arXiv subjects

Jürgen Dölz

Publications and source records attributed to Jürgen Dölz.

26 records · Page 2Linked to original sources

arXiv2019

A higher order perturbation approach for electromagnetic scattering problems on random domains

We consider time-harmonic electromagnetic scattering problems on perfectly conducting scatterers with uncertain shape. Thus, the scattered field will also be uncertain. Based on the knowledge of the two-point correlation of the domain boundary variations around a reference domain, we derive a perturbation analysis for the mean of the scattered field. Therefore, we compute the second shape derivative of the scattering problem for a single perturbation. Taking the mean, this leads to an at least third order accurate approximation with respect to the perturbation amplitude of the domain variations. To compute the required second order correction term, a tensor product equation on the domain boundary has to be solved. We discuss its discretization and efficient solution using boundary integral equations. Numerical experiments in three dimensions are presented.

math.NA↗

arXiv2019

Multipatch Approximation of the de Rham Sequence and its Traces in Isogeometric Analysis

We define a conforming B-spline discretisation of the de Rham complex on multipatch geometries. We introduce and analyse the properties of interpolation operators onto these spaces which commute w.r.t. the surface differential operators. Using these results as a basis, we derive new convergence results of optimal order w.r.t. the respective energy spaces and provide approximation properties of the spline discretisations of trace spaces for application in the theory of isogeometric boundary element methods. Our analysis allows for a straightforward generalisation to finite element methods.

math.NA↗

arXiv2019

Isogeometric Boundary Elements in Electromagnetism: Rigorous Analysis, Fast Methods, and Examples

We present a new approach to three-dimensional electromagnetic scattering problems via fast isogeometric boundary element methods. Starting with an investigation of the theoretical setting around the electric field integral equation within the isogeometric framework, we show existence, uniqueness, and quasi-optimality of the isogeometric approach. For a fast and efficient computation, we then introduce and analyze an interpolation-based fast multipole method tailored to the isogeometric setting, which admits competitive algorithmic and complexity properties. This is followed by a series of numerical examples of industrial scope, together with a detailed presentation and interpretation of the results.

math.NA↗

arXiv2018

A Numerical Comparison of an Isogeometric and a Classical Higher-Order Approach to the Electric Field Integral Equation

In this paper, we advocate a novel spline-based isogeometric approach for boundary elements and its efficient implementation. We compare solutions obtained by both an isogeometric approach, and a classical parametric higher-order approach via Raviart-Thomas elements to the solution of the electric field integral equation; i.e., the solution to an electromagnetic scattering problem, promising high convergence orders w.r.t. pointwise error. We discuss both, the obtained accuracy per DOF, as well as the effort required to solve the corresponding system iteratively, on three numerical examples of varying complexity.

cs.CE↗

arXiv2018

On the best approximation of the hierarchical matrix product

The multiplication of matrices is an important arithmetic operation in computational mathematics. In the context of hierarchical matrices, this operation can be realized by the multiplication of structured block-wise low-rank matrices, resulting in an almost linear cost. However, the computational efficiency of the algorithm is based on a recursive scheme which makes the error analysis quite involved. In this article, we propose a new algorithmic framework for the multiplication of hierarchical matrices. It improves currently known implementations by reducing the multiplication of hierarchical matrices towards finding a suitable low-rank approximation of sums of matrix-products. We propose several compression schemes to address this task. As a consequence, we are able to compute the best-approximation of hierarchical matrix products. A cost analysis shows that, under reasonable assumptions on the low-rank approximation method, the cost of the framework is almost linear with respect to the size of the matrix. Numerical experiments show that the new approach produces indeed the best-approximation of the product of hierarchical matrices for a given tolerance. They also show that the new multiplication can accomplish this task in less computation time than the established multiplication algorithm without error control.

math.NA↗

arXiv2017

A Fast Isogeometric BEM for the Three Dimensional Laplace- and Helmholtz Problems

We present an indirect higher order boundary element method utilising NURBS mappings for exact geometry representation and an interpolation-based fast multipole method for compression and reduction of computational complexity, to counteract the problems arising due to the dense matrices produced by boundary element methods. By solving Laplace and Helmholtz problems via a single layer approach we show, through a series of numerical examples suitable for easy comparison with other numerical schemes, that one can indeed achieve extremely high rates of convergence of the pointwise potential through the utilisation of higher order B-spline-based ansatz functions.

math.NA↗

arXiv2017

Recent Advances of Isogeometric Analysis in Computational Electromagnetics

In this communication the advantages and drawbacks of the isogeometric analysis (IGA) are reviewed in the context of electromagnetic simulations. IGA extends the set of polynomial basis functions, commonly employed by the classical Finite Element Method (FEM). While identical to FEM with Nédélec's basis functions in the lowest order case, it is based on B-spline and Non-Uniform Rational B-spline basis functions. The main benefit of this is the exact representation of the geometry in the language of computer aided design (CAD) tools. This simplifies the meshing as the computational mesh is implicitly created by the engineer using the CAD tool. The curl- and div-conforming spline function spaces are recapitulated and the available software is discussed. Finally, several non-academic benchmark examples in two and three dimensions are shown which are used in optimization and uncertainty quantification workflows.

cs.CE↗

arXiv2017

$\mathcal{H}$-matrix based second moment analysis for rough random fields and finite element discretizations

We consider the efficient solution of strongly elliptic partial differential equations with random load based on the finite element method. The solution's two-point correlation can efficiently be approximated by means of an $\mathcal{H}$-matrix, in particular if the correlation length is rather short or the correlation kernel is non-smooth. Since the inverses of the finite element matrices which correspond to the differential operator under consideration can likewise efficiently be approximated in the $\mathcal{H}$-matrix format, we can solve the correspondent $\mathcal{H}$-matrix equation in essentially linear time by using the $\mathcal{H}$-matrix arithmetic. Numerical experiments for three-dimensional finite element discretizations for several correlation lengths and different smoothness are provided. They validate the presented method and demonstrate that the computation times do not increase for non-smooth or shortly correlated data.

math.NA↗