+03 .62500E+03 .22042E+01

2 .44894E+02 .58593E+03 .10442E+01

3 .19618E+02 .53414E+03 .30080E+00

4 .12703E+02 .46602E+03 .92855E-02l

5 .12505E+02 .41028E+03 .13928E-01

6 .12225E+02 .56751E+03 .13928E-01

7 .12118E+02 .55516E+03 .16202E-01

8 .12019E+02 .44611E+03 .64808E-01

9 .11586E+02 .26951E+03 .49019E-01

10 .11035E+02 .88305E+04 .14217E-01

11 .10961E+02 .44519E+03 .18189E-01

12 .10866E+02 .26959E+03 .28420E-03

A-9


Table A.1-4 Design Iteration History for Deep Beam with RQP

Iteration Number

Objectivea

Function



b. Norm of Lagrangian gradient (ACS)


c. Maximum constraint violation (ACV)

1 .10000E+03 .25000E+02 .00000E+00

2 .50500E+02 .25000E+03 .00000E+00

3 .25750E+02 .33636E+03 .74580E+00

4 .14178E+02 .25445E+03 .74403E+00

5 .93795E+01 .22024E+02 .72888E+00

6 .82589E+01 .69012E+01 .72629E+00

7 .80199E+01 .18899E+00 .72206E+00

8 .82230E+01 .11374E+00 .56381E+00

9 .94884E+01 .51341E-01 .14896E+00

10 .99485E+01 .73899E-02 .17070E-01

11 .99993E+01 .12574E-03 .28127E-03

A-10

Example Problems

Table A.1-5 Design Iteration History for Deep Beam with OCB

Iteration Number

Objectivea Function



b. Norm of search direction vector (ACS)



1 .10000E+03 .10000E+01 .00000E+00

2 .75000E+02 .10000E+01 .00000E+00

3 .50000E+02 .10000E+01 .00000E+00

4 .29750E+02 .10393E+00 .51107E+00

5 .29750E+02 .46971E-01 .12905E+00

6 .29750E+02 .59729E-02 .13221E-01

7 .29750E+02 .96833E+00 .17002E-03

8 .18906E+02 .90984E-02 .44120E-01

9 .18906E+02 .92323E+00 .18633E-02

10 .12704E+02 .28173E-02 .10776E-01

11 .12704E+02 .64727E+00 .11125E-03

12 .89532E+01 .84546E-01 .15291E+01

13 .92507E+01 .33554E-01 .55246E+00

14 .98931E+01 .31969E-01 .15266E+00

15 .99830E+01 .30039E-02 .24199E-01

16 .10033E+02 .42042E-02 .58322E-03

A-11


Figure A.1-3 Objective function histories for deep beam

Results for Design Sensitivity Analysis

There are 16 elements with the same type of design variable. The design variable isthe constant thickness over the element. Therefore, there are a total of 16 (16*1)global design variables in this finite element model. For NKTP=1, NORDR=2,number of Gauss points are 4, thus, for INNATE=9, number of von Mises stresssensitivities for 16 elements, requested by *STRELM data group are 64 (4*16).

Sensitivity coefficients of stresses are calculated with respect to global designvariables and printed out for all elements at Gauss points. Printout selectionsdepend on the specified entries in *SETS and LABEL in *PRINTCNTL datagroups (see input data). Table A.1-6 on page 13 shows the relations between designvariable group number specified in *DVGROUP data group and global designvariable numbering scheme computed internally by the program. Sensitivitycoefficients are computed for all elements at Gauss points.

A-12

Example Problems

As an example sensitivity coefficients of von Mises stress for element 12 at 3rdGauss point is given in Table A.1-7 on page 14 and graphically illustrated inFigure A.1-4. Using Eq. Eqeation 2.5-9 and Figure A.1-4 one can conclude thatvon Mises stress for this element is more sensitive to thicknesses of elements 12and 16 and less sensitive to other element thicknesses. Thus, changes in designvariables 12 and 16 decrease von Mises stress. These type of graphs are very usefulin case of constraint corrections and can be plotted in DISPLAY III under an optioncalled post results and graphs.

Table A.1-6 Global design variable (D.V.) numberingD.V. Group

Number Type of D.V. Number of D.V.

Local D.V. Number

Global D.V. Number

1 thickness 1 1 12 thickness 1 1 23 thickness 1 1 34 thickness 1 1 45 thickness 1 1 56 thickness 1 1 67 thickness 1 1 78 thickness 1 1 89 thickness 1 1 9

10 thickness 1 1 1011 thickness 1 1 1112 thickness 1 1 1213 thickness 1 1 1314 thickness 1 1 1415 thickness 1 1 1516 thickness 1 1 16

A-13


Table A.1-7 Design sensitivity coefficients of Von mises stress for element 12 at gauss point 3

Global Design Variable No. Sensitivity Coefficients

1 0.18539E-022 0.41149E-013 0.16064E-014 -0.90574E-015 -0.25810E-016 0.78389E-017 0.33442E-018 -0.15730E-019 -0.47568E-01

10 -0.28790E-0111 0.17040E+0012 -0.72312E+0013 -0.15345E-0114 -0.61970E-0115 -0.14836E-00

A-14

Example Problems

Figure A.1-4 von Mises stress sensitivity coefficients for deep beam (Element 12 at gauss point number 3)

A-15

Optimum Design of a Simply Supported Beam

A.2 Optimum Design of a Simply Supported Beam

Title


Problem

A simply supported beam with dimensions shown in Figure A.2-1 is subjected to auniform pressure of 10 lb/in.

The design objective is to find element radii that minimize the beam volume whilesatisfying the constraints on 1) displacement at node 3 in Y-direction, 2) maximumstresses due to bending moments about Z-axis for all elements, 3) lowest naturalfrequency.

Element Type

3D General Beam (NKTP = 12, NORDR = 1)

Section Type

Solid Circular Section (ISECT = 3)

Material Properties

Elastic Modulus = 30,000 ksiPoisson’s Ratio = 0.3Mass Density = .000728 lb.s2/in2

A-16

Example Problems

Design Data

Figure A.2-1 Simply supported beam

No. of Design Variables = 4No. of Load Cases = 1No. of Elements = 4No. of Stress Constraints = 8No. of Displacement Constraints = 1No. of Natural Freq. Constraints = 1Upper Bound on Design Variables = 10.0 in.Lower Bound on Design Variables = 0.10 in.Stress Limit = 40.0 ksiDisplacement Limit = 0.1 in.Natural Frequency Lower Limit = 30 Hz.

A-17


ANALYSIS = STATICLOAD CASE NO = 1AUTO CONSTRAINT = OFFRESEQUENCING OF ELEMENT = OFFEIGEN EXTRACTION = SUBSPCAE,CONVENTIONALMASS FORMULATION = CONSISTANT*TITLE Problem TitleOPTIMUM DESIGN OF A SIMPLY SUPPORTED BEAM*ELTYPE Element Type Selection (3-D Beam)1,12,1*RCTABLE Real Constant Table1, 8.1775,.00251,.00251,.00501,0.,0.,.4754,.4754*E1 Element Connectivities1,1,2,$,1,1,1,4,1,1*NODES Nodal Coordinates1,0,$,0.0,0.05,0,1,0,80.0,0.0*MATERIAL Material Property DataEX,1,0,30.0E6NUXY,1,0,0.3DENS,1,0,7.28E-4*EIGCNTL Eigenvalue Analysis Control Card1,0,10*LDCASE Load Case Control Card0, 1, 2, 0, 0, 0, 0, .0000E+00, .0000E+00*SPDISP Specified Displacements1,UX,0.0,$,UY,UZ,ROTX,ROTY5,UY,0.0,$,UZ,ROTX,ROTY*L1 Pressure Data1,0,4,1,1,0,10.0,10.01*OPTPAR Control Parameters for Optimization1,1,20,2,10.1,0.01,0.01,0.01*DVGROUP Design Variable Groups and Section Type1,4,1,3*DVGELMT Design Variable Groups and Element Numbers1,1,1,1,4

A-18

Example Problems

*DVLOCAL Local Design Variable Information1,41,1.0,0.1,10.0*DISCON Displacement Constraint IDs1,UY,0.1*DISNOD Node Numbers For Displacement Constraint IDs1,13,3*STRCON Stress Constraint IDs1,BEND3,40000.*STRELM Element Numbers for Stress Constraint IDs1,1,911,4,1*FRQCON Natural Frequency Constraints1,1,30.0*ENDDATA Data Deck Terminator

A-19


Results

Table A.2-1 on page 20 shows optimum results obtained by three optimizationalgorithms (GRG, RQP and OCB). Computational effort required in this problem isgiven in Table A.2-2 on page 20 with the total numbers of structural and sensitivityanalyses. The convergence parameters ACV, ACS and ALS used in this exampleare 0.01, 0.01, and 0.001, respectively. Parameter ACT is assigned a value of 0.01for GRG and 0.1 for both RQP and OCB. Design iteration histories are given inTables Table A.2-3 - Table A.2-5. It is observed that initial design is infeasibleand there is an increase of 38% in volume. Figure A.2-2 shows graphically theobjective function histories.

Table A.2-1 Optimum results of simply supported beam

Table A.2-2 Comparison of optimization algorithms for simply supported beam

Design Group

Number




GRG RQP OCB

1 1 .10000E+01 .10737E+01 .10528E+01 .10553E+01

2 2 .10000E+01 .12851E+01 .12770E+01 .13107E+01

3 3 .10000E+01 .12529E+01 .12821E+01 .12608E+01

4 4 .10000E+01 .10735E+01 .10609E+01 .10552E+01


Algorithm NITa


NSAb


NDSc


CPU timed

(sec)


CPU time/Design

Iteration (sec)GRG 8 76 7 864.11 108.01RQP 8 22 8 268.33 33.54OCB 22 22 22 309.02 14.46

A-20

Example Problems

Table A.2-3 Design iteration history for simply supported beam with GRGIteration Number

Objectivea Func-tion

a. Material volume of the beam



Convergencec Param-eter


1 .12877E+04 .32363E+06 .20749E+012 .37781E+03 .30841E+05 .21748E+003 .34983E+03 .12935E+06 .54370E-014 .34812E+03 .12488E+06 .27185E-015 .34809E+03 .13016E+06 .27185E-016 .34758E+03 .54527E+06 .33981E-027 .34721E+03 .12616E+06 .27185E-018 .34711E+03 .12908E+06 .00000E+00

Table A.2-4 Design iteration history for simply supported beam with RQP

Iteration Number

Objectivea Function


Convergenceb Param-eter




1 .25133E+03 .13512E+03 .12635E+012 .23104E+03 .32700E+00 .13094E+013 .31606E+03 .23116E+01 .21227E+004 .31609E+03 .57854E+00 .21109E+005 .31785E+03 .17735E+00 .20133E+006 .34095E+03 .10647E+00 .64672E-017 .34201E+03 .29393E-01 .20854E-018 .34643E+03 .21026E-02 .10404E-02

A-21


Table A.2-5 Design iteration history for simply supported beam with OCBIteration Number

Objectivea Function




Convergencec

Parameter


1 .25133E+03 .26845E+00 .12635E+012 .31274E+03 .11362E+00 .27350E+003 .34358E+03 .17635E-01 .32023E-014 .34856E+03 .28086E-01 .18344E-035 .32771E+03 .17344E+00 .35349E+006 .32960E+03 .36189E+00 .11334E+007 .33783E+03 .17596E+00 .25305E+008 .33977E+03 .88021E+00 .62320E-019 .38845E+03 .24316E+00 .36931E+01

10 .35227E+03 .19662E+00 .14044E+0111 .35099E+03 .17264E+00 .47949E+0012 .35043E+03 .12013E+00 .14211E+00

13 .34947E+03 .64935E-01 .36268E-0114 .34882E+03 .61114E-01 .77639E-0215 .33356E+03 .18382E+00 .44012E+0016 .33568E+03 .15384E+00 .14043E+0017 .33717E+03 .34010E+00 .45707E-0118 .34444E+03 .15067E+00 .26610E+0019 .34586E+03 .92643E-01 .71338E-0120 .34640E+03 .28845E-01 .16623E-0121 .34861E+03 .20708E+00 .12692E-0222 .34750E+03 .14107E+00 .17877E-02

A-22

Example Problems

Figure A.2-2 Objective function histories for simply supported beam

A-23

Optimum Design of a Four-Bar Truss

A.3 Optimum Design of a Four-Bar Truss

Title


Problem

A four-bar truss with dimensions shown in Figure A.3-1 is subjected to the threedifferent loading cases.The design objective is to find areas of elements that minimize the truss volumewhile satisfying the constraints on 1) displacement at node 5 in Y- and Z-directionand 2) axial stresses for all elements.

Element Type

3D Spar (NKTP = 14, NORDR = 1)

Section Type

Cross-sectional Area (ISECT = 7)

Material Properties

Elastic Modulus = 10,000 ksi

A-24

Example Problems

Design Data

Load Data

No. of Design Variable Groups = 4


No. of Elements = 4

No. of Stress Constraints = 12

No. of Displacement Constraints = 6

Upper Bound on Design Variables = 10.0 in.2

Lower Bound on Design Variables = 0.10 in.2

Stress Limit = 25.0 ksi

Displacement Limit in Y-Dir. = 0.3 in.

Displacement Limit in Z-Dir. = 0.4 in.

Load Case Number

Node Num-ber

Load (Kips)

X Y Z

1 5 5.0 0.0 0.0

2 5 0.0 5.0 0.0

3 5 0.0 0.0 -7.5

A-25


Figure A.3-1 Geometry of four-bar truss

A-26

Example Problems

ANALYSIS = STATICLOAD CASE NO=3AUTO CONSTRAINT = OFFRESEQUENCING OF ELEMENT = OFF*TITLE Problem TitleOPTIMUM DESIGN OF A FOUR-BAR TRUSS*ELTYPE Element Type Selection (3-D Spar)1,14,1*RCTABLE Real Constant Table1,20.1,0.1*E1 Element Connectivities1,1,5$1,1,12,2,5$2,1,13,3,5$3,1,14,4,5$4,1,1*NODES Nodal Coordinates1,0 $ 0.,0.,0.2,0 $ 0.,192.,0.3,0 $ 204.,192.,0.4,0 $ 204.,0.,0.5,0 $ 60.,120.,96.*MATERIAL Material Property DataEX,1,0,1.0E4EX,2,0,1.0E4EX,3,0,1.0E4EX,4,0,1.0E4*LDCASE, ID = 10, 1, 2, 0, 0, 0, 0, .0000E+00, .0000E+00*SPDISP Specified Displacements1,UX,0.,4,1,UY,UZ*CFORCE Concentrated Nodal Forces5,FX,5.0*LDCASE, ID = 20, 1, 2, 0, 0, 0, 0, .0000E+00, .0000E+00*CFORCE Concentrated Forces5,FY,5.0*LDCASE, ID = 30, 1, 2, 0, 0, 0, 0, .0000E+00, .0000E+00

A-27


*CFORCE Concentrated Nodal Forces5,FZ,-7.5 1*OPTPAR Control Parameters for Optimization1,1,20,2,30.1,0.01,0.01,0.01,0.01*DVGROUP Design Variable Groups and Section Type1,4,1,7*DVGELMT Design Variable Groups and Element Numbers1,1,1,0,4*DVLOCAL Local Design Variable Information1,41,0.1,0.1,10.0*DISCON Displacement Constraint IDs1,UY,0.3,UZ,0.4*DISNOD Node Numbers for Displacement Constraint IDs1,1,1,2,35,5*STRCON Stress Constraints1,AXIAL,25.0*STRELM Element Numbers for Stress Constraint IDs1,1,91,1,2,31,4,1*ENDDATA Data Deck Terminator

A-28

Example Problems

Results

Table A.3-1 on page 29 shows optimum results obtained by three optimizationalgorithms (GRG, RQP and OCB). Computational effort required in this problem isgiven in Table A.3-2 on page 29 with the total numbers of structural and sensitivityanalyses. The convergence parameters ACV, ACS and ALS used in this exampleare 0.01, 0.01, and 0.001, respectively. Parameter ACT is assigned a value of 0.01for GRG and 0.1 for both RQP and OCB. Design iteration histories are given inTables Table A.3-3 - Table A.3-5. It is observed that initial design is infeasibleand there is an increase of 105% in volume. Figure A.3-2 shows graphically theobjective function histories.

Table A.3-1 Optimum results of Four-Bar Truss

Design Group Number

User Ele-ment Num-

bers

Initial Design (in.2)


GRG RQP OCB

1 1 .10000E+00 .21103E+00 .25608E+00 .24180E+00

2 2 .10000E+00 .33924E+00 .30008E+00 .31745E+00

3 3 .10000E+00 .16400E+00 .20856E+00 .18932E+00

4 4 .10000E+00 .15293E+00 .10000E+00 .11800E+00


Table A.3-2 Comparison of optimization algorithms for Four-Bar Truss

Algorithm NITa


NSAb


NDSc CPU timed (sec)

CPU time/Design

Iteration (sec)

GRG 3 32 3 145.59 48.53

RQP 9 24 9 128.58 14.29

OCB 16 16 16 108.32 6.77

A-29


c. Total number of sensitivity analysesd. ELXSI 6400 system

Table A.3-3 Design iteration history for Four-Bar Truss with GRG

Iteration Num-ber


a. Material volume of the four-bar truss





1 .15841E+03 .55995E+05 .62500E-01

2 .14843E+03 .21584E+05 .62500E-01

3 .14322E+03 .44856E+02 .00000E+00

Table A.3-4 Design iteration history for Four-Bar Truss with RQP

Iteration Num-ber




1 .69718E+02 .10835E+00 .12721E+01

2 .94779E+02 .15068E+00 .53506E+00

3 .11999E+03 .15963E+00 .21480E+00

4 .12532E+03 .78150E-01 .13984E+00

5 .12579E+03 .13293E+00 .13550E+00

6 .12585E+03 .99239E-01 .13497E+00

7 .13339E+03 .77007E-01 .73129E-01

8 .14159E+03 .17522E-01 .97369E-02

A-30

Example Problems

9 .14266E+03 .81385E-02 .12095E-02

a. Material volume of the four-bar trussb. Norm of Lagrangian gradient (ACS)c. Maximum constraint violation (ACV)

Table A.3-5 Design iteration history for Four-Bar Truss with OCB

Iteration Num-ber




1 .69718E+02 .10251E+00 .12721E+01

2 .10082E+03 .94741E-01 .47867E+00

3 .13158E+03 .41568E-01 .11912E+00

4 .14543E+03 .49082E-02 .11485E-01

5 .14707E+03 .32261E+00 .12912E-03

6 .14351E+03 .59254E-01 .12711E+00

7 .14351E+03 .47464E-01 .40703E-01

8 .14351E+03 .14972E+00 .94951E-02

9 .13668E+03 .92255E-01 .75360E-01

10 .13668E+03 .29130E+00 .46877E-01

11 .14130E+03 .73185E-01 .67286E+00

12 .14130E+03 .70681E-01 .18171E+00

13 .14130E+03 .54314E-01 .58977E-01

14 .14130E+03 .46494E-01 .19233E-01

15 .14130E+03 .43119E-02 .11024E-01


A-31


Figure A.3-2 Objective function histories for Four-Bar Truss

16 .14282E+03 .15233E-01 .14596E-03

a. Material volume of the four-bar trussb. Norm of search direction vector (ACS)c. Maximum constraint violation (ACV)


A-32

Example Problems

A.4 Optimum Design of a Cantilever Plate

Title

Optimum Design of a Cantilever Plate

Problem

A cantilever plate with dimensions shown in Figure A.4-1 is subjected to a tiploading of 500 lbs. at node 33 in z-direction. The plate is divided into three designvariable groups due to symmetry.

The design objective is to find element thicknesses that minimize the plate volumewhile satisfying separately three different constraint cases as listed below:

Element Type

3D General Shell (NKTP = 20, NORDR = 2)

Section Type

Uniform Element Thickness (ISECT = 1)

Material Properties

Design Data

Case (1) : Stress constraints onlyCase (2) : Stress and displacement constraintsCase (3) : Stress, displacement and natural frequency constraints



Mass Density = .000733 lb s2/in2

No. of Design Variables = 3


No. of Elements = 6

A-33


Figure A.4-1 Geometry of cantilever plate



No. of Natural Frequency Constraint = 1

Upper Bound on Design Variables = 0.0 in

Lower Bound on Design Variables = 0.10 in.


Displacement Limit at node 33 = 1.0 in.

Natural Frequency Limit = 30.0 Hz.


A-34

Example Problems

Case (1) : Stress constraint onlyANALYSIS = STATICAUTO CONSTRAINT = OFFRESEQUENCING_OF_ELEMENT = OFF** TOTAL NO.OF ELEMENTS = 6** TOTAL NO.OF NODES = 29*TITLE Problem TitleOPTIMUM DESIGN OF A CANTILEVER PLATE*ELTYPE Element Type Selection (3-D General Shell)1,20,2*RCTABLE Real Constant Table1, 81.0///////*E1 Element Connectivities$$,-3,10,21,1,6,11,12,13,8,3,2,1,1,1,2,2,1*NODES Nodal Coordinates-5,0,5,7,5.0,0.0,0.01,$,0.0,0.0,0.05,0,1,0,0.0,10.0,0.0*MATERIAL Material Property DataEX,1,0,.3E8NUXY,1,0,.3DENS,1,0,.000733*EIGNTRL Eigenvalue Analysis Control Card1,0,10*LDCASE Static Load Case Control Card0, 1, 2, 0, 0, 0, 0, .0000E+00, .0000E+00*SPDISP Specified Displacements1,UX,0.,5,1,UY,UZ,ROTX,ROTY,ROTZ6,ROTZ,0.,35,1*CFORCE Concentrated Nodal Forces33,FZ,500.*OPTPAR Control Parameters for Optimization1,1,10,2,40.1,0.01,0.01,0.01,0.001*DVGROUP Design Variable Groups and Section Type1,3,1,1,1*DVGELMT Design Variable Groups and Element Numbers

A-35


1,1,2,1,3,1,1*DVLOCAL Local Design Variable Information1,3,11,1.0,0.1,10.0*STRCON Stress Constraint IDs1,VON,30000.0*STRELM Element Numbers for Stress Constraint IDs1,1,341,6,1*ENDDATA Data Deck Terminator

Case (2) : Stress and displacement constraints*TITLE --- *DVLOCAL same as for Case (1)*DISCON Displacement Constraint IDs1,UZ,1.0*DISNOD Node Numbers for Displacement Constraint IDs1,133*STRCON Stress Constraint IDs1,VON,30000.0*STRELM Element Numbers for Stress Constraint IDs1,1,341,6,1*ENDDATA Data Deck Terminator

Case (3) : Stress, displacement and natural frequency constraintsANALYSIS = STATICAUTO CONSTRAINT = OFFRESEQUENCING OF ELEMENT = OFF** TOTAL NO.OF ELEMENTS = 6** TOTAL NO.OF NODES = 29EIGEN EXTRACTION = SUBSPACE,CONVENTIONALMASS FORMULATION = CONSISTANT*TITLE --- *CFORCE same as for Case (2)*I21,0,10*OPTPAR --- *STRELM same as for Case (2)*FRQCON Natural Frequency Constraints1,1,30.0

A-36

Example Problems

Results

Tables Table A.4-1 - Table A.4-3 show optimum results obtained by threeoptimization algorithms (GRG, RQP and OCB) for cases 1, 2 and 3, respectively.Computational effort required in this problem is given in tables Table A.4-4 -Table A.4-6 with the total numbers of structural and sensitivity analyses. Theconvergence parameters ACV, ACS and ALS used in this example are 0.01, 0.01,and 0.001, respectively. Parameter ACT is assigned a value of 0.01 for GRG and0.1 for both RQP and OCB. Design iteration histories are given in tables TableA.4-7 - Table A.4-15. It is observed that initial design is feasible and reductions of

56%, 51% and 51% in volume are achieved for cases 1, 2 and 3, respectively.tables Table A.4-2 - Table A.4-4 show graphically the objective function historiesfor cases 1, 2 and 3, respectively.

*ENDDATA Data Deck Terminator

Table A.4-1 Optimum results of cantilever plate (Case 1)

Design Group

Number




GRG RQP OCB

1 1,2 .10000E+01 0.54423E+00 0.54336E+00 0.54567E+00

2 3,4 .10000E+01 0.44031E+00 0.43957E+00 0.43935E+00

3 5,6 .10000E+01 0.30583E+00 0.30582E+00 0.30711E+00

Total Volume (in3) .30000E+03 0.12904E+03 0.12891E+03 0.12921E+03


Design Group

Number




GRG RQP OCB

1 1,2 .10000E+01 0.67156E+00 0.64550E+00 0.60973E+00

A-37


2 3,4 .10000E+01 0.49621E+00 0.51243E+00 0.53665E+00

3 5,6 .10000E+01 0.31045E+00 0.31724E+00 0.34003E+00



A-38

Example Problems


Design Group

Number




GRG RQP OCB

1 1,2 .10000E+01 0.67697E+00 0.65581E+00 0.66728E+00

2 3,4 .10000E+01 0.48964E+00 0.51274E+00 0.49992E+00

3 5,6 .10000E+01 0.31637E+00 0.31032E+00 0.31200E+00


Table A.4-4 Comparison of optimization algorithms for cantilever plate (Case 1)

Algorithm NITa


NSAb


NDSc


CPU timed (sec)


CPU time/Design Iter-ation (sec)

GRG 5 38 4 106.33 21.26

RQP 6 14 5 66.64 11.08

OCB 10 10 9 76.72 7.67

A-39



Algorithm NITa


NSAb


NDSc


CPU timed (sec)


CPU time/Design Itera-

tion (sec)

GRG 7 16 6 173.36 24.76

RQP 7 12 6 63.25 9.04

OCB 19 19 18 124.65 6.56


Algorithm NITa


NSAb


NDSc


CPU timed (sec)


CPU time/Design Itera-

tion (sec)

GRG 6 41 5 594.95 95.70

RQP 8 12 7 216.56 27.07

OCB 10 10 9 207.68 20.77

A-40

Example Problems

Table A.4-7 Design iteration history for cantilever plate with GRG (Case 1)

Iteration Num-ber


a. Material volume of the cantilever plate





1 0.30000E+03 0.30000E+05 0.79703.E+01

2 0.16195E+03 0.18973E+05 0.13902E+01

3 0.14280E+03 0.10730E+05 0.13147E+00

4 0.12918E+03 0.90552E+06 0.11844E-02

5 0.12904E+03 0.93271E+06 0.29611E-03

Table A.4-8 Design iteration history for Cantilever Plate with RQP (Case 1)

Iteration Num-ber







1 0.30000E+03 0.17321E+03 0.00000E+00

2 0.16500E+03 0.19529E+03 0.00000E+00

3 0.14374E+03 0.11877E+02 0.00000E+00

4 0.13542E+03 0.90552E+06 0.00000E+00

5 0.13227E+03 0.20354E-01 0.00000E+00

6 0.12891E+03 0.14117E-02 0.84026E-02

A-41


Table A.4-9 Design iteration history for Cantilever Plate with OCB (Case 1)

Iteration Num-ber







1 0.30000E+03 0.10000E+01 0.00000E+00

2 0.15000E+03 0.47586E-01 0.16424E+00

3 0.15000E+03 0.49841E-02 0.13958E-01

4 0.15000E+03 0.79348E+00 0.00000E+00

5 0.82500E+02 0.83166E-01 0.80414E+01

6 0.94163E+02 0.10151E+00 0.32855E+01

7 0.10832E+03 0.96887E-01 0.12189E+01

8 0.12167E+03 0.56518E-01 0.35966E+00

9 0.12921E+03 0.20393E-01 0.58494E-01

10 0.12921E+03 0.16437E-02 0.19272E-02


Iteration Num-ber




1 0.30000E+03 0.30000E+05 0.76299E+00

2 0.16785E+03 0.12465E+05 0.19075E+00

A-42

Example Problems

3 0.15230E+03 0.64554E+04 0.15851E+00

4 0.14857E+03 0.83159E+03 0.39626E-01

5 0.14823E+03 0.35297E+02 0.39626E-01

6 0.14807E+03 0.18218E+04 0.39626E-01

7 0.14774E+03 0.77654E+03 0.00000E+00

a. Material volume of the cantilever plateb. Squared norm of projected reduced gradient (ACS)c. Estimated stepsize (ALS)


Iteration Num-ber







1 0.30000E+03 0.17321E+03 0.00000E+00

2 0.16500E+03 0.13069E+03 0.47205E-01

3 0.13652E+03 0.11102E+00 0.31715E+00

4 0.14191E+03 0.21110E+00 0.15297E+00

5 0.14206E+03 0.10815E+00 0.14834E+00

6 0.14345E+03 0.31042E-01 0.80694E-02

7 0.14752E+03 0.88253E-02 0.80694E-02


A-43



Iteration Number

Objectivea Function



1 0.30000E+03 0.10000E+01 0.00000E+00

2 0.15000E+03 0.10299E+00 0.39343E+00

3 0.15000E+03 0.59045E-01 0.11327E+00

4 0.15000E+03 0.14995E-01 0.18816E-01

5 0.15000E+03 0.31714E+00 0.11299E-02

6 0.13499E+03 0.55559E-01 0.86968E+01

7 0.13499E+03 0.69861E-01 0.35672E+01

8 0.13499E+03 0.82395E-01 0.13315E+01

9 0.13499E+03 0.32743E+00 0.33717E+00

10 0.13499E+03 0.12545E+00 0.17233E+01

11 0.13499E+03 0.88414E-01 0.67363E+00

12 0.14728E+03 0.74352E-01 0.15972E+00

13 0.14728E+03 0.43944E-01 0.43743E-01

14 0.14728E+03 0.52387E-01 0.14819E-01

15 0.14728E+03 0.31205E-01 0.16480E+00

16 0.14728E+03 0.10039E+00 0.13165E-01

17 0.14728E+03 0.22543E-01 0.80201E-01

18 0.15000E+03 0.34287E+00 0.44779E-02

19 0.14864E+03 0.24588E+00 0.81726E-02

A-44

Example Problems

a. Material volume of the cantilever plateb. Norm of search direction vector (ACS)c. Maximum constraint violation (ACV)

Table A.4-13 Design iteration history for Cantilever Plate with GRG (Case 3)

Iteration Num-ber







1 0.30000E+03 0.30000E+05 0.33321E+00

2 0.24229E+03 0.24815E+05 0.33321E+00

3 0.19027E+03 0.26962E+05 0.25703E+00

4 0.14880E+03 0.44860E+04 0.16166E-01

5 0.14791E+03 0.32894E+01 0.32332E-01

6 0.14786E+03 0.62061E+03 0.00000E+00


Iteration Num-ber




1 0.30000E+03 0.17321E+03 0.00000E+00

2 0.23250E+03 0.25192E+03 0.88935E-01

3 0.16751E+03 0.17321E+03 0.77763E-01

A-45


4 0.13512E+03 0.11498E+00 0.39021E+00

5 0.14087E+03 0.59679E-01 0.16634E+00

6 0.14655E+03 0.30417E-01 0.29606E-01

7 0.14764E+03 0.17075E-01 0.70669E-02

8 0.14781E+03 0.16890E-02 0.19951E-02

a. Material volume of the cantilever plateb. Norm of Lagrangian gradient (ACS)c. Maximum constraint violation (ACV)


Iteration Num-ber




1 0.30000E+03 0.10000E+01 0.00000E+00

2 0.15000E+03 0.13653E+00 0.16109E+01

3 0.15000E+03 0.89785E-01 0.50273E+00

4 0.15000E+03 0.21311E-01 0.82247E-01

5 0.15000E+03 0.11386E+00 0.30483E-02

6 0.14509E+03 0.72034E-01 0.18934E+00

7 0.14509E+03 0.11589E+00 0.72722E-01

8 0.14509E+03 0.75069E-01 0.11558E+00

9 0.14509E+03 0.15908E-01 0.61579E-01

10 0.14784E+03 0.00000E+00 0.21208E-02


A-46

Example Problems


A-47


Figure A.4-2 Objective function histories for cantilever plate (Case 1)


A-48

Example Problems


A-49

Optimum Design of a Transmission Tower

A.5 Optimum Design of a Transmission Tower

Title


Problem

A transmission tower shown in Figure A.5-1 is subjected to two different loadingcases.

The design objective is to find cross-sectional areas of the members that minimizethe weight of tower while satisfying seperately two different constraint cases aslisted below:

Case 1. : Stress constraints onlyCase 2. : Stress and displacement constraints

Element Type


Section Type

Cross-sectional Area (ISECT = 7)

Material Properties


Weight Density = 0.1 lb/in.3

A-50

Example Problems

Design Data






Upper Bound on Design Variable = 100.0 in2

Lower Bound on Design Variable (Case 1) = 0.1 in2

Lower Bound on Design Variable (Case 2) = 0.01 in2


Displacement Limit (at nodes 1 to 6) = 0.35 in.

A-51


Load Data

Figure A.5-1 Geometry of 25-member transmission tower

Load Case

Number

Node Number

Load (kips)

x y z

1 1 1.0 10.0 -5.0

2 — 10.0 -5.0

3 0.5 — —

6 0.5 — —

2 1 — 20.00 -5.0

2 — -20.00 -5.0

A-52

Example Problems

Case (1) : Stress constraints onlyANALYSIS = STATICLOAD CASES = 2AUTO CONSTRAINT = OFFRESEQUENCING OF ELEMENT = ON** TOTAL NO.OF ELEMENTS = 25** TOTAL NO.OF NODES = 10*TITLE Problem TitleOPTIMUM DESIGN OF A TRANSMISSION TOWER*ELTYPE Element Type Selection (3-D Spar)1,14,1*RCTABLE Real Constant Table1, 80.1/*E1 Element Connectivities1,1,2,0,0,0,0,0,0,1,1,1,,,,02,1,4,0,0,0,0,0,0,1,1,1,,,,03,2,3,0,0,0,0,0,0,1,1,1,,,,04,1,5,0,0,0,0,0,0,1,1,1,,,,05,2,6,0,0,0,0,0,0,1,1,1,,,,06,2,4,0,0,0,0,0,0,1,1,1,,,,07,2,5,0,0,0,0,0,0,1,1,1,,,,08,1,3,0,0,0,0,0,0,1,1,1,,,,09,1,6,0,0,0,0,0,0,1,1,1,,,,010,3,6,0,0,0,0,0,0,1,1,1,,,,011,4,5,0,0,0,0,0,0,1,1,1,,,,012,3,4,0,0,0,0,0,0,1,1,1,,,,013,5,6,0,0,0,0,0,0,1,1,1,,,,014,3,10,0,0,0,0,0,0,1,1,1,,,,015,6,7,0,0,0,0,0,0,1,1,1,,,,016,4,9,0,0,0,0,0,0,1,1,1,,,,017,5,8,0,0,0,0,0,0,1,1,1,,,,018,4,7,0,0,0,0,0,0,1,1,1,,,,019,3,8,0,0,0,0,0,0,1,1,1,,,,020,5,10,0,0,0,0,0,0,1,1,1,,,,021,6,9,0,0,0,0,0,0,1,1,1,,,,022,6,10,0,0,0,0,0,0,1,1,1,,,,023,3,7,0,0,0,0,0,0,1,1,1,,,,024,5,9,0,0,0,0,0,0,1,1,1,,,,025,4,8,0,0,0,0,0,0,1,1,1,,,,0*NODE Nodal Coordinates

A-53


1,0,,,-37.50000,0.000,200.002,0,,,37.50000,0.000,200.003,0,,,-37.500,37.500,100.004,0,,,37.500,37.500,100.005,0,,,37.500,-37.500,100.006,0,,,-37.500,-37.500,100.007,0,,,-100.00,100.00,0.000008,0,,,100.00,100.00,0.000009,0,,,100.00,-100.00,0.00000

10,0,,,-100.00,-100.00,0.00000 1*MATERIAL Material Property DataEX,1,0,10E3DENS,1,0,0.1*LDCASE Static Load Case Control Card0, 1, 2, 0, 0, 0, 0, .0000E+00, .0000E+00*SPDISP Specified Displacements7,UX,0.0,10,1,UY,UZ*CFORCE Concentrated Nodal Forces1,FX,1.01,FY,10.01,FZ,-5.02,FY,10.02,FZ,-5.03,FX,0.56,FX,0.5*LDCASE Static Load Case Control Card0, 1, 2, 0, 0, 0, 0, .0000E+00, .0000E+00*CFORCE Concentrated Nodal Forces1,FY,20.01,FZ,-5.02,FY,-20.02,FZ,-5.0*OPTPAR Control Parameters for Optimization3,1,20,2,40.1,0.01,0.01,0.01,0.001,1.0*DVGROUP Design Variable Groups and Section Type1,8,1,7,1*DVGELMT Design Variable Groups and Element Numbers1,1,1,02,2,5,1

A-54

Example Problems

Results

Tables Table A.5-1 - Table A.5-2 show optimum results obtained by threeoptimization algorithms (GRG, RQP and OCB) for cases 1 and 2, respectively.Computational effort required in this problem is given in tables Table A.5-3 -Table A.5-4 with the total numbers of structural and sensitivity analyses. Theconvergence parameters ACV, ACS and ALS used in this example are 0.01, 0.01,

3,6,9,14,10,11,15,12,13,16,14,17,17,18,21,18,22,25,1*DVLOCAL Local Design Variable Information1,8,11,1.0,0.1,100.0*STRCON Stress Constraint IDs1,AXIAL,40.0*STRELM Element Numbers for Stress Constraint IDs1,1,911,25,1*ENDDATA Data Deck Terminator

Case (2) : Stress and Displacement Constraints*TITLE --- *DVGROUP same as for Case (1)*DVGELMT Local Design Variable Information1,8,11,1.0,0.01,100.0*DISCON Displacement Constraint IDs1,UX,0.35,UZ,0.35,UY,0.35*DISNOD Node Numbers for Displacement Constraint IDs1,11,6,1*STRCON Stress Constraint IDs1,AXIAL,40.0*STRELM Element Numbers for Stress Constraint IDs1,1,911,25,1*ENDDATA Data Deck Terminator

A-55


and 0.001, respectively. Parameter ACT is assigned a value of 0.01 for GRG and0.1 for both RQP and OCB. Design iteration histories are given in tables TableA.5-5 - Table A.5-10. It is observed that initial designs are infeasible and there are

increases of 176% and 65% in weight for cases 1 and 2, respectively. Tables TableA.5-2 - Table A.5-3 show graphically the objective function histories for cases 1

and 2, respectively.

Table A.5-1 Optimum results of transmission tower (Case 1)

Design Group

Number




GRG RQP OCB

1 1 .10000E+00 .10000E+00 .10000E+00 .10000E+00

2 2,3,4,5 .10000E+00 .38099E+00 .37612E+00 .37086E+00

3 6,7,8,9 .10000E+00 .46763E+00 .47072E+00 .49423E+00

4 10,11 .10000E+00 .10000E+00 .10000E+00 .10000E+00

5 12,13 .10000E+00 .10000E+00 .10000E+00 .10000E+00

6 14,15,16,17 .10000E+00 .10000E+00 .10000E+00 .10000E+00

7 18,19,20,21 .10000E+00 .27591E+00 .27731E+00 .28336E+00

8 22,23,24,25 .10000E+00 .38105E+00 .38007E+00 .37771E+00

Total Weight (lb.) .33072E+02

.91195E+02

.91122E+02

.92165E+02

A-56

Example Problems

Table A.5-2 Optimum results of transmission tower (Case 2)

Design Group Num-

ber




GRG RQP OCB

1 1 .10000E+01 .14995E+01 .10000E-01 .10000E-01

2 2,3,4,5 .10000E+01 .20420E+01 .20451E+00 .20529E+01

3 6,7,8,9 .10000E+01 .31983E+01 .30055E+00 .29935E+01

4 10,11 .10000E+01 .77842E+00 .10000E-01 .10000E-01

5 12,13 .10000E+01 .79470E+00 .10000E-01 .11235E-01

6 14,15,16,17 .10000E+01 .61716E+00 .68218E+00 .66849E+00

7 18,19,20,21 .10000E+01 .15126E+01 .16196E+00 .16387E+00

8 22,23,24,25 .10000E+01 .26919E+01 .267345+01 .26811E+01

Total Weight (lb.) .33072E+03 .57608E+03 .54503E+03 .54575E+03

Table A.5-3 Comparison of optimization algorithms for transmission tower (Case 1)

Algo-rithm NITa


NSAb


NDSc


CPU timed (sec)


CPU time/ Design Itera-

tion (sec)GRG 7 32 7 145.37 20.76RQP 6 6 6 48.06 8.01OCB 11 11 11 80.87 7.35

A-57


Table A.5-4 Comparison of optimization algorithms for transmission tower (Case 2)

Algorithm NITa


NSAb


NDSc


CPU timed (sec)



tion (sec)GRG 9 39 9 179.67 19.96RQP 13 16 13 131.61 10.12OCB 17 17 17 109.66 6.45

Table A.5-5 Design iteration history for transmission tower with GRG (Case 1)

Iteration Number

Objectivea Function

a. )Weight of the transmission tower





1 .15497E+03 .16042E+05 .27580E+002 .11976E+03 .10943E+05 .13520E-013 .11842E+03 .78936E+04 .40560E-014 .11497E+03 .52600E+04 .16224E+005 .10377E+03 .51040E+04 .16224E+006 .10151E+03 .51099E+03 .12979E+017 .91195E+02 .00000E+00 .12979E+01

A-58

Example Problems

Table A.5-7 Design iteration history for transmission tower with OCB (Case 1)

Table A.5-6 Design iteration history for transmission tower with RQP (Case 1)

Iteration Number


a. Weight of the transmission tower





1 .33072E+02 .14757E+00 .36859E+012 .49080E+02 .18751E+00 .16210E+013 .68786E+02 .16091E+00 .61697E+004 .84496E+02 .71125E-01 .16852E+005 .90447E+02 .10387E-01 .20747E-016 .91122E+02 .20780E-03 .40244E-03

Iteration Number

Objectivea Function






1 .33072E+02 .14751E+00 .36859E+012 .49392E+02 .18738E+00 .16216E+013 .69546E+02 .16086E+00 .61798E+004 .85573E+02 .71338E-01 .16928E+005 .91522E+02 .10476E-01 .20917E-016 .92165E+02 .49510E+00 .40597E-037 .93016E+02 .67910E-01 .72276E-018 .92165E+02 .35496E-01 .57196E-029 .88311E+02 .64303E-01 .19539E+00

10 .92165E+02 .16544E-01 .26867E-0111 .92165E+02 .25661E-01 .85120E-03

A-59


Table A.5-8 Design iteration history for transmission tower with GRG (Case 2)

Iteration Number







1 .73438E+03 .94729E+04 .16716E+002 .72016E+03 .76431E+04 .66865E+003 .67295E+03 .54553E+04 .66865E+004 .63793E+03 .35728E+04 .66865E+005 .61301E+03 .66855E+03 .66865E+006 .60750E+03 .37452E+04 .66865E+007 .60615E+03 .55872E+03 .13373E+018 .59029E+03 .29151E+04 .66865E+009 .57608E+03 .90117E+03 .00000E+00


Iteration Number

Objectivea Function



1 .33072E+03 .80506E+02 .12206E+012 .28335E+03 .53545E+02 .12713E+013 .26862E+03 .33909E+02 .13703E+014 .24739E+03 .37530E+02 .13278E+015 .24717E+03 .92161E+01 .13273E+016 .25090E+03 .13464E+01 .12584E+017 .38810E+03 .11458E+01 .50786E+008 .41525E+03 .88231E+00 .38087E+00

A-60

Example Problems

9 .51295E+03 .37946E+00 .86023E-0110 .54184E+03 .94173E-01 .83483E-0211 .54511E+03 .11664E+00 .25929E-0312 .54492E+03 .34178E-01 .32963E-0313 .54503E+03 .44893E-02 .36302E-04

a. Weight of the transmission towerb. Norm of Lagrangian gradient (ACS)c. Maximum constraint violation (ACV)


Iteration Number

Objectivea Function



1 .33072E+03 .11030E+01 .12206E+012 .45118E+03 .96447E+00 .44252E+003 .56277E+03 .38853E+00 .10584E+004 .60877E+03 .47781E+00 .93648E-025 .56866E+03 .36990E+00 .26280E-016 .56866E+03 .17577E+00 .28853E-027 .55593E+03 .18396E+00 .16613E-028 .54072E+03 .38474E+00 .22112E-019 .54072E+03 .34736E+00 .21040E-01

10 .54072E+03 .36544E+00 .18777E-0111 .54095E+03 .26752E+00 .16012E-0112 .54095E+03 .42527E+00 .13337E-0113 .54095E+03 .21183E+00 .32075E-0114 .54095E+03 .36357E+00 .13441E-0115 .54095E+03 .31097E+00 .20822E-0116 .54095E+03 .39368E-01 .12008E-01


A-61


17 .54575E+03 .90516E-01 .34228E-03

a. Weight of the transmission towerb. Norm of search direction vector (ACS)c. Maximum constraint violation (ACV)


A-62

Example Problems

Figure A.5-2 Objective function histories for transmission tower (Case 1)

Figure A.5-3 Objective function histories for transmission tower (Case 2)

A-63

Optimum Design of a Geodesic Dome

A.6 Optimum Design of a Geodesic Dome

Title


Problem

The geometry and the finite element model of Geodesic Dome are shown intables Table A.6-1-Table A.6-2. The design objective is to find the member cross-sectional areas that give the lightest structure under multiple load cases whilesatisfying constraints on nodal displacements, stresses and member sizes.

Element Type


Section Type

Cross-sectional area (ISECT = 7)

Properties

Elastic Modulus = 10,000 ksi.

Specific Weight = 0.1 lb/in3

Design Data






Lower Bound on Design Variables = 0.1 in2

A-64

Example Problems

Load Data

Upper Bound on Design Variables = 100 in2

Displacement Limit (at node No. 1, 3-5, 10-16, 24-32 and in global X, Y, Z directions) = 0.1 in.

Stress Limit (for element No. 1-3, 7-9, 12-20, 30-35, 42-55, 71-80, 89-108, 130-132)

= 25 ksi

Load Case Number Node Number Load (kip) in Z-direction

100 1 -1.0

200 1-4,7-13,19-28,37 -1.0

300 all -1.0


A-65


Figure A.6-1 Geometry of 132 bar geodesic dome

A-66

Example Problems

Figure A.6-2 132 Bar geodesic dome (element and node numbers)

ANALYSIS=STATICLOAD_CASES=3AUTO CONSTRAINT = OFFAUTO=ONBLANK=35000EXEC=GOMAXWAVE=750RESEQUENCE=ONSAVE=26,27

A-67


FILE=DOMEWARN=GO** TOTAL NO.OF ELEMENTS = 132** TOTAL NO.OF NODES =61*TITLE Problem TitleOPTIMUM DESIGN OF A GEODESIC DOME*ELTYPE Element Type Selection (3-D Spar)**NSRL,NKTP,NORDR1, 14, 1*RCTABLE Real Constant Table** INDEXRC,NRC1, 21.0, 1.0*ELEMENTS Element Connectivities (See Figure A.6.2)*NODES Nodal Coordinates (See Figure A.6.2)*MATERIAL Material Property DataEX,1,0,10000.DENS,1,0,0.1*LDCASE Static Load Case Control Data0, 0, 2, 0,-1, 0, 1, .0000E+00, .0000E+00*SPDISP Specified Displacements38,UX,0.0,61,1,UY,UZ*CFORCE Concentrated Nodal Forces1,FZ,-1.*LDCASE Static Load Case Control Data0, 0, 2, 0,-1, 0, 1, .0000E+00, .0000E+00*CFORCE Concentrated Nodal Forces1,FZ,-1.,4,17,FZ,-1.,13,119,FZ,-1.,28,137,FZ,-1.*LDCASE Static Load Case Control Data0, 0, 2, 0,-1, 0, 1, .0000E+00, .0000E+00*CFORCE Concentrated Nodal Forces1,FZ,-1.,37,1 1*OPTPAR Control Parameters for Optimization1,1,50,2,40.1,0.01,0.01,0.01,0.01,1.0

A-68

Example Problems

*DVGROUP Design Variable Groups and Section Type1,36,1,7,1*DVGELMT Design Variable Groups and Element Numbers1,0,41,3,4,62,0,22,53,0,47,8,10,114,0,29,125,0,413,19,22,286,0,414,18,23,277,0,415,17,24,268,0,216,259,0,420,21,29,3010,0,431,34,37,4011,0,432,33,38,3912,0,435,36,41,4213,0,443,53,58,6814,0,444,52,59,6715,0,445,51,60,6616,0,446,50,61,6517,0,447,49,62,6418,0,248,6319,0,454,57,69,72

A-69


20,0,455,56,70,7121,0,473,78,82,8722,0,474,77,83,8623,0,475,76,84,8524,0,479,81,88,9025,0,280,8926,0,491,105,112,12627,0,492,104,113,12528,0,493,103,114,12429,0,494,102,115,12330,0,495,101,116,12231,0,496,100,117,12132,0,497,99,118,12033,0,298,11934,0,4106,111,127,13235,0,4107,110,128,13136,0,4108,109,129,130*DVLOCAL Local Design Variable Information1,36,11,0.1,0.1,100.0*DISCON Displacement Constraint IDs1,UX,.10,UY,.10,UZ,.10*DISNOD Node Numbers for Displacement Constraint IDs1,4,100,200,300

A-70

Example Problems

Results

Table A.6-1 shows optimum results obtained by three optimization algorithms(GRG, RQP and OCB). Computational effort required in this problem is given inTable A.6-2 with the total numbers of structural and sensitivity analyses. Theconvergence parameters ACV, ACS and ALS used in this example are 0.01, 0.05,and 0.005, respectively. Parameter ACT is assigned a value of 0.01 for GRG and0.1 for both RQP and OCB. Design iteration histories are given in tables TableA.6-3 - Table A.6-5. It is observed that initial design is infeasible and there is an

increase of 267% in weight. Figure A.6-3 shows graphically the objective functionhistories.

1,1,0,3,5,110,16,1,24,32,1*STRCON Stress Constraint IDs1,AXIAL,25*STRELM Element Numbers for Stress Constraint IDs1,8,91,100,200,3001,3,1,7,9,112,20,1,30,35,142,55,1,71,80,189,108,1,130,132,1*ENDDATA Data Deck Terminator

Table A.6-1 Optimum Results of Geodesic Dome

Design Group

Number




GRG RQP OCB

1 1,3,4,6 .10000E+00 .10929E+01 .10290E+01 .10273E+01

2 2,5 .10000E+00 .94846E+00 .11498E+01 .11667E+01

3 7,8,10,11 .10000E+00 .80810E+00 .83244E+00 .82971E+00

4 9,12 .10000E+00 .83896E+00 .91797E+00 .96150E+00

5 13,19,22,28 .10000E+00 .39776E+00 .34070E+00 .32878E+00

6 14,18,23,27 .10000E+00 .57308E+00 .57554E+00 .56950E+00

7 15,17,24,26 .10000E+00 .45736E+00 .40672E+00 .40254E+00

A-71


8 16,25 .10000E+00 .64158E+00 .68671E+00 .67940E+00

9 20,21,29,30 .10000E+00 .37253E+00 .29825E+00 .30720E+00

10 31,34,37,40 .10000E+00 .34741E+00 .30555E+00 .29739E+00

11 32,33,38,39 .10000E+00 .36664E+00 .35480E+00 .34610E+00

12 35,36,41,42 .10000E+00 .49301E+00 .47437E+00 .47657E+00

13 43,53,58,68 .10000E+00 .37970E+00 .33857E+00 .31545E+00

14 44,52,59,67 .10000E+00 .29493E+00 .33761E+00 .33542E+00

15 45,51,60,66 .10000E+00 .32679E+00 .29315E+00 .28837E+00

16 46,50,61,65 .10000E+00 .51523E+00 .55091E+00 .54443E+00

17 47,49,62,64 .10000E+00 .30673E+00 .31900E+00 .31431E+00

18 48,63 .10000E+00 .56796E+00 .65395E+00 .66171E+00

19 54,57,69,72 .10000E+00 .20466E+00 .19171E+00 .20312E+00

20 55,56,70,71 .10000E+00 .37989E+00 .32475E+00 .32950E+00

21 73,78,82,87 .10000E+00 .10719E+00 .10000E+00 .10000E+00

22 74,77,83,86 .10000E+00 .15433E+00 .10000E+00 .10002E+00

23 75,76,84,85 .10000E+00 .30426E+00 .33203E+00 .32917E+00

24 79,81,88,90 .10000E+00 .17117E+00 .21038E+00 .19919E+00

25 80,89 .10000E+00 .48082E+00 .30206E+00 .30134E+00

26 91,105,112,126 .10000E+00 .13908E+00 .24082E+00 .24932E+00

27 92,104,113,125 .10000E+00 .34766E+00 .29925E+00 .31966E+00

28 93,103,114,124 .10000E+00 .17506E+00 .13725E+00 .14813E+00

29 94,102,115,123 .10000E+00 .47926E+00 .49680E+00 .49580E+00

30 95,101,116,122 .10000E+00 .19486E+00 .10000E+00 .10000E+00

31 96,100,117,121 .10000E+00 .34382E+00 .42030E+00 .42760E+00

32 97,99,118,120 .10000E+00 .27982E+00 .32750E+00 .33632E+00

Table A.6-1 Optimum Results of Geodesic Dome (Continued)

A-72

Example Problems

33 98,119 .10000E+00 .39900E+00 .10000E+00 .10036E+00

34 106,111,127,132 .10000E+00 .16692E+00 .10000E+00 .10497E+00

35 107,110,128,131 .10000E+00 .38057E+00 .49704E+00 .49821E+00

36 108,109,129,130 .10000E+00 .25790E+00 .10000E+00 .10101E+00

Weight (lb.) .43591E+02 .16536E+03 .15993E+03 .16029E+03

Table A.6-2 Comparison of Optimization Algorithms for Geodesic Dome

Algorithm NITa


NSAb


NDSc


CPU timed (sec)


CPU time/Design Iter-ation (sec)

GRG 29 175 29 2609.53 89.98

RQP 33 52 33 801.42 24.28

OCB 32 32 32 458.36 14.32

Table A.6-3 Design Iteration History of Geodesic Dome with GRG

Iteration Number

Objectivea Function



1 .35885E+03 .51597E+04 .10000E+01

2 .28761E+03 .50562E+04 .89399E+00

3 .22483E+03 .43777E+04 .55875E-01

4 .22129E+03 .37591E+04 .16762E+00

5 .21131E+03 .34547E+04 .17286E+00

Table A.6-1 Optimum Results of Geodesic Dome (Continued)

A-73


6 .20209E+03 .26800E+04 .17286E+00

7 .19655E+03 .24342E+04 .17286E+00

8 .19162E+03 .24565E+04 .17286E+00

9 .18755E+03 .28514E+04 .17286E+00

10 .18536E+03 .24861E+04 .17286E+00

11 .18083E+03 .39652E+04 .17286E+00

12 .17923E+03 .29111E+04 .55336E-01

13 .17709E+03 .26349E+04 .11067E+00

14 .17426E+03 .24233E+04 .37023E-01

15 .17358E+03 .16607E+04 .74047E-01

16 .17206E+03 .31654E+05 .92559E-02

17 .17181E+03 .13422E+04 .74047E-01

18 .17054E+03 .14650E+04 .74047E-01

19 .17037E+03 .13734E+04 .14809E+00

20 .16936E+03 .42739E+04 .14809E+00

21 .16826E+03 .16937E+05 .46279E-02

22 .16806E+03 .83736E+03 .37023E-01

23 .16782E+03 .55762E+03 .74047E-01

24 .16706E+03 .72998E+03 .74047E-01

25 .16665E+03 .17205E+04 .74047E-01

26 .16639E+03 .89473E+03 .14809E+00

27 .16579E+03 .30250E+05 .46279E-02


A-74

Example Problems

28 .16559E+03 .14939E+04 .37023E-01

29 .16537E+03 .79072E+03 .00000E+00

a. Weight of the geodesic domeb. Squared norm of projected reduced gradient (ACS)c. Estimated stepsize (ALS)

Table A.6-4 Design Iteration History of Geodesic Dome with RQP

Iteration Number

Objectivea Function



1 .43591E+02 .57234E+00 .72323E+01

2 .66999E+02 .17601E+01 .35461E+01

3 .67243E+02 .40102E+01 .30504E+01

4 .66156E+02 .38026E+01 .29388E+01

5 .66896E+02 .14636E+01 .28597E+01

6 .72883E+02 .12385E+01 .26127E+01

7 .74042E+02 .27880E+01 .22286E+01

8 .74904E+02 .11099E+01 .21217E+01

9 .81104E+02 .32446E+01 .16040E+01

10 .81334E+02 .39068E+01 .15496E+01

11 .81876E+02 .14338E+01 .15290E+01

12 .89093E+02 .16076E+01 .12755E+01

13 .91995E+02 .23823E+01 .11490E+01

14 .93297E+02 .22147E+01 .10784E+01


A-75


15 .96990E+02 .98936E+00 .98318E+00

16 .10684E+03 .36528E+01 .75183E+00

17 .10692E+03 .22497E+01 .71226E+00

18 .11092E+03 .20096E+01 .63260E+00

19 .11182E+03 .17620E+01 .61715E+00

20 .11476E+03 .24436E+01 .54149E+00

21 .11694E+03 .16877E+01 .49251E+00

22 .12080E+03 .87032E+00 .43929E+00

23 .13248E+03 .32436E+01 .30596E+00

24 .13270E+03 .15656E+01 .30185E+00

25 .13484E+03 .25306E+01 .27498E+00

26 .13538E+03 .63225E+00 .26715E+00

27 .15360E+03 .18682E+01 .12508E+00

28 .15348E+03 .11310E+01 .12170E+00

29 .15227E+03 .77742E+00 .12622E+00

30 .15363E+03 .27164E+00 .10511E+00

31 .15884E+03 .17441E+00 .16415E-01

32 .15977E+03 .10863E+00 .98467E-02

33 .15993E+03 .30328E-01 .39763E-02

a. Weight of the geodesic domeb. Norm of Lagrangian gradient (ACS)c. Maximum constraint violation (ACV)

Table A.6-4 Design Iteration History of Geodesic Dome with RQP

A-76

Example Problems

Table A.6-5 Design Iteration History of Geodesic Dome with OCB

Iteration Number

Objectivea Function



1 .43591E+02 .40605E+00 .72323E+01

2 .65386E+02 .53924E+00 .38272E+01

3 .98079E+02 .60543E+00 .17687E+01

4 .13372E+03 .52053E+00 .70968E+00

5 .15686E+03 .30942E+00 .22090E+00

6 .16797E+03 .71865E-01 .36302E-01

7 .17040E+03 .50892E+00 .13354E-02

8 .15503E+03 .29252E+00 .79088E+00

9 .15503E+03 .54059E+00 .49988E+00

10 .15635E+03 .26677E+00 .37727E+00

11 .15635E+03 .61424E+00 .17460E+00

12 .15661E+03 .21675E+00 .64884E+00

13 .15681E+03 .29688E+00 .12161E+00

14 .15682E+03 .56301E+00 .75840E-01

15 .15682E+03 .19292E+00 .26807E+00

16 .15682E+03 .24791E+00 .12883E+00

17 .15682E+03 .40758E+00 .81688E-01

18 .15712E+03 .19178E+00 .22105E+00

19 .15712E+03 .52712E+00 .82555E-01

20 .15862E+03 .20190E+00 .48546E+00

21 .15862E+03 .19800E+00 .22298E+00

A-77


22 .15862E+03 .38485E+00 .82671E-01

23 .15891E+03 .14290E+00 .51419E+00

24 .15891E+03 .20116E+00 .50264E-01

25 .15891E+03 .24921E+00 .28903E-01

26 .15971E+03 .10691E+00 .58897E-01

27 .15971E+03 .12612E+00 .16901E-01

28 .15971E+03 .10154E+00 .19333E-01

29 .15971E+03 .15976E+00 .18523E-01

30 .15971E+03 .89192E-01 .58730E-01

31 .15971E+03 .11098E-01 .24132E-01

32 .16029E+03 .53319E-01 .59257E-03

a. Weight of the geodesic domeb. Norm of Lagrangian gradient (ACS)c. Maximum constraint violation (ACV)

Table A.6-5 Design Iteration History of Geodesic Dome with OCB (Contin-

A-78

Example Problems

Figure A.6-3 Objective function histories for geodesic dome

A-79

Optimum Design of a Piston

A.7 Optimum Design of a Piston

Title


Problem

The geometry and the finite element model of a piston subjected to uniformpressure and body forces due to linear acceleration are shown in Figure Table A.7-1-Table A.7-2. Due to symmetry, only a quarter of the structure is modeled.

The design objective is to find piston thicknesses of the crown and skirt thatminimize piston volume while satisfying constraints on stresses and thicknesses.

Element Type

3D General Shell (NKTP = 20, NORDR = 2)

Section Type

Uniform thickness (ISECT = 1)

Material Properties

Design Data

Elastic Modulus = 10,000 ksiPoisson’s Ratio = 0.3Specific Weight = 0.098 lb/in3

No. of Design Variables = 2No. of Load Cases = 1No. of Elements = 124No. of Stress Constraints = 124Lower Bound on Design Variables = 0.1 in.

A-80

Example Problems

Load Data

Uniform pressure of 250 psi on piston head and a linear acceleration of -180gwhere g is gravitational acceleration in Z-direction.

Figure A.7-1 Geometry of the piston

Upper Bound on Design Variables = 1.0 in.Stress Limit (von Mises) = 14 ksi

A-81


Figure A.7-2 Finite element model of piston (element and node numbers)

ANALYSIS=STATICAUTO=ONBLANK=35000ELEMENT=OFFEXEC=GOFILE=PISTMAXWAVE=750NODE=OFFRESEQUENCE = ONSAVE=26,27WARN=GO*TITLE Problem TitleOPTIMUM DESIGN OF PISTON*ELTYPE Element Type Selection (3-D General Shell)1, 20, 2,*RCTABLE Real Constant Table

A-82

Example Problems

1 , 8.2500E+00///////2 , 8.2500E+00///////*ELEMENTS (See Figure A.7.2)*NODES Nodal Coordinates (See Figure A.7.2)*MATERIAL Material Property TableEX , 1,0, 0.100000E+08NUXY, 1,0, 0.300000E+00GXY , 1,0, 0.384615E+07DENS, 1,0, 0.098*LDCASE Static Load Case Control Card**KELFR KRCTN KSTR KSTN LQ1 LQ2 LQ7 TSFRE RCFORCE0, 0, 4, 0, 2, 4, 0, 0.000, 0.000,*SPDISP Specified Displacements10, UX , 0.00000 ,0,0, UY , UZ , ROTX, ROTY, ROTZ,15, UX , 0.00000 ,0,0, UY , UZ , ROTX, ROTY, ROTZ,20, UX , 0.00000 ,0,0, UY , UZ , ROTX, ROTY, ROTZ,21, UY , 0.00000 ,0,0, ROTX, ROTZ,22, UY , 0.00000 ,0,0, ROTX, ROTZ,23, UY , 0.00000 ,0,0, ROTX, ROTZ,24, UY , 0.00000 ,0,0, ROTX, ROTZ,25, UX , 0.00000 ,0,0, UY , UZ , ROTX, ROTY, ROTZ,69, UY , 0.00000 ,0,0, ROTX, ROTZ,73, UY , 0.00000 ,0,0, ROTX, ROTZ,77, UY , 0.00000 ,0,0, ROTX, ROTZ,81, UY , 0.00000 ,0,0, ROTX, ROTZ,85, UY , 0.00000 ,0,0, ROTX, ROTZ,89, UX , 0.00000 ,0,0, ROTY, ROTZ,93, UX , 0.00000 ,0,0, ROTY, ROTZ,97, UX , 0.00000 ,0,0, ROTY, ROTZ, 1101, UX , 0.00000 ,0,0, ROTY, ROTZ,105, UX , 0.00000 ,0,0, ROTY, ROTZ,109, UX , 0.00000 ,0,0, ROTY, ROTZ,113, UX , 0.00000 ,0,0, ROTY, ROTZ,117, UX , 0.00000 ,0,0, ROTY, ROTZ,121, UX , 0.00000 ,0,0, ROTY, ROTZ,139, UX , 0.00000 ,0,0, UY , UZ , ROTX, ROTY, ROTZ,148, UX , 0.00000 ,0,0, UY , UZ , ROTX, ROTY, ROTZ,157, UX , 0.00000 ,0,0, UY , UZ , ROTX, ROTY, ROTZ,158, UY , 0.00000 ,0,0, ROTX, ROTZ,

A-83


159, UY , 0.00000 ,0,0, ROTX, ROTZ,160, UY , 0.00000 ,0,0, ROTX, ROTZ,161, UY , 0.00000 ,0,0, ROTX, ROTZ,241, UY , 0.00000 ,0,0, ROTX, ROTZ,249, UY , 0.00000 ,0,0, ROTX, ROTZ,257, UY , 0.00000 ,0,0, ROTX, ROTZ,265, UY , 0.00000 ,0,0, ROTX, ROTZ,277, UX , 0.00000 ,0,0, ROTY, ROTZ,285, UX , 0.00000 ,0,0, ROTY, ROTZ,293, UX , 0.00000 ,0,0, ROTY, ROTZ,301, UX , 0.00000 ,0,0, ROTY, ROTZ,309, UX , 0.00000 ,0,0, ROTY, ROTZ,317, UX , 0.00000 ,0,0, ROTY, ROTZ,325, UX , 0.00000 ,0,0, ROTY, ROTZ,333, UX , 0.00000 ,0,0, ROTY, ROTZ,338, UX , 0.00000 ,0,0, ROTY, ROTZ,339, UX , 0.00000 ,0,0, ROTY, ROTZ,354, UY , 0.00000 ,0,0, ROTX, ROTZ,355, UY , 0.00000 ,0,0, ROTX, ROTZ,356, UX , 0.00000 ,0,0, ROTY, ROTZ,357, UX , 0.00000 ,0,0, ROTY, ROTZ,388, UY , 0.00000 ,0,0, ROTX, ROTZ,389, UY , 0.00000 ,0,0, ROTX, ROTZ,390, UX , 0.00000 ,0,0, ROTY, ROTZ,398, UY , 0.00000 ,0,0, ROTX, ROTZ,399, UX , 0.00000 ,0,0, UY , ROTX, ROTY, ROTZ,400, UX , 0.00000 ,0,0, ROTY, ROTZ,416, UY , 0.00000 ,0,0, ROTX, ROTZ,417, UX , 0.00000 ,0,0, ROTY, ROTZ,421, UY , 0.00000 ,0,0, ROTX, ROTZ,*L1 PRESSURE DATA97,,124,1,1,1,250.,250.,250.,250.,250.,250.$, 250.,250.,*BODYFORCE Body Forces0.0, ////, -180.0 1*OPTPAR Control Parameters for Optimization1,1,10,2,40.1,0.01,0.01,0.01,0.005*DVGROUP Design Variable Groups and Section Type1,2,1,1,1*DVGELMT Design Variable Groups and Element Numbers

A-84

Example Problems

Results

Table A.7-1 shows optimum results obtained by three optimization algorithms(GRG, RQP and OCB). Computational effort required in this problem is given inTable A.7-2 with the total numbers of structural and sensitivity analyses. Theconvergence parameters ACV, ACS and ALS used in this example are 0.01, 0.01,and 0.001, respectively. Parameter ACT is assigned a value of 0.01 for GRG and0.1 for both RQP and OCB. Design iteration histories are given in tables TableA.7-3 - Table A.7-5. It is observed that initial design is feasible and a reduction of

40% in volume is achieved. Figure A.7-3 shows graphically the objective functionhistories.

1,1,96,12,97,124,1*DVLOCAL Local Design Variable Information1,2,11,0.25,0.1,1.0*STRCON Stress Constraint IDs1,VON,14000.*STRELM Element Numbers for Stress Constraint IDs1,1,641,124,1*ENDDATA Data Deck Terminator

Table A.7-1 Optimum Results of Piston

Design Group

Number




GRG RQP OCB

1 1-96 .25000E+00 .14629E+00 .14614E+00 .14597E+00

2 97-124 .25000E+00 .16606E+00 .16816E+00 .17703E+00


A-85


Table A.7-2 Comparison of Optimization Algorithms for Piston

Algorithm NITa


NSAb


NDSc


CPU timed (sec)



tion (sec)

GRG 3 13 2 1544.86 514.95

RQP 5 5 4 924.74 184.95

OCB 6 6 5 1110.03 185.01

Table A.7-3 Design Iteration History for Piston with GRG

Iteration Num-ber Objectivea Function

a. Material volume of one quarter of the piston





1 .18596E+01 .34877E+02 .11003E+00

2 .12100E+01 .29664E+01 .49915E-01

3 .11241E+01 .94914E-02 .49915E-01

Table A.7-4 Design Iteration History for Piston with RQP

Iteration Num-ber




1 .18596E+01 .59057E+01 .00000E+00

2 .92854E+00 .30904E-01 .56098E+00

A-86

Example Problems

3 .11050E+01 .26495E-01 .13515E+00

4 .11395E+01 .12236E-01 .12478E-01

5 .11271E+01 .73515E-02 .12228E-03

a. Material volume of one quarter of the pistonb. Norm of Lagrangian gradient (ACS)c. Maximum constraint violation (ACV)

Table A.7-5 Design Iteration History for Piston with OCB

Iteration Num-ber


a. Material volume of one quarter of the piston





1 .18596E+01 .10000E+01 .00000E+00

2 .93026E+00 .30794E-01 .55589E+00

3 .11057E+01 .12986E-01 .13331E+00

4 .11790E+01 .50446E-02 .12314E-01

5 .11790E+01 .29202E-01 .67725E-04

6 .11423E+01 .96004E-02 .55467E-04

Table A.7-4 Design Iteration History for Piston with RQP

A-87


Figure A.7-3 Objective function histories for piston

A-88

Example Problems

A.8 Optimum Design of a Composite Plate

Title

Optimum Design of a Composite Plate

Problem

A simply supported composite plate shown in Figure A.8-1 is subjected to auniform normal pressure of 10.0 psi. The plate is composed of nine (9) layers. Eachlayer has a uniform thickness over an element. The layer orientation angles alter at0 and 90 degrees with respect to the global X-axis. Due to symmetry, only onequarter of the plate is modeled as shown in Figure A.8-2.

Element Type

3D Composite Shell (NKTP = 32, NORDR = 2)

Section Type

Laminated Composite Shell (ISECT =8)

Case (1) : The design objective is to find layer thicknesses that give the min-imum volume of the plate while satisfying constraints on Hill-Mises failure criteria and layer thicknesses.

Case (2) : The design objective is to find layer thicknesses and orientation angles that give the minimum value of the plate while satisfying constraints on Hill-Mises failure criteria and layer thicknesses.

A-89


Material Properties

Design Data

Elastic Moduli ExEy GxyGyzGxz

=====

40,000 ksi1,000 ksi600 ksi500 ksi600 ksi

Poisson’s Ratio xy = 0.25

Basic Ply Thickness = 0.005 in.

Failure Stresses FxtFytFxcFycFs

=====

154,000 psi10,000 psi100,000 psi15,000 psi20,000 psi



No. of Elements = 4

No. of Layers = 9


Lower Bound on Thicknesses = 0.005 in.

Lower Bound on Orientation Angles = 0.0 degree

Upper Bound on Thicknesses = 0.5 in.

Upper Bound on Orientation Angles = 360.0 degree

υ

A-90

Example Problems

Figure A.8-1 Geometry of the composite plate

A-91


Figure A.8-2 Finite element model of composite plate (element and node numbers)

Case 1: Layer thicknesses alone as design variablesANALYSIS = STATICAUTO_CONSTRAINT = ONFILE = COMPSAVE = 26,27WARN = GOEXEC = GONODE = ONELEMENT = ONRESEQUENCING OF ELEMENT = OFF** TOTAL NO.OF ELEMENTS = 4

A-92

Example Problems

** TOTAL NO.OF NODES = 25*TITLE Problem TitleOPTIMUM DESIGN OF COMPOSITE LAMINATES (NKTP = 32,NORDR = 2)*ELTYPE Element Type Selection (3-D Laminated Composite General Shell)1,32,2*RCTABLE Real Constant Table1, 80.01///////2, 80.025///////*LAMANGLE Rotation Angles for Composites1, 80.0///////2, 890.0///////*LAMSEQ Composite Lamination Sequence9,1, 1,2,1,2,1,2,1,2,1, 1,2,1,2,1,2,1,2,1,1,1,1,1,1,1,1,1,1*E1 Element Connectivities1,1,2,3,8,13,12,11,6, 1,1,0,,,,12,3,4,5,10,15,14,13,8, 1,1,0,,,,13,11,12,13,18,23,22,21,16,1,1,0,,,,14,13,14,15,20,25,24,23,18,1,1,0,,,,1*NODES Nodal Coordinates-5,0,5,5, 0.,1.25,0.1,0,,,0.,0.,0.5,0,1,,5.,0.,0.1*MATERIAL Material Property DataEX,1,,40.0E6EY,1,,1.0E6GXY,1,,0.6E6GXZ,1,,0.6E6GYZ,1,,0.5E6NUXY,1,,0.25FXT,1,,154.0E3FYT,1,,10.0E3FXC,1,,100.0E3FYC,1,,15.0E3FS,1,,20.0E3*LDCASE Static Load Case Control Card

A-93


0, 1, 2, 0, 0, 1, 0, .0000E+00, .0000E+00*SPDISP Specified Displacements1,UX,0.0,5,1,UY,UZ,ROTY1,UX,0.0,21,5,UY,UZ,ROTX22,UY,0.0,25,1,ROTX10,UX,0.0,25,5,ROTY1,ROTZ,0.0,25,1*L1 Pressure Load1,0,4,1,2,1,10.0/////$ 10.0/*OPTPAR Control Parameters for Optimization1,1,10,2,40.1,0.01,0.01,0.01,0.001*DVGROUP Design Variable Groups and Section Type1,1,0,8,18*DVELMT Design Variable Groups and Element Numbers1,1,4,1*DVLOCAL Local Design Variable Information1,1,01,0.01,0.005,0.500,0,0.0,0.0,0.01,0.025,0.005,0.50,0,90.0,90.0,90.01,0.01,0.005,0.500,0,0.0,0.0,0.01,0.025,0.005,0.50,0,90.0,90.0,90.01,0.01,0.005,0.500,0,0.0,0.0,0.01,0.025,0.005,0.50,0,90.0,90.0,90.01,0.01,0.005,0.500,0,0.0,0.0,0.01,0.025,0.005,0.50,0,90.0,90.0,90.01,0.01,0.005,0.500,0,0.0,0.0,0.0*STRCON Stress Constraint IDs1,RATIO*STRELM Element Numbers for Stress Constraint IDs1,1,371,4,1*ENDDATA Data Deck Terminator

Case 2: Both layer thicknesses and/or orientation angles as design vari-ables*TITLE --- *L1 Same as for Case (1)*OPTPAR Control Parameters for Optimization1,1,20,2,40.1,0.01,0.01,0.01,0.001

A-94

Example Problems

Results

Tables Table A.8-1 - Table A.8-2 show optimum results obtained by threeoptimization algorithms (GRG, RQP and OCB) for cases 1 and 2, respectively.Computational effort required in this problem is given in tables Table A.8-3 -Table A.8-4 with the total numbers of structural and sensitivity analyses. Theconvergence parameters ACV, ACS and ALS used in this example are 0.001, 0.01,and 0.001, respectively. Parameter ACT is given as 0.001 for GRG and 0.1 for bothRQP and OCB. Design iteration histories are given in tables Table A.8-5 - TableA.8-10. It is observed that initial design is feasible and reductions of 44% and 49%

in volume are achieved for cases 1 and 2, respectively. Table A.8-3 - Table A.8-4show graphically the objective function histories for cases 1 and 2, respectively.

*DVGROUP Design Variable Groups and Section Type1,1,0,8,18*DVGELMT Design Variable Groups and Element Numbers1,1,4,1*DVLOCAL LOcal Design Variable Information1,1,01,0.01,0.005,0.500,1,0.0,0.0,360.01,0.025,0.005,0.50,1,90.0,0.0,360.01,0.01,0.005,0.500,1,0.0,0.0,360.01,0.025,0.005,0.50,1,90.0,0.0,360.01,0.01,0.005,0.500,1,0.0,0.0,360.01,0.025,0.005,0.50,1,90.0,0.0,360.01,0.01,0.005,0.500,1,0.0,0.0,360.01,0.025,0.005,0.50,1,90.0,0.0,360.01,0.01,0.005,0.500,1,0.0,0.0,360.0*STRCON Stress Constraint IDs1,RATIO*STRELM Element Numbers for Stress Constraint IDs1,1,371,4,1*ENDDATA Data Deck Terminator

Table A.8-1 Optimum results of composite plate (Case 1)

Design Group

Number

User Ele-ment Num-

bers



GRG RQP OCB

A-95


1 1,2,3,4 .10000E-01 .500000E-02 .50000E-02 .5019E-02

.00000E+00 .000000E+00 .00000E+00 .0000E+00

.25000E-01 .247653E-01 .24761E-01 .2440E-01

.90000E+02 .900000E+02 .90000E+02 .9000E+02

.10000E-01 .500000E-02 .50000E-02 .5463E-02

.00000E+00 .000000E+00 .00000E+00 .0000E+00

.25000E-01 .500000E-02 .50000E-02 .5122E-02

.90000E+02 .900000E+02 .90000E+02 .9000E+02

.10000E-01 .500000E-02 .50000E-02 .5121E-02

.00000E+00 .000000E+00 .00000E+00 .0000E+00

.25000E-01 .500000E-02 .50000E-02 .5122E-02

.90000E+02 .900000E+02 .90000E+02 .9000E+02

.10000E-01 .500000E-02 .50000E-02 .5463E-02

.00000E+00 .000000E+00 .00000E+00 .0000E+00

.25000E-01 .247653E-01 .24761E-01 .2440E-01

.90000E+02 .900000E+02 .90000E+02 .9000E+02

.10000E-01 .500000E-02 .50000E-02 .5019E-02

.00000E+00 .000000E+00 .00000E+00 .0000E+00



A-96

Example Problems


Design Group

Number




GRG RQP OCB

1 1,2,3,4 .10000E-01 .50000E-02 .50000E-02 .50000E-02

.00000E+00 .18108E+02 .16179E+02 .00000E+00

.25000E-01 .18244E-01 .18215E-01 .24390E-01

.90000E+02 .98412E+02 .98541E+02 .90691E+02

.10000E-01 .50000E-02 .50000E-02 .52058E-02

.00000E+00 .52840E+00 .00000E+00 .00000E+00

.25000E-01 .50000E-02 .50000E-02 .50000E-02

.90000E+02 .90041E+02 .91779E+02 .90052E+02

.10000E-01 .50000E-02 .50000E-02 .50016E-02

.00000E+00 .13606E-01 .00000E+00 .00000E+00

.25000E-01 .50000E-02 .50000E-02 .50000E-02

.90000E+02 .10890E+03 .91779E+02 .90052E+02

.10000E-01 .50000E-02 .50000E-02 .50000E-02

.00000E+00 .52840E+00 .00000E+00 .00000E+00

.25000E-01 .18239E-01 .18215E-01 .24390E-01

.90000E+02 .98412E+02 .98541E+02 .90691E+02

.10000E-01 .50000E-02 .50000E-02 .50000E-02

.00000E+00 .14152E+02 .16179E+02 .00000E+00


A-97


Table A.8-3 Comparison of optimization algorithms for composite plate (Case 1)

Algorithm NITa


NSAb


NDSc


CPU timed (sec)



tion (sec)

GRG 3 14 2 48.92 16.31

RQP 7 8 5 63.85 9.12

OCB 11 11 7 76.62 6.96

Table A.8-4 Comparison of optimization algorithms for composite plate (Case 2)

Algorithm NITa


NSAb


NDSc


CPU timed (sec)

d. IBM RT-PC

CPU time/

Design Iteration

(sec)

GRG 20 137 17 3118.00 155.9

RQP 7 8 5 582.00 83.14

OCB 11 11 6 610.00 55.45

A-98

Example Problems

Table A.8-5 Design iteration history for composite plate with GRG (Case 1)

Iteration Number

Objectivea Function

a. Material volume of one quarter of the composite plate





1 .37500E+01 .56250E+04 .23030E-01

2 .23573E+01 .18293E+03 .23030E-01

3 .21133E+01 .00000E+00 .23030E-01

Table A.8-6 Design iteration history for composite plate with RQP (Case 1)

Iteration Number

Objectivea Function






1 .37500E+01 .75000E+02 .00000E+00

2 .26250E+01 .75000E+02 .00000E+00

3 .18750E+01 .12932E+02 .75255E+00

4 .17241E+01 .73417E-02 .73679E+00

5 .19837E+01 .32454E-02 .18202E+00

6 .20984E+01 .41474E-03 .18599E-01

7 .21131E+01 .43043E-05 .18781E-03

A-99


Table A.8-7 Design iteration history for composite plate with OCB (Case 1)

Iteration Number

Objectivea Function



1 .37500E+01 .10000E+01 .00000E+00

2 .22917E+01 .52593E-02 .70397E-01

3 .22917E+01 .15665E+00 .31641E-02

4 .34608E+01 .57735E+00 .00000E+00

5 .25264E+01 .57735E+00 .00000E+00

6 .19125E+01 .34947E-02 .30403E+00

7 .21435E+01 .21223E-02 .45362E-01

8 .21435E+01 .19496E+00 .16715E-02

9 .22509E+01 .57735E+00 .00000E+00

10 .21023E+01 .39679E-03 .24105E-01

11 .21284E+01 .00000E+00 .34811E-03

a. Material volume of one quarter of the composite plateb. Norm of Lagrangian gradient (ACS)c. Maximum constraint violation (ACV)


Iteration Number

Objectivea Function



1 .37500E+01 .56250E+04 .23030E-01

2 .23573E+01 .25000E+04 .11967E+00

3 .21200E+01 .20148E+03 .59833E-01

A-100

Example Problems

4 .19176E+01 .20268E-01 .59833E-01

5 .19071E+01 .42583E-01 .59833E-01

6 .18930E+01 .69609E-01 .59833E-01

7 .18758E+01 .98635E-01 .59833E-01

8 .18560E+01 .12725E+00 .59833E-01

9 .18340E+01 .15354E+00 .59833E-01

10 .18103E+01 .44355E+09 .29916E-01

11 .18016E+01 .89066E+08 .29916E-01

12 .17924E+01 .12500E+04 .37395E-02

13 .17923E+01 .33404E+15 .29916E-01

14 .17920E+01 .24674E+15 .23933E+00

15 .17902E+01 .69921E+14 .23933E+00

16 .17897E+01 .26983E+18 .74791E-02

17 .17897E+01 .37833E+15 .59833E-01

18 .17893E+01 .16763E+15 .23933E+00

19 .17882E+01 .59275E+09 .74791E-02

20 .17871E+01 .59275E+09 .74791E-02

a. Material volume of one quarter of the composite plateb. Squared norm of projected reduced gradient (ACS)c. Estimated stepsize (ALS)


A-101


Table A.8-9 Design iteration history for composite plate with RQP (Case 2)

Iteration Number

Objectivea Function






1 .37500E+01 .75000E+02 .00000E+00

2 .26250E+01 .75000E+02 .00000E+00

3 .18750E+01 .12941E+02 .75255E+00

4 .17239E+01 .12605E+00 .73879E+00

5 .18785E+01 .38177E+00 .12980E+00

6 .18201E+01 .65223E-01 .00000E+00

7 .17857E+01 .55523E-02 .39503E-02


Iteration Number

Objectivea Function



1 .37500E+01 .10000E+01 .00000E+00

2 .22917E+01 .51925E-02 .70397E-01

3 .22917E+01 .15899E+00 .29981E-02

4 .33925E+01 .66667E+00 .00000E+00

5 .27055E+01 .66667E+00 .00000E+00

6 .22125E+01 .66667E+00 .00000E+00

7 .18496E+01 .39432E-02 .38077E+00

A-102

Example Problems

Figure A.8-3 Objective function histories for composite plate (Case 1)

8 .21087E+01 .29701E-02 .65637E-01

9 .21087E+01 .19808E+00 .36028E-02

10 .22305E+01 .66667E+00 .00000E+00

11 .20997E+01 .24670E-00 .00000E+00

a. Material volume of one quarter of the composite plateb. Norm of search direction vector (ACS)c. Maximum constraint violation (ACV)


A-103


Figure A.8-4 Objective function histories for composite plate (Case 2)

A-104

Example Problems

A.9 Optimum Design of a Sandwich Plate

Title

Optimum Design of a Sandwich Plate

Problem

A simply supported sandwich plate shown in Figure A.9-1 is subjected to auniform normal pressure of 100 psi. The plate is composed of three (3) layers (2face sheets and 1 core material). Due to symmetry, only one quarter of the plate ismodeled as shown in Figure A.9-2.

The design objective is to find layer thicknesses and rotation angles that give theminimum volume of the plate while satisfying constraints on stress ratios and layerthicknesses.

Element Type

3D Sandwich Shell (NKTP = 33, NORDR = 2)

Section Type

Laminated Sandwich Shell (ISECT =8)

Material Properties

Face sheets:

Elastic Moduli Ex = 10,000 ksi

Ey = 4,000 ksi

Gxy = 1,875 ksi

Yield Stresses Fxt = 50 ksi

Fxc = 50 ksi

Fyt = 15 ksi

A-105


Design Data

Fyc = 20 ksi

Fs = 15 ksi


Core material:

Elastic Moduli Gxz = 30 ksi

Gyz = 12 ksi

Yield Stresses Fxz = 2 ksi

Fyz = 2 ksi



No. of Elements = 4


Limiting Values for Design Variables

D. V. Type Lower Limit Upper Limit

thickness

— face 0.005 in. 0.100 in.

— core 0.100 in. 2.000 in.

angle -90.0 deg 90.0 deg.

Face sheets:

υ

A-106

Example Problems

Figure A.9-1 Geometry of sandwich plate

A-107


Figure A.9-2 Finite element model of sandwich plate

ANALYSIS=STATICLOAD CASE NO= 1AUTO CONSTRAINT=OFFRESEQUENCING OF ELEMENT=ONSAVE=26,27FILE=SAND*TITLE Problem TitleOPTIMUM DESIGN OF A SANDWICH PLATE*ELTYPE Element Type Selection (3-D Sandwich Shell)1,33,2*RCTABLE Real Constant Table1, 8

A-108

Example Problems

.025///////2, 8.75///////*LAMANGLE Rotation Angle for Composites1, 80.0///////*LAMSEQ Composite Lamination Sequence3,0, 1,2,1, 1,1,1, 1,2,1*E1 Element Connectivities1,1,2,3,8,13,12,11,6,1,1,0,2,2,1,3,11,12,13,18,23,22,21,16,1,1,0,2,2,1,*NODES Nodal Coordinates-5,0,5,5, 0.,1.25,0.1,0,,, 0.,0.,0.5,0,1,, 5.,0.,0.*MATERIAL Material Property DataEX,1,0, 1.E7EY,1,0, .4E7NUXY,1,0, .3GXY,1,0, 1.875E6FXT,1,0, 50.E3FXC,1,0, 50.E3FYT,1,0, 15.E3FYC,1,0, 20.E3FS,1,0, 15.E3EX,2,0, 0.GXZ,2,0, 3.E4GYZ,2,0, 1.2E4F12,2,0, 2000.*LDCASE Static Load Case Control Card0,1,2,0, 0,1*SPDISP Specified Displacements1, UX,0., 5, 1, UZ,ROTY21, UY,0., 25, 1,ROTX1, UY,0., 21, 5, UZ,ROTX5, UX,0., 25, 5, ROTY1,ROTZ,0.,25,1*L1 Pressure Loads1,1,4,1,1,1, 100./////$ 100./*CFORCE Concentrated Nodal Forces

A-109


Results

Table A.9-1 shows optimum results obtained by three optimization algorithms(GRG, RQP and OCB). Computational effort required in this problem is given inTable A.9-2 with the total numbers of structural and sensitivity analyses. Theconvergence parameters ACV, ACS and ALS used in this example are 0.005, 0.01,and 0.001, respectively. Parameter ACT is assigned a value of 0.001 for GRG and0.1 for both RQP and OCB. Design iteration histories are given in tables TableA.9-3 - Table A.9-5. It is observed that initial design is not feasible and an

increase of 24% in volume is achieved by RQP after 17 iterations. Note that OCBsolution is based on only three design variables viz. layer thicknesses whereasother two algorithms have six design variables viz layer thicknesses and rotationangles. Refer to Section 3.3 for more information on OCB. Figure A.9-3 showsgraphically the objective function histories.

*SFDCOMP Stress Filtering for Composites0.0*OPTPAR Control Parameters for Optimization1,1,30,2,40.1,0.005,0.01,0.01*DVGROUP Design Variable Groups and Section Type1,1,0,8,6*DVGELMT Design Variable Groups and Element Numbers1,1,4,1*DVLOCAL Local Design Variable Information1,1,01,0.025,0.005,0.10,1,0.0,-90.0,90.01,0.75,0.10,2.0,1,0.0,-90.0,90.01,0.025,0.005,0.10,1,0.0,-90.0,90.0** NDST11 for Face Sheets** NDST13 for Core Material*STRCON Stress Constraint IDs1,RATIO*STRELM Element Numbers For Stress Constraint IDs1,1,371,4,1*ENDDATA Data Deck Terminator

A-110

Example Problems

Table A.9-1 Optimum results of sandwich plate

Design Group

Number


Initial Design

(in, deg)


GRG RQP OCB

1 1,2,3,4 2.500E-02 .17323E-01 .26496E-01 .22019E-01

.000E+00 -.18269E+01

-.71164E+02 .17257E+01

7.500E-01 .11999E+01 .94048E+00 .11552E+01

.000E+00 .91181E+01 .11743E+02 -.32951E+01

2.500E-02 .12790E-01 .25582E-01 .20493E-01

.000E+00 -.15005E+02 .90000E+02 .30982E+02

Total Volume (in3) 2.000E+02 .30750E+02 .24814E+02 .29944E+02

Table A.9-2 Comparison of optimization algorithms for sandwich plate algo-rithm

Algorithm NITa


NSAb


NDSc


CPU timed (sec)

d. LXSI 6400 system

CPU time/ Design

Iteration (sec)

GRG 15 84 15 298.66 13.24

RQP 17 27 17 180.04 10.59

OCB 15 15 15 163.03 10.87

A-111


Table A.9-3 Design iteration history for sandwich plate with GRG

Iteration Number

Objectivea Function

a. Material volume of one quarter of the plate


b. Square norm of projected reduced gradient (ACS)



1 .34523E+02 .18750E+04 .47413E-01

2 .32505E+02 .18750E+04 .87790E-02

3 .32487E+02 .12182E+07 .87812E-02

4 .32365E+02 .77870E+06 .87812E-02

5 .32320E+02 .78796E+06 .35125E-01

6 .32051E+02 .82204E+06 .35125E-01

7 .31758E+02 .83707E+06 .35125E-01

8 .31444E+02 .83753E+06 .35125E-01

9 .31113E+02 .82776E+06 .98771E-02

10 .31015E+02 .98024E+03 .39509E-01

11 .30987E+02 .28299E+04 .39509E-01

12 .30941E+02 .55927E+04 .39509E-01

13 .30878E+02 .93989E+04 .39509E-01

14 .30796E+02 .30434E+02 .18806E-01

15 .30750E+02 .34887E+02 .29385E-03

A-112

Example Problems

Table A.9-4 Design iteration history for sandwich plate with RQP

Iteration Number

Objectivea Function






1 .20000E+02 .33350E+00 .69737E+00

2 .23348E+02 .12019E+01 .40042E+00

3 .25773E+02 .66240E+00 .17061E+00

4 .27043E+02 .10325E+01 .11034E+00

5 .27013E+02 .53689E+00 .17098E-01

6 .26529E+02 .19078E+00 .30241E-01

7 .26642E+02 .18811E+00 .14930E-01

8 .26431E+02 .89632E+00 .14265E-01

9 .26215E+02 .33750E+00 .84734E-02

10 .26250E+02 .80266E+00 .70197E-02

11 .26213E+02 .25302E+01 .70442E-02

12 .25929E+02 .12834E+00 .13923E-01

13 .25176E+02 .56614E+00 .13935E-01

14 .24572E+02 .14104E+00 .30888E-01

15 .24788E+02 .59268E-01 .18149E-01

16 .24800E+02 .23960E-01 .15707E-02

17 .24814E+02 .42573E-02 .74908E-04

A-113


Table A.9-5 Design iteration history for sandwich plate with OCB

Iteration Number

Objectivea Function






1 .20000E+02 .28824E+00 .69737E+00

2 .26752E+02 .14925E+00 .19015E+00

3 .30384E+02 .23959E-01 .23424E-01

4 .30968E+02 .22671E-01 .45511E-03

5 .30957E+02 .27460E-01 .37764E-03

6 .30726E+02 .55140E-03 .25903E-01

7 .30726E+02 .29420E-01 .65471E-03

8 .29711E+02 .26389E-01 .31270E+00

9 .30640E+02 .13081E-01 .52212E-01

10 .30640E+02 .50754E+00 .22693E-02

11 .30176E+02 .31831E-01 .45603E+00

12 .30176E+02 .22469E-01 .12025E+00

13 .30176E+02 .21559E-02 .13927E-01

14 .30176E+02 .53746E-01 .21872E-03

15 .29944E+02 .58789E-01 .22118E-02

A-114

Example Problems

Figure A.9-3 Objective function histories for sandwich plate

A-115

Optimum Design of a Circular Plate

A.10 Optimum Design of a Circular Plate

Title


Problem

The geometry and the finite element model of an axisymmetric circular plate areshown in Figure A.10-1. The plate is subjected to a uniform annular line load andbody forces due to gravity.

The design objective is to find element thicknesses that give the minimum volumewhile satisfying constraints on stresses and element thicknesses.

Element Type

Axisymmetric Shell (NKTP = 36, NORDR = 3)

Section Type

Uniform Thickness (ISECT=1)

Material Properties

Design Data

Elastic Modulus Ex = 30,000 ksi


Mass Density = 0.283 lb s2/in4



No. of Elements = 5

υ

A-116

Example Problems

Load Data

Annular line load of 10 lb at a radius of 3 inches and gravitational acceleration of386.4 in/sec2.

Figure A.10-1 Geometry of circular plate


Stress Limit (von Mises) = 30 ksi

Lower Limit on all Design Variables = .1 in.

Upper Limit on all Design Variables = 10 in.

ANALYSIS=STATICAUTO_CONSTRAINT=OFFRESEQUENCING_OF_ELEMENT=OFF** TOTAL NO.OF ELEMENTS=5** TOTAL NO.OF NODES =16*TITLE Problem Title0 OPTIMUM DESIGN OF CIRCULAR PLATE*ELTYPE Element Type Selection (Axisymmetric Shell)


A-117


1,36,3*RCTABLE Real Constant Table1, 81.0///*E1 Element Connectivities1,1,2,3,4,$1,1,1,5,3,1*NODES Nodal Coordinates1,,,,0.,0.,0.4,,1,,3.0,0.,0.5,,,,3.58333,0.,0.16,,1,,10.,0.,0.*MATERIAL Material Property DataEX,1,0,.30E6NUXY,1,0,.3DENS,1,0,.283*LDCASE Static Load Case Control Card0, 0, 2, 0, 0, 0, 0, .0000E+00, .0000E+00*SPDISP Specified Displacements1,UX,0.,1,0,ROTZ16,UX,0.0,16,0,UY*BODYFORCE Body Forces0,0,0,0,-386.4,0*CFORCE Concentrated Nodal Forces4,FY,-10.0,4,0*OPTPAR Control Parameters for Optimization1,1,10,2,4.1,.01,.01,.01,.001*DVGROUP Design Variable Groups and Section Type1,5,1,1,1*DVGELMT Design Variable Groups and Element Numbers1,1,1,0,5,1,1*DVLOCAL Local Design Variable Information1,5,11,1.0,0.1,10.*STRCON Stress Constraint IDs1,VON,30000.*STRELM Element Numbers For Stress Constraint IDs1,1,341,5,1*ENDDATA Data Deck Terminator

ANALYSIS=STATIC

A-118

Example Problems

Results

Table A.10-1 shows optimum results obtained by three optimization algorithms(GRG, RQP and OCB). Computational effort required in this problem is given inTable A.10-2 with the total numbers of structural and sensitivity analyses. Theconvergence parameters ACV, ACS and ALS used in this example are 0.01, 0.01,and 0.001, respectively. Parameter ACT is assigned a value of 0.01 for GRG and0.1 for both RQP and OCB. Design iteration histories are given in tables TableA.10-3 - Table A.10-5. It is observed that initial design is feasible and a reduction

of 65% in volume is achieved. Figure A.10-2 shows graphically the objectivefunction histories.

Table A.10-1 Optimum results of circular plate

Design Group

Number




GRG RQP OCB

1 1 .10000E+01 .48569E+00 .48311E+00 .48319E+00

2 2 .10000E+01 .45252E+00 .45172E+00 .45150E+00

3 3 .10000E+01 .40569E+00 .40476E+00 .40475E+00

4 4 .10000E+01 .33644E+00 .33591E+00 .33596E+00

5 5 .10000E+01 .23818E+00 .23785E+00 .23784E+00


Table A.10-2 Comparison of optimization algorithms for circular plate

Algorithm NITa NSAb NDSc CPU timed (sec)

CPU time/ Design Iteration (sec)

GRG 8 44 7 83.83 10.48

RQP 10 16 7 45.26 4.52

OCB 10 10 8 38.55 3.85

A-119


a. Total number of design iterationsb. Total number of structural analysesc. Total number of sensitivity analysesd. ELXSI 6400 system

Table A.10-3 Design iteration history for circular plate with GRG

Iteration Number

Objectivea Function

a. Material volume of the circular plate





1 .31416E+03 .48003E+05 .10860E+01

2 .14916E+03 .34081E+05 .34825E-01

3 .14477E+03 .23537E+05 .95156E-01

4 .13487E+03 .15184E+05 .16482E+00

5 .12129E+03 .96459E+04 .11536E+00

6 .11418E+03 .90579E+04 .82184E-01

7 .10955E+03 .16304E+04 .12112E-01

8 .10924E+03 .00000E+00 .12112E-01

Table A.10-4 Design iteration history for circular plate with RQP

Iteration Number

Objectivea Function



1 .31416E+03 .21909E+03 .00000E+00

2 .21063E+03 .21909E+04 .00000E+00

3 .15887E+03 .21909E+05 .00000E+00

A-120

Example Problems

4 .13298E+03 .25264E+03 .62566E+00

5 .10577E+03 .21865E+03 .87554E+00

6 .86446E+02 .96934E+02 .82350E+00

7 .77243E+02 .20270E+00 .56848E+00

8 .10079E+03 .77439E-01 .13393E+00

9 .10810E+03 .69764E-02 .23234E-01

10 .10904E+03 .12666E-03 .14232E-03

a. Material volume of the circular plateb. Norm of Lagrangian gradient (ACS)c. Maximum constraint violation (ACV)

Table A.10-5 Design iteration history for circular plate with OCB

Iteration Number

Objectivea Function



1 .31416E+03 .10000E+01 .00000E+00

2 .20523E+03 .10000E+01 .00000E+00

3 .14119E+03 .34124E-01 .24073E+00

4 .14119E+03 .47419E-02 .27384E-01

5 .14119E+03 .68637E+00 .17372E-03

6 .10154E+03 .13904E+00 .19028E+01

7 .10154E+03 .16923E+00 .52250E+00

8 .10154E+03 .69081E-01 .12506E+00

9 .10811E+03 .10100E-01 .24432E-01

Table A.10-4 Design iteration history for circular plate with RQP

A-121


Figure A.10-2 Objective function histories for circular plate

10 .10904E+03 .24354E-03 .48146E-03

a. Material volume of the circular plateb. Norm of search direction vector (ACS)c. Maximum constraint violation (ACV)

Table A.10-5 Design iteration history for circular plate with OCB

A-122

Example Problems

A.11 Optimum Design of a Hemispherical Dome

Title

Optimum Design of a Hemispherical Dome

Problem

The geometry and the finite element model of a hemispherical shell with anopening at the crown, subjected to edge loading, are shown in Figure A.11-1 andFigure A.11-2.

The design objective is to find element thicknesses that give the minimum volumeof the structure while satisfying constraints on stresses and displacements.

Element Type


Section Type

Uniform Thickness (ISECT= 1)

Material Properties

Design Data

Elastic Modulus Ex = 107psi

Poisson’s Ratio xy = .33




No. of Stress constraints = 36

υ

A-123


Load Data

Axisymmetric edge loading of 1000 lb/in at the free edge in Y-direction(downward).

Figure A.11-1 Geometry of hemispherical shell



Displacement Limit for Top Edge = 0.1 in.




A-124

Example Problems

Figure A.11-2 Hemispherical shell (node and element numbers)

ANALYSIS=STATICAUTO_CONSTRAINT=OFFRESEQUENCING_OF_ELEMENT=OFF** TOTAL NO.OF ELEMENTS=12** TOTAL NO.OF NODES =25*TITLE Problem TitleOPTIMUM DESIGN OF A HEMISPHERICAL DOME*ELTYPE Element Type Selection (Axisymmetric Shell)1,36,2

A-125


*RCTABLE Real Constant Table1, 81.0//*E1 Element Connectivities1,1,2,3$1,1,1,12,2,1*NODES Nodal Coordinates1,1$100.25,1,1,0,100.,60.*MATERIAL Material Property DataEX,1,0,.10E8NUXY,1,0,.33*LDCASE Static Load Case Control Card0, 0, 2, 0, 0, 0, 0, .0000E+00, .0000E+00*SPDISP Specified Displacements1,UX,0.,$UY,ROTZ*CFORCE Nodal Concentrated Forces25,FY,-3.1416E+05*OPTPAR Control Parameters for Optimization1,1,20,3,40.1,0.01,0.01,0.01*DVGROUP Design Variable Groups and Section Type1,12,1,1,1*DVGELMT Design Variable Groups and Element Numbers1,1,1,0,12,1,1*DVLOCAL Local Design Variable Information1,12,11,1.0,0.1,10.0*DISCON Displacement Constraint IDs1,UY,0.1*DISNOD Node Numbers for Displacement Constraint IDs1,125*STRCON Stress Constraint IDs1,VON,15000.*STRELM Element Numbers For Stress Constraint IDs1,1,341,12,1*ENDDATA Data Deck Terminator

ANALYSIS=STATIC

A-126

Example Problems

Results

Table A.11-1 shows optimum results obtained by three optimization algorithms(GRG, RQP and OCB). Computational effort required in this problem is given inTable A.11-2 with the total numbers of structural and sensitivity analyses. Theconvergence parameters ACV, ACS and ALS used in this example are 0.01, 0.01,and 0.001, respectively. Parameter ACT is assigned a value of 0.01 for GRG and0.1 for both RQP and OCB. Design iteration histories are given in tables TableA.11-3 - Table A.11-5. It is observed that initial design is infeasible but a

reduction of 75% in volume is achieved. Note that GRG fails in line search after 7iterations. Figure A.11-3 shows graphically the objective function histories.

Table A.11-1 Optimum results of hemispherical dome

Design Group

Number




GRG RQP OCB

1 1 .10000E+01 0.10000E+00 0.10000E+00 0.10000E+00

2 2 .10000E+01 0.10000E+00 .10000E+00 .10000E+00

3 3 .10000E+01 0.10000E+00 .10000E+00 .10000E+00

4 4 .10000E+01 0.10000E+00 .10000E+00 .10000E+00

5 5 .10000E+01 0.10000E+00 .10000E+00 .10000E+00

6 6 .10000E+01 0.10265E+00 .10000E+00 .10000E+00

7 7 .10000E+01 0.10000E+00 .10000E+00 .10000E+00

8 8 .10000E+01 0.10000E+00 .10000E+00 0.10000E+00

9 9 .10000E+01 0.11213E+00 0.10788E+00 0.11252E+00

10 10 .10000E+01 0.20165E+00 0.14525E+00 0.13985E+00

11 11 .10000E+01 0.82827E+00 0.75707E+00 0.84781E+00

12 12 .10000E+01 0.17936E+01 0.18826E+01 0.18311E+01


A-127


Table A.11-2 Comparison of optimization algorithms for hemispherical dome

Algorithm NITa


NSAb


NDSc


CPU timed (sec)


CPU time/ Design

Iteration (sec)

GRG 7 33 6 209.06 29.86

RQP 12 14 12 133.06 9.50

OCB 13 13 13 128.33 9.07

Table A.11-3 Design iteration history for hemispherical dome with GRG

Iteration Number

Objectivea Function

a. Material volume of the hemispherical dome





1 0.79317E+05 0.39169E+09 0.88384E+00

2 0.65715E+05 0.36769E+09 0.88384E+00

3 0.52218E+05 0.36410E+09 0.88384E+00

4 0.38788E+05 0.36005E+09 0.88384E+00

5 0.25439E+05 0.35585E+09 0.88384E+00

6 0.15808E+05 0.10024E+09 0.44249E+00

7 0.13507E+05 0.12028E+10 0.00000E+00

A-128

Example Problems

Table A.11-4 Design iteration history for hemispherical dome with RQP

Iteration Number

Objectivea Function

a. Material volume of the hemisperical dome





1 0.53893E+05 0.19125E+05 0.47886E+00

2 0.14704E+05 0.18987E+05 0.65380.E+00

3 0.13134E+05 0.16192E+05 0.65367E+00

4 0.12146E+05 0.12273E+05 0.65213E+00

5 0.11805E+05 0.42971E+01 0.64975E+00

6 0.10965E+05 0.21758E+01 0.62908E+00

7 0.10738E+05 0.41551E+00 0.60381E+00

8 0.11632E+05 0.23803E+00 0.21643E+00

9 0.12666E+05 0.94050E-01 0.97098E-01

10 0.12923E+05 0.93983E-01 0.50874E-02

11 0.12936E+05 0.15016E-01 0.40657E-02

12 0.12947E+05 0.92085E-02 0.27463E-02

A-129


Table A.11-5 Design iteration history for hemispherical dome with OCB

Iteration Number

Objectivea Function

a. Material volume of the hemispherical dome





1 0.53893E+05 0.23184E+00 0.47886E+00

2 0.54733E+05 0.14493E+00 0.17178E+00

3 0.55314E+05 0.43488E-01 0.39326E-01

4 0.55497E+05 0.96251E+00 0.52881E-02

5 0.40859E+05 0.16602E-01 0.12489E-01

6 0.40858E+05 0.95790E+00 0.11757E-02

7 0.26337E+05 0.95361E+00 0.98788E-02

8 0.17712E+05 0.88586E+00 0.86474E-02

9 0.14651E+05 0.50539E+00 0.75079E-02

10 0.13194E+05 0.39387E-01 0.21666E+00

11 0.13298E+05 0.17383E+00 0.78299E-03

12 0.12974E+05 0.49436E-01 0.54812E+00

13 0.13095E+05 0.43536E-01 0.79724.E-02

A-130

Example Problems

Figure A.11-3 Objective function histories for hemispherical dome

A-131

Optimum Design of a Water Tank

A.12 Optimum Design of a Water Tank

Title


Problem

A cylindrical concrete wall with dimensions as shown in Figure A.12-1 issubjected to a triangular pressure load of 0 Pa. at top and 98100 Pa. at bottom of thewall. The wall is fixed at bottom and free at top.

The design objective is to find wall thicknesses in each of the four regions (2.5 mheight) that minimize the volume of the wall without exceeding limiting tensilestresses in the concrete.

Element Type


Section Type


Material Properties

Design Data

Elastic Modulus = 28,000 MPa (Concrete)

Poisson’s Ratio = 1/6

Specific Weight = 24.465 kN/m3

Number of Design Variables = 4

Number of Load Cases = 1

Number of Elements = 4

A-132

Example Problems

Remarks

It can be seen from Figure A.12-1 that σYY represents normal stress due tobending moment about Z-axis and the weight of the wall. Also σZZ represents thehoop stress.

Figure A.12-1 Geometry of cylindrical concrete wall

Number of Stress Constraints = 16

Lower Bound on Design Variables = 50 mm

Upper Bound on Design Variables = 1500 mm

Limit on Hoop Tensile Stress = 1.40 MPa

Limit on Bending Tensile Stress = 1.88 MPa


A-133


FILE=CYLTSAVE=26,27*TITLE Problem TitleOPTIMUM DESIGN OF A WATER TANK*ELTYPE Element Type Selection (Axisymmetric Shell)1, 36 , 1*RCTABLE Real Constant Table1 , 2900.0/2 , 2700.0/3 , 2600.0/4 , 2200.0/*ELEMENTS Element Connectivities1, 1, 1, 1, 01, 22, 1, 1, 2, 02, 33, 1, 1, 3, 03, 44, 1, 1, 4, 04, 5*NODES Nodal Coordinates1 , 0 , , ,20000., 0.0 , 0.0 , 02 , 0 , , ,20000., 2500.0 , 0.0 , 03 , 0 , , ,20000., 5000.0 , 0.0 , 04 , 0 , , ,20000., 7500.0 , 0.0 , 05 , 0 , , ,20000.,10000.0 , 0.0 , 0*MATERIAL Material Property DataEX, 1, 0, 28000.000 ,,,,,NUXY, 1, 0, 0.1666667 ,,,,,DENS, 1, 0, 2.4465E-9 ,,,,,ALPX, 1, 0, 0.0000000 ,,,,,*LDCASE Static Load Case Control Card0, 1, 2, 0, 1, 0, 1, 0.0, 0.0*I4 LOAD CASE TITLETRIANGULAR WATER PRESSURE AND SELF WEIGHT OF THE WALL

A-134

Example Problems

Results

Table A.12-1 shows optimum results obtained by three optimization algorithms(GRG, RQP and OCB). Computational effort required in this problem is given inTable A.12-2 with the total numbers of structural and sensitivity analyses. The

*SPDISP Specified Displacements1, UX, 0.0, 1, 0, UY, ROTZ*BODYFORCE Body Forces0.0, 0.0, 0.0, 0.0,-9810.0, 0.0*PRESSURE Pressure Loads1 , 0 , 0 , 2 , 198.1E-3 ,73.575E-32 , 0 , 0 , 2 , 173.575E-3,49.05E-33 , 0 , 0 , 2 , 149.05E-3,24.525E-34 , 0 , 0 , 2 , 124.525E-3,0.0*OPTPAR Control Parameters for Optimization1,1,25,3,20.4,0.01,0.005,0.005,0.005,9810.0*DVGROUP Design Variable Groups and Section Type1,4,1,1,1*DVGELMT Design Variable Groups and Element Numbers1,1,1,0,4,1,1*DVLOCAL Local Design Variable Information1,1,01,900.0,50.0,1500.02,2,01,700.0,50.0,1500.03,3,01,600.0,50.0,1500.04,4,01,200.0,50.0,1500.0*STRCON Stress Constraint IDs1,NDST22,1.88,NDST33,1.40*STRELM Element Numbers for Stress Constraint IDs1,1,311,4,1*ENDDATA Data Deck Terminator

A-135


convergence parameters ACV, ACS and ALS used in this example are 0.01, 0.01,and 0.001, respectively. Parameter ACT is assigned a value of 0.01 for GRG and0.1 for both RQP and OCB. Design iteration histories are given in tables TableA.12-3 - Table A.12-5. It is observed that initial design is infeasible but a

reduction of 11% in volume and a feasible design is achieved with RQP.Figure A.12-2 shows graphically the objective function histories.

Table A.12-1 Optimum results for water tank

Design Group

Number




GRG RQP OCB

1 1 .90000E+03 .9761E+03 .82041E+03 0.92690E+03

2 2 .70000E+03 .75924E+03 .37588E+03 0.21620E+03

3 3 .60000E+03 .63017E+03 .76884E+03 0.86740E+03

4 4 .20000E+03 .23221E+03 .16964E+03 0.10760E+03

Total Volume (mm3) .75398E+12 0.81612E+12 0.67066E+12 0.66514E+12

Table A.12-2 Comparison of optimization algorithms for water tank

Algorithm NITa


NSAb


NDSc


CPU timed (sec)


CPU time/ Design Iteration

(sec)

GRG 5 44 4 73.93 14.76

RQP 4 22 4 37.30 9.32

OCB 25 25 25 104.10 4.16

A-136

Example Problems

Table A.12-3 Design iteration history for water tank with GRG

Iteration Number

Objectivea Function

a. Material volume of the water tank





1 0.81779E+12 0.23675E+21 0.80000E+01

2 0.81753.E+12 0.58037E+21 0.80000E+01

3 0.81717E+12 0.42213E+22 0.20000E+01

4 0.81658E+12 0.11298E+22 0.80000E+01

5 0.81612E+12 0.39478E+18 0.0000E+00

Table A.12-4 Design iteration history for water tank with RQP

Iteration Number

Objectivea Function

a. Material volume of the water tank





1 0.75398E+12 0.20966E+09 0.84626E-01

2 0.67709E+12 0.10034E+04 0.00000E+00

3 0.67215E+12 0.55601E+09 0.00000E+00

4 0.67066E+12 0.62832E+09 0.19241E-02

A-137


Table A.12-5 Design iteration history for water tank with OCB

Iteration Number

Objectivea Function



1 0.75398E+12 0.73896E+02 0.84626E-01

2 0.79693E+12 0.10000E+01 0.00000E+00

3 0.66788E+12 0.56133E+03 0.44254E+00

4 0.66788E+12 0.11992E+03 0.16486E+00

5 0.66788E+12 0.37949E+00 0.47390E+00

6 0.66776E+12 0.38106E+00 0.47913E+00

7 0.66764E+12 0.38263E+00 0.48442E+00

8 0.66752E+12 0.38418E+00 0.48976E+00

9 0.66740E+12 0.38572E+00 0.49516E+00

10 0.66728E+12 0.38724E+00 0.50063E+00

11 0.66716E+12 0.38875E+00 0.50616E+00

12 0.66704E+12 0.39025E+00 0.51174E+00

13 0.66691E+12 0.39173E+00 0.51739E+00

14 0.66679E+12 0.39319E+00 0.52310E+00

15 0.66667E+12 0.39465E+00 0.52888E+00

16 0.66654E+12 0.39608E+00 0.53472E+00

17 0.66642E+12 0.39750E+00 0.54062E+00

18 0.66629E+12 0.39891E+00 0.54659E+00

19 0.66617E+12 0.40030E+00 0.55263E+00

20 0.66604E+12 0.40167E+00 0.55873E+00

A-138

Example Problems

Figure A.12-2 Objective function histories for water tank

21 0.66592E+12 0.40302E+00 0.56490E+00

22 0.66579E+12 0.40436E+00 0.57114E+00

23 0.66566E+12 0.40568E+00 0.57745E+00

24 0.66553EE+12 0.40698E+00 0.58383E+00

25 0.66541E+12 0.40827E+00 0.59028E+00

a. Material volume of the water tankb. Norm of search direction vector (ACS)c. Maximum constraint violation (ACV)

Table A.12-5 Design iteration history for water tank with OCB

A-139

Optimum Design and Design Sensitivity Analysis of a Swept Wing

A.13 Optimum Design and Design Sensitivity Analysis of a Swept Wing

Title


Problem

The geometry and the finite element model of the Swept Wing structure are shownin tables Table A.13-1 and Table A.13-2. Since the structure is symmetrical aboutthe horizontal X-Y plane, only half of the structure above X-Y plane is modeled.Figure A.13-1 shows nodes from 1 to 44 only. However, there are 44 more nodeswhich are the projections of node numbers 1 to 44 on the X-Y plane of symmetry.These node numbers (45 to 88) are not shown on the Figure A.13-1. Skin panel ismodeled by triangular shell elements, Webs by quadrilateral shell elements, andSpar caps by beam elements, respectively.

The following objectives are desired.

1. To find shell element thicknesses and beam radii that give the lightestWing structure while satisfying constraints on displacements, stresses,and member sizes.

2. To compute sensitivity coefficients of i) ratio of von Mises stress toallowable stress sensitivities for all the shell elements at all the nodes, ii)ratio of axial stress to allowable stress of bending stress sensitivities forall the beam elements at all the nodes, and iii) ratio of displacement (Z-direction) to allowable displacement sensitivities for nodes 41 to 44.Note that the sensitivities are computed for both load cases.

Element Type

3D Thin Shell (NKTP = 40, NORDR = 1)

3D Thin Shell (NKTP = 40, NORDR = 10)


A-140

Example Problems

Section Type

Uniform Thickness (ISECT =1) for 3D Thin ShellSolid Circular Cross-section (ISECT=3) for Beam

Material Properties

Design Data

Elastic Modulus = 10,600 ksi (Aluminum alloy)Specific Weight = 0.096 lb/in3


No. of Design Variables = 32No. of Load Cases = 2No. of Nodes = 88No. of Elements = 150No. of Constraints = 1008Displacement Limit for Tip Nodes (41-44) in Z-directionStress Limit (von Mises) at all Nodes of Shell ElementsStress Limit (Axial) for all Beam ElementsStress Limit (BEND3) for all Beam Elements

= 60 in.= 25 ksi.= 25 ksi.= 25 ksi.

Limiting values for design variables

Design Variable

TypeElement Starting Value Lower

LimitUpper Limit

Radius 3D Beam 0.08 in. (for sensitivity analysis 0.02 in.) 0.050 in. 1.00 in.

Thickness 3D Shell 0.20 in. 0.020 in. 10.00 in

A-141


Figure A.13-1 Geometry of swept wing (node numbering)

A-142

Example Problems

Figure A.13-2 Swept wing (element numbers)

A-143


Load Data

All loads are in kips and are along Z direction (upward).

Node Number Load Case 1 Load Case 2

5 1.282 2.361

6 2.581 3.876

7 3.398 2.308

8 2.380 0.793

9 0.978 1.772

10 2.013 2.895

11 2.593 1.705

12 1.764 0.582

13 0.727 1.310

14 1.386 2.135

15 1.906 1.258

16 1.297 0.433

17 0.570 1.047

18 1.190 1.719

19 1.453 1.025

20 1.057 0.355

21 0.459 0.843

22 0.958 1.374

23 1.251 0.825

24 0.852 0.284

A-144

Example Problems

25 0.362 0.665

26 0.756 1.092

27 0.986 0.651

28 0.671 0.224

29 0.282 0.518

30 0.589 0.851

31 0.768 0.508

32 0.522 0.175

33 0.206 0.402

34 0.431 0.646

35 0.563 0.398

36 0.383 0.154

37 0.144 0.311

38 0.302 0.482

39 0.395 0.306

40 0.269 0.135

41 0.620 0.133

42 0.129 0.206

43 0.169 0.131

44 0.116 0.580

ANAL=STATIC***For sensitivity analysis add following line onlyDESIGN=SENSITIVITY

A-145


***AUTO=ONBLANK=35000EXEC=GOMAXWAVE=750RESEQUENCE=ONWARN=GO*TITLE Problem TitleOPTIMUM DESIGN OF SWEPT WING*ELTYPE Element Type Selections (3-D Thin Shell and 3-D Beam)1, 40, 10,2, 40, 1,3, 12, 1,*RCTABLE Real Constant Table1 , 30.2//2 , 40.2///3 , 80.02,0.0000318,0.0000318,0.0000636,0.,0.,0.15958,0.15958*ELEMENTS (See Figure A.13.2)*NODES Nodal Coordinates (See Figure A.13.1)*VECTORS Definition of Vectors1,0.0000E+00,0.0000E+00,0.1000E+022,0.0000E+00,0.0000E+00,0.1300E+02*MATERIAL Material Property DataEX,1,0,10.6E3NUXY,1,0,0.3*sets555,s,43,23777,s,40,45888,r,105,109,1*********************************** LOAD CASE : 1 ************************************LDCASE Static Load Case Control Card**KELFR KRCTN KSTR KSTN LQ1 LQ2 LQ7 TSFRE RCFORCE0, 0, 2, 0, 0, 4, 0, 0.000, 0.000,*SPDISP Boundary Conditions and Specified Displacements1,UX , 0.000000,4,1,UY,UZ,ROTX,ROTY,ROTZ45,UX , 0.000000,$,UY,UZ,ROTX,ROTY,ROTZ

A-146

Example Problems

56,UX , 0.000000,$,UY,UZ,ROTX,ROTY,ROTZ67,UX , 0.000000,$,UY,UZ,ROTX,ROTY,ROTZ78,UX , 0.000000,$,UY,UZ,ROTX,ROTY,ROTZ46,UX, 0.000000,55,1,UY57,UX, 0.000000,66,1,UY68,UX, 0.000000,77,1,UY79,UX, 0.000000,88,1,UY 1*CFORCE Concentrated Nodal Forces5,FZ, 1.282006,FZ, 2.581007,FZ, 3.398008,FZ, 2.380009,FZ, 0.9780010,FZ, 2.0130011,FZ, 2.5930012,FZ, 1.7640013,FZ, 0.7270014,FZ, 1.3860015,FZ, 1.9060016,FZ, 1.2970017,FZ, 0.57018,FZ, 1.1900019,FZ, 1.4530020,FZ, 1.0570021,FZ, 0.4590022,FZ, 0.9580023,FZ, 1.2510024,FZ, 0.8520025,FZ, 0.3620026,FZ, 0.7560027,FZ, 0.9860028,FZ, 0.6710029,FZ, 0.2820030,FZ, 0.5890031,FZ, 0.76832,FZ, 0.5220033,FZ, 0.2060034,FZ, 0.4310035,FZ, 0.5630036,FZ, 0.3830037,FZ, 0.14400

A-147


38,FZ, 0.3020039,FZ, 0.3950040,FZ, 0.2690041,FZ, 0.6200042,FZ, 0.1290043,FZ, 0.1690044,FZ, 0.11600*********************************** LOAD CASE : 2 ************************************LDCASE Static Load Case Control Card**KELFR KRCTN KSTR KSTN LQ1 LQ2 LQ7 TSFRE RCFORCE0, 0, 2, 0, 0, 4, 0, 0.000, 0.000, 1*CFORCE Concentrated Nodal Forces5,FZ, 2.361006,FZ, 3.876007,FZ, 2.308008,FZ, 0.793009,FZ, 1.7720010,FZ, 2.8950011,FZ, 1.7050012,FZ, 0.5820013,FZ, 1.3100014,FZ, 2.1350015,FZ, 1.2580016,FZ, 0.4330017,FZ, 1.0470018,FZ, 1.7190019,FZ, 1.0250020,FZ, 0.3550021,FZ, 0.8430022,FZ, 1.3740023,FZ, 0.8250024,FZ, 0.2840025,FZ, 0.6650026,FZ, 1.0920027,FZ, 0.6510028,FZ, 0.2240029,FZ, 0.5180030,FZ, 0.8510031,FZ, 0.50800

A-148

Example Problems

32,FZ, 0.1750033,FZ, 0.4020034,FZ, 0.6460035,FZ, 0.3980036,FZ, 0.1540037,FZ, 0.3110038,FZ, 0.4820039,FZ, 0.3060040,FZ, 0.1350041,FZ, 0.1330042,FZ, 0.2060043,FZ, 0.1310044,FZ, 0.58000*printsedi,555*OPTPAR Control Parameters for Optimization1,20,3,2,0,00.4,0.005,0.01,0.01,0.005*DVGROUP Design Variable Groups and Section Type1,18,1,1,119,32,1,3,1 1*DVGELMT Design Variable Groups and Element Numbers1,1,6,12,7,12,13,13,18,14,19,24,15,25,36,16,37,48,17,49,60,18,61,64,19,65,70,110,71,74,111,75,80,112,81,84,113,85,90,114,91,94,115,95,100,116,101,109,117,110,118,118,119,130,119,131,131,0

A-149


20,132,132,021,133,133,022,134,134,023,135,136,124,137,138,125,139,140,126,141,141,027,142,142,028,143,143,029,144,144,030,145,146,131,147,148,132,149,150,1*DVLOCAL Local Design Variable Information (for sensitivity analysis starting radius and thick-ness are set to 0.02.)1,18,11,0.2,0.02,10.019,32,11,0.08,0.05,1.0*DISCON Displacement Constraint IDs1,UZ,60.0*DISNOD Node Numbers for Displacement Constraint IDs1,1,1,241,44,1*STRCON Stress Constraint IDs1,VON,25.02,AXIAL,25.0,BEND3,25.0*STRELM Element Numbers for Stress Constraint IDs1,1,34,1,21,130,12,1,91,1,2131,150,1*ENDDATA Data Deck Terminator

A-150

Example Problems

Results for Structural Optimization

Table A.13-1 shows optimum results obtained by three optimization algorithms(GRG, RQP and OCB). Computational effort required in this problem is given inTable A.13-2 with the total numbers of structural and sensitivity analyses. Theconvergence parameters ACV, ACS and ALS used in this example are 0.01, 0.01,and 0.001, respectively. Parameter ACT is assigned a value of 0.01 for GRG and0.1 for both RQP and OCB. Design iteration histories are given in tables TableA.13-3 - Table A.13-5. It is observed that initial design is infeasible but a

reduction of 30% in volume is achieved with RQP and OCB. Figure A.13-3 showsgraphically the objective function histories.

Table A.13-1 Optimum results of swept wing

Design Group

Number




GRG RQP OCB

1 1-6 .20000E+00 .19519E+00 .19917E+00 .22398E+00

2 7-12 .20000E+00 .19622E+00 .16467E+00 .18569E+00

3 13-18 .20000E+00 .19742E+00 .14277E+00 .16772E+00

4 19-24 .20000E+00 .19788E+00 .11730E+00 .13936E+00

5 25-36 .20000E+00 .19692E+00 .10453E+00 .11565E+00

6 37-48 .20000E+00 .19810E+00 .63532E-01 .64754E-01

7 49-60 .20000E+00 .19903E+00 .20157E-01 .20000E-01

8 61-64 .20000E+00 .19931E+00 .20009E-01 .20000E-01

9 65-70 .20000E+00 .19974E+00 .20011E-01 .20000E-01

10 71-74 .20000E+00 .19899E+00 .14351E+00 .14464E+00

11 75-80 .20000E+00 .19963E+00 .10569E+00 .12809E+00

12 81-84 .20000E+00 .19915E+00 .22585E+00 .20318E+00

13 85-90 .20000E+00 .19970E+00 .87844E-01 .56444E-01

A-151


14 91-94 .20000E+00 .19967E+00 .55057E-01 .64796E-01

15 95-100 .20000E+00 .19986E+00 .22772E-01 .20000E-01

16 101-109 .20000E+00 .19955E+00 .20006E-01 .20000E-01

17 110-118 .20000E+00 .19938E+00 .21474E-01 .20739E-01

18 119-130 .20000E+00 .19947E+00 .20008E-01 .20000E-01

19 131 .80000E-01 .80000E-01 .50028E-01 .50000E-01

20 132 .80000E-01 .80000E-01 .58261E-01 .50882E-01

21 133 .80000E-01 .80000E-01 .50000E-01 .68945E-01

22 134 .80000E-01 .80000E-01 .50010E-01 .66040E-01

23 135,136 .80000E-01 .80000E-01 .10599E+00 .58516E-01

24 137,138 .80000E-01 .80000E-01 .56568E-01 .56242E-01

25 139,140 .80000E-01 .80000E-01 .55738E-01 .50000E-01

26 141 .80000E-01 .86262E-01 .10000E+01 .13073E+00

27 142 .80000E-01 .86262E-01 .84898E+00 .74977E-01

28 143 .80000E-01 .86264E-01 .10000E+01 .87903E-01

29 144 .80000E-01 .86264E-01 .88403E+00 .92307E-01

30 145,146 .80000E-01 .86261E-01 .49467E+00 .79533E-01

31 147,148 .80000E-01 .86262E-01 .14644E+00 .63620E-01

32 149,150 .80000E-01 .86264E-01 .69484E-01 .59225E-01

Total Volume (in3) .38835E+05 0.19159E+05

.266122E+05 .27150E+05

Table A.13-1 Optimum results of swept wing (Continued)

A-152

Example Problems

Table A.13-2 Comparison of optimization algorithms for swept wing

Algorithm NITa


NSAb


NDSc


CPU timed (sec)*

d. ELXSI 6400 system* Finite difference method for design sensitivity analysis is used in this run


(sec)

GRG 3 15 3 693.00 231

RQP 30 33 30 2355.83 78.52

OCB 14 14 14 854.87 61.06

A-153


Table A.13-3 Design iteration history for swept wing with GRG

Iteration Number

Objectivea Function

a. Material volume of one half of the swept wing





1 0.19417E+05 0.10113E+10 0.50048E-02

2 0.19258E+05 0.10113E+10 0.25024E-02

3 0.19159E+05 0.10113E+10 0.62560E-03

Table A.13-4 Design iteration history for swept wing with RQP

Iteration Number

Objectivea Function



1 .19417E+05 .20253E+05 .78356E-01

2 .16346E+05 .12061E+05 .17410E+00

3 .15605E+05 .56980E+05 .17358E+00

4 .15284E+05 .53618E+04 .17293E+00

5 .14814E+05 .16781E+04 .16985E+00

6 .14501E+05 .87775E+03 .16689E+00

7 .14436E+05 .98959E+02 .16765E+00

8 .14422E+05 .48241E+02 .16786E+00

9 .14398E+05 .16949E+03 .16746E+00

10 .14348E+05 .18771E+03 .16735E+00

A-154

Example Problems

11 .14067E+05 .19331E+03 .15502E+00

12 .14019E+05 .28922E+02 .12604E+00

13 .14008E+05 .25789E+03 .12437E+00

14 .13996E+05 .20369E+01 .12282E+00

15 .13967E+05 .26728E+02 .11834E+00

16 .13963E+05 .15234E+02 .11811E+00

17 .13930E+05 .62393E+02 .11142E+00

18 .13912E+05 .33409E+02 .10989E+00

19 .13944E+05 .33690E+01 .10381E+00

20 .13860E+05 .21823E+02 .95535E-01

21 .13850E+05 .16371E+02 .92566E-01

22 .13829E+05 .79406E+01 .87471E-01

23 .13808E+05 .37007E+01 .84367E-01

24 .13784E+05 .14947E+01 .81390E-01

25 .13738E+05 .10919E+01 .72227E-01

26 .13620E+05 .36494E+00 .20386E-01

27 .13336E+05 .17829E+01 .60782E-02

28 .13329E+05 .10493E+01 .45488E-02

29 .13323E+05 .46869E+00 .27696E-02

30 .13306E+05 .43099E+00 .18233E-02

a. Material volume of one half of the swept wingb. Norm of Lagrangian gradient (ACS)c. Maximum constraint violation (ACV)

Table A.13-4 Design iteration history for swept wing with RQP (Contin-

A-155


Table A.13-5 Design iteration history for swept wing with OCB

Iteration Number

Objectivea Function

a. Material volume of one half of the swept wing





1 .19417E+05 .19601E-01 .78356E-01

2 .19822E+05 .63842E+00 .60703E-02

3 .11804E+05 .14082E+00 .35785E+01

4 .11804E+05 .18419E+00 .15312E+01

5 .12256E+05 .35140E+00 .59940E+00

6 .13011E+05 .40871E+00 .15334E+00

7 .13247E+05 .23527E+00 .17552E+00

8 .13247E+05 .20791E+00 .22692E+00

9 .13365E+05 .51134E-01 .68188E+00

10 .13388E+05 .12128E+00 .45853E-01

11 .13389E+05 .11402E+00 .32973E-01

12 .13396E+05 .69775E-01 .79557E-01

13 .13396E+05 .83044E-01 .18118E-01

14 .13575E+05 .24444E-01 .16253E-02

A-156

Example Problems

Figure A.13-3 Objective function histories for swept wing

Results for Design Sensitivity Analysis

There are 150 elements with two types of design variables. Design variables arethicknesses of shell elements and radius of the solid circular section of beamelements. Elements are grouped in 32 different design variable groups as given in*DVGELMT data group and depicted in Table A.13-5 Design variables for groups1 to 18 are the thickness of thin shell elements and for groups 19 to 32 are theradius of the solid circular sections consisting of beam elements. As Figure A.13-2shows for NKTP=40, NORDR=10 the number of nodes for each triangular elementis 3, thus, for INNATE=34 number of von Mises stress sensitivities for 60 elements(elements 1 through 60) are 180 (3*60). Similarly, for NKTP=40, NORDR=1 thenumber of nodes for each quadrilateral element is 4, thus, for INNATE=34, numberof von Mises stress sensitivities for 70 elements (elements 61 through 130) are 280

A-157


(4*70). Also, for NKTP=12, NORDR=1 the number of nodes for each Sparelement is 2, thus, for INNATE=91, number of axial as well as bending momentstress sensitivities for 20 elements (elements 131 through 150) are 80 (2*2*20).Consequently total number of stress sensitivities for 1 load case are 540 (that is180+280+80) and for 2 load cases as requested by *STRELM data group are 1080(540*2). Displacement sensitivities are calculated at nodes 41 to 44 (4 nodes) in Z-direction (1 direction) which implies total number of displacement sensitivitiesrequested by *DISNOD data group for 2 load cases are 8 (4*1*2).

Sensitivity coefficients of ratio of displacements to allowable displacements andratio of stresses to allowable stresses are calculated with respect to global designvariables and printed out selectively. Printout selections depend on the specifiedentries in *SETS and LABEL in *PRINTCNTL data groups (see input data). TableA.13-6 shows the relations between design variable group number specified in

*DVGROUP data group and global design variable numbering scheme computedinternally by the program. Sensitivity coefficients are randomly computed asfollows:

1. for displacements, nodes 41 to 44 in Z-direction for both load cases.

2. for stresses, von Mises stress for elements 1 to 130 and stress due to axialforce and bending moments for elements 131 to 150 for both load cases.

Sensitivity coefficients are randomly printed out as follows:

1. displacement at node 43 for load case ID number 200.

The sensitivity coefficients are graphically shown for some cases.Figure A.13-4 shows displacement sensitivity for node number 43 in Z-direction for load case ID number 200. Using Eqeation 2.5-9 andFigure A.13-4, one can conclude that the ratio of displacement to allow-able displacement is more sensitive to design variables 1 and 2 (thick-nesses of design variable groups 1 and 2) and the most critical globaldesign variable is number 1 with sensitivity coefficients of 72.25. Graphtypes such as Figure A.13-4 are quite useful in case of constraint correc-tions. They can be plotted easily in DISPLAY III by invoking post resultsand graph options.

A-158

Example Problems

Table A.13-6 Design variable group number and associated user element numbers

Design Group Number

User Element Numbers D.V. Type Value of Design

(in.)

1 1-6 Thickness .20000E+00

2 7-12 Thickness .20000E+00

3 13-18 Thickness .20000E+00

4 19-24 Thickness .20000E+00

5 25-36 Thickness .20000E+00

6 37-48 Thickness .20000E+00

7 49-60 Thickness .20000E+00

8 61-64 Thickness .20000E+00

9 65-70 Thickness .20000E+00

10 71-74 Thickness .20000E+00

11 75-80 Thickness .20000E+00

12 81-84 Thickness .20000E+00

13 85-90 Thickness .20000E+00

14 91-94 Thickness .20000E+00

15 95-100 Thickness .20000E+00

16 101-109 Thickness .20000E+00

17 110-118 Thickness .20000E+00

18 119-130 Thickness .20000E+00

19 131 Radius .80000E-01

A-159


20 132 Radius .80000E-01

21 133 Radius .80000E-01

22 134 Radius .80000E-01

23 135,136 Radius .80000E-01

24 137,138 Radius .80000E-01

25 139,140 Radius .80000E-01

26 141 Radius .80000E-01

27 142 Radius .80000E-01

28 143 Radius .80000E-01

29 144 Radius .80000E-01

30 145,146 Radius .80000E-01

31 147,148 Radius .80000E-01

32 149,150 Radius .80000E-01

Table A.13-7 Global design variable (D.V.) numbering

D.V. Group Number Type of D.V. Number of D.V. Local D.V.

NumberGlobal D.V.

Number

1 thickness 1 1 1

2 thickness 1 1 2

3 thickness 1 1 3

Table A.13-6 Design variable group number and associated user element numbers (Continued)

Design Group Number

User Element Numbers D.V. Type Value of Design

(in.)

A-160

Example Problems

4 thickness 1 1 4

5 thickness 1 1 5

6 thickness 1 1 6

7 thickness 1 1 7

8 thickness 1 1 8

9 thickness 1 1 9

10 thickness 1 1 10

11 thickness 1 1 11

12 thickness 1 1 12

13 thickness 1 1 13

14 thickness 1 1 14

15 thickness 1 1 15

16 thickness 1 1 16

17 thickness 1 1 17

18 thickness 1 1 18

19 radius 1 1 19

20 radius 1 1 20

21 radius 1 1 2 1

22 radius 1 1 22

23 radius 1 1 23

24 radius 1 1 24

25 radius 1 1 25

26 radius 1 1 26



NumberGlobal D.V.

Number

A-161


Figure A.13-4 Displacement sensitivity coefficients for swept wing (Node 43 in Z-direction for load case ID 200)

27 radius 1 1 27

28 radius 1 1 28

29 radius 1 1 29

30 radius 1 1 30

31 radius 1 1 31

32 radius 1 1 32



NumberGlobal D.V.

Number

A-162

Example Problems

A.14 Optimum Design of a Control Arm

Title

Optimum Design of a Control Arm

Problem

The geometry and the finite element model of a control arm are shown intables Table A.14-1-Table A.14-2.

The design objective is to find the thicknesses of the model that minimize volumeof the control arm, while maximum stress in control should be less than 240 MPa.

Element Type

3D Thin Shell (NKTP = 40, NORDR = 1)3D General Shell (NKTP = 20, NORDR = 10)

Section Type


Material Properties

Design Data

Elastic Moduli Ex = 207,000 MPa

Gxy = 79,615 MPa





υ

A-163


Load Data

Concentrated Nodal Forces


Lower Bound on Design Variable = 1.0 mm

Upper Bound on Design Variables = 8.0 mm

Stress Limit (von Mises) = 240 MPa

NodeNumber

Fx(N)

Fy(N)

Fz(N)

Mx(N-mm)

My(N-mm)

Mz(N-mm)

508 -641. 4810. 0. -43750. 4300. 19140.

534 -1361. 0. 0. -43420. 8760. 21230.


A-164

Example Problems

Figure A.14-1 Geometry of control arm

Figure A.14-2 Finite element model of control arm

ANAL=STATICAUTO=ONFILE=armoptRESEQUENCE=ONSAVE=26,27*TITLE Problem TitleOPTIMUM DESIGN OF A CONTROL ARM*ELTYPE Element Type Selections (3-D General Shell and 3-D Thin Shell)1, 40, 1,2, 20, 10,*RCTABLE Real Constant Table5 , 8.2000E+01///////*ELEMENTS Element Connectivities (See Figure A.14.2)*F1 See Figure A.14.2

A-165


*MATERIAL Material Property DataEX , 1,0, 0.207000E+06NUXY, 1,0, 0.300000E+00GXY , 1,0, 0.796154E+05*LDCASE Static Load Case Control Card**KELFR KRCTN KSTR KSTN LQ1 LQ2 LQ7 TSFRE RCFORCE4, 1, 4, 0, 0, 4, 1, 0.000, 0.000,*LCTITLE Load Case TitleCONTROL ARM : 6G VERTICAL*SPDISP Boundary Conditions and Specified Displacements** SPECIFIED DISPLACEMENT SET:1534,UY , 0.000000534,UZ , 0.000000** SPECIFIED DISPLACEMENT SET:2508,UZ , 0.000000536,UXYZ, 0.000000*CFORCE Concentrated Nodal Forces508,FX, -641.00000508,FY, 4810.00000508,MX,-43750.00000508,MY, 4300.00000508,MZ, 19140.00000534,FX, -1361.00000534,MX,-43420.00000534,MY, 8760.00000534,MZ, 21230.00000 1*OPTPAR Control Parameters for Optimization1,1,10,3,40.1,0.01,0.01,0.01,0.005*DVGROUP Design Variable Groups and Section Type1,1,0,1,1*DVGELMT Design Variable Groups and Element Numbers1,1,238,11,243,266,11,271,290,11,299,306,11,315,322,11,331,338,11,343,346,11,351,354,11,359,362,1

A-166

Example Problems

Results

Table A.14-1 shows optimum results obtained by three optimization algorithms(GRG, RQP and OCB). Computational effort required in this problem is given inTable A.14-2 with the total numbers of structural and sensitivity analyses. Theconvergence parameters ACV, ACS and ALS used in this example are 0.01, 0.01,and 0.001, respectively. Parameter ACT is assigned a value of 0.01 for GRG and0.1 for both RQP and OCB. Design iteration histories are given in tables TableA.14-3 - Table A.14-5. It is observed that initial design is infeasible and there is

an incease of 70% in volume. Figure A.14-3 shows graphically the objectivefunction histories.

1,367,374,11,383,438,11,461,464,1*DVLOCAL Local Design Variable Information1,1,01,2.0,1.0,8.0*STRCON Stress Constraint IDs1,VON,240.0*STRELM Element Numbers for Stress Constraint IDs1,12,641,238,1, 243,266,1271,290,1, 299,306,1315,322,1, 331,338,1343,346,1, 351,354,1359,362,1, 367,374,1383,438,1, 461,464,1*ENDDATA Data Deck Terminator

Table A.14-1 Optimum results of control arm

Design Group

Number


Initial Design (mm.)


GRG RQP OCB

1 1-238 .20000E+01 .34246E+01 .34159E+01 .34116E+01

243-266

271-290

A-167


299-306

315-322

331-338

343-346

351-354

359-362

367-374

383-438

461-464

Total Volume (mm3) .54460E+06 .93254E+06 .93016E+06 .92898E+06

Table A.14-2 Comparison of optimization algorithms for control arm

Algorithm NITa


NSAb


NDSc


CPU timed (sec)



tion (sec)

GRG 2 6 1 801.65 400.83

RQP 5 7 5 1184.58 236.92

OCB 4 4 4 824.80 206.20

Table A.14-1 Optimum results of control arm

A-168

Example Problems

Table A.14-3 Design iteration history for control arm with GRG

Iteration Number

Objectivea Function

a. Material volume of the control arm


b. Square norm of projected reduced gradient (ACS)



1 .14063E+07 .74148E+11 .17398E+01

2 .93254E+06 .00000E+00 .17398E+01

Table A.14-4 Design iteration history for control arm with RQP

Iteration Number

Objectivea Function






1 .54460E+06 .77415E+00 .15822E+01

2 .59730E+06 .73353E+00 .11633E+01

3 .79704E+06 .40783E+00 .32272E+00

4 .90810E+06 .81007E-01 .45778E-01

5 .93016E+06 .15538E-02 .82428E-03

A-169


Figure A.14-3 Objective function histories for control arm

Table A.14-5 Design iteration history for control arm with OCB

Iteration Number

Objectivea Function






1 .54460E+06 .77415E+00 .15822E+01

2 .75541E+06 .50130E+00 .45392E+00

3 .89191E+06 .13612E+00 .80636E-01

4 .92898E+06 .00000E+00 .31567E-02

A-170

Example Problems

A.15 Optimum Design of a Pressure Vessel

Title

Optimum Design of a Pressure Vessel

Problem

A pressure vessel with dimensions as shown in Figure A.15-1 is subjected tointernal pressure load of 100 psi. The end caps (elements 1-100, 201-300) have thesame thickness and entire cylindrical section has the same thickness (elements 101-200).

The design objective is to find the thicknesses of the center and end sections of thepressure vessel that minimize volume of the vessel, while maximum stress in tankshould be less than 40 ksi.

Element Type

3D Thin Shell (NKTP = 40, NORDR = 1, 10)

Section Type


Material Properties

Elastic Moduli Ex = 30,000 ksi

Gxy = 11,250 ksi


Specific Weight = 0.29 lb/in3

υ

A-171


Design Data

Number of Design Variable Groups = 2







A-172

Example Problems

Figure A.15-1 Geometry of pressure vessel

A-173


ANALYSIS = STATICAUTO = ONELEM = OFFNODE = OFFRESE = ONMAXW = 750SAVE = 26,27FILE = VESS*TITLE Problem Title4 OPTIMUM DESIGN OF PRESSURE VESSEL*ELTYPE Element Type Selection (3-D Thin Shell)

1, 40, 10,2, 40, 1,3, 40, 10,

*RCTABLE Real Constant Table1, 8.250E+00///////2, 8.250E+00///////3, 8.250E+00///////

*ELEMENTS Element Connectivities (See Figure A.15.1)*NODES Nodal Coordinates (See Figure A.15.1)*MATERIAL Material Property Data** MATERIAL PROPERTY NO: 29EX , 29,0, .300000E+05NUXY, 29,0, .330000E+00GXY , 29,0, .112500E+05DENS, 29,0, .290000E+00*LDCASE Static Load Case Control Card0, 1, 4, 0, 0, 0, 0, .0000E+00, .0000E+00*SPDISP Specified Displacements** CONSTRAINTS FOR SET ID = 100

91,UY , .00000E+00191,UX , .00000E+00191,UY , .00000E+00191,UZ , . 00000E+00201,UX , .00000E+00

A-174

Example Problems

Results


of 66% in volume with GRG is obtained. Figure A.15-2 shows graphically theobjective function histories.

201,UZ , . 00000E+00*L1 PRESSURE LOADS1,0,300,1,1,0,0.1/// 1*OPTPAR Control Parameters for Optimization1,1,10,3,40.1,0.01,0.01,0.01,0.01*DVGROUP Design Variable Groups and Section Type1,2,1,1,1*DVGELMT Design Variable Groups and Element Numbers1,1,100,11,201,300,12,101,200,1*DVLOCAL Local Design Variable Information1,2,11,0.25,0.01,10.0*STRCON Stress Constraint IDs1,VON,40.*STRELM Element Numbers For Stress Constraint IDs1,1,641,300,1*ENDDATA Data Deck Terminator

A-175


Table A.15-1 Optimum results of pressure vessel

Design Group

Number




GRG RQP OCB

1 1-100, 201-300 .25000E+00 .18901E+00 .19057E+00 .19482E+00

2 101-200 .25000E+00 .34086E-01 .37289E-01 .34091E-01


Table A.15-2 Comparison of optimization algorithms for pressure vessel

Algorithm NITa


NSAb


NDSc


CPU timed (sec)


CPU time/ Design

Iteration (sec)

GRG 3 57 2 5326.32 1775.44

RQP 8 16 6 1876.69 234.58

OCB 8 8 7 1262.66 157.83

Table A.15-3 Design iteration history for pressure vessel with GRG

Iteration Number

Objectivea Function



1 .13445E+04 .16139E+08 .16172E+00

2 .69486E+03 .58580E+17 .42705E-01

A-176

Example Problems

3 .45808E+03 .13341E+21 .16682E-03

a. Material volume of the pressure vesselb. Squared norm of projected reduced gradient (ACS)c. Estimated stepsize (ALS)

Table A.15-4 Design iteration history for pressure vessel with RQP

Iteration Number

Objectivea Function

a. Material volume of the pressure vessel





1 .13445E+04 .40173E+04 .00000E+00

2 .80727E+03 .40173E+05 .00000E+00

3 .53864E+03 .10373E+05 .26740E+00

4 .43786E+03 .19836E+08 .25271E+00

5 .41195E+03 .19914E-01 .25289E+00

6 .47320E+03 .33575E-02 .32792E-01

7 .48142E+03 .25372E+03 .36904E-03

8 .47238E+03 .38781E-02 .00000E+00

Table A.15-3 Design iteration history for pressure vessel with GRG

A-177


Table A.15-5 Design iteration history for pressure vessel with OCB

Iteration Number

Objectivea Function

a. Material volume of the pressure vessel





1 .13445E+04 .10000E+01 .00000E+00

2 .67227E+03 .46505E-02 .35313E-01

3 .67227E+03 .78722E+00 .13229E-03

4 .39072E+03 .69608E-02 .35476E+01

5 .41584E+03 .79445E-02 .13324E+01

6 .44452E+03 .66136E-02 .46355E+00

7 .46838E+03 .58273E-02 .10565E+00

8 .46838E+03 .67351E-02 .83765E-02

A-178

Example Problems

Figure A.15-2 Objective function histories for pressure vessel

A-179

Optimum Design of a Simply Supported Beam with a Generalized Thin Walled Section

A.16 Optimum Design of a Simply Supported Beam with a Generalized Thin Walled Section

Title

Optimum Design of a Simply Supported Beam with a Generalized Thin WalledSection

Problem

A simply supported beam with dimensions shown in Figure A.16-1 is subjected toa uniform pressure of 5 N/mm. The geometry (nodes & members) of the beamcross section is as shown in Figure A.16-2.

The design objective is to find component gages that minimize the beam volumewhile satisfying the constraints on 1) maximum stresses due to bending momentsabout Z-axis for all elements, and 2) maximum shear stresses for all elements.

Element Type

3D General Beam (NKTP=12, NORDR=1)

Section Type

Generalized Thin Walled Section (ISECT=22)

Material Properties

Elastic Modulus Ex = 200,000 N/mm2


Mass Density = 7.86E-06 Kg/mm2

υ

A-180

Example Problems

Design Data

Figure A.16-1 Geometry of simply supported beam with a generalized thin-walled section

No. of Design Variable Groups = 2


No. of Elements = 4


Lower Bound on Design Variables = 1.0 mm

Upper Bound on Design Variables = 6.0 mm.

Bending Stress Limit = 200.0 N/mm2

Shear stress Limit = 120.0 N/mm2

A-181


Figure A.16-2 Section geometry (nodes and members) of generalized thin-walled beam section

ANALYSIS=STATICLOAD_CASE_NO=1AUTO_CONSTRAINT=OFFRESEQUENCING_OF_ELEMENT=OFF*TITLE Problem Title0SIMPLY SUPPORTED BEAM SUBJECTED TO A UNIFORM PRESSURE LOAD*ELTYPE Element Type Selection (3-D General Beam)1,12,1*RCTABLE Real Constant Table1, 8

7.04,14.407467,184.951467,0.2858667,0.0,0.0,12.0,6.0*E1 Element Connectivities1,1,2,$,1,1,1,4,1,1*NODES Nodal Coordinates1,0,$,0.0,0.05,0,1,0,1600.0,0.0*MATERIAL Material Property Data

A-182

Example Problems

EX,1,0,20.0E4NUXY,1,0,0.3DENS,1,0,7.86E-6*LDCASE Static Load Case Control Card0, 0, 2, 0, -1, 0, 1, .0000E+00, .0000E+00*SPDISP Specified Displacements1,UX,0.0,$,UY,UZ,ROTX,ROTY5,UY,0.0,$,UZ,ROTX,ROTY*L1 Pressure Data1,0,4,1,1,0,5.0,5.0*OPTPAR Control Parameters for Design Optimization1,-1,30,3,2,20.1,0.05,0.05*DVGROUP Design Variable Groups and Section Type1,2,1,22,11*DVGELMT Design Variable Groups and Element Numbers1,0,21,42,0,22,3*DVLOCAL Local Design Variable Information1,2,11,3.0,1.0,6.0,1,3.0,1.0,6.01,3.0,1.0,6.0,1,3.0,1.0,6.00,3.0,3.0,3.0,1,3.0,1.0,6.01,3.0,1.0,6.0,1,3.0,1.0,6.01,3.0,1.0,6.0,0,3.0,3.0,3.01,3.0,1.0,6.012,0,0,1,01 ,.5000E+03,.5000E+03,2 ,.5299E+03,.4974E+033 ,.6606E+03,.4859E+03,4 ,.6711E+03,.4735E+035 ,.7011E+03,.4735E+03,6 ,.7011E+03,.4705E+037 ,.6711E+03,.4705E+03,8 ,.6711E+03,.4635E+039 ,.5311E+03,.4635E+03,10,.5299E+03,.4944E+0311,.5000E+03,.4970E+03,12,.6606E+03,.4859E+031,1,2,0,0,0,02,2,12,0,1,0,03,3,4,0,1,0,04,4,5,0,0,0,05,4,7,0,1,0,06,6,7,0,0,0,0

A-183


Results


of 33% in volume is achieved. Figure A.16-3 shows graphically the objectivefunction histories.

7,7,8,0,1,0,08,8,9,0,1,0,09,9,10,0,1,0,010,10,2,0,1,0,011,10,11,0,0,0,0*DVLINK Local Design Variable Linking1,1,3A1,A2,A1,A3,A1,A41,1,4A6,A7,A6,A8,A6,A9,A6,A112,2,3B1,B2,B1,B3,B1,B42,2,4B6,B7,B6,B8,B6,B9,B6,B11*STRCON Stress Constraint IDs1,BEND3,200.0,SHEAR,120.0*STRELM Element Numbers For Stress Constraint IDs1,1,911,4,1*ENDDATA Data Deck Terminator

A-184

Example Problems

Table A.16-1 Optimum results for simply supported beam with generalized thin walled section

Design Group

Number


Initial Design (mm)


GRG RQP OCB

1 1, 4

.30000E+01 .10000E+01 .10000E+01 .10000E+01

.30000E+01 .10000E+01 .10000E+01 .10000E+01

.30000E+01 .10000E+01 .10000E+01 .10000E+01

.30000E+01 .10000E+01 .10000E+01 .10000E+01

.30000E+01 .30000E+01 .30000E+01 .30000E+01

.30000E+01 .23539E+01 .23530E+01 .23531E+01

.30000E+01 .23539E+01 .23530E+01 .23531E+01

.30000E+01 .23539E+01 .23530E+01 .23531E+01

.30000E+01 .23539E+01 .23530E+01 .23531E+01

.30000E+01 .30000E+01 .30000E+01 .30000E+01

.30000E+01 .23539E+01 .23530E+01 .23531E+01

A-185


2 2, 3

.30000E+01 .10333E+01 .10378E+01 .10503E+01

.30000E+01 .10333E+01 .10378E+01 .10503E+01

.30000E+01 .10333E+01 .10378E+01 .10503E+01

.30000E+01 .10333E+01 .10378E+01 .10503E+01

.30000E+01 .30000E+01 .30000E+01 .30000E+01

.30000E+01 .33205E+01 .33030E+01 .32947E+01

.30000E+01 .33205E+01 .33030E+01 .32947E+01

.30000E+01 .33205E+01 .33030E+01 .32947E+01

.30000E+01 .33205E+01 .33030E+01 .32947E+01

.30000E+01 .30000E+01 .30000E+01 .30000E+01

.30000E+01 .33205E+01 .33030E+01 .32947E+01

Total Volume (mm3) .21667E+07 .14463E+07 .14436E+07 .14441E+07

Table A.16-2 Comparison of optimization algorithms for simply supported beam

Algorithm NITa


NSAb NDSc CPU timed (sec)


GRG 6 45 5 170.19 28.36

RQP 13 56 13 208.50 16.04

OCB 27 27 27 173.97 6.44

Table A.16-1 Optimum results for simply supported beam with generalized thin walled section

Design Group

Number


Initial Design (mm)


GRG RQP OCB

A-186

Example Problems

b. Total number of structural analysesc. Total number of sensitivity analysesd. ELXSI 6400 system

Table A.16-3 Design iteration history for simply supported beam with GRG

Iteration Number

Objectivea Function






1 .21667E+07 .12756E+12 .50000E+00

2 .19882E+07 .82413E+11 .10625E+01

3 .16899E+07 .34549E+11 .10625E+01

4 .15157E+07 .19666E+11 .10625E+01

5 .14653E+07 .33057E+12 .13281E+00

6 .14463E+07 .32185E+12 .00000E+00


Iteration Number

Objectivea Function



1 .21667E+07 .28586E+06 .00000E+00

2 .13082E+07 .23330E+01 .99831E+00

3 .12631E+07 .13558E+01 .75616E+00

4 .11799E+07 .76204E+00 .31878E+00

5 .13965E+07 .57208E+00 .65313E-01

A-187


6 .14263E+07 .57041E+00 .28089E-01

7 .14176E+07 .10959E+00 .33860E-01

8 .14440E+07 .68591E-01 .11804E-02

9 .14437E+07 .51785E-01 .91175E-03

10 .14436E+07 .48604E-01 .85687E-03

11 .14436E+07 .48230E-01 .85036E-03

12 .14436E+07 .48045E-01 .84712E-03

13 .14436E+07 .47953E-01 .86576E-03

a. Material volume of the beamb. Norm of Lagrangian gradient (ACS)c. Maximum constraint violation (ACV)

Table A.16-5 Design iteration history for simply supported beam with OCB

Iteration Number

Objectivea Function



1 .21667E+07 .86807E+00 .00000E+00

2 .17258E+07 .22482E+00 .64272E-01

3 .17258E+07 .52745E+00 .44257E-02

4 .14755E+07 .36029E+00 .75175E-01

5 .14755E+07 .77576E-01 .10960E-01

6 .14755E+07 .29669E+00 .44394E-03

7 .15064E+07 .48949E+00 .63624E-02

8 .12569E+07 .23042E+01 .20855E+00

9 .15251E+07 .32738E+00 .22717E-01


A-188

Example Problems

10 .14755E+07 .40026E+00 .63884E-03

11 .13090E+07 .54774E+00 .25722E+00

12 .14396E+07 .42691E+00 .44288E-01

13 .14396E+07 .22577E+00 .11307E-01

14 .14465E+07 .45241E-01 .14163E-01

15 .14465E+07 .39824E+00 .24317E-03

16 .13778E+07 .13000E+01 .10770E+00

17 .15144E+07 .40617E+00 .00000E+00

18 .13778E+07 .27414E+00 .99303E-01

19 .14459E+07 .14456E+00 .86150E-02

20 .14457E+07 .10170E+00 .13253E-02

21 .15505E+07 .46892E+00 .73408E-12

22 .14118E+07 .12904E+00 .42812E-01

23 .14441E+07 .91898E-01 .19051E-02

24 .15520E+07 .47748E+00 .58535E-11

25 .14279E+07 .87165E-01 .27930E-01

26 .14279E+07 .99724E-01 .23490E-01

27 .14441E+07 .00000E+00 .76844E-03

a. Material volume of the beamb. Norm of search direction vector (ACS)c. Maximum constraint violation (ACV)

Table A.16-5 Design iteration history for simply supported beam with OCB

Iteration Number

Objectivea Function



A-189


Figure A.16-3 Objective function histories for simply supported beam with general-ized thin walled section

A-190

Example Problems

A.17 Optimum Design of a Two-Bay, Two-Story Frame

Title

Optimum Design of a Two-Bay, Two-Story Frame

Problem

The geometry and the finite element model of a two-bay, two-story frame is shownin Figure A.17-1. The design objective is to find thicknesses of generalized thinwalled sections that give the minimum volume while satisfying constraints onstresses and member thicknesses.

Element Type

3-D Beam Element (NKTP = 12, NORDR = 4)

Section Type

Generalized Thin Walled section with Uniform Wall Thickness (ISECT = 23)

Material Properties

Elastic Modulus = 29000 Ksi


A-191


Design Data

Load Data

Concentrated nodal forces at various node points are shown in Figure A.17-2.





Upper Bound on Design Variable = 2.0 in.


Stress Limit (von Mises at Critical points on the Cross Sections) = 27.0 Ksi

A-192

Example Problems

Figure A.17-1 Geometry and FEM of the Two-Story frame

PROB=EXAM17ANALYSIS=STATICAUTO_CONSTRAINT=OFFRESEQUENCING_OF_ELEMENT=OFF** STROPT Example Problem #17** TOTAL NO.OF ELEMENTS=18** TOTAL NO.OF NODES =17******************************************************************************************** ANALYSIS INPUT DATA STARTS HERE **************************************************************************************************TITLE Problem Title0Optimum design of a Two-Bay, Two Story Frame*ELTYPE Element Type Selection (3-D General Beam Element)1,12,1*RCTABLE Real Constant Table1,865.717424,1218.1539,9650.2219,1.,0.,0.,36.,10.*E1 Element Connectivities1,1,4$1,1,12,4,6$1,1,13,6,9$1,1,14,9,11$1,1,15,11,13$1,1,16,13,15$1,1,17,1,2$1,1,18,6,7$1,1,19,15,16$1,1,110,2,5$1,1,111,5,7$1,1,112,7,10$1,1,113,10,12$1,1,114,12,14$1,1,115,14,16$1,1,116,2,3$1,1,1

A-193


17,7,8$1,1,118,16,17$1,1,1*NODES Nodal Coordinates1,0//0.,300.,0.2,0//0.,180.,0.3,0//0.//4,0//240.,300.,0.5,0//240.,180.,0.6,0//480.,300.,0.7,0//480.,180.,0.8,0//480.,0./9,0//720.,300.,0.10,0//720.,180.,0.11,0//960.,300.,0.12,0//960.,180.,0.13,0//1200.,300.,0.14,0//1200.,180.,0.15,0//1440.,300.,0.16,0//1440.,180.,0.17,0//1440.,0./*MATERIAL Material Property DataEX,1,0,29000.NUXY,1,0,.3*LDCASE Static Load Case Control Card0, 1, 2, 0, 0, 0, 0, .0000E+00, .0000E+00*SPDISP Specified Displacements1,UZ,0.,18,1,ROTX,ROTY3,UX,0.,8,5,UY17,UX,0. $ UY*CFORCE Concentrated Nodal Forces1,FY,-10.,15,141,FX,3.572,FY,-20.,6,22,FX,8.929,FY,-20.,13,216,FY,-20.5,FY,-40.,7,210,FY,-40.,14,2***************************************************************************

A-194

Example Problems

Results

Table A.17-1 shows optimum results obtained by three optimization algorithms(GRG, RQP and OCB). Computational effort required in this problem is given inTable A.17-2 with the total numbers of structural and sensitivity analyses.

The convergence parameters ACT, ACV, ACS and ALS used in this example are0.1, 0.01, 0.01 and 0.01, respectively. GRG and OCB results are with thesensitivity method 2 and the RQP results with the method 4. Design iterationhistories are given in tables Table A.17-3-Table A.17-5. It is observed that theinitial design is highly infeasible and an immense increase in volume is required to

***************** OPTIMIZATION INPUT DATA STARTS HERE ******************OPTPAR Control Parameters for Optimization1,1,50,3,20.1,0.01,0.01,0.01,0.01*DVGROUP Design Variable Groups and Section Type1,18,1,23,1*DVGELMT Design Variable Groups and Element Numbers1,1,1,1,18,1,1*DVLOCAL Local Design Variable Information1,18,11,0.01,0.01,2.069.08,4175.76,4175.76,8351.52,0.0,380.63,11.,0.0*STRCON Stress Constraint IDS17897,VON,27.0*STRELM Element Numbers for Stress Constraint IDS17897,1,931,18,1***************** OPTIMIZATION INPUT DATA ENDS HERE ****************************************************************************************ENDDATA Data Deck Terminator

A-195


achieve optimum design. Figure A.17-2 shows graphically the objective functionhistories.

Table A.17-1 Optimum results of Two-Bay, Two-Story frame (Thicknesses given in inch)

Design Group

Number




GRG RQP OCB

1 1 0.01 .13489E+00 .13231E+00 .13222E+00

2 2 0.01 .13303E+00 .13247E+00 .13382E+00

3 3 0.01 .57538E+00 .56476E+00 .56568E+00

4 4 0.01 .43814E+00 .40869E+00 .40851E+00

5 5 0.01 .43226E+00 .40866E+00 .40847E+00

6 6 0.01 .50034E+00 .49282E+00 .49227E+00

7 7 0.01 .10760E+00 .94598E-01 .94401E-01

8 8 0.01 .96994E+00 .90390E+00 .90190E+00

9 9 0.01 .12870E+01 .11047E+01 .11032E+01

10 10 0.01 .20279E+00 .18629E+00 .18463E+00

11 11 0.01 .48357E+00 .47371E+00 .48311E+00

12 12 0.01 .15075E+01 .14783E+01 .14804E+01

13 13 0.01 .40549E+00 .44853E+00 .44863E+00

14 14 0.01 .39180E+00 .44845E+00 .44860E+00

15 15 0.01 .13948E+01 .13735E+01 .13718E+01

16 16 0.01 .10290E+00 .27154E-01 .27081E-01

17 17 0.01 .19273E+00 .11202E+00 .11191E+00

18 18 0.01 .83960E-01 .16552E+00 .16524E+00


A-196

Example Problems

Table A.17-2 Comparison of optimization algorithms for Two-bay, Two-Story frame

Algorithm NITa


NSAb


NDSc


CPU timed (sec)


CPU time/ Design

Iteration (sec)

GRG 12 77 12 190.56 15.88

RQP 11 11 11 62.62 5.69

OCB 31 31 31 167.40 5.40

Table A.17-3 Design history for the Two-Bay, Two-Story frame with GRG

Iteration Number

Objectivea Function



1 .31155E+06 .31983E+10 .22481E+01

2 .18611E+06 .20853E+10 .57521E+00

3 .16119E+06 .11886E+10 .25676E+00

4 .15310E+06 .89057E+09 .12838E+00

5 .14933E+06 .56510E+09 .50615E-01

6 .14818E+06 .21884E+10 .40492E+00

7 .14574E+06 .21759E+12 .40492E+00

8 .14263E+06 .36947E+09 .80983E+00

A-197


9 .13656E+06 .49879E+10 .15000E+00

10 .13552E+06 .62683E+09 .13386E+00

11 .13381E+06 .42788E+08 .26771E+00

12 .13295E+06 .20879E+08 .00000E+00

a. Material volume of the two-bay, two-story frameb. Squared norm of projected reduced gradient (ACS)c. Estimated stepsize (ALS)

Table A.17-4 Design iteration history for the Two-Bay, Two-Story with RQP

Iteration Number

Objectivea Function



1 .26112E+04 .40566E-01 .11831E+03

2 .51041E+04 .77075E-01 .58770E+02

3 .97969E+04 .14302E+00 .29140E+02

4 .18262E+05 .24695E+00 .14329E+02

5 .32506E+05 .40148E+00 .69281E+01

6 .53727E+05 .57697E+00 .32383E+01

7 .80445E+05 .65672E+00 .14097E+01

8 .10643E+06 .54746E+00 .52560E+00

9 .12309E+06 .24935E+00 .13802E+00

10 .12904E+06 .33462E-01 .15519E-01

Table A.17-3 Design history for the Two-Bay, Two-Story frame with GRG

Iteration Number

Objectivea Function



A-198

Example Problems

11 .12979E+06 .50911E-03 .24088E-03

a. Material volume of the two-bay, two-story frameb. Norm of Lagrangian gradient (ACS)c. Maximum constraint violation (ACV)

Table A.17-5 Design iteration history for the Two-Bay, Two-Story with OCB

Iteration Number

Objectivea Function



1 .26112E+04 .40520E-01 .11831E+03

2 .39168E+04 .59799E-01 .76909E+02

3 .58753E+04 .87728E-01 .50064E+02

4 .88129E+04 .12798E+00 .32230E+02

5 .13219E+05 .18502E+00 .20430E+02

6 .19829E+05 .26485E+00 .12636E+02

7 .29743E+05 .37159E+00 .75305E+01

8 .44615E+05 .49905E+00 .42303E+01

9 .66923E+05 .60861E+00 .21142E+01

10 .97039E+05 .57790E+00 .85674E+00

11 .12220E+06 .34104E+00 .27014E+00

12 .13482E+06 .81428E-01 .46601E-01

Table A.17-4 Design iteration history for the Two-Bay, Two-Story with RQP

Iteration Number

Objectivea Function



A-199


13 .13745E+06 .25638E+00 .19097E-02

14 .13248E+06 .10074E+00 .19719E+01

15 .13248E+06 .21899E-01 .78380E+00

16 .13248E+06 .89207E-02 .25517E+00

17 .13248E+06 .25531E-02 .48453E-01

18 .13248E+06 .15841E+00 .24562E-02

19 .12795E+06 .25282E+00 .25569E+01

20 .12795E+06 .30640E+00 .23081E+00

21 .12846E+06 .17657E+00 .32647E+01

22 .13598E+06 .24068E+00 .99426E+00

23 .12846E+06 .54349E-01 .76391E+00

24 .13071E+06 .28610E-01 .22897E+00

25 .13071E+06 .52881E-02 .35004E-01

26 .13071E+06 .13291E+00 .10788E-02

27 .12992E+06 .13242E-01 .16538E+01

28 .12992E+06 .73639E-02 .63614E+00

29 .12992E+06 .42429E-02 .17834E+00

30 .12992E+06 .77355E-03 .23519E-01

31 .12992E+06 .10538E-01 .53090E-03

a. Material volume of the two-bay, two-story frame

Table A.17-5 Design iteration history for the Two-Bay, Two-Story with OCB (Continued)

Iteration Number

Objectivea Function



A-200

Example Problems

Figure A.17-2 Objective function histories for Two-Bay, Two-Story frame

b. Norm of search direction vector (ACS)c. Maximum constraint violation (ACV)

A-201

Dynamic Response Optimization of a Simply Supported Beam

A.18 Dynamic Response Optimization of a Simply Supported Beam

Title

Dynamic Response Optimization of a Simply Supported Beam.

Problem

A 50 inch long simply supported beam shown in Figure A.18-1 is subjected to asinusoidal concentrated load of unit amplitude and zero phase lag at its mid span.The amplitudes of displacements are to be constrained for an exciting frequency of10 Hz with zero damping assumption. Due to symmetry, only half the beam ismodeled with 8-noded plane stress elements as shown in Figure A.18-2.

The design objective is to find thickness of the beam that minimizes the beamvolume while satisfying the constraints on dynamic amplitude at node numbers 53and 63 in global Y-direction.

Element Type

Plane Stress Element (NKTP = 1, NORDR = 2)

Section Type


Material Properties

DESIGN DATA

Elastic Modulus = 7.5 x 107 psi


Mass Density (DENS) = 1.0 lb.sec2/in.4


A-202

Example Problems

Figure A.18-1 Geometry of a simply supported beam under harmonic load


No. of Freq. Resp. Constraints = 2

Upper Bound on Design Variable = 10.0 in.


Amplitude Limit = 0.0001 in.

A-203


Figure A.18-2 Finite element model of a simply supported beam under harmonic load

PROB=EXAM18ANALYSIS=EIGENVALUEEIGEN EXTRACTION=SUBSPACE, CONVENTIONALMASS FORMULATION=CONSISTENTRESEQUENCING OF ELEMENT=ON,LISTFILE=FR01SAVE=26,27FLOWER=0.50FUPPER=100.0INPHASE=LINEARINPOLATION=LINEARGENFREQUENCY=OFFMRES=OFF** STROPT EXAMPLE PROBLEM #18 **** TOTAL NO. OF ELEMENTS=20 **** TOTAL NO. OF NODES=105**************************************************************************************** ANALYSIS INPUT DATA STARTS HERE *********************************************************************************************TITLE Problem TitleFREQUENCY RESPONSE OPTIMIZATION OF A SIMPLY SUPPORTED BEAM*ELTYPE Element type Selection (2-D Plane Stress Element)1, 1, 2*RCTABLE Real Constant Table1, 8,1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0*ELEMENT Element Connectivities$ -2,42,10,101,1,1,1,0,10,2,1,11,2,3,24,45,44,43,22*NODES Nodal Coordinates-21,0,21,5,0.0,0.25,0.01,0, 0,0, 0.0,0.00,0.0

A-204

Example Problems

21,0, 1,0,25.0,0.00,0.0*MATERIAL Material Property DataEX, 1, 0, 0.75000+08NUXY, 1, 0, 0.0000E+00DENS, 1, 0, 0.1000E+01*EIGCNTL Eigenvalue Analysis Control Card3,-1,10,0,0.0,0.0,1.0E-04*MODESELECTION Mode Selection Data** The first three modes are selected.1,3,1,1001*ADDFREQ Additional Exciting Frequency** User specified exciting frequency10.0*DAMPING Dampling Value1001,0.05*SPECTRUM Spectrum Definition Data** Two flat spectra with ID nos 1001, and 1002 are defined.AMPLITUDE SPECTRUM1001,0,3,1.0,1.0PHASE SPECTRUM1002,0,3,1.0,0.0***DCFORCE Concentrated Nodal Forces Data** A harmonic force defined by the amplitude spectrum ID 1001 and** phase spectrum ID 1002 of magnitude 1.0 is applied in the Y direction ** at node 63.63,FY,1001,1002,$,1.0***RSET Response Set Data** Nodal response set ID 1001 for Y displacement components.1001, 53, UY, 63,10***SPOUT Output Spectra Request Data** The amplitude and phase of the responses specified in ID 1001.1001, 0,1,1001*SPDISP Specified Displacement43,UX,0.0$UY21,UX,0.0,105,21**************************************************************************************** OPTIMIZATION INPUT DATA STARTS HERE *****************

A-205


Results

Table A.18-1 shows optimum results obtained by three optimization algorithms(GRG, RQP and OCB). Computational effort required in this problem is given inTable A.18-2 with the total numbers of structural and sensitivity analyses.Structural response quantities (amplitudes) at initial and optimum designs aregiven in Table A.18-3. The convergence parameters ACV, ACS and ALS used inthis example are 0.01, 0.01 and 0.005, respectively. Parameter ACT is assigned avalue of 0.1 for RQP, OCB, and GRG. Design iteration histories are given in tablesTable A.18-4-Table A.18-6. It is observed that the initial design is feasible and areduction of 67.5% in volume is achieved at optimum design. Figure A.18-3 showsgraphically the objective function histories.

************************************************************************OPTPAR Control Parameters for Optimization1,-1,10,3,40.1,0.01,0.01,0.01*DVGROUP Design Variable Groups and Section Type1,1,1,1,1*DVGELMT Design Variable Groups and Element Numbers1,1,20,1*DVLOCAL Local Design Variable Information1,11,1.0,0.1,10.0*FRSCON Frequency Response Constraints (Amplitude Limit) IDS1001,UY,0.0001*FRSNOD Node Numbers for Amplitude Constraints IDS100153,63,10***************** OPTIMIZATION INPUT DATA ENDS HERE *****************************************************************************************ENDDATA

A-206

Example Problems

Table A.18-1 Optimum results for the simply supported beam (Thicknesses Given in inch)

Design Group

Number




GRG RQP OCB

1 1 - 20 .10000E+01 .3002E+00 .3250E+00 .3014E+00


Table A.18-2 Comparison of optimization algorithms for the simply sup-ported beam design

Algorithm NITa


NSAb


NDSc


CPU timed (sec)


CPU time/ Design

Iteration (sec)

GRG 2 8 1 92.87 46.44

RQP 3 5 1 65.43 21.81

OCB 5 5 3 78.22 15.64

Table A.18-3 Structural response quantities (amplitudes) for the simply sup-ported beam

Max Amplitude at Node Number Initial Design (in.)


GRG RQP OCB

A-207


53 3.005E-05 1.0011E-04 9.247E-05 9.9729E-05

63 3.400E-06 1.1324E-05 1.046E-05 1.1281E-05

Table A.18-4 Design iteration history for the beam with GRG

Iteration Number

Objectivea Function

a. Material volume of one half of the beam





1 .25000E+02 .62500E+03 .69979E+00

2 .75052E+01 .00000E+00 .69979E+00

Table A.18-5 Design iteration history for the beam with RQP

Iteration Number

Objectivea Function






1 .25000E+02 .25000E+02 .00000E+00

2 .13750E+02 .25000E+02 .00000E+00

3 .81250E+01 .59674E-03 .00000E+00

Table A.18-3 Structural response quantities (amplitudes) for the simply sup-ported beam

A-208

Example Problems

Table A.18-6 Design iteration history for the beam with OCB

Iteration Number

Objectivea Function






1 .25000E+02 .10000E+01 .00000E+00

2 .12500E+02 .10000E+01 .00000E+00

3 .68750E+01 .52701E-01 .92854E-01

4 .81925E+01 .66814E-01 .00000E+00

5 .75338E+01 .18452E-02 .00000E+00

A-209


Figure A.18-3 Objective function histories for simply supported beam under speci-fied harmonic load

A-210

Example Problems

A.19 Dynamic Response Optimization of a Simply Supported Beam Under a Sinusoidal Pressure Load

Title

Dynamic Response Optimization of a Simply Supported Beam Under a SinusoidalPressure Load

Problem

A 50 inch long simply supported beam shown in Figure A.19-1 is subjected to asinusoidal pressure load of unit amplitude and zero phase lag. The amplitudes ofdisplacements are to be constrained for exciting frequency of 10 Hz. Modal viscousdamping of 5 percent is assumed. Due to symmetry, only half the beam is modeledwith solid rectangular beam elements.

The design objective is to find width and height of the beam that minimizes thebeam volume while satisfying the constraints on dynamic amplitude for entireexciting frequency range at node numbers 6 and 11 in global Y-direction and onlowest natural frequency.

Element Type

General Beam Element (NKTP = 12, NORDR = 1)

Section Type

Solid Rectangular Cross-Section (ISECT = 5)

Material Properties

Elastic Modulus = 7.5 x 107 psi


Mass Density = 1.0 lb.sec2/in.4

A-211

Dynamic Response Optimization of a Simply Supported Beam Under a Sinusoidal Pressure Load

Design Data

Figure A.19-1 Simply supported beam subjected to sinusoidal pressure load



No. of Freq. Resp. Constraints = 2

No. of Natural Freq. Constraints = 1



Amplitude Limit = 0.1 in.

Natural Frequency Limit = 3.0 Hz

A-212

Example Problems

PROB=EXAM19ANALYSIS=EIGENVALUEEIGEN EXTRACTION=SUBSPACE, CONVENTIONALMASS FORMULATION=CONSISTENTRESEQUENCING OF ELEMENT=OFFFILE=FR03SAVE=26,27DAMPING=VISCOUSFILE=FR03FLOWER=0.5FUPPER=10.E2INPHASE=LINEARINPOLATION=LINEARGENFREQUENCY=ONMRESPONSE=OFF** STROPT EXAMPLE PROBLEM #19 **** TOTAL NO. OF ELEMENTS=10 **** TOTAL NO. OF NODES=11 **** TOTAL NO. OF D. V.=10 ********************************************************************************************** ANALYSIS INPUT DATA STARTS HERE *************************************************************************************************TITLEFREQUENCY RESPONSE OPTIMIZATION OF A SIMPLY SUPPORTED BEAM*ELTYPE Element type Selection (3-D General Shell)1,12,1*RCTABLE Real Constant Table1,8,1.0,.08333333,.08333333,.16, 0.0, 0.0, 1.0, 1.0*ELEMENT Element Connectivities1,1,1,1,0,10,1,1,11,2*NODES Nodal Coordinates1,0,0,0, 0.0,0.0,0.011,0,1,0,25.0,0.0,0.0*MATERIAL Material Property Data

A-213


EX,1,0,0.7500E+08NUXY,1,0,0.0000E+00DENS,1,0,0.1000E+01*EIGCNTL Eigenvalue Analysis Control Card3,0,20,0,0.0,0.0,1.0E-05*MODESELECTION Mode Selection Data** The first three modes are employed with damping ID 10011,3,1,1001*DAMPING Damping Value** Damping ID 1001 has percentage damping ratio of 0.05 specified.1001,0.05*ADDFREQ Additional Exciting Frequency** User specified exciting frequency10.0*SPECTRUM Spectrum Definition DataAMPLITUDE SPECTRUM1001,0,3,1.0,1.0PHASE SPECTRUM1002,0,3,1.0,0.0*DPRESSURE Pressure Loading Data** A harmonic pressure loading with amplitude spectrum 1001 and** phase spectrum 1002 is applied in the Y direction.1,1001,1002,1,0,10,1,1.0*RSET Response Set Data1001, 6,UY,11,5*SPOUT Output Spectra Request Data1001, 0,1,1001*SPDISP Specified Displacement1, UX,0.0,11,1, UZ,ROTX,ROTY1, UY,0.011,ROTZ,0.0******************************************************************************************** OPTIMIZATION INPUT DATA STARTS HERE *********************************************************************************************OPTPAR Control Parameters for Optimization1,1,20,1,40.1,0.01,0.1,0.01*DVGROUP Design Variable Groups and Section Type

A-214

Example Problems

Results

Table A.19-1 shows optimum results obtained by three optimization algorithms(GRG, RQP and OCB). Computational effort required in this problem is given inTable A.19-2 with the total numbers of structural and sensitivity analyses.Structural response quantities (amplitude) at initial and optimum designs are givenin Table A.19-3. Natural frequencies at initial and optimum design are 1.57 Hz and2.994 Hz respectively. The convergence parameters ACV, ACS and ALS used inthis example are 0.01, 0.1 and 0.005, respectively. Parameter ACT is assigned avalue of 0.1 for RQP, OCB, and 0.01. Design iteration histories are given in tablesTable A.19-4 - Table A.19-6. It is observed that the initial design violates all theconstraints but reduction of 109.40% in volume is achieved at optimum design.Figure A.19-2 shows graphically the objective function histories.

1,1,0,5,2*DVGELMT Design Variable Groups and Element Numbers1,1,10,1*DVLOCAL Local Design Variable Information11,1.0,0.3,3.0,1,1.0,0.3,3.0*FRQCON Natural frequency Constraints IDS1,1,3.0*FRSCON Frequency Response Constraints (Amplitude Limit) IDS1001,UY,0.1*FRSNOD Node Numbers for Amplitude Constraints IDS10016,11,5***************** OPTIMIZATION INPUT DATA ENDS HERE *******************************************************************************************ENDDATA

A-215


Table A.19-1 Optimum results of the simply supported beam (Thickness given in inch)

Design Group

Number




GRG RQP OCB

1 1 - 10 1.000E+01 3.000E-01 3.000E-01 3.000E-01

1.000E+01 1.918E+01 1.911E+00 1.911E+00

Total Volume (in3) 5.000E+02 2.877E+02 2.866E+02 2.866E+02

Table A.19-2 Comparison of optimization algorithms for beam design

Algorithm NITa


NSAb


NDSc


CPU timed (sec)



tion (sec)

GRG 3 9 2 114.07 51.28

RQP 6 6 6 99.93 22.26

OCB 7 7 7 112.507 23.30

A-216

Example Problems

Table A.19-3 Structural response quantities (amplitudes) of the beam

Max Amplitude at Node Number



GRG RQP OCB

6 9.254E-02 4.372E-02 4.420E-02 4.421E-02

11 1.309E-01 6.182E-02 6.250E-02 6.252E-02

Table A.19-4 Design iteration history for the beam with GRG

Iteration Number

Objectivea Function






1 .22500E+03 .11250E+05 .15301E+01

2 .91971E+02 .22993E+04 .20000E+01

3 .14385E+02 .00000E+00 .20000E+01


Iteration Number

Objectivea Function



1 .25000E+02 .22164E+02 .26488E+01

2 .80601E+01 .37298E+00 .21595E+01

3 .10857E+02 .31313E+00 .74169E+00

A-217


4 .13206E+02 .13499E+00 .17769E+00

5 .14218E+02 .15276E-01 .16112E-01

6 .14333E+02 .47521E-04 .00000E+00

a. Material volume of one half of the beamb. Norm of Lagrangian gradient (ACS)c. Maximum constraint violation (ACV)

Table A.19-6 Design iteration history for the beam with OCB

Iteration Number

Objectivea Function






1 .25000E+02 .36860E+00 .26488E+01

2 .34215E+02 .33838E+00 .94858E+00

3 .42675E+02 .17513E+00 .25303E+00

4 .47053E+02 .28669E-01 .30900E-01

5 .47770E+02 .88600E+00 .23348E-03

6 .23883E+02 .96744E+00 .99715E-04

7 .14332E+02 .00000E+00 .76175E-04


Iteration Number

Objectivea Function



A-218

Example Problems

Figure A.19-2 Objective function histories for the simply supported beam under sinusoidal pressure load

A-219

Optimum Design of a Passenger-Compartment-Structure

A.20 Optimum Design of a Passenger-Compartment-Structure

Title


Problem

A two dimensional baseline FEM model claimed* to simulate response of acomplete passenger compartment under beaming type of loading is shown inFigure A.20-1. Generalized thin walled beam sections used for different regions ofthe model are shown in Figure A.20-3 .

The design objective is to find the optimal thicknesses of the beam sections thatminimize the material volume of the model while satisfying constraints ondisplacements and stresses under the action of frontal (L1 = 5000N) and beaming(L2 = 3500N) loads. The problem is solved by employing ISECT = 22 first andthen 23 for all the generalized beam elements.

Element Type


Section Type

Generalized Thin-Walled Section (ISECT=22,23)Tubular Rectangular Section (ISECT=6)

Material Properties

Design Data

Elastic Modulus = 2.0E+007 N/CM2Poisson’s Ratio = 0.3Mass Density = 0.00786 N.S2/CM4

No. of Design Variables = 6No. of Load Cases = 1

A-220

Example Problems

* Chang, D.C., “Effects of Flexible Connections on Body Structural Response”, SAE Transactions, Val. 83, Paper No. 740071, pp. 233-244, 1974

Figure A.20-1 Baseline FEM of the passenger-compartment-structure

No. of Elements = 39No. of Displacement Constraints = 25No. of Stress Constraints = 46Upper Bound on Design Variables = 1.0 CMLower Bound on Design Variables = 0.1 CMDisplacement Limit in X-direction = 0.5 CMDisplacement Limit in Y-direction = 0.6 CMOctahedral Shear Stress Limit = 125 KN/CM2

A-221


Figure A.20.1a Beam section models with ISECT = 22

A-222

Example Problems

Figure A.20.1b Beam section models with ISECT = 22

A-223


Complete Input Data for ISECT = 22**FRAME DESIGN WITH GEN. BEAM ELEMENTSANALYSIS=STATICLOAD CASE NO= 1AUTO CONSTRAINT=OFFRESEQUENCING OF ELEMENT=OFF*TITLE0 TYPICAL SEDAN STRUCTURE*ELTYPE ELEMENT TYPES1,12,1*RCTABLE REAL CONSTANT TABLE1, 8.1775,.00251,.00251,.00501,0.,0.,.4754,.4754*E1 ELEMENT CONNECTIVITIES1, 1, 2, $,1,1,12, 2, 3, $,1,1,13, 1, 4, $,1,1,14, 4, 5, $,1,1,1,26,1,1,030,30,37, $,1,1,131,26,31, $,1,1,132,31,32, $,1,1,1,5,1,1,037,36,12, $,1,1,138,37, 4, $,1,1,139, 3, 7, $,1,1,1*NODES NODAL COORDINATES (ALL COORDINATES IN CM.)1,0,0,0, 10.0, 10.0, 0.2,0,0,0, 10.0, 58.0, 0.3,0,0,0, 10.0, 74.0, 0.4,0,0,0, 38.0, 10.0, 0.5,0,0,0, 38.0, 38.0, 0.6,0,0,0, 38.0, 58.0, 0.7,0,0,0, 38.0, 74.0, 0.8,0,0,0, 56.0, 89.0, 0.9,0,0,0, 71.0,102.0, 0.10,0,0,0, 81.0,117.0, 0.11,0,0,0,114.0,117.0, 0.12,0,0,0,145.0,117.0, 0.13,0,0,0,175.0,117.0, 0.14,0,0,0,208.0,114.0, 0.

A-224

Example Problems

15,0,0,0,221.0,102.0, 0.16,0,0,0,243.0, 81.0, 0.17,0,0,0,229.0, 71.0, 0.18,0,0,0,220.0, 58.0, 0.19,0,0,0,217.0, 46.0, 0.20,0,0,0,215.0, 31.0, 0.21,0,0,0,208.0, 10.0, 0.22,0,0,0,198.0, 10.0, 0.23,0,0,0,185.0, 10.0, 0.24,0,0,0,165.0, 10.0, 0.25,0,0,0,145.0, 10.0, 0.26,0,0,0,127.0, 10.0, 0.27,0,0,0,114.0, 10.0, 0.28,0,0,0, 92.0, 10.0, 0.29,0,0,0, 72.0, 10.0, 0.30,0,0,0, 58.0, 10.0, 0.31,0,0,0,127.0, 25.0, 0.32,0,0,0,127.0, 38.0, 0.33,0,0,0,131.0, 53.0, 0.34,0,0,0,133.0, 63.0, 0.35,0,0,0,138.0, 74.0, 0.36,0,0,0,143.0, 96.0, 0.37,0,0,0, 51.0, 10.0, 0.*MATERIAL PROPERTIES (EX=2.0E7N/SQ.CM.,NUXY=NUXZ=0.3,DENS=.00786N*S*2/CM*4)EX,1,0, 2.0E7NUXY,1,0, 0.3NUXZ,1,0, 0.3DENS,1,0, 0.00786*LDCASE CASE CONTROL CARD0, 1, 2, 0, 1, 0, 1, .0000E+00, .0000E+00*SPDISP BOUNDARY CONDITIONS1,UX,0.0,$,UY,UZ,ROTX,ROTY23,UY,0.0,$,UZ,ROTX,ROTY*CFORCE2,FX, 5000.027,FY,-3500.0*OPTPAR1,1,50,3,40.1,0.01,0.01,0.01,0.001*DVGROUP

A-225


1,1,0,22,212,2,0,22,203,3,0,22,134,4,0,22,145,5,0,6,46,6,0,6,4*DVGELMT1,21,301,-1,23,382,31,373,10,134,4,95,14,206,-1,31,2,39*DVLOCAL1,1,01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.022,0,0,1,01,30.00,48.10, 2,31.15,48.053,34.20,47.90, 4,34.90,47.305,37.20,47.20, 6,37.80,47.907,41.10,47.90, 8,42.40,47.909,42.40,47.80,10,41.10,47.8011,39.80,47.80,12,39.80,45.0013,36.15,45.00,14,36.15,44.9015,32.30,44.90,16,32.30,47.9017,31.15,47.95,18,30.00,48.0019,33.50,45.00,20,33.30,45.5021,37.50,44.90,22,31.15,48.05

A-226

Example Problems

1, 1, 2,0,0,0,02, 2, 3,0,1,0,03, 3, 4,0,1,0,04, 4, 5,0,1,0,05, 5, 6,0,1,0,06, 6, 7,0,1,0,07, 7, 8,0,0,0,08, 7,10,0,1,0,09,10, 9,0,0,0,010,10,11,0,1,0,011,11,12,0,1,0,012,12,13,0,1,0,013,13,14,0,1,0,014,14,21,0,0,0,015,13,19,1,1,0,016,19,20,1,1,0,017,14,15,0,1,0,018,15,16,0,1,0,019,16,17,0,1,0,020,17,22,0,1,0,01,17,18,0,0,0,02,2,01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.021,0,0,1,01, 0.0, 0.7, 2, 0.8, 0.73, 1.6, 0.7, 4, 2.1, 1.85, 3.7, 2.0, 6, 4.5, 4.67, 6.5, 4.6, 8, 6.9, 2.09, 8.4, 1.8,10, 8.8, 0.711, 9.6, 0.7,12,10.4, 0.713,10.4, 0.5,14, 9.6, 0.515, 8.8, 0.5,16, 8.1, 0.0

A-227


17, 2.3, 0.0,18, 1.6, 0.519, 0.8, 0.5,20, 0.0, 0.521, 0.8, 0.71, 1, 2,0,0,0,02, 2, 3,0,1,0,03, 3, 4,0,1,0,04, 4, 5,0,1,0,05, 5, 6,0,1,0,06, 6, 7,0,1,0,07, 7, 8,0,1,0,08, 8, 9,0,1,0,09, 9,10,0,1,0,010,10,11,0,1,0,011,11,12,0,0,0,012,11,14,0,1,0,013,14,13,0,0,0,014,14,15,0,1,0,015,15,16,0,1,0,016,16,17,0,1,0,017,17,18,0,1,0,018,18,19,0,1,0,019,19,20,0,0,0,020,19,21,0,1,0,03,3,01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.014,0,0,1,01, 0.00, 1.20, 2, 2.70, 4.103,10.00, 6.00, 4,12.15, 6.355,14.30, 6.70, 6,14.30, 5.807,12.15, 5.45, 8,10.00, 5.109, 4.90, 0.90,10, 4.60, 0.4011, 3.40, 0.00,12, 0.90, 1.80,13, 0.40, 0.80,14, 0.00, 1.201, 1, 2,0,1,0,02, 2, 3,0,1,0,0

A-228

Example Problems

3, 3, 4,0,1,0,04, 4, 5,0,0,0,05, 4, 7,0,1,0,06, 7, 6,0,0,0,07, 7, 8,0,1,0,08, 8, 9,0,1,0,09, 9,10,0,1,0,010,10,11,0,1,0,011,11,12,0,1,0,012,12,13,0,1,0,013,13,14,0,1,0,04,4,01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.01,0.15,0.1,1.0,1,0.15,0.1,1.015,0,0,1,01, 2.5, 3.2, 2, 4.2, 3.23, 4.2, 0.6, 4, 3.5, 0.45,5, 2.8, 0.3, 6, 3.8, 0.1,7, 3.5, 0.1, 8, 2.2, 0.09, 0.0, 2.4,10, 1.7, 3.711, 1.3, 4.3,12, 0.9, 4.913, 1.2, 5.1,14, 1.6, 4.515, 2.5, 3.21, 1, 2,0,1,0,02, 2, 3,0,1,0,03, 3, 4,0,1,0,04, 4, 5,1,1,0,05, 4, 7,0,1,0,06, 7, 6,0,0,0,07, 7, 8,0,1,0,08, 8, 9,0,1,0,09, 9,10,0,1,0,010,10,11,0,1,0,011,11,14,0,1,0,012,11,12,0,0,0,013,14,13,0,0,0,0

A-229


14,14,15,0,1,0,05,6,10,4.0,4.0,4.0,0,4.,4.0,4.01,0.15,0.1,1.0,1,0.15,0.1,1.0*DVLINK1,1,7A1,A2,A1,A3,A1,A4,A1,A5A1,A6,A1,A7,A1,A81,1,6A9,A10,A9,A11,A9,A12,A9,A13A9,A15,A9,A161,1,5A14,A17,A14,A18,A14,A19,A14,A20A14,A212,2,11A1,A2,A1,A3,A1,A4,A1,A5A1,A6,A1,A7,A1,A8,A1,A9A1,A10,A1,A11,A1,A122,2,7A13,A14,A13,A15,A13,A16,A13,A17A13,A18,A13,A19,A13,A203,3,4A1,A2,A1,A3,A1,A4,A1,A53,3,7A6,A7,A6,A8,A6,A9,A6,A10A6,A11,A6,A12,A6,A13 4,4,6 A1,A2,A1,A3,A1,A4,A1,A5 A1,A13,A1,A144,4,6A6,A7,A6,A8,A6,A9,A6,A10A6,A11,A6,A125,5,1A3,A46,6,1A3,A4*DISCON1,UX,0.52,UY,0.6*DISNOD1,12,16,1

A-230

Example Problems

2,210,16,1,21,30,1*STRCON1,TAUOCT,125000.0*STRELM1,2,953,6,1,17,35,1*ENDDATA DATA DECK TERMINATOR

Input Data for ISECT = 23 is same as for ISECT = 22 except for the following Groupids.*DVGROUP1,1,0,23,12,2,0,23,13,3,0,23,14,4,0,23,15,5,0,6,46,6,0,6,4*DVLOCAL1,1,01,0.15,0.1,1.036.0,416.6,64,133.3,7.0,29.75,47.9,37.82,2,01,0.15,0.1,1.027.33,222.6,61.84,74.86,3.88,21.18,4.6,6.53,3,01,0.15,0.1,1.034.33,608.6,141.0,155.7,235.06,34.16,6.0,10.04,4,01,0.15,0.1,1.017.19,29.2,40.7,39.71,-11.9,12.09,0.0,2.25,6,10,4,4,4,0,4,4,41,.15,.1,1.0,1,.15,.1,1.0*DVLINK5,5A3,A46,6A3,A4

A-231


Results

Tables Table A.20-1 and Table A.20-2 show optimum results whereas tablesTable A.20-3 and Table A.20-4 show the computational efforts required by thethree algorithms (GRG, RQP and OCB) for ISECT = 22 and 23 respectively. Theconvergence parameters ACV, ACS, and ALS used are 0.01, 0.01, and 0.001respectively. However, the parameter ACT is assigned a value of 0.01 for GRG and0.1 for both RQP & OCB. Design iteration histories are given in tables TableA.20-5 - Table A.20-10. Figures Figure A.20-2 and Figure A.20-3 show

graphically the objective function histories.

Table A.20-1 Optimum results for passenger-compartment-structure with ISECT = 22

Design Group

Number


Initial Design (CM)


GRG RQP OCB

1 3, 21-30, 38 0.15 .24404E+00 .13347E+00 .12295E+00

0.15 .24822E+00 .10000E+00 .10304E+00

0.15 .25164E+00 .36006E+00 .30971E+00

2 31-37 0.15 .38810E+00 .41886E+00 .45193E+00

0.15 .26184E+00 .27089E+00 .29306E+00

3 10-13 0.15 .24227E+00 .10000E+00 .10000E+00

0.15 .24166E+00 .10000E+00 .10000E+00

4 4-9 0.15 .25616E+00 .23019E+00 .24348E+00

0.15 .25616E+00 .50509E+00 .49982E+00

5 14-20 0.15 .21257E+00 .31236E+00 .33423E+00

6 1, 2, 39 0.15 .25191E+00 .10000E+00 .10000E+00

Total Volume (cm3) .30076E+04 .51409E+04 .43957E+04 .44020E+04

A-232

Example Problems

Table A.20-2 Optimum results for passenger-compartment-structure with ISECT = 23

Design Group

Number


Initial Design (CM)


GRG RQP OCB

1 3, 21-30, 38 0.15 .19096E+00 .19096E+00 .19908E+00

2 31-37 0.15 .36435E+00 .36435E+00 .36794E+00

3 10-13 0.15 .10000E+00 .10000E+00 .10000E+00

4 4-9 0.15 .38795E+00 .38795E+00 .38923E+00

5 14-20 0.15 .31452E+00 .31452E+00 .29336E+00

6 1, 2, 39 0.15 .10000E+00 .10000E+00 .10000E+00

Total Volume (cm3) .30103E+04 .50496E+04 .44731E+04 .45069E+04

Table A.20-3 Comparison of optimization algorithms for passenger-com-partment-structure with ISECT = 22

Algorithm NITa


NSAb


NDSc


CPU timed (sec)


CPU time/ Design

Iteration (sec)

GRG 4 30 3 429.69 107.42

RQP 15 32 15 929.10 61.94

OCB 24 24 24 1077.13 44.88

A-233


Table A.20-4 Comparison of optimization algorithms for passenger-compart-ment-structure with ISECT = 23

Algorithm NITa


NSAb


NDSc


CPU timed (sec)

d. IBM RT-PC


(sec)

GRG 5 46 5 277 55.40

RQP 9 16 9 154 17.11

OCB 12 12 12 151.00 12.58

Table A.20-5 Design iteration history for passenger-compartment-structure with ISECT=22 and GRG

Iteration Number

Objectivea Function

a. Material Volume





1 .52946E+04 .30159E+08 .12500E+00

2 .52396E+04 .30627E+08 .12500E+00

3 .51867E+04 .24434E+08 .12500E+00

4 .51409E+04 .38659E+08 .00000E+00

A-234

Example Problems

Table A.20-6 Design iteration history for passenger-compartment-structure with ISECT=22 and RQP

Iteration Number

Objectivea Function

a. Material Volume




c. Maximum Constraint Violation (ACV)

1 .30076E+04 .47568E+00 .74148E+00

2 .30424E+04 .59713E+00 .70087E+00

3 .30769E+04 .90911E+03 .53084E+00

4 .29340E+04 .89044E+02 .57837E+00

5 .29198E+04 .11047E+01 .57966E+00

6 .29566E+04 .88500E+01 .55764E+00

7 .29278E+04 .28138E+00 .56647E+00

8 .33661E+04 .45050E+01 .33603E+00

9 .33681E+04 .49527E+00 .33514E+00

10 .34714E+04 .56552E+00 .29896E+00

11 .37628E+04 .41661E+00 .18105E+00

12 .38738E+04 .18103E+00 .14277E+00

13 .43296E+04 .13841E+00 .20060E-01

14 .43773E+04 .13342E-01 .57516E-02

15 .43957E+04 .74170E-03 .87909E-04

A-235


Table A.20-7 Design iteration history for passenger-compartment-structure with ISECT=22 and OCB

Iteration Number

Objectivea Function



1 .30076E+04 .19138E+00 .74148E+00

2 .40123E+04 .12864E+00 .23677E+00

3 .47242E+04 .30727E-01 .39401E-01

4 .48955E+04 .43096E+00 .15215E-02

5 .47256E+04 .72387E-01 .84507E-01

6 .47254E+04 .26308E-01 .18549E-01

7 .47254E+04 .28630E+00 .15504E-02

8 .35611E+04 .23290E+00 .90597E+00

9 .37516E+04 .35605E+00 .38602E+00

10 .38882E+04 .14368E+01 .14559E+00

11 .49412E+04 .21514E+00 .92018E+00

12 .50072E+04 .34495E+00 .37031E+00

13 .47254E+04 .15637E+00 .13736E+00

14 .47290E+04 .58264E-01 .33014E-01

15 .47251E+04 .29399E+00 .28462E-02

16 .47085E+04 .23444E+00 .00000E+00

17 .35599E+04 .20051E+00 .80476E+00

18 .36882E+04 .35147E+00 .30999E+00

19 .38931E+04 .58653E+00 .15993E+00

A-236

Example Problems

20 .40977E+04 .16425E+00 .65888E+00

21 .41201E+04 .22564E+00 .24733E+00

22 .41180E+04 .28755E+00 .87703E-01

23 .44021E+04 .37286E-01 .13120E-01

24 .44020E+04 .88534E-01 .31567E-02

a. Material Volumeb. Norm of Search Direction (ACS)c. Maximum Constraint Violation (ACV)

Table A.20-8 Design iteration history for passenger-compartment-structure with ISECT=23 and GRG

Iteration Number

Objectivea Function

a. Material Volume




c. Estimated step size (ALS)

1 .53013E+04 .57514E+10 .62500E-01

2 .50975E+04 .15339E+11 .31250E-01

3 .50602E+04 .47387E+16 .15625E-01

4 .50523E+04 .25828E+10 .15625E-01

5 .50496E+04 .32524E+12 .00000E+00

Table A.20-7 Design iteration history for passenger-compartment-structure with ISECT=22 and OCB

Iteration Number

Objectivea Function



A-237


Table A.20-9 Design iteration history for passenger-compartment-structure with ISECT=23 and RQP

Iteration Number

Objectivea Function

a. Material Volume




c. Maximum Constraint Violation (ACV)

1 .30103E+04 .30001E+00 .74254E+00

2 .32134E+04 .44241E+00 .62465E+00

3 .31826E+04 .68752E+03 .48739E+00

4 .31077E+04 .15034E+00 .50605E+00

5 .35341E+04 .34124E+00 .30441E+00

6 .36911E+04 .11391E+00 .24036E+00

7 .39890E+04 .12115E+00 .13801E+00

8 .44149E+04 .46228E-01 .25968E-01

9 .44731E+04 .43461E-02 .57290E-02

Table A.20-10 Design iteration history for passenger-compartment-struc-ture with ISECT=23 and OCB

Iteration Number

Objectivea Function



1 .30103E+04 .13634E+00 .74254E+00

2 .41313E+04 .91448E-01 .23593E+00

3 .48879E+04 .21820E-01 .39197E-01

A-238

Example Problems

4 .50684E+04 .47604E+00 .14815E-02

5 .45104E+04 .61311E-01 .15898E+00

6 .46559E+04 .26197E-01 .24447E-01

7 .46558E+04 .18568E+00 .29792E-02

8 .43624E+04 .16051E+00 .78388E+00

9 .44562E+04 .16903E+00 .29452E+00

10 .44543E+04 .11751E+00 .92215E-01

11 .44536E+04 .76102E-01 .28128E-01

12 .45069E+04 .37138E-01 .28281E-03

a. Material Volumeb. Norm of Search Direction (ACS)c. Maximum Constraint Violation (ACV)

Table A.20-10 Design iteration history for passenger-compartment-struc-ture with ISECT=23 and OCB

Iteration Number

Objectivea Function



A-239


Figure A.20-2 Objective function histories for passenger-compartment-structure with ISECT = 23

A-240

Example Problems

Figure A.20-3 Objective function histories for passenger-compartment-structure with ISECT = 22

A.21 Optimum Design of a Cantilever Plate with Buckling Constraint

Title

Optimum Design of a Cantilever Plate with Buckling Constraint

Problem

A cantilever plate with dimensions shown in Figure A.21-1 is subjected to aninplane load of 500 lbs. at node 33 in negative x-direction. The plate is divided intothree design variable groups using symmetry.

The design objective is to find element thicknesses that minimize the plate volumewhile satisfying constraint on buckling load factor.

Element Type

3D General Shell (NKTP=20, NORDR=2)

Section Type

Uniform Element Thickness (ISECT=1)

Material Properties

Design Data



Mass Density = 0.000733 lb. s2/ in2


A-241


Figure A.21-1 Geometry of cantilever plate


No. of Elements = 6

No. of Buckling Load Factor Constraint = 1



PROBLEM=EXAM21ANALYSIS=BUCKLINGAUTO CONSTRAINT=OFFRESEQUENCING OF ELEMENT=OFF

A-242

Example Problems

eigen extraction=inverse** STROPT Example Problem #21** TOTAL NO.OF ELEMENTS=6** TOTAL NO.OF NODES=29*TITLEOPTIMUM DESIGN OF CANTILEVER PLATE WITH BUCKLING CONSTRAINT ONLY*ELTYPE Element Type Section (3-D General Shell)1,20,2*RCTABLE Real Constant Table1, 81.0///////*E1 Element Connectivities$$,-3,10,21,1,6,11,12,13,8,3,2,1,1,1,2,2,1*NODES Nodal Coordinates-5,0,5,7,5.0,0.0,0.01,$,0.0,0.0,0.05,0,1,0,0.0,10.0,0.0*MATERIAL Material Property DataEX,1,0,.3E8NUXY,1,0,.3DENS,1,0,.000733*LDCASE Load Case Control Card0, 1, 2, 0, 0, 0, 0, .0000E+00, .0000E+00*EIGCNTL Eigenvalue Analysis Control Card2,0,40,0,0.0,3.0e6,1.0e-4*SPDISP Specified Displacements (Boundary Conditions)1,UX,0.,5,1,UY,UZ,ROTX,ROTY,ROTZ6,ROTZ,0.,35,1*CFORCE Concentrated Nodal Forces33,FX,-500.0*OPTPAR Control Parameters for Optimization1,-1,30,2,20.1,0.01,0.01,0.01,0.001*DVGROUP Design Variable Groups and Section Type1,3,1,1,1*DVGELMT Design Variable Groups and Element Numbers1,1,2,1,3,1,1*DVLOCAL Local Design Variable Information1,3,1

A-243


1,1.0,0.1,10.0*BUKCON Buckling Load Factor Constraints1,1.0*ENDDATA

A-244

Example Problems

Results

Table A.21-1 shows optimum results obtained by three optimization algorithms(GRG, RQP, OCB). Computational effort required in this problem is given in TableA.21-2 with total number of structural and sensitivity analyses. The convergence

parameters ACV, ACS and ALS used in this example are 0.01, 0.01 and 0.001,respectively. Parameter ACT is assigned a value of 0.01 for GRG and 0.1 for bothRQP and OCB. Design iteration histories are given in tables Table A.21-3 - TableA.21-5. It is observed that with all the three algorithms a reduction of 85% in

volume is achieved. Table A.21-5 shows that none of the convergence criteria issatisfied within the 30 number of iterations. However, STROPT selected thefeasible design with minimum cost at iteration number 23 which may be taken asthe best available design. The value of the volume for best design is 54.07 in3.Figure A.21-2 shows graphically the objective function histories for the threeoptimization algorithms (GRG, RQP, OCB).

Table A.21-1 optimum results of cantilever plate

Design Group

Number




GRG RQP OCB

1 1,2 0.10000E+01 0.22512E+00 0.22527E+00 0.2048E+00

2 3,4 0.10000E+01 0.17735E+00 0.20465E+00 0.2028E+00

3 5,6 0.10000E+01 0.13842E+00 0.11552E+00 0.1330E+00


Table A.21-2 Comparison of optimization algorithms for cantilever plate

Algorithm NITa NSAb NDSc CPU timed (sec)


GRG 4 48 3 800.70 200.175

A-245


RQP 8 14 7 249.9 31.23

OCB 30 30 17 580.76 19.36

a. Total number of design iterationsb. Total number of structural analysesc. Total number of sensitivity analysesd. ELXSI 6400 system

Table A.21-3 Design iteration history for cantilever plate with GRG

Iteration Number

Objectivea Function






1 0.21330E+03 0.30000E+05 0.89835E+00

2 0.57701E+02 0.19152E+05 0.28074E-01

3 0.55292E+02 0.17886E+05 0.28074E-01

4 0.54051E+02 0.17757E+05 0.00000E+00

Table A.21-4 Design iteration history for cantilever plate with RQP

Iteration Number

Objectivea Function



1 0.30000E+03 0.17321E+03 0.00000E+00

2 0.30000E+02 0.36161E+00 0.85935E+00

3 0.34520E+02 0.28696E+00 0.77028E+00

Table A.21-2 Comparison of optimization algorithms for cantilever plate

A-246

Example Problems

4 0.72791E+02 0.63599E+00 0.00000E+00

5 0.59141E+02 0.28640E+00 0.67667E-01

6 0.54074E+02 0.15023E-01 0.60769E-01

7 0.55790E+02 0.14138E-01 0.00000E+00

8 0.54544E+02 0.36748E-02 0.91985E-02

a. Material volume of the cantilever plateb. Norm of Lagrangian gradient (ACS)c. Maximum constraint violation (ACV)

Table A.21-5 Design iteration history for cantilever plate with OCB

Iteration Number

Objectivea Function



1 0.30000E+03 0.10000E+01 0.00000E+00

2 0.15000E+03 0.10000E+01 0.00000E+00

3 0.82500E+02 0.10000E+01 0.00000E+00

4 0.49088E+02 0.67974E-01 0.38395E+00

5 0.49088E+02 0.86944E-01 0.25913E+00

6 0.54066E+02 0.47011E-01 0.17459E+00

7 0.54066E+02 0.70465E+00 0.00000E+00

8 0.45797E+02 0.21217E+00 0.44820E+00

9 0.49568E+02 0.19612E+00 0.78625E+00

Table A.21-4 Design iteration history for cantilever plate with RQP

Iteration Number

Objectivea Function



A-247


10 0.49568E+02 0.14992E+00 0.35379E+00

11 0.49568E+02 0.11643E+00 0.57248E+00

12 0.49568E+02 0.26346E+00 0.29409E+00

13 0.57017E+02 0.17560E+00 0.78512E+00

14 0.54066E+02 0.75443E-01 0.56819E+00

15 0.54066E+02 0.98674E-02 0.63738E-01

16 0.54066E+02 0.47235E-02 0.28846E-01

17 0.54066E+02 0.25813E-02 0.15135E-01

18 0.54066E+02 0.71631E+00 0.86268E-02

19 0.45829E+02 0.15584E+01 0.40038E+00

20 0.15311E+03 0.11757E+01 0.68086E+00

21 0.76557E+02 0.33168E+00 0.17942E+00

22 0.45829E+02 0.94385E-01 0.57262E+00

23 0.54066E+02 0.65989E+00 0.00000E+00

24 0.50002E+02 0.21326E+00 0.27585E+00

25 0.50002E+02 0.31617E+00 0.77896E+00

26 0.58366E+02 0.12099E+00 0.82201E-01

27 0.45829E+02 0.70239E-01 0.43461E+00

28 0.53417E+02 0.15990E-01 0.38793E-01

29 0.53417E+02 0.10050E+00 0.10827E+00

30 0.53417E+02 0.40334E-01 0.28662E+00

Table A.21-5 Design iteration history for cantilever plate with OCB (Con-

Iteration Number

Objectivea Function



A-248

Example Problems

Figure A.21-2 Objective function histories for cantilever plate


A-249

Optimum Design of a Four-Bar Truss with Buckling, Stress, and Displacement Constraints

A.22 Optimum Design of a Four-Bar Truss with Buckling, Stress, and Displace-ment Constraints

Title

Optimum Design of a Four-Bar Truss with Buckling, Stress, and DisplacementConstraints

Problem

A four-bar truss with dimensions shown in Figure A.22-1 is subjected to aconcentrated force of 7.5 Kips at node 5 in negative z-direction.

The design objective is to find areas of elements that minimize the truss volumewhile satisfying constraints on buckling load factor, stress, and displacement.

Element Type

3D Spar (NKTP=14, NORDR=1)

Section Type

Cross-sectional area (ISECT=7)

Material Properties


A-250

Example Problems

Design Data



No. of Elements = 4



No. of Buckling Constraint = 1

Upper Bound on Design Variables = 100.0 in2.

Lower Bound on Design Variables = 0.10 in2.


Displacement Limit in Y-Dir. = 0.3 in.

Displacement Limit in Z-Dir. = 0.4 in.

Limiting Value of Buckling Load

Factor = 100.0

A-251


Figure A.22-1 Geometry of Four-Bar Truss

PROBLEM=EXAM22ANALYSIS=BUCKLINGAUTO CONSTRAINT=OFFEIGEN EXTRACTION = SUBSPACE,CONVENTIONAL

A-252

Example Problems

RESEQUENCING OF ELEMENT=OFF* STROPT Example Problem #22*TITLEOPTIMUM DESIGN OF FOUR-BAR TRUSS UNDER BUCKLING, STRESS, AND STRESS CONSTRAINTS*ELTYPE Element Type Section (3-D Spar)1,14,1*RCTABLE Real Constant Table1,2.1,0.1*E1 Element Connectivities1,1,5$1,1,12,2,5$2,1,13,3,5$3,1,14,4,5$4,1,1*NODES Nodal Coordinates1,0 $ 0.,0.,0,2,0 $ 0.,192.,0.3,0 $ 204.,192.,0.4,0 $ 204.,0.,0.5,0 $ 60.,120.,96.*MATERIAL Material Property DataEX,1,0,1.0E4EX,2,0,1.0E4EX,3,0,1.0E4EX,4,0,1.0E4*LDCASE, ID = 1 Load Case Control Card0, 1, 2, 0, 0, 0, 0, .0000E+00, .0000E+00*EIGCNTL Eigenvalue Analysis Control Card2,0,40,0,0.0,3.0e6,1.0e-4*SPDISP Specified Displacements (Boundary Conditions)1,UX,0.,4,1,UY,UZ*CFORCE Concentrated Nodal Forces5,FZ,-7.5*OPTPAR Control Parameters for Optimization1,1,20,2,40.1,0.01,0.01,0.01,0.001*DVGROUP Design Variable Groups and Section Type1,4,1,7*DVGELMT Design Variable Groups and Element Numbers1,1,1,0,4

A-253


Results

Table A.22-1 shows optimum results obtained by three optimization algorithms(GRG, RQP, OCB). Computational effort required in this problem is given in TableA.22-2 with total number of structural and sensitivity analyses. The convergence

parameters ACV, ACS and ALS used in this example are 0.01, 0.01 and 0.001,respectively. Parameter ACT is assigned a value of 0.01 for GRG and 0.1 for bothRQP and OCB. Design iteration histories are given in tables Table A.22-3 - TableA.22-5. It is observed that initial design is infeasible and there is an increase of

103% in volume. Figure A.22-2 shows graphically the objective function historiesfor the three optimization algorithms (GRG, RQP, OCB).

*DVLOCAL Local Design Variable Information1,41,0.1,0.1,100.0*DISCON Displacement Constraint IDs1,UY,0.3,UZ,0.4*DISNOD Node Numbers for Displacement Constraint IDs1,1,15,5*STRCON Stress Constraint IDs1,AXIAL,25.0*STRELM Element Numbers for Stress Constraint IDs1,1,91,11,4,1*BUKCON Buckling Load Factor Constraints1,100.0*ENDDATA

Table A.22-1 Optimum Results of Four-Bar Truss

Design Group

Number




GRG RQP OCB

1 1 0.10000E+00 0.21699E+00 0.15502E+00 0.18431E+00

2 2 0.10000E+00 0.25347E+00 0.38071E+00 0.36193E+00

3 3 0.10000E+00 0.22509E+00 0.10000E+00 0.13417E+00

4 4 0.10000E+00 0.18028E+00 0.21932E+00 0.18369E+00

A-254

Example Problems


Table A.22-2 Comparison of optimization algorithms for Four-Bar Truss

Algorithm NITa


NSAb


NDSc


CPU timed (sec)



(sec)

GRG 9 48 3 91.52 10.20

RQP 8 14 8 14.85 1.85

OCB 12 12 12 20.36 1.70


Iteration Number

Objectivea Function



1 0.15841E+03 0.32564E+05 0.62500E-01

2 0.15470E+03 0.19479E+06 0.31250E-01

3 0.15341E+03 0.70503E+04 0.31250E-01

4 0.15127E+03 0.11888E+07 0.19531E-02

5 0.15086E+03 0.41620E+04 0.39063E-02

Table A.22-1 Optimum Results of Four-Bar Truss

Design Group

Number




GRG RQP OCB

A-255


6 0.15067E+03 0.16049E+06 0.39063E-02

7 0.15006E+03 0.28380E+06 0.19531E-02

8 0.14939E+03 0.11751E+05 0.19531E-02

9 0.14930E+03 0.71198E+07 0.00000E+00

a. Material volume of the four-bar trussb. Squared norm of projected reduced gradient (ACS)c. Estimated stepsize (ALS)


Iteration Number

Objectivea Function






1 0.69718E+02 0.12258E+00 0.12721E+01

2 0.98832E+02 0.38041E+00 0.52397E+00

3 0.94803E+02 0.22684E+01 0.56998E+00

4 0.94606E+02 0.12147E+00 0.57033E+00

5 0.11950E+03 0.12039E+01 0.19587E+00

6 0.11988E+03 0.10690E+00 0.19180E+00

7 0.12949E+03 0.56698E-01 0.10339E+00

8 0.14159E+03 0.43475E-02 0.89159E-02


Iteration Number

Objectivea Function



A-256

Example Problems


Iteration Number

Objectivea Function






1 0.69718E+02 0.10975E+00 0.12721E+01

2 0.10371E+03 0.92174E-01 0.46305E+00

3 0.13230E+03 0.40791E-01 0.11644E+00

4 0.14576E+03 0.47235E-02 0.11048E-01

5 0.14733E+03 0.34218E+00 0.11974E-03

6 0.14305E+03 0.61878E-01 0.13130E+00

7 0.14305E+03 0.48081E-01 0.43603E-01

8 0.14305E+03 0.27462E-01 0.12510E-01

9 0.14305E+03 0.96042E-01 0.27982E-02

10 0.14017E+03 0.66597E-01 0.32671E-01

11 0.14017E+03 0.74544E-02 0.19364E-01

12 0.14280E+03 0.28444E-01 0.42825E-03

A-257


Figure A.22-2 Objective function histories for Four-Bar Truss

A-258

Appendix

B
Comparison of Design Sensitivity Coefficients
B.1 Comparison of Design Sensitivity Coefficients Between Exact Solution and Numerical Solution using STROPT

Comparison of Design Sensitivity Coefficients for a Three-bar Truss by TwoMethods: STROPT and Exact Solution

1. Problem

A small truss with dimensions shown in Figure B.1 is subjected to a concentratedforce of 100 kips at node1. The objective is to compute design sensitivitycoefficients for displacements and stresses, analytically (exact solution) andnumerically using STROPT and then compare the results.

2. Method 1: Numerical Method Using STROPT

(a) Finite Element Input:

The truss is modeled by two-noded, 3-D spar elements (NKTP=14,NORDR=1). All members of the truss are assumed to be subjected to axialforce only. The material to be used is steel with modulus of elasticity of30,000 ksi. Load of 100 kips is applied at node 1 as shown in Figure B.1.

(b) Structural Optimization Input Data:

There are 2 design variable groups having cross-sectional area as designvariable (see *DVGROUP data group). There are three elements. Elements 1and 3 are linked to design variable group 1 and element 2 with design

B-1

Comparison of Design Sensitivity Coefficients Between Exact Solution and Numerical Solution using STROPT

variable group 2. The cross-sectional area for elements 1 and 3 is 1.0 in2 andfor element 2 it is 2.0 in2 (see *DVLOCAL data group). Displacementsensitivities are calculated at node 1 (see *DISNOD data group) in global xand y directions (see *DISCON data group). Stress sensitivities due to axialforce (see *STRCON data group) are computed for all elements (see*STRELM data group).

(c) STROPT Input Data:

For a STROPT-run, the input data groups for the structural optimization partof analysis (“STROPT Data Block”) should be appended to standard NISAinput file. Input file is given in Table B.1.

(d) Results Interpretation:

Sensitivity coefficients for displacements at node 1 in x and y directions, andfor axial stresses for all members are calculated with respect to global designvariables 1 and 2 and printed. Printout selections depend on the specifiedentries in *SETS and LABEL in *PRINTCNTL data groups. Table B.2 showsdesign sensitivity coefficients w.r.t. design variable numbers 1 and 2 for

B-2


displacements at node 1 in x and y directions (Ux and Uy) and axial stressesfor members 1, 2, and 3, i.e. s1, s2, and s3.

Fig B.1 Three bar truss

Table B.1 STROPT Input Data for Three Bar Truss

**

********************** EXECUTIVE COMMANDS STARTS HERE ***********************

**********************************************************************************

**

analysis = static

design = sensitivity

LOAD_CASE_NO= 1

AUTO_CONSTRAINT=OFF

RESEQUENCING_OF_ELEMENT=OFF

B-3


**

**********************************************************************************

************************ ANALYSIS INPUT DATA STARTS HERE ***********************

**********************************************************************************

**

*title Problem title

Design sensitivity analysis of a 3-bar truss

*eltype Real constant table

1,14,1

*elements Element connectivities

1,1,1,1

1,2

2,1,1,2

1,3

3,1,1,1

1,4

*rctable Real constant table

1,2

1.0,1.0

2,2

2.0,2.0

*nodes Nodal coordinates

1,,,,0.0,0.0,0.0

2,,,,-10.0,-10.0,0.0

B-4


3,,,,0.0,-10.0,0.0

4,,,,10.0,-10.0,0.0

*material Material property data

EX,1,0,30000.0

*ldcase, id=1 Load case data card

0,0,1,0,1

*spdisp Specified displacement (boundary conditions)

1,uz,0.0

2,ux,0.0,4,1,uy,uz

*cforce Concentrated nodal forces

,fx,70.71

1,fy,70.71

*printcntl Selective printout control

elst,0

disp,0

sest,0

sedi,0

**

**********************************************************************************

*************** STRUCTURAL OPTIMIZATION INPUT DATA STARTS HERE **************

**********************************************************************************

*OPTPAR

1,1,20,2,1

B-5


0.1,0.01,0.01,0.01

*dvgroup Design variable groups and section type

1,2,1,7

*dvgelmt Design variable groups and element numbers

1,1,3,2

2,2,2

*dvlocal Local design variable information

1

1,1.0,0.1,10.0

2

1,2.0,0.1,10.0

*discon Displacement sensitivity IDs

1,ux,1.0,uy,1.0

*disnod Node numbers for displacement sensitivity IDs

1

1*

strcon Stress sensitivity IDs

1,axial,1.0

*strelm Element numbers for stress sensitivity IDs

1,1,91

1,3,1

**

**********************************************************************************

*************** STRUCTURAL OPTIMIZATION INPUT ENDS STARTS HERE **************

B-6


Table B.2 Design Sensitivity Coefficients of Nodal Displacements and Axial Stresses for Three Bar Truss with STROPT

3. Method 2: Exact Solution

This simple structural model problem is used to illustrate the designsensitivity analysis using analytical method. Objective is to computesensitivity coefficients analytically with respect to cross-sectional areas x1,x2, and x3 (the design variables). Sensitivity coefficients are calculated fornormal stresses in elements 1, 2, and 3 (S1, S2, S3) and displacements at node 1in x- and y-directions (Ux and Uy).

**********************************************************************************

*enddata Data deck terminator

Nodal DisplacementSensitivity Coefficients*

Elemental Axial StressSensitivity Coefficients*

Uxw.r.txl

Uxw.r.tx2

Uyw.r.t.

xl

Uyw.r.t.x2

slw.r.t.x1

slw.r.t.x2

s2w.r.t.x1

s2w.r.t.x2

s3w.r.t.x1

s3w.r.t.x2

-0.033 0.0 -0.0023

-0.0032

53.41 4.82 6.82 9.65 -46.59 4.83

* NOTATION:

x1 = Design variable number 1, i.e. cross-sectional area for member elements 1 or 3.

x2 = Design variable number 2, i.e. cross-sectional area for member element 2.

Ux = Displacement along x-axis of node number 1

Uy = Displacement along y-axis of node number 1

s1 = Normal stress due to Axial load for member (element) number 1



B-7


(a) State Equations

The horizontal and vertical nodal displacements Ux and Uy of the commonnode 1 are obtained from linear elastic structural equations ( 2.1-1). For thisstructure the equations are

K(x)U = P(x) (B.1-1)

where, U is the Nodal displacement vector at node 1 and x is the vector ofcross-sectional areas and are shown as

(B.1-2)

K(x) is positive definite stiffness matrix for the three bar truss, and P is theload vector,

(B.1-3)

where E is Young’s Modulus and is the angle of load application from thehorizontal. K(x) is obtained by the assembly of element stiffness matrix fortruss elements 1, 2, and 3 and imposing proper displacement boundaryconditions, i.e. Ux=0, Uy=0 at nodes 2, 3, and 4. Element stiffness matrix k,for a 2-Dimensional truss element in a local coordinate system can be writtenas:

(B.1-4)

where A , E, and L are member cross-sectional area, elastic modulus, andlength of the member, respectively.

U UxUy

= x,x1x2x3

=

K x( ) 2 E40

-------------------- x1 x3+( ) x1 x3–( )

x1 x3–( ) x1 x3 2 2 x2+ +( ) P, P θ( )cos

P θ( )sin= =

θ

k A EL

----------- 1 1–1– 1

=

B-8


(b) Displacement and Stress Calculations

Considering a symmetric structure, after some manipulation Eq. B.1-1becomes

(B.1-5)

The solution for [U] is therefore easily obtained as

(B.1-6)

The stress in each member is obtained by calculating strain in terms of nodaldisplacement and using Hooke’s law (stress = E * strain)

(B.1-7)

(c) Displacement and Stress Sensitivities

Sensitivity coefficients of Ux and Uy with respect to design variables 1 and 2are obtained by differentiation of Eq. B.1-6 w.r.t. x1 and x2. These are

For Ux:

x1 0

0 x1 2 x2+( )UxUy

10 2E

------------- P θ( )cosP θ( )sin

=

Ux 10 2 P θ( )cosx1 E

-------------------------------------=

Uy 10 2 P θ( )sinx1 2 x2+( ) E

----------------------------------------=

s1 E Ux Uy+( )20

----------------------------------=

s2 E Uy10

-------------=

s3 E Uy Ux–( )20

-------------------------------=

B-9


(B.1-8)

For Uy:

(B.1-9)

Substituting expressions for Ux and Uy from ( B.1-6) into ( B.1-7) one obtains

(B.1-10)

Also, total Differentiation of s1, s2, and s3 w.r.t. design variables x1 and x2,gives sensitivity coefficients for s1, s2, and s3 respectively. These are,

d Ux( )d x1( )--------------- 10 2 P θ( )cos

x1( )2 E-------------------------------------–=

d Ux( )d x2( )--------------- 0.0=

d Uy( )d x1( )--------------- 10 2 P θ( )sin

x1 2 x2+( )2 E-------------------------------------------–=

d Uy( )d x2( )--------------- 20 P θ( )sin

x1 2 x2+( )2 E-------------------------------------------–=

s1 22

------- P θ( )cosx1

-------------------- P θ( )sinx1 2 x2+( )

---------------------------------+=

s2 2 P θ( )sinx1 2 x2+( )

---------------------------------=

s3 22

------- P θ( )cosx1

--------------------– P θ( )sinx1 2 x2+( )

---------------------------------+=

B-10


For s1:

(B.1-11)

For s2:

(B.1-12)

For s3:

(B.1-13)

(d) Sensitivity Coefficients:

For the design problem, P=100 Kips, θ=45°, x1=1.0 in2, x2=2.0 in2, x3=1.0in2, and E=30,000 ksi. Equations B.1-8, B.1-9, B.1-11, B.1-12, and B.1-13after appropriate substitutions yield the required sensitivity coefficients. Thesecoefficients are tabulated in Table B.3.

d s1( )d x1( )-------------- 2

2------- P θ( )cos

x1( )2--------------------– P θ( )sin

x1 2 x2+( )2-----------------------------------–=

d s1( )d x2( )-------------- P θ( )sin

x1 2 x2+( )2-----------------------------------–=

d s2( )d x1( )-------------- 2 P θ( )sin

x1 2 x2+( )2-----------------------------------–=

d s2( )d x2( )-------------- 2 P θ( )sin

x1 2 x2+( )2-----------------------------------–=

d s3( )d x1( )-------------- 2

2------- P θ( )cos

x1( )2-------------------- P θ( )sin

x1 2 x2+( )2-----------------------------------–=

d s3( )d x2( )-------------- 2 P 0( )sin

x1 2 x2+( )2-----------------------------------–=

B-11


Table B.3 Exact design sensitivity coefficients of nodal displacements and axial stresses for three bar truss

4. Sensitivity Coefficients Comparisons: Stropt Vs. Exact Solution

Let’s compare response sensitivity coefficients obtained by the two previousmethods. Comparing Table B.2 and Table B.3 show, identical response sensitivitycoefficients obtained analytically and numerically using STROPT. Figure B.2shows displacement sensitivity for node number 1 in x and y directions and FigureB.3 shows sensitivity of axial stresses for all 3 elements.

Fig B.2 Displacement sensitivity coefficients for four-bar truss (Node 1 in x and y directions)

Nodal DisplacementSensitivity Coefficients

Elemental Axial StressSensitivity Coefficients

Uxw.r.txl

Uxw.r.tx2

Uyw.r.t.

xl

Uyw.r.t.x2

slw.r.t.x1

slw.r.t.x2

s2w.r.t.x1

s2w.r.t.x2

s3w.r.t.x1

s3w.r.t.x2

-0.033 0.0 -0.0023 -0.0032 -53.41 -4.82 -6.82 -9.65 46.59 -4.83

B-12


Fig B.3 Axial stress sensitivity coefficients for 3-bar truss (Elements 1, 2, and 3)

B-13


B-14

Appendix

C
Output Format for Example Problem A.1 Using Optimum Cost Bounding Algorithm
C-1


C-3


C-5


C-7


C-9


C-11


C-13


C-15


C-17


C-19


C-21


C-23


C-25


C-27

Appendix

D
Important Hints for Using STROPT
Structural Optimization Hints

Structural optimization is an important and at the same time sensitive subject.Care should be taken to obtain a reasonable solution. Hence, the guidelines givenhere are to assist users to conquer the rugged road encountered in the process ofstructural optimization.

Structural optimization is a design process which cannot be accomplished withouta preliminary analysis. Since adopted structural analysis is based on finiteelements, one should know geometric modeling and finite element analysis beforeconsidering conceptual design, and in particular structural design optimization.we recommend the users to study and practise modeling and other analysis hintsgiven in DISPLAY III and NISA user manuals before getting started withstructural optimization. Guidelines given here are mostly based on our users whoexperienced difficulties in efficient use of structural optimization.. The followingare recommended by staff involved in design and implementation of the programSTROPT.

1. Creation of Optimization Input File with DISPLAY III

DISPLAY III uses a very user-friendly, forms-based input format. It offers theoptions for defining, listing, modifying and deleting structural optimization inputdata. First, user needs to create NISA related data blocks such as executive,model, and analysis block. In the second step, insert structural optimizationrelated data by following appropriate path in the menu mode. As an example,

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control parameters and convergence criteria can be defined by the following pathin the menu mode as shown.

Results can be plotted utilizing DISPLAY III for initial and optimum (final) design.For example, design variable for shell structure, thickness, can be plotted for initialand optimum values after STROPT run by reading files 26 and 27. Responses suchas displacements, stresses or others can also be plotted easily with DISPLAY III forinitial and final (optimum) designs.

2. Experiences with Lengthy/Moderate/Least Printout in STROPT Output File

In STROPT there is a variable called IPRINT defined in *OPTPAR data group.IPRINT and other variables given in *PRINCNTL data group control bulk ofprintouts either specified by user or generated/computed by the program.Sometimes, one desires to view detailed diagnostic of the output for the followingreasons: oscillation, divergence, catching optimum point, constraint violations,number of active constraints, feasible solutions, convergence scheme towardsoptimum, direction of solution. For example, by inspecting cost function history,maximum constraint violation, step size, and convergence parameters againstconvergence criteria and limiting values, user has a good knowledge about theprogress of the solution sequence. Convergence criteria and limiting values arespecified in the second card set in *OPTPAR data group.

3. Experiences with Selection of Optimization Method

Currently, there are three (3) optimization algorithms based on nonlinearprogramming techniques in STROPT. Supported optimization algorithms are:Generalized Reduced Gradient (GRG), Recursive Quadratic programming (RQP),and Optimum Cost Bounding (OCB).

Though, all three calculate change in designs, their directional behavior may bequite different. GRG and RQP are expensive approaches because extensive linesearch is required. Line search involves additional structural analysis run which istime consuming and costly. On the contrary, OCB method does not employ linesearch, and thus, is a less expensive approach. However, it may encounter

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oscillations around optimum and may not converge within maximum number ofiterations specified in *OPTPAR data group. In general, OCB is the recommendedalgorithm if one is not interested in optimum solution and is looking for a feasibledesign better than the starting design. For example if one is interested in reducingweight of the structure by 50% and simultaneously satisfying subjected constraints,OCB can be selected as the prime candidate. One must be particularly careful incase of composite structures if angle orientations are prime design variables. Insuch a situation, OCB is not recommended because volume/mass/ weight are notfunctions of angles and thus, no convergence will be achieved. RQP and GRG arerecommended in case of composite elements such as 3-D laminated compositeshell (NKTP=32) and 3-D sandwich element (NKTP=33).

4. Experiences with Selection of Design Sensitivity Analysis Method

There are three design sensitivity analysis methods in STROPT as given in*OPTPAR data group. These are: Direct Differentiation, Adjoint Variable andHybrid. Depending on the number of design variables (NDV), number of activeconstraints (more important constraints; NAC), and number of load cases (NLC),one should select one of these approaches. If hybrid method is chosen, STROPTselects the best approach based on NDV, NAC, and NLC automatically. Hybrid iseither direct differentiation or adjoint variable method, thus, STROPT recommendshybrid method.

5. Experiences with Convergence Criteria and Limiting Values

Convergence criteria and limiting values are suitable for monitoring optimizationprocess and trends towards optimum solution. Five entries in 2nd card set in*OPTPAR data group should be specified to determine active constraints,acceptable constraint violation, acceptable convergence criterion for optimumsolution, design perturbation, and tolerance on line search convergence criterion.There are pre-assigned default values for each one in the program. To specifynon-default values, optimization knowledge and significant experiences areessential. Thus, for a beginner, we strongly recommend not to specify any valuesfor these criteria and let the STROPT program select default values.

6. Experiences with Design Variable Linking

Design variable linking is employed to impose equality of design variables in someregions of the optimized structure. Many design variables can be linked, and then

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on treated as one design variable in the process of structural optimization. Let’stake a simple plate model with four shell elements as shown in Figure D.1.Suppose elements 1 & 3 and elements 2 & 4 have same thicknesses. We maytherefore, link elements 1 & 3 and elements 2 & 4. Linking can be done easily byemploying *DVLINK data group. We specify one design variable group numberfor elements 1 & 3 and another design variable group number for elements 2 & 4.This type of linking is called local linking. Let’s consider the problem given inFigure D.1 again. Suppose we require uniform thickness over the entire model. Insuch a case, let’s define one (1) design variable group ID number say 100 in*DVGROUP, for all four elements. Now, relate all four elements (1, 2, 3, and 4) todesign variable group number 100 using *DVGELMT data group. This type oflinking is called global linking, because more designs (here more elements) arelinked together.

Fig D.1 Four shell-element model

Linking contributes significantly to design uniformity in particular and process ofdesign optimization in general. For example, linking common node-thickness oftwo shell elements not only makes smooth surfaces but also reduces process ofdesign sensitivity analysis considerably. Thus, STROPT highly recommends localand global linking to satisfy fabricational constraints and other important designfactors.

7. Experiences with Displacement and Amplitude Constraints

Displacement constraints are imposed at nodes. In general each node has six (6)degrees of freedom. In STROPT, one can impose displacement constraint at each/all degree(s) of freedom for each/all node(s) and for every/all load case(s). Twodata groups are required in conjunction with displacement constraint. They are*DISCON and *DISNOD. In *DISCON, user defines label corresponding to aparticular degrees of freedom and the limiting value associated to them. In

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*DISNOD, user defines nodes where displacement constraints are imposed.Symbolically, *DISCON is similar to *RCTABLE data group in NISA where userdefines an ID for real constant table related to physical properties and then relatesthe ID to several element(s) holding same physical properties in *ELEMENTSdata group. Here, we define an ID in *DISCON for transitional or rotationaldisplacements and associated limiting values. We then relate the ID to nodesholding same allowable limit specified in *DISNOD data group. Note thatimposing unnecessary displacement constraints or defining too many displacementconstraints can markedly slow down process of structural optimization in STROPT.One should try to impose displacement constraints for the nodes at degrees offreedom which are crucial in design. For example, suppose there are 10,000 nodesin a structure and 2 nodes are critical (important) in actual design environment. Insuch a situation, one should attempt to restraint the two (2) nodes only.

Amplitude constraints are identical to displacement constraints with the exceptionof no load case involvement. There are two data groups corresponding to amplitudeconstraints. They are *FRSCON and *FRSNOD. They are identical to *DISCONand *DISNOD, respectively. The guidelines that apply to displacement constraintsare applicable to amplitude constraints as well.

8. Experiences with Stress Constraints

Stress constraints are at nodes, Gauss points, and centroids of elements. Forexample, in case of a 4-noded shell element, there are 4 nodes corresponding toeach element, thus, imposing stress constraints at node points imply imposition ofstress constraints at all 4 nodes. For beam elements, STROPT imposes stressconstraint at critical points of beam cross sections at both end nodes of the beamelement. Depending on the element type, nodal stresses can be imposed on: state ofstress, principal, von Mises, octahedral shear, maximum shear, Hill Mises, Tsai-Wu, and AISC type of stress due to axial force, bending moment, and shear forces.Stress types and associated elements are classified under Stress Library and areshown in Table 4.3-1 . There are two data groups used in conjunction with stressconstraints. They are *STRCON and *STRELM. In *STRCON, user defines stresstypes and limiting values associated with them. In *STRELM, user defineselements where stress constraints are imposed on. Symbolically, *STRCON is like*RCTABLE in NISA where user defines an ID for real constant table related tophysical properties. The ID is referred to in several elements having same physicalproperties in *ELEMENTS data group. In *STRCON, we define an ID for stress

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types/limiting values and then elements holding same stress type and allowablelimit referred to the ID number. Note that imposing unnecessary stress constraintsor defining too many stress constraints can slow down process of structuraloptimization in STROPT. Because more constraints form smaller feasible region,this restrains the program from progressing towards optimum solution. Let’s take avery simple example. Suppose we want to constrain all six stress components viz.Sxx, Syy, Szz, Sxy, Sxz, Syz, at every point of the shell element shown inFigure D.1. That means for this simple problem there are 24 i.e. (6*4) stressconstraints for each element. There are 4 such elements in this model , thus, thereare 96 i.e. (4*24) stress constraints yielding significant number of constraints forsuch a small model. Question: how can one reduce number of stress constraints?.The Answer: number of constraints can be reduced significantly by imposing vonMises stress constraints at the centroid of each of 4 elements instead of stresscomponents. Thus, there are a total of 4 stress constraints. Note that less number ofconstraints implies less run time and faster solution.

9. Experiences with Natural Frequency Constraints

The vibration of structural and mechanical systems is an important factor inpractical design environment. Sometimes requirements are such that naturalfrequencies should not lie within a specified range as shown in Figure D.2

Fig D.2 Desirable ranges in natural frequencies.

STROPT can be employed to obtain optimum structural design based on the user’sdesirable ranges. User can specify lower bound or upper bound of naturalfrequency depending on design applications. Since STROPT is totally incorporatedwithin NISA, care should be taken. If in *EIGCNTL data group certain frequenciesare not calculated, then one cannot impose natural frequency constraints for thosefrequencies. STROPT detects the inconsistencies, issues an error message andstops.

10. Experiences with Buckling Load Factor Constraints

STROPT supports buckling load factor constraints for only one load case and canonly be imposed on the lowest buckling load factor computed by NISA II.

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Buckling load factor can be imposed together with displacement and/or stressconstraints but not with natural frequency and amplitude constraints. In case ofbuckling load factor constraint together with displacement and/or stressconstraints, extraction of eigenvalues must be done by using ConventionalSubspace Method.

11. Experiences with Laminated Composite and Sandwich Shell Elements

STROPT supports 3-D laminated composite (NKTP=32) and 3-D sandwich(NKTP=33) elements. Associated design variables are layer thicknesses and layerorientations. Optimum Cost Bounding (OCB) algorithm is not recommended iflayer orientations are the only design variables. Thus, the algorithm can createsignificant obstacles if employed. OCB algorithm is based on the gradient of thecost function. If angles are the only design variables, then volume/mass/weight arenot functions of the design variables, thus cost-gradient will have zero-components. In such a situation, the problem should be understood as the one withreduced number of design variables.

In general, response constraints such as displacement/stress are more sensitive tolayer thicknesses than to layer orientations. Thus, if both thicknesses andorientations are design variables, user may observe larger relative changes in layerthicknesses than relative changes in layer orientations angles. This problem can beresolved by assigning different DELs (design perturbation for gradient calculationof functions by finite difference method) given in *OPTPAR data group. Presently,more than one DEL cannot be defined, though this will be available in the nearfuture to circumvent the difficulties.

12. Experiences with Printing and Plotting Design Sensitivity Coefficients

In any optimization process there are three major steps, so-called three-tree. Theyare: structural analysis, design sensitivity analysis, and optimization algorithms.Design sensitivity coefficients are very important and may be utilized as trends indesign improvement. Discussions regarding key points are already given inChapter 2 and Appendix B. The coefficients vs. design variables can be plottedwithin DISPLAY III for all active (important constraints) in one unique graph.Examining signs of the coefficients for different active constraints graphically can

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enhance users’ knowledge for design improvement. Numerical values can beprinted in STROPT output if desirable.

Let’s take same example problem shown in Figure D.1. The objective is tominimize weight of the model subjected to von Mises stress constraints at centroidof each element and displacement constraint in z-direction at node 5. The structurehas 4 stress constraints and 1 displacement constraint. Also, assume each elementhas different thickness implying total of 4 design variables. After STROPT run, useDISPLAY III to plot sensitivity coefficients of 5 constraints vs. all 4 designvariables. Suppose, the graph obtained resembles Figure D.3. The figure revealsseveral facts about the structure and associated constraints. For example,sensitivity coefficients of all 5 constraints are negative with respect to designvariable number 4 i.e. thickness of element number 4. Design variables 1 and 3shows positive and negative signs and design variable number 2 shows smallernegative number in comparison with design variable number 4. Thus, to correct theconstraints in the model, increase thickness of design variable number 4 (seetheoretical overview given in section 2.5.2). Note that design variable number 4 inconjunction with other design changes may give better constraint corrections. Theexample and discussion presented here is solely to show the application of designsensitivity coefficients in practical design environment as a design approach

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.

Fig D.3 Sensitivity coefficients of stress and displacement constraints vs. global design variable number

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Appendix

E
PCG Iterative Solver in STROPT

STROPTS incorporates an iterative solver. It is an alternative equation solverto frontal solver. The PCG solver can solve a positive definite matrix with lessdisk space requirement and is up to 200 times faster in solution time than thefrontal solver.

Introduction

1. The iterative solver in STROPT is a PCG (Preconditioned Conjugate Gradi-ent) solver.

2. The PCG solver pre-conditions the global matrix instead of decomposing theglobal matrix, and then iterates the solution based on the pre-conditioned glo-bal matrix until the solution converges to a certain accuracy.

The pre-conditioning process takes very little time compared to the PCG iter-ation process and is based on the underlying physics of the discretization of acontinuous problem, geometry of the elements, and characteristics of the dif-ferent type of elements.

3. The PCG solver STROPT is applicable only to linear static problems.

4. The PCG iterative solver has been incorporated into the STROPT executable.There is no separate entity for the iterative solver, and the STROPT execut-able contains both the solvers, frontal and iterative. The default option isdirect (frontal) solver.

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5. The PCG solver is available on all major computer systems ranging from PCto supercomputers.

Iterative Solver in Structural Analysis vs. Structural Optimization

The iterative solver can be more efficient for large scale problems in linear staticanalysis. Direct (frontal) solvers are very efficient for small to medium sizedproblem. Please refer to the document in NISA user manual "PCG Iterative Solverin NISA II" for further details.

In STROPT, the above discussion may or may not be valid and depends on severalother factors such as number of load cases, same or different boundary conditionfor different load cases, number of design variables in the model, number ofconstraints and/or number of active constraints, selection of optimizationalgorithm, method of design sensitivity analysis. In structural design optimization,each of the above factors plays an important role in the section of iterative vs.frontal solver.

The main idea behind using iterative vs. direct is to minimize solution time. AsFigure 4.4-1 shows structural optimization splits up into two major stages. Stage Iis structural analysis and stage II is design sensitivity analysis. In the 2nd stage,iterative vs. frontal would play a major role from CPU time consumption point ofview.

Choosing direct differentiation method as the design sensitivity analysis method(MDS=2 in *OPTPAR data group) and frontal solver as the solution technique, thechange of deformation with respect to design variables are solved for all designvariables at once. That is for every load case, STROPT solves equations with multiright hand sides simultaneously. In case of iterative solver, change of deformationwith respect to design variables are carried out for one design variable at a time.Which indicates that a structural model with several design variables can betremendously time consuming in stage II. On the contrary, in frontal decomposedstiffness matrix is already available from analysis stage and it is only a matter ofdecomposition of the new right hand side i.e. pseudo load vectors and backsubstitution. In iterative case the strategy is totally different. Because not onlydecomposed structural matrix (through not needed) is not available, solutionshould start from scratch for every design variable and each load case as well.

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Following table summarizes the key differences between iterative vs. frontal inSTROPT.

PCG Iterative Solver Direct Solver

1. Needs pre-conditioned global matrix in DSA i.e. stage II again.

1. Decomposed global stiffness matrix is available for DSA in stage II from structural analysis i.e. stage I.

2. For each design variable solution should start all over in DSA stage using DDM.

2. For all design variables solution starts by decomposing right hand side of equation in DSA stage using DDM.

3. For each additional load case item No. 2 repeats itself using DDM.

3. For each additional load case item No.2 repeats itself using DDM.

4. In DSA using AVT, each active constraint repeat item No. 1 which can be very time consuming

4. In DSA using AVT, each active constraint repeat item No. 1 which is not time consuming.

5. GRG and RQP involves line search techniques which in turn involves stage I. CPU time for large scale may be less than frontal solver for each solution.

5. GRG and RQP involves line search techniques which in turn involves stage I. CPU time for large scale may be much more than iterative solver for each solu-tion.

Nomenclature:

DSA: Design Sensitivity Analysis

DDM: Direct Differentiation Method

AVT: Adjoint Variable Technique

Stage I: Structural Analysis Stage

Stage II: Design Sensitivity Analysis Stage

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Important Restrictions and Application Notes

1. User should have access right to iterative solver otherwise STROPT switchesto frontal solver automatically with proper warning messages.

2. Adjoint variable technique is not available for iterative solver with Version7.0. If selected, STROPT switches to direct differentiation automatically.

3. Imposing natural frequency, amplitude and buckling load factor constraints arenot supported with iterative solver and STROPT switches to frontal techniqueautomatically.

4. In general very large scale problems with much less number of design vari-ables (1 or 2) in the model, iterative solver may be faster than frontal solver.More design variables are certainly a big no with iterative solver and can beextremely time consuming. Also, user not only should have enough knowl-edge about the size of the model, but also No. of design variables, number ofload cases and associated B.C. for each load case.

5. Every restriction and limitation in NISA II using iterative, are apply inSTROPT as well. Please refer to NISA II user manuals.

6. Memory and disk space requirements in STROPT follows NISA II guidelinesfor iterative solver and frontal solver.

Usage

A executive card, SOLV, should be entered into the STROPT input file.

SOLVER =

where,

FRON : Use wavefront solver. (This is the default option).ITER : Use iterative solverTOLE : convergence tolerance for iterative solver (default = 1.e-6)

FRON{ }

ITER, TOLE, NITR, CORE

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NITR : Maximum number of allowable iterations for a iterative solution.(default value = 2500)

MPIVT :

===

Options to modify a bad pivot element matrix to avoid negative pivot in linear analysis since PCG iterative solver can only solve a positive definite global matrix.0 no modification (default)1 modify negative pivot to a small positive number2 change negative to a positive number

CORE :==

Activate the out-of-core scheme instead of the in-core scheme.IN use in-core scheme for iteration (default)OUT use out-of-core scheme for iteration

Note:"iteration" in the context of iterative solver (NITR) has a totally different meaning than maximum number of iterations (ITRS) in *OPTPAR data group. ITRS points out to the maximum number of allowable iterations within structural optimization loop and NITR refers to the maximum number of iterations within each solution. here "Solution" refers to structural analysis or structural design sensitivity analysis.

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Example

Let’s select example problem number A.1 from Appendix A. The problem is toacquire optimum design of a deep beam. Following shows STROPT input fileusing frontal solver. Please note that SOLV=FRON is not specified because defaultis wavefront solver.

PROBLEM=EXAM01ANALYSIS=STATICLOAD = 1**design=sensAUTO = ONMAXWAVE = 750WARN = GO**TOTAL NO. ELEMENTS = 16**TOTAL NO. OF NODES = 81*TITLEOPTIMUM DESIGN OF DEEP BEAM*ELTYPE1,1,2*RCTABLE1, 81.0///////*E1$$,-4,18,4 1,1,2,3,12,21,20,19,10,1//4,2,11,1,2,3,12,21,20,19,10,1//4,2,1*NODES-9,0,9,9,0.0,1.251,0//,0.0/*MATERIALEX,1,0,300000NUXX,1,0,0.3*LDCASE0, 0, 3, 0, 0, 0, 1, .0000E+00, .0000E+00*SPDISP9,UX,0.0,81,91,UY,0.0

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*CFORCE73,FY,-0.625,7374,FY,-125,80,181,FY,-0625,81*PRINTELST,0sest,0*OPTPAR1,-1,20,3,20.1,0.01,0.01,0.001*DVGROUP1,16,1,1,1*DVGELMT1,1,0,0,16,1,1*DVLOCAL1,16,11,1.0,.01,3.0*STRCON1,VON,30.0*STRELM1,1,91,16,1*ENDDATATo change solution scheme to ITERATIVE SOLVER just add the line SOLV=ITER anywhere between executive cards i.e. somewhere after ANALY-SIS=STATIC and before *TITLE data group. Following is the only change needed:PROBLEM=EXAM01ANALYSIS=STATICLOAD = 1**design=sensSOLV=ITER THIS IS THE ONLY CHANGEAUTO = ONMAXWAVE = 750WARN = GO**TOTAL NO. OF ELEMENTS=16**TOTAL NO. OF NODES=81

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**STROPT Example Problem #01*TITLEOPTIMUM DESIGN OF DEEP BEAM*ELTYPE1,1,2*RCTABLE1, 81.0///////*E1$$,-4,18,4 1,1,2,3,12,21,20,19,10,1//4,2,1*NODES-9,0,9,9,0.0,1.251,0//,0.0/9,0,1,0,10.0,0.0*MATERIALEX,1,0,30000.NUXY,1,0,0.3*LDCASE0, 0, 3, 0, 0, 0, 1, .0000E+00, .0000E+00*SPDISP9,UX,0.0,81,91,UY,0.0*CFORCE73,FY,-0.625,7374,FY,-1.25,80,181,FY,-0.625,81*PRINTELST,0sest,0*OPTPAR1,-1,20,3,20.1,0.01,0.01,0.01,0.001*DVGROUP1,16,1,1,1*DVGELMT1,1,0,0,16,1,1*DVLOCAL

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1,16,11,1.0,.01,3.0*STRCON1,VON,30.0*STRELM1,1,91,16,1*ENDDATA

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