SYSTEMATIC DESIGN OF COMPLIANT MORPHING STRUCTURES WITH STIMULUS AS DESIGN AND STATE VARIABLE

Abstract

Advances in additive manufacturing and synthesis of complex “responsive” materials whose properties can be altered through external stimuli are opening the door to a new generation of integrated devices and materials. While manufacturing such structures or materials has received a considerable attention (see for instance [72, 85]), their actual design remains challenging. Starting from the pioneering works of [74, 57, 52], topology optimization has established itself as a powerful tool for systematic design of micro-devices, Micro Electro Mechanical Systems (MEMS), or materials microstructures. Topology optimization aims to answer the question, what is the optimal distribution of materials in a ground domain in order to optimize a given objective function subject to some constraints? Mathematically, topology optimization is formulated as a PDE-constrained optimization, conventionally employing Finite Elements Methods (FEM) to solve the underlying PDE constraints. In this thesis, we study optimal design of responsive structures made of several materials, with at least one of the materials is responsive material, though topology optimization. The objective of the present work is to algorithmically find the distribution of materials in a ground domain that optimizes an objective function [26]. It is well-known that such problems are generally ill-posed (see [5] for instance) resulting in optimal designs consisting of an infinitely fine mixture of multiple materials. Homogenization approaches [36, 8, 5, 7] tackle this problem directly by extending admissible designs to such mixtures. This type of approach is mathematically well grounded and leads to well posed problems that can be implemented efficiently. However, it is often criticized for leading to designs that cannot be manufactured. Several other classes of techniques aim at restricting the class of admissible designs in such a way that avoids fine mixtures. The combination of material interpolation (SIMP) and filters [22, 28] is a commonly employed approach. Shape parameterization by level set functions [10, 12] also limits the complexity of designs. Finally, by penalizing the length (or surface) of interfaces between materials, perimeter penalization [16, 50, 67] also produces designs with limited complexity. Additionally, perimeter penalization can be efficiently implemented using a phase-field approach [29, 30, 82]. In this work, we propose a phase-field algorithm for the systematic design of responsive structures achieving prescribed deformations under some unknown distributions of a stimulus. Our focus is on linear elastic materials in which an external stimulus can generate an isotropic inelastic strain, similar to linear thermo-elastic materials. We begin by providing mathematical analysis of the problem and review classical optimal design methods and finally we detail the phase-field approach to optimal design. We introduce the responsive minimimum compliance problem of linear elastic structures. After giving the intricacies of this seemingly simple problem, we introduce the phase-field model to prove the existence of a solution and provide a numerical implementation. We then turn to the design of compliant morphing linear elastic structures. Here we begin by considering design of responsive structure that can move in a prescribed direction upon activation by a stimulus. We demonstrate the stregth of our approach by studying the optimal design of 2D structures consisting of void, one non-responsive material and one responsive material. Next, we explore the design of time-dependent compliant morphing linear elastic structures. Here we consider the stimulus to be a state variable controlled by the transient heat equations. We conclude by summarizing the presented work and discuss the its contribution towards the overarching goal of optimal design for responsive structure.