Design optimization of interconnected porous structures using extended triply periodic minimal surfaces
Introduction
Porous structures appear extensively in natural objects such as bones, wood, cork, and bird beaks [1]. These structures manifest exceptional properties at relatively low densities, such as shock resistance [2], damping enhancement [3], and defect tolerance [4]. The supreme properties of porous structures arise from their complex geometries and topologies, particularly the structural heterogeneity and anisotropy, which have been tailored through thousand-year evolution given the competing constraints in nature [5]. These porous structures usually have fully interconnected pore networks within their interiors so as to enhance nutrition transfer or attachment, or to provide sufficient space for regeneration in tissue engineering [6]. The interconnection property is also mandatory when fabricating such porous structures by means of additive manufacturing, so that the unnecessary support structures during fabrication can be cleaned during postprocessing [7]. However, approaches for the efficient design of such interconnected heterogeneous porous structures have seldom been presented.
We study the problem of designing a heterogeneous interconnected porous structure while simultaneously maintaining a high degree of mechanical and structural rigidity. The macro-design mesh is assumed fixed in the study so that we have restricted the design space and thereby formulated a well-posed design optimization problem. The challenges of this problem lie in its proper geometric modeling, efficient performance simulation, and reliable design optimization.
In terms of the geometric modeling aspect, most conventional methods have mainly assumed a homogeneous element distribution, which only allows for rough shape tuning using parameters relating to the porosity and pore size. However, to utilize the design potentiality of porous structures maximally, it is highly necessary to model and control both the directional anisotropy and distributional heterogeneity. To achieve this goal, the concept of the extended triply periodic minimal surface (ETPMS) is introduced, whereby the structural volume is prescribed using an implicitly defined surface under shape control parameters. In contrast to its counterpart the triply periodic minimal surface (TPMS), the ETPMS exhibits additional design freedom along three axis directions, further enhancing the element anisotropy. Moreover, instead of using discrete constant design variables in each cell, the ETPMS controls the element distribution using certain continuous controlling parameter fields, thereby enabling smooth transition between adjacent elements. Consequently, the consistent implicit definition of the ETPMS is expected to offer significant advantages in terms of continuous connectivity, structural integrity, and the distinctive properties of porous structures; compared to conventional lattice- or strut-based architectures, the ETPMS does not have joints and struts along the body.
The modeling advantages of the ETPMS come at the cost of challenges in terms of its efficient simulation. Conventional approaches [8], [9], [10], [11], [12], [13], [14] generally simulated the performance of such biscale structures via numerical homogenization, which computes an effective elasticity tensor for each microstructure via finite element (FE) analysis and substantially reduces the overall simulation costs. However, the costs remain high as the homogenization process must be conducted for each microstructure during each optimization iteration. We resolve the issue using the strategy of offline pre-computed parametric homogenization, which explicitly expresses the material property as a function of the ETPMS design parameters and can thus instantly simulate the property of an ETPMS element at any specific parameter. It ultimately not only avoids the complex remeshing process, but also facilitates the process by building on a previous study on proper generalized decomposition (PGD) [15].
A further disadvantage lies in the effective and efficient optimization owing to the implicitly defined shape description. A set of validity constraints must be prescribed for each ETPMS element for the ETPMS-based porous structure to be solid and/or core connected. This induces a significant number of constraints for the design optimization problem and makes it difficult to derive a converged solution efficiently. The numerical techniques of constraint aggregation and adjoint-based sensitivity computation are introduced to resolve the issue. Ultimately, the proposed approach can produce smooth and heterogeneous porous structures with scaled anisotropy and fully interconnected pore networks for performance optimization, as demonstrated via a suite of three-dimensional (3D) benchmark examples.
The main contributions of this study are as follows: An approach is proposed using implicit surfaces to design heterogeneous and anisotropic porous structures with fully interconnected pore networks, as well as a systematic approach for the implementation. The concept of ETPMS is presented for modeling the interconnected heterogeneous porous structures with consistent anisotropy and heterogeneity control, and a sufficient condition on its geometric validity is provided. Offline parametric homogenization is conducted based on PGD to predict the properties of a range of microstructures with high accuracy. A specially tailored biscale optimization approach for the ETPMS, in combination with the techniques of offline pre-computation and sensitivity computation, is demonstrated via various 3D numerical examples.
The remainder of this paper is organized as follows: pertinent studies are briefly reviewed in Section 2. The concept of the ETPMS is proposed, and its analytical and geometrical properties are described in Section 3. Thereafter, the problem of designing a heterogeneous porous structure for physical performance optimization is formulated in Section 4. The approach for computing the parametric homogenized material properties using PGD is presented in Section 5. The overall numerical optimization approach is outlined in Section 6. The numerical results from benchmark examples are discussed in Section 7. Finally, we conclude the paper in Section 8.
Section snippets
Review of pertinent studies
The current study is closely related to porous structure modeling, the TPMS, biscale topology optimization, and model reduction. Existing work on these topics is briefly reviewed in this section.
Extended triply periodic minimal surface (ETPMS)
We first introduce the following notations used throughout the paper. Let be a fixed 3D macro-design domain consisting of cubic elements , , and and are the fixed boundary and loading boundary, respectively. Let be the spatial Cartesian coordinate of a point in Ω, and is the vector of control parameters defined on each cell ; I is the range of the parameters so that a connected ETPMS is generated. Accordingly, a set of
Problem statement and approach overview
We aim to optimize an interconnected porous structure using the c-ETPMS for performance optimization. The problem is first described within the context of the well-studied compliance minimization problem of linear elasticity analysis, which is followed by an overview of the approach.
Computing parametric homogenized material properties using PGD
The homogenized property of a microstructure inside a cell is determined by its configuration; specifically, the shape parameter . We construct the property explicitly in terms of by building on previous work on PGD [15].
Numerics behind optimization
The modeling advantages of the ETPMS come at the cost of challenges in terms of its efficient optimization of Problem in Eq. (15) owing to its implicitly defined and varied computational domain and the large number of validity constraints on the ETPMS cells. A numerical approach for resolving this issue is explained in this section.
Numerical examples
The approach was implemented in MATLAB 2018b, and run on a PC with a 3.6 GHz Intel Core i7 CPU and 16 GB RAM. Its performance was tested in various cases, as outlined in this section. For all cases, the base material was assumed to have a Young's modulus of 1 Pa and Poisson's ratio of 0.33.
Conclusions and future work
In this study, the concept of ETPMS was introduced to model heterogeneous and anisotropic porous structures via an implicit function of additional shape control parameters. The concept results in a perfectly smooth and fully connected porous network, but also comes at the cost of challenges in the effective simulation and optimization. We further resolved these issues using a combination of offline and online computations, particularly by using a PGD-based offline parametric homogenization
CRediT authorship contribution statement
Ming Li: Conceptualization, Methodology, Software, Visualization. Liangchao Zhu: Data curation, Software, Visualization. Jingzhi Li: Funding acquisition, Investigation, Methodology, Supervision, Writing - review & editing. Kai Zhang: Software, Validation, Writing - original draft, Writing - review & editing.
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.
Acknowledgements
The valuable comments from the anonymous reviewers are greatly appreciated. The work of M. Li is supported by the NSF of China (No. 61872320). The work of Li was partially supported by the NSF of China No. 11971221 and 11731006, the Shenzhen Sci-Tech Fund No. JCYJ20190809150413261 and JCYJ20170818153840322, and Guangdong Provincial Key Laboratory of Computational Science and Material Design No. 2019B030301001. The work of K. Zhang is supported by the NSF of China under the grant No. 11871245,
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