Adaptive Concurrent Topology Optimization of Coated Structures with Nonperiodic Infill for Additive Manufacturing☆
Introduction
Recent advances in material science and high-precision printing technologies allow fabricating porous structures that are inspired by optimal materials in nature such as plant stems, butterfly wings and human bones. Topology optimization for AM becomes a significantly important tool in designing such porous structures that are fabricated by 3D printing techniques. The combination of topology optimization and 3D printing in the porosity-material field has been extensively studied in recent years, for example, honeycomb structures [1], [2], lattice structures [3], [4], and infill designs [5], [6]. Up to now, most multiscale topology optimization methods of porous structures have been conducted by traditional methods including solid isotropic material with penalization (SIMP) [7], [8], level set [9], [10], and evolutionary structural optimization (ESO) [11], in which optimal structures are implicitly described by element density fields (SIMP/ESO method) or level set functions (level set method). The traditional methods require the number of design variables equal to the number of grid elements/nodes. In addition to the aforementioned well-known methods, geometric component-based methods such as material mask overlay strategy [12], moving morphable components [13], geometry projection [14], and moving morphable bars [15] have been proposed for structural optimization and gained much interest from the research community recently. These methods are also known as explicit topology optimization methods that consider geometric parameters as design variables. The explicit methods possess outstanding advantages, e.g., using a few design variables (namely hundreds to thousands of parameters), explicitly presenting structural boundaries that allow accurately capturing optimal geometries for the final design extraction [13], easily controlling structural features [15], and simply modeling multilayer materials [16]. Although many explicit methods have been proposed for topology optimization of many homogeneous-material applications, e.g., coated structures [16], embedded components [17], self-supporting structures [18], and ribs-stiffened structures [19], [20], they have not been widely applied in the design of porous materials. Related to the design of porous materials using explicit methods, Liu et al. used perturbed basis functions to design graded material [21], Deng et al. employed a geometric component linkage scheme to ensure the continuity between neighboring microstructures of the infill [22], and Hoang et al. modeled and designed honeycomb-like structures by projecting macro-solid bars and micro-void circles onto the same analytical grid system [23].
Cellular materials are referred to as high-porosity materials with high strength to weight ratio [24], [25], [26], energy absorption [27], [28], and heat transfer enhancement [29]. Concurrent topology optimization of cellular materials is typically performed at multiple scales: at the macroscale, the overall structural topology is optimized and at the microscale, the configuration of microstructures is optimized. Sivapuram et al. proposed an approach for simultaneous material and structural optimization by assigning the positions of microstructures for local areas of the design domain [30]. Xia and Breitkopf simultaneously designed the macrostructure and microstructures using nonlinear multiscale analysis [31]. Li et al. employed multi-patch microstructures for concurrent design [32]. Concurrent designs were also investigated under mechanical and thermal loads [33], nonlinear structures [34], and frequency responses [35]. In the concurrent topology optimization, the macrostructure is discretized with finite macro-elements/cells (microstructures) that need to be independently designed with finer meshes. Connector and local volume constraints may be necessary to ensure the continuity of neighboring macro-elements and constraint the local material volume in the microstructures. The homogenization technique [7], [36], [37] is often used to approximate the material properties of macro-elements. FEA and design variable updates are required at both macro- and microstructural levels.
The majority of current multiscale topology optimization methods [30], [31], [32], [33], [34], [35] only focus on designing the internal architectures of the infill rather than the overall structure including the coating and infill. No coating skin is coupled with the macrostructure. In many cases, however, porous materials are often protected by an outer thin skin/shell which plays the main load-bearing part. Hence, it is necessary to propose a multiscale design method that concurrently optimizes the overall structure including the external skin of the macrostructure and the internal architectures of the microstructures. Although a few topology optimization methods of coated structures have been proposed recently, such as using a two-step filtering approach [38], coating filter [39] and two distinct level sets of a single level set function [40], [41], they are just mono-scale designs, of which the coating and infill are assumed to be separate solid materials. The skin in these studies is defined with uniform thickness by using filtering techniques or level set functions. A framework of moving morphable sandwich bars proposed by Hoang et al. [16] allows designing coated structures with flexible length scale control of the coating and infill. Also like [38], [39], [40], [41], the coating and infill are also assumed to be separate solid materials. This assumption limits design freedoms and structural performance.
More recently, multiscale topology optimization methods of coated structures with orthogonal infill have been proposed, allowing us to design the cellular structures with higher design freedoms. Wu et al. [42] used multi-step filtering and projection approaches, and local volume constraints to design shell-infill structures with non-uniform infill. These approaches were later combined with homogenization-based optimization by Groen et al. [43] to design coated structures with orthotropic infill. Non-linear filters were introduced to realize the coating layers and length scale control of multiscale design with solid coating and periodic infill patterns [44]. The shell and periodic infill were realized by two distinct level sets of a single level set function [45].
In this work, we address an adaptive concurrent topology optimization method for cellular structures with the solid coating and nonperiodic infill without material homogenization at microscale, using AGCs including a framework of macro-sandwich bars and a network of micro-solid bars. The AGCs are mapped onto a fixed element grid with three auxiliary density functions, which are combined to realize the coating and infill. The coated structure is simultaneously optimized at both macro-and microscales without material homogenization or any additional constraints by straightforwardly optimizing a set of geometric parameters of the AGCs. Because of not using the homogenization, it is possible to avoid inherent errors in homogenizing material especially when the macro-element width is relatively large compared with the overall structure. In addition, local connector and volume constraints are often required in other methods but not necessary for the proposed method.
Section snippets
AGCs for coated structures with non-periodic infill
In this investigation, AGCs are geometrically modeled to realize the coating and infill of a cellular structure. The AGCs consist of a framework of macro-sandwich bars and a network of micro-solid bars, which can morph and operate logical boolean in the design domain to shape optimal structures with the separate coating and non-periodic/periodic infill material. These components are projected onto two separate element density fields using an adaptive mapping technique, as briefly described
Optimization formulation
The geometric parameters of AGCs including the parameters of macro-sandwich bars and micro-solid bars are considered as design variables. The design variable vector consists of macro- and micro-parameters, ; is a set of the macro-parameters, ; and is a set of the micro-parameters, .
The optimization formulation for minimum structural compliance problems considered in this study can be expressed as
Numerical examples
In this section, we perform several numerical experiments to verify the effectiveness of the proposed method. For all numerical problems in this paper, we used the design parameters , , , and . It was assumed that the infill and coating had the same Poisson’s ratio . The design domain was discretized with square plane-stress elements. The upper/lower bounds for the coordinates of macro-parameters were set so that the coordinates of macro-sandwich bars moved within the
Conclusion and discussion
In this work, we modeled AGCs with geometric parameters for direct multiscale topology optimization of cellular structures with solid coating and nonperiodic infill. The cellular structures are simultaneously optimized at both the macro- and microscales without material homogenization. The proposed method allows modeling coated designs with porous infill on a coarse finite element grid and explicitly presenting the boundaries of both the macro- and microstructures.
The macrostructure, including
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.
Acknowledgments
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.01-2019.317. The last author would like to thank the supports from the the Alexander von Humboldt Foundation for a renewed research stay in Germany. The authors acknowledge the facilities and technical assistance of the RMIT Advanced Manufacturing Precinct and Vietnam Maritime University.
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This paper has been recommended for acceptance by Prof M Wang.