prof.sudi july27 apss2010

8/6/2019 Prof.sudi July27 APSS2010

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Fundamentals of FiniteElement Analysis

Di SuResearch assistant professorBridge & Structure Laboratory

Department of Civil EngineeringThe University of Tokyo

2010 Asia 2010 Asia - - Pacific Summer School in Smart Structures Technology Pacific Summer School in Smart Structures Technology July 27, 2010 July 27, 2010

Outline

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Discussion of Some Key Problems

Application of FEA

Bar Element and Beam Element

FEA Concept

Introduction and History

Introduction and History

About this short courseFundamentals of finite element analysisNo a textbook of FEA, no tensor, no Galerkin methodOnly focus in Civil EngineeringRealized by Matlab, Abaqus and Ansys

Try to study FEA byMathematical principle + Analysis modeling + Software application

Try to use FEA bySoftware + Practical problem + Self development

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FEA & Structure

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Finite ElementFinite ElementMethodMethodDemolish

Maintenance

Construction

Design

Structure


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FEA & Structure

Beijing National Stadium (40,000 tons)

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Beijing National Stadium

Mode shape

6From Herzog and de Meuron, Arup, CAG .


Failure verification



Truss column design



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Construction process



Construction process


Finite Element Method Defined

Complexities in the geometry, properties and in the boundaryconditions that are seen in most real-world problems usuallymeans that an exact solution cannot be obtained or obtained ina reasonable amount of time.

Engineers are content to obtain approximate solutions that canbe readily obtained in a reasonable time frame, and withreasonable effort. The FEM is one such approximate solutiontechnique.The FEM is a numerical procedure for obtaining approximatesolutions to many of the problems encountered in engineeringanalysis.

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Discretized approximation

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Rayleigh-Ritz principle Approximation in the wholedomain Higher-order continuousfunction Fewer base functions

Another method Pieces functionapproximation in sub-domain Linear or polynomial function More base functions

Basic idea of FEM

Describe one complexfunction


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Finite Element Method Definition

The continuum has an infinite number of degrees-of-freedom(DOF), while the discretized model has a finite number of DOF. This is the origin of the name, finite element method.

The number of equations is usually rather large for most real-world applications of the FEM, and requires the computationalpower of the digital computer. The FEM has little practicalvalue if the digital computer were not available .

Solution of FEM gives the approximate behavior of thecontinuum or system.

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The concept of FINITE

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Finite NumberThere is only finitenumber of elementsin your analysis

model, not infinite.

FINITE

Finite AccuracyThe accuracy of your

analysis is finite. Evenfor very fine model, it

is not accuratesolution.

History of FEM

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Engineering Mathematics

Trial function Finite differencemethodVariational

methodSimilar structure

replacement

Method of Weighted

Residuals

Continuous trialfunction

Direct continuumelements

Variable finitedifference method

Present FiniteElement Method

Rayleigh 1870Ritz 1909 Gauss 1795Galerkin 1915

Biezeno-Koch 1923

Richardson 1910Liebman 1918Southwell 1946

Hrenikoff 1941Mchenry 1943Newmark 1949

Courant 1943Prager-Synge 1947Zienkiewicz 1964

Argyris 1955Turner et al. 1956

Varga 1962

First coined by Clough 1960

History of FEM

It is difficult to document the exact origin of the FEM, because the basicconcepts have evolved over a period of 150 or more years. The first book on the FEM by Zienkiewicz and Chung was published in 1967.

Most commercial FEM software packages originated in the 1970s and1980s.

The FEM is one of the most important developments in computationalmethods to occur in the 20th century. Advances in and ready availability of computers and software has brought the FEM within reach of engineersworking in small industries, and even students.

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FEA Concept

Example1: One dimension problem

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Try to solve this problem?

FEA Concept

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FEA Concept

Use u A , u B , u C as unknowns

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FEA Concept

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FEA Concept

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Load matrix Inner forcematrix

Nodaldisplacement

The equilibrium equation for whole structure, not for each component

Lets derive more

FEA Concept

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More general form

The equilibrium for each node has turned into the relationship of each component. This component description is generalized and standard; i.e. ELEMENT . In this example, it is Bar Element.

General description of 1D bar element

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Nodal displacement

External force

Inner force

Equilibrium equation

Stiffness matrix

Application of bar element

Could you solve this three-link structure problem usingthe bar element you just learned?

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P3=50N

Example2:


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FEM Solution process

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Element 1 Element 2 Element 3

Assembly

Stiffness matrix

Nodal force

Boundary condition

FEM Solution process

Solve the linear equations

Derived other parameters

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Very standard, very simple solution, right?

Analysis modeling process

1D model 272D model 3D model

FEM Solution

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Step 1: Discretization Step 2: Stiffness matrix foreach element

Step 3: Assembly Step 4: Solution (nodal disp.)Step5: Other parameters (strain,

stress, et al. )

Simple element

Complex structure


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Element type in FEM software

Abaqus

Ansys

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Lets discuss the process more generally.

It will be very difficult to derive the stiffness matrix ofelement by the mechanical equations in most cases. Inthis section two general methods will be introduced toobtain the basic equation for bar element and beamelement.

Principle of virtual work

Principle of minimum potential energy

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Bar Element

The basic parameters in x axisDisplacement: u( x)Strain: x( x)Stress: x( x)

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Example 3: 1D problem

Bar Element

Basic equation of 1D problemEquilibrium equation or (c1 is constant)

Geometric equation

Physical equation

Boundary condition

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How to solve?1.Direct solution: 3 unknownsfor 3 equations2.Indirect solution: Trialfunction?


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For this equilibrium system

If a small disturbance happens,but still remains equilibrium

Principle of virtual work based on the virtual displacement

When a rigid body that is in equilibrium is subject to virtual compatible displacements, the total virtual work of all external forces is zero.

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Virtual displacement

--Johann (Jean) Bernoulli (1667-1748) and Daniel Bernoulli (1700-1782)


Principle of virtual work for a deformable body

If U is virtual strain energy, and W is the virtual work byexternal force

External virtual work is equal to internal virtual strain energy when equilibrated forces and stresses undergo

unrelated but consistent displacements and strains .

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Application of principle of virtual work

Assume the displacement field as

(Trial function, c is unknown)The strain, virtual displacement, and virtual strain is

The virtual work and virtual strain energy

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Application of principle of virtual work

From the principle of virtual work

Final solution

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Principle of minimum potential energy

It asserts that a structure or body shall deform or displace to a position that minimizes the total potential energy.Assume the displacement field as u( x)

Potential energy(U is the strain energy, W is the external work)

For bar element

The true displacement field should satisfy

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Application of principle of minimum potential energy

Again,

Potential energy

From the minimum value

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Bar element

Description of one elementGeometrics and node descriptionDisplacement field (Trial function)Strain field

Stress fieldPotential energy

Obtain the stiffness equation of element by principle ofvirtual work or principle of minimum potential energy

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Bar element in local coordinate system

Geometrics and node description

Nodal displacementNodal forceDisplacement field

Assume the linear functionFrom the nodal displacement

Then

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Shape function

matrix

Nodaldisplacement

vector


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Strain field

Stress field

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Strain-displacementmatrix

Stress-displacementmatrix


Potential energy

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Stiffness equation of bar element

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Stiffness matrix of element

Nodal force vector

Bar element in global coordinate system

Local coordinate system

Global coordinate system

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Transformation matrix


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Bar element in global coordinate system

Potential energy

Stiffness equation for global coordinate system

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Bar element in space

Transformation matrix

Stiffness equation for bar element in space

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Bar Element in MATLAB

MATLAB program for 1D bar element

Bar1D2Node _Stiffness(E,A,L)Calculate the stiffness matrix k(2 2)

Bar1D2Node _Assembly(KK,k,i,j)Assemble the stiffness matrix

Bar1D2Node _Stress(k,u,A)Calculate the stress of element

Bar1D2Node_Force(k,u)Calculate the nodal force vector

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All the codes can be downloaded inhttp://www.bridge.t.u-tokyo.ac.jp/apss/downloads/FEM%20code.zip

Bar Element in MATLAB

MATLAB program for 2D bar element

Bar2D2Node _Stiffness(E,A,x1,y1,x2,y2,alpha)Calculate the stiffness matrix k(4 4)

Bar2D2Node _Assembly(KK,k,i,j)Assemble the stiffness matrix

Bar2D2Node _Stress(E,x1,y1,x2,y2,alpha,u)Calculate the stress of element

Bar2D2Node_Force(E,A,x1,y1,x2,y2,alpha,u)Calculate the nodal force vector

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Example 4: Four-bar truss structure

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Stiffness matrix for each element

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Assemble to whole stiffness equation

Boundary conditions

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Results

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Compare with the results from MATLAB, ANSYS and ABAQUS


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Basic equation of beam element

Equilibrium equation

Geometric equation

Physical equation

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Basic equation of beam element

Choose deflection v as the fundamental unknown

Boundary conditions

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Equilibrium in y

Equilibrium in x

Physical

Geometric

Beam element

Description of one elementGeometrics and node descriptionDisplacement field (Trial function)Strain field

Stress fieldPotential energy

Obtain the stiffness equation of element by principle ofvirtual work or principle of minimum potential energy

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Beam element in plane in local coordinate system

Geometrics and node description

Nodal displacementNodal force

Displacement fieldAssume the polynomial functionFrom the nodal displacement

60Shape function matrix


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Strain field

Stress field

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Strain-displacement matrix

Stress-displacement matrix


Strain energy

External work

Stiffness equation 62

Stiffness matrix of element

Nodal force vector

General beam element in local coordinate system

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Bending beam + axial deformation

Nodal displacement

Nodal force

Stiffness equation of beam element

Equivalent nodal force

How to obtain the nodal force?

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Uniform loadDifferent BC



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Displacement field

External work


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Shape function

No relationship with BC, it is a universal expression for uniform load.


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Application of beam element

Example 6: Cantilever-continuous beam

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How to obtain structural responses?


Modeling using 2 beam elements

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69No need to solve the differential equations or partialdifferential equations, just linear equations

Plane beam element in global coordinate system

Need coordinate transferLocalGlobal

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Beam element in space

Local coordinate system

For u1 and u2 , the same with bar element

For x1 and x2, similar with bar element

For v1, v2, z1 and z2, the same with pure bending beam

For w1, w2, y1 and y2, similar as above equation71

Beam element in space

Stiffness matrix for beam element in space (localcoordinate system)

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Coordinate transfer in space

Transfer to global coordinate system

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Beam Element in MATLAB

MATLAB program for 1D beam element

Beam1D2Node_Stiffness(E,I,L)Calculate the stiffness matrix k(4 4)

Beam1D2Node _Assembly(KK,k,i,j)Assemble the stiffness matrix

Beam1D2Node_ Strain(x,L,y)Calculate the geometric matrix B(1 4)

Beam1D2Node _Stress(E,B,u)Calculate the stress of element

Beam1D2Node_Deflection(x,L,u)Calculate the deflection of element

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Beam Element in MATLAB

MATLAB program for 2D beam element

Beam2D2Node_Stiffness(E,I,A,L)Calculate the stiffness matrix k(6 6)

Beam2D2Node_Assemble(KK,k,i,j)Assemble the stiffness matrix

Beam2D2Node_Forces(k,u)Calculate the nodal force of element

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Example 7: One frame structure

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Modeling using 3 beam elements

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For element 1, stiffness matrix is

For element 2 and 3,

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Transfer matrix for element 2 and 3

Stiffness matrix for element 2 and 3 in global coordinatesystem

Assemble the whole stiffness matrix

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After considering the BC,

Final solution

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MATLAB Program

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ANSYS Program

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Discretization for continuum elements

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The real power of Finite Element method is that it successfullysolved the continuum problem.

Application of FEM

General-purpose FEM software packages are available atreasonable cost, and can be readily executed onmicrocomputers, including workstations and PCs.

The FEM can be coupled to CAD programs to facilitate solidmodeling and mesh generation.

Many FEM software packages feature GUI interfaces, auto-meshers, and sophisticated postprocessors and graphics tospeed the analysis and make pre and post-processing moreuser-friendly.

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Commercially available general FEM software

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Year Software Company Website1965 ASKA (PERMAS) IKOSS GmbH, (INTES),Germany www.intes.de

STRUDL MCAUTO, USA www.gtstrudl.gatech.edu1966 NASTRAN MacNeal-Schwendler Corp., USA www.macsch.com

1967 BERSAFE CEGB, UK (restr uctured in 1990)SAMCEF Univer. of Liege, Belgium www.samcef.com

1969 ASAS Atkins Res.&Devel., UK www.wsasoft.com

MARC MARC Anal. Corp., USA www.marc.com

PAFEC PAFEC Ltd, UK now SER Systems

SESAM DNV, Norway www.dnv.no1970 ANSYS Swanson Anal. Syst., USA www.ansys. com

SAP NISEE, Univ. of California, Berkeley, USA www.eerc.berkeley.edu

1971 STARDYNE Mech. Res. Inc., USA www.reiusa.comTITUS (SYSTUS) CITRA, France; ESI Group www.systus.com

1972 DIANA TNO, The Netherla nds www.diana.n l

WECAN Westinghouse R&D, USA1973 GIFTS CASA/GIFTS Inc., USA

1975 ADINA ADINA R&D, Inc., USA www.adina.c omCASTEM CEA, France www.castem.org:8001/ HomePage.html

FEAP NISEE, Univ. of California, Berkeley, USA www.eerc.berkeley.edu1976 NISA Eng. Mech. Res. Corp., USA www.emrc .com1978 DYNA2D, DYNA3D Livermore Softw. Tech. Corp., USA www.lstc.com

1979 ABAQUS Hibbit, Karlsson & Sorensen, Inc., USA www.abaqus.c om1980 LUSAS FEA Ltd., UK www.lusas. com1982 COSMOS/M Structural Res. & Anal. Corp., USA www.cosmosm.com

1984 ALGOR Algor Inc., USA www.algor. com

Information Available from Various Types of FEM Analysis

Static analysis Deflection Stresses Strains Forces Energies

Dynamic analysis Frequencies Deflection (mode

shape) Stresses Strains Forces Energies

Heat transfer analysisTemperature

Heat fluxes

Thermal gradients

Heat flow fromconvection faces

Fluid analysis

Pressures

Gas temperatures

Convection coefficients Velocities

Example FEM Application AreasAutomotive industry

Static analyses Modal analyses Transient dynamics Heat transfer

Mechanisms Fracture mechanics Metal forming Crashworthiness

Aerospace industry

Static analyses

Modal analyses

Aerodynamics

Transient dynamics Heat transfer

Fracture mechanics

Creep and plasticity analyses

Composite materials Aeroelasticity

Metal forming

Crashworthiness

Architectural

Soil mechanics

Rock mechanics

Hydraulics

Fracture mechanics

Hydroelasticity

Variety of FEM Solutions is Wide and Growing Wider

The FEM has been applied to a richly diverse array of scientificand technological problems.

The next few slides present some examples of the FEM appliedto a variety of real-world design and analysis problems.


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89 90

Several examples

91From Mr. M., Chingthaka and Dr. Pellegrino, S. @Caltech

Joint expansion of aerospace structure

Sever examples

92From Jaesung Eom. et al @ Rensselaer Polytechnic Institute

Lung cancer analysis


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Several examples

Balloon inflation

From Mr. XW. Deng @Caltech

Several examples

Heat transfer analysis

94From Mr. Hida@ the University of Tokyo

Several example

Electromagnetic analysis

95From Mr. Mizutani@ the University of Tokyo

Classification of Solid-Mechanics Problems

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Analysis of solids

Static Dynamics

Behavior of Solids

Linear Nonlinear

Material

FractureGeometric

Large Displacement

Instability

Plasticity

ViscoplasticityGeometric

Classification of solids

Skeletal Systems1D Elements

Plates and Shells2D Elements

Solid Blocks3D Elements

TrussesCablesPipes

Plane StressPlane StrainAxisymmetricPlate BendingShells with flat elementsShells with curved elements

Brick ElementsTetrahedral ElementsGeneral Elements

Elementary Advanced

Stress Stiffening


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Application of FEM

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Example 8: Elastic-plastic analysis

Application of FEM

Example 9: Multibody system

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How can the FEM Help the Design Engineer?

The FEM offers many important advantages to the design engineer :

Easily applied to complex, irregular-shaped objects composed of several different materials and having complex boundary conditions.

Applicable to steady-state, time dependent and eigenvalueproblems.

Applicable to linear and nonlinear problems.

One method can solve a wide variety of problems, includingproblems in solid mechanics, fluid mechanics, chemical reactions,electromagnetics, biomechanics, heat transfer and acoustics, to namea few.

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How can the FEM Help the Design Organization?

Simulation using the FEM also offers important business advantages tothe design organization :

Reduced testing and redesign costs thereby shortening the productdevelopment time.

Identify issues in designs before tooling is committed.

Refine components before dependencies to other componentsprohibit changes.

Optimize performance before prototyping.

Discover design problems before litigation.

Allow more time for designers to use engineering judgment, and less

time turning the crank.


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Disadvantages of the Finite Element Method

Numerical problems: Computers only carry a finite number of significant digits. Round off and error accumulation. Can help the situation by not attaching stiff (small) elements

to flexible (large) elements.Susceptible to user-introduced modeling errors:

Poor choice of element types. Distorted elements. Geometry not adequately modeled.

Certain effects not automatically included: Buckling Large deflections and rotations. Material nonlinearities . Other nonlinearities.

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Sources of Error in the FEM

The three main sources of error in a typical FEM solution arediscretization errors, formulation errors and numerical errors.

Discretization error results from transforming the physical system(continuum) into a finite element model, and can be related tomodeling the boundary shape, the boundary conditions, etc.

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Formulation error results from the use of elements that don't precisely describe thebehavior of the physical problem.Elements which are used to model physical problems for which they are not suited aresometimes referred to as ill-conditioned or mathematically unsuitable elements.For example a particular finite element might be formulated on the assumption thatdisplacements vary in a linear manner over the domain. Such an element will produceno formulation error when it is used to model a linearly varying physical problem (linearvarying displacement field in this example), but would create a significant formulationerror if it used to represent a quadratic or cubic varying displacement field.

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Numerical error occurs as a result of numericalcalculation procedures, and includes truncation errors andround off errors.

Numerical error is therefore a problem mainly concerningthe FEM vendors and developers.

The user can also contribute to the numerical accuracy,for example, by specifying a physical quantity, sayYoungs modulus, E, to an inadequate number of decimalplaces.

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Stiffening and lower bound

The finite element method (FEM) provides a lower boundin energy norm for the exact solution, i.e., theapproximation solution (displacement field) from FEM issmaller than actual case.

This is simply explained like this. FEM uses a finitenumber of DOF to describe the continuum which has aninfinite number of DOF. This will made the stiffness ofsystem increase (stiffening), therefore, displacement willbecome small for the same external force.

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High-order element

Using a different set of shape functions of high-order polynomials willexpect to reduce the computational effort and increase the accuracyof the results. It can provide that an increase of polynomial degree iscombined with a proper mesh design.

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2 nodes, linear function

3 nodes, quadratic function 4 nodes, cubic function

h-method vs p-method

h-methodThe basis functions for each finite element can be refined and thediameter of the largest element, hmax , allowed to approach zero. Thismode is called h-convergence and its computer implementation theh-version or h-method of the finite element method.

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Defined in Ivo Babuska, Barna Szabo, On the rates of convergence of the finite element method, International Journal for Numerical Methods in Engineering, 18(3):323-341, 2005 .


p-methodThe finite element mesh can be refined and the minimal order of (polynomial) basis functions, pmin , allowed to approach infinity.This mode is called p-convergence and its computer implementationthe p-version or p-method of the finite element method.

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Defined in Ivo Babuska, Barna Szabo, On the rates of convergence of the finite element method, International Journal for Numerical Methods in Engineering, 18(3):323-341, 2005 .


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Which method is better? No conclusionIn the p-version of the finite element method the rate of convergence cannot be slower than in the h-version.Numerical oscillation problem would happen for p-version of thefinite element method.For obvious practical reasons, finite element analyses should beboth efficient and reliable.

My personal viewFor structural analysis, h-method is more popular. Two-orderelement is a good application considering the efficiency andreliability.p-method seems to act against the original goal of FEM.

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Really? From p-version FEM software Stresscheck

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Up to 8-order element???

From http://www.ada.co.jp/products/StressCheck/sc_pfem.html

General software vs specific software

My personal view:It is very important to make the FEM program by oneselfwhen studying the FEM.

For normal use of FEA, general software is morerecommendable. The current software has been well-developed and ready to handle all the problems

Even for very specific problems, plenty of user-definedsubroutines can be used.

Comparing with maintenance of one whole analysisprogram, just to maintain one specific part of the programwill be more focused and efficient. 115

General software vs specific software

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User Subroutine in ABAQUS


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The FEM in particular, and simulation in general, are becomingintegrated with the entire product development process (rather than

just another task in the product development process).

A broader range of people are using the FEM.

Increased data sharing between analysis data sources (CAD, testing,FEM software, ERM software.)

FEM software is becoming easier to use:Improved GUIs, automeshers.Increased use of sophisticated shellscripts and wizards.

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Future Trends in the FEM and Simulation

Enhanced multiphysics capabilities are coming:Coupling between numerous physical phenomena.

Ex: Fluid-structural interaction is the most common example.

Increasing use of non-deterministic analysis and design methods:Statistical modeling of material properties, tolerances, and anticipated loads.Sensitivity analyses.

Faster and more powerful computer hardware. Massively parallel processing. Ex: ADVENTURE PROJECT @ the University of Tokyo.

Decreasing reliance on testing.

FEM and simulation software available freely. Ex: OpenSees @ University of California, Berkeley .

Ex: ADVENTURE PROJECT @ the University of Tokyo.118

Future Trends in the FEM and Simulation

Suggested reference

Chandrupatla, T. R. and Ashok D. Belegundu, 1997. Introduction to Finite Elementsin Engineering , Prentice Hall, Upper Saddle River, New Jersey.Kardestuncer, H., 1987. Finite Element Handbook , McGraw-Hill, New York.Segerlind, L. J., 1984. Applied Finite Element Analysis , John Wiley and Sons, New

York.Chandrupatla, Tirupathi R., 2002 . Introduction to finite elements in engineering ,Prentice Hall, Third Edition.R2. O. C. Zienkiewicz, R. L. Taylor and J. Z. Zhu, 2005. The Finite Element Method:

Its Basis and Fundamentals , Elsevier Butterworth Heinemann, Sixth Edition.Pan Zeng, 2008. Fundamentals of Finite Element Analysis , Tsinghua University.

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prof.sudi july27 apss2010

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