A polytree-based adaptive polygonal finite element method for multi-material topology optimization

https://doi.org/10.1016/j.cma.2017.07.035Get rights and content

Highlights

  • A polytree-based adaptive mesh strategy is proposed for multi-material topology optimization.

  • We present an adaptive filter radius to improve the efficiency of filters and optimized results.

  • Error indicator is determined by the density discrepancy through edges of polygonal elements.

  • Numerical results show high effectiveness of the present approach.

Abstract

This study presents a polytree-based adaptive methodology for multi-material topology optimization (MMTOP). Polytree data structure is introduced as a general recursive multi-level mesh that is automatically refined in processing based on error analysis. In order to resolve hanging nodes in element edges, the Wachspress coordinate is employed on a reference element before using a mapping scheme to obtain shape functions and their derivatives for any polygons. A new definition of filter radius is also proposed to improve the efficiency of filters and optimized results. The combination of polytree meshes and adaptive filters not only clarifies the interfaces between material phases (including void phase), but also decreases the computing time of the overall process in comparison to using the regular fine meshes. Several benchmark and practical problems are considered to show distinct features of the proposed method.

Introduction

Topology optimization is an intelligent approach to seek the best design in many fields of computational mechanics. By analyzing the information inside the determined domain, it changes the topology of such domain into a new design to satisfy the given criterion. Since the first research about generating optimal topology using a homogenization method published by Bendsøe and Kikuchi [1], many numerical methods for topology optimization have been developed and innovated to solve multiphysics problems [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. These papers almost always used an ideal material to implement the topology optimization problems. Therefore, the optimal designs are still considered as reference or theoretical solutions while the practical problems need more than that. The multi-material topology optimization (MMTOP) is established to evaluate the behavior of many materials in a problem and give an applicable solution for design. In the beginning, several approaches have been utilized to investigate multi-material problems. The homogenization technique is applied to maximize the structural stiffness of isotropic materials which are modeled as composites [18]. The three-phase composites are also considered using this aspect for extremal thermal expansion and bulk modulus [19], [20]. The unification of parameterized formulation as an auxiliary of SIMP (Solid Isotropic Material with Penalization) and RAMP (Rational Approximation of Material Properties) interpolation schemes was suggested for multi-material problems with given anisotropic properties [21]. The continuous peak function is introduced to model multi-materials in optimization problems while maintaining the number of design variables [22]. On the other hand, the level-set method has been adapted for modeling multi-materials without an interpolation scheme but it showed the mesh and initial design dependencies of the final topology [23], [24]. Based on Cahn–Hilliard formulation, the multi-material topology optimization using phase field model was developed in [25], [26]. Although this approach exhibits its advantage in solving the partial differential equations with volume preservation, the slow convergence is one of shortcomings which makes it become uneconomical in computational processes. Furthermore, optimal designs decidedly depend on many phenomenological correction constants that must be adjusted to obtain the best solution. The alternating active phase algorithm was presented by Tavakoli and Mohseni to solve the multi-material optimization problems [27]. Inheriting the single-phase topology optimization algorithm and the similar Gauss–Seidel iteration scheme, the framework can be proceeded by decomposing the original problem into many binary phase sub-problems. The traditional single-material optimization approach can be extended to generalize the optimizer for a material set with minimal modification.

The efficiency of topological optimizers depends on many factors: finite element solver, number of degrees of freedom, material modeling, data processing, etc. The topology optimization framework is always required to get high resolution while reducing the computational cost for the overall process. There are several works focusing on enhancing the solver performance related to saving computational cost and time-consumption. Borrvall and Petersson [28] solved the large-scale topology optimization by combining parallel computing with domain decomposition. Kim et al. [29] applied this algorithm for large-scale eigenvalue topology optimization. Wang et al. [30] suggested a fast iterative solver for 3D topology optimization problems with over a million unknowns. Amir et al. [31], [32] offered reanalysis techniques after a number of intervals in optimization procedures. Nguyen et al. [33], [34] developed a multiresolution topology optimization (MTOP) by using different mesh levels for finite element analysis and design variables high resolution designs are then harvested corresponding with the fine mesh while the displacement solutions are obtained from the coarser mesh with lower computational cost. The notation of isogeometric analysis has been introduced by Hughes et al. [35] that produces a smooth approximation to exclude the limitations of the conventional FEM and makes a direct connection between the exact geometry and analysis. Therefore, it is able to reduce the overall cost in converting processes for complex geometries [36], [37], [38]. However, several shortcomings of this approach in local refinement, patch continuity and interior domain parameterization should be carefully resolved in the computational implementation. These difficulties can be allayed by advanced techniques such as T-splines [39] and IGABEM [40]. The meshless methods were combined in the optimization problems to avoid the burdensome meshing cost [41], [42], [43], [44] but they showed the drawbacks in imposing the essential boundary conditions, expensive cost for shape functions computing and numerical integral evaluation. An efficient approach based on a so-called adaptive mesh refinement scheme has been developed to improve the precision of the solution and also the computational efficiency of the topology optimizer. For example, Maute and Ramm [45] utilized an adaptive design space in each design circle to describe the physical density. Lin and Chou [46] proposed a two-stage optimization algorithm that employed a large size mesh in the equational solving stage and a finer mesh during the topology configuration. Costa and Alves [47] focused on smoothing domain boundaries and diminishing strain errors by combining topology optimization and an h-adaptive refinement strategy. Stainko [48] carried out an adaptive scheme which only refined interface elements and added filters to maintain the stability of solutions. Guest and Genet [49] performed continuous optimization using adaptive design variables with genetic algorithms while preserving the finite element mesh. Bruggi and Verani [50] proposed a fully adaptive algorithm for triangular elements that interface elements are detected by two kinds of error estimators. However, the majority of above mentioned schemes in adaptive refinement field are valid for triangular or quadrilateral elements. Both of these element types are still recognized for both their advantages and drawbacks in computational engineering. Although triangular elements are widely employed in the numerical methods and applied science due to their easy generation, their convergent rate in many problems is suboptimal [51]. Conversely, quadrilateral elements always get high precision in a large number of problems because of higher order approximation in the basis function, but it is still hard to get purely or regularly quadrilateral meshes when the geometry becomes highly complicated. The polygonal element is an alternative selection which achieves the balance between the meshing cost and computational accuracy. In addition, the polygonal element can reach stable solutions even in nonlinear finite element analysis without any additional handling of entities [51]. It has also been utilized to replace traditional element types in many special analyses such as computational fracture mechanics [52], [53], [54], nonlinear behavior of polycrystalline materials [55], [56], [57] and incompressible materials [58]. In the common framework of topology optimization, many researchers have published a large number of significant contributions for polygonal mesh-generators [59], [60], efficient topology optimization frameworks [61], [62], multiresolution scheme [33], [34], [63] and so on.

By taking the advantages of polygonal elements and quadtree decomposition, polytree meshes were firstly introduced to implement limit analysis of cracked structures and single-phase topology optimization in [64], [65] that was developed from the work on quadtree meshes suggested by Tabarraei and Sukumar [66]. Thereby, an arbitrary polygon with n edges can be recursively subdivided into (n+1) children polygons. Subsequently, the shape functions will be employed to acquire the approximations along element sides without any auxiliary technique to resolve the hanging node issues between neighbor elements. Galvanized by polytree meshes, we develop this algorithm for multi-phase structural topology optimization with several remarkable improvements to minimize the adaptive regions between different phases. Compared to our previous work [50], our key contribution in this study is how to manipulate efficiently both an improved error indicator and a new adaptive filter which allow us to decrease significantly the dimension of interface areas in multi-phase topology optimization. Therefore, the computational cost of the overall framework is appreciably reduced in comparison with the previous adaptive polytree mesh algorithm.

The remaining of this paper is organized as follows. The polygonal finite element approach for multi-phase topology optimization problems with modified filtering technique is formulated in Section 2 which focuses on compliance minimization. An elemental selection criterion and adaptive procedure using a polytree are presented in Section 3. Section 4 shows several numerical examples to examine the applicability of the proposed method. Section 5 summarizes and closes our paper with some assertions.

Section snippets

A polygonal finite element method (PFEM) for topology optimization

The physical domain Ω is decomposed into computing (approximate) domain Ωh by a polygonal mesh generator. The primeval mesh then contains nn nodes, ne elements, ned edges. Due to the unpredictable property of the polygonal element shape, it is quite hard to define the basis functions Nx in finite dimension space Vh. Up to now, many barycentric coordinate theories have been proposed to construct the basis functions for polygonal elements[67], [68], [69], [70], [71], [72], [73], [74], [75], [76]

Implementation of adaptive refinement on polytree mesh

In general refinement strategies, the error indicators and an adaptive mesher are two agents which decide the performance of a topology optimizer. An error indicator using the convergent results from coarser mesh level is utilized to clarify the elements to be refined. Then the poly-mesher would be called to discretize these elements into smaller cells for an optimization loop.

Numerical examples

In this section, six numerical examples are considered to illustrate the performance of the proposed approach. The efficient of adaptive filter and its mesh independence property are proved in the first example. Then the filter is utilized in a series of problem with many kinds of boundary condition and loading state. In the second and third examples, the single load-simple domain test cases are released as well-known topology problems. Subsequently, two cases with complex geometries are

Conclusions

We presented for the first time a novel adaptive filter in association with a polytree-based adaptive mesh strategy for multi-material topology optimization problems. It has perfectly combined adaptive topology optimization into AAPA as a very efficient adaptive approach for many designs. This scheme supplies smooth boundaries for optimal structures while reducing the computational cost compared with the regularly fine mesh and takes fewer essential iterations to gain the solution convergence

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      Citation Excerpt :

      However, convergent rates of triangular elements in terms of many problems are unsatisfactory (Chi et al., 2014). Meanwhile, quadrilateral elements could be very awkward when dealing with highly complex structures (Chau et al., 2017). The polygonal element has been used as a substitution recently due to its flexibility in mesh generation and high quality in computational result and convergence degree.

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