Topology Optimization for Cellular Material Design
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Topology Optimization for Cellular Material Design Reza Lotfi, Seunghyun Ha, Josephine V. Carstensen and James K. Guest Johns Hopkins University, Civil Engineering Department, 3400 N Charles St., Baltimore, MD 21218-2682, U.S.A. ABSTRACT Topology optimization is a systematic, computational approach to the design of structure, defined as the layout of materials (and pores) across a domain. Typically employed at the component-level scale, topology optimization is increasingly being used to design the architecture of high performance materials. The resulting design problem is posed as an optimization problem with governing unit cell and upscaling mechanics embedded in the formulation, and solved with formal mathematical programming. This paper will describe recent advances in topology optimization, including incorporation of manufacturing processes and objectives governed by nonlinear mechanics and multiple physics, and demonstrate their application to the design of cellular materials. Optimized material architectures are shown to (computationally) approach theoretical bounds when available, and can be used to generate estimations of bounds when such bounds are unknown. INTRODUCTION As the ability to control material architecture continues to improve in quality and decrease in length scale, an increased emphasis is being placed on improving design to leverage this processing control. The key questions are: (1) how does material architecture influence bulk material properties defined at the macroscale? (the forward problem); and (2) what is the material architecture(s) that optimize these effective properties? (the inverse problem). Here we focus on the inverse design problem and illustrate how topology optimization, a free form approach to designing architecture, can be used to explore challenging inverse problems. Topology optimization is a systematic approach to optimizing the distribution of materials across a design domain, which for periodic material architectures is the characteristic unit cell (Figure 1a). As it does not require an educated guess, topology optimization is capable of identifying new, high performance architectures. A typical strategy is to discretize the design domain with finite elements (Figure 1b) and then determine whether each finite element contains material or is a void (for two-phase solid-void design) (Figure 1c). This binary condition on design variable, however, is relaxed to allow use of gradient-based optimizers and some form of penalization is used to give preference to 0-1 solutions. The solutions contained herein, for example, use the well-known SIMP [1] penalization for solid mechanics, Darcy-Stokes interpolation [2] (and penalization) for fluid mechanics, and Method of Moving Asympotes [3,4] for the gradient-based optimizer. While homogenization approaches for upscaling unit cell response to bulk effective properties are well established for elastic properties, inverse homogenization via topology optimization is relatively new idea, first achieved for negative Poisson’s ratio ma
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