Christophe Geuzaine

Prescribed error tolerances within fixed computational times for scattering problems of arbitrarily high frequency: the convex case

We present a new algorithm for the numerical solution of problems of electromagnetic or acoustic scattering by large, convex obstacles. This algorithm combines the use of an ansatz for the unknown density in a boundary-integral formulation of the scattering problem with an extension of the ideas of the method of stationary phase. We include numerical results illustrating the high-order convergence of our algorithm as well as its asymptotically bounded computational cost as the frequency increases.

Robust untangling of curvilinear meshes

This paper presents a technique that allows to untangle high-order/curvilinear meshes. The technique makes use of unconstrained optimization where element Jacobians are constrained to lie in a prescribed range through moving log-barriers. The untangling procedure starts from a possibly invalid curvilinear mesh and moves mesh vertices with the objective of producing elements that all have bounded Jacobians. Bounds on Jacobians are computed using the results of Johnen et al. (2012, 2013) [1,2]. The technique is applicable to any kind of polynomial element, for surface, volume, hybrid or boundary layer meshes. A series of examples demonstrate both the robustness and the efficiency of the technique. The final example, involving a time explicit computation, shows that it is possible to control the stable time step of the computation for curvilinear meshes through an alternative element deformation measure.

A 3-D magnetic vector potential formulation taking eddy currents in lamination stacks into account

A method is developed to take the eddy currents in lamination stacks into account with the finite-element method using the three-dimensional (3-D) magnetic vector potential magnetodynamic formulation. It consists in converting the stacked laminations into continuums with which terms are associated for considering the eddy-current loops produced by both parallel and perpendicular fluxes. Two levels of accuracy are proposed. The best one is based on an accurate analytical expression of the eddy currents and makes the method adapted to a wide-frequency range, i.e., even for low skin depths in the laminations.

A Galerkin projection method for mixed finite elements

The aim of the proposed method is the projection of an electromagnetic field belonging to a given function space (continuous or not) onto a discrete one spanned by finite element basis functions. This technique is useful for imposing inhomogeneous boundary conditions or volumic source fields, for calculating a dual field given the primal one or for mesh to mesh interpolation.

On the O(1) solution of multiple-scattering problems

In this paper, we present a multiple-scattering solver for nonconvex geometries such as those obtained as the union of a finite number of convex surfaces. For a prescribed error tolerance, this algorithm exhibits a fixed computational cost for arbitrarily high frequencies. At the core of the method is an extension of the method of stationary phase, together with the use of an ansatz for the unknown density in a combined-field boundary integral formulation.

Harmonic-balance finite-element modeling of electromagnetic devices: a novel approach

In this paper, a novel and easy-to-implement approach to the harmonic-balance finite-element modeling of electromagnetic devices is presented. The governing system of nonlinear algebraic equations is derived assuming an arbitrary (anisotropic) magnetic constitutive law. It is solved by means of the Newton-Raphson (NR) method, the elaboration of which is very simple thanks to the introduction of the differential reluctivity tensor. The method is validated by applying it to a three-dimensional and a two-dimensional voltage-driven model of a three-phase inductor. The convergence of the NR scheme and the accuracy of the obtained harmonic-balance current waveforms are studied.

A quasi-optimal non-overlapping domain decomposition algorithm for the Helmholtz equation

This paper presents a new non-overlapping domain decomposition method for the Helmholtz equation, whose effective convergence is quasi-optimal. These improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the Dirichlet to Neumann operator. A convergence theorem of the algorithm is established and numerical results validating the new approach are presented in both two and three dimensions.

Geometrical validity of curvilinear finite elements

In this paper, we describe a way to compute accurate bounds on Jacobian determinants of curvilinear polynomial finite elements. Our condition enables to guarantee that an element is geometrically valid, i.e., that its Jacobian determinant is strictly positive everywhere in its reference domain. It also provides an efficient way to measure the distortion of curvilinear elements. The key feature of the method is to expand the Jacobian determinant using a polynomial basis, built using Bezier functions, that has both properties of boundedness and positivity. Numerical results show the sharpness of our estimates.

Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem

This paper presents a preconditioner for non-overlapping Schwarz methods applied to the Helmholtz problem. Starting from a simple analytic example, we show how such a preconditioner can be designed by approximating the inverse of the iteration operator for a layered partitioning of the domain. The preconditioner works by propagating information globally by concurrently sweeping in both directions over the subdomains, and can be interpreted as a coarse grid for the domain decomposition method. The resulting algorithm is shown to converge very fast, independently of the number of subdomains and frequency. The preconditioner has the advantage that, like the original Schwarz algorithm, it can be implemented as a matrix-free routine, with no additional preprocessing.

Dual magnetodynamic formulations and their source fields associated with massive and stranded inductors

The treatment of massive and stranded inductors is studied in the frame of dual magnetodynamic finite element h- and /spl alpha/-formulations. On both sides, edge finite elements are used and source fields are defined when needed as mathematical tools to be used directly in each formulation to lead to circuit relations for inductors. Simplified expressions of source fields are proposed. The accuracy obtained on local solutions and circuit parameters is pointed out.