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.
ii
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)
iv
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.
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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.
10 3 Optimizing the shape of a cantilever beam
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.
12 3 Optimizing the shape of a cantilever beam
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.
3.2 Results and discussion 13
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.
14 3 Optimizing the shape of a cantilever beam
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.
3.2 Results and discussion 15
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.
16 3 Optimizing the shape of a cantilever beam
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).
5 Click the Add button
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
2 Select Boundary 4 only.
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
18 3 Optimizing the shape of a cantilever beam
2 Select Domain 1.
Prescribed Mesh Displacement 2
1 On the Physics toolbar, click Boundaries and choose Prescribed Mesh Displacement
2 Select Boundary 1 only.
3 In the Prescribed Mesh Displacement settings window, locate the Prescribed MeshDisplacement section
4 In the dY field type q1*s
Prescribed Mesh Displacement 3
1 On the Physics toolbar, click Boundaries and choose Prescribed Mesh Displacement
2 Select Boundary 3 only.
3 In the Prescribed Mesh Displacement settings window, locate the Prescribed MeshDisplacement section
4 In the dY field type q1*(1-s)+q3*s+q2*sin(pi*s)
Prescribed Mesh Displacement 4
1 On the Physics toolbar, click Boundaries and choose Prescribed Mesh Displacement
2 Select Boundary 4 only.
3 In the Prescribed Mesh Displacement settings window, locate the Prescribed MeshDisplacement section
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
3.3 Modeling instructions 19
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”
20 3 Optimizing the shape of a cantilever beam
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
3.3 Modeling instructions 21
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
1 On the Study toolbar, click Show Default Solver
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
22 3 Optimizing the shape of a cantilever beam
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
26 4 Optimization of a plate with a hole
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 discussion 27
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.
28 4 Optimization of a plate with a hole
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.
σvM,max [MPa] U [J/m3] Utot [J] Area [m2]6.20 12.32 107.70 87.43
Table 4.7: Reference values for the plate with σ1 = 1000 kN and σ2 = 2000 kN
Figure 4.8: Standard plate with σ2/σ1 = 2
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.
σvM,max [MPa] U [J/m3] Utot [J] Area [m2]3.27 11.10 97.05 87.43
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
4.2 Results and discussion 29
(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.
σvM,max [MPa] U [J/m3] Utot [J] Area [m2]10.41 28.86 252.34 87.43
Table 4.11: Reference values for the plate with σ1 = 1000 kN and σ2 = 3000 kN
30 4 Optimization of a plate with a hole
Figure 4.12: Standard plate with σ2/σ1 = 3
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.
σvM,max [MPa] U [J/m3] Utot [J] Area [m2]4.48 24.01 209.94 87.43
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
4.2 Results and discussion 31
(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 4.14: 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.
32 4 Optimization of a plate with a hole
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
(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 4.15: 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.
4.3 Modeling instructions 33
4.3 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).
5 Click the Add button
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 bracket_shape_optimization_parameters.txt and double-click it to load the parameters
GEOMETRY 1
Square 1
34 4 Optimization of a plate with a hole
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
4 Click the Build All Objects button
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
4 Click the Build All Objects button
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)
6 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
4.3 Modeling instructions 35
SOLID MECHANICS (SOLID)
Roller 1
1 On the Physics toolbar, click Boundaries and choose Roller
2 Select Boundary 1 and 3 only.
Boundary Load 1
1 On the Physics toolbar, click Boundaries and choose Boundary Load
2 Select Boundary 2 only.
3 In the Boundary Selection window, locate the Force section
4 Under Load type, change it to Total force
5 Let the y component remain 0, but change the x component to P1
Boundary Load 2
1 On the Physics toolbar, click Boundaries and choose Boundary Load
2 Select Boundary 4 only.
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 P2
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
36 4 Optimization of a plate with a hole
2 Select Boundary 1 only.
3 In the Prescribed Mesh Displacement settings window, locate the Prescribed MeshDisplacement section
4 In the dY field type p1*(1-s)
Prescribed Mesh Displacement 3
1 On the Physics toolbar, click Boundaries and choose Prescribed Mesh Displacement
2 Select Boundary 5 only.
3 In the Prescribed Mesh Displacement settings window, locate the Prescribed MeshDisplacement section
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
Prescribed Mesh Displacement 4
1 On the Physics toolbar, click Boundaries and choose Prescribed Mesh Displacement
2 Select Boundary 3 only.
3 In the Prescribed Mesh Displacement settings window, locate the Prescribed MeshDisplacement section
4 In the dX field type p3*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
4.3 Modeling instructions 37
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 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
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
38 4 Optimization of a plate with a hole
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 sideˆ2-pi*rˆ2/4
6 The Integral Inequality Constraint will automatically be added to the optimization inStudy 2
Optimization stress constraint
1 On the Physics toolbar, click Optimization
2 Locate the Domains button, and click Global Inequality Constraint
3 In the Global 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 3[MPa]
6 The Global Inequality Constraint will automatically be added to the optimization inStudy 2
4.3 Modeling instructions 39
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
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 bracket_shape_optimization_control_parameters.txt
9 Locate Output While Solving section and make sure that Plot is checked
Solver 2
1 On the Study toolbar, click Show Default Solver
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
40 4 Optimization of a plate with a hole
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
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
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, q1, q2 andq3.
4.3 Modeling instructions 41
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)
5 To see the probes in effect, compute Study 2 again
42
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:
5.2 Results and discussion 45
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).
(a) von Mises surface stress in 2D (b) von Mises surface stress in 3D
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.
46 5 Optimizing the shape of a fillet
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.
5.2 Results and discussion 47
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.
48 5 Optimizing the shape of a fillet
(a) von Mises surface stress in 2D (b) von Mises surface stress in 3D
(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.
5.2 Results and discussion 49
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.
50 5 Optimizing the shape of a fillet
5.3 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).
5 Click the Add button
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 fillet_shape_optimization_parameters.txt and double-click it to load the parameters
GEOMETRY 1
Rectangle 1
5.3 Modeling instructions 51
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 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
8 Click the Build All Objects button
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
6 Find the Added segments subsection. Click the Add Linear button
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
9 Find the Added segments subsection. Click the Add Linear button
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
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
52 5 Optimizing the shape of a fillet
3 Activate the Object to add window by clicking Activate, and then click on the rectangle(Rectangle 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 theBézier polygon (Bézier Polygon 1)
6 Click the Build All Objects button
Rectangle 2
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 b
4 In the Height edit field, type a
5 Click the Build All Objects button
Difference 2
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 newrectangle (Rectangle 2)
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 theDifference domain created before (Difference 1)
6 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
5.3 Modeling instructions 53
SOLID MECHANICS (SOLID)
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
1 On the Physics toolbar, click Boundaries and choose Boundary Load
2 Select Boundary 3 only.
3 In the Boundary Selection window, locate the Force section
4 Under Load type, change it to Total force
5 Let the r component remain 0, but change the z component to P
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
2 Select Boundary 4 only.
3 In the Prescribed Mesh Displacement settings window, locate the Prescribed MeshDisplacement section
54 5 Optimizing the shape of a fillet
4 In the dZ field type k*p1*s
Prescribed Mesh Displacement 3
1 On the Physics toolbar, click Boundaries and choose Prescribed Mesh Displacement
2 Select Boundary 5 only.
3 In the Prescribed Mesh Displacement settings window, locate the Prescribed MeshDisplacement section
4 In the dR field type k*p2*s
5 In the dZ field type k*(p1*(1-s)+q1*sin(pi*s))
Prescribed Mesh Displacement 4
1 On the Physics toolbar, click Boundaries and choose Prescribed Mesh Displacement
2 Select Boundary 6 only.
3 In the Prescribed Mesh Displacement settings window, locate the Prescribed MeshDisplacement section
4 In the dR field type k*p2*(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
5.3 Modeling instructions 55
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
5 Repeat the procedure for the 3D plot
ADD STUDY
1 To add a new study for the optimization, go to the Home toolbar and click Add Study
56 5 Optimizing the shape of a fillet
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 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
8 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
5.3 Modeling instructions 57
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
1 On the Study toolbar, click Show Default Solver
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 von Mises stress 3D plot
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
58 5 Optimizing the shape of a fillet
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, p1, p2, andq1.
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)
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.
60
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
64 A Cantilever Beam
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:
66 A Cantilever Beam
(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
68 A Cantilever Beam
A.3 Two summations
1 ============================================================2 Optimization Solver 1 in Solver 2 started at 22-maj-2014 11:17:17.3 Optimization solver (SNOPT)4 Analytic gradient with the adjoint method.5 Itns Major Minor Step nPDE Error Objective6 3 0 3 - 1 3.23 1.315e+047 6 1 3 0.40 2 1.18 1.425e+048 10 2 4 0.21 4 0.608 1.088e+049 11 3 1 1.00 5 0.349 1.086e+04
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
(a) von Mises Stress (b) von Mises Stress along the upper boundary
Figure A.6:
70 A Cantilever Beam
(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.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
72 A Cantilever Beam
A.4 Three summations
1 ============================================================2 Optimization Solver 1 in Solver 2 started at 22-maj-2014 11:02:24.3 Optimization solver (SNOPT)4 Analytic gradient with the adjoint method.5 Itns Major Minor Step nPDE Error Objective6 4 0 4 - 1 3.23 1.268e+047 8 1 4 0.38 2 1.05 1.496e+048 13 2 5 0.18 4 0.61 1.102e+049 14 3 1 1.00 5 0.267 1.099e+04
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
(a) von Mises Stress (b) von Mises Stress along the upper boundary
Figure A.9:
74 A Cantilever Beam
(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.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
76 A Cantilever Beam
A.5 Four summations
1 ============================================================2 Optimization Solver 1 in Solver 2 started at 22-maj-2014 10:51:35.3 Optimization solver (SNOPT)4 Analytic gradient with the adjoint method.5 Itns Major Minor Step nPDE Error Objective6 5 0 5 - 1 3.23 1.24e+047 10 1 5 0.36 2 1.22 1.546e+048 17 2 7 0.17 4 0.671 1.119e+049 18 3 1 1.00 5 0.843 1.107e+04
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
78 A Cantilever Beam
(a) von Mises Stress (b) von Mises Stress along the upper boundary
Figure A.12:
A.5 Four summations 79
(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.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
A.5 Four summations 8131
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
82 A Cantilever Beam
A.6 Five summations
1 ============================================================2 Optimization Solver 1 in Solver 2 started at 22-maj-2014 10:57:45.3 Optimization solver (SNOPT)4 Analytic gradient with the adjoint method.5 Itns Major Minor Step nPDE Error Objective6 6 0 6 - 1 3.23 1.242e+047 12 1 6 0.37 2 1.08 1.549e+048 20 2 8 0.18 4 0.657 1.122e+049 21 3 1 1.00 5 1.01 1.108e+04
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
(a) von Mises Stress (b) von Mises Stress along the upper boundary
Figure A.15:
84 A Cantilever Beam
(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.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
86
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
90 B Plate with a hole
Figure B.4: Surface v. Mises stress
Figure B.5: Surface v. Mises stress showing the whole plate
B.1 Tensile ratio of 1 91
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
92 B Plate with a hole
Figure B.8: The error of the solution as a function of iteration number for the nonlinearsolver
B.2 Tensile ratio of 2Standard plate
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
B.2 Tensile ratio of 2 93
Figure B.9: v. Mises stress
Figure B.10: The error of the solution as a function of iteration number for the nonlinearsolver
94 B Plate with a hole
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
Table B.11: Values for each iteration during optimization
B.2 Tensile ratio of 2 95
Figure B.12: Surface v. Mises stress
Figure B.13: Surface v. Mises stress showing the whole plate
96 B Plate with a hole
Figure B.14: Graph comparing the v. Mises stress for the original shape with the optimizeddesign
Figure B.15: The error of the solution as a function of iteration number for the optimizationsolver
B.3 Tensile ratio of 3 97
Figure B.16: The error of the solution as a function of iteration number for the nonlinearsolver
B.3 Tensile ratio of 3Standard plate
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
98 B Plate with a hole
Figure B.17: v. Mises stress
Figure B.18: The error of the solution as a function of iteration number for the nonlinearsolver
B.3 Tensile ratio of 3 99
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
100 B Plate with a hole
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
Table B.19: Values for each iteration during optimization
B.3 Tensile ratio of 3 101
Figure B.20: Surface v. Mises stress
Figure B.21: Surface v. Mises stress showing the whole plate
102 B Plate with a hole
Figure B.22: Graph comparing the v. Mises stress for the original shape with the optimizeddesign
Figure B.23: The error of the solution as a function of iteration number for the optimizationsolver
B.3 Tensile ratio of 3 103
Figure B.24: The error of the solution as a function of iteration number for the nonlinearsolver
104
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
Figure C.2: v. Mises stress in 3D
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
Figure C.4: v. Mises stress in 1D
Figure C.5: v. Mises stress in 2D
C.2 One summation 109
Figure C.6: v. Mises stress in 3D
Figure C.7: The error of the solution as a function of iteration number for the nonlinearsolver
110 C Fillet
Figure C.8: The error of the solution as a function of iteration number for the optimizationsolver
C.2 One summation 111
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
1 ============================================================2 Optimization Solver 1 in Solver 2 started at 24-jun-2014 01:25:27.3 Optimization solver (SNOPT)4 Analytic gradient with the adjoint method.5 Itns Major Minor Step nPDE Error Objective6 3 0 3 - 1 0.569 1.919e+057 4 1 1 0.15 2 0.521 1.913e+058 5 2 1 0.17 3 0.29 1.886e+059 6 3 1 0.62 4 0.174 1.875e+05
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
Figure C.10: v. Mises stress in 1D
Figure C.11: v. Mises stress in 2D
114 C Fillet
Figure C.12: v. Mises stress in 3D
Figure C.13: The error of the solution as a function of iteration number for the nonlinearsolver
C.3 Two summations 115
Figure C.14: The error of the solution as a function of iteration number for the optimizationsolver
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
Figure C.16: v. Mises stress in 1D
118 C Fillet
Figure C.17: v. Mises stress in 2D
Figure C.18: v. Mises stress in 3D
C.4 Three summations 119
Figure C.19: The error of the solution as a function of iteration number for the nonlinearsolver
120 C Fillet
Figure C.20: The error of the solution as a function of iteration number for the optimizationsolver
C.4 Three summations 121
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
1 ============================================================2 Optimization Solver 1 in Solver 2 started at 24-jun-2014 01:03:00.3 Optimization solver (SNOPT)4 Analytic gradient with the adjoint method.5 Itns Major Minor Step nPDE Error Objective6 5 0 5 - 1 0.521 1.919e+057 6 1 1 0.13 2 0.506 1.912e+058 7 2 1 0.17 3 0.399 1.888e+059 8 3 1 0.49 4 0.192 1.875e+05
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
Figure C.22: v. Mises stress in 1D
Figure C.23: v. Mises stress in 2D
124 C Fillet
Figure C.24: v. Mises stress in 3D
Figure C.25: The error of the solution as a function of iteration number for the nonlinearsolver
C.5 Four summations 125
Figure C.26: The error of the solution as a function of iteration number for the optimizationsolver
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
1 ============================================================2 Optimization Solver 1 in Solver 2 started at 24-jun-2014 00:57:39.3 Optimization solver (SNOPT)4 Analytic gradient with the adjoint method.5 Itns Major Minor Step nPDE Error Objective6 6 0 6 - 1 0.487 1.919e+057 7 1 1 0.12 2 0.494 1.911e+058 8 2 1 0.18 3 0.429 1.888e+059 9 3 1 0.49 4 0.164 1.875e+05
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
Figure C.28: v. Mises stress in 1D
Figure C.29: v. Mises stress in 2D
C.6 Five summations 129
Figure C.30: v. Mises stress in 3D
Figure C.31: The error of the solution as a function of iteration number for the nonlinearsolver
130 C Fillet
Figure C.32: The error of the solution as a function of iteration number for the optimizationsolver
C.6 Five summations 131
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
132
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.