A Topology Optimization Method for Three-dimensional ... - CiteSeerX

6th World Congresses of Structural and Multidisciplinary Optimization Rio de Janeiro, 30 May - 03 June 2005, Brazil

A Topology Optimization Method for Three-dimensional Continuum Structures Jianhua Rong 1, 2 , Guojing Tang 2 , Qing Quan Liang3, Zhenxing Yang1 (1) School of

Automobile and Mechatronic Engineering, Changsha University of Science & Technology, Hunan Province,

410076,P.R. China (2) School of Airspace and Material Engineering, National University of Defense Technology, Hunan Province, 410073, P. R. China (3) Australian Postdoctoral Fellow, School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, Australia.

Abstract: In the optimization process of a complex three-dimensional continuum structure by using the conventional evolutionary structural optimization method, some isolated groups of elements or a few elements being of rigid movements, often appear in the optimized structure, which becomes singular so that the optimization process can’t continue. In order to overcome this problem, this paper proposes an improved evolutionary structural optimization method for topology design of three-dimensional continuum structures with a stress uniform requirement. In the proposed method, a few of finite elements with artificial material property are added around the cavities and boundaries of the current structure to approximately transform the singular topology optimization model into a non-singular one, an idea is proposed that all datum information in the optimization process of a structure are grouped as a set of structural optimization datum flow and a set of structural characteristics computation datum flow, and a simple transferring procedure between them is given. Because there not is any ill phenomenon in the global stiffness matrix of the new transferred structural model, an iteration solving method is adopted in structural nodal displacement computations. Element addition and deletion are based on Ishai stress criterion. Continuum structures constructed with isotropic material and tension/compression dominant material can be optimized by the proposed method. Two examples of typical and complex three-dimensional continuum structures are given. It is demonstrated that the proposed method is very effective in generating optimal topology designs of complex three-dimensional continuum structures that are modeled using more than several ten thousands of brick elements. Keywords: Engineering mechanics, structural optimization, structural topology optimization, three-dimension structural optimization, Ishai criterion; 1. Introduction Topology optimization is of considerable practical interest due to the fact that it can achieve much more material savings and higher performance designs than the pure sizing optimization. Extensive research efforts have been made in structural topology optimization and many optimization methods have been developed during the past decade. As the state-of-the-art, homogenization-based methods have become the main approach to structural optimization [1-3], in which a material model with micro-scale voids is introduced and the topology optimization problem is defined by seeking the optimal porosity of such a porous medium using one of the optimality criteria. By transforming the difficult topology design problem into a relatively easier “sizing” problem, the homogenization technique is capable of producing internal holes without prior knowledge of their existence. That is, it offers a tool for simultaneous shape and topology optimization. A number of variations of the homogenization method have been investigated to deal with these issues by penalizing intermediate densities, the “solid isotropic material with penalization” (SIMP) approach, in particular, for its conceptual and practical simplicity [2,4]. Material properties are assumed constant within each element used to discretize the design domain and the design variables are the element densities. The material properties are modeled to be proportional to the relative material density raised to some power. The power-law-based approach to topology optimization has been widely applied to the problems with multiple constraints, multiple physics, and multiple materials [3, 4]. However, the homogenization method may not yield the intended results for some objectives in the mathematical modeling of structural design. It often produces designs with infinitesimal pores in the materials that make the structure not feasible. Further,

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numerical instabilities may introduce “non-physical” artifacts in the results and make the designs sensitive to variations in the loading [3,4,5]. Numerical instability and computational complexity remain major difficulties and are encountered in every realistic application. In the end, the designer must interpret the resulting material distribution and extract a boundary and topological description, which is essential for obvious reasons [6]. These fundamental issues are still argued in the literature [4,5]. Another approach is called the “bubble method”, proposed by Eschenauer and Schumacher [7,8]. In the method, so-called characteristic functions of the stresses, strains and displacements are employed to determine the placement or insertion of holes of known shape at optimal positions in the structure, thus modifying the structural topology in a prescribed fashion. In this case, the design for a given topology is settled before its further changes. A new approach, so-called the level set method, has been proposed (Michael Yu Wang etc.)[9,10]. As a boundary-based optimization problem, the structure is implicitly represented with a level-set model that is embedded in a scalar function. The dynamics of the level-set function is governed by a simple Hamilton-Jacobi convection equation. The movement of the moving boundaries of the structure is driven by a transformation of the objective and the constraints into a speed function that defines the level-set propagation. In this method, the structural boundaries are viewed as moving during the optimization process – interior boundaries (or holes) may merge with each other or with the exterior boundary and new holes may be created out as a moving front of the level sets driven by the dynamics of the interior region under optimization conditions. The results obtained by this approach, have demonstrated outstanding flexibility in handling topological changes, the fidelity of boundary representation, and the degree of automation, comparing favorably with other methods in the literature based on explicit boundary variation or homogenization. However, the mentioned-above speed function in this level-set approach must be constructed by a series of derivations dealing with the Fre ′chet derivatives of the objective and the constraints [9,10]. If we extend it to more complex cases including stress, dynamic response and etc. requirements, the Fre ′chet derivative formula of stresses and dynamic responses must be derived. Because stress and dynamic response quantities all belong to structural local quantities, it is very difficult to build the relation of zero level-set variations with these derivative formula in a simple way by existing methods. Thus, it is very complex to derive the Fre ′chet derivative formula of stresses and dynamic responses so that it is very difficult to build the required speed function. Another approach, so-called Evolutionary Structural Optimization (ESO), has been developed based on the idea that by systematically removing the unwanted material, the residual shape of the structure evolves towards an optimum [11,12]. It has the advantages of clear concept and easy mathematical operation compared to conventional analytical and numerical optimization methods. Extensive research has been done on the ESO method for various types of structures and the optimality constraints can be stresses based, stiffness / displacements based, frequencies or bulking loads based [11,12]. Although it cannot be assured that the optimum structural topology may be obtained by the use of the ESO method, this approach can improve design by making more efficient use of the material. Its wide coverage demonstrates that it is capable of solving all kinds of structural optimization problems in practical engineering. Recently, a so-called bi-directional structural optimization (BESO) algorithm presented by Querin et al. [13] allows not only the removal of lowly stressed elements, but also the addition of elements to the highly stressed portions. It implies that, if the only removing element’s ESO algorithm is used and subsequent high stress elements appear, then the adding element’s algorithm acts to alleviate them. The bi-directional capability thereby allows a more thorough search of all the design space and offers a greater ability to find the optimum. Comparing favorably with other methods, such as the SIMP approach, this BESO method requires more iteration steps. But, when a lot of elements are removed, because the finite element analysis is only based on remained elements in the structure by the BESO method, its dimension becomes smaller, and lesser computation time is needed in each iteration step for the BESO approach. Moreover, due to its conceptual and practical simplicity, this approach still is of practical engineering application value. However, only simple three-dimensional continuum structures and space frame structures are optimized by the ESO and BESO methods [14]. The main cause is that the ESO and BESO methods often produce some singular structural topologies that consist of some isolated groups of elements and a few elements being of rigid movements, so that the optimization process cannot be continued. Currently, there are no effective schemes of preventing singular problems from occurring in complex three-dimensional continuum structures that are modeled using more than several thousands of brick elements. To overcome the singular problem in topology optimization, an effective ESO method based on von Mises and Ishai stress criteria for

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three-dimensional continuum structures is proposed in this paper. The proposed ESO method transforms the singular topology optimization models into non-singular ones by employing finite elements with artificial material property, which are added around the cavities and boundaries of the structure. The Ishai stress criterion is incorporated in the bi-directional optimization algorithms to optimize continuum structures with tension and compression material properties. Because there not is any ill phenomenon in the global stiffness matrix of the new transferred structural model, an iteration solving method (Being only used for the non-ill situation of the global structural stiffness matrix) is adopted in structural nodal displacement computations. The modified ESO method can be used to solve real-world complex optimization problems with more than several ten thousands of brick elements with good computation efficiency. The optimization algorithms developed are tested on two examples.

2. Topology Optimization Model In optimal structural design, a general goal is to minimize the structural weight while satisfying the requirements of structural working characteristics. Here, the topology optimization problem of a continuum three-dimensional structure with a uniform structural stress requirement is considered.

(a)

(b)

Figure 1. Element categories in the fixed finite mesh domain: (a) a structure including only removed elements (with a property value of zero, the white area) and retained elements (the black and red area); (b) the transforming structure with retained elements (the black and red area), removed elements (the white area) and elements with artificial material property (the red-white area). In this paper, before each optimization iteration step, the current structure as depicted in Fig. 1(a) is transformed into a new structure by adding elements with artificial material property as shown in Fig. 1(b). All elements, which include those with property values of zero and non-zero in the fixed mesh domain, can be categorized into three kinds of elements: (a) temporarily retained elements; (b) temporarily removed elements; and (c) elements with artificial material property. Elements with artificial material property are added to the place of some removed elements to the singular problem of the structure and are considered in the structural analysis and optimization process. Their elasticity modulus is set to E m = 10 −4 × E × r 2 , where E = the elasticity modulus of the real material;

w

2

and rw = the ratio of the weight of the retained elements in the current structure to that of the initial structure with a fixed finite element mesh fully filled with material. Elements with artificial material property are denoted as S 3 and the retained elements are denoted as

S1 . The set of other temporary removed elements is denoted as S 2 . In the proposed method, an initial design domain that is big enough to cover the optimal topology is considered. Similar to the conventional ESO method, the relations between the effective data, which consist of the retained elements and elements with artificial material property, and the data in the initial design domain are established by a mapping transference. All structural characteristics can then be computed. These results are again fed back to the datum space of the initial design domain, and can be used, shown and recorded in the optimization process. Obviously, only few elements with artificial material property are needed to reduce or escape the ill phenomenon of the global stiffness matrix for the current structure. A lot of simulation examples show that there not is any ill phenomenon in the global stiffness matrix of the new transferred structural model, thus an iteration solving method can be adopted in structural nodal displacement computations. And the impact of elements with artificial material property on the results of structural characteristics is minor and can be ignored. Further, the information platform of an operation dealing with adding elements can be set by the use of the characteristic data (such as stresses). As a result, the optimization problem approximately is equivalent to the original optimization problem.

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3. The basic principle for adding artificial material elements and their mechanical function description In the maximum design domain mesh model for the current structural topology (as shown in Fig.1 (a)), before each optimization iteration step, the material property value of temporarily remained elements is set to 1, and the material property value of temporarily removed elements is set to zero. A general principle for adding artificial material elements, is: if an element with a material property value of zero, is of three nodes and more than three nodes (for a three-dimensional structure) or two nodes and more than two nodes (for a plane structure) belonging to the set of all the remained nodes in the current structure, the material property value of this element is changed into 2, namely, this element being set as an artificial material element. Thus, for a two-dimensional or three-dimensional structure, here only a layer of elements, being non-existing elements and surrounding the boundaries and cavities of the current structure, which only consists of remained elements, is set to artificial material elements. In the process of optimization iterations, the material property parameters for the material property value of 2 vary in a way described in chapter 2. In order to assure that the current structural topology (Fig. 1(a)) approximately is equivalent to the new structural topology (Fig. 1(b)), the main problem is to make sure that the impact of elements with artificial material property on the results of structural characteristics is minor and can be ignored. Similar to the SIMP method, artificial material elements are treated as new material elements and are considered in the structural analysis and optimization process. By way of a lot of simulation computations for a structure being of a three thousand elements, it is shown that if the Young’s modulus of all artificial material elements is selected as

E m = 10 −4 × E × rw2 , and other material property parameters are the same as corresponding parameters of the real material elements removed, the relative errors of the displacements and stresses at all retained nodes in the new structural topology, to those at corresponding nodes in the current structure, can be controlled within 0.3% , and there not is any ill phenomenon of the global structural stiffness matrix and structural singular problem for the new structural topology. If the current topology structure is singular (Namely, a few isolated groups of elements appear in the optimized structure), the stress distribution of the structural topology (being a non- singular structure) obtained at the prior iteration step shows that the stresses at the places corresponding to those isolated elements and the elements surrounding them are very low, except removing those elements surrounding the isolated elements, removing the isolated groups of elements is also reasonable. For a complex three-dimensional structural model with more than several thousand elements, it is very difficult to directly judge whether the current structure is singular by using the geometrical data in the structural finite element model, and its calculation quantity also is very big. By using the information of the retained elements and elements with material property value of zero for the current structure, and by the principle of adding artificial material elements, it is easy to construct artificial material elements for the current structure. Generally speaking, the isolated groups of elements and the principal structural part can be connected into a non-singular structure by adopting those artificial material elements. Because those isolated groups of elements and artificial material elements do not locate at principal load-supported roads, for the new topology structure, those isolated groups of elements still are of very low stress level, and can be removed by the removal operation of the next optimization iteration. Although the filtering function of the SIMP method can make the ratio of the elastic modulus dropping into the vicinity of 0 or 1 for each element, and can solve the structural topology singular problem, because the matrix (such as the global structural stiffness matrix) size for the structural analysis is invariable, its characteristic computation quantity remains invariable at each iteration step, in particular, when a lot of elements is removed (as shown in Fig.1(a), those removed elements exist in the structure with a very small elastic modulus, and are considered in the structural analysis and optimization process. ), ill phenomena easily occur in the global structural stiffness matrix, thus their numerical instability and computational complexity remain major difficulties and are encountered in every realistic application. In the proposed method, first, a fixed mesh finite element model for a design domain, which is big enough to include the optimum topology, is set up, and its node and element information is storied. Second, in order to carry out three-dimensional structural topology optimization, two sets of datum information flows, i.e. an information flow for the structural optimization and an information flow for the structural analysis, are set up. Structural optimization information data are based on the fixed maximum design domain mesh finite

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element model. And data for structural characteristic computations are based on a structural effective datum mesh model, which only consists of remained and artificial material elements, and varies with optimization iteration evolution. Namely, before each optimization iterative step, based on the fixed maximum design domain mesh finite element model, all nodes, not belonging to the set of nodes of all effective elements in the optimized structure, are deleted, and the nodes of all effective elements are renumbered one by one according to their numbering orders in the fixed maximum design domain mesh finite element model. All effective element numbering is treated in a similar way. Thus, it is easy to form an effective datum mesh model for the structural analysis, and its computation quantity is small. If the bandwidth minimum requirement of the global structural stiffness matrix is considered, this measure approximately is equivalent to the automatic re-mesh of the new structural topology (In this case, the mesh size of the structure is of a same mesh size of the fixed maximum design domain mesh finite element model.), and makes it more easy to construct a mapping relation between the structural optimization data (such as required stresses etc.) and the structural characteristic data. Comparing favorably with the SIMP approach, the proposed method requires more iteration steps, but, when a lot of elements are removed, because the finite element analysis is only based on effective elements in the new structure, there not occurs any ill phenomenon in the global structural stiffness matrix, an iteration solving method can be adopted in structural nodal displacement computations, its dimension becomes smaller, and lesser computation time is needed in each iteration step. Moreover, the characteristic data of artificial material elements provide a quantitative information platform of an operation dealing with adding elements.

4. Element Deletion/Addition Based on von Mises and Ishai Stress Criteria For structures made by isotropic material, the von Mises stress is one of the criteria often used in engineering applications. The von Mises stress for three-dimensional problems is defined by

σ vm =

1 2

2 2 2 [(σ xx − σ yy ) 2 + (σ yy − σ zz ) 2 + (σ zz − σ xx ) 2 + 6 (σ xy + σ yz + σ zx )]

(1)

For a structure with different tension and compression material property, different material strength failure criteria should be used. It is hoped that a smooth material strength criterion can be used not only for isotropic materials, but also for different tension and compression property materials. The Ishai stress criterion is incorporated in the proposed ESO algorithms to describe the features of different type materials. The method is particularly applicable to the optimization of concrete structures. The stress invariable J and the stress bias quantity J 2 D may be given as follows

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J 1 = σ xx + σ yy + σ zz J 2D =

(2)

1 2 2 2 [(σ xx − σ yy ) 2 + (σ yy − σ zz ) 2 + (σ zz − σ xx ) 2 + 6 (σ xy + σ yz + σ zx )] 6

The relations between the stress bias quantity ( J 2 D ), the twist tangential stress ( τ 0 ) and the von Mises stress ( σ

3 J 2 D = τ 0 2 = (σ vm ) 2

vm

(3)

) are defined by

(4)

Eq. (4) is also applicable to the stress states of isotropic materials. The introduction of the stress invariable J 1 is very important because it gives the stress sign. Summing up the above descriptions，the Ishai stress may be expressed as

σ ISH =

(S + 1)

3 J 2 D + (S − 1)J 1 2S

(5)

where S = T / C ; T = tension stress limit; and C = the compression stress limit. Ideally, the stress in every part of a structure should be near the same safe level. This concept leads to the rejection criterion based on the local stress level. It means that lowly stressed material is assumed to be under-utilized and can be removed subsequently. The element removal criteria can be expressed as

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σ e ISH ISH σ max

where σ e

ISH

< RRi

(6)

= the Ishai stress of the eth element; σ max = the maximum Ishai stress in the current structure; and RRi = the rejection ISH

ratio at the ith steady state. By gradually removing material with lower stresses that satisfy the Eq. (6) from the model, the stress levels in the new designs become more and more uniform. The cycle of the finite element analysis and element removals is repeated using the same value of RRi until a steady state is reached. At the steady state, no more elements can be deleted from the structure. At this stage an evolutionary ratio ( ER ) is introduced and added to the rejection ratio, which becomes

RRi = RRi + ER

i = 0,1,2, LL

(7)

With this increased rejection ratio, the cycle of the finite element analysis and element removal takes place again until a new steady state is reached. The evolutionary optimization process continues until a desired optimum is obtained. The evolutionary procedure requires two parameters to be prescribed. The first is the initial rejection ratio RR0 and the second is the evolutionary rate ER . Typical values of RR0 = 1% and ER = 1% have been used for many test examples. But for some problems much lower values need to be used in order to achieve a smooth solution. For any new model, after a few trials, it is not difficult to choose the suitable values for these parameters. For example, if too much material has been removed from the structure within one iteration or one steady state, it indicates that smaller values should be used for RR0 or ER . The average Ishai stress of elements with artificial material property is used as the criteria for element addition in the proposed method. After several iterations, a small number (m) of elements with artificial material property are grouped as the potential candidates for element addition if they satisfy the following condition:

γ σ e ISH ≥ RRi ISH σ max The number of elements with maximum Ishai stress among these

γ = E / Em

m

(8)

elements is determined as l = min ( Int ( IR × n), m) , where

IR is a ratio of added elements to the number ( n ) of elements removed in the current iteration step; and Int is an integral function. The values of IR = 0.3 to 0.4 have been used for some examples. The element addition criterion can be expressed by ISH ISH σ mean , m < JR × σ mean, l

(9)

where σ mean, m = the average Ishai stress of the m (m ≠ 0) elements; σ mean , l = the average Ishai stress of these l (l ≠ 0) elements； ISH

ISH

and JR = element addition ratio that representing a stress uniformity of elements around the boundaries and cavities of the structure. The values of JR = 0.8 to 0.85 have been used for some examples in this paper. If the Eq. (9) is not satisfied or

l = 0 , no element is

added in the current iteration step. Otherwise, these l elements are added to the structure by changing the artificial material property

Em

to real material property E.

5. Performance Index Formulae Similar to the idea in reference [15], a modified performance index is introduced here. The modified performance index PI d at the ith iteration can then be expressed by ISH ) ISH PI d = ( (σ max d ,0 V d ,0 ) / ((σ max ) d , i V d , i )

(10)

* ISH where Vd ,0 = the volume of the original design domain; (σ max ) d ,0 = the maximum Ishai stress; σ = the maximum stress limit;

vm Vd , i = the volume of the current design domain at the ith iteration; and (σ max ) d , i = the maximum Ishai stress at the ith iteration.

With the performance index formulae, the efficiency of the resulting topologies can be evaluated during the optimization process. The performance index PI d also reflects the changes in the volume and the Ishai stress level in the design domain. The optimal topology

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can then be selected as the one corresponding to the highest PI d value. 6. A Bi-directional Algorithm Based on the Ishai Stress Criterion A bi-directional evolutionary structural optimization algorithm incorporating the Ishai stress criterion is developed here for topology design of three-dimensional continuum structures with stress constraints. In the optimization process, elements with the lowest Ishai stresses are removed from the structure while elements with the highest Ishai stresses are added to the structure. The main steps of the evolutionary optimization procedure is given as follows: (1) Model the initial design domain with fine finite element mesh; (2) Apply the boundary constraints, loads, and element properties to the structure; (3) Specify the initial rejection ratio RR0 and the evolutionary parameter ER ; (4) Add elements with artificial material property around the boundaries and cavities of the structure according to the information of the retained elements in the current structure, and set the elasticity modulus of these elements with artificial material property to

E m = 10 −4 × E × rw2 ; (5) Perform a FEA on the current structure, and compute each element’s Ishai stresses and the maximum stress; (6) Remove a small number of elements from the structure according to the element stress levels. For the symmetry structure, there is a symmetry requirement; (7) Add a small number of elements with artificial material property to the structure to prevent singular topology; (8) Repeat Steps (4) to (7) until a steady state is reached. Increase the rejection ratio by an evolutionary rate; and (9) Repeat Steps (4) to (8) until the performance index is maximized.

6. Examples 6.1 Topology Design of A Vehicle Frame The topology design of a postal transporting vehicle frame is considered here. The initial design domain with eight point pin-jointed supports is shown in Fig. 2. The thickness of the initial design domain was 0.016m. The maximum vehicle frame design domain as a three-dimensional continuum structure was divided into a regular mesh of 10032 8-node brick finite elements. The engine weight acting on the first three small circle points from the left side of the structure depicted in Fig. 2 was in total of 3.9 kN. The merchandise and passengers’ weights acting on the rest of the small circle points depicted in Fig. 2 were in total of 7.2 kN. The vehicle body-weight of 7.2 kN was applied to the six small circle points of the top and bottom sides of the structure. The Young’s modulus E = 207 GPa, Poisson’s ratio ν = 0.3 and mass density ρ = 7800 kg/m3 were assumed in the analysis. The precribed stress limt σ * was 0.7 MPa. In the initial structural model used in the analysis and optimization，elements with loads or supporting points were set as non-design elements. The BESO parameters RR0 = 0.01, ER = 0.01, IR = 0.35 and JR = 0.80 were specified in the optimization process.

Figure 2 The initial structural model and the static loading case of a vehicle frame structure (unit: mm) The topology optimization history of the vehicle frame by the proposed method is shown in Figs. 3 and 4. Fig. 3(c) and Fig. 4(c)

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present the optimal topology that satisfies the stress constraint and its volume is reduced by 51.8% compared to the initial structure. The vertical directional stress contours of the optimal structure are depicted in Fig. 5, which shows the structure in different points of view. From Fig. 5, it is noted that the stress distribution within the optimum was approximately uniform. Fig. 6 depicts the variation history of the maximum von Mises stresses in the optimization process. The performance index and weight reduction histories are demonstrated in Fig. 7. From the topologies obtained in the optimization process by the proposed method, the bi-directional algorithm is a more reliable and can recovery some elements removed by mistake that is caused by numerical computation errors, such as the static displacement iteration calculation errors. The optimal topology obtained by the proposed method is very resemble to practical engineering structures, and demonstrate its feasibility of engineering applications.

(a)

(b)

(c)

(d)

Figure 3. The evolutionary topology optimization history of the vehicle frame ( front views)： (a) The volume ratio 0.9816, maximum von Mises stress 0.4850 MPa , and PI d =1.02；(b) The volume ratio 0.6675, maximum von Mises stress 0.5960 MPa , and

PI d =1.23；(c) The volume ratio 0.4824, maximum von Mises stress 0.6210 MPa , and PI d =1.62；(d) The finite element model of the structural topology corresponding to figure 3 (a).

(a)

(b)

(c)

(d)

Figure 4. Topology optimization history (side-view): (a), (b) and (e) correspond to Fig. 3; (d) is the finite element model of the structural topology corresponding to Fig. 4(c).

(a)

(b)

Figure 5. The vertical directional stress contours of the optimum topology of the vehicle frame structure observed in different points of view.

Stress /Mpa

1.40E+00

2.0 Maximum Mises Stress

PI value

Weight ratio

1.5

9.00E-01

1.0 4.00E-01

0.5 0.0

-1.00E-01 0

10

20

30

40

50

Number of Iteration

Figure 6. The evolutionary histories of the maximum von Mises stress of the vehicle frame structure

0

10

20

30

40

50

Number of Iteration

Figure 7. The evolutionary histories of the volume ratio and performance index PI d of the vehicle frame structure

6.2 Topology Design of an Arch Bridge Figure 8 shows the three-dimensional design domain of a bridge where the two ends at the bottom of the bridge were fixed. A uniform static pressure of p = 266.7 kN/m2 was applied on the surface of the bridge desk. The gravity load was considered in the structural analysis. The bridge was assumed to be constructed by reinforced concrete. The Young’s modules E = 21 GPa and Poisson’s ratio ν = 0.2 were specified in the analysis. The design domain was divided into a regular mesh of 20000 8-node brick finite elements. The thickness of the bridge desk was 0.4 m and the bridge desk was defined as a non-design domain. The Ishai stress criterion was

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employed in the optimization process. The BESO parameters RR0 = 0.01, ER = 0.01, IR = 0.35 and JR = 0.80 were specified in the optimization process. Fig. 9 shows the finite element model of the arch bridge structure at the iteration 75. The topology optimization history of the three-dimensional bridge structure is presented in Fig. 10. The optimal topology that satisfies the stress constraint is given in Fig.10(c). The stress contours of the optimal structure in the y direction is demonstrated in Fig. 11. From Figs. 11, it is noted that the stress distribution within the optimum was uniform. The performance index and weight reduction histories of the bridge structure are depicted in Fig. 12. The maximum Ishai stress history is shown in Fig. 13. The optimum topology structure obtained by the proposed method is very reasonable, and is of much resemblance, but is of some reasonable creative parts different from practical arch bridge structures. Of course, for practical bridge structures, it is required that all loading cases must be considered in design. However, the above result demonstrates the validity, effectiveness and wide engineering application prospects of the proposed method.

Figure 8. The initial structural model and static loading case of an arch structure

(a)

Figure 9. The finite element model of the arch bridge structure at iteration 75

(b)

(c)

(d)

Figure 10. The evolutionary history of the arch bridge structural topology： (a) Topology at iteration 75, the volume ratio 0.5889, maximum Ishai stress 55.626 MPa , and PI d =1.3406；(b) Topology at iteration 104, the volume ratio 0.4233, maximum Ishai stress 72.39 MPa , and PI d =1.4333；(c) Topology at iteration 139, the volume ratio 0.2802, maximum Ishai stress 118.96 MPa , and

PI d =1.4811；(d) The finite element model of the arch bridge structure at iteration 139.

(a)

(b)

Figure 11. The vertical directional stress contours of the optimum topology observed in different points of view, obtained by the proposed method. 150000000

2.00

PI value

Stress /Pa

1.50 1.00 0.50 0.00 0

Maximum Ishai Stress

Weight ratio

50

100

150

Number of Iteration

Figure 12. The evolutionary histories of the volume ratio and

100000000 50000000 0 0

50 100 Number of Iteration

150

Figure 13. The evolution histories of the maximum Ishai stress of the arch bridge structure

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performance index

PI d

for the arch bridge structure

7. Conclusions An effective evolutionary structural optimization method incorporating the Ishai stress criteria for topology design of three-dimensional continuum structures with the full stress requirement has been presented in this paper. The proposed method allows lowly stressed elements to be removed from the structure and elements to be added to the highly stressed portions in the structure. Elements with artificial material property are added to the cavities and boundaries of the structure to suppress the formation of singular topologies in the optimization process. A performance index in terms of Ishai stresses has been proposed for the evaluation of the performance of structural topology. A bi-directional evolutionary optimization procedure has been described. Examples presented demonstrate that the proposed method is effective in generating optimal topologies of complex three-dimensional continuum structures. The performance index developed is able to indicate the optimum of three-dimensional continuum structures from the optimization history. The method can be used to optimize not only continuum structures made by isotropic materials but also those with different tension and compression property materials. Therefore, the optimization technique presented is an effective design tool for structural design.

8. Acknowledgements This work is supported by the National Natural Science Foundation of China under the grant No. 10472016 and by the Application Fundamental Research Funds of the Communication Ministry of China under the grant No. 200331982509.

9. REFERENCES 1. G. Allaire 1997 :The homogenization method for topology and shape optimization. In: Rozvany, G. (ed.) Topology Optimization in structural Mechanics, pp.101-133.

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for topology and shape optimization of plate bending problems with displacement constraints.

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