Different effects of economic and structural performance indexes on model construction of structural topology optimization

Transcription

1 Acta Mech. Sin. (2015) 31(5): DOI /s RESEARCH PAPER Different effects of economic and structural performance indexes on model construction of structural topology optimization G. L. Yi 1,2 Y. K. Sui 1 Received: 6 January 2014 / Revised: 25 August 2014 / Accepted: 5 February 2015 / Published online: 21 September 2015 The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2015 Abstract The objective and constraint functions related to structural optimization designs are classified into economic and performance indexes in this paper. The influences of their different roles in model construction of structural topology optimization are also discussed. Furthermore, two structural topology optimization models, optimizing a performance index under the limitation of an economic index, represented by the minimum compliance with a volume constraint (MCVC) model, and optimizing an economic index under the limitation of a performance index, represented by the minimum weight with a displacement constraint (MWDC) model, are presented. Based on a comparison of numerical example results, the conclusions can be summarized as follows: (1) under the same external loading and displacement performance conditions, the results of the MWDC model are almost equal to those of the MCVC model; (2) the MWDC model overcomes the difficulties and shortcomings of the MCVC model; this makes the MWDC model more feasible in model construction; (3) constructing a model of minimizing an economic index under the limitations of performance indexes is better at meeting the needs of practical engineering problems and completely satisfies safety and economic requirements in mechanical engineering, which have remained unchanged since the early days of mechanical engineering. B Y. K. Sui yksui@bjut.edu.cn G. L. Yi yigl2007@ s.bjut.edu.cn 1 College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing , China 2 Department of Mechanical and Aerospace Engineering, Seoul National University, Seoul, South Korea Keywords Economic index Performance index Structural topology optimization models MCVC model MWDC model Safety and economy 1 Introduction The purpose of structural topology optimization is to seek the optimum layout of structural components or subdomains within a given design space under a given set of loads and boundary conditions such that the resulting layout meets a prescribed set of structural performance targets. As an important branch of structural optimization research, topology optimization has been enriched by many solution methods. An exhaustively detailed summary of the development of topology optimization was made by Bendsøe and Sigmund [1], and the application and development of numerical modeling methods used in structural topology optimization were also reviewed in detail by Rozvany [2]. Looking back at the research of truss structure topology optimization led by Michell [3], great progress has been made since then. Not only does Michell truss theory bring development through the researches of Rozvany et al. [4 8], but also the research objects of topology optimization have developed from skeleton structures such as truss and frame into continuum structures. Although Bendsøe and Kikuchi s paper [9] superficially stated the homogenization method, they broadened the research objects of structural topology optimization in that they presented the concept of topological optimization for continuum structures. The ground structure approach [10,11], which was only used in the topological optimization of skeleton structures, has become the foundation of the homogenization method in the topological optimization of continuum structures and the foundation of those methods

2 778 G.L.Yi,Y.K.Sui seeking the optimum layout of subregions for a given domain, such as, for example, the variable thickness method [12], artificial material method [13 16], evolutionary structural optimization method [17 19], independent continuous mapping (ICM) method [20 23], and the level set method [24 26]. Recently, several new methods have been developed that are quite different from the traditional methods. For example, projection methods that use a relaxed Heaviside function [27,28] and morphology-based operators [29,30] have been developed to project filtered densities into void and solid space. A new computational framework for structural topology optimization based on the concept of moving morphable components was also proposed by Guo et al. [31]. Because of length constraints on this paper, we are unable to list all of the research here and can only mention the aforementioned methodologies, which represent just a few out of thousands. Can we conduct deep research into topology optimization as new numerical methods are being continuously proposed? This is a difficult topic indeed. However, for research in different subjects and different areas, a similar methodology may be applied to improve the methods development called the classification-explicitation method. This method looks for a perspective of classification for current research and highlights the breakthroughs related to their use, which will be studied in greater depth subsequently. For example, the ICM method proposed by the second author of this paper is a classification-explicitation method based on notions regarding structural optimization levels. It frees the design variables from their dependence on the lower level of physical variables and raises them to an independent level of design variables. In addition to the expansion of discrete variables of 0/1 into continuous variables between [0, 1], mapping functions are adopted in this method. To develop topology optimization research further, it is necessary to propose a classification perspective that has not been considered previously. Therefore, this paper aims to choose objective and constraint functions in topology optimization model construction. Furthermore, the parameters that may be chosen as the objective and constraint functions are identified, and these actually comprise only two large categories, economic indexes and performance indexes. Using this classification as a perspective, regardless of the number of structural topology optimization methods, they can only be classified into two categories: optimizing structural performance given the limitations of structural economic indexes, and minimizing the structural economic indexes while satisfying structural performance indexes. Examples of the so-called economic indexes include structural volume, weight, and cost. Examples of the so-called structural performance indexes are structural performances, such as structural compliance, displacements at points of interest, elemental stresses, and structural frequencies and amplitudes. For structural and multidisciplinary optimization problems, multifield physical, chemical, and engineering performance also belongs to the category of structural performance indexes. This paper benefits from the use of the classificationexplicitation method. First, the classification of economic and performance indexes is the perspective to be observed. Second, explicitation for two categories of optimization problems is proposed. To compare the effects of the economic and performance indexes on the model construction of structural topology optimization, two simple models based on the artificial material method are utilized to express the two categories specifically. The solid isotropic material with penalization (SIMP) model [1] in the artificial material method is based on a penalty function, E e = E(ρ e ) = ρ p e E 0, ρ e [0, 1], (1) where the subscript e denotes the number of elements, E e and E 0 are the Young s modulus of artificial and real material, respectively, ρ e is the elementary artificial relative density, and p is the penalty factor. Based on the finite-element method formula, two simple optimization models are expressed as follows: For ρ : min c(ρ) = U T KU = N u T e K eu e, MCVC model: (2) s.t. KU = F, V (ρ) V = f, 0 <ρ min ρ e 1 (e = 1, 2,..., N); For ρ : minw (ρ) = N ρ e we 0, MWDC model: (3) s.t. KU = F, u(ρ) =ū, 0 <ρ min ρ e 1 (e = 1, 2,...,N), where MCVC stands for minimum compliance with a volume constraint, and MWDC stands for minimum weight with a displacement constraint; N is the total number of elements; and ρ is the artificial relative density vector. To avoid a singularity of the stiffness matrix, the minimum relative density ρ min is usually taken as 0.001; K, U, and F denote the structural global stiffness matrix, total displacement vector, and total force vector, respectively; u e is the displacement vector of the e-th element; K e = K(ρ e ) = E(ρ e )k 0 e is the elemental stiffness matrix of the eth element; k 0 e is the elemental stiffness matrix for the unit Young s modulus; V(ρ) and V are the total volumes of the designed structure and the ground structure, respectively; f is the preset volume ratio; we 0 is the element weight when the density of the e-th element equals 1; W(ρ) and u(ρ) are the structural total weight and the dis-

3 Different effects of economic and structural performance 779 placement at a point of interest, respectively; and ū is the allowable displacement value. To compare the two MCVC and MWDC models, the strategy of this paper is to use existing and effective codes or software to solve the models. For the solutions of the MCVC model, the 99-line topology optimization code written in MATLAB by Sigmund [32] is used. The code is published online ( and can be downloaded for free. However, since no code exists that can be used to solve the MWDC model, its specialized derivation and code will be described in the third section of this paper. Note that Rozvany [2] gave a good evaluation of the 99- line code, saying it played an important role in SIMP s general acceptance. In our opinion, an important role has two meanings: it is not simply extracurricular material to be used by students to master structural topology optimization; it is also an introductory tool for engineers to understand and use as a method of structural topology optimization. The 99-line code is short and includes finite-element analysis and optimizer subroutines, where the user can change the definition of structural sizes, set different loading and boundary conditions, and solve problems. Beginners can obtain great inspiration and help from it. Before comparing the MCVC and MWDC models through detailed numerical examples, it is necessary to assess the pros and cons of the two models. The pros of the MCVC model are that its objective function and constraints are both analytical functions of the artificial density variables, and the sensitivity calculations are free. Its cons are as follows: (1) appropriate economic index bounds are difficult to preset because no one knows whether or not the material used is sufficient; (2) though different optimal results provide engineers with more choices, the final results might be unfeasible, and it is very costly to run optimization calculations with somewhat arbitrarily specified material volumes; (3) an optimization problem involving the objective function of a performance index in multiple loading cases and multiple performances becomes a multiple-objective problem; and (4) the economic indexes are usually taken as the objective functions in sizing optimization and shape optimization, but the structural performance indexes are taken as the objective functions in topology optimization, which illustrates the lack of uniformity in the different levels of optimization. The pros and cons of the MWDC model are the exact opposite of those of the MCVC model. It has three pros: (1) the appropriate bounds of structural performance indexes are easily provided by engineers; (2) when an economic index is taken as an objective function, the optimization problem is always accompanied by a single objective function; (3) different levels of structural optimization are unified in the stage of model construction. The model s con is that the approximate explicit functions of structural performance index constraints are required for sensitivity analyses. In structural topology optimization, even though many researches are based on the model construction of minimum structural weight with a displacement constraint or multiple displacement constraints, we have seen no research providing a detailed comparison of the MWDC and MCVC models. Therefore, detailed discussions will be presented in the following sections. 2 Results of MCVC model and its inspiration While trying to solve the MCVC model with the 99-line topology optimization code, a number of examples were computed. We found that the preset volume ratio had a decisive influence on the results of structural topology optimization: different volume ratios would produce different topological configurations. This prompted us to think more deeply about the relationship between the preset volume ratios and optimal topological configurations. This study will be limited to changes in structural topological configurations from the MCVC model and focus on structures that can bear external forces rather than force inverters or compliant mechanisms. Therefore, four simple structures are used as examples in this paper (Fig. 1). Example 1 (Fig. 1a) is half of an MBB beam with a ground structure of 60 mm 20 mm 1 mm, a unit force of F = 1, a unit Young s modulus of E 0 = 1.0, and Poisson s ratio v = 0.3. Example 2 (Fig. 1b) is a short cantilever beam with a ground structure of 32 mm 20 mm 1 mm, a unit force of F = 1, a unit Young s modulus of E 0 = 1.0, and Poisson s ratio v = 0.3. (a) (b) (c) (d) Fig. 1 Ground structures and boundary conditions for four test examples

4 780 G.L.Yi,Y.K.Sui Example 3 (Fig. 1c) is a short cantilever beam with a fixed hole and a ground structure of 45 mm 30 mm 1mm, a unit force of F = 1, a unit Young s modulus of E 0 = 1.0, and Poisson s ratio v = 0.3. The center of the hole is located at the intersection of one-third the horizontal length and onehalf the vertical length from left to right, while the radius is equal to one-third the vertical length. Example 4 (Fig. 1d) is a cantilever beam with a ground structure of 80 mm 50 mm 1 mm, a loading force of F = 9 kn, a Young s modulus of E 0 = MPa, and Poisson s ratio v = 0.3. The first three examples are from the paper [32]. There are nine subexamples for each example, where the preset volume ratios are nine equally distributed numbers between 0.1 and 0.9. A total of 36 subexamples are computed using the 99- line topology optimization code, and the results are shown in Table 1. Meanwhile, the optimal results of four examples with a preset volume ratio of 0.1 are examined in more detail and shown in Table 2, where in the Number of elements column, Black represents the number of black elements with a relative density of 1.0, White represents the number of white elements with a relative density of 0.001, and Gray represents the number of gray elements with various relative densities on the open interval (0.001, 1.0). From the results of the 36 subexamples in Table 1, we observe the dependency of the structural optimal topological configurations on the preset volume ratios and summarize the following three aspects: (1) Three or four different kinds of typical structural configurations are obtained in all four examples. (2) When the preset volume ratio is too small, the structure will be broken off or disconnected and degenerate into a mechanism. According to the data shown in Table 2, when the preset volume ratio is 0.1, the number of black elements in the optimal structural topological configurations is very small or even zero, and the average densities of the gray elements are 0.20, 0.25, 0.29, and 0.31, respectively. It indicates that a very low specified volume ratio means that there is insufficient material to construct a clear structural configuration. This shows that the structures have already degenerated into mechanisms. (3) Along with the reduction in preset volume ratios, all four examples go through an evolution from cumbersome structures (heavyweight), into Michell-truss-like structures, into light structures (lightweight), and, finally, into degenerating mechanisms. Since optimal topological configurations are really dependent on the preset volume ratios, there is a question as to how to select a reasonable preset volume ratio at the stage of conceptual design. Can the most appropriate volume ratio be found logically? In other words, rather than using a presumed volume ratio, why not determine an optimal volume ratio while searching for an optimal topological configuration? Many of the aforementioned examples have inspired us to seek the optimal volume ratio while searching for an optimal topological configuration. This means that there will be a new volume ratio objective function in addition to the compliance objective function if we use the MCVC model to solve structural topology optimization problems. If there are a compliance objective function and a volume ratio objective function at the same time, the optimization problem would not be one with a single objective function and the optimum solutions would not be determined. Could the volume ratio be considered only as a single objective function? Since compliance is not taken as the objective function, it should be transformed into a constraint condition. However, how should the compliance constraint value be selected? Thus, there is a difficulty similar to that in the MCVC model, in which a constraint value of the most appropriate volume ratio is not determinate. How can the difficulties of selecting an appropriate compliance constraint value be overcome? In fact, if we jump out of the box of focusing on compliance, there is a solution. Actually, rather than the compliance constraint, the strength or stiffness constraint is considered for practical engineering problems. That is to say, we can use the strength or stiffness constraint instead of the compliance constraint. This is exactly the MWDC model: a structural topology optimization model of minimum weight under the constraint of the displacement at the point of interest. Why is the volume ratio objective function changed into a structural weight objective function? Because minimizing the structural weight is equivalent to minimizing the structural volume with the same material, while the minimum structural volume is equivalent to the minimum structural volume ratio for the same ground structure. Thus minimizing the structural weight should be equivalent to minimizing the volume ratio. The advantage of using the structural weight objective function is that the formulation of topology optimization is in line with formulations of section optimization and shape optimization. 3 Derivation of solutions for MWDC model and program implementation 3.1 Derivation for explicit displacement function at point of interest Using the unit virtual load method [33], the general displacement at a point of interest can be expressed by calculating the virtual work as follows:

5 Different effects of economic and structural performance 781 Table 1 Optimum topological configurations for different preset volume ratios Volume ratio Example 1 Example 2 Example 3 Example 4 Iterations Iterations Iterations Iterations Iterations Iterations Iterations Iterations Iterations

6 782 G.L.Yi,Y.K.Sui Table 2 Results of four examples with preset volume ratio of 0.1 Examples Number of structure analysis Compliance Number of elements Total volume of gray elements Average density of gray elements Black White Gray Example Example Example Example u = A e = e (σ V ) T (ε R )dv, (4) Substituting Eq. (8) into Eq. (7), and substituting Eq. (7) into Eq. (4), we obtain where the superscripts V and R denote, respectively, the virtual and real loads. A e = (σ V ) T (ε R )dv is the elemental e virtual work and the contribution of the eth element to the general displacement at the point of interest. σ V and ε R denote the element stress vectors associated to the virtual load and the element strain vectors associated to the real load, respectively. According to the work energy theorem, the virtual work in Eq. (4) is equal to the virtual work obtained from nodal forces and nodal displacements. Therefore, the displacement contribution could be expressed as follows: A e = (σ V ) T (ε R )dv = (F R e )T (u V e ), (5) e where F R e and uv e are, respectively, the nodal force vector associated to the real load and the corresponding nodal displacement vector associated to the virtual load. Because of the element stiffness equation, we have K e u V e = FV e, (6) where F V e is the nodal force vectors associated to the virtual load. Substituting Eq. (6) into Eq. (5), we obtain A e = (σ V ) T (ε R )dv = (F R e )T (K e ) 1 (F V e ). (7) e Since the element stiffness is proportional to the Young s modulus, the element stiffness can be written as K e = k 0 e E e = k 0 e E 0ρ p e = K 0 e ρ p e, (8) where K 0 e is the elemental stiffness matrix when ρ e = 1 with the Young s modulus E 0. u = A e = (F R e )T (K 0 e ) 1 F V e ρ p e D 0 e ρ p e, (9) where De 0 (FR e )T (K 0 e ) 1 F V e, and it is the constant coefficient of the displacement contribution component of the eth element to the point of interest. In Eq. (9), the expression De 0 (FR e )T (K 0 e ) 1 F V e is given approximately by using the hypothesis of static determination, which assumes that the deformation of a statically determined structure barely changes compared to its dimensions. It is also proved that the hypothesis of static determination has no effect on the optimal solutions for an optimization problem with displacement constraints [34]. De 0 is constant and independent of FR e and F V e because the displacements of the statically determinate structure have no relationship with the nodal force vectors and the structure is regarded as a statically determinate one in each iteration. 3.2 Solutions of MWDC model Substituting Eq. (9) into Eq. (3), an explicit topology optimization model is obtained as follows: For ρ : min W (ρ) = N ρ e we 0, s.t. D 0 e ρ p e =ū, 0 <ρ min ρ e 1 (e = 1, 2,...,N). According to the Lagrangian multiplier method, we obtain L(ρ,λ)= ( N ρ e we 0 + λ D 0 e ρ p e (10) ) ū. (11)

7 Different effects of economic and structural performance 783 The first-order derivatives of the Lagrangian function are determined to be L = we 0 ρ λp D0 e e ρe = 0, (12) L N λ = De 0 ū = 0. (13) ρ p e The design variables can be obtained from Eq. (12) as follows: ( λpd 0 ρ e = e w 0 e ) 1. (14) Substituting Eq. (14) into Eq. (13), we obtain (D (λp) 1 e 0 ) 1 (we 0) p = u 1 p. (15) Then the Lagrangian multiplier λ is eliminated by substituting Eq. (15) into Eq. (14), and the solutions are obtained as follows: (Dk 0 ) 1 (wk 0) p k=1 ρ e = u 1 p ( D 0 ) 1 e we 0. (16) Considering the contribution of element densities to the displacement at the point of interest, the partial derivative of Eq. (9) with respect to the densities are obtained as follows: u = pd0 e. (17) ρ e ρe When De 0 u > 0, we have ρ e > 0. That is to say, if the element density ρ e is increased, then the displacement u at the point of interest will increase; if the element density ρ e is decreased, then the displacement at the point of interest will decrease. Thus increasing the element density ρ e is not economical in terms of design. In this case, the element density values must be decreased to attain the minimum value; therefore, the displacement contribution and stiffness contribution of the corresponding element can attain their minimum and maximum values, respectively. We need not design the density of this particular element in this case, so it is called a passive design variable. When De 0 u 0, we have ρ e < 0. That is to say, if the element density ρ e is increased, the displacement u at the point of interest will decrease; if the element density ρ e is decreased, the displacement at the point of interest will increase. In other words, if the element density ρ e is increased, then the displacement and stiffness contributions of the corresponding element will decrease and increase, respectively. This is a reasonable return on an investment in increasing the element density ρ e. In this case, the density should be designed, and it is called an active design variable. The preceding discussion is summarized as follows. The elements satisfied by De 0 0 are designable elements, and the corresponding density variables are active variables whose element numbers are included in the set of I a = { e D 0 e 0, e = 1, 2,...,m } ; the elements satisfied by De 0 < 0 are not designable, and their corresponding element numbers are in the set of passive design variables. The values of the density variables in the passive design variable set are to remain constant during each iteration while optimum solutions are sought. Therefore, the topology optimization model in Eq. (10) can be rewritten as follows: For ρ : minw (ρ) = m ρ e we 0 + W 0, s.t. m D 0 e ρ p e + u 0 =ū, 0 <ρ min ρ e 1(e = 1, 2,...,N), (18) where W 0 = ρ e we 0 and u 0 = D0 e ρ p. W 0 and u 0 are e e / I a e / I a constants. Thus the constant terms, which have nothing to do with seeking the optimum, can be eliminated from the objective function. Like the previous derivation, we can obtain the following equation to replace Eq. (16): ˆρ e = m k=1 ( ) 1 D 0 ( k wk 0 ) p ū u 0 ρ min or 1 (e / I a ). 1 p ( D 0 ) 1 e we 0 (e I a ), (19) The active design variables calculated from the preceding equation are not guaranteed to have a value smaller than 1 or larger than the minimum density ρ min. Then the following adjustment should be made: 1, where e / I a (if ˆρ e 1), ρe = ˆρ e, where e I a (if ρ min < ˆρ e < 1), ρ min, where e / I a (if ˆρ e ρ min ). (20) Now that the active/passive design variable sets have been updated, the calculation according to Eq. (19) must be carried out again; this is called an inner optimization iteration. The classification of the active/passive design variable sets

8 784 G.L.Yi,Y.K.Sui by the sign of De 0 is a global prejudgment following a structural analysis prior to constructing the topology optimization model for iterative solutions, while Eq. (20) is a local adjustment to the reclassification of the active/passive design variable sets following an inner optimization iteration. This kind of inner optimization iteration should be computed continuously according to Eq. (19) until the active/passive design variable sets are no longer changed and the process of inner optimization iterations stops. Thus the optimal solutions of the model Eq. (18) for the corresponding structural analysis are obtained. Here we say that an outer optimization iteration is finished. Before the next outer optimization iteration, a new structural finite-element analysis should be conducted. However, the whole optimization process needs sequential reconstructions of optimization models corresponding to sequential outer optimization iterations. To ensure mesh independency, the elemental displacement contribution component filtering scheme is used in this paper. It has a formulation that is similar to the sensitivity filtering scheme used in the solution of the MCVC model. But the elemental displacement contribution component, A e, is filtered here rather than the sensitivity of the constraint function, which refers to the coefficient of the displacement constraint function. The former filtering scheme seems to improve the numerical stability more than the latter one. Then the coefficients of the displacement constraint function are updated according to its relationship with the elemental displacement contribution component in Eq. (9). The elemental displacement contribution component filtering and the new coefficients of the displacement constraint function are shown, respectively, in Eqs. (21) and (22): Ã e = w(x j )ρ j A j j N e, (21) w(x j ) j N e D 0 e = Ã e ρ p e, (22) where x j is the spatial location of the geometric center for the j-th element; N e = { j } x j x e r min denotes the neighborhood of the e-th element within a given filter radius r min of the center of the eth element; w(x j ) = max ( 0, r min ) x j x e is a weight function and a linearly decaying weighting function, which is linearly reduced along the distances of the elements neighboring the center element. To retain the comparability of the MCVC and MWDC models, the same convergence criterion as shown in what follows is used in both models. The optimal solutions ρ (k+1) for the (k + 1)-th iteration and ρ (k) for the k-th iteration should satisfy a relationship as follows: [ ( )] max max ρ (k+1) ρ (k) (23) No Initialize design variables in each element Finite element analysis Calculate coefficients of displacement function Active and passive variable sets Obtain design variables by Eq. (19) Update active and passive variables Do active and passive variable sets change? No Update design variables and results visualization Convergence? Yes Stop Yes Fig. 2 Optimization flow chart for 120-line topology optimization code The preceding solution derivation process for the MWDC model in Eq. (3) is implemented on the MATLAB platform and programmed into a 120-line topology optimization code, which can be downloaded at yiguilian/link. The optimization flow chart of this code is shown in Fig Comparison of MCVC and MWDC model results To compare the MCVC model solved by the 99-line code and the MWDC model solved by the 120-line code, four examples in Fig. 1 are solved by the two codes. The loading points are taken as the points of interest. Since a small-volume structure has little material and small stiffness, it causes a large displacement at the point of interest. In contrast, a large-volume structure has a large stiffness and a small displacement at the point of interest.

9 Different effects of economic and structural performance 785 Table 3 Results for four examples Examples Number of structural analysis Total weight Displacement at point of interest Number of elements Black White Gray Weight of gray elements Ratio of gray elements (%) M nd (%) Example 1 MCVC MWDC Example 2 MCVC MWDC Example 3 MCVC MWDC Example 4 MCVC MWDC Therefore, the whole calculation for each example has two stages. In the first stage, the MCVC model in Eq. (2) under a preset volume ratio constraint is solved using the 99-line topology optimization code, and the optimum topological configuration and the displacement value at the loading point are obtained. This displacement value is taken as the allowable displacement value in the MWDC model, which moves on to the second stage. The MWDC model in Eq. (3) under a particular allowable displacement value is solved using the 120-line topology optimization code. During the calculation, there is a measure of discreteness [29] that is used to determine whether an optimum solution converges to a discrete one. This measure of discreteness is defined as M nd = 4ρe (1 ρ e ) N 100 %, (24) where ρe is the optimum relative density of the e-th element. Once the design has converged, M nd with a value of 0 % denotes that all element-relative densities are equal to 0 or 1, and M nd with a value of 100 % denotes that all elementrelative densities are equal to 0.5. Therefore, a smaller value of M nd means there are fewer gray elements and is expected to give a better design. All the data and optimal topological configurations are shown in Table 3 and Fig. 3, respectively. The meanings of Black, White, and Gray in the Number of elements column of Table 3 are the same as those in Table 2. Total weight denotes the structural weight without the white elements with a relative density of 0.001, including the weight of all black and gray elements. Weight of gray elements denotes the total weight of all gray elements, and Ratio of grey elements is the percentage ratio of the number of gray elements to the number of all elements. In Fig. 3, the five parameters in parentheses in each of the eight subcaptions for the configurations all have their own meaning on the left-hand side and right-hand side, respectively. For example, on the left-hand side in Fig. 3a1 there is a caption MCVC (60,20,0.5,3.0,1.5), which represents the concrete computing data of an example using the MCVC model, with 60-element grids in the x-direction, 20-element grids in the y-direction, a volume ratio constraint value of 0.5, a penalty factor of 3.0, and a filter radius of 1.5. For example on the right side, in Fig. 3a2, there is a caption MWDC (60,20,203.30,3.0,1.5), which represents concrete computing data of an example by using the MWDC model, where 60 elements grids in x-direction, 20 elements grids in y-direction, displacement constraint value , penalty factor 3.0 and filter radius1.5. The computational results show that the MWDC model can basically meet the displacement constraint bounds at the loading points provided by the MCVC model. The optimal configurations from the two models are almost the same because they bear the same external forces and produce the same displacements. However, the MWDC model obtains smaller discreteness values than the MCVC model, which indicates that constructing structural topology optimization models using an economic index as the objective and performance indexes as the constraints does in fact make sense. The computational speeds of two codes for one optimization iteration are about the same. The optimization code for the MWDC model converges faster in Examples 2 and 3 and slower in Examples 1 and 4 than that for MCVC model.

10 786 G.L.Yi,Y.K.Sui Optimal configurations of the MCVC model Optimal configurations of the MWDC model a1) a2) b1) b2) c1) c2) d1) d2) Fig. 3 Optimum topological configurations for four examples. a1 MCVC (60,20,0.5,3.0,1.5), a2 MWDC (60,20,203.30,3.0,1.5), b1 MCVC (32,20,0.4,3.0,1.2), b2 MWDC (32,20,57.35,3.0,1.2), c1 MCVC (45,30,0.5,3.0,1.5), c2 MWDC (45,30,52.10,3.0,1.5), d1 MCVC (80,50,0.5,3.0,1.5), d2 MWDC (80,50,0.35,3.0,1.5) 5 Conclusion The emphasis in this paper on classifying objectives and constraints into economic and performance indexes essentially highlights the conflict between safety and economy in engineering problems, which is reflected by the relationship between the economic and performance indexes. The presentation of the MCVC and MWDC models describes in a deep way two approaches to dealing with the conflict between safety and economy. In these two models, the different locations of economic and performance indexes in the process of selecting objectives and constraints determine their different roles and effects on the constructions of structural topology optimization models. Using numerical example calculations and comparing the two models, it is found that optimal topological configurations depend on the volume ratios that are chosen for the constraint. However, using the same external force and displacement performance requirements, the MWDC model obtains the same results as the MCVC model. But from the perspective of model construction, the MWDC model overcomes the difficulties and shortcomings of the MCVC model. It realizes and satisfies the demands of the economic and performance indexes at the same time and completely fulfills the unvarying requirements for safety and economy in mechanical engineering. The solutions for the two models show two ways of searching for the equilibrium point in the tension between safety and economy. Numerical results indicate that the equilibrium point obtained from solving the MWDC model is more reasonable than that obtained from solving the MCVC model because constructing a structural topology optimization model for optimizing an economic index under the limitations of performance indexes meets the requirements of practical engineering problems much more satisfactorily. Meanwhile, the discussion on the different roles of the economic and performance indexes in model construction resolves the following three issues: (1) Based on engineers experiences, the issue of apriorism in presetting the volume ratio constraint in the MCVC model is overcome. Since empirical evidence is a basic requirement of science, the amount of material to be used in the MCVC model is hardly preset, nor can the amount of material that is needed in the structural design to achieve the best performance be determined

11 Different effects of economic and structural performance 787 in advance. However, in contrast, the values of the structural performance can be preset because the safety requirements represented by the performance indexes is the result of long-term experience in the empirical sciences and technologies are are, in addition, in the form of standard structural design specifications. A variety of requirements for performance indexes are based on design specifications. To satisfy mechanical performance indexes, the appropriate cost will naturally be determined by pursuing the minimum economic index. (2) The model construction of a minimum economic index with constraints on performance indexes is indicated to be more in line with actual engineering practices because such a minimum is sufficient for guaranteeing the structural safety represented by mechanical performance indexes, whereas maximizing mechanical performance has no significance. The mechanical performance indexes, including, for example, allowable displacements and allowable stresses, are the main points that should be satisfied directly in design, and they are being refined to perfection on a daily basis as a result of the accumulated experience of engineers. But academic performance indexes, such as compliance, are sometimes what researchers are concerned with. If an academic performance is referenced in a design process, it is only an indirect consideration and not of direct consequence. (3) Multiple-objective problems are avoided. Since mechanical performance indexes relate to loading cases, if a mechanical performance index is taken as the objective function in a multiloading case, the optimization problem will become a multiobjective optimization problem, which would be difficult to solve. 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