shape optimization of mechanical systems in comsol 4.4

B.Sc.Eng. ThesisBachelor of Science in Engineering

Shape optimization of mechanical systems inCOMSOL Multiphysics 4.4

Thomas Agger (s113357)

Kongens Lyngby 2014

DTU Mechanical EngineeringDepartment of Mechanical EngineeringTechnical University of Denmark

Nils Koppels AlléBuilding 4042800 Kongens Lyngby, DenmarkPhone +45 4525 [email protected]

AbstractThis paper investigates parametric shape optimization in COMSOLMultiphysics. The strengthsand weaknesses of the parametric and non-parametric methods are discussed. Furthermore,three classic shape optimization problems are parameterized and analyzed with the use ofCOMSOL. The models investigated are a beam, a hole in a plate under biaxial stress and afillet on a bar.The results of these optimizations are compared to analytical work as well as numerical workfrom other shape optimization methods. Furthermore the difficulties regarding shape optimiza-tion in COMSOL are discussed.

PrefaceThis bachelor’s thesis was prepared at the Department of Mechanical Engineering at the Tech-nical University of Denmark in fulfillment of the requirements for acquiring a B.Sc.Eng. degreein Mechanical Engineering. The project was realized in the period between the 4th Februaryand 24th June 2014 with the supervision of Ole Sigmund and Niels Aage. The project iscredited 20 ECTS points.

Kongens Lyngby, December 2, 2014

Thomas Agger (s113357)

Acknowledgements

First and foremost I would like to thank my two supervisors, Professor Ole Sigmund andResearcher Niels Aage, for their support and help throughout my thesis.Furthermore I would like to thank Rune Westin and Thure Ralfs from COMSOL for theirsupport when COMSOL was misbehaving.

Contents

Abstract i

Preface iii

Acknowledgements v

Contents vii

1 Introduction 1

2 Theory 32.1 Shape optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Parameter free shape optimization . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Parametric shape optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Optimizing the shape of a cantilever beam 73.1 Model definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Modeling instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Optimization of a plate with a hole 234.1 Model definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3 Modeling instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Optimizing the shape of a fillet 435.1 Model definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 Modeling instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6 Conclusion 596.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Appendix A Cantilever Beam 61A.1 Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61A.2 One summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64A.3 Two summations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68A.4 Three summations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72A.5 Four summations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76A.6 Five summations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

viii Contents

Appendix B Plate with a hole 87B.1 Tensile ratio of 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87B.2 Tensile ratio of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92B.3 Tensile ratio of 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Appendix C Fillet 105C.1 Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105C.2 One summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107C.3 Two summations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112C.4 Three summations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117C.5 Four summations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122C.6 Five summations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Bibliography 133

Abbreviations and Nomenclature

Abbrevation MeaningσvM von Mises StressDisp. DisplacementC Stress optical coefficient, compli-

anceU Elastic strain energyFE Finite element

CHAPTER 1Introduction

The 27th of November 2013 COMSOL released a new version of their FEM software. WithCOMSOL Multiphysics 4.4 they offered new opportunities for doing shape and topology opti-mization. With version 4.4 they introduced two new optimization solvers: BOBYQA, whichis a gradient-free optimization solver, and MMA, which is gradient-based; thus giving newabilities for optimization solving.

For a long time COMSOL have had great capabilities for doing topology optimization, buthas been lacking in the area of shape optimization. As of right now is are only one model inCOMSOL’s model library that shows how to do shape optimization, but this model is verylimited in its application to more complex problems as well as mechanical problems. The modelshows how the sound pressure from a horn can be optimized for a given angle by parameter-izing one boundary. However, there is no good documentation on geometric parameterizationin COMSOL and the complexity rises when more than one boundary has to be parameterized.Therefore the main goal of this project has been to explore the strengths and weaknesses forshape optimization, map the capabilities for doing shape optimization and apply these capa-bilities to mechanical problems in COMSOL.

The first focus of the project is to optimize the shape of a cantilever beam. A simple problemthat is much more troublesome than anticipated. The first real obstacle is to figure out how toconfigure the parameterizations using prescribed mesh displacement when having more thanone boundary. Another obstacle is figuring out the right scale factors in such a way thatinverted elements do not occur in the model.

The next focus of the project is to investigate a more advanced problem. A plate with ahole in the middle is to be investigated and benchmarked in accordance with former analyticalstudies .

The last focus of the project will be to look at a 3D model. The object will be to opti-mize the shape of a fillet on a 2D axisymmetric bar. The parameterization technique will besimilar to the one used in the two other models, but will be able to produce results in 3D.

2 1 Introduction

A question that might arise is ”why is it so interesting to do shape optimization in COMSOLwhen there is a lot of other software capable of doing the same thing?”The other shape optimization software around typically has a lot of limitations compared toCOMSOL. For instance, some of the software only has 2D capabilities, whereas COMSOLoffers both 2D and 3D shape optimization. COMSOL also offers built-in CAD tools. Thisenables the user to create 1D, 2D and 3D models directly in the software. If the user have anadvanced geometry it is also possible to import CAD models from separate CAD software.In addition to this COMSOL also offers very good options for topology optimization which alot of other software also lacks, which is a huge advantage seeing as it can be interesting todo both shape and topology optimization on the same model. Furthermore COMSOL has theability to combine multiple physics, hence the name COMSOL Multiphysics. This means thatwhile doing shape optimization on a mechanical structure it is also possible to see how otherphysics will affect the model, e.g. COMSOL is capable of doing FE calculations for acoustics,electrochemistry, heat transfer, and many other branches of physics.

CHAPTER 2Theory

2.1 Shape optimizationStructural shape optimization can be formulated with mathematical terms by the followingequation [1]:

minx∈Rn

f(x) with{xl ≤ x ≤ xug(x) ≤ 0

Here f(x) is the objective function, meaning the parameter sought to be either minimized ormaximized in the optimization. For mechanical problems it is common practice to minimize theobjective function for the total volume or the total strain energy of the model. By minimizingthe total strain energy, the compliance is minimized, thus increasing the stiffness. The vector,x, is made up of the design variables, xj , which control the size, shape and topology of themodel. These design variables are optimized to get the best structure within their constraints.The design variables’ constraints are defined by a lower and upper bound, xl and xu, respec-tively. Behavior inequality constraints, g(x), are typically used for applying limitations onstresses, deflections, and natural frequencies or to make sure the structure meets its demands.

Shape optimization is divided into two main groups, which deal with the change of the shapein different ways; parametric and non-parametric shape optimization. They both have theirstrengths and weaknesses which will be looked into in this chapter.

2.2 Parameter free shape optimizationThere are a lot of variations to the parameter free approach, but the general idea is that it usesFE-based data as design variables, for instance nodal coordinates, nodal thickness, elementthickness etc.The advantage of using the non-parametric approach is that it allows the greatest degree offreedom regarding the shape change; it is independent of the function describing the boundariesto be optimized. Another advantage is that the user completely avoids parameterizing theboundaries, which can be very time-consuming and demands thorough knowledge of geometricfunctions and linear combinations when optimizing complex structures.The main weakness of this method is its lack of a length scale control [2]. Without a lengthscale control the method doesn’t produce any meaningful results seeing as the boundariesbecome jagged, see Figure 2.1. These jagged boundaries are neither optimal nor desirable asthey result in immense stress concentrations that are very sensitive to the local shape change.Furthermore the shape seen in Figure 2.1(a) isn’t a realistic shape for manufacturing.

4 2 Theory

(a) The jagged boundaries when length scalecontrol is not applied

(b) When using the length scale control thejagged boundaries are smoothed out

Figure 2.1: Illustrations borrowed from [2]

Another problem occurs when using the nodal coordinates as design variables. The designupdates can change the mesh of the structure without changing the shape of the structure [1,Fig. 1]. This makes the solution of the optimization problem non-unique seeing as a surfacecan be represented by an infinite number of finite element meshes [1].However, both of these problems can be solved using regularization. The problem with non-uniqueness can be solved by achieving a regular mesh without element distortions; to achievethis in-plane regularization is used. Laplace regularization is widely used for mesh regulariza-tion. The way that Laplace regularization works is by attempting to change the mesh in sucha way that the mesh will consist of square elements with the same area content. This is doneby iteratively moving each mesh node to the center of gravity for its adjacent nodes [1][3].The problem with the smoothness of the boundaries are solved with out-of-plane regularization(sensitivity filtering). The out-of-plane regularization controls the local curvature of the result,thus giving the boundary the smoothness required. The jagged boundaries occurs because ofthe discretization errors from the sensitivity fields. In short these sensitivity errors are filteredby modifying the response gradients. This is where the use of a length scale control comes in.

Despite its problems, parameter-free optimization still has its strengths. This approach isable to explore a large number of solutions with a considerable degree of freedom for changingthe model while greatly reducing the time required for modeling the structure.

2.3 Parametric shape optimization 5

2.3 Parametric shape optimizationIn this approach, instead of using the FE data as design variables, the boundaries of the originalgeometry is parameterized such that the shape of the structure can be changed by adjusting aset of parameters. To avoid long computational time it is preferred to describe the geometrywith as few parameters as possible, but with a parameterization that gives the geometry thegreatest degree of freedom for shape change. A number of different parameterization methodsexist but the basis is the same; describing the boundaries by means of geometric functions withparameters that can be tweaked. A few of the methods will be described below.

Discrete approachThis approach uses the coordinates of the boundary points as design variables, see Figure 2.2.This approach gives a great degree of freedom seeing as it is only limited by the design variablesdescribing the structure. In addition to this it is also easy to implement.

Figure 2.2: An airfoil described using the boundary points. Illustration borrowed from [4]

The disadvantages of using this approach are that for complex structure a large number ofdesign variables are necessary to describe the geometry thus leading to high computationaltime, moreover the smoothness of the geometry is hard to obtain if using too few designvariables. The smoothness problem can be solved by using multi-point constraints [4] whichmakes it possible to ”connect” different nodes and degrees of freedom [5] and by adding dynamicadjustments of the upper and lower bounds.

Natural design approachA varitation of the discrete approach is the natural design approach. The natural approachuses an auxiliary structure similar to the original structure. Fictitious loads are then added tothe auxiliary structure and the mesh is updated by adding the auxiliary nodal displacementto the current nodal coordinates. These fictitious shape loads are chosen using nonlinearprogramming methods [6], e.g. Pedersen et. al [7] use eigenvectors from a fictitious modalanalysis to define orthogonal shape basis functions [2].

Polynomial and spline approachWith the use of polynomial and spline functions the number of design variables can be greatlyreduced compared to the discrete and natural design approach seeing as they can describe awhole boundary; this makes them particularly suitable for shape optimization. The optimiza-tion parameters are incorporated in the expressions, and therein there is no other need fordesign variable constraints than the ones set for the optimization parameters.One of the advantages of this approach is that a nice smooth boundary is obtained. The poly-nomial has a very compact form for describing geometries. Unfortunately it’s only good fordescribing simple curves seeing as it has a tendency for round-off errors when the coefficientshave too large variation.

6 2 Theory

The Bézier curves are based of on Bernstein polynomials [8] and are very similar, in a math-ematical point of view, to the polynomial form. A general form for a Bézier curve is givenby:

B(t) =n∑

i=0

Pibi,p(t) s ∈ [0, 1] (2.1)

Pi are denominated the control points and act as design variables in the optimization, n is thenumber of control points, and bi,p(t) are the Bernstein polynomials of the pth degree. Despitetheir resemblance, the Bézier curves are a much better representation than the polynomial form,and the control points used in the expression are very closely related to the curve positioning,which is beneficial when setting the geometric constraints [4]. The Bézier curve is great forshape optimization for simple geometries, like the polynomial form. The more complex acurve is, the higher the degree of the Bernstein polynomials needs to be in order to describeit properly. Unfortunately the round-off error increases when the degree of the Bernsteinpolynomials increases, on top of that the computational time for high-degree Bézier curves ishigh [9].A better way to describe complex curves is to use a series of low-degree Bézier curves calledBasis spline, or B-spline. The advantages of these splines is their ability to describe complexcurves accurately and efficiently.The weakness of the polynomial and spline approach is its inability to describe complex 3Dstructures entirely from polynomial forms and splines as they need a lot of control points.The strengths, however, are numerous: it has the ability to handle surfaces; to handle largegeometry changes; to handle local shape changes; and to give a smooth surface/boundary.

CAD-based approachThe CAD-based approach uses the benefits of having the structure already drawn and is a fullyintegrated solution saving time from geometric modeling. The CAD part can be an integratedpart of the FE software or a separate program and is either based on boundary representationor a constructive solid geometry method to represent the physical, solid object [4]. In theCAD software it is possible to describe the boundaries or the surfaces of the structure withlinear combinations of basic functions. Some of the parameters of these functions are thenused as design variables for the optimization procedure. The use of the CAD approach allowsfor optimization for both 2D and 3D, and it’s limitations are dependent on the user’s abilityto parameterize. In order to get good results when parameterizing this way, it is crucial thatthe initial guess of the basic functions is good, seeing as the basic functions can be what limitsthe optimization. Another strength of this method is that the smoothness of the boundary issuch that you get smooth boundaries with little effort. The drawback is that it can be quitetime consuming to implement the linear combinations, and the optimization is greatly reliableon how good the initial guess for shape function is.

CHAPTER 3Optimizing the shape of a cantilever beam

This model shows how to apply boundary shape optimization to a cantilever beam. The modelwill focus on optimization when more than one boundary has to be parametrized. Optimalobjective function and the use of probes will also be investigated.

3.1 Model definitionThe cantilever beam is a classic mechanical problem. It is a beam anchored at one end andcarries the load along its length or in the other end. Thus the upper half of the beam is sub-jected to tensile stress whereas the lower half is subjected to a compression. It’s a structure thatis mostly employed in construction; among these are especially cantilever bridges and balconies.

The beam to be studied is made of a linear elastic material; structural steel. The dimen-sions of the beam are 10m×1m×0.5m, meaning it has a total weight of 39,250 kg. The beamis fixed at x = 0 and has a boundary load in x = a with a total magnitude of P = 1000 kN,see Figure 3.1.

P

a

b

Figure 3.1: The beam with dimensions a× b× d

The parameterization of the upper part of the beam is going to be described by the followingfunction from [10]:

dy =

N∑i=1

qici sin(iπx) (3.1)

Where x is the parameterization parameter that varies from 0 to 1 along the boundary, ci arescale factors and qi are the optimization variables. By increasing the number of optimizationvariables the optimization obtains a higher degree of freedom to optimize, thus potentiallyachieving a better value of the objective function. However, this will also increase sensitivityand if the scale factors aren’t right the solution can continue into the undefined region, thuscreating an unfeasible model. The magnitude of the scale factors depend on how many termsfrom equation 3.1 are included, but generally speaking ci should be R ∈ [0, 1]. If i → ∞ thenci → 0 meaning that the parameterization becomes more and more sensitive when more termsare included, thus the terms have to be scaled more to make sure that the model is still feasible.

8 3 Optimizing the shape of a cantilever beam

Equation (3.1) will be modified slightly to fit the situation in question. The parameterizationdescribes the change in the y-direction for the upper boundary, for simplicity N = 1 and thescale factors ci are removed to begin with, thus equation (3.1) becomes:

dy = q1 sin(πx) (3.2)

The boundaries to be parametrized can be seen on Figure 3.2. This figure also shows the nameand direction of the parameterizations.

0

10 1

0

11 0

s1(x) s

3(x)

s2(x)

s4(x)

Figure 3.2: Boundaries to be parametrized and the direction of the parameterization param-eters

In order to make the height of parameterization s2(x) change according to the height of pa-rameterization s1(x) and s3(x) the terms (1− x)s1(1) + xs3(0) are added to equation 3.2.In Figure 3.2 the boundaries to be parametrized can be seen plus the direction of the param-eterization parameters. The boundary s4(x) isn’t to change during the optimization thereforethe parameterization will be s4(x) = 0. The parameterization of the four boundaries can beseen below:

s1(x) = p1x (3.3)s2(x) = (1− x)s1(1) + xs3(0) + q1 sin(πx) (3.4)s3(x) = p2(1− x) (3.5)s4(x) = 0 (3.6)

With the optimization parameters set to p1 = p2 = q1 = 0 the model will be a regular beam.By changing the parameters to p1 = q1 = 1 and p2 = 0 the beam seen in Figure 3.3 will appear.It is quite obvious that this beam has an increase in the area. Whereas the original had anarea of 10m2 this new beam has an area of 21.4m2.

Figure 3.3: Beam with optimization parameters p1 = q1 = 1 and p2 = 0

3.1 Model definition 9

Before the optimization can begin the objective function has to be defined. The objectivefunction is what defines the criterion for optimality. In this model the goal is to minimize thedisplacement of the beam in the y-direction. The displacement can be minimized through thetotal elastic strain energy in the point where the total load is acting on, since the equation forstrain energy is equal:

U =1

2Pδ (3.7)

Where P is the force acting in the end and δ is the displacement produced by the force . It canbe seen that the elastic strain energy is directly proportional to the displacement, therefore byminimizing the strain energy the compliance of the structure is minimized, thus maximizingthe stiffness.

Another condition for the optimization is an area constraint. If an area constraint isn’t appliedto the model, the optimization parameters will just reach their upper bound as that wouldgive the strongest structure, but also the heaviest. The area constraint for this model will bethe original area, a · b = 10m2. If the wish was to get a lighter structure and still maintain thestrength, that could be accommodated by multiplying the constraint with a scale factor.

In the model the parameterizations are implemented by the use of the prescribed boundarydisplacement in the Deformed Geometry interface. The beam is set to have free deformation.In order to avoid inverted elements in the model when it changes form due to the optimization,a mapped quad mesh is used instead of a tri mesh seeing as quad and mapped meshes areless likely to become inverted [11, p. 471]. Another thing to help avoiding inverted meshelements are the scale factors, ci, when using a higher order summation. It is also necessary totake great care when setting the boundaries for the optimization variables; if given too muchfreedom inverted elements will occur.

The Deformed Geometry interface is primarily used to study how the physics of the modelchange when the geometry of the model changes; it is therefore ideal for shape optimziation. Itis worth noticing that the model isn’t remeshed when using the Deformed Geometry interface,instead the mesh is deformed. Typically when looking at multiple versions of the same model,a new mesh is created per each new iteration, but by using Deformed Geometry the meshelements are instead being ”stretched”. The original mesh can be seen on Figure 3.4 whereasthe morphed mesh of Figure 3.2 can be seen on Figure 3.5.


Figure 3.4: The original mesh of the standard beam

Figure 3.5: The original mesh of the standard beam morphed to fit the new model

The advantage of deforming the mesh elements, instead of remeshing, is speed. For thisparticular model the difference between remeshing and morphing isn’t so distinct, but forlarger models and models with finer meshing there is a notable difference seeing as creating anew mesh each time can be quite time-consuming.However, if the deformations in the mesh becomes too large, inverted mesh elements can occur.There are a number of ways to avoid this[11, p. 861]:

• Changing the mesh. A good method for changing the mesh is to use a predefined meshand then changing the maximum element size. As stated earlier it is also useful to usequad meshes to prevent inverted mesh elements.

• Another mesh smoothing type can also be used. By standard it is set to Laplace smooth-ing which is the least time-consuming to use seeing as it is linear and uses one uncoupledequation for each coordinate direction. This mesh smoothing type is most suitable forsmall, linear deformations.The other smoothing types are non-linear and use a single coupled system of equationsfor all coordinate directions; this makes it more demanding for computational power,they are nevertheless better at avoiding inverted mesh elements.

3.2 Results and discussion 11

• One of the advantages of the Deformed Geometry is that it isn’t necessary to remesh.However, the mesh quality can become too poor, if the mesh deformations become suf-ficiently large, thus making it necessary to remesh. This can be done with an ”adaptivemesh refinement” in COMSOL, but the use of this feature also eliminates the use ofmapped quad meshes seeing as it only works on tri meshes.

3.2 Results and discussionBy changing the shape of a standard beam without altering the volume, the displacement ofthe right end can be lowered by 65%. In addition to this the distribution of the stresses willbe more evenly spread out and the peak value of the stresses will be only half as big!On Figure 3.6 the standard beam can be seen. It is quite clear that the stresses are highlyconcentrated at the end where the beam is fixed, whereas the stresses in the other end are afraction of that.

Figure 3.6: Standard beam with no optimization done, q1 = q2 = p1 = 0

Table 3.7 shows the key values from the solution of the standard beam, these values will seta benchmark for the optimizations. The main goal of the optimization is to increase stiffnessand decrease stress peaks.

σvM,max [MPa] U [J/m3] Utot [J] Area [m2] Y-disp. [m]137.78 3580 17900 10 -0.036

Table 3.7: Values for the standard beam, reference points for the optimization

Figure 3.8 shows an optimized beam with optimization parameters p1 = 0.33, q1 = 0.44 andp2 = −0.9. The parameterization of the upper boundary only has one summation (N = 1)meaning it’s a single sine curve.


Figure 3.8: Optimized beam with one summation and p1 = 0.33, q1 = 0.44 and p2 = −0.9

From the scale of the von Mises stress it is obvious that the peak stress has decreased con-siderably, it is in fact 89% lower. By comparing Table 3.9 and Table 3.7 it can also be seethat the elastic strain energy has been reduced a great amount, which comes to show in thedisplacement which has been reduced by almost 56%. Another thing worth noticing is that thestress is much more evenly distributed on Figure 3.8. The stress along the axis of the beam isalmost constant ensuring that the fixed end isn’t exposed to stresses significant higher than atthe other end.

σvM,max [MPa] U [J/m3] Utot [J] Area [m2] Y-disp. [m]73.01 2298 11490 10 -0.023

Table 3.9: Values for a optimized beam with N = 1, p1 = 0.33, q2 = 0.44 and p2 = −0.9

It should also be noted that there are stress singularities at the corners of the fixed end (largerversions of the stress plots can be found in Appendix A). The reason for stress singularitiesis that the area of the corners is very small and approaching zero meaning that the stressesare approaching infinite. These singularities will not appear in the real world because when abeam is fixed, the material will still yield a bit and/or the support material will move slightlyto allow the point stress to remain finite [12].

To see if the beam can be optimized further with the sine parameterization, the optimiza-tion has been run for N = [1, 5], where N ∈ Z. When going higher than N = 1, it is necessaryto remember the scale factors, otherwise there’s a high possibility that the model will go out ofbounds and return an unfeasible solution. Figure 3.10 shows the solution to the optimizationwhen N = 5. The plot doesn’t look that much different to Figure 3.8, and it’s a bit hard to seeall five sine curves but the stresses are lower. In Table 3.11 the key values of all optimizationsare compared.


Figure 3.10: Optimized beam with five summations and p1 = 0.47, p2 = −0.84, q1 = 0.27,q2 = −0.61, q3 = 0.36, q4 = −0.17, and q5 = 0.13

σvM,max [MPa] U [J/m3] Utot [J] Area [m2] Y-disp. [m]

Std. 137.78 3580 17900 10 -0.036N = 1 73.01 2298 11490 10 -0.023N = 2 71.13 2161 10806 10 -0.022N = 3 69.73 2156 10781 10 -0.022N = 4 69.73 2154 10771 10 -0.022N = 5 70.53 2153 10765 10 -0.022

Table 3.11: Comparison of key values for optimized beams

As mentioned earlier there’s a big difference between the standard beam and the first optimiza-tion with N = 1. The difference between one summation and five summations for the param-eterization of the upper boundary shows an improvement of 6,7%. However, the improvementfrom N = 2 to N = 5 is only 0,4% which shows it is a bit excessive to have multiple sine curves.

Figure 3.12 and Figure 3.13 support the von Mises stress surface plots seen on Figure 3.6,3.8, and 3.10 and give a clearer image of how the stress is distributed along the upper bound-ary. The stress on the standard beam, Figure 3.12, shows a very high stress singularity atx = 0m and then a drop before slightly increasing again, the stress then decreases linearlyuntil it reaches x = 10m with a magnitude of ≈ 2− 3MPa.


Figure 3.12: von Mises stress along the upper boundary for the standard beam

Figure 3.13 shows the von Mises stress along the upper boundary too. The same stress sin-gularity can be seen, but the peak stress is approximately 90% lower. Another thing worthnoticing is that stress along the boundary is almost constant, if excluding the ends. The stressvaries from ≈ 45− 50MPa, which is a great improvement compared to before where it variedfrom ≈ 10− 100MPa.

Figure 3.13: von Mises stress along the upper boundary for an optimized beam with fivesummation and p1 = 0.47, p2 = −0.84, q1 = 0.27, q2 = −0.61, q3 = 0.36,q4 = −0.17, and q5 = 0.13

In appendix A the stresses along the upper boundary can be found for the standard beam andthe beams with N = 1..5 in higher resolution. The higher the order, the more constant thestress variation becomes, which makes sense seeing it has more ”degrees of freedom” to change.

The results found for this model also make sense according to Pedersen’s work regardingbeams’ analytic optimal designs [13]. For a beam similar to the one examined in this project,Pedersen was able to reduce the compliance to 65%, see Figure 3.14.


Figure 3.14: Left: optimal design with obtained compliance. Right: the corresponding can-tilever elementary load cases. Illustration and text borrowed from [13]

The compliance of the COMSOL model was also 65%. However, there are a few differencesbetween Pedersen’s analytical work and this model. Pedersen uses a point load at the end,whereas in this model a boundary load has been used. Pedersen also parameterizes both theupper and lower boundary; this model only focuses on the upper.


3.3 Modeling instructionsThe following will describe how to create the model. From the File menu, choose New

NEW

1 In the New window, click the Model Wizard button

MODEL WIZARD

1 In the Model Wizard window, click the 2D button

2 In the Select Physics tree, selectMathematics>Deformed Mesh>Deformed Geom-etry (dg).

3 Click the Add button

4 In the Select Physics window, select Structural Mechanics>Solid Mechanics (solid).


6 In the Select Physics tree, selectMathematics>Optimization and Sensitivity>Optimization(opt).

7 Click the Add button.

8 Click the Study button

9 In the tree, select Preset Studies for Selected Physics>Stationary

10 Click the Done button

GLOBAL DEFINITIONS

Parameters1 On the Home toolbar, click Parameters

2 In the Parameters settings windows, locate the Parameters section

3 Click Load from file

4 Browse to find the file called beam_shape_optimization_parameters.txt and double-clickit to load the parameters

GEOMETRY 1

Rectangle 1

3.3 Modeling instructions 17

1 In the Model Builder window, right-click Geometry 1 and choose Rectangle

2 In the Rectangle settings window, locate the Size section

3 In the Width edit field, type a

4 In the Height edit field, type b

5 Click the Build All Objects button

MATERIALS

Add material

1 Go to the Add Material window

2 In the tree, select Built-In>Structural Steel

3 In the Add material window, click Add to Component and choose Component 1

SOLID MECHANICS (SOLID)

Fixed Constraint 1

1 On the Physics toolbar, click Boundaries and choose Fixed Constraint

2 Select Boundary 1 only.

Boundary Load 1

1 On the Physics toolbar, click Boundaries and choose Boundary Load


3 In the Boundary Selection window, locate the Force section

4 Under Load type, change it to Total force

5 Let the x component remain 0, but change the y component to -F_T

DEFORMED GEOMETRY (DG)

Free Deformation 1

1 On the Physics toolbar, click Domains and choose Free Deformation


2 Select Domain 1.

Prescribed Mesh Displacement 2

1 On the Physics toolbar, click Boundaries and choose Prescribed Mesh Displacement


3 In the Prescribed Mesh Displacement settings window, locate the Prescribed MeshDisplacement section

4 In the dY field type q1*s





4 In the dY field type q1*(1-s)+q3*s+q2*sin(pi*s)





4 In the dY field type q3*(1-s)

MESH 1

Free Quad 1

1 In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 andchoose Free Quad

2 In the Free Quad settings window, locate the Domain Selection section

3 From the Geometric entity level list, choose Domain


4 Select Domain 1

Size

1 In the Model Builder window, under Component 1 (comp1)>Mesh 1 click Size

2 In the Size settings window, locate the Element size section

3 Under Predefined, select Extra fine from the list

4 Click the Build All button

STUDY 1Before starting the actual optimization it can be a good idea to check the model by solving forthe default parameters. In this way you have a good reference point when doing the optimiza-tion later.

Solver 1

1 On the Study toolbar, click Show Default Solver

2 In the Model Builder window, expand the Study 1>Solver Configurations node

3 In the Model Builder window, expand the Solver 1 node, then click Stationary Solver1

4 In the Stationary Solver settings window, locate the General section

5 From the Linearity list, choose Nonlinear

6 On the Home toolbar, click Compute

RESULTS

von Mises StressThe default plot in the main window shows the von Mises stress surface distribution in thebeam. Note that stress reaches its maximum near the fixed constraint and is practically zerowhere we apply the force. This is as expected.Right now the deformation is being displayed as well for the stress plot, to disable this do thefollowing

1 In the Model Builder window, locate Results>Stress (solid)

2 Right-click Stress (solid) and clickRename, rename it to ”von Mises stress, solution1”


3 Expand the von Mises stress, solution 1 node, then the Surface 1 node

4 Right-click Deformation 1 and click delete. The plot should automatically be replottedwithout showing the deformation

Displacement

1 On the Results toolbar, click 2D Plot Group

2 Right-click the new 2D Plot Group and clickRename. Change the name to Y-displacement,solution 1

3 Right-click the plot group and click Surface

4 The Surface window will open, locate the Expression section. The standard expression issolid.disp, change this to v either by editing the field or by cliking the button Replaceexpression and then navigating to Solid Mechanics>Displacement>DisplacementField (material)>Displacement Field, Y component (v)

5 To show the actual deformation (although scaled), right-click Y-displacement, solution1>Surface 1 and click Deformation. The Deformation window will show the scalingfactor under the section Scale

ADD STUDY

1 To add a new study for the optimization, go to the Home toolbar and click Add Study

2 Go to the Add Study window

3 Find the Studies subsection. In the tree, select Preset Studies>Frequency Domain

4 In the Add study window, click Add study

5 On the Home toolbar, click Add Study to close the Add Study window

STUDY 2

Optimization area constraint

1 On the Physics toolbar, click Optimization

2 Locate the Domains button, and click Integral Inequality Constraint

3 In the Integral Inequality Constraint window, select Domain 1

4 Locate the Constraint section and under Constraint expression type 1


5 Locate the Bounds section, don’t change the Lower Bound, but edit the Upper Bound,type a*b

6 The Integral Inequality Constraint will automatically be added to the optimization inStudy 2

Optimization

1 On the Study toolbar, click Optimization. This will add the Optimization module toStudy 2

2 In the Optimization settings window, locate the Optimization Solver section

3 From the Method list, choose SNOPT

4 In the Optimality tolerance edit field, type 1e-4

5 In the Maximum number of objective evaluations edit field, type 200

6 Locate the Objective Function section. In the table, enter following settings:

Expression Descriptioncomp1.solid.Ws_tot Total elastic strain energy

7 Locate the Control Variables and Parameters section. Click Load from File

8 Browse to find the file called beam_shape_optimization_control_parameters.txt

9 Locate Output While Solving section and make sure that Plot is checked

Solver 2


2 In the Model Builder window, expand the Study 2>Solver Configurations>Solver2>Optimization Solver 1 node, then click Stationary 1

3 In the Stationary settings window, locate the General section

4 In the Relative tolerance edit field, type 1e-6

5 From the Linearity list, choose Automatic

6 Right-click Study 2 and press the Compute button


RESULTS

PlotsThe plots from solution 1 can be duplicated relatively easy thus avoiding the same procedureall over again.

1 In the Model Builder window under Results, locate the von Mises stress, solution 1

2 Right-click it and choose Duplicate

3 Right-click the new plot group and rename it to von Mises stress, solution 2

4 In the 2D Plot Group window, locate the Data section

5 In the drop down menu change it from Solution 1 to Solution 2

6 Click Plot

7 Follow the same procedure for the Y-displacement

ProbesIt is possible to see the parameter values for each iteration by using probes. In addition it’salso possible to show the values of stresses, displacement and the total elastic strain energy.

1 In the Model Builder window under Component 1, locate the Definitions menu

2 Right-click it and choose Probes>Domain Probe

3 The Domain Probe window will open, locate the Expression section. The expression willby default be set to solid.disp, change this to solid.mises, which is the von Mises stress

4 Right-click Domain Probe 1 and click Rename

5 Rename the probe to von Mises

6 Repeat step 1-5 for the following parameters: strain energy, total strain energy, displacement(Displacement field, Y component (v)), q1, q2 and q3.

Area Probe

1 In order to create an Area Probe, first right-click Definitions, choose Component Cou-plings>Integration

2 The Integration window will open, select Domain 1

3 Now create a new Domain Probe using the steps from above. Rename it Area

4 In the Expression field type intop1(1)

CHAPTER 4Optimization of a plate with a hole

This model will investigate how to apply boundary shape optimization to a simple bracket. Themodel will focus on shape optimization when more than one boundary has to be parameterizedin both the x and the y direction.

4.1 Model definitionThe quadratic plate with a hole in the middle and biaxially loaded is a typical problem withinshape optimization. The model can be seen in Figure 4.1(a), it has dimensions S × S × d =10m× 10m× 0.1m and radius, r = 4m. The plate is made in structural steel meaning it hasa total weight of 68633 kg. The boundaries are loaded with σ1 in the x direction and σ2 in they direction in such a way that they both experience tensile stress.In this model a survey of how the relation between the stresses σ1 and σ2 will influence theshape optimization parameters will be conducted. First for σ2/σ1 = 1, then for σ2/σ1 = 2and at last for σ2/σ1 = 3. It is expected for the first case, σ1 = σ2, that the shape is alreadyoptimal.In geometric modeling it is only necessary to model a quarter of the model because of symmetry,see Figure 4.1(b). The boundaries, where the cut is made, is replaced with rollers to representthe missing sides. Figure 4.1(b) also shows which sides are to be parameterized.The left side,s1(y) and s3(x) are only to parameterized in one direction, respectively the y and x direction,whereas s2(x, y) is to be parameterized in both directions. The parameterization will changethe shape of the hole by changing the curve of the hole and by changing the coordinates of theendpoints of the curve.The parameterizations describing s1 and s3 are very similar to the parameterizations of thebeam in Chapter 3:

s1,y(s) = p1(1− s)

s3,x(s) = p3s(4.1)

Where p1 and p3 are both optimization parameters and s ∈ [0, 1]. The boundary, s2, is a bitmore difficult to parameterize. At first a parameterization similar to the one from Chapter 3was used, but with the addition of a parameterization in the x direction:

s2,x(s) = p3(1− s)

s2,y(s) = s1,y(0)s+ p2 sin(πs)(4.2)

This solution, however, proved unsuccessful due to slope at the end points which createdunfeasible results. This can be seen in Figure 4.2, the stress concentration for this resultwas evenly distributed but the transition between the joints was not smooth. Therefore theresulting hole looked the number 8, which is not desirable and created high stress peaks at thejoints.

24 4 Optimization of a plate with a hole

S

S

σ2

σ1

r

σ2

σ1

Symmetry

Symmetry

(a) Mechanical analysis setup with dimensionsS = 10m, r = 4m, and d = 0.1m

σ2

σ1

s1(y)

s2(x,y)

s3(x)

S/2

S/2

(b) Shape optimization setup. Due to symmetryonly 1/4 of the plate needs to be analysed

Figure 4.1: Plate with a hole

Figure 4.2: v. Mises stress showing the parameterization of the plate with equation (4.2) andσ2/σ1

In order to get a nice transition between the joints, the curve needs to have zero slope at bothendpoints. The equation for the dY displacement will be found first. The slope at the endsmust be zero, meaning that the derivative of the parameterization must be zero at s = 0 ands = 1. At the right end point the parameterization should be zero when s = 0, and at the left

4.1 Model definition 25

end point equal to p1 when s = 1. These demands can be expressed mathematically in thefollowing way:

s2,y(0) = 0

s2,y(1) = p1

s′2,y(0) = 0

s′2,y(1) = 0

(4.3)

In order to get zero slope at the end points a cosine curve forms the basis of the parameteriza-tion. In order to get the curve to begin and end in the same y-coordinate only equal integersenter in the first cosine term, furthermore the whole term is subtracted from 1 to shift thewhole function above y = 0. This equals the following expression where p2/2 controls theheight of the curve:

p22(1− cos(nks))

In the above expression k = 2π, n ∈ Z, and p2 is an optimization parameter.The left end point of s2,y needs to match its y-coordinate with the s1,y(s) expression fromequation (4.1). This can be done by adding the expression p1

2 (1− cos(πs)). When combiningthe terms the following equation is generated:

s2,y(s) =p22(1− cos(nks)) + p1

2(1− cos(πs)) (4.4)

Where k = 2π, n ∈ Z, and p1 and p2 are both optimization parameters.The boundary, s2,x was at first parameterized with the expression p3(1 − s), although thisexpression did not yield zero slope due to the fact that the parameterization in COMSOL isa parameterization of the boundary arc length and not the original x-coordinate. Therefore itwas necessary to make a transformation to a trigonometric expression. The problem is outlinedin Figure 4.3.

s=0, ϕ=0

s=1, ϕ = π/2

s

r

ϕ

x(ϕ)

y(ϕ)

Figure 4.3: Illustration of the transformation of the x-coordinates radial to the boundary

The boundary curve can be described in terms of s with equation (4.5) and the x-coordinatecan be described in terms of ϕ with equation (4.6). By combining these two equations the


transformation for describing the x-coordinate in terms of s is complete:

ϕ(s) =πs

2(4.5)

x(ϕ) = r cos (ϕ) (4.6)

x(s) = r cos(πs2

)(4.7)

By combining the above with the parameterization of s2,x(s) it becomes:

s2,x(s) = p3 cos(πs2

)(4.8)

So to repeat, the whole parameterization of the model is:

s1,y(s) = p1(1− s)

s2,x(s) = p3 cos(πs2

)s2,y(s) =

p22(1− cos(nks)) + p1

2(1− cos(πs))

s3,x(s) = p3s

(4.9)

Figure 4.4 shows the comparison between the original mesh and the mesh after the parame-terization. The figure also illustrates that there are now zero slope at the end points of thecurve.

(a) The mesh of the orignal plate with parameters p1 =0, p2 = 0, and p3 = 0

(b) The mesh after the parameterization, here withparameters p1 = 1, p2 = 0.5, and p3 = −3

Figure 4.4:

The objective function for this model will also be the total elastic strain energy seeing as thestiffest design is equal to the design with lowest stress concentration[14].


4.2 Results and discussionIn this section the results of the optimization will be presented and discussed. Three differentcases will be looked into; σ2/σ1 = 1, σ2/σ1 = 2, and σ2/σ1 = 3. The objective of the shapeoptimization is to even out the stresses and reduce the peak stress without increasing the area.Throughout the analysis an integral inequality constraint will be used to make sure the areadoesn’t exceed the original area domain.

First case: σ2/σ1 = 1In the first case the plate is loaded with stress of the same magnitude on the right and upperboundary in the x and y direction, respectively. When the tension rate is equal to 1, theoptimal shape is expected to be a circular hole to have evenly distributed stresses. Referencevalues for the non-optimized solution can be found in Table 4.5, and the stress distributioncan be seen on Figure 4.6.

σvM,max [MPa] U [J/m3] Utot [J] Area [m2]2.12 4.08 35.69 87.43

Table 4.5: Reference values for the plate with σ1 = σ2 = 1000 kN

Figure 4.6: Standard plate with σ2/σ1 = 1

As seen on Figure 4.6 the stress is already evenly distributed. The optimization solver shouldreturn the optimization parameters p1 = p2 = p3 = 0, this is however not the case. Theparameters are slightly optimized but still practically zero. The reason for this is due tonumerical approximation and is explained in the third case.


Second case: σ2/σ1 = 2The results for the non-optimized plate can be seen on Table 4.7 and Figure 4.8. The rightboundary is loaded with 1000 kN and the upper boundary with 2000 kN.


Table 4.7: Reference values for the plate with σ1 = 1000 kN and σ2 = 2000 kN


It is quite clear from Figure 4.8 that the shape of the hole is not optimal when the tensile ratiois 2. The shape optimization is performed resulting in Table 4.9 and Figure 4.10. Completetables, log files and higher-resolution plots can be found in Appendix B.2.


Table 4.9: Optimized values for the plate with σ1 = 1000 kN and σ2 = 2000 kN with opti-mization parameters p1 = 3.01, p2 = 0.40, and p3 = 2.09


(a) The circular hole has become an ellipse and the vonMises stress is now evenly distributed along the bound-ary

(b) The von Mises stress as a function of the arc lengthfor s2. The blue curve is the original design and thegreen is the optimized shape

Figure 4.10: Optimized plate with optimization parameters p1 = 3.01, p2 = 0.40, and p3 =2.09

By comparing the peak stresses of the original model and the optimized one it can be seen thatthe peak stress has been reduced by 89.6% without increasing the area. From Figure 4.10(a) itcan be seen that the stress concentration in the top of the hole has been distributed along theboundary of the hole. Figure 4.10(b) shows how the stress has changed from being a ”linearfunction” to being almost constant.Take note of the form of the hole, it has changed from being circular to being an ellipse. InPedersen’s analytical results [13] of holes subjected to biaxial stress he finds out that therelationship between the tensile stresses is equal to the relationship between the vertices of theellipse.

Third case: σ2/σ1 = 3Table 4.11 and Figure 4.12 shows the non-optimized results when the tensile ratio is 3. Theupper boundary is loaded with 3000 kN and the right boundary with 1000 kN. The stress plotis very similar to Figure 4.8 but due to the higher load the stress concentration has becomeeven higher.


Table 4.11: Reference values for the plate with σ1 = 1000 kN and σ2 = 3000 kN



Table 4.13 and Figure 4.14 show the optimized results. According to [13] the ratio of the verticesfor the elliptic should be equal to the ratio of the tensile stresses. This is not the case though. Bylooking at Figure 4.14(a) the vertex ratio does not seem to match with 3. When investigatingfurther it turns out that the vertex ratio is a/b = 6.56m/2.49m = 2.63. The reason for this isbecause the analytical solution from [13] is based on a hole in an infinite plane domain whichis subjected to stress far away from the hole. This is supported by numerical results, if thesides of the plate is changed from 10 m to 100 m, thus making the domain closer to infiniteand moving the loads farther way, the vertex ratio increases to a/b = 6.94m/2.34m = 2.97.By making the domain larger and larger, the more close the numerical method gets to theanalytical method. This explanation follows for approximation error for all three cases.


Table 4.13: Optimized values for the plate with σ1 = 1000 kN and σ2 = 3000 kN with opti-mization parameters p1 = 5.12, p2 = 0.75, and p3 = −3.02


(a) The circular hole has become an ellipse and the vonMises stress is now evenly distributed along the bound-ary


Figure 4.14: Optimized plate with optimization parameters p1 = 5.12, p2 = 0.75, and p3 =−3.02

In this case the maximum equivalent stress has been reduced from 10.41 MPa to 4.48 MPa, areduction of 132%. As in case 2, not only have the peak stresses been lowered, but the stressis evenly distributed along the arc boundary. This is also clearly show in Figure 4.14(b) whichshows little variation in the stress; it only spans from 4.12 MPa to 4.48 MPa.


Other casesIn order to save material and therein cost it is possible to use a lower upper limit for the areaconstraint by multiplying with a constant. E.g. how does the plate look if the material cost isto be cut down by 20%? In Figure 4.15 the results of this reduction can be seen. The modelhas become 13727 kg lighter and the maximum equivalent stress has still been reduced from6.20 MPa to 5.61 MPa (10.5%). The result, however, is not optimal. The stress is evenlydistributed, but a large stress concentration at the top left corner is not desirable. The wholedesign domain should be larger in order to distribute the stresses better in the domain.

(a) Stress is still evenly distributed along the boundary,but the design is not optimal. The design domain shouldbe larger


Figure 4.15: Optimized plate with a 20% reduction in material and optimization parametersp1 = 8.58, p2 = 0.80, and p3 = 1.67

This model shows that COMSOL Multiphysics 4.4 currently has some scaling problems. Thismodel was originally supposed to be 25mm×25mm with a hole with a radius of 10mm, but dueto these scaling problems when COMSOL tried to optimize the shape of the plate it stoppedafter one iteration. This problem occurred even though the model was setup with variousscaling factors in the parameterizations to take this into account. The COMSOL support hasbeen contacted and normally they’re rather quick at responding (usually less than 24 hours),but they have been working on this problem for more than a week.



NEW


MODEL WIZARD











GLOBAL DEFINITIONS




4 Browse to find the file called bracket_shape_optimization_parameters.txt and double-click it to load the parameters

GEOMETRY 1

Square 1


1 In the Model Builder window, right-click Geometry 1 and choose Square

2 In the Square settings window, locate the Size section

3 In the Side length edit field, type s


Circle 1

1 In the Model Builder window, right-click Geometry 1 and choose Circle

2 In the Circle settings window, locate the Size and Shape section

3 In the Radius edit field, type r


Difference 1

1 In the Model Builder window, right-click Geometry 1 and choose Boolean Opera-tions>Difference

2 In the Difference settings window, locate the Objects to add section

3 Activate the Object to add window by clicking Activate, and then click on the square(Square 1)

4 In the Difference settings window, locate the Objects to subtract section

5 Activate the Object to subtract window by clicking Activate, and then click on the circle(Square 1)


MATERIALS

Add material






Roller 1

1 On the Physics toolbar, click Boundaries and choose Roller

2 Select Boundary 1 and 3 only.

Boundary Load 1





5 Let the y component remain 0, but change the x component to P1

Boundary Load 2





5 Let the x component remain 0, but change the y component to P2


Free Deformation 1


2 Select Domain 1.






4 In the dY field type p1*(1-s)





4 In the dX field type p3*cos(pi*s/2)

5 In the dY field type p2*(1-cos(k*s))/2+p1*(1-cos(pi*s))/2





4 In the dX field type p3*s

MESH 1

Free Quad 1




4 Select Domain 1

Size







Solver 1







RESULTS

von Mises StressThe default plot in the main window shows the von Mises stress surface distribution in theplate. Note that the stress is concentrated at the right end of arc boundary. This is as ex-pected. The peak stress for the model is 6.22MPa.Right now the deformation is being displayed as well for the stress plot, to disable this do thefollowing






ADD STUDY






STUDY 2






5 Locate the Bounds section, don’t change the Lower Bound, but edit the Upper Bound,type sideˆ2-pi*rˆ2/4


Optimization stress constraint


2 Locate the Domains button, and click Global Inequality Constraint

3 In the Global Inequality Constraint window, select Domain 1


5 Locate the Bounds section, don’t change the Lower Bound, but edit the Upper Bound,type 3[MPa]

6 The Global Inequality Constraint will automatically be added to the optimization inStudy 2


Optimization




4 In the Optimality tolerance edit field, type





8 Browse to find the file called bracket_shape_optimization_control_parameters.txt


Solver 2







RESULTS








6 Click Plot

1D PlotsIt can also be interesting to see the stress as a function of the boundary. This can be done inthe following

1 In the Model Builder window, right-click Results and locate 1D Plot Group, click it

2 The 1D Plot Group window will open, locate the Data section, under Data set chooseNone

3 Right-click the 1D Plot Group under Results, click Line Graph to add a graph

4 The Line Graph window will open, under Data set choose Solution 1

5 Locate the Selection section, toggle the on/off switch and select Boundary 5

6 Locate the y-Axis Data section and replace the expression with solid.mises, also changethe Unit from N/mˆ2 to MPa

7 In order to compare the value with Study 2, right-click Line Graph 1 and clickDuplicate

8 In the new Line Graph window, change the Data set to Solution 2







6 Repeat step 1-5 for the following parameters: strain energy, total strain energy, q1, q2 andq3.


Area Probe





5 To see the probes in effect, compute Study 2 again

CHAPTER 5Optimizing the shape of a fillet

This model will look into how COMSOL can be used for 3D shape optimization. The focus ofthe optimization will be of a fillet on a bar in tension. The model will show how to model aaxisymmetric geometry and will look into some of the problems that may arise when performingshape optimization in COMSOL.

5.1 Model definitionThis model investigates the optimization of a fillet in the transition region of a bar in tension.The bar is modeled in structural steel, has a volume of 39573m3 and thereby a total weight of310,648 tonne. The model can be seen on Figure 5.1 with dimensions.

σ

10

12

.5

27

.5

5 17.5

s1

s2

s3

5

Ax

is o

f ro

tati

on

Figure 5.1: The bar with dimensions

The figure is rotational symmetric around the left boundary. When modeling in 2D axisymme-try in COMSOL the axis of rotation is automatically replaced with roller boundary conditions.The upper boundary is loaded with a uniaxial tension, σ, with a magnitude of σ = 1000MN.

44 5 Optimizing the shape of a fillet

The goal of the optimization is to get a better stress distribution along the fillet boundarywhile still maintaining the volume.Because of the rotational symmetry it is only necessary to model half of the model in 2D. Theparameterization of this problem is very similar to the ones seen in Chapter 3 and 4 and willnot be explained further:

s1(s) = p1s

s2,R(s) = p2s

s2,Z(s) = p1(1− s) + q1sin(πs)

s3(s) = p2(1− s)

(5.1)

In the above parameterization p1, p2, and q1 are all optimization parameters. The parameter-ization for s2,Z is the first term of equation (3.1) and can be expanded with more terms as inChapter 3.Due to the current scaling problems in COMSOL, this model is also heavily enlarged. I amwell aware of the size is unrealistic, but the basis of the optimization is still valid.

5.2 Results and discussionOn Figure 5.2 the standard bar can be seen both in 2D and 3D. It is clear that there is a stressconcentration due to a singularity. If the mesh is too fine when there’s a singularity in themodel, the von Mises stress will not reflect a proper image of the stress distribution. In orderto get a better result of the stress distribution a fillet with radius 1m has been added.

(a) von Mises surface stress in 2D (b) von Mises surface stress in 3D

Figure 5.2: The non-optimized fillet

Table 5.3 shows the key for the non-optimized model. These values will set the reference pointsfor the benchmark:


σvM,max [MPa] U [J/m3] Utot [kJ] Area [m2]6.73 10.35 192.61 525

Table 5.3: Values for the standard bar, reference points for the optimization

Figure 5.4 shows an optimized fillet with optimization parameters p1 = 26.00, p2 = −7.60, andq1 = −11.24. From the von Mises stress plot it is clear that high stress concentration has beenflattened out with the sine curve. Figure 5.4 is parameterized with a single sine curve (N = 1).


Figure 5.4: Optimized fillet with N = 1, and optimization parameters p1 = 26.00, p2 =−7.60, and q1 = −11.24

The stress is now nicely distributed in the upper part of the bar. By comparing Table 5.5 toTable 5.3, it can be seen that the peak stress has been lowered from 6.73 MPa to 4.44 MPa.That is a reduction of 52%.

σvM,max [MPa] U [J/m3] Utot [kJ] Area [m2]4.44 9.85 186.57 525

Table 5.5: Values for the optimized fillet withN = 1, and optimization parameters p1 = 26.00,p2 = −7.60, and q1 = −11.24

Figure 5.6 shows the stress along the boundary for s2. It can be seen that the stress has becomemore constant. Due to the form of the parameterization for s2 the boundary becomes longerwhen optimizing the shape, while s1 becomes shorter.


Figure 5.6: The von Mises stress as a function of the arc length for s2. The blue curveis the original design and the green is the optimized design with optimizationparameters p1 = 26.00, p2 = −7.60, and q1 = −11.24

When the parameterizations change the boundaries by such a big amount, e.g. making themtwice as long as, the mesh is more likely to get inverted, because the mesh elements arestretched too much. To take this into account a mesh refinement study is added to the model.In COMSOL a mesh refinement can only be added if the mesh is made of triangular meshelements, so the mesh is changed from mapped quads to free triangular mesh elements. Themesh refinement allows for bigger mesh displacements in the optimization and thus gives abetter solution.Too see if the fillet can be optimized further with the sine parameterization more terms areadded. The optimization is run for N = [1, 5], where N ∈ Z. In order to avoid the optimiza-tion going into the undefined region of the domain, it is very important to choose the rightboundaries for the optimization parameters as well as scaling the displacement. This cannotbe stressed enough seeing as it is as important as having a high quality mesh. Choosing theright parameters can be a very time consuming task and the only way is by trial-and-error. Ifthe boundaries or scaling factors are not well chosen, inverted mesh elements are certain to de-velop. The parameterization for this model has shown very troublesome regarding this subject,because the joining of the parameterizations apparently is not optimal, so the parameters arevery sensitive.Table 5.7 shows the results of all five optimization studies.


σvM,max [MPa] U [J/m3] Utot [kJ] Area [m2]

Std. 6.73 10.35 192.61 525N = 1 4.44 9.85 186.57 525N = 2 4.14 9.84 186.49 525N = 3 4.02 9.84 186.44 525N = 4 3.91 9.84 186.43 525N = 5 3.86 9.84 186.42 525

Table 5.7: Comparison of key values for optimized fillets

The table shows that with each terms added the von Mises maximum stress is reduced as well asthe strain energy. The reduction of the stress from the standard fillet to the parameterizationwith five terms is 74% while the reduction from four terms to five terms is only 1.3%. Figure5.8 shows the von Mises stress for the best optimization, N05. The distribution shows a slightimprovement compared to Figure 5.4.



(c) The von Mises stress as a function of the arc lengthfor s2. The blue curve is the original design and thegreen is the optimized design

Figure 5.8: Optimized fillet with N = 5, and optimization parameters p1 = 40.00, p2 =−7.60, q1 = −21.33, q2 = −5.27, q3 = −2.46, q4 = −1.03, and q5 = −0.39

In [2] an analysis of a similar bar has been made with the use of the parameter-free approach.Le et al. obtained identical results regarding the stress image, see Figure 5.9. Riehl alsoobtained similar results with the parameter-free approach [14]. The different between themodel analyzed in this paper and Riehl’s and Le’s models is that they have a larger degree offreedom plus they are maximizing the von Mises stress while reducing the volume as well.


Figure 5.9: The obtained optimized result from [2]

Therefore they get a curve in lower region of the model. In order to get the same curved resultsfor boundary s3 for this model, the boundaries have to be reparameterized plus the volumeconstraint should be stricter. This will not be looked into in this thesis. In Appendix C all ofthe plots, tabels and logs of the optimization can be found.



NEW


MODEL WIZARD











GLOBAL DEFINITIONS




4 Browse to find the file called fillet_shape_optimization_parameters.txt and double-click it to load the parameters

GEOMETRY 1

Rectangle 1




3 In the Width edit field, type d

4 In the Height edit field, type c

5 Locate the Position section

6 In the r field, type b-d

7 In the z field, type a-c


Bézier Polygon 1

1 In the Model Builder window, right-click Geometry 1 and choose Bézier Polygon

2 In the Bézier Polygon settings window, locate the Polygon Segments section

3 Find the Added segments subsection. Click the Add Linear button

4 Find the Control points subsection. In row 1, set r to b-d and z to a-c

5 In row 2, set r to b-d+radius and z to a-c


7 Find the Control points subsection. In row 1, set r to b-d+radius and z to a-c

8 In row 2, set r to b-d and z to a-c+radius


10 Find the Control points subsection. In row 1, set r to b-d and z to a-c+radius

11 In row 2, set r to b-d and z to a-c

12 Click the Build All Obecjts button

13 Click the Zoom Extents button on the Graphics toolbar

Difference 1




3 Activate the Object to add window by clicking Activate, and then click on the rectangle(Rectangle 1)


5 Activate the Object to subtract window by clicking Activate, and then click on theBézier polygon (Bézier Polygon 1)


Rectangle 2



3 In the Width edit field, type b

4 In the Height edit field, type a


Difference 2



3 Activate the Object to add window by clicking Activate, and then click on the newrectangle (Rectangle 2)


5 Activate the Object to subtract window by clicking Activate, and then click on theDifference domain created before (Difference 1)


MATERIALS

Add material






Roller 1

1 On the Physics toolbar, click Boundaries and choose Roller

2 Select Boundary 2 only

Fixed Constraint 1

1 On the Physics toolbar, click Points and choose Fixed Constraint

2 Select Point 1 only (the point in the lower left corner)

Boundary Load 1





5 Let the r component remain 0, but change the z component to P


Free Deformation 1


2 Select Domain 1.






4 In the dZ field type k*p1*s





4 In the dR field type k*p2*s

5 In the dZ field type k*(p1*(1-s)+q1*sin(pi*s))





4 In the dR field type k*p2*(1-s)

MESH 1

Free Quad 1




4 Select Domain 1

Size







Solver 1







RESULTS

von Mises StressThe default plot in the main window shows the von Mises stress surface distribution in thebeam. Note that stress reaches its maximum near the fixed constraint and is practically zerowhere we apply the force. This is as expected.Right now the deformation is being displayed as well for the stress plot, to disable this do thefollowing





5 Repeat the procedure for the 3D plot

ADD STUDY







STUDY 2






5 Locate the Quadrature settings section and expand it

6 Make sure that Multiply by 2πr isn’t checked

7 Locate the Bounds section, don’t change the Lower Bound, but edit the Upper Bound,type 525 mˆ2


Optimization




4 In the Optimality tolerance edit field, type 1e-4






8 Browse to find the file called beam_shape_optimization_control_parameters.txt


Solver 2







RESULTS







6 Click Plot

7 Follow the same procedure for the von Mises stress 3D plot








6 Repeat step 1-5 for the following parameters: strain energy, total strain energy, p1, p2, andq1.

Area Probe





Adapative Mesh Refinement

1 Expand Study 1 and locate Step 1: Stationary, click it

2 Locate Study Extensions and expand it

3 Go to the bottom and check Adaptive mesh refinement

4 Locate Stationary Solver 1 and expand it

5 Click Adaptive Mesh Refinement

6 Locate the General section

7 Change Maximum number of refinements from 2 to 3

8 Compute both studies again

CHAPTER 6Conclusion

Throughout this thesis shape optimization has been investigated. The strengths and weak-nesses of a number of different shape optimization methods have been discussed. Shape op-timization in COMSOL has been implemented on three classic shape optimization problemsand has resulted in solutions similar analytical results and to what others have obtained. Al-though COMSOL has improved their software for shape optimization there are still obstaclesto tackle. First and foremost it is critical to have the mathematical knowledge of how to createwell-connected linear combinations. Furthermore it is important that the ”guess” for the linearcombinations is good seeing as the solution is only as good as the guess.And even though the parameterization is well-connected in COMSOL, there is still a lot ofwork with tweaking both the boundaries for the optimization parameters, but also for findingthe right scaling factors for the linear combinations.

6.1 Future workThere are still a number of aspects to be investigated in COMSOL with regards to shapeoptimization. It could be of great interested to implement Bézier curves and B-splines in theparameterization as they offer a great degree of freedom with few design variables. Anotherinteresting aspect is to work further with 3D optimization, COMSOL offers great possibilitiesfor 3D structures.

APPENDIX ACantilever Beam

A.1 Standard

1 ============================================================2 Stationary Solver 1 in Solver 1 started at 8-maj-2014 18:13:56.3 Nonlinear solver4 Number of degrees of freedom solved for: 19268.5 Nonsymmetric matrix found.6 Scales for dependent variables:7 Displacement field (Material) (comp1.u): 2.8e-058 Material coordinates (Geometry) (comp1.XY): 109 Iter ErrEst Damping Stepsize #Res #Jac #Sol LinErr LinRes

10 1 2.7e-11 1.0000000 0.63 2 1 2 1.5e-08 2.9e-1111 2 6.4e-16 1.0000000 2.7e-11 3 2 4 3e-08 4e-1612 Stationary Solver 1 in Solver 1: Solution time: 1 s13 Physical memory: 877 MB14 Virtual memory: 5213 MB

62 A Cantilever Beam

(a) von Mises stress (b) von Mises Stress along the upper boundary

Figure A.1:

A.1 Standard 63

Figure A.2: The error of the solution as a function of iteration number for the nonlinearsolver


A.2 One summation

1 ============================================================2 Optimization Solver 1 in Solver 2 started at 22-maj-2014 11:42:41.3 Optimization solver (SNOPT)4 Analytic gradient with the adjoint method.5 Itns Major Minor Step nPDE Error Objective6 2 0 2 - 1 2.99 1.331e+047 4 1 2 0.45 2 1.54 1.538e+048 7 2 3 0.25 4 0.589 1.09e+049 8 3 1 1.00 5 0.478 1.086e+04

10 9 4 1 1.00 6 0.179 1.088e+0411 11 5 2 0.25 16 0.61 1.149e+0412 Warning: Current point cannot be improved.13 Optimization Solver 1 in Solver 2: Solution time: 63 s (1 minute, 3 seconds)14 Physical memory: 867 MB15 Virtual memory: 5200 MB

A.2 One summation 65

(a) von Mises Stress (b) von Mises Stress along the upper boundary

Figure A.3:


(a) The error of the solution as a function of itera-tion number for the optimization solver

(b) The error of the solution as a function of iter-ation number for the nonlinear solver

Figure A.4:

A.2 One summation 67

Iteration#

σvM

,max[M

Pa]

U[J/m

3]

Utot[J]

Area[m

2]

Y-disp

[m]

p1[-]

q 1[-]

p2[-]

178

.45

2662

1330

810

.00

-0.026

60.92

14-0.405

3-0.405

32

160.26

3076

1538

110

.00

-0.030

8-0.046

70.34

16-0.388

33

67.00

2181

1090

310

.00

-0.021

80.52

190.15

91-0.724

44

72.26

2171

1085

510

.00

-0.021

70.45

520.21

14-0.724

45

75.37

2176

1088

110

.00

-0.021

80.41

860.27

98-0.774

86

73.01

2298

1149

010

.00

-0.023

00.33

400.44

45-0.900

0

TableA.5:Va

lues

fore

achite

ratio

ndu

ringop

timiza

tion


A.3 Two summations


10 12 4 1 1.00 6 0.094 1.086e+0411 13 5 1 0.59 8 0.0624 1.085e+0412 14 6 1 1.00 9 0.0493 1.084e+0413 16 7 2 1.00 10 0.212 1.102e+0414 18 8 2 0.29 12 0.239 1.082e+0415 19 9 1 1.00 13 0.153 1.081e+0416 20 10 1 1.00 14 0.01 1.081e+0417 22 11 2 0.00 20 0.0472 1.081e+0418 24 12 2 1.00 21 0.0492 1.081e+0419 26 13 2 1.00 23 0.0489 1.081e+0420 28 14 2 0.00 27 0.0493 1.081e+0421 29 15 1 0.50 29 0.0493 1.081e+0422 30 16 1 1.00 30 0.0493 1.081e+0423 32 17 1 0.00 54 0.00687 1.081e+0424 33 18 1 1.00 56 0.0306 1.081e+0425 Warning: Current point cannot be improved.26 Optimization Solver 1 in Solver 2: Solution time: 107 s (1 minute, 47 seconds)27 Physical memory: 856 MB28 Virtual memory: 5189 MB

A.3 Two summations 69


Figure A.6:




Figure A.7:

A.3 Two summations 71

Iteration#

σvM

,max[M

Pa]

U[J/m

3]

Utot[J]

Area[m

2]

Y-disp

[m]

p1[-]

q 1[-]

p2[-]

q 2[-]

179

.27

2630

1315

010

.00

-0.026

30.84

88-0.361

0-0.389

10.40

112

127.94

2850

1425

010

.00

-0.028

50.05

870.25

39-0.381

9-0.186

73

70.22

2177

1088

410

.00

-0.021

80.47

970.18

51-0.715

30.07

954

73.54

2172

1085

910

.00

-0.021

70.43

970.23

38-0.737

40.02

725

74.59

2172

1086

210

.00

-0.021

70.42

810.22

51-0.714

70.00

176

73.91

2171

1085

310

.00

-0.021

70.43

580.23

22-0.731

4-0.008

97

73.20

2168

1083

910

.00

-0.021

70.44

440.23

89-0.748

5-0.107

58

87.82

2204

1102

110

.00

-0.022

00.52

240.28

83-0.889

4-0.900

09

70.99

2164

1082

110

.00

-0.021

60.47

220.24

58-0.785

2-0.366

710

68.41

2161

1080

710

.00

-0.021

60.50

650.21

68-0.782

6-0.578

411

68.08

2162

1080

710

.00

-0.021

60.51

120.21

71-0.787

6-0.613

412

68.19

2161

1080

710

.00

-0.021

60.50

960.21

46-0.782

8-0.591

313

68.21

2161

1080

710

.00

-0.021

60.50

930.21

47-0.782

7-0.590

114

68.25

2161

1080

710

.00

-0.021

60.50

880.21

51-0.782

7-0.587

615

68.21

2161

1080

710

.00

-0.021

60.50

930.21

47-0.782

7-0.590

116

68.21

2161

1080

710

.00

-0.021

60.50

930.21

47-0.782

7-0.590

117

68.21

2161

1080

710

.00

-0.021

60.50

930.21

47-0.782

7-0.590

118

68.27

2161

1080

710

.00

-0.021

60.50

860.21

34-0.780

2-0.590

019

68.16

2161

1080

710

.00

-0.021

60.51

000.21

45-0.783

2-0.589

7

TableA.8:Va

lues

fore

achite

ratio

ndu

ringop

timiza

tion


A.4 Three summations


10 15 4 1 1.00 6 0.0912 1.096e+0411 16 5 1 1.00 8 0.244 1.089e+0412 17 6 1 1.00 9 0.432 1.083e+0413 18 7 1 1.00 10 0.216 1.08e+0414 19 8 1 1.00 11 0.042 1.078e+0415 20 9 1 1.00 12 0.0511 1.078e+0416 21 10 1 1.00 13 0.00975 1.078e+0417 22 11 1 1.00 14 0.00499 1.078e+0418 23 12 1 1.00 15 0.00814 1.078e+0419 24 13 1 1.00 17 0.00979 1.078e+0420 25 14 1 1.00 19 0.0123 1.078e+0421 26 15 1 1.00 20 0.0046 1.078e+0422 27 16 1 1.00 22 0.0033 1.078e+0423 28 17 1 1.00 23 0.00824 1.078e+0424 29 18 1 1.00 25 0.00892 1.078e+0425 30 19 1 1.00 26 0.00664 1.078e+0426 31 20 1 1.00 28 0.00762 1.078e+0427 32 21 1 1.00 29 0.00501 1.078e+0428 33 22 1 1.00 31 0.00651 1.078e+0429 34 23 1 1.00 32 0.00365 1.078e+0430 35 24 1 1.00 34 0.00482 1.078e+0431 36 25 1 1.00 36 0.00701 1.078e+0432 37 26 1 0.19 38 0.00171 1.078e+0433 Warning: Current point cannot be improved.34 Optimization Solver 1 in Solver 2: Solution time: 89 s (1 minute, 29 seconds)35 Physical memory: 854 MB36 Virtual memory: 5185 MB

A.4 Three summations 73


Figure A.9:




Figure A.10:

A.4 Three summations 75

Iteration#

σvM

,max[M

Pa]

U[J/m

3]

Utot[J]

Area[m

2]

Y-disp

[m]

p1[-]

q 1[-]

p2[-]

q 2[-]

q 3[-]

174

.41

2535

1267

710

.00

-0.025

40.82

56-0.345

6-0.372

50.38

40-0.308

42

126.51

2993

1496

410

.00

-0.029

90.05

140.25

81-0.355

7-0.192

0-0.573

83

67.81

2204

1102

010

.00

-0.022

00.51

700.12

67-0.658

00.10

66-0.477

54

70.50

2197

1098

710

.00

-0.022

00.48

210.15

62-0.661

80.06

31-0.452

35

71.96

2193

1096

310

.00

-0.021

90.46

340.19

38-0.693

00.01

11-0.404

16

74.64

2179

1089

310

.00

-0.021

80.42

970.26

07-0.754

0-0.160

5-0.181

37

73.21

2167

1083

410

.00

-0.021

70.44

490.25

77-0.772

5-0.243

9-0.012

78

72.05

2159

1079

510

.00

-0.021

60.45

450.27

19-0.824

6-0.572

10.56

319

72.22

2156

1078

210

.00

-0.021

60.45

390.26

48-0.806

4-0.457

80.36

1510

71.99

2156

1078

210

.00

-0.021

60.45

650.26

42-0.809

4-0.473

50.38

8011

71.85

2156

1078

210

.00

-0.021

60.45

810.26

33-0.809

4-0.470

10.37

9612

71.62

2156

1078

110

.00

-0.021

60.46

110.25

97-0.807

0-0.460

30.35

6713

71.62

2156

1078

110

.00

-0.021

60.46

120.25

97-0.807

0-0.462

40.35

7214

71.64

2156

1078

110

.00

-0.021

60.46

090.26

00-0.807

3-0.465

70.36

0315

71.69

2156

1078

110

.00

-0.021

60.46

030.26

02-0.806

8-0.468

10.35

9916

71.67

2156

1078

110

.00

-0.021

60.46

060.26

01-0.806

9-0.470

10.35

6417

71.66

2156

1078

110

.00

-0.021

60.46

080.26

01-0.806

9-0.472

50.35

1718

71.59

2156

1078

110

.00

-0.021

60.46

170.25

97-0.807

1-0.475

70.34

8619

71.51

2156

1078

110

.00

-0.021

60.46

270.25

92-0.807

4-0.479

50.34

4520

71.44

2156

1078

110

.00

-0.021

60.46

360.25

86-0.807

5-0.483

90.34

2121

71.34

2156

1078

110

.00

-0.021

60.46

480.25

79-0.807

6-0.489

50.33

8622

71.28

2156

1078

110

.00

-0.021

60.46

570.25

73-0.807

6-0.493

80.33

7723

71.19

2156

1078

110

.00

-0.021

60.46

680.25

65-0.807

6-0.499

50.33

6224

71.16

2156

1078

110

.00

-0.021

60.46

710.25

62-0.807

6-0.501

30.33

6825

71.13

2156

1078

110

.00

-0.021

60.46

760.25

58-0.807

6-0.503

80.33

7526

71.13

2156

1078

110

.00

-0.021

60.46

750.25

57-0.807

4-0.505

50.33

9727

71.13

2156

1078

110

.00

-0.021

60.46

760.25

58-0.807

6-0.504

00.33

78

TableA.11:

Values

fore

achite

ratio

ndu

ringop

timiza

tion


A.5 Four summations


10 19 4 1 1.00 6 0.158 1.102e+0411 20 5 1 1.00 8 0.234 1.103e+0412 21 6 1 0.40 10 0.113 1.096e+0413 22 7 1 1.00 11 0.33 1.082e+0414 23 8 1 1.00 12 0.11 1.08e+0415 24 9 1 1.00 13 0.0568 1.08e+0416 25 10 1 0.10 16 0.0267 1.079e+0417 26 11 1 1.00 17 0.0312 1.079e+0418 27 12 1 1.00 18 0.0405 1.078e+0419 28 13 1 1.00 19 0.0437 1.078e+0420 29 14 1 1.00 20 0.0523 1.078e+0421 30 15 1 1.00 21 0.0305 1.078e+0422 31 16 1 1.00 22 0.0194 1.078e+0423 32 17 1 1.00 23 0.0215 1.077e+0424 34 18 2 1.00 24 0.0239 1.09e+0425 36 19 2 0.17 26 0.0273 1.077e+0426 37 20 1 1.00 27 0.0159 1.077e+0427 38 21 1 1.00 28 0.0055 1.077e+0428 39 22 1 1.00 29 0.00936 1.077e+0429 40 23 1 1.00 31 0.0129 1.077e+0430 41 24 1 1.00 32 0.00454 1.077e+0431 42 25 1 1.00 34 0.00531 1.077e+0432 43 26 1 1.00 35 0.00227 1.077e+0433 44 27 1 1.00 37 0.00233 1.077e+0434 45 28 1 1.00 39 0.00219 1.077e+0435 46 29 1 1.00 41 0.00215 1.077e+0436 47 30 1 1.00 43 0.00256 1.089e+0437 48 31 1 0.00 48 0.00686 1.077e+0438 49 32 1 1.00 50 0.00764 1.077e+0439 50 33 1 1.00 51 0.00195 1.077e+0440 51 34 1 1.00 52 0.00473 1.077e+0441 52 35 1 0.24 56 0.00561 1.077e+0442 53 36 1 0.09 60 0.00653 1.077e+0443 54 37 1 1.00 61 0.00779 1.077e+0444 55 38 1 1.00 63 0.0118 1.077e+0445 56 39 1 0.00 67 0.00531 1.077e+0446 57 40 1 1.00 69 0.00566 1.077e+0447 58 41 1 1.00 70 0.00338 1.077e+0448 59 42 1 1.00 72 0.00477 1.077e+0449 60 43 1 0.14 78 0.00397 1.077e+0450 61 44 1 1.00 80 0.00456 1.077e+0451 62 45 1 1.00 82 0.00335 1.077e+0452 63 46 1 1.00 83 0.0117 1.077e+0453 64 47 1 0.01 90 0.00372 1.077e+0454 Warning: Current point cannot be improved.55 Optimization Solver 1 in Solver 2: Solution time: 162 s (2 minutes, 42 seconds)

A.5 Four summations 77

56 Physical memory: 854 MB57 Virtual memory: 5176 MB



Figure A.12:




Figure A.13:

80 A Cantilever BeamIteration#

σvM

,max[M

Pa]

U[J/m

3]

Utot[J]

Area[m

2]

Y-disp

[m]

p1[-]

q 1[-]

p2[-]

q 2[-]

q 3[-]

q 4[-]

165

.05

2480

1239

910

.00

-0.024

80.77

18-0.323

1-0.348

20.35

90-0.288

30.35

902

148.33

3093

1546

510

.00

-0.030

9-0.010

60.29

62-0.342

1-0.230

3-0.574

60.52

503

76.13

2238

1118

910

.00

-0.022

40.49

460.17

25-0.694

20.10

49-0.471

30.44

714

71.66

2214

1107

010

.00

-0.022

10.46

140.10

28-0.574

80.09

74-0.414

30.40

025

71.81

2204

1102

210

.00

-0.022

00.45

920.13

84-0.618

90.08

15-0.392

20.37

966

73.20

2205

1102

710

.00

-0.022

10.44

180.25

05-0.749

7-0.001

6-0.261

70.27

527

72.18

2191

1095

710

.00

-0.021

90.45

450.18

75-0.681

50.03

05-0.277

40.28

938

71.97

2165

1082

410

.00

-0.021

70.45

790.22

08-0.744

1-0.099

60.11

91-0.010

59

73.32

2159

1079

710

.00

-0.021

60.44

240.26

40-0.793

4-0.208

10.35

19-0.180

410

72.93

2160

1080

110

.00

-0.021

60.44

680.24

80-0.773

3-0.177

00.25

18-0.099

911

73.49

2159

1079

410

.00

-0.021

60.44

040.26

57-0.792

3-0.220

80.32

26-0.148

112

73.03

2157

1078

710

.00

-0.021

60.44

540.26

54-0.795

8-0.274

90.29

35-0.093

713

72.16

2157

1078

410

.00

-0.021

60.45

560.26

05-0.799

5-0.335

10.28

76-0.058

314

71.42

2156

1078

110

.00

-0.021

60.46

450.25

50-0.801

9-0.399

10.29

94-0.042

415

70.84

2156

1077

810

.00

-0.021

60.47

180.24

93-0.803

1-0.464

10.32

73-0.049

116

70.76

2155

1077

710

.00

-0.021

60.47

300.24

87-0.803

9-0.480

40.33

77-0.062

917

70.71

2155

1077

510

.00

-0.021

60.47

450.24

98-0.807

8-0.517

00.35

94-0.111

318

70.68

2155

1077

410

.00

-0.021

60.47

500.25

15-0.810

2-0.524

00.35

46-0.121

119

88.26

2181

1090

410

.00

-0.021

80.50

620.31

32-0.914

2-0.900

00.21

85-0.579

120

70.11

2154

1077

110

.00

-0.021

50.48

380.25

97-0.828

4-0.602

80.32

91-0.186

521

69.96

2154

1077

110

.00

-0.021

50.48

550.25

63-0.825

9-0.603

30.33

18-0.170

322

69.97

2154

1077

110

.00

-0.021

50.48

550.25

71-0.827

0-0.606

50.33

12-0.179

323

69.93

2154

1077

110

.00

-0.021

50.48

610.25

68-0.827

2-0.610

40.33

10-0.183

624

69.88

2154

1077

110

.00

-0.021

50.48

690.25

66-0.827

6-0.615

40.33

12-0.190

925

69.84

2154

1077

110

.00

-0.021

50.48

730.25

61-0.827

4-0.617

60.33

04-0.188

726

69.79

2154

1077

110

.00

-0.021

50.48

800.25

54-0.827

2-0.621

30.32

94-0.185

327

69.78

2154

1077

110

.00

-0.021

50.48

810.25

53-0.827

1-0.621

00.32

85-0.185

828

69.78

2154

1077

110

.00

-0.021

50.48

820.25

53-0.827

1-0.620

70.32

76-0.186

529

69.77

2154

1077

110

.00

-0.021

50.48

830.25

52-0.827

1-0.619

10.32

64-0.187

730

69.76

2154

1077

110

.00

-0.021

50.48

850.25

51-0.827

1-0.617

00.32

52-0.188

9


69.37

2178

1088

810

.00

-0.021

80.57

010.22

27-0.864

7-0.575

30.26

15-0.216

132

69.76

2154

1077

110

.00

-0.021

50.48

850.25

51-0.827

1-0.617

00.32

52-0.188

933

69.79

2154

1077

110

.00

-0.021

50.48

810.25

53-0.826

9-0.617

20.32

55-0.188

834

69.79

2154

1077

110

.00

-0.021

50.48

810.25

53-0.826

9-0.617

20.32

55-0.188

835

69.68

2154

1077

110

.00

-0.021

50.48

950.25

42-0.826

9-0.616

50.32

40-0.189

236

69.74

2154

1077

110

.00

-0.021

50.48

880.25

47-0.826

9-0.616

90.32

47-0.189

037

69.76

2154

1077

110

.00

-0.021

50.48

850.25

50-0.826

9-0.617

00.32

50-0.188

938

69.67

2154

1077

110

.00

-0.021

50.48

960.25

37-0.826

3-0.616

60.32

32-0.189

339

69.45

2154

1077

110

.00

-0.021

50.49

250.25

04-0.824

8-0.615

50.31

82-0.190

340

69.68

2154

1077

110

.00

-0.021

50.48

960.25

37-0.826

3-0.616

60.32

32-0.189

341

69.71

2154

1077

110

.00

-0.021

50.48

910.25

42-0.826

5-0.616

80.32

39-0.189

142

69.72

2154

1077

110

.00

-0.021

50.48

900.25

44-0.826

6-0.616

90.32

41-0.189

143

69.74

2154

1077

110

.00

-0.021

50.48

880.25

46-0.826

6-0.616

90.32

44-0.189

044

69.73

2154

1077

110

.00

-0.021

50.48

890.25

44-0.826

5-0.616

90.32

41-0.189

145

69.73

2154

1077

110

.00

-0.021

50.48

890.25

43-0.826

5-0.616

90.32

41-0.189

146

69.73

2154

1077

110

.00

-0.021

50.48

890.25

43-0.826

5-0.616

90.32

41-0.189

147

70.11

2154

1077

210

.00

-0.021

50.48

410.25

39-0.821

1-0.617

90.32

03-0.188

948

69.73

2154

1077

110

.00

-0.021

50.48

890.25

43-0.826

5-0.616

90.32

40-0.189

1

TableA.14:

Values

fore

achite

ratio

ndu

ringop

timiza

tion


A.6 Five summations


10 22 4 1 1.00 6 0.197 1.103e+0411 23 5 1 1.00 8 0.287 1.105e+0412 24 6 1 0.38 10 0.0967 1.098e+0413 25 7 1 1.00 11 0.283 1.084e+0414 26 8 1 1.00 12 0.116 1.08e+0415 27 9 1 1.00 13 0.0445 1.079e+0416 29 10 2 1.00 14 0.0424 1.116e+0417 31 11 2 0.14 17 0.0727 1.078e+0418 32 12 1 1.00 18 0.0872 1.077e+0419 33 13 1 1.00 19 0.117 1.077e+0420 34 14 1 1.00 20 0.057 1.077e+0421 35 15 1 1.00 21 0.0175 1.077e+0422 36 16 1 1.00 22 0.00669 1.077e+0423 37 17 1 1.00 23 0.00439 1.077e+0424 38 18 1 1.00 24 0.00568 1.077e+0425 39 19 1 1.00 25 0.00632 1.077e+0426 40 20 1 1.00 26 0.0046 1.077e+0427 41 21 1 0.21 28 0.00424 1.077e+0428 42 22 1 1.00 29 0.0241 1.077e+0429 43 23 1 1.00 30 0.0336 1.077e+0430 44 24 1 1.00 31 0.0255 1.077e+0431 45 25 1 1.00 32 0.00682 1.077e+0432 46 26 1 1.00 34 0.0159 1.077e+0433 47 27 1 1.00 35 0.00234 1.077e+0434 48 28 1 1.00 37 0.00285 1.077e+0435 49 29 1 1.00 39 0.00359 1.077e+0436 Warning: Current point cannot be improved.37 Optimization Solver 1 in Solver 2: Solution time: 104 s (1 minute, 44 seconds)38 Physical memory: 857 MB39 Virtual memory: 5185 MB

A.6 Five summations 83


Figure A.15:




Figure A.16:

A.6 Five summations 85

Iteration#

σvM

,max[M

Pa]

U[J/m

3]

Utot[J]

Area[m

2]

Y-disp

[m]

p1[-]

q 1[-]

p2[-]

q 2[-]

q 3[-]

q 4[-]

q 5[-]

165

.64

2483

1241

710

.00

-0.024

80.79

95-0.333

4-0.359

40.37

05-0.297

40.37

05-0.115

02

134.14

3098

1548

810

.00

-0.031

00.01

800.27

97-0.337

7-0.213

7-0.574

50.36

34-0.476

03

82.26

2245

1122

410

.00

-0.022

50.51

170.15

46-0.679

90.10

79-0.480

40.37

53-0.326

04

70.42

2216

1107

910

.00

-0.022

20.48

510.06

52-0.543

10.10

92-0.423

60.33

54-0.279

85

70.38

2205

1102

710

.00

-0.022

10.48

510.10

11-0.589

40.09

45-0.407

90.31

64-0.276

16

72.41

2209

1104

610

.00

-0.022

10.47

030.20

98-0.717

90.01

68-0.313

00.22

96-0.247

67

70.63

2195

1097

610

.00

-0.022

00.48

070.14

63-0.646

90.04

97-0.327

40.24

99-0.241

28

71.02

2168

1083

910

.00

-0.021

70.47

200.19

76-0.724

0-0.100

10.04

09-0.024

6-0.054

19

72.87

2159

1079

510

.00

-0.021

60.44

700.26

64-0.803

4-0.256

80.34

86-0.248

70.09

4810

73.52

2158

1078

910

.00

-0.021

60.43

910.28

22-0.817

2-0.293

90.37

86-0.259

10.10

5811

225.97

2231

1115

510

.00

-0.022

30.37

870.43

51-0.970

0-0.699

00.73

00-0.376

60.25

0312

73.41

2156

1077

810

.00

-0.021

60.43

970.29

60-0.836

0-0.372

90.39

28-0.214

50.10

8313

72.34

2155

1077

310

.00

-0.021

50.45

210.28

25-0.829

4-0.408

90.35

74-0.141

00.09

1514

71.95

2154

1077

010

.00

-0.021

50.45

650.27

73-0.827

4-0.445

20.36

05-0.116

00.09

4515

71.68

2153

1076

710

.00

-0.021

50.45

910.27

84-0.834

1-0.545

30.40

95-0.100

30.12

2016

71.66

2153

1076

710

.00

-0.021

50.45

950.27

59-0.830

8-0.524

40.39

88-0.108

30.11

8517

71.72

2153

1076

710

.00

-0.021

50.45

890.27

66-0.831

3-0.519

10.40

19-0.123

80.12

2818

71.64

2153

1076

710

.00

-0.021

50.45

980.27

68-0.832

8-0.530

20.40

62-0.133

60.12

9419

71.39

2153

1076

610

.00

-0.021

50.46

290.27

79-0.837

9-0.563

50.41

19-0.152

50.14

3920

71.13

2153

1076

610

.00

-0.021

50.46

620.27

95-0.843

4-0.594

70.41

13-0.167

30.15

4621

70.96

2153

1076

610

.00

-0.021

50.46

850.28

03-0.846

7-0.613

00.40

63-0.174

60.15

8522

71.00

2153

1076

610

.00

-0.021

50.46

800.27

88-0.843

9-0.600

50.40

00-0.166

40.15

1723

70.78

2153

1076

610

.00

-0.021

50.47

110.27

61-0.842

4-0.604

20.37

90-0.164

80.14

4724

70.64

2153

1076

510

.00

-0.021

50.47

300.27

40-0.841

2-0.608

50.36

84-0.167

40.14

1925

70.52

2153

1076

510

.00

-0.021

50.47

480.27

17-0.839

7-0.614

40.36

11-0.172

70.13

9926

70.53

2153

1076

510

.00

-0.021

50.47

480.27

15-0.839

3-0.614

20.36

20-0.173

60.13

8727

70.52

2153

1076

510

.00

-0.021

50.47

500.27

06-0.838

4-0.614

70.36

17-0.175

60.13

6728

70.54

2153

1076

510

.00

-0.021

50.47

460.27

10-0.838

5-0.612

80.36

28-0.174

00.13

5929

70.58

2153

1076

510

.00

-0.021

50.47

420.27

13-0.838

5-0.609

50.36

32-0.171

10.13

4130

70.53

2153

1076

510

.00

-0.021

50.47

480.27

13-0.838

9-0.606

30.36

18-0.166

60.13

22

TableA.17:

Values

fore

achite

ratio

ndu

ringop

timiza

tion

APPENDIX BPlate with a hole

B.1 Tensile ratio of 1Standard plate

1 ============================================================2 Stationary Solver 1 in Solver 1 started at 19-jun-2014 12:48:43.3 Nonlinear solver4 Number of degrees of freedom solved for: 34708.5 Nonsymmetric matrix found.6 Scales for dependent variables:7 comp1.u: 6.1e-058 comp1.XY: 109 Iter ErrEst Damping Stepsize #Res #Jac #Sol LinErr LinRes

10 1 5.5e-13 1.0000000 0.7 2 1 2 7.4e-11 8.7e-1411 2 2.1e-16 1.0000000 5.5e-13 3 2 4 4.2e-11 3.4e-1612 Stationary Solver 1 in Solver 1: Solution time: 2 s13 Physical memory: 864 MB14 Virtual memory: 5254 MB

88 B Plate with a hole

Figure B.1: v. Mises stress

Figure B.2: The error of the solution as a function of iteration number for the nonlinearsolver

B.1 Tensile ratio of 1 89

Optimized plate

1 ============================================================2 Optimization Solver 1 in Solver 2 started at 19-jun-2014 12:36:08.3 Optimization solver (SNOPT)4 Analytic gradient with the adjoint method.5 Warning: New constraint force nodes detected: These are not stored.6 Itns Major Minor Step nPDE Error Objective7 2 0 2 - 1 0.0151 35.698 4 1 2 1.00 2 0.0126 35.889 6 2 2 0.06 4 0.00824 35.69

10 8 3 2 0.00 10 0.00462 35.6911 10 4 2 1.00 12 0.00325 35.8512 12 5 2 0.01 15 0.00313 35.6913 13 6 1 0.03 17 0.00114 35.6914 14 7 1 1.00 19 0.00136 35.6915 15 8 1 1.00 20 0.000671 35.6916 16 9 1 1.00 22 0.000951 35.6917 17 10 1 0.15 24 0.000883 35.6918 18 11 1 0.01 28 0.000118 35.6919 19 12 1 1.00 30 0.000108 35.6920 20 13 1 0.65 32 0.000105 35.6921 21 14 1 1.00 33 0.000104 35.6922 Number of optimization variables: 3.23 Number of objective function evaluations: 57.24 Number of Jacobian evaluations: 55.25 Final objective function value: 35.69020753.26 Warning: Requested accuracy could not be achieved.27 Optimization Solver 1 in Solver 2: Solution time: 179 s (2 minutes, 59 seconds)28 Physical memory: 910 MB29 Virtual memory: 5265 MB

Iteration # σvM,max [MPa] U [J/m3] Utot [J] Area [m2] q1 [-] q2 [-] q3 [-]1.00 2.11 4.08 35.69 87.43 0.01 -0.02 0.012 2.94 4.10 35.88 87.44 0.56 -0.99 0.223 2.16 4.08 35.69 87.43 0.11 -0.12 -0.014 2.08 4.08 35.69 87.43 0.05 -0.09 0.025 2.68 4.10 35.85 87.48 0.20 -1.00 0.546 2.13 4.08 35.69 87.43 -0.03 -0.10 0.107 2.09 4.08 35.69 87.43 0.05 -0.10 0.038 2.08 4.08 35.69 87.43 0.05 -0.10 0.039 2.08 4.08 35.69 87.43 0.05 -0.09 0.0310 2.08 4.08 35.69 87.43 0.04 -0.09 0.0311 2.08 4.08 35.69 87.43 0.05 -0.09 0.0312 2.08 4.08 35.69 87.43 0.05 -0.09 0.0313 2.08 4.08 35.69 87.43 0.05 -0.09 0.0314 2.08 4.08 35.69 87.43 0.05 -0.09 0.0315 2.08 4.08 35.69 87.43 0.05 -0.09 0.03

Table B.3: Values for each iteration during optimization


Figure B.4: Surface v. Mises stress

Figure B.5: Surface v. Mises stress showing the whole plate


Figure B.6: Graph comparing the v. Mises stress for the original shape with the optimizeddesign

Figure B.7: The error of the solution as a function of iteration number for the optimizationsolver




1 ============================================================2 Stationary Solver 1 in Solver 1 started at 19-jun-2014 11:17:14.3 Nonlinear solver4 Number of degrees of freedom solved for: 34708.5 Nonsymmetric matrix found.6 Scales for dependent variables:7 comp1.u: 6.1e-058 comp1.XY: 109 Iter ErrEst Damping Stepsize #Res #Jac #Sol LinErr LinRes

10 1 7.1e-13 1.0000000 0.68 2 1 2 5.3e-11 6.6e-1411 2 1e-15 1.0000000 7.1e-13 3 2 4 2.9e-11 5.4e-1612 Stationary Solver 1 in Solver 1: Solution time: 2 s13 Physical memory: 1.07 GB14 Virtual memory: 5.45 GB


Optimized plate

1 ============================================================2 Optimization Solver 1 in Solver 2 started at 19-jun-2014 11:57:41.3 Optimization solver (SNOPT)4 Analytic gradient with the adjoint method.5 Itns Major Minor Step nPDE Error Objective6 2 0 2 - 1 0.892 102.37 3 1 1 1.00 2 0.649 98.568 4 2 1 0.64 3 0.617 96.69 5 3 1 1.00 4 0.077 97.06

10 6 4 1 1.00 5 0.0449 97.0611 7 5 1 1.00 6 0.0235 97.0512 8 6 1 1.00 7 0.00767 97.0513 9 7 1 1.00 8 0.00385 97.0514 10 8 1 1.00 9 0.000274 97.0515 11 9 1 1.00 11 0.000305 97.0516 12 10 1 0.07 14 3.39e-05 97.0517 Number of optimization variables: 3.18 Number of objective function evaluations: 16.19 Number of Jacobian evaluations: 14.20 Final objective function value: 97.05107473.21 Optimality conditions satisfied.22 Optimization Solver 1 in Solver 2: Solution time: 16 s23 Physical memory: 712 MB24 Virtual memory: 5048 MB

Iteration # σvM,max [MPa] U [J/m3] Utot [J] Area [m2] q1 [-] q2 [-] q3 [-]1 5.20 11.69 102.32 87.53 0.62 0.23 -0.692 4.25 11.25 98.56 87.58 1.42 0.48 -1.373 3.35 11.03 96.60 87.61 2.70 0.64 -2.124 3.35 11.10 97.06 87.44 3.03 0.61 -2.185 3.29 11.10 97.06 87.43 2.98 0.56 -2.136 3.27 11.10 97.05 87.43 2.96 0.46 -2.097 3.28 11.10 97.05 87.43 2.98 0.42 -2.088 3.28 11.10 97.05 87.43 3.00 0.40 -2.089 3.28 11.10 97.05 87.43 3.00 0.40 -2.0810 3.28 11.10 97.05 87.43 3.00 0.40 -2.0811 3.28 11.10 97.05 87.43 3.00 0.40 -2.08





1 Stationary Solver 1 in Solver 1 started at 19-jun-2014 11:06:38.2 Nonlinear solver3 Number of degrees of freedom solved for: 34708.4 Nonsymmetric matrix found.5 Scales for dependent variables:6 comp1.u: 6.1e-057 comp1.XY: 108 Iter ErrEst Damping Stepsize #Res #Jac #Sol LinErr LinRes9 1 6.9e-13 1.0000000 0.67 2 1 2 7.6e-11 8.8e-14

10 2 5.7e-16 1.0000000 6.9e-13 3 2 4 3.8e-11 5.4e-1611 Stationary Solver 1 in Solver 1: Solution time: 2 s12 Physical memory: 1.03 GB13 Virtual memory: 5.45 GB


Optimized plate

1 ============================================================2 Number of vertex elements: 53 Number of boundary elements: 1924 Number of elements: 21215 Minimum element quality: 0.42946 Number of vertex elements: 57 Optimization Solver 1 in Solver 2 started at 19-jun-2014 03:18:29.8 Optimization solver (SNOPT)9 Analytic gradient with the adjoint method.

10 Warning: New constraint force nodes detected: These are not stored.11 Itns Major Minor Step nPDE Error Objective12 2 0 2 - 1 1.1 236.613 3 1 1 1.00 2 0.828 225.114 4 2 1 0.62 3 1.62 207.615 5 3 1 1.00 4 0.296 209.616 6 4 1 1.00 5 0.134 21017 7 5 1 1.00 6 0.107 21018 8 6 1 0.27 8 0.0319 21019 9 7 1 1.00 9 0.0258 209.920 10 8 1 1.00 10 0.0033 209.921 12 9 2 1.00 11 0.00604 210.322 15 10 1 0.00 22 0.000981 209.923 16 11 1 0.00 26 0.00081 209.924 17 12 1 1.00 27 0.00286 21025 18 13 1 0.01 31 0.000622 209.926 19 14 1 0.24 33 0.000438 209.927 20 15 1 1.00 34 0.000443 209.928 21 16 1 1.00 36 0.000618 209.929 22 17 1 0.20 38 0.000366 209.930 23 18 1 1.00 39 0.000501 209.931 24 19 1 1.00 40 0.000561 209.932 25 20 1 1.00 42 0.000719 209.933 26 21 1 0.00 46 0.000225 209.934 27 22 1 1.00 48 0.000232 209.935 28 23 1 1.00 50 0.00138 209.936 29 24 1 1.00 51 0.000972 209.937 30 25 1 1.00 52 0.000245 209.938 31 26 1 1.00 54 0.000248 209.939 32 27 1 1.00 56 0.000504 209.940 33 28 1 1.00 57 0.000228 209.941 34 29 1 1.00 58 0.000186 209.942 35 30 1 1.00 59 0.000776 209.943 36 31 1 0.05 63 6.46e-05 209.944 37 32 1 1.00 64 0.000142 209.945 38 33 1 0.07 67 1.23e-05 209.946 40 34 1 0.50 92 1.6e-05 209.947 41 35 1 0.25 95 7.98e-06 209.948 Number of optimization variables: 3.49 Number of objective function evaluations: 120.50 Number of Jacobian evaluations: 118.51 Final objective function value: 209.9367235.52 Warning: Requested accuracy could not be achieved.53 Optimization Solver 1 in Solver 2: Solution time: 377 s (6 minutes, 17 seconds)54 Physical memory: 1.09 GB55 Virtual memory: 5.45 GB


Iteration # σvM,max [MPa] U [J/m3] Utot [J] Area [m2] q1 [-] q2 [-] q3 [-]1 8.82 27.03 236.65 87.54 0.62 0.30 -0.732 7.46 25.71 225.14 87.59 1.37 0.61 -1.403 4.88 23.57 207.62 88.10 3.67 1.10 -2.904 4.92 23.95 209.63 87.53 5.20 1.35 -3.245 4.77 24.02 210.02 87.44 5.11 1.23 -3.156 4.88 24.00 210.01 87.49 4.51 0.68 -2.827 4.53 24.01 209.96 87.43 5.01 1.00 -3.058 4.50 24.01 209.93 87.43 5.13 0.72 -3.019 4.48 24.01 209.94 87.43 5.12 0.74 -3.0210 5.78 24.04 210.34 87.48 5.03 2.00 -3.3411 4.48 24.01 209.94 87.43 5.12 0.74 -3.0212 4.48 24.01 209.94 87.43 5.12 0.75 -3.0213 4.72 24.01 209.97 87.44 4.94 0.52 -2.9014 4.49 24.01 209.94 87.43 5.12 0.74 -3.0215 4.49 24.01 209.94 87.43 5.12 0.74 -3.0216 4.49 24.01 209.94 87.43 5.12 0.75 -3.0217 4.48 24.01 209.94 87.43 5.12 0.75 -3.0218 4.49 24.01 209.94 87.43 5.12 0.75 -3.0219 4.49 24.01 209.94 87.43 5.12 0.75 -3.0220 4.49 24.01 209.94 87.43 5.12 0.75 -3.0221 4.49 24.01 209.94 87.43 5.12 0.74 -3.0222 4.49 24.01 209.94 87.43 5.12 0.75 -3.0223 4.49 24.01 209.94 87.43 5.12 0.75 -3.0224 4.49 24.01 209.94 87.43 5.12 0.75 -3.0225 4.49 24.01 209.94 87.43 5.12 0.75 -3.0226 4.49 24.01 209.94 87.43 5.12 0.75 -3.0227 4.49 24.01 209.94 87.43 5.12 0.75 -3.0228 4.49 24.01 209.94 87.43 5.12 0.75 -3.0229 4.49 24.01 209.94 87.43 5.12 0.75 -3.0230 4.49 24.01 209.94 87.43 5.12 0.75 -3.0231 4.48 24.01 209.94 87.43 5.12 0.75 -3.0232 4.49 24.01 209.94 87.43 5.12 0.75 -3.0233 4.48 24.01 209.94 87.43 5.12 0.75 -3.0234 4.49 24.01 209.94 87.43 5.12 0.75 -3.0235 4.49 24.01 209.94 87.43 5.12 0.75 -3.0236 4.49 24.01 209.94 87.43 5.12 0.75 -3.02


APPENDIX CFillet

C.1 Standard

1 ============================================================2 Number of vertex elements: 73 Number of boundary elements: 964 Number of elements: 10005 Minimum element quality: 0.8936 Stationary Solver 1 in Solver 1 started at 24-jun-2014 00:18:20.7 Nonlinear solver8 Number of degrees of freedom solved for: 8388.9 Nonsymmetric matrix found.

10 Scales for dependent variables:11 comp1.RZ: 3212 comp1.u: 1.2e-0513 Iter ErrEst Damping Stepsize #Res #Jac #Sol LinErr LinRes14 1 1.5e-14 1.0000000 0.67 2 1 2 2e-12 1.4e-1415 2 3.2e-16 1.0000000 1.5e-14 3 2 4 3.7e-11 5.1e-1616 Stationary Solver 1 in Solver 1: Solution time: 0 s17 Physical memory: 821 MB18 Virtual memory: 5224 MB

106 C Fillet

Figure C.1: v. Mises stress in 2D


C.2 One summation 107

Figure C.3: The error of the solution as a function of iteration number for the nonlinearsolver

C.2 One summation

1 ============================================================2 Optimization Solver 1 in Solver 2 started at 24-jun-2014 01:21:15.3 Optimization solver (SNOPT)4 Analytic gradient with the adjoint method.5 Itns Major Minor Step nPDE Error Objective6 2 0 2 - 1 0.569 1.92e+057 3 1 1 0.18 2 0.507 1.914e+058 4 2 1 0.19 3 0.326 1.89e+059 5 3 1 0.60 4 0.204 1.88e+05

10 6 4 1 1.00 5 0.0499 1.87e+0511 7 5 1 1.00 6 0.0547 1.867e+0512 9 6 2 1.00 7 0.0354 1.866e+0513 10 7 1 1.00 8 0.0143 1.866e+0514 11 8 1 1.00 9 0.00237 1.866e+0515 12 9 1 1.00 10 0.000995 1.866e+0516 13 10 1 1.00 11 5.62e-05 1.866e+0517 Number of optimization variables: 3.18 Number of objective function evaluations: 13.19 Number of Jacobian evaluations: 11.20 Final objective function value: 186574.0434.21 Optimality conditions satisfied.22 Optimization Solver 1 in Solver 2: Solution time: 12 s23 Physical memory: 881 MB24 Virtual memory: 5296 MB

108 C Fillet



110 C Fillet

Figure C.8: The error of the solution as a function of iteration number for the optimizationsolver


Iteration#

σvM

,max[M

Pa]

Utot[kJ]

U[J/m

3]

Area[m

2]

p1[-]

p2[-]

q 1[-]

16.57

191.98

10.30

524.99

0.92

-0.46

-0.36

26.40

191.39

10.25

524.99

1.84

-0.93

-0.70

35.37

189.02

10.06

524.78

7.44

-3.69

-2.73

44.75

187.95

9.97

524.46

14.90

-7.32

-4.15

54.62

187.00

9.88

524.90

19.99

-8.38

-5.23

64.49

186.69

9.85

525.00

23.45

-8.55

-7.37

74.41

186.60

9.85

524.99

26.00

-8.12

-10.28

84.43

186.59

9.85

524.99

26.00

-7.79

-10.90

94.43

186.58

9.85

525.00

26.00

-7.66

-11.14

104.44

186.57

9.85

525.00

26.00

-7.60

-11.24

114.44

186.57

9.85

525.00

26.00

-7.60

-11.24

TableC.9:Va

lues

fore

achite

ratio

ndu

ringop

timiza

tion

112 C Fillet

C.3 Two summations


10 7 4 1 1.00 5 0.0644 1.871e+0511 8 5 1 1.00 6 0.0386 1.868e+0512 9 6 1 1.00 7 0.0243 1.866e+0513 10 7 1 1.00 8 0.0342 1.865e+0514 11 8 1 1.00 9 0.0162 1.865e+0515 12 9 1 1.00 10 0.01 1.865e+0516 13 10 1 1.00 11 0.00133 1.865e+0517 14 11 1 1.00 13 0.00252 1.865e+0518 15 12 1 0.01 17 0.000688 1.865e+0519 16 13 1 0.10 21 8.22e-05 1.865e+0520 Number of optimization variables: 4.21 Number of objective function evaluations: 23.22 Number of Jacobian evaluations: 21.23 Final objective function value: 186485.925.24 Optimality conditions satisfied.25 Optimization Solver 1 in Solver 2: Solution time: 18 s26 Physical memory: 880 MB27 Virtual memory: 5301 MB

C.3 Two summations 113



114 C Fillet



C.3 Two summations 115


116 C FilletIteration#

σvM

,max[M

Pa]

Utot[kJ]

U[J/m

3]

Area[m

2]

p1[-]

p2[-]

q 1[-]

q 2[-]

16.75

191.90

10.29

525.00

0.78

-0.39

-0.31

0.53

26.73

191.26

10.24

524.99

1.60

-0.79

-0.62

0.99

35.49

188.62

10.02

524.83

8.03

-3.21

-3.62

1.14

45.07

187.51

9.93

524.82

13.91

-5.30

-5.74

1.79

54.76

187.06

9.89

524.83

19.44

-7.14

-7.27

0.92

64.41

186.81

9.87

524.88

24.64

-8.49

-8.68

-0.25

74.32

186.58

9.85

525.00

26.49

-8.55

-9.75

-0.69

84.15

186.51

9.84

524.99

30.33

-8.25

-13.39

-1.96

94.13

186.49

9.84

525.00

31.57

-8.11

-14.64

-2.12

104.14

186.49

9.84

525.00

31.76

-8.06

-14.89

-2.01

114.14

186.49

9.84

525.00

31.72

-8.06

-14.85

-1.97

124.15

186.49

9.84

525.00

31.67

-8.08

-14.77

-1.91

134.15

186.49

9.84

525.00

31.70

-8.06

-14.84

-1.97

144.14

186.49

9.84

525.00

31.72

-8.06

-14.85

-1.97

TableC.15:

Values

fore

achite

ratio

ndu

ringop

timiza

tion

C.4 Three summations 117

C.4 Three summations


118 C Fillet



120 C Fillet



Iteration#

σvM

,max[M

Pa]

Utot[kJ]

U[J/m

3]

Area[m

2]

p1[-]

p2[-]

q 1[-]

q 2[-]

q 3[-]

16.84

191.89

10.29

525.00

0.71

-0.40

-0.33

0.51

0.28

26.86

191.23

10.24

524.99

1.55

-0.84

-0.68

0.96

0.44

35.43

188.88

10.04

524.83

7.78

-3.26

-2.98

1.81

-1.18

45.09

187.57

9.93

524.76

14.53

-5.69

-6.22

1.00

1.26

54.90

187.00

9.88

524.87

19.43

-7.20

-7.29

1.47

0.63

64.64

186.81

9.86

524.91

23.44

-8.38

-8.16

0.72

0.74

74.49

186.64

9.85

524.99

25.50

-8.66

-8.95

0.20

0.63

84.23

186.56

9.84

525.00

28.60

-8.76

-10.98

-1.05

0.14

94.05

186.50

9.84

525.00

31.92

-8.52

-13.89

-2.46

-0.55

103.98

186.48

9.84

524.99

34.73

-8.09

-16.79

-3.57

-1.18

113.99

186.45

9.84

525.00

35.35

-7.88

-17.64

-3.67

-1.25

124.01

186.45

9.84

525.00

35.33

-7.71

-17.99

-3.52

-1.16

134.02

186.44

9.84

525.00

35.24

-7.65

-18.02

-3.48

-1.13

144.02

186.44

9.84

525.00

35.14

-7.64

-17.97

-3.47

-1.12

154.03

186.45

9.84

525.00

34.86

-7.50

-18.02

-3.43

-1.10

164.02

186.44

9.84

525.00

35.09

-7.63

-17.95

-3.46

-1.12

174.02

186.44

9.84

525.00

35.05

-7.63

-17.91

-3.45

-1.13

184.02

186.44

9.84

525.00

35.01

-7.61

-17.93

-3.45

-1.13

194.02

186.44

9.84

525.00

35.07

-7.63

-17.93

-3.46

-1.12

TableC.21:

Values

fore

achite

ratio

ndu

ringop

timiza

tion

122 C Fillet

C.5 Four summations


10 9 4 1 1.00 5 0.0653 1.871e+0511 10 5 1 1.00 6 0.138 1.868e+0512 11 6 1 1.00 7 0.048 1.867e+0513 12 7 1 1.00 8 0.0251 1.866e+0514 13 8 1 1.00 9 0.0278 1.865e+0515 14 9 1 1.00 10 0.022 1.865e+0516 15 10 1 1.00 11 0.025 1.867e+0517 16 11 1 0.38 13 0.0182 1.865e+0518 17 12 1 1.00 14 0.0153 1.864e+0519 18 13 1 1.00 15 0.018 1.864e+0520 19 14 1 1.00 16 0.0107 1.864e+0521 20 15 1 1.00 17 0.000933 1.864e+0522 21 16 1 1.00 19 0.00347 1.864e+0523 22 17 1 1.00 20 0.0013 1.864e+0524 23 18 1 1.00 22 0.0031 1.864e+0525 24 19 1 0.32 24 0.000521 1.864e+0526 25 20 1 1.00 26 0.00098 1.864e+0527 26 21 1 0.01 30 0.0004 1.864e+0528 27 22 1 0.00 34 7.01e-05 1.864e+0529 Number of optimization variables: 6.30 Number of objective function evaluations: 36.31 Number of Jacobian evaluations: 34.32 Final objective function value: 186426.2261.33 Optimality conditions satisfied.34 Optimization Solver 1 in Solver 2: Solution time: 28 s35 Physical memory: 879 MB36 Virtual memory: 5299 MB

C.5 Four summations 123



124 C Fillet



C.5 Four summations 125


126 C FilletIteration#

σvM

,max[M

Pa]

Utot[kJ]

U[J/m

3]

Area[m

2]

p1[-]

p2[-]

q 1[-]

q 2[-]

q 3[-]

q 4[-]

17.01

191.86

10.29

525.00

0.63

-0.36

-0.30

0.45

0.25

0.46

27.10

191.17

10.23

524.99

1.47

-0.78

-0.64

0.91

0.35

0.75

35.66

188.84

10.04

524.83

7.60

-3.18

-3.06

2.65

-0.77

-0.16

45.31

187.46

9.92

524.86

13.48

-5.01

-5.81

1.70

0.30

1.04

55.18

187.13

9.89

524.90

18.07

-6.43

-7.81

1.50

1.77

0.21

64.96

186.83

9.87

524.91

22.70

-7.69

-9.07

1.25

1.17

0.75

74.78

186.67

9.85

524.98

25.01

-8.23

-9.71

0.59

1.24

0.76

84.57

186.61

9.85

524.99

26.90

-8.54

-10.43

-0.07

1.01

0.65

94.18

186.54

9.84

525.00

30.08

-8.75

-12.16

-1.69

0.07

0.22

104.04

186.50

9.84

525.00

31.46

-8.60

-13.35

-2.37

-0.54

-0.11

113.95

186.66

9.86

524.93

37.63

-7.60

-19.39

-5.69

-3.37

-1.58

123.88

186.45

9.84

525.00

34.59

-8.11

-16.44

-3.64

-1.69

-0.75

133.90

186.44

9.84

525.00

35.80

-7.94

-17.73

-3.71

-1.69

-0.71

143.91

186.43

9.84

525.00

36.79

-7.80

-18.74

-4.04

-1.76

-0.64

153.92

186.43

9.84

525.00

37.58

-7.73

-19.50

-4.40

-1.80

-0.56

163.92

186.43

9.84

525.00

37.75

-7.75

-19.58

-4.47

-1.83

-0.57

173.91

186.43

9.84

525.00

37.87

-7.87

-19.44

-4.49

-1.85

-0.60

183.91

186.43

9.84

525.00

38.03

-7.85

-19.59

-4.53

-1.86

-0.60

193.92

186.43

9.84

525.00

38.25

-7.77

-19.91

-4.62

-1.90

-0.60

203.91

186.43

9.84

525.00

38.07

-7.83

-19.67

-4.54

-1.87

-0.60

213.91

186.43

9.84

525.00

37.99

-7.84

-19.58

-4.50

-1.86

-0.60

223.91

186.43

9.84

525.00

38.04

-7.82

-19.66

-4.54

-1.87

-0.60

233.91

186.43

9.84

525.00

38.07

-7.83

-19.67

-4.54

-1.87

-0.60

TableC.27:

Values

fore

achite

ratio

ndu

ringop

timiza

tion

C.6 Five summations 127

C.6 Five summations


10 10 4 1 1.00 5 0.13 1.876e+0511 11 5 1 0.44 7 0.0877 1.869e+0512 12 6 1 1.00 8 0.0705 1.868e+0513 13 7 1 1.00 9 0.037 1.866e+0514 14 8 1 1.00 10 0.0204 1.866e+0515 15 9 1 1.00 11 0.0247 1.866e+0516 16 10 1 1.00 12 0.0477 1.865e+0517 18 11 2 1.00 13 0.0358 1.867e+0518 20 12 2 0.46 15 0.0186 1.864e+0519 21 13 1 1.00 16 0.0069 1.864e+0520 22 14 1 1.00 17 0.00557 1.864e+0521 24 15 2 1.00 18 0.00301 1.864e+0522 25 16 1 1.00 19 0.00166 1.864e+0523 26 17 1 1.00 20 0.00122 1.864e+0524 28 18 2 1.00 22 0.00488 1.864e+0525 30 19 2 0.32 24 0.00105 1.864e+0526 31 20 1 1.00 26 0.0016 1.864e+0527 32 21 1 0.23 28 0.000186 1.864e+0528 33 22 1 1.00 30 0.00141 1.864e+0529 34 23 1 0.30 32 0.000146 1.864e+0530 35 24 1 1.00 34 0.000281 1.864e+0531 36 25 1 0.06 39 5.23e-05 1.864e+0532 Number of optimization variables: 7.33 Number of objective function evaluations: 41.34 Number of Jacobian evaluations: 39.35 Final objective function value: 186418.0779.36 Optimality conditions satisfied.37 Optimization Solver 1 in Solver 2: Solution time: 33 s38 Physical memory: 860 MB39 Virtual memory: 5283 MB

128 C Fillet



130 C Fillet



Iteration#

σvM

,max[M

Pa]

Utot[kJ]

U[J/m

3]

Area[m

2]

p1[-]

p2[-]

q 1[-]

q 2[-]

q 3[-]

q 4[-]

q 5[-]

17.11

191.85

10.29

525.00

0.58

-0.36

-0.31

0.43

0.22

0.43

0.32

27.22

191.15

10.23

524.99

1.43

-0.82

-0.67

0.91

0.32

0.68

0.45

35.53

188.82

10.04

524.81

7.90

-3.38

-3.06

2.72

-0.56

-0.03

-0.61

45.38

187.47

9.92

524.85

13.96

-5.29

-6.05

1.92

0.42

0.46

0.77

56.36

187.56

9.92

524.83

19.92

-7.10

-8.09

0.60

1.66

2.39

-0.77

65.18

186.87

9.87

524.91

20.51

-7.34

-8.19

0.87

1.61

0.89

0.32

74.88

186.84

9.87

524.86

25.93

-8.80

-9.59

0.43

0.99

1.07

0.91

84.86

186.60

9.85

525.00

26.07

-8.64

-9.75

0.16

1.10

1.01

0.80

94.73

186.58

9.85

525.00

26.30

-8.43

-10.30

-0.26

1.00

0.87

0.62

104.36

186.56

9.85

524.98

27.89

-7.90

-12.36

-1.54

0.36

0.37

0.18

114.19

186.49

9.84

525.00

29.70

-7.82

-13.69

-2.17

-0.18

0.13

0.09

123.80

186.66

9.86

524.90

40.00

-6.55

-22.74

-6.44

-4.01

-1.84

-0.87

133.85

186.43

9.84

525.00

36.59

-7.45

-19.04

-4.45

-2.13

-0.97

-0.41

143.84

186.42

9.84

525.00

38.36

-7.60

-20.09

-4.81

-2.31

-1.06

-0.43

153.84

186.42

9.84

525.00

39.80

-7.69

-20.99

-5.19

-2.48

-1.11

-0.42

163.85

186.42

9.84

525.00

40.00

-7.69

-21.16

-5.28

-2.49

-1.09

-0.41

173.86

186.42

9.84

525.00

40.00

-7.68

-21.20

-5.27

-2.44

-1.03

-0.38

183.86

186.42

9.84

525.00

40.00

-7.65

-21.24

-5.27

-2.44

-1.03

-0.38

193.86

186.42

9.84

525.00

39.71

-7.49

-21.31

-5.25

-2.46

-1.01

-0.40

203.86

186.42

9.84

525.00

40.00

-7.60

-21.33

-5.28

-2.46

-1.04

-0.39

213.86

186.42

9.84

525.00

40.00

-7.67

-21.22

-5.23

-2.43

-1.04

-0.38

223.86

186.42

9.84

525.00

40.00

-7.61

-21.31

-5.27

-2.46

-1.03

-0.39

233.86

186.42

9.84

525.00

40.00

-7.56

-21.41

-5.28

-2.48

-1.04

-0.40

243.86

186.42

9.84

525.00

40.00

-7.60

-21.34

-5.27

-2.46

-1.04

-0.39

253.86

186.42

9.84

525.00

40.00

-7.67

-21.22

-5.25

-2.42

-1.02

-0.37

263.86

186.42

9.84

525.00

40.00

-7.60

-21.33

-5.27

-2.46

-1.03

-0.39

TableC.33:

Values

fore

achite

ratio

ndu

ringop

timiza

tion

Bibliography

[1] S. Arnout, M. Firl, and K.-U. Bletzinger, “Parameter free shape and thickness optimisationconsidering stress response,” STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZA-TION, vol. 45, no. 6, pp. 801–814, 2012.

[2] C. Le, T. Bruns, and D. Tortorelli, “A gradient-based, parameter-free approach to shapeoptimization,” COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEER-ING, vol. 200, no. 9-12, pp. 985–996, 2011.

[3] Y. Ohtake, A. Belyaev, and I. Bogaevski, “Mesh regularization and adaptive smoothing,”COMPUTER-AIDED DESIGN, vol. 33, no. 11, pp. 789–800, 2001.

[4] J. A. Samareh, “A survey of shape parameterization techniques,” CASI, 1999.

[5] Autodesk Inc., “Multi-Point Constraints (Linear.” http://download.autodesk.com/us/algor/userguides/mergedProjects/setting_up_the_analysis/linear/loads_and_constraints/multi-point_constraints_(linear).htm, 2014. [Online; accessed16-June-2014].

[6] S. D. Rajan, A. D. Belegundu, and J. Budiman, “An integrated system for shape optimaldesign,” Computers and Structures, vol. 30, no. 1-2, pp. 337–346, 1988.

[7] P. Pedersen, L. Tobiesen, and S. Jensen, “Shapes of orthotropic plates for minimum en-ergy concentration,” MECHANICS OF STRUCTURES AND MACHINES, vol. 20, no. 4,pp. 499–514, 1992.

[8] S. Lewanowicz and P. Wozny, “Generalized bernstein polynomials,” BIT NUMERICALMATHEMATICS, vol. 44, no. 1, pp. 63–78, 2004.

[9] M. H. Straathof, Shape Parameterization in Aircraft Design: A Novel Method, Based onB-Splines. Phd thesis, Delft Universty of Technlogy.

[10] COMSOL, Optimizing the Shape of a Horn. Version 4.4, COMSOL, 2013.

[11] COMSOL, COMSOL Multiphysics - reference manual. Version 4.4, COMSOL, 2013.

[12] J. Akin, What is a Fixed Support? Rice University, 2006.

[13] P. Pedersen, Optimal Desings - Structures and Materials - Problems and Tools. TechnicalUniversity of Denmark, 2003.

[14] S. Riehl, J. Friederich, M. Scherer, R. Meske, and P. Steinmann, “On the discrete variantof the traction method in parameter-free shape optimization,” Comput. Methods Appl.Mech. Engrg., 2014.

http://download.autodesk.com/us/algor/userguides/mergedProjects/setting_up_the_analysis/linear/loads_and_constraints/multi-point_constraints_(linear).htm



shape optimization of mechanical systems in comsol 4.4

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