Topology Optimization of Single- and Multi-material Structures: From Single-physics to Multi-physics Designs

Topology optimization is a computational design method used to find the optimized geometry of materials or structures meeting some performance criteria while satisfying constraints applied either globally (as usual) or locally (a focus of this work). Topology optimization can be used, for instance, to find lightweight structures that safely carry loads without failing. All you need is a design objective (e.g., minimize the weight) and constraints (e.g., material strength) and, through a nonlinear programming technique, the computer explores the solution space to find the optimized design. Despite the design freedoms afforded by topology optimization, its widespread adoption has primarily been hindered by the inability of current formulations to efficiently handle problems involving, for instance, multi-physics, multiple materials, and local material failure constraints. Thus, this thesis contributes to theoretical formulations, computer algorithms, and numerical implementations for topology optimization with an emphasis on problems subjected to either global constraints (e.g., energy-type constraints) or local constraints (e.g., material failure constraints), and for applications involving single or multiple physical phenomena and single- or multi-material designs. This work can be divided into two parts. In the first part, we present a general multi-material formulation that can handle an arbitrary number of materials and volume constraints (i.e., global-type constraints), and any type of objective function. To handle problems with such generality, we adopt a special linearization of the original optimization problem using a non-monotonous convex approximation of the objective function written in terms of positive and negative components of its gradient. The outcome is a scheme that updates the design variables associated with one constraint independently of the others, leading to an efficient, parallelizable formulation. The new update scheme allows us to design multi-phase viscoelastic microstructures, thermoelastic structures, and structures subjected to general dynamic loading. In the second part of this thesis, we introduce an augmented Lagrangian formulation to solve problems with local stress constraints correctly—a dilemma that has been unresolved thus far. First, we create a formulation to solve stress-constrained problems both for linear and nonlinear structures and provide an educational open-source code aiming to bridge the gap between research and education. Next, to extend the range of applications to structures that can be made of materials other than ductile metals, we introduce a function that unifies several classical strength criteria to predict the failure of a wide spectrum of materials, including either ductile metals or pressure-dependent materials, and use it to solve topology optimization problems with local stress constraints. We then extend the framework to time-dependent problems and address stress-constrained problems for structures subjected to general dynamic loading, in which the stress constraints are satisfied both in space (i.e., locally at every point of the discretized domain) and time (i.e., throughout the duration of the dynamic event). Unlike most work in the literature, this augmented Lagrangian framework leads to a scalable formulation that solves the optimization problem consistently with the local definition of stress and handles thousands or even millions of constraints efficiently. In summary, all components of this work are aimed to address critical challenges that have prevented topology optimization from being embraced as a practical design tool for industry-relevant applications.Ph.D