New higher-order basis functions for Curvilinear finite elements

The first contribution is a fast calculation method for tetrahedral finite element matrices which is applicable to curvilinear geometries and inhomogeneous material properties. The element matrices are obtained at a low computational cost via scaled additions of universal matrices. The proposed technique is more efficient than competing approaches and provides well-defined lower and upper bounds for the required number of matrices. In the case of tetrahedral H(div) elements, a new set of basis functions is proposed for the mixed-order Nédélec space. The specialty of the functions is a high level of orthogonality which applies to arbitrary straight-sided tetrahedra. The resulting condition numbers, compared to competing bases, are significantly lower. The remaining contributions concern hexahedral elements, where a new, mixed-order serendipity element is proposed for H(curl)-conforming functions. It allows the construction of a single set of hierarchical basis functions that can also be used to span various other finite element spaces. Therefore, it is possible to use different finite element spaces within the same mesh while maintaining conformity. In the curvilinear case, a special yet versatile way of mesh refinement is proposed along with serendipity basis functions for the interpolation of the geometry. The main advantage of the proposed methods is the resulting algebraic rate of convergence in H(curl)-norm with the least possible number of unknowns.Der erste Beitrag ist eine schnelle Berechnungsmethode von Finite-Elemente-Matrizen für Tetraeder, die auf krummlinige Geometrien und inhomogene Materialeigenschaften anwendbar ist. Die Elementmatrizen werden mit geringem Rechenaufwand durch skalierte Addition vorgefertigter Matrizen erstellt. Die vorgeschlagene Methode ist effizienter als vergleichbare Ansätze und liefert wohldefinierte obere und untere Schranken für die Anzahl der benötigten Matrizen. Für H(div)-konforme Elemente auf Tetraedern werden neue Ansatzfunktionen für den N´ed´elec-Raum gemischter Ordnung vorgestellt. Die Besonderheit dieser Funktionen ist ein hohes Maß an Orthogonalität für beliebige geradlinige Tetraeder. Im Vergleich zu anderen Ansatzfunktionen sind die resultierenden Konditionszahlen deutlich kleiner. Die übrigen Beiträge betreffen Hexaeder, für die ein neues Serentipity-Element gemischter Ordnung vorgestellt wird. Es ermöglicht die Konstruktion hierarchischer Ansatzfunktionen, die auch zum Aufspannen anderer Finite-Elemente-Räume angewandt werden kann. Daher ist es möglich, verschiedene Finite-Elemente-Räume auf dem gleichen Netz zu verwenden und dabei Konformität zu bewahren. Für den krummlinigen Fall wird eine spezielle aber vielseitige Methode zur Netzverfeinerung mit Serentipity-Ansatzfunktionen zur Interpolation der Geometrie vorgestellt. Der Hauptvorteil der vorgestellten Methoden ist die algebraische Konvergenz in der Norm des H(rot) mit der kleinstmöglichen Anzahl an Unbekannten

Hybrid explicit-implicit FDTD-FEM time-domain solver for electromagnetic problems

The Finite-Difference Time-Domain (FDTD) method and Finite-Element (FEM) method are numerical techniques used for solving Maxwell\u27s electromagnetic equations. FDTD-FEM hybrid methods opt for combining the advantages of both FDTD and FEM. In this dissertation, signal processing techniques were used to analyze the FDTD stability condition. A procedure, which reduces time-sampling error yet preserves the stability of algorithm is proposed. Both explicit and implicit time-stepping schemes were treated in the framework of the developed method. An improved version of the implicit-explicit FEM-FDTD hybrid method was developed. The new method minimizes reflection from the interface between different types of grids. A class of transfer functions with low reflection error for stable hybrid time-stepping was derived. The stability of the method is rigorously proven for a general three-dimensional case

Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p

Eigenvalue problems for Beltrami fields arising in a three-dimensional toroidal magnetohydrodynamic equilibrium problem

A generalized energy principle for finite-pressure, toroidalmagnetohydrodynamic(MHD) equilibria in general three-dimensional configurations is proposed. The full set of ideal-MHD constraints is applied only on a discrete set of toroidal magnetic surfaces (invariant tori), which act as barriers against leakage of magnetic flux, helicity, and pressure through chaotic field-line transport. It is argued that a necessary condition for such invariant tori to exist is that they have fixed, irrational rotational transforms. In the toroidal domains bounded by these surfaces, full Taylor relaxation is assumed, thus leading to Beltrami fields ∇×B=λB, where λ is constant within each domain. Two distinct eigenvalue problems for λ arise in this formulation, depending on whether fluxes and helicity are fixed, or boundary rotational transforms. These are studied in cylindrical geometry and in a three-dimensional toroidal region of annular cross section. In the latter case, an application of a residue criterion is used to determine the threshold for connected chaos.This work was supported in part by the U.S. Department of Energy Contract No. DE-AC02-76CH03073 and Grant No. DE-FG02-99ER54546 and the Australian Research Council

Interpolation Based Parametric Model Order Reduction

In this thesis, we consider model order reduction of parameter-dependent large-scale dynamical systems. The objective is to develop a methodology to reduce the order of the model and simultaneously preserve the dependence of the model on parameters. We use the balanced truncation method together with spline interpolation to solve the problem. The core of this method is to interpolate the reduced transfer function, based on the pre-computed transfer function at a sample in the parameter domain. Linear splines and cubic splines are employed here. The use of the latter, as expected, reduces the error of the method. The combination is proven to inherit the advantages of balanced truncation such as stability preservation and, based on a novel bound for the infinity norm of the matrix inverse, the derivation of error bounds. Model order reduction can be formulated in the projection framework. In the case of a parameter-dependent system, the projection subspace also depends on parameters. One cannot compute this parameter-dependent projection subspace, but has to approximate it by interpolation based on a set of pre-computed subspaces. It turns out that this is the problem of interpolation on Grassmann manifolds. The interpolation process is actually performed on tangent spaces to the underlying manifold. To do that, one has to invoke the exponential and logarithmic mappings which involve some singular value decompositions. The whole procedure is then divided into the offline and online stage. The computation time in the online stage is a crucial point. By investigating the formulation of exponential and logarithmic mappings and analyzing the structure of sums of singular value decompositions, we succeed to reduce the computational complexity of the online stage and therefore enable the use of this algorithm in real time

On a Polyanalytic Approach to Noncommutative de Branges–Rovnyak Spaces and Schur Analysis

In this paper we begin the study of Schur analysis and of de Branges–Rovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like polynomials. This approach is very efficient from various points of view, for example in operator theory, and allows us to make connections with the recently developed theory of slice polyanalytic functions. We tackle a number of problems: we describe a Hardy space, Schur multipliers and related results. We also discuss Blaschke functions, Herglotz multipliers and their associated kernels and Hilbert spaces. Finally, we consider the counterpart of the half-space case, and the corresponding Hardy space, Schur multipliers and Carathéodory multipliers