Int J Adv Manuf Technol (2007) 32: 787–796 DOI 10.1007/s00170-005-0387-0
ORIGINA L ARTI CLE
Kong-Tian Zuo . Li-Ping Chen . Yun-Qing Zhang . Jingzhou Yang
Study of key algorithms in topology optimization
Received: 30 April 2004 / Accepted: 22 April 2005 / Published online: 21 March 2006 # Springer-Verlag London Limited 2006
Abstract The theory of topology optimization based on the solid isotropic material with penalization model (SIMP) method is thoroughly analyzed in this paper. In order to solve complicated topology optimization problems, a hybrid solution algorithm based on the method of moving asymptotes (MMA) approach and the globally convergent version of the method of moving asymptotes (GCMMA) approach is proposed. The numerical instability, which always leads to a non-manufacturing result in topology optimization, is analyzed, along with current methods to control it. To eliminate the numerical instability of topology results, a convolution integral factor method is introduced. Meanwhile, an iteration procedure based on the hybrid solution algorithm and a method to eliminate numerical instability are developed. The proposed algorithms are verified with illustrative examples. The effect and function of the hybrid solution algorithm and the convolution radius in optimization are also discussed. Keywords Topology optimization . MMA series algorithms . Hybrid solution algorithm . Numerical instability . Convolution integral factor method
1 Introduction Topology optimization is considered to be one of the most challenging fields in structural optimization. Topology optimization has been an interesting research topic since Bendsoe and Kikuchi [1] introduced the homogenization K.-T. Zuo . L.-P. Chen . Y.-Q. Zhang Center for Computer-Aided Design, Huazhong University of Science & Technology, Wuhan, Hubei, 430074, People’s Republic of China e-mail:
[email protected] J. Yang (*) Center for Computer-Aided Design, The University of Iowa, Iowa City, IA 52246, USA e-mail:
[email protected] Tel.: +1-319-3532249 Fax: +1-319-384-0542
method into the field. After more than 20 years of development, researchers have made a great deal of progress in theories as well as in real engineering applications. Recent research in topology optimization has focused on such topics as: material interpolation methods, optimal algorithms suitable for topology optimization, methods to eliminate numerical instabilities, and the application of topology optimization in engineering. Material interpolation used in topology optimization mainly include the following two types of methods: the homogenization method [1, 2], primarily used in theory derivations, such as the existence of solutions, investigating numerical instabilities, etc.; and the material density method, sometimes named the solid isotropic material with penalization model (SIMP) method, proposed by Mlejnek and Schirrmacher [3], Sigmund [4], and Bendsoe and Sigmund [5]. The SIMP method, which is used in this paper, has been accepted by most researchers and engineers, and has been successfully used in many engineering fields. At present, two types of optimization algorithms are used in topology optimization: the optimality criteria (OC) approach and the method of moving asymptotes (MMA) series approach. The former [6, 2] is usually deduced by a Lagrange function composed of objective and constraint functions according to the Kuhn-Tucker condition. This approach has good convergence because it is based on heuristic formulation and does not need the derivatives of functions. However, its disadvantage is that it has a narrow restriction in the single-constraint condition and is difficult to use in problems with multiple constraints. The MMA approach [7–9, 12] can fit single and multiple constraints conditions, so it has wider engineering applications, such as the design of a compliant mechanism, microelectromechanical systems (MEMS), and multidiscipline optimization. The disadvantage of the MMA approach exists in the bad convergence of calculation. The globally convergent version of the method of moving asymptotes (GCMMA) has a better convergence as compared to the MMA approach. However, the calculation speed of GCMMA is not perfect. A new hybrid algorithm based on MMA and GCMMA is proposed. This hybrid algorithm combines the
788
advantages of both MMA and GCMMA together and can be widely used in complicated engineering problems. One important focus of topology optimization is developing a method to eliminate numerical instability, since it may result in a non-manufacturable structure in engineering. Many methods, such as the perimeter control method [16], density slope control [17], and meshindependent filter [18], have been proposed to deal with this problem. But most of them have some shortcomings. This paper first reviews the SIMP method in topology optimization. Then, the MMA approach and GCMMA approach based on SIMP is applied to topology optimization, and a hybrid solution algorithm based on both MMA and GCMMA is proposed. To eliminate numerical instability in the topology optimization, a convolution integral factor method is developed, and an iteration procedure including the hybrid solution algorithm and the convolution integral factor method is presented. Finally, an illustrative example is used to discuss the effect and function of the hybrid algorithm and the convolution integral factor method.
and E are the elastic moduluses of an element before and after optimization, respectively, then E ¼ ðxe Þp E0 : If k0 and ke are the element’s initial stiffness and the after-optimization stiffness matrices, respectively, the following relationship exists: ke ¼ ðxe Þp k0 : The parameter p is a penalty factor, and it is important to penalize the middle density in order to decrease the number of middle-density elements and ensure that most element densities are close to zero or one. Given the above preconditions, every element has only one design variable. Compared to the homogenization method, the SIMP approach makes excellent progress on decreasing the number of design variables. Another advantage of this SIMP method is that the material characteristic after the change can be written as the exponent function of the initial element density and the initial material characteristic, so that this approach greatly simplifies the solution of topology optimization. Since a relationship exists in the discrete element: V ¼ N P
f V0 ¼
xe ve ; where v e is the element volume after
e¼1
2 SIMP method in topology optimization The general topology optimization problem of minimal compliance can be defined as: minimize : C ¼ F U T
(1a)
optimization, the topology optimization formulation of the minimal compliance problem based on the SIMP approach will be: minimize : C ¼ U T KU ¼
N X
ue k e ue ¼
e¼1
subject to : V ¼ f V0 ¼
N X
xv V e e
subject to : V ¼ f V0 ¼ (1b)
e¼1
N X
N X
ðxe Þp ue k0 ue
e¼1
xe ve V
e¼1
F ¼ KU 0 < xmin xe xmax
F ¼ KU 0x 1 e
(2)
(1c)
(1d)
where C is the compliance of the structure, F is the force vector, U is the displacement vector, K is the stiffness matrix of the structure, V0 is the initial volume of the structure, V is the structure’s volume after optimization, and f is the ratio of the volume after optimization with the initial volume. The volume constraint and equilibrium equation of the structure are included. In the SIMP approach, the preconditions include: 1. The material characteristics, such as the elastic modulus, in a discrete element are constant. The design variable is the density of the element, represented with xe. If ρ0 is the initial element density and ρ is the element density after optimization, then ρ ¼ xe ρ0 exists. 2. The material characteristics in an element should be changed with the exponent of element density. If E0
where xmin is the lower bound of the density, which is introduced to prevent singularity of the equilibrium problem, and xmax is the upper bound of the density, ue is the element displacement, and N is the total number of discrete elements.
3 A new hybrid solution algorithm for topology optimization 3.1 MMA and GCMMA approach based on the SIMP method The MMA approach, which was initially proposed by Svanberg [7, 8], is based on the first-order Taylor series expansion of the objective and constraint functions. With this method, an explicit convex subproblem is generated to approximate the implicit nonlinear problem. Because the subproblem is separable and convex, a dual approach or a primal-dual interior-point method can be used to solve it
789 Construct the MMA/GCMMA form of initial problem
where:
Choose a initial calculation point
ðkÞ e fi ðxÞ
ðkÞ
¼fi x
Construct MMA/GCMMA subproblem
n X j¼1
Convergence? Yes Output result The end
ðkÞ
αj
Fig. 1 Flow chart of the MMA/GCMMA approach
ðk Þ
βj [10, 13, 14]. The solutions of a sequence of subproblems can converge towards the original problem. A general formulation of nonlinear optimization can be written as:
i¼1
1 ðci yi þ di y2i Þ 2
subject to : fi ðxÞ ai z yi 0 xmin j
xj
yi 0; z 0
i ¼ 1; . . . ; m
(3)
j ¼ 1; ; n
xmax j
i ¼ 1; . . . ; m
where the design variable x ¼ ðx1 ; . . . ; xn ÞT 2
