572dbae4766fa-Topology Optimization Design Of 3D
JestrJOURNAL OFJournal of Engineering Science and Technology Review 9 (2) (2016) 121-128Engineering Science andTechnology ReviewResearch Articlewww.jestr.orgTopology Optimization Design of 3D Continuum Structure with Reserved Hole Basedon Variable Density MethodBai Shiye and Zhu JiejiangThe Department of Civil engineering, Shanghai University, Shanghai 200072, ChinaReceived 25 February 2016; Accepted 22 May 2016AbstractAn objective function defined by minimum compliance of topology optimization for 3D continuum structure wasestablished to search optimal material distribution constrained by the predetermined volume restriction. Based on theimproved SIMP (solid isotropic microstructures with penalization) model and the new sensitivity filtering technique,basic iteration equations of 3D finite element analysis were deduced and solved by optimization criterion method. All theabove procedures were written in MATLAB programming language, and the topology optimization design examples of3D continuum structure with reserved hole were examined repeatedly by observing various indexes, includingcompliance, maximum displacement, and density index. The influence of mesh, penalty factors, and filter radius on thetopology results was analyzed. Computational results showed that the finer or coarser the mesh number was, the largerthe compliance, maximum displacement, and density index would be. When the filtering radius was larger than 1.0, thetopology shape no longer appeared as a chessboard problem, thus suggesting that the presented sensitivity filteringmethod was valid. The penalty factor should be an integer because iteration steps increased greatly when it is a noninteger. The above modified variable density method could provide technical routes for topology optimization design ofmore complex 3D continuum structures in the future.Keywords: Topology Optimization, Reserved Hole, Multiple Loading Conditions, Variable Density Method, MATLAB1. IntroductionTopology optimization design is a computational method forachieving optimal material distribution without knowing thespecial shape of the structure in advance. Therefore,topology optimization can be used to develop great potentialof new structure with high performance. The history of thetopology optimization dates back to the truss theoryproposed by Michell in 1904. At that time, the theory canonly be applied to single working condition that dependedon the strain filed and was not applicable in practicalengineering [1]. In 1964, Dorn et al. proposed groundstructure approach [2] that was applied into topologyoptimization. From then on, topology optimization hadbecome a more active research field. In recent years,topology optimization theory of continuum structure [3] hasdeveloped rapidly. Many topology optimization methodsincluding variable thickness method, variable densitymethod, asymptotic structure optimization, independentcontinuum mapping method (ICM), level set, and nodaldensity method [4] have been proposed. Topologyoptimization methods have been applied in many fields.Especially the variable density method has beensuccessfully applied in many practical engineering projectsbecause of its simple programming procedure and highefficiency in calculation competence [5]. Yang et al.transformed topology optimization problems into linearprogramming problems and then used the variable densitymethod to design engine components [6]. With the* E-mail address: bai880602@sina.comISSN: 1791-2377 2016 Eastern Macedonia and Thrace Institute of Technology.All rights reserved.development of many CAE software based on the variabledensity method, the engineering examples of topologyoptimization that are solved by numerical simulationmethods have increased daily [7], [8], [9], [10], [11], [12].2. State of the ArtNumerical instability problems including chessboard andmesh dependence phenomenon are ubiquitous in topologyoptimization design based on variable density method [13].Sigmund proposed a filtering radius method to solve thisproblem [14]. Zuo modified the previous filtering method[15]. Chen used an adjacent entropy filtering method basedon graph theory to eliminate the chessboard and meshdependence problems [16].The essence of topology optimization is to solve theextrema problem. Therefore, the development of topologyoptimization is inseparable from optimal mathematicalalgorithms. The advantage of the optimization criterionmethod (OC method) is fast convergence speed, however,this method might be difficult to deal with the complexstructure under the conditions of different constraints [17].In 1960, Schmit adopted mathematical programming theoryto solve optimization problems of elastic structure under thecondition of multiple load [18]. His research pushed thedevelopment and application of topology optimizationalgorithms. Traditional optimization methods are notapplicable for these problems because mathematical modelsare complicated, nonlinear, random, and blurry in realoptimization design problems. Therefore, many newalgorithms, including simulated annealing method [19],
Bai Shiye and Zhu Jiejiang/ Journal of Engineering Science and Technology Review 9 (2) (2016) 121-128genetic algorithm [20], evolutionary algorithm [21], andneural network algorithm [22], [23], were presented. Basedon the above modern methods, the global or approximatelyglobal solutions can generally be obtained. However, thiswill cost huge amount of calculation time. The minimumcompliance problem in nonlinear programming can be dealtwith by sequential quadratic programming (SQP) [24], [25]and moving asymptotes method (MMA) [26], [27].Compared with SQP and MMA methods, the OC methodhas many advantages including good computationconvergence and fast computation speed in topologyoptimization design. Hence, in this study, the OC method isused to solve the minimum compliance problem of thetopology optimization model.The remainder of this study is organized as follows.Section 3 establishes a 3D topology optimization model ofcontinuum structure with the object function of minimumcompliance and deduces the 3D finite element formulationsby OC method. Section 4 studied the topology optimizationdesign examples of 3D continuum structure with reservedhole and discusses the influence of mesh numbers, penaltyfactor, and filtering radius on results of topologyoptimization. Section 5 presents conclusions.where Emin is the elasticity modulus of void material. Toavoid the singularity of the stiffness matrix, the value is notzero as usual. The value is set to “0.001.” The improvedSIMP model makes the penalty factor and elasticity modulusof void material mutually independent. Compared with theformer model, the improved SIMP model has betterconvergence in computation [28].3. MethodologyNi { j : dist (i, j ) R}3.3 Sensitivity FilteringTopology optimization model based on variable densitymethod is always accompanied with numerical problems,such as mesh-dependence, checkerboard pattern, localextremum, and so on. To solve these problems, the commonway is to introduce density-filtering method shown asfollows:xi wherej N i(4)H ij v jis the volume of element.H ij isthe weighting(5)Hij R dist (i, j )(6)(1)Filtering density xi is modified density. This factor isintroduced into the SIMP model, and the formula can bededuced as follows:where C(x) is compliance of structure. x is material pseudodensity as the design variable. Ω is the given design domain.V* is the required optimal design volume. V0 is the originaldesign volume.α is the volume fraction.pEi ( xi ) Emin xi ( E0 Emin ), xi [0,1](7)3.4 Element Stiffness Matrix and Formulation of FiniteElementBased on improved SIMP model, as indicated in Formula (7),sensitivity filtering method, and Hooke's Law, the 3D stressmatrix of isotropic material element i is expressed as:3.2 Newly Improved SIMP ModelThe basic approach used to handle the problem of discretevariables in numerical calculation is to substitute continuousfunction for discrete function. The continuous function cangenerate many elements, and the continuous density ofwhich is between 0 and 1. However, making this kind ofstructure material practical is difficult. To solve this problem,the penalty factor is usually introduced to suppress theappearance of intermediate-density element. By usingvariable density method and introducing penalty factor, therelationship between variable density xi and elasticitymodulus Ei based on SIMP model is expressed as follows:Di ( xi ) Ei ( xi ) Di0 , xi [0,1](8)where Di0 is the stress matrix composed of unit Young’smodulus and can be expressed as:Di0 1 (1 υ )(1 - 2υ )υ000 1 υ υ υ 1 υ υ 000 υ υ 1 υ000 00 (1 - 2υ ) / 200 0 0 000(1 - 2υ ) / 20 00000(12υ)/2 (2)where E0 is elasticity modulus of solid materials. p is thepenalty factor. The evolved solid isotropic microstructureswith penalization (SIMP) model are expressed as follows:Ei Ei ( xi ) Emin xip ( E0 Emin ), xi [0,1]viH ij v j x jwhere the operator dist (i , j ) is the center distancebetween element i and element j, and R is the filtering radius.The weighting factor H ij is:min C ( x)Ei Ei ( xi ) xip E0 , xi [0,1]j N ifactor. Ni is the element set adjacent to element i and can bedefined as3.1 Minimum Compliance ProblemThe minimum compliance problem of topology optimizationdesign of the continuum structure constrained by volumefraction is expressed as follows: V [ x] xdΩ V * Ω s.t. 0 x 1, V * αV0 x 0 or 1 (9)where υ is Poisson ratio of isotropic material. Based onfinite element theory, the element stiffness matrix of elastic(3)122
Bai Shiye and Zhu Jiejiang/ Journal of Engineering Science and Technology Review 9 (2) (2016) 121-128solid is the volume integral of stress matrix Di ( xi ) and strainmatrix B, which can be expressed as follows: 1 1 1 T(10)ki ( xi ) B Di ( xi )Bdξ1dξ2dξ31 11nni 1i 1K ( x) K i ( x i ) Ei ( x i ) K i0where Ki0 is the global constant stiffness matrix that iscomposed of the element stiffness matrix. According toFormula (7), the matrix can be expressed as:where ξ e (e 1,2,3) is the natural coordinate system ofhexahedron element. Strain matrix B describes therelationship between the node displacement of elements andthe strain. Based on SIMP model, the element stiffnessmatrix can be expressed as:nki ( xi ) Ei ( xi )k(11)(18). 1 1 1 1 1 1BiT Di0 Bdξ1dξ 2 dξ3K ( x)U ( x) F(12) k1 k T1 2ki0 (1 υ)(1 2υ) k T3 k 4where k1 k 2 kk1 2 k3 k5 k5k2k1k7k6k8k2 k6 k 7 kk3 5 k9 k12 k k6k3k3k6k T5k2k5 k9 kk7 8k6 k k 10k8 2 k6 k4k2 k1 k11 k14k8 kk10 11k9 k k11 4 k5 k13 k10k4 k3 k10k4 k T4 k T2 k 1T k10k14k7k2k7k14k2k9k11k7k4 k14 kk11 11 k7k6 k 6 k2 k13 k10k8 k1 k123.5 Topology Optimization Model Based on ImprovedSIMP methodThe solution of minimum compliance problem is to find thedistribution form of material density, which makes structuraldeformation minimum under the action of the specified loadand constraint. Therefore, compliance of structure can bedefined as:(13)k8k9c( x) F TU ( x)k7 k5 k6 k10 k12 k13 k10 k8 k9 k7 k11 k14 k12 k 2 k9 k11 k7 k14 x [ x 1 , x 2 ,.x n ]TFindc( x) F T U ( x)Minimizen U T ( x) [ E min x i ( E 0 E min )]K i0U ( x)p(21)i 1Ts.t. v( x) x v V * 0,(14){}x χ, χ x R n : 0 x 1wheredensityxisdeterminedbyFormulav [v1 ,., vn ] is the volume vector of elements.(4).T3.6 The Sensitivity Analysis of StructureBased on the improved SIMP model, sensitivity analysisshould be indispensable in obtaining the solution toobjective function.In Formula (21), the derivative of volume constraintk1 (6υ 4) / 9, k 2 1 / 12,k3 1 / 9, k 4 (4υ 1) / 12,k5 (4υ 1) / 12, k6 1 / 18,function(15)k9 (6υ 5) / 36, k10 (4υ 1) / 24,v(x)with respect to design variable is: v( x) v( x) x i xei N e x i xek11 1 / 24, k12 (4υ 1) / 24,k13 (3υ 1) / 18, k14 (3υ 2) / 18The global stiffness matrix is the collection of elementstiffness matrix, which can be expressed as:K ( x) Αin 1ki ( xi ) Αin 1Ei ( xi )ki0(20)By introducing the volume restriction, the minimumcompliance problem can be further expressed as follows:wherek7 1 / 24, k8 1 / 12,(19)where F is the force vector of nodes.Substituting Formula (9) into Formula (12), it can befurther organized as: k1 k 2 kk5 8 k3 k5 k 4]pBy solving Formula (19), we can obtain the displacementvector of nodes U (x)whereki0 [K ( x) Emin x i ( E0 Emin ) K i0i 10i(17)where(22)H ieve v( x) xi vi , . The mesh element used xe j N H ij v j xiiin the study is the cube element with unit volume. That is,vi v j ve 1.In Formula (21), the derivative of compliance with(16)where n is the amount of elements. Based on the definitionof the global stiffness matrix, Formula (16) can be furtherexpressed as:respect to design variable123xe is:
Bai Shiye and Zhu Jiejiang/ Journal of Engineering Science and Technology Review 9 (2) (2016) 121-128 c( x) c( x) x i xei N e x i xe(24)The large change of relative density from void to solid isnot allowed. Therefore, moving limit m should beintroduced into the design variable x. The iterative densitycan be further expressed as follows:In Formula (19), the derivative of total stiffness withrespect to design variable x i is: K ( x) U ( x)U ( x) K ( x) 0 xi xixenew(25) xi n xii 1 pxp 1i] x i (E0 E min ) K i0pmin(E0 E min )K xi(27)x new x0i U ( x)T p x i ( E0 - Emin ) K i0 U ( x) p 1Given thatKi0where(28) xi[p 1]whereui (x) isBecauseki0(29)the displacement vector of element node.is positive definite, c( x) 0 . xi3.7 Optimization algorithmOC method is an indirect optimization method because itdoes not optimize the object function directly. The methodmakes K-T condition, which the optimal solution shouldmeet in math as the guideline the most optimal structureshould satisfy. The outstanding characteristic is with fastconvergence speed and less iteration number. The K-Tcondition of optimization criteria method should satisfy c( x) v( x) λ 0 xe xe(33)ε ε(34)is the error limitation that is set to “0.001.”(1) Input of original data: maximum iteration number,material parameters (elastic modulus and Poissonratio), coordinate of force acting point, coordinatesof constraint node and freedom numbers(2) Definition of elemental stiffness matrix andintegration of total stiffness matrix(3) Finite element analysis and calculation of theelement nodal displacement(4) According to the above calculated displacement,calculating sensitivity, and objection function (compliance)(4) Sensitivity filtering method(5) To obtain the solution (Lagrange multiplier) ofFormula (33) by bisection method and then obtainthe new design variable density.(6) Convergence test is performed by Formula (34). Ifsatisfied, the results (compliance, displacement, andnephogram of density distribution) will be theoutput; otherwise the solving step returns to Step 3,in which the density variable is updated to performthe finite element analysis.is the collection of element stiffness u i ( x) T p x i ( E 0 - E min )k i0 u i ( x)can be obtained3.8 Procedure of Obtaining SolutionsBased on MATLAB programming, we depict the wholesolving flow chart shown in Figure 1 and provide specificexplanations for each step.matrix, Formula (28) can be expressed as: c( x)(32)In addition, the condition that convergence rule shouldmeet isCombined with Formula (24), Formula (26) and Formula(27), it can be expressed as: c( x)λx T ( λ) V * 0 [E max(0, xe m), if xe Beη max(0, xe m), min(1, xe m), if xe Beη min(1, xe m), ηotherwise xe Be(32), λ is the only unknown quantity.in Formula (33) by dichotomy method.(26)Based on Formula (18), the equation can be expressed as: K( x )(31)where m is the moving limit.η is the damping coefficientranging from 0 to 1. Introducing the damping coefficient andthe moving limit aims to improve iteration convergence.Sigmund proposed m 0.2 and η 0.5 [23]. In FormulaFormula (25) can be further organized as: U ( x) K ( x) K ( x) 1U ( x) xi xi 1Be where c( x) U ( x) U ( x) FT U ( x )T K ( x ) xi xi xi c( x) v( x) λ xe xe (23)(30)where λ is Lagrangian multiplier. Formula (30) can befurther expressed as:124
Bai Shiye and Zhu Jiejiang/ Journal of Engineering Science and Technology Review 9 (2) (2016) 121-128function is to minimize the compliance of the entirestructure. The iteration results are shown in Table 1.StarInput originaldataInitialization (elementstiffness matrix and totalstiffness matrix)1. Iteration parameters;2. Material parameters;3. Loading nodal coordinatesand freedom numbers;4. Constrains coordinates andfreedom numbers.Finite elementanalysisObtain compliance andsensitivitySensitivityfilteringLagrange multiplier andupdated design variableNoFig. 2. Initial design domain of the 3D continuum cantilever beamConvergencetestTable 1 presents the topology optimization results in theiteration process. From the table, we can see that when theiteration step reaches step 5, the shape of the topologystructure is unformed. When the iteration process reachesstep 50, the structure is fully shaped into many supportingbars in the local region. With the increase of iterationnumbers, the structure form has only a subtle change.To reflect the change trend of compliance, maximumdisplacement, and density index with the increase of theiteration numbers, Figures 3(a), (b), (c) are drawnrespectively. Herein, the density index is defined as the ratioof element numbers of which density is less than 0.01 andgreater than 0.99 to the total element numbers. This indexreflects the extent that the whole element density tends to be0 or 1. When the iteration number begins to reach 20, thecompliance and the maximum displacement decline sharply,whereas the density index climbs rapidly. As the iterationnumber continues to increase, all the results (compliance andmaximum displacement) have only subtle changes. From theabove density nephogram change and iteration results ofvarious index, we can conclude that the presented numericalmethod is stable, reasonable, and with good convergence.YesOutput resultsFig.1. Solving flow chart of 3D continuum structure topologyoptimization4. Result Analysis and Discussion4.1 Example VerificationAs is shown in Figure 2, the design domain is a 3Dcontinuum cantilever beam with reserved hole. The length inx direction is 40 mm; the width in y direction is 20 mm; andthe thickness in z direction is 10 mm. A cylinder hole withradius of 7 mm is in the center of the entire structure. Thediscrete structure is partitioned into 32 16 8 hexahedralelements. The left side boundary condition is fully fixed andthe right side at the bottom of the beam is subjected to aconcentrated line load 1 kN/m. The elastic modulus ofmaterial E0 1 GPa, the volume fraction is 0.3, the penaltyfactor p 3.0, and the filtering radius r 1.5. The objectiveTable 1. Topology optimization results in the process of iterationsIterationnumbers51020501001753D cubicnephogramSide-viewnephogramNote:is density scale, which is applicable to all the density nephogram in thisstudy.125
Bai Shiye and Zhu Jiejiang/ Journal of Engineering Science and Technology Review 9 (2) (2016) 121-128Compliance C/ (N.mm)400030004.2 Mesh Dependency AnalysisTo analyze the influence of element numbers onoptimization results and computation time, five cases withmesh numbers 16 8 4, 20 10 5, 32 16 8, 40 20 10,48 24 12 are discussed. The penalty factor is p 3.0 andfiltering radius is r 1.5.From Table 2, we can see that if the mesh numbers aretoo coarse or too fine, the calculated compliance and themaximum displacement are both too large. In special cases,when the mesh is 32 16 8, the optimal compliance is theminimum in five cases. As the mesh numbers increase,iteration steps increase and computation time becomeslonger. Meanwhile, the density index becomes larger, that is,the element number ratio of the intermediate densitydecreases. In addition, the mesh numbers do not affect thestructure form significantly. Based on overall considerationof the optimization results and calculating cost, thereasonable mesh numbers in this study are 32 16 8.2000020 40 60 80 100 120 140 160 180Iteration number NMaximum displacement U/(mm)(a) Compliance0.450.400.350.300.254.3 Penalty Factor AnalysisTo analyze the influence of different penalty factors on theoptimization results and computation time, we studied fivecases with penalty factors 2.0, 2.5, 3.0, 3.5, and 4.0. Herein,the mesh number is 32 16 8 and filtering radius r 1.5 infive cases.As is shown in Table 3, with the increase of penaltyfactor, the calculated compliance, maximum displacement,and density index become larger. Furthermore, when thepenalty factor is non-integer, iteration steps grow apparentlyand computation cost increases. When the penalty factor isan integer, the iteration number is relatively small, and thelarger the penalty factor is, the faster the convergence timewill be. However, the optimal compliance and the maximumdisplacement both increase, which suggests that theoptimization effect declines. Based on an overallconsideration of optimization results and calculating cost,the reasonable penalty factor in this example should be 3.0.0.20020 40 60 80 100 120 140 160 180Iteration number N(b) Maximum displacement1.00.9Density index0.80.70.60.50.40.30.2020406080 100 120 140 160 180Iteration numbers N(c) Density indexFig. 3. Variation of compliance, maximum displacement, and densityindex with iteration numbersTable 2. Final topology optimization results of different numbers of mesh elementsMesh partitionIteration stepCompliance (N·m)Maximumdisplacement (m)Density index16 8 4491836.01 10-320 10 51591787.01 10-332 16 81751774.25 10-340 20 101701931.22 10-348 24 124752077.22 10-30.2345 10-30.2267 10-30.1998 10-30.2400 10-30.2530 10-30.92190.94500.98190.99150.99423D view nephogramSide-view nephogram126
Bai Shiye and Zhu Jiejiang/ Journal of Engineering Science and Technology Review 9 (2) (2016) 121-128Table 3. Final topology optimization results of different penalty factorsPenalty factorIteration stepCompliance (N·m)Maximumdisplacement (m)Density index2.02851472.35 10-32.52491635.32 10-33.01751774.25 10-33.53151895.25 10-34.01782006.01 10-30.1649 10-30.1835 10-30.1998 10-30.2158 10-30.2270 10-30.96630.98240.98190.98880.99173D view nephogramSide-viewnephogramincrease. When filtering radius is 1.0, the local region of thestructure appears as chessboard phenomenon.As the filtering radius is larger than 1.0, the optimalstructure form no longer appears as a chessboard problem.When filtering radius is too large or too small, the densityindex will increase. As the filtering radius is 3.5, the shapeof the structure changes sharply. Meanwhile, the complianceand the maximum displacement of structure become larger.Based on overall consideration of results and calculatingcost, the reasonable filtering radius in this example is 1.5.4.4 Filtering Radius AnalysisTo analyze the influence of different filtering radius on theoptimization results and computation time, four cases withthe filtering radius 1, 1.5, 2.0, 3.5 are discussed. Herein, themesh number is 32 16 8 and penalty factor p 2.0 in fivecases.Table 4 shows that with the increase of filtering radius,compliance, maximum displacement, iteration numberTable. 4. Final topology optimization results of different filtering radiusFiltering radiusIteration stepCompliance (N·m)Maximum displacement(m)Density index1.0221472.35 10-31.51751774.25 10-32.02612059.27 10-33.53063096．51 10-30.1649 10-30.1998 10-30.2331 10-30.3510 10-30.96630.98190.97900.98003D view nephogramSide-view nephogram(2) Too fine or too coarse mesh number will greatlymake the compliance and the maximum displacement larger.In addition, the finer the mesh is, the more the iteration stepis, and the computation cost will increase. Meanwhile, thedensity index will become larger, which means that theintermediate density decreases. Furthermore, the meshnumber affects the shape form of structure less.(3) As the filtering radius is larger than 1.0, the topologyshape will no longer appear as a chessboard problem, whichsuggests that the presented sensitivity filtering method isvalid. However, the larger the filtering radius is, the longerthe computation time will be. Therefore, selecting anappropriate filtering radius can not only improve the optimaleffect but also save the computation cost.(4) The penalty factor should be an integer because theiteration step increases greatly when the value is a noninteger.5. ConclusionIn this study, the topology optimization model of 3Dcontinuum structure with objective function of minimumcompliance and design variable of material distribution isestablished. By introducing the sensitivity filtering methodand the improved SIMP model, the OC iteration equation offinite element analysis is deduced. By MATLABprogramming, the 3D continuum structure with reservedhole is designed to obtain optimal topology structure shape.Based on the above analysis, we obtain the followingconclusions:(1) Compared with the previous topology optimization of2D continuum structures, the presented method of topologyoptimization of 3D continuum structure with reserved holecan enlarge the application for more complicated structures.References1.Michell A G M., “The limit of economy of material in framestructures”. Philosophical Magazine, 8(6), 1904, pp. 589-597.2.127Dorn W., Gomory R., Greenberg H., “Automatic design of optimalstructures”. Design Magazine, 3(1), 1964, pp. 25-52.
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The new 2nd grade Reading Standard 6 has been created by merging two separate reading standards: “Identify examples of how illustrations and details support the point of view or purpose of the text. (RI&RL)” Previous standards: 2011 Grade 2 Reading Standard 6 (Literature): “Acknowledge differences in the
