SEMDOT: Smooth-edged material distribution for optimizing topology algorithm

Highlights

• Smooth-Edged Material Distribution for Optimizing Topology (SEMDOT) algorithm is presented.

• Effects of Heaviside smooth and step functions on SEMDOT are investigated.

• The benefits and distinctions of SEMDOT compared to well-established element-based algorithms are shown.

• The source code of SEMDOT is provided.

Abstract

Element-based topology optimization algorithms capable of generating smooth boundaries have drawn serious attention given the significance of accurate boundary information in engineering applications. The basic framework of a new element-based continuum algorithm is proposed in this paper. This algorithm is based on a smooth-edged material distribution strategy that uses solid/void grid points assigned to each element. Named Smooth-Edged Material Distribution for Optimizing Topology (SEMDOT), the algorithm uses elemental volume fractions which depend on the densities of grid points in the Finite Element Analysis (FEA) model rather than elemental densities. Several numerical examples are studied to demonstrate the application and effectiveness of SEMDOT. In these examples, SEMDOT proved to be capable of obtaining optimized topologies with smooth and clear boundaries showing better or comparable performance compared to other topology optimization methods. Through these examples, first, the advantages of using the Heaviside smooth function are discussed in comparison to the Heaviside step function. Then, the benefits of introducing multiple filtering steps in this algorithm are shown. Finally, comparisons are conducted to exhibit the differences between SEMDOT and some well-established element-based algorithms. The validation of the sensitivity analysis method adopted in SEMDOT is conducted using a typical compliant mechanism design case. In addition, this paper provides the Matlab code of SEMDOT for educational and academic purposes.

Introduction

Topology optimization basically aims to distribute a given amount of material within a predefined design domain such that optimal or near optimal structural performance can be obtained [1], [2], [3]. It often provides highly efficient designs that could not be obtained by simple intuition without assuming any prior structural configuration. Topology optimization as a design method has been greatly developed and extensively used since the pioneering paper on numerical topology optimization by Bendsøe and Kikuchi [4]. A number of topology optimization algorithms have been proposed based on different strategies: homogenization of microstructures [4], using elemental densities as design variables [5], evolutionary approaches [6], topological derivative [7], level-set (LS) [8], [9], phase field [10], moving morphable component (MMC) [11], moving morphable void (MMV) [12], elemental volume fractions [13], and using the floating projection [14], [15]. In recent years, these topology optimization approaches have been applied in a wide range of distinct engineering problems, including frequency responses [16], [17], stress problems [18], [19], convection problems [20], [21], [22], structural failure problems [23], [24], large-scale problems [25], [26], [27], nanophotonics [28], metamaterial design [29], and manufacturing oriented methods [30], [31], [32], [33], [34] have been presented in recent years.

Early proposed topology optimization algorithms are mainly element-based such as solid isotropic material with penalization (SIMP) algorithm [35] and bi-directional evolutionary structural optimization (BESO) algorithm [36]. SIMP uses the artificial power-law function between elemental densities and material properties to suppress intermediate elements to a solution with black and white elements, and BESO heuristically updates design variables using discrete values (0 and 1). As elements are not only involved in finite element analysis (FEA) but the formation of topological boundaries, zigzag (for example, BESO) or both zigzag and blurry boundaries (for example, SIMP) will be inevitably generated. Therefore, shape optimization or other post-processing methods have to be used to obtain accurate boundary information after topology optimization [37], [38], [39]. Given the significance of the accurate boundary representation, some proposed algorithms such as the level-set method, MMC-based method, elemental volume fractions based method, and using the floating projection have successfully solved the boundary issue. Even though there are a number of algorithms capable of forming smooth or high resolution boundaries, element-based algorithms that could combine the benefits of different methods are generally preferred because of their advantages of easy implementation and ability of creating holes freely across the design domain and obtaining final topologies that are not heavily dependent on the initial guess [14].

Elemental volume fractions based methods are originally from the evolutionary topology optimization (ETO) algorithms using the continuation method on the volume and BESO-based optimizer [13], [40], [41], [42]. To provide a more easy-to-use, flexible, and efficient optimization platform, the authors proposed a new smooth continuum topology optimization algorithm through combining the benefits of the smooth representation in ETO and density-based optimization in SIMP [43], [44], [45], [46]. The proposed algorithm is termed Smooth-Edged Material Distribution for Optimizing Topology (SEMDOT) based on its optimization mechanism. The basic idea of elemental volume fractions based algorithms (for example, ETO and SEMDOT) is to pursue the solid/void design of grid points that are assigned to each element through which smooth boundaries can be obtained. The concept of using design points within an element has been proposed and studied before ETO and SEMDOT. Nguyen et al. [47] presented a multiresolution topology optimization (MTOP) scheme that can generate high resolution designs with relatively low computational cost using three different meshes: the displacement mesh, density mesh, and design variable mesh. In MTOP, Gauss points were used for the integration of the stiffness matrix. Afterwards, Park and Sutradhar [48] first used MTOP to obtain high resolution designs for 3D multi-material problems. However, Gupta et al. [49] pointed out that the complexity of MTOP would restrict its attractiveness to students and scholars. Instead of using design points within each element, Kang and Wang [50] proposed a pointwise density-based interpolation method where the density field is constructed from design points within a certain circular influence domain of the point. Unlike ETO and SEMDOT, design points in the method proposed by Kang and Wang [50] are not involved in forming smooth boundaries, so the non-smooth boundary issue persists in it.

Compared with some newly developed or improved algorithms capable of generating smooth boundaries, SEMDOT can be easily integrated with some existing methods that were established based on SIMP to achieve specific performance goals. An example is the combination of SEMDOT and Langelaar’s additive manufacturing (AM) filter [51] which can successfully generate smooth self-supporting topologies [43], [45], [52]. Other than Langelaar’s AM filter, some other strategies regarding AM restrictions proposed by van de Ven et al. [53] and Zhang et al. [54] can also be considered in SEMDOT for the support-free design. However, extra efforts have to be made for MMCs-based methods to obtain self-supporting designs [55]. In addition, the effectiveness of SEMDOT in solving 3D optimization problems is recently demonstrated by Fu et al. [46], through solving a number of benchmark problems and a comparison with a well-established large-scale topology optimization framework, TopOpt (proposed by Aage et al. [25]).

Even though the theoretical framework of SEMDOT was built and demonstrated by authors, other benefits and distinctions of SEMDOT compared with some current element-based algorithms have not been thoroughly discussed. Furthermore, details of SEMDOT algorithm and some of its subtle differences with methods like ETO which translate into more robust performance have not been discussed before.

In this work, the reason behind using the Heaviside smooth function in SEMDOT instead of the Heaviside step function that is extensively used in ETO algorithms is explained. Effects of different combinations of filter radii on performance, convergence, and topologies, which have not been discussed in authors’ previous works, are investigated, and the rationality of the sensitivity analysis strategy used in SEMDOT is proved through a compliant mechanism design case. Numerical comparisons with other element-based topology optimization methods are thoroughly conducted. Finally, the Matlab code of SEMDOT is released to the topology optimization community facilitating the replication of the results presented in this paper, and also for future use.

An overview of this paper is as follows. Section 2 explains the mathematical framework of SEMDOT. Section 3 conducts several numerical examples to exhibit the benefits and distinctions of SEMDOT compared with a number of element-based topology optimization algorithms. Concluding remarks are drawn in Section 4. Finally, the Matlab code of SEMDOT is presented in the Appendix.