title: On structure preserving simulations in nonlinear electromagnetics, electric circuits, and efficient treatment of systems with memory
Creators: name: Shashkov, Vsevolod
Publisher: name: UNSPECIFIED
Year: 2024

description: This thesis is dedicated to the modelling and numerical treatment of electromagnetic
phenomena governed by (nonlinear) field and circuit equations, which are the fundamental topics in electrical engineering. The main focus is on energy transformation principles and the construction of discretization schemes that preserve these principles. While the numerical treatment of linear problems has been extensively studied in the literature over the years, a systematic treatment of nonlinear problems is not yet fully established.

In the first chapter of the thesis, Maxwell’s equations in nonlinear media are discussed.
We consider an energy-based modelling approach for material description and present two formulations that lead to systems of certain generalized (port-) Hamiltonian and gradient structures. To preserve the underlying structure, we employ variational techniques based on Galerkin approximations in space and discontinuous- or Petrov-Galerkin methods in time. This approach allows a systematic construction of higher-order schemes based on implicit time-stepping. The discrete energy balance can be derived under relatively general assumptions. For energy-conserving systems, the two approaches enable the construction of dissipative and energy-conserving schemes, ensuring the passivity of the discretizations.

The second chapter is dedicated to electric circuit problems. The state-of-the-art approach to modelling electric circuits is Modified Nodal Analysis (MNA). This formulation leads to differential-algebraic systems with an index of ν ≤ 2, which presents challenges in the analysis and numerical treatment. We present an alternative magnetic-oriented nodal analysis formulation (MONA) that results in differential-algebraic systems with an index ν ≤ 1, which is much simpler to handle. We demonstrate that both formulations result in finite-dimensional systems of a certain port-Hamiltonian or gradient structure, similar to the field problems. Therefore, variational time-stepping methods can again be utilized to construct passivity-preserving higher-order time-discretization schemes.

In the last chapter, the systems with memory described by a Volterra-integro-differential equation are considered. Such systems arise in the context of dispersive media or reduced order models for field circuit coupling. The numerical treatment of such problems requires an efficient realization of the integral term in an evolutionary manner. After an appropriate discretization, the Volterra integral term can be interpreted as a matrix-vector product with a densely populated matrix. For a sufficiently fine approximation, the size of the system becomes large, leading to storage and complexity issues. We present a fast, oblivious, and evolutionary algorithm based on the H2−matrix compression technique. The approach can be applied to Volterra integrals of convolution type. Further, it shares some similarities with the fast and oblivious quadrature methods of Schädle, Lopez-Fernandez, and Lubich. The latter can be interpreted as a particular realization of the H-matrix approximation.