Three-field floating projection topology optimization of continuum structures

https://doi.org/10.1016/j.cma.2022.115444Get rights and content

Highlights

  • A three-field floating projection topology optimization method is proposed.

  • The algorithm can use the linear material interpolation scheme.

  • The algorithm is extended straightforwardly for robust formulation.

  • A simple approach for the topology optimization of shell-infill structures is proposed.

Abstract

Topology optimization using the variable substitution among three fields can achieve a design with desired solid and/or void features. This paper proposes a three-field floating projection topology optimization (FPTO) method using the linear material interpolation. The implicit floating projection constraint is used as an engine for generating a 0/1 solution at the design field. The substitution filtering and projection schemes enhance the length scale and solid/void features to accelerate the formation of structural topology in the physical field. Meanwhile, the three-field FPTO method can be extended to robust formulation, which obtains the eroded, intermediate, and dilated designs with the same topology. The most distinct feature of the FPTO method lies in the adoption of the linear material interpolation scheme, which makes many topology optimization problems straightforward. As an example, the proposed three-field FPTO algorithm is further applied to the design of shell-infill structures using the linear multi-material interpolation scheme. The distribution of the shell material is generated through a simple filtering scheme, and the shell thickness is accurately controlled by the filter radius. Numerical examples are presented to demonstrate the effectiveness and advantage of the proposed three-field FPTO method.

Introduction

Topology optimization is a mathematical approach to achieving the best distribution of materials within the design domain so that the resulting structure maximizes its performance [1]. The mathematical formulation for a topology optimization problem includes three key components: objective functions, constraint functions, and design variables, which remain unchanged during optimization unless some assumptions are made. Among them, the definition of design variables is the most important one and divides the topology optimization method into two categories: the boundary-based approach and the element-based approach.

The boundary-based topology optimization approach defines design variables at the structural boundary, and structural topology can be naturally formed and updated through the boundary evolution in an iterative manner. The well-established boundary-based topology optimization approach includes the level-set (LS) method [2], [3], [4], moving morphable components (MMC) method [5], [6], the feature-driven optimization method [7], and other LS variants [8], [9]. However, digging new holes in the boundary-based topology optimization approach means the addition of new design variables and is rigorously prohibited. Therefore, the boundary-based topology optimization approach may depend on the initial guess design, e.g., the number and location of initial holes, and the adopted regularization technique [10], [11]. Instead, the element-based topology optimization approach provides a convenient way to dig new holes as design variables are defined over the whole design domain.

The typical topology optimization problem, e.g., compliance minimization, for the element-based topology optimization approach can be naturally stated as follows [12], [13]: min:C=fTus.t.:f=KuVf=exeVeeVeV¯fxe=0or1Where C is structural compliance. f and u are the force and displacement vectors, and K is the global stiffness matrix. Vf and V¯f are the volume (or weight) fraction of the structure and its constraint value. xe(e=1,2,) is the design variable and xe=0 or 1 denotes two different statuses of an element, void or solid, respectively. It can be seen that digging a new hole requires assigning the corresponding design variables with the value of 0 only, without any need to add or delete any design variables. In some cases, the design variables can be defined on nodes [14] or unstructured points [15], but this does not change the nature of the problem.

The main challenge for solving a topology optimization problem lies in the discrete nature of the design variables. A natural approach for this problem is a continuous relaxation, and the relaxed topology optimization problem can be stated as [12] min:C=fTus.t.:f=KuVf=exeVeeVeV¯fxminxe1where xmin is a small positive value, e.g., 10−3, to avoid the singularity. One should note that the relaxed continuous problem is not naturally equivalent to the original discrete problem. There are physical and numerical ways to make them equal.

In a physical way, the solid isotropic material penalization (SIMP) method [12], [16], [17], following the homogenization method proposed by Bendsœand Kikuchi [1], was established by assuming the relaxed xe as the density (ρe) of isotropic porous materials. The relationship between the material properties such as Young’s modulus (E) and the density could follow the power law or other material penalization schemes [18], [19]. The principle of employing a material penalization scheme is to make intermediate density less efficient than solid and void. So that the optimization would physically drive the solution toward either solid or void. To this end, the resulting 0/1 solution of the relaxed problem is also the solution of the original discrete problem, and the trivial 0/1 constraints of design variables can be naturally ignored. Given the premise of the existence of a 0/1 solution under material penalization, the bi-directional evolutionary structural optimization (BESO) method [13], [20], [21], [22], [23] introduced a unique way to achieve the 0/1 solution by using discrete design variables only. The concept of material penalization was also employed implicitly or equivalently in other topology optimization methods [24], [25], [26], [27]. Physically, the low efficiency of intermediate density (E/ρ) is valid in terms of compliance minimization under the volume constraint. So far, topology optimization has been applied to many engineering fields for various problems with different objective and constraint functions. However, the physical nature of the material penalization for those new problems was rarely explored, and the optimized solution could depend on the selection of the material penalization scheme [28], [29]. In such cases, the material penalization scheme becomes a heuristic way to achieve a 0/1 solution, whose difference from the one of the original 0/1 problem remains mysterious.

Compared with the physical way, the numerical way is more straightforward by imposing 0/1 constraints of design variables for the relaxed problem. However, a large number of 0/1 constraints makes the problem unsolvable. Instead, Huang [30], [31] proposed the floating projection topology optimization (FPTO) method, where 0/1 constraints of design variables are simulated by an implicit floating projection constraint. Such a numerical way can conveniently ensure a 0/1 solution by using any material interpolation scheme, even the linear one. Apart from frequency optimization and compliant mechanism [30], the FPTO method has demonstrated its advantages for the design of chiral metamaterials [32], where the maximization and minimization of the objective function are equivalent except for opposite handedness, and the design of acoustic-mechanical structures [33] to achieve the numerical consistency between the mixed p/u formulation and the segregated formulation. Furthermore, 0/1 constraints of design variables for multiple materials can be simulated by multiple floating projection constraints [34]. The study overcame a challenging issue in designing multi-material structures where one multi-material penalization interpolation equation hardly ensures a 0/1 solution of multiple design variables [35], [36], [37], especially when the number of materials increases.

The three-field SIMP method [38], [39], [40], [41] introduced the design field (x¯e), the filtered field (x̃e), and the physical (or projected) field (xe), as shown in Fig. 1. Instead of directly solving the problem in the physical field, topology optimization seeks the optimal solution in the design field by using the concept of variable substitution. Due to the definitions of substitution functions, the final physical design has desired features, including eliminating thin members and small holes, and clear 0/1 characteristics, as demonstrated in Fig. 1. The three-field method can be extended further to robust formulation to achieve the eroded, intermediate, and dilated designs simultaneously and control the minimum length scale of solids and voids [42], [43], [44]. In summary, the three-field method and its extensions have extensively promoted the capability of topology optimization techniques in achieving desired features for easy fabrication or other purposes.

Although the formation mechanism of structural topology in the FPTO and the SIMP methods is different, as explained above, both methods belong to the element-based topology optimization approach. Thus, one critical question is whether the FPTO method can also be developed using three fields to achieve desired solid and void features, similar to the three-field SIMP method. Meanwhile, the FPTO method can use the linear material interpolation scheme, and whether this feature brings us a simple way to solve new topology optimization problems? With these questions in mind, the rest of the paper is organized as follows. Section 2 will develop a three-field FPTO method and illustrate its numerical implementation procedure. Section 3 will extend the three-field FPTO method to robust formulation to show its capability of obtaining the same topology for the eroded, intermediate, and dilated designs. Section 4 will apply the developed topology optimization algorithm to the design of shell-infill structures and demonstrate the simplicity of generating the shell with a specified thickness. Finally, some conclusions are drawn.

Section snippets

Three-field FPTO method for compliance minimization

This section proposes a three-field FPTO method for the compliance minimization problem by using a linear material interpolation scheme. The numerical implementation of the proposed topology optimization algorithm is illustrated in detail, and some numerical examples are presented to demonstrate the effectiveness of the proposed algorithm.

Robust formulation for the eroded, intermediate and dilated designs

The three-field FPTO method accelerates the formation of structural topology through the substitution filtering and projection schemes. In terms of optimized solutions, the three-field FPTO method has no particular advantages over the one-field FPTO method. This is since the one-field FPTO method already considers the length scale and 0/1 features of the solution by the filter and floating projection constraints. In the three-field FPTO, those features formed in the design field are just

Topology optimization of shell-infill structures

The most distinct feature of the FPTO method lies in the utilization of the linear material interpolation scheme, which makes many topology optimization problems simple. In the section, we extend the three-field FPTO method for topology optimization of shell-infill structures [48], [49], [50], [51], [52], [53], where the stiff shell material uniformly coats the soft infill material with a specified thickness, and show the simplicity of the algorithm.

Conclusions

When a topology optimization problem formulates with a linear material interpolation scheme, it cannot be solved by the classical element-based topology optimization approaches, such as SIMP or BESO. Nevertheless, the linear material interpolation scheme makes many topology optimization problems straightforward, especially when the physical nature of the traditional material penalization is unclear. This paper proposes a three-field FPTO method using the linear material interpolation scheme.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The author wishes to acknowledge the financial support from the Australian Research Council (DP210103523).

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