Computers and Structures 228 (2020) 106146
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Efficient topology optimization of multicomponent structure using substructuring-based model order reduction method
Hyeong Seok Koh, Jun Hwan Kim, Gil Ho Yoon ⇑ Mechanical Engineering, Hanyang University, Republic of Korea
article info
Article history:
Received 12 July 2019
Accepted 28 October 2019 Available online 18 November 2019
Keywords:
Topology optimization Model reduction schemes Substructure design Krylov subspace
Ritz vector method
1. Introduction
This study developed a novel model reduction (MR) scheme and investigated how the dynamic characteristics of multicomponent structures could be efficiently improved through the density- based topology optimization. For a long time improving the vibra- tion or noise characteristics of structure by changing its geometry has been a subject of research and discussion for engineers and sci- entists [1,2]. To improve these characteristics systematically via finite element (FE) method, topology optimizations have been developed and applied to various engineering problems [2,5,6,10,11,14,17–20,22–26,28–33]. However, despite the interest in topology optimization for dynamic structures, the very high computation time required for the optimization process often becomes a huge obstacle for practical applications. FE method must frequently consider complex manifold structures with many degrees of freedom (DOFs), which can easily become unsupport- able for step-by-step frequency or time-domain analyses, in spite of technological developments in high-performance computer hardware and computer-aided engineering (CAE) software. When we consider topology optimization for complex manifold struc- tures shown in Fig. 1, a large computation time would be required
⇑ Corresponding author at: School of Mechanical Engineering, Hanyang Univer- sity, Seoul, Republic of Korea.
E-mail address: ghy@hanyang.ac.kr (G.H. Yoon).
https://doi.org/10.1016/j.compstruc.2019.106146
0045-7949/! 2019 Published by Elsevier Ltd.
abstract
This study develops a novel model reduction (MR) scheme called the multi-substructure multi-frequency quasi-static Ritz vector (MMQSRV) method to compute dynamic responses and sensitivity values with adequate efficiency and accuracy for topology optimization (TO) of dynamic systems with multiple sub- structures. The calculation of structural responses of dynamic excitation using the framework of the finite element (FE) procedure usually requires a significant amount of computation time. The ever-increasingly complex phenomena of FE models with many degrees of freedom make it difficult to calculate FE responses in the time or frequency domain. To overcome this difficulty, model reduction schemes can be utilized to reduce the size of the dynamic stiffness matrix. This paper presents a new model order reduction method called MMQSRV, based on the quasi-static Ritz vector method, with Krylov subspaces spanned at multiple angular velocities for efficient TO. Through several analysis and design examples, we validate the efficiency and reliability of the model reduction schemes for TO.
! 2019 Published by Elsevier Ltd.
to calculate FE procedure with fine incremental frequencies or times.
One of the efficient and effective approaches to reduce the com- putation time in the FE procedure is reducing the system size prior to calculating the structural responses by using model reduction schemes. Many innovative model order reduction (MOR) methods, such as the Guyan reduction method [9,13], the mode superposi- tion method (MS method) [22], the proper orthogonal decomposi- tion method [20,27], the Ritz vector method (RV method) [15,22,26,28], the quasi-static Ritz vector method (QSRV method) [14,31], the multi-frequency quasi-static Ritz vector (MQSRV) method [30], and the transient quasi-static Ritz vector (TQSRV) method [31], have been developed. In [26], comparative transient dynamic analyses were performed using the load-dependent and the mode-superposition method based on the superposition of eigenvectors. In [8], they employed static recurrence procedures to generate the Ritz vectors. As such, these vector methods are best suited for low-frequency problems with the so-called quasi-static Ritz vector. In [3], the mode acceleration method and the mode superposition method have been employed to solve structural engineering problems. Especially, the time reduction effect of MOR methods come to remarkable when we have to deal with large scale problem [7]. Furthermore, MOR methods play an impor- tant role in connection to structural optimization, because many optimization and analysis iterations can be accelerated by a MOR approach in frequency domain [19,20,29,32]. The first MOR
2 H.S. Koh et al. / Computers and Structures 228 (2020) 106146
approach in topology optimization can be found in [29] which investigated three types of MOR approaches, i.e., the MS, the RV, and the QSRV methods in topology optimization and showed that the mode superposition approach using the eigenvector can be troublesome due to the local mode issue. In [17], a MOR approach was applied to transient analysis in TO using the mode acceleration method (MAM) and mode displacement method (MDM). In [32,33], three kinds of objective functions (the mean dynamic com- pliance, the mean strain energy, and the mean squared displace- ment) were considered for TO. using MAM and MDM. Recently, Mediante et al. applied the projection-based parametric MOR method to reduce the computational cost of material or size opti- mization in large vibroacoustic models [21].
To contribute this research subject, the present study develops a new model order reduction for multiple components shown in Fig. 2. and applies it to the structural topology optimization with multiple components. To our best knowledge, the model order reductions for multiple components have been developed [4,12,18,20]. However, its application to structural topology optimization has not been studied or developed before this research. The existing model order reduction shows high efficiency with multi-components connected through points or shallow regions [18]. Indeed by presenting the model order reduction with Ritz vectors for multi-components con- nected through line or surface, the approximately structural responses are efficiently predicted. As the number of the Ritz vectors is proportional to the number of nodes along the interface lines, the numerical efficiency is also influenced. After that, its applications for efficient TO have been proposed.
The layout of the paper is organized as follows: First, we explain the basic concepts of density-based topology optimization and MR schemes. Then, a new MOR method called the multi-substructure multi-frequency quasi-static Ritz vector (MMQSRV) method for calculating the reduction bases for multiple substructures is pre- sented. Using some analysis examples with arbitrary chosen mate- rial properties and boundary conditions, the efficiency and accuracy of the MMQSRV method will be checked. Then, the TO with the MMQSRV method will be solved. Finally, we summarize our findings and discuss some topics for future research in the conclusion.
2. Optimization formulation
2.1. Frequency response analysis of FE
Without the loss of generality, Newton’s second equation is solved for the time-varying response of linear solid structure with time-varying force, Ft as follows:
MX€t þCX_t þKXt 1⁄4Ft; ð1Þ
where M, C, and K are the mass matrix, the damping matrix, and the stiffness matrix. The time-varying displacements, velocities, and accelerations of the structure are denoted by Xt , X_ t , and X€ t , respec- tively. For the sake of simplicity, it is assumed that the following Rayleigh damping with damping coefficients ar and br in Eq. (2).
C1⁄4arMþbrK ð2Þ
Fig. 1. Complex manifold structure.
Fig. 2. Application of MMQSRV method in simple model.
H.S. Koh et al. / Computers and Structures 228 (2020) 106146 3
For the frequency response analysis, the following harmonic excitation is assumed [1].
Xt 1⁄4 Xeixt ; Ft 1⁄4 Feixt ð3Þ The dynamic stiffness matrix, S, can be derived as follows:
SX1⁄4F; S1⁄4x2MþixCþK ð4Þ 2.2. Statement of topology optimization formulation
Optimized design for given static or dynamic external loads has been an important issue in many structural applications. In this study, the topology optimization problem involves a volume con- straint and strives to minimize the dynamic compliance defined by Ma et al. [19] and Jensen [11]. The topology optimization of a linear problem minimizing the dynamic compliance and subject to a volume constraint can be formulated as follows:
R T Minimize U 1⁄4 xe F Xdx
Zxe
dU1⁄4 2Real kT dSX dx ð12Þ
dc xs dc aXTF
k1⁄4 2 X;a1⁄4 T ; kk1 1⁄4k2 ð13Þ X F
c xs
3. Model reduction (MR) schemes for topology optimization
3.1. Introduction of model reduction schemes
For precise response calculation, the number of DOFs in a com- putational model is increased significantly over time. Therefore, the solutions of refined FE meshes are difficult for even the most advanced and state-of-the-art computational systems within a moderate computation time. In frequency response analysis, these limits are often overcome by applying a MOR scheme to reduce the size of the assembled stiffness and mass matrices. Many relevant studies have developed MOR methods, such as the Guyan reduc- tion method [9,13], MS method [22], RV method [22,26,28], QSRV method [8], MQSRV method [30], TQSRV method [31], and proper orthogonal decomposition method [16]. By approximating the original structural response X with the reduced response wQ, the size of the linear algebra system can be decreased by transforming a large set of system equations into a small set of equations [29,30]. From a mathematical point of view, the approximated response XA of the original response X can be defined as follows:
AX 1⁄4 B; ð14Þ X 1⁄4 W Q þ R ffi W Q 1⁄4X; ð15Þ
nsnd nd1 ns1 hi
ð16Þ
where A and B denote an arbitrary ns ns system matrix and an ns 1 force vector, respectively; the number of DOFs in the system is denoted by ns, and the number of reduced DOFs is denoted by nd. In (15), the frequency-dependent basis vectors of the order nd, the reduced unknown variables, and the residual matrix are denoted by W, Q, and R, Respectively. By pre-multiplying WT into (14), the following reduced equation with order nd is obtained.
Subject to
PNE
civi 6 V ð5Þ MX€t þCX_t þKXt 1⁄4Ft
0