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ScienceDirect Comput. Methods Appl. Mech. Engrg. 326 (2017) 694–712 www.elsevier.com/locate/cma
Explicit isogeometric topology optimization using moving morphable components Wenbin Hou a , b , Yundong Gai a , Xuefeng Zhu a , b , ∗, Xuan Wang a , Chao Zhao a , Longkun Xu a , Kai Jiang a , Ping Hu a , b a School of Automotive Engineering, Dalian University of Technology, Dalian, 116024, China b State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, China
Received 4 June 2017; received in revised form 11 August 2017; accepted 14 August 2017 Available online 6 September 2017
Highlights • • • •
We proposed an isogeometric topology optimization approach using moving morphable components. NURBS-based Isogeometric Analysis (IGA) is adopted for structural response analysis and sensitivity analysis. The presented method inherits the explicit representation of MMC and the higher accuracy of IGA. The proposed approach improves the numerical stability significantly.
Abstract We propose an explicit isogeometric topology optimization approach based on Moving Morphable Components (MMCs). The prescribed design domain is discretized using a NURBS patch and NURBS-based Isogeometric Analysis (IGA) method is adopted for structural response analysis and sensitivity analysis. We employ the MMCs to represent the geometries of structural components (a subset of the design domain) with use of explicit design parameters. The central coordinates, half-length, half-width, and inclined angles of MMCs are taken as design variables. The proposed method not only inherits the explicitness of the MMC-based topology optimization, but also embraces the merits of the Isogeometric Analysis (IGA) such as a tighter link with Computer-Aided Design (CAD) and higher-order continuity of the basis functions. Several numerical examples illustrate that the presented method based on IGA is more robust and stable than FEM-based topology optimization using MMCs. c 2017 Elsevier B.V. All rights reserved. ⃝
Keywords: Isogeometric analysis; NURBS; Topology optimization; Moving morphable components; Topology description function; Sensitivity analysis
∗ Corresponding author at: School of Automotive Engineering, Dalian University of Technology, Dalian, 116024, China.
E-mail address:
[email protected] (X. Zhu). http://dx.doi.org/10.1016/j.cma.2017.08.021 c 2017 Elsevier B.V. All rights reserved. 0045-7825/⃝
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1. Introduction In general, structural topology optimization aims at finding the appropriate material distribution in a prescribed design domain so that the optimal structure has some exceptional properties. Since the pioneering works of Prager and Rozvany [1], Cheng and Olhoff [2], Bendsøe and Kikuchi [3,4], and Zhou and Rozvany [5], topology optimization has received considerable research attention. Numerous topology optimization approaches were proposed and applied successfully to a wide class of problems in various physical disciplines such as acoustics, electromagnetics, and optics. Among these approaches, the most well-established approaches are the solid isotropic material with penalization (SIMP) approach [4–7] and level set approach [8–11]. The SIMP approach, within a pixel-based solution framework, is widely studied among researchers owing to the easy-to-understand 99- or 88-lines MATLAB code that can be used [6,7]. The level set approach, which is a representative node point-based topology optimization approach, was first introduced to topology optimization by Wang [8]. The readers are referred to [12–15] and the references therein for a state-of-the-art review of structural topology optimization. Recently, Guo et al. [16–18] proposed a novel topology optimization approach based on the concept of MMC. Their approach optimizes the central position, length, inclined angles, and some other geometrical features of components in the design domain and allows these MMCs to overlap and merge so that the optimal structural components can describe the desired topology structure. The MMC-based topology optimization framework can not only incorporate more geometric and mechanical information into the topology optimization process as compared to that in traditional structural topology optimization approaches, but also reduce the number of design variables and the computational burden substantially during the optimization process. In their MMC-based approach, the topology structure can be described by the MMCs, while the design domain is approximated using the shape functions of finite element meshes. However, low-order shape functions often lead to numerical instabilities and slow convergence rate. Moreover, the gap and barrier between geometric representation and FEA still exist. Isogeometric analysis (IGA) is expected to address these issues. IGA was developed by Hughes et al. [19], with the aim of unifying the fields of computer aided design(CAD) and FEA. In the process of traditional FEA, mesh generation and the creation of the analysis-suitable geometry account for approximately 80% of the overall analysis time mainly because of the entirely different geometrical representation in engineering and designs. Hughes hoped to transform this situation by reconstituting the analysis procedure within the geometric framework of CAD technologies—NURBS. This new analysis process is referred to as IGA. The most remarkable advantages of IGA are its potential to break down the barrier between engineering design and analysis and its compatibility with the existing practices. IGA has been applied in a variety of fields such as fluid and fluid–solid interaction [20–22], beams and shells [23–26], fracture [27], nonconforming structures [28,29] and structural vibration [30]. Many new modeling techniques such as T-splines [31–37], hierarchical B-Splines [38–40], PHT-Splines [41,42], Web-splines [43–46], B++ splines [47] and Subdivision Surfaces [48–50] have been used for IGA and finite element analysis. IGA also attracts attention from researchers in the field of structural optimization because the NURBS basis functions for IGA have higher accuracy than the shape functions in the traditional FEA for an identical number of DOF and because these basis functions have natural compatibility with the boundary geometry of objects. Youn et al. applied IGA to the structural shape optimization of two-dimensional (2D) and shell problems [51,52] and extended IGA to topology optimization by trimming the spline [53,54]. Ded`e et al. employed IGA to topology optimization with a phase field model [55]. Hassani et al. proposed an isogeometric SIMP topology optimization approach using optimality criteria [56], and employed an implicit function to replace the density function [57]. Wang et al. applied IGA to the conventional level set method where the level set function is updated by the optimality criterion [58,59]. Recently, Jahangiry et al. proposed an isogeometric level set topology optimization where the control mesh (controlling the level set function) is updated by calculating the Hamilton–Jacobi equation and then employed it to tackle stress optimization problems [60]. All the aforementioned isogeometric topology optimization approaches employ an implicit optimization framework. In the present work, we propose an explicit isogeometric topology optimization approach by combining IGA with the MMC-based topology optimization approach; our approach is denoted as TOP-IGA-MMC. The proposed method inherits both the advantages of MMC and IGA. In fact, the proposed method not only can incorporate substantial geometric and mechanical information into topology optimization explicitly and directly but also render the process of analysis and optimization more flexible. Hence, it has the potential to integrate CAD, CAE, and topology optimization into a unified framework. Several numerical benchmark examples demonstrate the effectiveness, stability
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Fig. 1. Geometry description of quadratically variable components with straight skeletons.
and robustness of the proposed approach. For the sake of simplicity, only 2D plane stress problems and components with straight skeletons (central lines) are considered in this work. The rest of the paper is organized as follows. In Section 2, we briefly review the MMC-based topology optimization approach and IGA. The numerical implementation aspects of TOP-IGA-MMC are discussed in Section 3. In Section 4, some benchmark numerical examples are presented to demonstrate the efficiency, stability and robustness of the proposed approach. Finally, some concluding remarks are provided in Section 5. 2. Moving morphable components and isogeometric analysis In this section, the basic idea of the MMC-based topology optimization approach and IGA is reviewed briefly. We refer the readers to [16–18,61], and [19,62,63] and the references therein for more details about the MMC-based topology optimization approach, NURBS, and IGA, respectively. 2.1. Topology optimization based on moving morphable components Traditional topology optimization is accomplished by updating the pixel-element densities in the SIMP approach or by evolving structural boundaries in the level set approach. An explicit topology optimization approach based on MMC was proposed in [16]. 2.1.1. Geometry description of a structural component The proposed MMC-based topology optimization approach employs MMCs as primary building blocks. In this approach, the region Ωi occupied by a primary structural component can be implicitly described as follows: ⎧ ⎨φi (x) > 0, i f x ∈ Ωi φi (x) = 0, i f x ∈ ∂Ωi (1) ⎩ φi (x) < 0, otherwise, where φi (x) is the topology description function: ) ( cos θi (x − x0i ) + sin θi (y − y0i ) p φi (x) = φi (x, y) = Li ( ) − sin θi (x − x0i ) + cos θi (y − y0i ) p + − 1, f (x ′ )
(2)
and p is a relatively large even number (for example, p = 6). In Eq. (2), (x0 , y0 ), L, and θ represent the coordinates of the center, half-length and inclined angle (measured from the horizontal axis anti-clockwise) of a component, ( ) respectively, and the subscript i represents the serial number of the component. Note that f x ′ controls the shape of the components. Here, we consider quadratically variable components with straight skeletons for simplicity: ( ) t1 + t2 − 2t3 ( ′ )2 t2 − t1 ′ f x′ = x + x + t3 . (3) 2L 2 2L ( ′) Fig. 1 depicts the shape of the corresponding component when f x takes the form of Eq. (3). We refer the readers to [17] for more types of straight skeleton components, and to [18] for curved skeletons with absolutely different topology description functions.
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The structural components can move, dilate, shrink, and overlap. When different components overlap, the region occupied by two and more components can be described by the maximum value of the topology description function corresponding to the components. This representation is compatible with the region occupied by only one specific component. Therefore, the structural topology can be described as follows: ⎧ s ⎨φ (x) > 0, i f x ∈ Ω s φ s (x) = 0, i f x ∈ ∂Ω s (4) ⎩ s φ (x) < 0, otherwise, where φ s (x) = max (φ1 (x) , . . . , φnc (x)) with φi (x) , i = 1, . . . , nc denoting the TDF of the ith component. In Eq. (4), Ω s is a subset of the prescribed design domain D and denotes the region occupied by the solid structural materials. Thus, the structural topology can be described by a group of structural components. For 2D quadratically variable components, a structural component can be determined by Di = (x0i , y0i , L i , t1i , t2i , t3i , θi )T . Hence, the structural topology can be explicitly and uniquely described by the vector of the design variables: ( ) T T D = D1T , . . . , Dnc . (5) 2.1.2. Problem formulation based on MMC As shown in [16], the topology optimization problem in the MMC-based approach can be formulated mathematically as follows: ( ) T T Find D = D1T , . . . , Dnc Minimi ze I = I (D) (6) s.t. g j (D) ≤ 0, j = 1, . . . , m, D ⊆ U (D) . In Eq. (6), I is the objective function/functional, g j , j = 1, . . . , m, are constraint functions/functionals and U (D) is the admissible set to which D belongs. If the compliance minimization problem under the available volume constraint is considered and the Galerkin numerical method is employed, the optimization problem can be formulated as follows: ( ) T T Find D = D1T , . . . , Dnc , u (x) ∫ nc ∫ ∑ i Minimi ze C = f · ud V + T · ud S ⋃ ⋂ s.t.
i=1
nc ∫ ∑ i=1
Ωi \( 1≤ j
