Two-scale concurrent topology optimization of lattice structures with connectable microstructures

Abstract

Two-scale concurrent design of lattice material microstructures and their macroscale distributions can effectively enlarge the design space, and thus achieve lightweight structures with desirable mechanical and multi-physics performances. Most of the existing inverse homogenization-based design methods do not take into consideration the connectivity issues of the microstructures. In practice, this may hinder the manufacturing and application of the optimized two-scale designs. To handle this issue, the present paper proposes a designable connective region method to obtain connectable microstructures in the context of two-scale concurrent structural topology optimization, considering structures composed of lattice materials with repetitive unit cells and prescribed porosity. On the microscale, the microstructures topologies are represented by the density model, and the effective material properties are computed with the homogenization method. On the macroscale, the distribution of different lattice materials is described with the discrete material optimization method, which can effectively reduce the amounts of macroscale elements with intermediate densities. The connectivity between any two types of the microstructures is naturally ensured by introducing pre-defined connective regions in the microstructural unit cells and keeping these regions sharing the same topology. This method can be conveniently implemented through microscale design variable linking and requires no evaluation of extra connectivity constraints and the corresponding sensitivities. It is exemplified by two-scale concurrent structural topology optimization for compliance minimization problems in two- and three-dimensional design domains. Numerical results show that this method is able to generate connectable lattice structures, which exhibits improved stiffness as compared with their uniform-lattice counterparts.

Introduction

Two-scale structures composed of lattice materials have great potentials in achieving lightweight property and desirable mechanical/multi-physics performances, such as high specific stiffness [1], energy absorption [2], and heat exchange capability [3]. For instance, lattice structures with periodic unit cells can provide both load-bearing and active cooling functionalities [4]; biodegradable scaffolds composed of lattice materials are required to possess certain porosity for nutrient delivery while still maintaining reasonable stiffness to support the tissues [5,6]. Particularly, the geometry and spatial distribution of the microstructures, and the topology of the global structure are crucial to the overall structural performance. However, the systematic design of such multiscale lattice structures is still a challenging task because of their high geometrical complexity.

Topology optimization [[7], [8], [9], [10], [11]] is recognized as a highly effective method to obtain the optimal structural layouts with desired performances. Extensive researches on topology optimization-based lattice microstructural design have been carried out. For instance, Sigmund and Torquato [12] achieved material designs with positive, zero and negative thermal expansion with multiple base materials. Larsen et al. [13] designed and fabricated compliant micromechanisms and materials with a negative Poisson ratio. Sigmund [14] optimized material microstructures to obtain prescribed mechanical constitutive parameters. Guest and Prévost [15] achieved multifunctional material design with maximum stiffness and fluid permeability. Kazemi et al. [16] studied the design of the multi-material truss lattice materials.

A hot topic that has attracted increasing attention is to use topology optimization to design high-performance multiscale lattice structures. Therein, both the macroscale and microscale geometries are to be simultaneously optimized under the assumption of scale separation. Rodrigues et al. [17] proposed a hierarchical optimization procedure to iteratively design macroscale structure with point-wise different microstructures. Liu et al. [18] developed a so-called Porous Anisotropic Material Penalization (PAMP) method to interpolate anisotropic effective properties in the stiffness maximization problem. The PAMP method can be used to design the distribution of structured materials. It decouples the topology description of the macroscale structure and the microstructure and enables simultaneous optimization of the material distribution and the microstructural topologies. On the basis of nonlinear multiscale analysis, Xia and Breitkopf [19] proposed an optimization scheme to obtain microstructure topologies. Design schemes concerning the grouping of lattice materials have been studied by Li et al. [20] and Xu and Cheng [21]. Besides the studies based on the scale separation assumption, some works also dealt with lattice structure design problems accounting for the scale effects [[22], [23], [24], [25]]. Numerical results have proven the performance benefits earned by the optimized multiscale structures.

Because of the rapid advance of additive manufacturing techniques [26,27], novel microstructure configurations with high geometric complexity can be readily realized [[28], [29], [30], [31]]. Though certain manufacturing limits such as the overhang constraints exist (see e.g. Gaynor and Guest [32], Langelaar [33], Liu and To [34]. and Wang et al. [35]), the combination of topology optimization and additive manufacturing [36,37] can greatly enlarge the design space of structures. Topology methods utilizing B-splines (see e.g [38,39]) have been proposed, which can effectively ease the difficulties when rebuilding the CAD models of the optimized designs. However, the above-mentioned inverse homogenization-based approaches may yield optimized microstructural unit cells that are not connectable, as shown in Fig. 1. Though such designs provide meaningful material configurations of a lattice structure, they cannot be manufactured in practice.

There have been a number of studies to resolve the microstructure connectivity issues when designing structures composed of graded lattice materials. Zhou and Li [40] compared three schemes to enforce the connectivity of microstructures. One of these schemes that is convenient to implement is to set up non-design domains in the microstructures. However, designating such non-design domains may pose a limitation to the design space and thus lead to sub-optimal microstructural geometries. Rodman et al. [41] proposed a progressive optimization scheme that optimizes three base unit cells at a stage to keep the connectivity of microstructures. Garner et al. [42] considered the assembly of adjacent cells along with the optimization of individual cells to enforce the microstructure connectivity, which may demand a large computational cost in 3D cases. Besides the above-mentioned methods, some constraints on the design variables have also been proposed to achieve microstructure connectivity: Alexandersen and Lazarov [22] imposed constraints on PDE-filtered densities on the boundaries between layers of graded lattice materials to generate connectable microstructures; Du et al. [43] proposed a so-called “connectivity index” constraint on the basis of the level set method for fixed distributions of lattice materials. For such methods, a large number of constraints, as well as their design sensitivities, need to be evaluated if the number of considered lattice materials is large. Design methods on the basis of metamorphosis techniques to enforce the microstructure connectivity have also investigated. For instance, Wang et al. [44] proposed an optimization framework to obtain a series of connectable microstructures with similar topologies from a prototype. Zhou et al. [45] presented a post-processing scheme to enforce microstructure connectivity, which may introduce a performance deviation between the optimized design and the post-processed one.

Besides, some works based on the so-called de-homogenization concept aim to achieve near-optimal stiffness of lattice structures without restricting the repeatability of unit cells. Pantz and Trabelsi [46,47] and Allaire et al. [48] projected the designs obtained with the homogenization method to connectable solid/void designs with a finite length scale. Groen et al. [49,50] and Groen and Sigmund [51] simplified and further improved the method [46,47] and implemented control of the size and shape of the projected designs. With their method, connectable lattice designs with near-optimal stiffness can be obtained with relatively high efficiency, on the basis of the homogenization results and a postprocessing step to project them onto a finer mesh. The optimized designs obtained with these de-homogenization approaches usually have spatially varying porosities ranging from 0 to 1.

The present study focuses on the microstructure connectivity in the two-scale concurrent topology optimization of structures with multiple lattice materials. Specifically, we consider lattice materials that are composed of repetitive unit cells with specified porosity (which is beneficial for possible applications with multi-physics requirements such as active cooling). In this context, as the distribution of the lattice materials on the macroscale is not known a priori but is updated during the optimization process, the locations (i.e. the macroscale multi-material interfaces) to enforce microstructure connectivity keep changing. To resolve the microstructure connectivity issue under this circumstance, this paper proposes a designable connective region method, which ensures the microstructural connectivity by keeping the connective regions in all types of microstructures sharing a common topology. This is conveniently achieved through microscale design variable linking and avoids imposing any connectivity constraints. Therein, the microstructure topologies are described with the Solid Isotropic Material with Penalization (SIMP) model [52], and their effective material properties are computed with the asymptotic homogenization method. The macroscale material distribution is described with the Discrete Material Optimization (DMO) method [53]. For both the macroscale and microscale design problems, the density filter technique [54] is adopted to avoid the checkerboard and other numerical instability in the optimization process. The sensitivities of the objective function with respect to the design variables are derived using the adjoint method, and the optimization problem is solved with the mathematical programming method MMA [55].

The present paper is organized as follows: first, the proposed microstructure connectivity method is introduced in Section 2. Then the asymptotic homogenization method employed to compute the effective material properties, the topology optimization formulation, and sensitivity analysis procedures are discussed in Section 3. Numerical examples are given in Section 4, which is followed by conclusions.