Stress Constraints Sensitivity Analysis in Structural Topology Optimization J. Par s, F. Navarrina, I. Colominas , M. Casteleiro Group of Numerical Methods in Engineering, Dept. of Applied Mathematics, Universidade da Coru~ na, Campus de Elvi~ na, 15071 A Coru~ na, SPAIN Abstract Sensitivity Analysis is a essential issue in the structural optimization eld. The calculation of the derivatives of the most relevant magnitudes (displacements, stresses, strains, ...) in optimum design of structures allows to estimate the structural response when changes in the design variables are introduced. This essential information is used by the most frequent conven- tional optimization algorithms (SLP, MMA, Feasible directions, ...) in order to reach the optimal solution. According to this idea, the Sensitivity Analy- sis of the stress constraints in Topology Optimization problems is a crucial aspect to obtain the optimal solution when stress constraints are considered. This is not the usual statement since minimum compliance approaches does not usually involve neither stress constraints nor displacement constraints. However, in the topology optimization problem with stress constraints, e- cient and accurate computation of the derivatives is needed in order to reach appropriate optimal solutions. In this paper, a complete analytic and ecient procedure to obtain the Sensitivity Analysis of the stress constraints in topol- ogy optimization of continuum structures is analyzed. First order derivatives and second order directional derivatives are analyzed and included in the op- timization procedure. In addition, topology optimization problems usually involve a large number of design variables and constraints. Thus, an ecient implementation of the algorithms used in the computation of the Sensitiv- ity Analysis is developed in order to reduce the large computing resources required. Finally, the sensitivity analysis techniques presented in this paper are tested by solving some application examples. Corresponding author, e-mail: [email protected]Preprint submitted to Computer Methods in Applied Mechanics and EngineeringDecember 2, 2009
Transcript
Stress Constraints Sensitivity Analysis in Structural
Topology Optimization
J. Parıs, F. Navarrina, I. Colominas∗, M. Casteleiro
Group of Numerical Methods in Engineering, Dept. of Applied Mathematics,Universidade da Coruna, Campus de Elvina, 15071 A Coruna, SPAIN
Abstract
Sensitivity Analysis is a essential issue in the structural optimizationfield. The calculation of the derivatives of the most relevant magnitudes(displacements, stresses, strains, ...) in optimum design of structures allowsto estimate the structural response when changes in the design variables areintroduced. This essential information is used by the most frequent conven-tional optimization algorithms (SLP, MMA, Feasible directions, ...) in orderto reach the optimal solution. According to this idea, the Sensitivity Analy-sis of the stress constraints in Topology Optimization problems is a crucialaspect to obtain the optimal solution when stress constraints are considered.This is not the usual statement since minimum compliance approaches doesnot usually involve neither stress constraints nor displacement constraints.However, in the topology optimization problem with stress constraints, effi-cient and accurate computation of the derivatives is needed in order to reachappropriate optimal solutions. In this paper, a complete analytic and efficientprocedure to obtain the Sensitivity Analysis of the stress constraints in topol-ogy optimization of continuum structures is analyzed. First order derivativesand second order directional derivatives are analyzed and included in the op-timization procedure. In addition, topology optimization problems usuallyinvolve a large number of design variables and constraints. Thus, an efficientimplementation of the algorithms used in the computation of the Sensitiv-ity Analysis is developed in order to reduce the large computing resourcesrequired. Finally, the sensitivity analysis techniques presented in this paperare tested by solving some application examples.
Since topology optimization of continuum structures problems was stud-ied by Bendsøe and Kikuchi in 1988 [1, 2], important tasks have been ex-tensively analyzed (e.g. the Sensitivity Analysis and the Optimization Algo-rithms used to obtain the optimal solution). Although there are optimizationalgorithms that do not require the Sensitivity Analysis (e.g. GA, Optimalitycriteria [26, 32]), most of the conventional optimization algorithms requirespecific information about the derivatives of the objective function and theconstraints [7, 9, 17, 20, 21, 29]. In fact, when the problem involves a largenumber of design variables and constraints, high order derivatives are moreappropriate to reach the optimal solution [15, 16, 17].
Structural Topology Optimization problems with stress constraints usu-ally introduce a large number of design variables (usually one design variableper element of the mesh of finite elements: the relative density [1, 2, 3, 7, 18,21, 24, 31]) and a large number of non-linear constraints (usually one stressconstraint in the central node of each element [7, 8, 10, 20, 21, 24, 31]). Inaddition, if several load cases are considered, the number of stress constraintsincreases considerably.
According to that, the resulting optimization problems require efficientand specific optimization algorithms [15, 21, 28, 29]. These algorithms usu-ally require first order derivatives of the objective function and the con-straints. In some cases, higher order derivatives need to be considered in theoptimization process in order to avoid unexpected effects [15].
In this paper we have used a Sequential Linear Programming with QuadraticLine Search optimization algorithm. This procedure requires first orderderivatives of the stress constraints and the objective function, and first andsecond order directional derivatives of the stress constraints and the objectivefunction [15, 20].
The computation of these derivatives can be carried out by using differentnumerical algorithms (finite differences, theoretical analysis, ...). When itis possible and suitable, the most usual technique to obtain the sensitivityanalysis is the theoretical analysis. Thus, the derivation algorithm consistsin the implementation of the theoretical functions previously obtained byapplying analytical differentiation techniques [16].
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The theoretical analysis of the derivatives requires an important and rigor-ous mathematical study in order to obtain the expressions to be implementedin the optimization source code.
In this paper we develop the whole derivation process of the stress con-straints in topology optimization of structures. This theoretical analysis al-lows to obtain exact mathematical expressions of the derivatives of the stressconstraints and good numerical approximations. Finally, some applicationexamples are studied. The proposed examples are analyzed in order to checkthe validity of the sensitivity analysis procedure presented in this paper sinceanalytical solutions exist (see [11, 12, 27]).
2. Topology optimization methodology
The optimization methodology developed to solve the minimum weightstructural topology optimization problem with stress constraints is based onan iterative process where the sensitivity analysis is an essential issue to ob-tain adequate results. The entire methodology is represented schematicallyin figure 1. Each iteration of the method is developed in a number of stagesthat need to be analyzed individually. The first stage is the structural anal-ysis, by means of the FEM, incorporating the effect of the relative density.The second stage is the computation of the objective function, which is basedon the weight of the structure. This objective function may optionally in-clude a perimeter penalization. Then, stress constraints are computed andchecked to verify the active constraints. Three different formulations can beused to deal with stress constraints. These formulations have been previouslystudied by the authors in [20, 21]. The next stage is the computation of thefirst order derivatives of the active constraints. This stage is extensively ana-lyzed in this paper in sections 3.2.1, 5.1, 6.1 and 7.1. The sensitivity analysisobtained is used in the next stage of the method to obtain the search direc-tion. This direction is obtained with three different algorithms (see figure 1)depending on the number of active constraints (steepest descent method orSLP) and the existence of violated constraints (back to the feasible regionalgorithm). Note that, the search direction has been already obtained byusing one of the algorithms presented before. Thus, first and second ordersensitivity analysis can be obtained by using directional derivatives in thesearch direction previously obtained. The analysis and computation of thesedirectional derivatives is extensively analyzed in this paper in sections 3.2.2,5.2, 6.2 and 7.2. These directional derivatives are included in the second or-
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der Taylor expansions of the objective function and the constraints to obtainthe most adequate advance step in the search direction. Finally, the designvariables are modified according to the search direction and the advance step,and convergence is checked. If convergence is achieved, the optimal solutionhas been obtained. Otherwise, the process begins again in the first stagewith the modified values of the design variables.
3. Sensitivity analysis of constraints depending on displacementsand design variables
The computation of the Sensitivity Analysis of constraints depending ondesign variables can be easily developed by applying theoretical derivationtechniques (chain rule, ...). However, when the constraints of the problemdepend on the displacements of the structure, the computation of the sen-sitivities is much more complicated since it is necessary to obtain a priorithe derivatives of the displacements. These derivatives can not be computedin a direct way since the displacements are obtained by means of a systemof linear equations (FEM). Consequently, the computation of the derivativesof the displacements requires to apply theoretical derivation techniques to asystem of linear equations.
Next, we introduce some previous definitions used in the derivation algo-rithms.
3.1. Previous definitions
We define the vector of design variables as
�������������� =
⎧⎨⎩�1...�n
⎫⎬⎭ . (1)
We also define a generic function P : ℝn → ℝ as
P (��������������) = P (�1, �2, . . . , �n). (2)
The derivatives of this function P over the design variables can be definedas
dP
d��������������= ∇P =
{∂P
∂�1
∂P
∂�2
⋅ ⋅ ⋅ ∂P
∂�n
}. (3)
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Figure 1: Scheme of the topology optimization methodology.
5
The Hessian matrix can be obtained as
HHHHHHHHHHHHHH(P ) =d
d��������������
(dPd��������������
)T, (4)
HHHHHHHHHHHHHH(P ) =
⎧⎨⎩∂2P∂�12
∂2P∂�1∂�2
⋅ ⋅ ⋅ ∂2P∂�1∂�n
∂2P∂�2∂�1
∂2P∂�22
⋅ ⋅ ⋅ ∂2P∂�2∂�n
......
. . ....
∂2P∂�n∂�1
∂2P∂�n∂�2
⋅ ⋅ ⋅ ∂2P∂�n2
⎫⎬⎭ . (5)
On the other hand, let a function QQQQQQQQQQQQQQ : ℝn → ℝm,
QQQQQQQQQQQQQQ(��������������) =
⎧⎨⎩Q1(�1, . . . , �n)Q2(�1, . . . , �n)
...Qm(�1, . . . , �n)
⎫⎬⎭ . (6)
The first order derivatives can be stated as
dQQQQQQQQQQQQQQ
d��������������= ∇QQQQQQQQQQQQQQ =
⎧⎨⎩∂Q1
∂�1
∂Q1
∂�2⋅ ⋅ ⋅ ∂Q1
∂�n∂Q2
∂�1
∂Q2
∂�2⋅ ⋅ ⋅ ∂Q2
∂�n...
.... . .
...∂Qm∂�1
∂Qm∂�2
⋅ ⋅ ⋅ ∂Qm∂�n
⎫⎬⎭ (7)
and the second order derivatives as
HHHHHHHHHHHHHH(Qj) =d
d��������������
(dQj
d��������������
)T, j = 1, . . . ,m. (8)
HHHHHHHHHHHHHH(Qj) =
⎧⎨⎩
∂2Qj∂�12
∂2Qj∂�1∂�2
⋅ ⋅ ⋅ ∂2Qj∂�1∂�n
∂2Qj∂�2∂�1
∂2Qj∂�22
⋅ ⋅ ⋅ ∂2Qj∂�2∂�n
......
. . ....
∂2Qj∂�n∂�1
∂2Qj∂�n∂�2
⋅ ⋅ ⋅ ∂2Qj∂�n2
⎫⎬⎭(9)
If we consider that the vector �������������� is obtained iteratively, we can define animproved value of this vector as:
3.2. Sensitivity analysis of constraints that depend on displacements
The sensitivity analysis of constraints that depend on displacements canbe theoretically analyzed by taking into account the previous definitions.Thus, we define gj(qqqqqqqqqqqqqq(��������������)), (j = 1, . . . ,m), as a set of functions depending on qqqqqqqqqqqqqq(e.g. vector of stresses or displacements in predefined points of the structure),where m is the number of constraints of the problem. This vector qqqqqqqqqqqqqq generallydepends on the nodal displacements. In addition, these nodal displacementsdepend on the design variables (��������������). In order to simplify the notation, we are
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going to use qqqqqqqqqqqqqq(��������������) and ��������������(��������������) instead of the complete expressions qqqqqqqqqqqqqq ≡ qqqqqqqqqqqqqq(��������������) and�������������� ≡ ��������������(��������������). Thus, we may define
where the first order derivatives of the vector of displacements (��������������) can be ob-tained from the finite element formulation. In a conventional finite elementformulation, under small displacements and small displacements gradientshypotheses, the nodal displacements of the structure can be obtained bysolving a system of linear equations. If the trial and test functions used torepresent the displacement field and the geometry satisfy the essential bound-ary conditions (prescribed displacements), the system of linear equations canbe stated as
The matrix KKKKKKKKKKKKKK is the stiffness matrix of the structure, �������������� is the vector ofnodal displacements and ffffffffffffff is the vector of applied loads. Each one of theterms Kji of the stiffness matrix KKKKKKKKKKKKKK can be obtained as
KKKKKKKKKKKKKKji =Ne∑e=1
KKKKKKKKKKKKKKeji, j = 1, . . . , N i = 1, . . . , N (22)
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where Ne is the number of elements and N is the number of nodes of theFEM mesh used.
where the matrices ΦΦΦΦΦΦΦΦΦΦΦΦΦΦj and ΦΦΦΦΦΦΦΦΦΦΦΦΦΦi contain the values of the shape functionscorresponding to the nodal displacements j and i in a predefined point (rrrrrrrrrrrrrro)of the discretization mesh used.
where �i is the value of the shape function associated to node i and d isthe dimension of the problem. LLLLLLLLLLLLLL is the differential operator that, applied tothe displacement field, gives the strains and DDDDDDDDDDDDDD is the elasticity tensor of thematerial.
The vector of applied forces ffffffffffffff j can be obtained as
ffffffffffffff j =
∫∫Γo�
ΦΦΦΦΦΦΦΦΦΦΦΦΦΦTj tttttttttttttt dΓ +
Ne∑e=1
ffffffffffffff ej . (25)
Each contribution ffffffffffffff ej to the vector of applied forces ffffffffffffff j can be obtained as
where bbbbbbbbbbbbbb is the vector of forces per unit of volume, tttttttttttttt is the vector of forces perunit of area and uuuuuuuuuuuuuup is the vector of prescribed displacements.
This FEM approach may be modified according to [18, 20, 21] in order toconsider the design variables of the topology optimization problem (the rel-ative densities). Thus, the contributions KKKKKKKKKKKKKKe
ji can be obtained by consideringthe relative density (�e) of element e as:
As it can be observed in (20), reactions are not involved in this formulationsince the trial functions are forced to satisfy the essential boundary conditions[18, 21]. Consequently, first order derivatives of the nodal displacements (��������������)versus the design variables (��������������) can be obtained by differentiating (20) as:
KKKKKKKKKKKKKKd��������������
d��������������=
dffffffffffffff
d��������������− dKKKKKKKKKKKKKK
d����������������������������. (29)
The terms on the right side of (29) can be directly obtained from a con-ventional approach of the FEM. If we take into account that the externaldistributed forces per unit of area do not usually depend on the design vari-ables (in the topology optimization problem), the derivatives of the nodalforces (ffffffffffffffk, k = 1, ..., N) can be obtained, according to (28), as:
dffffffffffffffkd�e
=
∫∫∫Ee
(ΦΦΦΦΦΦΦΦΦΦΦΦΦΦTk bbbbbbbbbbbbbb) dΩ (30)
Note that the sensitivity analysis of the nodal forces (ffffffffffffff ek) can be obtainedby computing the vector of nodal forces per unit of volume with �e = 1. Theexternal forces per unit of area applied in this problem do not depend onthe design variables and, consequently, their derivatives are null. So, theseexternal forces will not be considered in the analysis.
On the other hand, the derivatives of the stiffness matrix can be obtainedby applying similar techniques as
Thus, the derivatives of the stiffness matrix (KKKKKKKKKKKKKKji) versus the design vari-able �e can be computed by assembling the original stiffness matrix of theelement e with �e = 1. The derivatives of the elemental stiffness matrix ofthe rest of the elements are obviously null.
These derivatives of the stiffness matrix and the nodal forces are requiredto obtain the derivatives of the nodal displacements according to (29). Twodifferent procedures can be developed to obtain the derivatives of the nodaldisplacements: the so called “Direct Differentiation” algorithms and “Adjointvariable” algorithms. Both procedures are exact from an analytic point ofview. The differences rely on the sequence of operations involved.
The “Direct Differentiation” technique proposed performs the computa-tion of the nodal displacement derivatives (29). Then, these derivatives are
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used to calculate the sensitivities of the stress constraints (19). Thus, withthis formulation it is necessary to save the whole set of nodal displacementderivatives. In fact, it is not strictly necessary to storage these derivativesbut it is highly advisable in order to reduce the computing time. In addition,a large number of systems of linear equations, like the ones proposed in (29),need to be solved. In practice, the number of systems of linear equations tobe solved is equal to the number of degrees of freedom of the structure.
The “Adjoint Variable” algorithm introduces some modifications in thecalculation of the stress constraints derivatives. The term d��������������
d�������������� in (19) can be
substituted according to (29) in order to avoid the computation of the nodaldisplacements derivatives. Thus,
Consequently, the derivatives of the stress constraints can be obtained,by using the “Adjoint variable” ��������������j, as:
dgjd��������������
= ��������������Tj
(dffffffffffffffd��������������− dKKKKKKKKKKKKKK
d����������������������������). (35)
In the case that we are analyzing in this paper, the matrix of the sys-tem of linear equations proposed in (34) is the stiffness matrix due to thesymmetry. This fact means an important advantage since it is possible toapply a factorization algorithm (e.g. Cholesky) in order to solve the struc-tural problem. Thus, the factorized matrix can be used to solve as manyadditional systems of linear equations as required, like the ones proposed in(34), in order to obtain the sensitivities. This technique reduces drastically
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the computing time since only forward and backward substitutions are re-quired to calculate the “Adjoint Variable” and, consequently, the sensitivityanalysis of each constraint. In addition, with the “Adjoint Variable” tech-nique, intermediate derivatives are not computed neither stored. This is avery important advantage since the number of systems of linear equations tobe solved is much smaller, for a reduced number of load cases (e.g. 2 or 3),than the number of systems required by using the “Direct Differentiation”technique ((19) and (29)).
Furthermore, the “Adjoint Variable” differentiation methodology allowsto compute the derivatives of a set of constraints that can be previouslydefined (e.g. a set of active constraints). Thus, it is not necessary to computethe whole set of constraints derivatives. This fact also means an importantreduction in computing time. Following with the same idea, the derivativesof each constraint can be computed individually. The computation of thederivatives of one stress constraints does not involve any interference withthe computation of the derivatives of the rest of the constraints. This factis crucial to perform an efficient computation of these derivatives in parallel(it is possible to use high performance parallelization procedures [6]). Thetechniques proposed were specifically developed to reduce as much as possiblethe computing effort without loosing precision. Otherwise the computationof the sensitivity analysis will become unaffordable in practical applicationsthat involve a large number of constraints.
3.2.2. First and second order directional derivatives
First order derivatives are usually required by the most frequent con-ventional optimization algorithms used in topology optimization. The op-timization algorithm proposed in this paper is based on Sequential LinearProgramming techniques. This algorithm requires first order derivatives ofthe constraints and first order derivatives of the objective function in orderto obtain an adequate search direction. Then, a line search in the directionobtained can be performed to define the most appropriate advance factor.This “line search” may be obtained by using directional Taylor expansions ofthe constraints and the objective function. In this paper, we propose to usea second order directional reconstruction of the constraints and the objectivefunction to obtain the advance factor (Quadratic Line Search). These secondorder directional reconstructions allow to avoid “zig-zag” phenomena aroundthe optimal solution [16, 17].
Thus, the directional derivatives of the constraints along the direction ssssssssssssss
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can be obtained, according to (11), as:
Ds gj =dgjd��������������ssssssssssssss, (36)
and by substituting (dgjd�������������� ) according to (19):
It is important to remark that, as it was proposed in (34), the matrix ofthe resulting system of linear equations is the stiffness matrix of the struc-tural problem. If we take into account that this matrix has been previouslyfactorized by means of a Cholesky algorithm, the solution of (38) only re-quires to update the vector of the right-hand side and to perform the suitableforward and backward substitutions.
The calculation of the directional derivatives is stated by means of a“Direct Differentiation” algorithm. This technique is computationally moreefficient that the “Adjoint variable” algorithm to compute high order deriva-tives. The “Adjoint variable” method requires to compute and store the fullset of “adjoint variables” (��������������j, j = 1, ...,m) in order to obtain first and sec-ond order directional derivatives. Note that, according to 3.2.1, the full set ofstress constraints derivatives is not computed since only active constraints areinvolved in the computation of the search direction. Consequently, the “ad-joint variables” obtained correspond to the set of active constraints. Thus,the directional derivatives of the whole set of constraints can not be obtainedwith this technique. On the other hand, the “Direct Differentiation” methoddoes not require to compute any adjoint variable and allows to compute thedirectional derivatives of the whole set of stress constraints. According to thisidea, the computation of the first order directional derivatives of the stressconstraints requires to solve the system of linear equations (38) and includethese derivatives in (37). Once again, the algorithms proposed have beenspecifically developed to reduce as much as possible the computing effortrequired (both memory and CPU time).
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Second order directional derivatives can be obtained by using again the“Direct Differentiation” technique proposed. Thus, these derivatives can beobtained by differentiating (37) as:
and Ds�������������� is obtained by means of the system of linear equations proposedin (38). The second order directional derivatives of the nodal displacements(D2
ss��������������) can be obtained by differentiating (38) as:
KKKKKKKKKKKKKK D2ss�������������� = D2
ss ffffffffffffff −D2ssKKKKKKKKKKKKKK ��������������− 2DsKKKKKKKKKKKKKK Ds��������������. (43)
Note that the matrix of the system of linear equations matches up with thestiffness matrix of the structural problem. Thus, the second order directionalderivative of the constraint (D2
ss gj) can be obtained by solving the systemof linear equations proposed in (43) and by replacing D2
ss�������������� in (41).
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4. Stress constraints derivatives
In the previous section we have analyzed the general case with constraintsthat depend on the nodal displacements. Now we focus our attention in thesensitivity analysis of stress constraints. Thus, we define a stress constraintgj(qqqqqqqqqqqqqq(��������������), ��������������) that depends on the design variables and on a vector qqqqqqqqqqqqqq. In thiscase, vector qqqqqqqqqqqqqq corresponds to a material failure criterion like, for example,the Von Mises reference stress:
and m is the number of functions that depend on the nodal displacements.In this case, m is the number of points of the structure where values of thereference stress are checked.
In Topology Optimization problems with stress constraints, the value ofthe reference stress is usually verified in the central node of each element ofthe FEM mesh. Thus,
qe(��������������) = �e(��������������e)
∣∣∣∣∣��������������e(��������������)
e = 1, . . . , Ne (46)
where ��������������e is the stress tensor and �e is the Von Mises reference stress in thecentral point of element e. Consequently,
If we take into account the previous considerations (being qe a referencestress) all the derivatives of the stress constraints can be easily obtained byusing the techniques proposed in section 3.2.
The previous algorithms and procedures allow to obtain analytically thederivatives of the stress constraints. However, a final step in the sensitivityanalysis process is required in order to complete the whole sensitivity analysisprocedure: it is necessary to compute the derivatives of the reference stressand the derivatives of the stress tensor.
4.1. Sensitivity analysis of the Von Mises failure criterion
The most usual reference stress used in topology optimization is the VonMises stress since most of the structural problems involved are made of steel.The Von Mises failure criterion defines a reference stress by assuming an equalresponse of the material under tension and compression configurations. Thisreference stress is defined by:
�VM =
√1
2
[(Δ�xy)2 + (Δ�yz)2 + (Δ�zx)2 + 6�
](51)
being
Δ�xy = �x − �yΔ�yz = �y − �zΔ�zx = �z − �x
� = � 2xy + � 2
yz + � 2zx
(52)
where
�������������� = {�x �y �z �xy �yz �zx}T (53)
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are the components of the stress tensor �������������� in a predefined point stated ac-cording to a Cartesian coordinate system (x, y, z).
In this paper, the topology optimization problems solved are two-dimensional.Thus, the sensitivity analysis and the corresponding stress tensor are also 2D.Consequently, the Von Mises reference stress can be defined, in plane stress,as:
�VM =√�2x + �2
y − �x�y + 3� 2xy. (54)
The sensitivity analysis of the Von Mises reference stress (54) can beobtained as:
d�VM
d��������������=(∂�VM
∂�x
∂�VM
∂�y
∂�VM
∂�xy
), (55)
where:
∂�VM
∂�x=
2�x − �y2�VM
∂�VM
∂�y=
2�y − �x2�VM
∂�VM
∂�xy=
6�xy2�VM
(56)
Second order derivatives can be obtained by differentiating (56) as:
d
d��������������
(d�VM
d��������������
)T=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
∂
∂�x
(∂�VM
∂�x
) ∂
∂�x
(∂�VM
∂�y
) ∂
∂�x
(∂�VM
∂�xy
)∂
∂�y
(∂�VM
∂�x
) ∂
∂�y
(∂�VM
∂�y
) ∂
∂�y
(∂�VM
∂�xy
)∂
∂�xy
(∂�VM
∂�x
) ∂
∂�xy
(∂�VM
∂�y
) ∂
∂�xy
(∂�VM
∂�xy
)
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(57)
and, by calculating the terms of the matrix, second order derivatives can beobtained as:
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d
d��������������
(d�VM
d��������������
)T=
3
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
�2y + 4� 2
xy
4�3VM
−�x�y − 2� 2xy
4�3VM
�xy(�y − 2�x)
2�3VM
−�x�y − 2� 2xy
4�3VM
�2x + 4� 2
xy
4�3VM
�xy(�x − 2�y)
2�3VM
�xy(�y − 2�x)
2�3VM
�xy(�x − 2�y)
2�3VM
�2VM − 3� 2
xy
�3VM
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(58)
4.2. Sensitivity Analysis of the stress tensor
The sensitivity analysis of the stress tensor is essential to obtain thederivatives of the stress constraints proposed. Thus, once we have obtainedthe sensitivity analysis of the reference stress, it is necessary to describe theprocedure to obtain the required derivatives of the stress tensor (��������������(��������������)) in apredefined point of the structure (rrrrrrrrrrrrrro). If we assume linear elasticity, smalldisplacements and small displacements gradient hypotheses and we use aGalerkin type FEM formulation, the computed stress tensor in FEM is givenby
where uuuuuuuuuuuuuu(rrrrrrrrrrrrrro) is the vector of computed displacements at point (rrrrrrrrrrrrrro). In fact,expression (59) is applied over the vector of theoretical displacements but, inpractice, we are only able to obtain computed values of the displacements.According to the discretization developed in the finite element formulation[18], the stress tensor in point rrrrrrrrrrrrrro can be obtained by considering the nodaldisplacements (��������������) as:
where the differential operator LLLLLLLLLLLLLL is applied over the matrix ΦΦΦΦΦΦΦΦΦΦΦΦΦΦi(rrrrrrrrrrrrrro). This
matrix contains the values of the shape function corresponding to node i inthe point rrrrrrrrrrrrrro. According to [18] the matrix ΦΦΦΦΦΦΦΦΦΦΦΦΦΦi(rrrrrrrrrrrrrr
where �i contains the value of the shape function associated to node i in thepoint rrrrrrrrrrrrrro.
The matrix DDDDDDDDDDDDDD and the matrix obtained by applying LLLLLLLLLLLLLL over ΦΦΦΦΦΦΦΦΦΦΦΦΦΦi do notdepend on the design variables according to the FEM formulation proposedin [18, 20, 21]. Thus,
d
d�jk
( d��������������
d�iℓ
)= 0,
{∀ i = 1, ..., N, ∀ ℓ = 1, 2∀ j = 1, ..., N, ∀ k = 1, 2
(64)
Note that, the full procedure to obtain stress constraints derivatives intopology optimization problems has been developed by stating the derivativesof the stress tensor and the derivatives of the reference stress proposed. Thus,specific sensitivities of the stress constraints approaches proposed in [20, 21]can be developed by applying the previous algorithms.
5. Sensitivity Analysis of Local Stress Constraints
According to [7, 8, 18, 20, 21, 24], the local stress constraints approachimposes one local constraint in the central point of each element of the finiteelement mesh. This local stress constraint is defined, according to [18, 20, 21],as
and " is the so called “relaxation factor” and usually takes the values " ∈(0.001, 0.1) [4, 5, 7, 21, 30].
In 2D problems and by assuming the plane stress hypothesis, the VonMises stress criterion is defined as
�VM,e =√�2x,e + �2
y,e − �x,e�y,e + 3� 2xy,e (67)
where
��������������e =
⎧⎨⎩�x,e�y,e�xy,e
⎫⎬⎭ (68)
is the stress tensor in the central point of element e.
5.1. First order derivatives
First order derivatives of the local stress constraints can be obtained byapplying the “Adjoint Variable” method and by taking into account sec-tions 4.1 and 4.2 as:
All the terms in the previous equation can be directly obtained by dif-ferentiating or by applying (55), (56) and (62). In addition, the “AdjointVariable” can be obtained by solving the system of linear equations proposedin (72). Thus,
∂ge∂�VM,e
= �qe, (73)
and
∂ge∂�i
= 0 if i ∕= e. (74)
If i = e then
∂ge∂�e
= q �q−1e [�VM,e − �max'e] + �max"�
q−2e . (75)
5.2. First and Second Order Directional Derivatives
According to the general optimization procedure proposed in [18, 20, 21],first order derivatives are required to obtain the search direction. This searchdirection is used now to obtain first and second order directional derivativessince they are required to develop the Quadratic Line Search. First and
21
second order directional derivatives of the stress constraints can be obtainedby following the ideas previously proposed in 3.2 as
where the term Ds�������������� can be previously obtained by applying (38). As it canbe observed, the rest of the terms match up with the ones previously studiedfor the first order analysis in (69), (73), (74) and (75).
Second order directional derivatives of the stress constraints can be ob-tained according to (41). In addition, the term D2
ss�������������� must be obtained bysolving the system of linear equations proposed in (43). Thus,
6. Sensitivity Analysis of a Global Stress Constraint
The local approach of stress constraints is the most usual formulationin topology optimization of structures. However, other different approachesthat consider stress constraints can be analyzed in order to reduce the com-puting effort required. The most usual alternative formulation is the socalled “Global Approach of stress constraints” [20, 21]. In this kind of for-mulations, one global constraint that aggregates the effect of all the localstress constraints is imposed. The global approach assembles the contribu-tion of all the local constraints in only one function. This approach does notguarantee the feasibility of all the local constraints. However, the use of theglobal approach of the stress constraints means a great reduction in com-puting time since only one constraint is involved in the optimization process.Furthermore, this formulation is usually based on the aggregation of the localstress constraints contributions. Thus, the Sensitivity Analysis required canbe obtained by applying the techniques proposed in the previous sectionswith some slight modifications.
6.1. First order derivatives
First order sensitivity analysis can be obtained by applying the techniquesand the algorithms proposed in (34) and (35). In order to establish thesealgorithms more specifically, it is necessary to define
23
��������������VM =
{�VM,e
}e=1,...,Ne
(83)
Now, if we take into account the previous consideration, the global stressconstraint function proposed in [20, 21] can be written, according to [13, 25],as
is a normalized stress associated to element e, the first term on the right of(87) can be obtained as
∂GKS
∂�VM,e
=1
S
exp �(�∗e−1)
�max 'e. (90)
In addition, each component of the vector defined in the last term on theright of (86) can be stated as
∂GKS
∂�e=
1
S
⎛⎜⎝ �VM,e"
�2e
�max'2e
⎞⎟⎠ exp � (�∗e − 1) . (91)
6.2. First and second order directional derivatives
First order derivatives of the global stress constraint have been alreadycalculated in the previous section. Thus, first and second order directionalderivatives of the global stress constraint can be obtained by applying the“Direct Differentiation” procedure proposed in section 4. First order direc-tional derivatives can be obtained, according to (38), as:
The rest of the terms in (93) can be directly obtained from the generalprocedures proposed in section 4.
7. Sensitivity Analysis of the Block Aggregated Approach of theStress Constraints
In the previous sections we have studied the sensitivity analysis proce-dures for the local and the global approach of stress constraints. Theseformulations of the stress constraints have been previously studied in somepapers of the authors [18, 21]. More recently, the authors have also pub-lished a different formulation of the stress constraints that defines groups ofelements and states one global constraint per group by aggregating the con-tributions of the elements of each group. This formulation is called “Block
27
aggregation of stress constraints” and it was implemented successfully intopology optimization problems by the authors [19, 20, 22, 23]. Thus, eachconstraint is defined for each block as:
and N be the number of elements contained in block b.
The constraints obtained in (102) are essentially identical to the onesproposed for the global constraint approach but they only aggregate thecontribution of the local constraints related to the elements contained ineach block. According to this idea, the sensitivity analysis of the stressconstraints can be obtained by using the algorithms proposed in section 6.1by taking into account the elements of the finite element mesh contained ineach global constraint. Thus, the sensitivity analysis procedure proposed insection 6.1 may be used to obtain the first order sensitivity analysis and firstand second order directional derivatives by taking into account the previousconsiderations.
7.1. First order derivatives
First order Sensitivity Analysis of the stress constraint associated to blockb can be obtained as:
the first term on the right of (104) can be obtained as:
∂GbKS
∂�VM,e
=
⎧⎨⎩1
Sb
exp � (�∗e − 1)
�max 'eif e ∈ Bb
0 if e /∈ Bb
(107)
and
∂GbKS
∂�e=
⎧⎨⎩�VM,e " exp � (�∗e − 1)
Sb �max '2e �
2e
if e ∈ Bb
0 if e /∈ Bb
(108)
7.2. First and second order directional derivatives
First and second order directional derivatives can be obtained by usingthe methodology proposed for the global constraint in section 6.2 by takinginto account that only the contributions of the set of elements Bb containedin block b must be considered. Thus,
where Ds�������������� can be obtained according to (38) and the rest of the involvedterms have been previously analyzed in (107) and (108).
Second order directional derivatives can be obtained by applying the tech-niques used to obtain the second order directional derivatives of the globalstress constraint by taking into account the elements contained in each blockas:
The terms in (110) can be obtained like it was proposed for (96), (97),(98) and (99) as:
∂
∂�VM,k
(∂Gb
KS
∂�VM,e
)= − �
(Sb)2
exp �(�∗k+�∗
e−2)
�2max'k'e
+
+ �ke1
Sb
� exp �(�∗e−1)
�2max'
2e
,
(113)
∂
∂�k
(∂Gb
KS
∂�VM,e
)= − �"
(Sb)2
�VM,k exp �(�∗k+�∗
e−2)
�2k�
2max'
2k'e
+
+ �ke"
Sb
exp �(�∗e−1)
�max�2e'
2e
(��VM,e
�max'e+ 1
),
(114)
∂
∂�VM,e
(∂Gb
KS
∂�k
)=
∂
∂�k
(∂Gb
KS
∂�VM,e
), (115)
31
∂
∂�k
(∂Gb
KS
∂�e
)= − �"2
(Sb)2�2max
(�VM,k�VM,e
(�e 'e �k 'k)2
)exp �(�∗
e+�∗k−2) +
+ �ke1
Sb
2"�VM,e
�max
"− �e'e�4e'
3e
exp �(�∗e−1) +
+ �ke�
Sb
("�VM,e
�2e�max'
2e
)2
exp �(�∗e−1)
(116)
{k ∈ Bb
e ∈ Bb
The rest of the derivatives required to obtain the directional derivativesof the block aggregated approach of the stress constraints have been analyzedin the previous sections.
8. Numerical examples
In this section we present some examples solved by using the previousmethodology in order to show its feasibility. The first example is the problemof the Michell beam. The second example analyses the topology optimizationof a cantilever beam.
8.1. Michell beam
The first example performs the analysis of the topology optimizationproblem associated to the well-known Michell beam [14]. This example corre-sponds to a cantilever beam supported on the border of a circle (see figure 2).This example was previously analyzed in 1904 by Michell [14], who obtainedthe optimal solution theoretically. According to the theoretical solution pro-posed by Michell in [14] (shown in figure 4), the optimal volume of materialcan be obtained as:
V = F a ln
(a
r0
) (1
P+
1
Q
)(117)
where F is the vertical force applied, a is the distance between the verticalforce and the center of the circle, r0 is the radius of the circle and P and Qare the tension and compression elastic limits of the material being used.
32
Figure 2 shows the dimensions of the domain of the structure proposedin this example. This domain is discretized with a structured mesh of 6000quadrilateral 8-node elements (figure 3). The structure is 0.01 m thick.
The external force (6 kN) has been distributed on the surface of twocontiguous elements in order to avoid stress accumulation. The materialbeing used is steel with density mat = 7650 kg/m3, Poisson ratio � = 0.3,Young Modulus E = 2.1 105 MPa and elastic limit �max = 230 MPa.
Figure 2: Michell beam dimensions (units in centimeters).
Figure 3: FEM mesh used in the Michell beam problem.
Figure 4 shows the theoretical solution of this problem proposed byMichell [14].
Figure 5 shows the optimal solution obtained by using the local approachof the stress constraints and the sensitivity analysis techniques proposed in
33
Figure 4: Theoretical solution of the Michell beam (Michell [14]).
section 5. Figure 6 shows the final stress configuration of the optimal solutionpresented in figure 5.
Figure 7 shows the optimal solution obtained by using the global approachof the stress constraints and the sensitivity analysis techniques proposed insection 6. Figure 8 shows the normalized stress configuration of the optimalsolution presented in figure 7.
Figure 9 shows the optimal solution obtained by using the block aggre-gated approach of the stress constraints and the sensitivity analysis tech-niques proposed in section 7. Figure 10 shows the normalized stress config-uration of the optimal solution presented in figure 9. The number of blocksused is 100.
Table 1 shows the most significant parameters of the solutions proposed(local, global and block aggregated). It also compares the amount of ma-terial obtained in these solutions (local, global and block aggregated) withthe optimal theoretical values proposed by Michell. As it can be observed,the optimal volume obtained with the three formulations is slightly smallerthan the theoretical one. This fact can be easily understood by consideringthat the proposed topology optimization formulation uses continuum designvariables to approach the discrete design variables of the original problem.
8.2. Cantilever beam
The second example is the optimal design of a cantilever beam with nulldisplacements in the left border and with a vertical force applied in themiddle of the right border. Figure 11 shows the dimensions of the domain
34
Figure 5: Michell beam solution obtained by using the local approach of the stress con-straints.
Figure 6: Normalized stress state of the Michell beam solution obtained by using the localapproach.
and the position of the vertical forces applied. In this example, self-weightof the structure has also been included as a structural load.
The domain of the structure has been discretized by using a homogeneousmesh with 120× 60 = 7200 8-node quadrilateral elements. The thickness ofthe structure is 0.2 m.
The external force applied (2 103 kN) has been distributed on four con-tiguous elements in order to avoid stress accumulation phenomena.
The material being used in this problem is steel with density mat = 7650 kg/m3,Young Modulus E = 2.1 105 MPa, Poisson ratio � = 0.3 and elastic limit�max = 230 MPa.
35
Figure 7: Michell beam solution obtained by using the global approach of stress constraints.
Figure 8: Normalized stress state of the Michell beam solution obtained by using theglobal approach.
Figures 12 and 13 show the optimal solution and the normalized stressstate for the cantilever beam problem obtained by using the local approachof stress constraints.
Figures 14 and 15 show the optimal solution and the normalized stressstate for the cantilever beam problem obtained by using the global approachof stress constraints.
Figures 16 and 17 show the optimal solution and the normalized stressstate for the cantilever beam problem obtained by using the block aggregatedapproach of stress constraints. The number of blocks used is 120.
The solution obtained must be symmetric since the material being used
36
Figure 9: Michell beam solution obtained by using the block aggregated approach of stressconstraints.
Figure 10: Normalized stress state of the Michell beam solution obtained by using theblock aggregated approach.
Figure 11: Scheme of the cantilever beam problem (units in m).
37
Table 1: Summary of the Michell beam solutions
q " p �Volume
(m3)Theor. Vol.
(m3)
Local appr.(Fig. 5)
1 0.03 4 - 5.23 10−6 5.61 10−6
Global appr.(Fig. 7)
− 0.03 4 40 5.00 10−6 5.61 10−6
Block aggr.appr. (Fig. 9)
− 0.03 4 40 5.27 10−6 5.61 10−6
Figure 12: Optimal solution of the cantilever beam problem by using the local approachof stress constraints.
Figure 13: Normalized stress state of the cantilever beam problem by using the localapproach of stress constraints.
presents an equal behaviour in tension and compression. However, we haveanalyzed the entire domain and do not consider this fact in order to verifythe methodology proposed. Note that the optimal material distribution ob-
38
Figure 14: Optimal solution of the cantilever beam problem by using the global approachof stress constraints.
Figure 15: Normalized stress state of the cantilever beam problem by using the globalapproach of stress constraints.
tained is symmetric although this issue has not been forced. This symmetricdistribution remarks the validity of the techniques proposed. In addition, therelaxation parameter also introduces a small effect on the stress constraintsfor intermediate values of the relative densities. This small decrease of thestress constraints due to the relaxation allows a larger weight reduction thanthe expected one without any relaxation.
Table 2 shows the most important parameters of this problem in order tofix their value and better understand them.
9. Conclusions
In this paper we propose a complete analytical procedure to obtain highorder sensitivity analysis of stress constraints in topology optimization ofstructures. The Sensitivity Analysis of the objective function and the stressconstraints is essential to obtain adequate solutions since the resulting op-
39
Figure 16: Optimal solution of the cantilever beam problem by using the global approachof stress constraints.
Figure 17: Normalized stress state of the cantilever beam problem by using the blockaggregated approach of stress constraints.
Table 2: Summary of the Cantilever Beam solutions
q " p �Final Weight
(%)
Local approach(Fig. 12)
1 0.01 4 - 18.27 %
Global approach(Fig. 14)
− 0.01 4 40 16.22 %
Block aggr. approach(Fig. 16)
− 0.01 4 40 16.61 %
timization problem is very complicated due to the large number of designvariables and constraints involved. Consequently, it is necessary to use suit-able optimization algorithms that involve, at least, first order derivatives. In
40
this paper we have developed first order derivatives and second order direc-tional derivatives of the stress constraints. This information is required bythe Sequential Linear Programming with Quadratic Line Search algorithmthat we have used.
All the derivatives were developed analytically in order to reduce thecomputing time required by numerical approximations (finite differences) andin order to obtain reliable approximations. The minimum weight approachwith stress constraints involves a very large number of constraints (contraryto the most usual maximum stiffness formulations) and consequently requiresvery large computing time.
Full set of first order derivatives can be obtained via an “Adjoint Vari-able” procedure in order to reduce the computing requirements. When thenumber of load cases is small, which is the most frequent situation, this tech-nique considerably reduces the computing effort required in comparison with“Direct Differentiation” algorithms. Furthermore, this technique allows tocompute the first order derivatives of a set of active constraints. Thus, itis possible to avoid the computation of the stress constraints derivatives incase they are not necessary. In addition, these calculations can be easily per-formed in parallel since the operations required to obtain each one of thesederivatives are independent from a computational point of view. Thus, firstorder derivatives of stress constraints can be easily computed in parallel [6].
On the other hand, full set of first and second order directional derivativescan be easily obtained with small computing resources by applying a “DirectDifferentiation” procedure. This technique allows to compute the full set offirst and second order directional derivatives of the stress constraints althoughfull set of first order derivatives were not previously obtained. This factallows to include the full set of constraints of the problem in the QuadraticLine Search algorithm although the full set of first order derivatives were notobtained neither used in the SLP algorithm.
The computation of these sensitivities requires to solve a large numberof systems of linear equations, specially in the computation of the full setof first order derivatives. However, the matrix of these systems of linearequations is the stiffness matrix of the structural problem. Thus, a Choleskyfactorization procedure is the right choice in order to reduce the computingeffort devoted to solve additional systems of linear equations.
The whole sensitivity analysis procedure proposed in this paper has demon-strated to work properly even in large structural optimization problems witha large number of design variables and a large number of non-linear con-
41
straints. The algorithms proposed and implemented in a topology optimiza-tion code are exact from an analytical point of view. In addition, compu-tational aspects have been addressed in order to improve and speed up thesensitivity analysis methodology. All the techniques proposed are devoted toreduce as much as possible the required computation effort without loosingprecision. Otherwise the computation of the sensitivity analysis will becomeunaffordable in practical applications due to the large number of stress con-straints involved.
Acknowledgements
This work has been partially supported by the Ministerio de Educacion yCiencia of the Spanish Government (#DPI2007-61214 and #DPI2009-14546-C02-01) cofinanced with FEDER funds, by the Autonomous Government ofthe Xunta de Galicia (Grant #PGDIT06TAM11801PR and file #2007/09),by the Universidade da Coruna and by the Fundacion de la Ingenierıa Civilde Galicia.
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