第 6 卷第 6 期 2002 年 12 月 Article ID: 1007 7294( 2002) 06 0107 07
船舶力学 Journal of Ship Mechanics
Vol. 6 No. 6 Dec. 2002
Study on Topology Optimum Desi gn of E
ngineering Structures
XIA Li  juan, WUJia meng , JIN Xian ding
( School of Naval Architecture and Ocean Eng., Shanghai Jiao Ton g Univ., Shanghai 200030, China)
Abstract: The topology optimization models of the static problem and the eigenvalue problem for the continu ums based on the homogenization method are proposed in this paper. At present the study on the application of the topology optimum design in eng ineeing structures is still under development. In this paper the homoge nization method is applied to the topology design of the actual engineering structures, which can provide a valuable conceptual design scheme for the selection of structural pattern. By applying the topology o ptimum design to a classical example and a frame structure of the satellite, the homog enization method and the topology o ptimization models presented are proved to be reliable and practical at the conceptual design stage of engineering structures. Key words: topology optimization; homogenization method; engineering structure CLC number: TU318 Document code: A
1 The homogenization method in topology optimization
The design process is generally divided into two stages: conceptual design and detailed design. The structural pattern can be obtained in the conceptual design stage by using the topology optimum design, then the dimensions of the structure are acquired in the detailed design stage. The homoge nization method is an effective technique for the topology optimization of continuums . In topology optimization based on the homogenization method, the design domain whose entire area consists of porous media is assumed to have a virtually infinite number of periodic micropores, shown in Fig. 1. The hole shape in the unit cell is assumed to be a rectangle in a two dimensional or three dimensional plate structure, or rectangular solid in a three dimensional solid structure, shown in Fig. 2.
[1 3]
Fig. 1 The micro structure with micropores
Fig. 2 Types of unit cells
By applying the homogenization method, the average macroelastic tensor of the respective ele ments having holes of different sizes can be calculated. Since the repetition of the calculation for each element requires an enormous number of calculations, it is convenient to create a database
Received date: 2001 06 25 Biography: XIA Li juan( 1975 ) , female, Dr. of Shanghai Jiao Tong University.
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beforehand that stores calculation results for several representative hole sizes. In this way, once the hole size is determined for each element in the optimization process, the value of elastic tensor can be easily obtained via an interpolation approximation using the values of multiple representative points provided in the database. The average macroelastic constant, density, body force of microstructures having periodic holes can be expressed by the following equations:
Ei j k l =
H H
1 Y 1 Y 1 Y
Y
Y
E i j k l  Ei j p q
l xk p dy yq
( 1) ( 2) ( 3) and f are the density
=
dy f dy
f =
H
H
Y
where Ei j k l represents the elastic tensor of the material comprising unit cells.
and the body force respectively. The superscript H and represent the material after homogenization and the unit cell, res pectivel y. Y denotes the domain of the cell. x is called Characteristic Defor mation and denotes the eigen deformation mode of the unit cell. It can be obtained as a solution to the following equation being satisfied within the unit cell:
Y
Ei j k l Ei j p q
xp yq
kl
vi dy = 0 yj
( 4)
where y j and vi represent the coordinate s ystem describing the cell and virtual deformation, re spectively. A topology optimization model with the homogenization method can be set as follows: ( 1) Design variables: the design variables are assumed to be the hole size of each element, which is the vertical and horizontal lengths in the case of a rectangular hole and the vertical length, horizontal length and height in the case of a rectangular solid. If the hole becomes larger and occu pies the entire cell, material becomes no longer necessary. If the hole becomes smaller to a point where there is no size, the resulting object becomes a solid completely filled with material. The number of the design variables depends on the number of the finite elements of the design domain. ( 2) Design constraints: the design constraints are the total volume ( mass) of the material that can be used within the domain. ( 3) Objective functions: the objective functions depend on the type of the optimization prob lem. For the static stress problem, the objective function is the minimization of mean compliance ( strain energy across the entire design domain) . For the eigenvalue problem, the objective function is the maximization of mean eigenvalue or the minimization of distance between the given eigenvalue of each order and the calculated eigenvalue. After topology optimization, a size distribution of the hole is yielded which is satisfied with the minimization or maximization of the objective function within the constraint material range. On the basis of the size distribution, some materials can be deleted and an obviously different resultant topology can be obtained [4].
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2 Description of various topology optimization analysis problems
2. 1 Static problem The topology optimization model for static problem can be expressed as follows:
min s . t. =
1 2 d
T
D
H
d
( 5)
Vc
That is, when the volume of the material used within the domain is constrained and the mean compliance within the domain is minimized, a topology having the maximum stiffness within the range of the constrained material Vc can be obtained. , D ,
H
and Vc represent the element' s strain,
equivalent stress strain matrix calculated by the homogenization method, material filling ratio and constraint volume over the entire domain, respectively. In this case, the sensitivity related to the stiffness of each element can be expressed as follows:
Ke d =
Be
T
De d Be d
H
( 6)
where B, Ke and d denote the element 's strain deformation matrix, stiffness matrix and design variable, respectively. Generally, structural design is performed under various complex conditions. There are many load cases and it is necessary to calculate a topology that can accommodate these complex combina tions. In such cases, use the unequal equation below to select conditions that yield the maximum strain energy density for each element, from among all constraints and load cases, and minimize the new problem thus defined.
1 Min i =Max 1, , n 2 Min 1 2
i = 1,
( ui ) D Max, n {
T
H
( ui ) d
H
( ui ) D
T
( ui ) } d
( 7)
2. 2 Eigenvalue problem For the eigenvalue problem, the objective function can be chosen for the maximization of mean eigenvalue or the minimization of distance between the given eigenvalue of each order and the cal culated eigenvalue. First, let us maximize a mean eigenvalue m e a n, calculated from the first eigenvalue 1 to the nth eigenvalue
n
via weighting by wi using the following equation:
m
wi
m e an
=
i = 1 m
wi
i
( 8)
i = 1
where wi ( i = 1,
max s . t.
, n) represent the weight coefficient of each eigenvalue. The topology optimization model can be expressed as follows:
m e an
d
Vc
( 9)
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In this case, the sensitivity of the mean eigenvalue is obtained as follows:
2 me a n
m
d
=
m ea n
wi
2 i
i
m
i = 1
wi
i = 1
d
( 10)
where
i
d
=
T i
K d
H
i
M d
i
( 11)
the sensitivity related to the stiffness and mass of each element can be expressed as follows:
Ke = d Me d = K e , Me , N e ,
i
Be
T
De Bd d e
e
( 12) ( 13)
d
N e Ne d
T
and
herein represent the element stiffness matrix, element mass matrix, element shape
function, eigenmode and element mass density, respectively. Now consider the objective function of minimizing the distance between the given eigenvalue of each order 0 i( i = 1, , m ) and the calculated eigenvalue i ( i= 1, , m) , the topology opti mization model can be expressed as follows:
m
min
i= 1
1
2 0i
(
i

2 0i
)
( 14)
Vc
s . t.
m
d
The sensitivity of the objective function is obtained as follows:
d=
2
2 0i
( i
0i
)
i
i= 1
d
( 15)
3 Calculation results
3. 1 Static problem with multiple load cases For a two dimensional rectangular flat plate being fixed at two points and subj ected to two cases of concentrated loads separatel y ( shown in Fig. 3) , the optimal shape will be obtained by using the homogenization method [5]. As we know, the optimal topological shape is dependent on the load case. Different load case will result in the different optimal topological shape. The study on the topology optimization subjected to multiple load cases is under development because of the difficulty and complexity of the topology optimization problem. When the topology optimization problem with multiple load cases is solved by the homogenization method, its objective function ( the mean compliance) is relevant to the load case, which results in a multi objective optimization problem.
Fig . 3 Schematic of the model
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The optimal topological shape in the first scenario subjected to the load case 1 only is shown in Fig . 4. The optimal topological shape in the second scenario subjected to load case 2 only is shown in Fig . 5. The optimal topological shape in the third scenario subjected to all the loads at the same time is shown in Fig. 6. The optimal topological shape in the fourth scenario subj ected to multiple load cases is shown in Fig. 7, in which the weighted mean compliance of two load cases being the ob jective function.
Fig. 4 To pological sha pe under load case one
Fig . 5 Topolog ical shape under load case two
Fig. 6 Topological sha pe under all loads
Fig. 7 To pological sha pe under multi load cases
3. 2 Eigenvalue problem An example of topology optimization is presented for a frame structure of a satellite. In this scenario, we make the fundamental frequency as close to the specified frequency as possible. Also we hope to reduce the number of trusses and joints as small as possible and then obtain an optimal structure with simpler shape, smaller weight and the same fundamental frequency characteristic after optimization. The frame structure is shown in Fig. 8. First the finite element model of the frame structure is modeled by using shell elements. In order to avoid simulating the complex intersect curves, MPC element of MSC. NASTRAN software is adopted in the joints, shown in Fig. 9. In Fig. 9 the instru ments of the satellite are added to the finite element model by using the point elements and stiffened trusses elements. When analyzing this optimization problem the instruments should be considered because of their heavy mass and large stiffness, while the stiffened truss elements relevant to the instruments are not inside the design domain. The fundamental frequency of the frame structure is 76. 5Hz. The objective function is to make the fundamental frequency of the frame structure as close to 76. 5Hz as possible. The optimal topological shape by using the homogenization method is shown in Fig . 10. The fundamental frequency of the optimal topological structure is 76. 52Hz; the corres pond ing mode shape is shown in Fig. 11.
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Fig. 8 Schematic of the frame structure
Fig. 9 The FE model with shell elements
Fig. 10 Topological shape of the frame structure
Fig. 11 The first mode shape after optimization
On the basis of the structure shown in Fig. 10, about 10 joints can be deleted because of the reduction of the intersect trusses. The weight of the frame structure ( not including the mass of the instruments) is decreased from 57. 45kg to 43. 47kg. That is to say, after topology optimization the frame structure has a weight 24. 3% smaller than before but with the same fundamental frequency characteristic as before.
4 Conclusions
Through this study the following conclusions can be drawn : ( 1) From Fig. 4 to Fig. 7 one can see that the different load case will result in the different optimal topological shape; the decision of the load cases in the design of the engineering structures is very important and has great effect on the final optimal topological shape. ( 2) From Fig. 8 and Fig. 10 one can see that about 14 trusses and 10 joints are deleted from the original frame structure; the aim for simplifying the structural shape and decreasing the structural mass has been reached and the effect of optimization is very obvious.
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( 3) From the calculation results, one can see that the topology optimization models presented in this paper for the static problem and eigenvalue problem are reasonable and can be applied to the topology optimum design of engineering structures.
References
[ 1] Hassani B, Hinton E. A review of homogenization and topology optimization I homogenization theory for media with periodic structure[ J] . Comp. & Struc., 1998, 69: 707 717. [ 2] Hassani B, Hinton E. A review of homogenization and topology optimization II analytical and numerical solution of homogeniza tion equation[ J] . Comp. & Struc., 1998, 69: 719 738. [ 3] Hassani B, Hinton E. A review of homogenization and top ology op timization III topology optimization using optimality criteria [ J] . Comp. & Struc., 1998, 69: 739 756. [ 4] MSC. Nastran Optishape Release Guide[ CP] . MSC. Software Japan Ltd., 2001. [ 5] Xia Lijuan. Study on the global dynamic optimum design of the satellite structure[ D] . Ph. D. dissertation, Shanghai Jiao Tong Universit y, 2002( in Chinese) .
工程结构的拓扑优化设计研究
夏利娟, 吴嘉蒙, 金咸定
( 上海交通大学船舶与海洋工程学院结构力学研究所, 上海 200030) 摘要: 本文讨论了连续体结构拓扑优化的均匀化方法及其相关理论, 分别针对静力问题和特征值问题建立了相应的结 构拓扑优化模型。 目前有关结构拓扑优化的工程应用研究还很不成熟, 尤其在国内尚属于起步阶段。 本文将结构拓扑优 化设计理论应用于实际工程结构的概念设计之中, 并取得了较好的优化效果。通过对经典算例和某卫星构架子结构的 拓扑优化计算, 表明本文建立的结构拓扑优化模型能够有效地应用于工程结构的拓扑优化设计, 从而为工程结构的结 构型式选取提供了有价值的概念设计方案。 关键词: 拓扑优化; 均匀化方法; 工程结构 中图分类号: TU318 文献标识码: A 作者简介: 夏利娟( 1975 ) , 女, 上海交通大学船舶与海洋工程学院博士; 吴嘉蒙( 1977 ) , 男, 中国船舶及海洋工程设计研究院硕士; 金咸定( 1940 ) , 男, 上海交通大学船舶与海洋工程学院教授。