arXiv subjects

Martin Schanz

Publications and source records attributed to Martin Schanz.

6 recordsLinked to original sources

arXiv2026

Proton Radiography with Isotrajectory Optics

We consider a compact isotrajectory-optics scheme for proton radiography. The calculation uses a 30-60 MeV proton interval as a reference example, typical of high-energy short-pulse laser acceleration in the TNSA regime. The idea is to use synchronized time-dependent fields such that, for one selected energy-arrival-time branch, the proton energy changes mainly the arrival time and not the transverse image position. The reference lattice consists of four pulsed electric quadrupoles with horizontal sign pattern + - + -, magnification Mx = My = -3.0 at a detector plane of 2.57 m, a common Fourier plane at zF = 0.962 m, and equal outer and equal inner lens lengths. The ideal closure residual is calculated from second central moments. Fourth central moments are kept separately as a diagnostic of non-Gaussian tails. The quoted residual widths are numerical closure tests of the ideal map, not experimental resolution predictions. Real electrode fields, waveform errors, scattering, detector blur, and space charge define the next level of the problem.

physics.plasm-ph↗

arXiv2026

A 3D-ACA accelerated time domain boundary element method for elastodynamics using FMM and $\mathcal{H}$-matrix techniques

The time-domain Boundary Element Method (BEM) for linear elastodynamics with vanishing initial conditions is considered. Spatial discretization uses standard low-order boundary elements, while temporal discretization employs the generalized Convolution Quadrature (gCQ) method. The gCQ framework requires evaluating BEM matrices in the Laplace domain at several complex frequencies along a chosen contour, producing a three-dimensional tensor with one spatial matrix slice per frequency. To reduce storage and computational cost, a low rank approximation of the tensor is computed using 3D-Adaptive Cross Approximation (3D-ACA), extending the classical ACA to handle both the additional frequency dimension and the tensorial structure of elastodynamics. Within each frequency slice, the BEM matrices are further compressed using either the classical ACA algorithm using the $\mathcal{H}$-matrix approach or a Chebyshev interpolation based Fast Multipole Method (FMM). A comparative study of all proposed methods is carried out using two academic examples, and the structural vibration of an induction machine is analyzed.

math.NA↗

arXiv2025

A Higher-Order Time Domain Boundary Element Formulation based on Isogeometric Analysis and the Convolution Quadrature Method

An isogeometric boundary element method (BEM) is presented to solve scattering problems in an isotropic homogeneous medium. We consider wave problems governed by the scalar wave equation as in acoustics and the Lam\'e-Navier equations for elastodynamics considering the theory of linear elasticity. The underlying boundary integral equations imply time-dependent convolution integrals and allow us to determine the sought quantities in the bounded interior or the unbounded exterior after solving for the unknown Cauchy data. In the present work, the time-dependent convolution integrals are approximated by multi-stage Runge-Kutta (RK) based convolution quadratures that involve steady-state solutions in the Laplace domain. The proposed method discretizes the spatial variables in the framework of isogeometric analysis (IGA), entailing a patchwise smooth spline basis. Overall, it enables high convergence rates in space and time. The implementation scheme follows an element structure defined by the non-empty knot spans in the knot vectors and local, uniform Bernstein polynomials as basis functions. The algorithms to localize the basis functions on the elements are outlined and explained. The solutions of the mixed problems are approximated by the BEM based on a symmetric Galerkin variational formulation and a collocation method. We investigate convergence rates of the approximative solutions in a mixed space and time error norm.

cs.CE↗

arXiv2025

Comparison of $\mathcal{H}$-matrix- and FMM-based 3D-ACA for a time-domain boundary element method

The homogeneous wave equation is solved by a time-domain boundary element method (BEM) using low-order shape functions for spatial, and the generalised convolution quadrature method (gCQ) by Lopez-Fernandez and Sauter for temporal discretisation. The three-dimensional array of BEM matrices according to a set of complex frequencies in Laplace domain is approximated by generalised Adaptive Cross Approximation (3D-ACA). Its rank is increased adaptively until a prescribed accuracy is reached, relying on a pure algebraic error criterion. The data slices for the selected frequency points are further processed by either the standard $\mathcal{H}$-matrices approach with ACA or by a fast multipole method (FMM). This paper compares both approaches with respect to their demands in storage and computing time. Both techniques are illustrated for calculating the sound scattered by an electric machine, for which the proposed algebraic compression techniques make time-domain BEM feasible for the first time.

math.NA↗

arXiv2025

Partial integration based regularization in BEM for 3D elastostatic problems: The role of line integrals

The Boundary Element Method (BEM) is a powerful numerical approach for solving 3D elastostatic problems, particularly useful for crack propagation in fracture mechanics and half-space problems. A key challenge in BEM lies in handling singular integral kernels. Various analytical and numerical integration or regularization techniques address this, including one that combines partial integration with Stokes' theorem to reduce hyper-singular and strong singular kernels to weakly singular ones. This approach typically assumes a closed surface, omitting the boundary integrals from Stokes' theorem. In this paper, these usually neglected boundary line integrals are introduced and their significance is demonstrated, first in a pure half-space problem, and then shown to be redundant in fast multipole method (FMM) based BEM, where geometry partitioning produces pseudo open surfaces.

math.NA↗

arXiv2021

On the space-time discretization of variational retarded potential boundary integral equations

This paper discusses the practical development of space-time boundary element methods for the wave equation in three spatial dimensions. The employed trial spaces stem from simplex meshes of the lateral boundary of the space-time cylinder. This approach conforms genuinely to the distinguished structure of the solution operators of the wave equation, so-called retarded potentials. Since the numerical evaluation of the arising integrals is intricate, the bulk of this work is constituted by ideas about quadrature techniques for retarded layer potentials and associated energetic bilinear forms. Finally, we glimpse at algorithmic aspects regarding the efficient implementation of retarded potentials in the space-time setting. The proposed methods are verified by means of numerical experiments, which illustrate their capacity.

math.NA↗