Multi-material topology optimization considering joint stiffness using a two-step filtering approach

Highlights

• A new multi-material topology optimization for considering joint stiffness between dissimilar materials is proposed.

• A modified discrete material optimization (DMO) approach for defining joint area is developed.

• The two-step filtering process for controlling the structural feature size and the joint thickness are adopted.

• Via numerical examples with various cases: joint thickness, material property, and mesh type, the proposed method is validated.

Abstract

In this paper, a new method is presented to concurrently determine the structural layout and joint interface during the multi-material topology optimization (MMTO) process. Although the development of additive manufacturing techniques allows the fabrication of multi-material structures for soft materials with graded properties, joint materials for joining metals or composites are still needed. This paper proposes a novel material interpolation scheme for defining the joint area between dissimilar materials using a two-step filtering process. The general MMTO process is performed in the first filtering step. Furthermore, the filtered variables generate the joint area, and the filter radius controls the thickness of the area in the second filtering step. A modified discrete material optimization (DMO) approach is developed to control the different materials independently and to expand the applicability of the method to cases with more than two design materials. To demonstrate the performance of the proposed method, a compliance minimization problem is formulated for various volume constraints, joint thicknesses, material properties, mesh types, and number of materials. To show the scalability, 3-D design and compliant mechanism design examples are adopted. Based on numerical examples, it was confirmed that the proposed method performs well in various cases; moreover, the results demonstrate that the concurrent designing of the structural layout and joint interface leads to better performance than when joint stiffness is not considered.

Introduction

Multi-material design using topology optimization to achieve a high performance and multiple functionalities such as damping structure [1] and four-dimensional (4-D) printing [2], has been employed successfully in various applications. The development of additive manufacturing (AM) enables the simultaneous fabrication of a multi-material structure [[2], [3], [4]]. However, there are some technical difficulties in fabricating dissimilar metallic or composite materials concurrently. Therefore, joining techniques are used for assembling components that are made with different materials [5,6]. Because the joint property on the interface between dissimilar materials is not simply the average of the properties of the structure material, the assumption of a perfect bonding condition in the early stages of the design process using multi-material topology optimization (MMTO) can negatively affect the performance of the final design. Although numerous studies have been conducted on MMTO, only a few have considered the interface or joint between materials.

To consider joint properties in MMTO, Yildiz and Saitou [7] first introduced an approach that considers a joint area in multi-component structural assemblies by defining the joint elements between structural elements. The genetic algorithm (GA) was used to determine the structural layout, and heuristic approaches were adopted to prevent unrealistic and infeasible results. Zhou and Saitou [8] expanded this concept based on the gradient method by using a solid isotropic material penalization (SIMP), such as an interpolation scheme with a membership variable where the density determines the existence of a material and the membership variables decide the type of material. Woischwill and Kim [9] adopted the additional joint element approach, but focused on the joint design by applying the sequential process of multi-material topology optimization. In this approach, a general MMTO is first performed on a cell element comprising one structure element and three joint elements; then, the joint MMTO design is performed on the joint elements of the interface area that is generated after the general MMTO process. Florea et al. [10] expanded this sequential process to the three-dimensional (3-D) realm, including tooling accessibility. This sequential process might achieve a stable convergence by not considering the layout design and joint location concurrently; however, design results obtained by considering joint stiffness dominate the first step of a general MMTO because the joint location appears only in the interface area. Besides, the structured joint element approach is not suitable for complex geometries. To overcome these limitations, Florea et al. [11] proposed a single-loop design process that defines the interface area during the MMTO process using the spatial gradient of the design variable. Chu et al. [12] first proposed a similar approach, in which the interface area is defined by a nodal gradient with a two-step filtering process [13]; however, the graded interface instead of the discrete joint property. In this multiphase SIMP [14] based approaches, however, the topological layouts of the joint design do not deviate significantly from the design results without joint material. In regard to the level set method, Vermaak et al. [15] presented the concept of a graded interface using a level set function characteristic. The material properties between the dissimilar materials varies depending on the distance from zero level set values by using the signed distance function for the re-initialization process. Liu et al. [16], using an extended finite element method (XFEM), proposed a method that can consider a cohesive model. Elements cut by the bonding interface of the zero level set are defined as having cohesive properties, and interface separation discontinuity is described via shape function modification using the XFEM.

This paper presents a new approach for considering the joint stiffness at the interface between dissimilar materials during the MMTO process. A density-based interpolation scheme with discrete material optimization (DMO) [17] is adopted instead of the multiphase SIMP (or multiphase approach) employed in Refs. [11,12]. Adopting the DMO approach means that multiple materials can be controlled by individual design variables and that the proposed method can be easily expanded to beyond four-phase materials [18]; in contrast, multiphase SIMP has generally been limited to less than four-phase materials because it entails a high complexity and recursive multiplications [17]. For defining the joint area, instead of using gradient calculations [11,12], a different form of interface definition that employs a projection-based approach with a two-step filtering process is proposed. This projection-based approach can express the joint area more easily and robustly than gradient methods because the interface area defined by the gradient is highly dependent on the result of the first filtering process [13]. Additionally, this paper compares the performance of the proposed method with that of the general MMTO, which manually replaces the interface with the joint material after optimization. Unlike the SIMP based approach, the proposed method provides significantly different design approaches while considering the joint stiffness. The remainder of this paper is organized as follows. In Section 2, we introduce the methodology for defining the joint area using a modified two-step filtering approach. The numerical test cases are presented in Section 3, and finally, our main conclusions are outlined in Section 4.