Novel Techniques for Non-Convex Optimization and Time-Domain Simulations of Electromagnetic Systems

This dissertation explores novel solutions to two fundamental problems with broad implications: explicit time-domain simulations of partial differential equations in unstructured meshes and non-convex optimization.

The first enables the fast simulation of complex geometries and materials in the time domain. Previously proposed unstructured finite-difference-based methods suffered from instability or limited accuracy. The inverse approximation and mass lumping of finite-element-based methods also compromise their accuracy. Matrix-free discrete-exterior-calculus-based methods impose stringent requirements on mesh generation to allow a naturally diagonal discrete Hodge star operator. Volumetric interpolating basis matrix-free time-domain methods employed convoluted schemes for perfectly matched layer truncation and modeling of anisotropic material properties. A faster and geometrically consistent patch-based construction of the curl operators for arbitrary unstructured meshes is proposed to overcome these challenges. A novel time-domain discretization for unsymmetrical systems is set forth with theoretically proven accuracy and stability properties. Within the proposed framework, modeling of complex media, perfectly matched layer truncation for open domain simulations, and lumped elements are studied and successfully realized.

The second part of this dissertation presents a novel rank-revealing algorithm designed to solve non-convex optimization problems with embarrassingly parallel function space exploration.

The resultant low-rank representation of function spaces describes the underlying system's unique responses and the points where such unique responses occur.

By evaluating system responses at these rank points, a function's global maximum, minimum, and other features can be identified rapidly.

The algorithm sampling efficiency is further enhanced with a dynamic sampling scheme.

The proposed optimization algorithm's efficacy is demonstrated in the most challenging synthetic functions and real-world optimization problems related to electronic packaging for heterogeneous integration and antenna design.

Furthermore, surrogate assistance with machine learning methods and multi-objective optimization under the proposed framework are developed and demonstrated.

Finally, a custom optimizer for arbitrarily shaped functional blocks in floorplanning during integrated circuit physical design is proposed. The floorplanning paradigm combines multiple optimization concepts to manipulate the shape of functional blocks in the fixed outline of a chip, significantly outperforming state-of-the-art methods.

The outcome of this thesis work enhances and accelerates the modeling, simulation, and optimization of a broad range of engineering problems.