arXiv subjects

Herbert Egger

Publications and source records attributed to Herbert Egger.

At least 37 records · Page 2Linked to original sources

arXiv2024

On the convergence of higher order finite element methods for nonlinear magnetostatics

The modeling of electric machines and power transformers typically involves systems of nonlinear magnetostatics or -quasistatics, and their efficient and accurate simulation is required for the reliable design, control, and optimization of such devices. We study the numerical solution of the vector potential formulation of nonlinear magnetostatics by means of higher-order finite element methods. Numerical quadrature is used for the efficient handling of the nonlinearities and domain mappings are employed for the consideration of curved boundaries. The existence of a unique solution is proven on the continuous and discrete level and a full convergence analysis of the resulting finite element schemes is presented indicating order optimal convergence rates under appropriate smoothness assumptions. For the solution of the nonlinear discretized problems, we consider a Newton method with line search for which we establish global linear convergence with convergence rates that are independent of the discretization parameters. We further prove local quadratic convergence in a mesh-dependent neighborhood of the solution which becomes effective when high accuracy of the nonlinear solver is demanded. The assumptions required for our analysis cover inhomogeneous, nonlinear, and anisotropic materials, which may arise in typical applications, including the presence of permanent magnets. The theoretical results are illustrated by numerical tests for some typical benchmark problems.

math.NA↗

arXiv2024

A second-order structure-preserving discretization for the Cahn-Hilliard/Allen-Cahn system with cross-kinetic coupling

We study the numerical solution of a Cahn-Hilliard/Allen-Cahn system with strong coupling through state and gradient dependent non-diagonal mobility matrices. A fully discrete approximation scheme in space and time is proposed which preserves the underlying gradient flow structure and leads to dissipation of the free-energy on the discrete level. Existence and uniqueness of the discrete solution is established and relative energy estimates are used to prove optimal convergence rates in space and time under minimal smoothness assumptions. Numerical tests are presented for illustration of the theoretical results and to demonstrate the viability of the proposed methods.

math.NA↗

arXiv2024

Error analysis for a viscoelastic phase separation model

We consider systematic numerical approximation of a viscoelastic phase separation model that describes the demixing of a polymer solvent mixture. An unconditionally stable discretisation method is proposed based on a finite element approximation in space and a variational time discretization strategy. The proposed method preserves the energy-dissipation structure of the underlying system exactly and allows to establish a fully discrete nonlinear stability estimate in natural norms based on the concept of relative energy. These estimates are used to derive order optimal error estimates for the method under minimal smoothness assumptions on the problem data, despite the presence of various strong nonlinearities in the equations. The theoretical results and main properties of the method are illustrated by numerical simulations which also demonstrate the capability to reproduce the relevant physical effects observed in experiments.

math.NA↗

arXiv2024

A reduced scalar potential approach for magnetostatics avoiding the coenergy

The numerical solution of problems in nonlinear magnetostatics is typically based on a variational formulation in terms of magnetic potentials, the discretization by finite elements, and iterative solvers like the Newton method. The vector potential approach aims at minimizing a certain energy functional and, in three dimensions, requires the use of edge elements and appropriate gauging conditions. The scalar potential approach, on the other hand, seeks to maximize the negative coenergy and can be realized by standard Lagrange finite elements, thus reducing the number of degrees of freedom and simplifying the implementation. The number of Newton iterations required to solve the governing nonlinear system, however, has been observed to be usually higher than for the vector potential formulation. In this paper, we propose a method that combines the advantages of both approaches, i.e., it requires as few Newton iterations as the vector potential formulation while involving the magnetic scalar potential as the primary unknown. We discuss the variational background of the method, its well-posedness, and its efficient implementation. Numerical examples are presented for illustration of the accuracy and the gain in efficiency compared to other approaches.

math.NA↗

arXiv2024

A magnetic oriented approach to the systematic coupling of field and circuit equations

A novel strategy is proposed for the coupling of field and circuit equations when modeling power devices in the low-frequency regime. The resulting systems of differential-algebraic equations have a particular geometric structure which explicitly encodes the energy storage, dissipation, and transfer mechanisms. This implies a power balance on the continuous level which can be preserved under appropriate discretization in space and time. The models and main results are presented in detail for linear constitutive models, but the extension to nonlinear elements and more general coupling mechanisms is possible. The theoretical findings are demonstrated by numerical results.

math.NA↗

arXiv2024

Stability and convergence of the penalty formulation for nonlinear magnetostatics

The magnetostatic field distribution in a nonlinear medium amounts to the unique minimizer of the magnetic coenergy over all fields that can be generated by the same current. This is a nonlinear saddlepoint problem whose numerical solution can in principle be achieved by mixed finite element methods and appropriate nonlinear solvers. The saddlepoint structure, however, makes the solution cumbersome. A remedy is to split the magnetic field into a known source field and the gradient of a scalar potential which is governed by a convex minimization problem. The penalty approach avoids the use of artificial potentials and Lagrange multipliers and leads to an unconstrained convex minimization problem involving a large parameter. We provide a rigorous justification of the penalty approach by deriving error estimates for the approximation due to penalization. We further highlight the close connections to the Lagrange-multiplier and scalar potential approach. The theoretical results are illustrated by numerical tests for a typical benchmark problem

math.NA↗

arXiv2024

Global convergence of iterative solvers for problems of nonlinear magnetostatics

We consider the convergence of iterative solvers for problems of nonlinear magnetostatics. Using the equivalence to an underlying minimization problem, we can establish global linear convergence of a large class of methods, including the damped Newton-method, fixed-point iteration, and the Kacanov iteration, which can all be interpreted as generalized gradient descent methods. Armijo backtracking isconsidered for an adaptive choice of the stepsize. The general assumptions required for our analysis cover inhomogeneous, nonlinear, and anisotropic materials, as well as permanent magnets. The main results are proven on the continuous level, but they carry over almost verbatim to various approximation schemes, including finite elements and isogeometric analysis, leading to bounds on the iteration numbers, which are independent of the particular discretization. The theoretical results are illustrated by numerical tests for a typical benchmark problem.

math.NA↗

arXiv2024

On the velocity-stress formulation for geometrically nonlinear elastodynamics and its structure-preserving discretization

We consider the dynamics of an elastic continuum under large deformation but small strain. Such systems can be described by the equations of geometrically nonlinear elastodynamics in combination with the St. Venant-Kirchhoff material law. The velocity-stress formulation of the problem turns out to have a formal port-Hamiltonian structure. In contrast to the linear case, the operators of the problem are modulated by the displacement field which can be handled as a passive variable and integrated along with the velocities. A weak formulation of the problem is derived and essential boundary conditions are incorporated via Lagrange multipliers. This variational formulation explicitly encodes the transfer between kinetic and potential energy in the interior as well as across the boundary, thus leading to a global power balance and ensuring passivity of the system. The particular geometric structure of the weak formulation can be preserved under Galerkin approximation via appropriate mixed finite elements. In addition, a fully discrete power balance can be obtained by appropriate time discretization. The main properties of the system and its discretization are shown theoretically and demonstrated by numerical tests.

eess.SY↗

arXiv2023

On the exponential stability of uniformly damped wave equations

We study damped wave propagation problems phrased as abstract evolution equations in Hilbert spaces. Under some general assumptions, including a natural compatibility condition for initial values, we establish exponential decay estimates for all mild solutions using the language and tools of Hilbert complexes. This framework turns out strong enough to conduct our analysis but also general enough to include a number of interesting examples. Some of these are briefly discussed. By a slight modification of the main arguments, we also obtain corresponding decay results for numerical approximations obtained by compatible discretization strategies.

math.AP↗

arXiv2023

On existence, uniqueness and stability of solutions to Cahn-Hilliard/Allen-Cahn systems with cross-kinetic coupling

A system of phase-field equations with strong-coupling through state and gradient dependent non-diagonal mobility matrices is studied. Existence of weak solutions is established by the Galerkin approximation and a-priori estimates in strong norms. Relative energy estimates are used to derive a general nonlinear stability estimate. As a consequence, a weak-strong uniqueness principle is obtained and stability with respect to model parameters is investigated.

math.AP↗

arXiv2023

An efficient mixed finite element method for nonlinear magnetostatics and quasistatics

We consider systems of nonlinear magnetostatics and quasistatics that typically arise in the modeling and simulation of electric machines. The nonlinear problems, eventually obtained after time discretization, are usually solved by employing a vector potential formulation. In the relevant two-dimensional setting, a discretization can be obtained by H1-conforming finite elements. We here consider an alternative formulation based on the H-field which leads to a nonlinear saddlepoint problem. After commenting on the unique solvability, we study the numerical approximation by H(curl)-conforming finite elements and present the main convergence results. A particular focus is put on the efficient solution of the linearized systems arising in every step of the nonlinear Newton solver. Via hybridization, the linearized saddlepoint systems can be transformed into linear elliptic problems, which can be solved with similar computational complexity as those arising in the vector or scalar potential formulation. In summary, we can thus claim that the mixed finite element approach based on the $H$-field can be considered a competitive alternative to the standard vector or scalar potential formulations for the solution of problems in nonlinear magneto-quasistatics.

math.NA↗

arXiv2023

On implicit interpolation models for nonlinear anisotropic magnetic material behavior

Implicit models for magnetic coenergy have been proposed by Pera et al. to describe the anisotropic nonlinear material behavior of electrical steel sheets. This approach aims at predicting magnetic response for any direction of excitation by interpolating measured of B--H curves in the rolling and transverse directions. In an analogous manner, an implicit model for magnetic energy is proposed. We highlight some mathematical properties of these implicit models and discuss their numerical realization, outline the computation of magnetic material laws via implicit differentiation, and discuss the potential use for finite element analysis in the context of nonlinear magnetostatics.

math.NA↗

arXiv2023

A second-order fully-balanced structure-preserving variational discretization scheme for the Cahn-Hilliard Navier-Stokes system

We propose and analyze a structure-preserving space-time variational discretization method for the Cahn-Hilliard-Navier-Stokes system. Uniqueness and stability for the discrete problem is established in the presence of concentration dependent mobility and viscosity parameters by means of the relative energy estimates and order optimal convergence rates are established for all variables using balanced approximation spaces and relaxed regularity conditions on the solution. Numerical tests are presented to demonstrate the proposed method is fully practical and yields the predicted convergence rates. The discrete stability estimates developed in this paper may also be used to analyse other discretization schemes, which is briefly outlined in the discussion.

math.NA↗

arXiv2023

Variational quantitative phase-field modeling of nonisothermal sintering process

In this work, we present a variational and quantitative phase-field model for non-isothermal sintering processes. The model is derived via an extended non-diagonal phase-field model. The model evolution equations have naturally cross-coupling terms between the conserved kinetics (i.e., mass and thermal transfer) and the non-conserved one (grain growth). These terms are shown via asymptotic analysis to be instrumental in ensuring the elimination of interface artefacts, while also examined to not modify the thermodynamic equilibrium condition (characterized by dihedral angle). Moreover, we demonstrate that the trapping effects and existence of surface diffusion in conservation laws are direction-dependent. An anisotropic interpolation scheme of the kinetic mobilities which differentiates the normal and the tangential directions along the interface is discussed. Numerically, we demonstrate the importance of the cross-couplings and the anisotropic interpolation via presenting thermal-microstructural evolutions.

cond-mat.mtrl-sci↗

arXiv2023

A Yee-like finite-element scheme for Maxwell's equations on unstructured grids

A novel finite element scheme is studied for solving the time-dependent Maxwell's equations on unstructured grids efficiently. Similar to the traditional Yee scheme, the method has one degree of freedom for most edges and a sparse inverse mass matrix. This allows for an efficient realization by explicit time-stepping without solving linear systems. The method is constructed by algebraic reduction of another underlying finite element scheme which involves two degrees of freedom for every edge. Mass-lumping and additional modifications are used in the construction of this method to allow for the mentioned algebraic reduction in the presence of source terms and lossy media later on. A full error analysis of the underlying method is developed which by construction also carries over to the reduced scheme and allows to prove convergence rates for the latter. The efficiency and accuracy of both methods are illustrated by numerical tests. The proposed schemes and their analysis can be extended to structured grids and in special cases the reduced method turns out to be algebraically equivalent to the Yee scheme. The analysis of this paper highlights possible difficulties in extensions of the Yee scheme to non-orthogonal or unstructured grids, discontinuous material parameters, and non-smooth source terms, and also offers potential remedies.

math.NA↗

arXiv2023

Regularity and long time behavior of a doubly nonlinear parabolic problem and its discretization

We study a doubly nonlinear parabolic problem arising in the modeling of gas transport in pipelines. Using convexity arguments and relative entropy estimates we show uniform bounds and exponential stability of discrete approximations obtained by a finite element method and implicit time stepping. Due to convergence of the approximations to weak solutions of the problem, our results also imply regularity, uniqueness, and long time stability of weak solutions of the continuous problem.

math.AP↗

arXiv2022

A Yee-like finite element scheme for Maxwell's equations on hybrid grids

A novel finite element method for the approximation of Maxwell's equations over hybrid two-dimensional grids is studied. The choice of appropriate basis functions and numerical quadrature leads to diagonal mass matrices which allow for efficient time integration by explicit methods. On purely rectangular grids, the proposed schemes coincide with well-established FIT and FDTD methods. Additional internal degrees of freedom introduced on triangles allow for mass-lumping without the usual constraints on the shape of these elements. A full error analysis of the method is developed and numerical tests are presented for illustration.

math.NA↗

arXiv2022

An asymptotic preserving hybrid-dG method for convection-diffusion equations on pipe networks

We study the numerical approximation of singularly perturbed convection-diffusion problems on one-dimensional pipe networks. In the vanishing diffusion limit, the number and type of boundary conditions and coupling conditions at network junctions changes, which gives rise to singular layers at the outflow boundaries of the pipes. A hybrid discontinuous Galerkin method is proposed, which provides a natural upwind mechanism for the convection-dominated case. Moreover, the method automatically handles the variable coupling and boundary conditions in the vanishing diffusion limit, leading to an asymptotic-preserving scheme. A detailed analysis of the singularities of the solution and the discretization error is presented, and an adaptive strategy is proposed, leading to order optimal error estimates that hold uniformly in the singular perturbation limit. The theoretical results are confirmed by numerical tests.

math.NA↗