20160208
The efficient numerical solution of eddy current problems, whether formulated with differential or integral techniques, requires some topological preprocessing. Kotiuga, in his pioneering papers in the eighties, showed that computational topology is something that one should get familiar with when bumps into problems related with the design of (electromagnetic) potentials. In particular, one needs to construct a cohomology basis that can be computed in polynomial time.
Yet, in practice to compute this basis remained an open problem for more than twenty years given that all available implementations were too slow to be of any practical use, at least in computational electromagnetics.
We first survey the techniques used for the computation of the first cohomology group generators, which are essential for solving eddy current problems with formulations based on the magnetic scalar potential. We start by reviewing standard algebraic techniques and we end up with the novel paradigm of "lazy" (co)homology generators and the fast combinatorial techniques to compute them. They simplify considerably the topological preprocessing required by electromagnetic simulations to the point that the computational effort is reduced by five orders of magnitude. Recent efforts are oriented to optimize the cohomology basis, for example by reducing the support of representatives of the generators with combinatorial algorithms based on network flow.
Then, we emphasize the recent revival of integral formulations to solve eddy current problems, thanks to stateoftheart sparsification techniques based on hierarchical matrices and Adaptive Cross Approximation (ACA). The suitable (co)homology generators for volumetric and boundary integral formulations are introduced. We also extend the computation of eddy currents to thin conductors represented by nonmanifold triangulated surfaces, which are singular spaces (i.e. they have points whose neighborhoods are not Euclidean) where even Poincaré duality does not hold. Thanks to the theory of stratified spaces, the singular space is decomposed (or stratified) into manifold parts called strata. Certain conditions have to be imposed where the strata meet and we show that these conditions have a clear physical interpretation.
The second part of the presentation shows the first results on threedimensional Poisson problems of the recently introduced discretization technique called hybrid high order (HHO) method. These methods allow to obtain discretization techniques for general polyhedral grids and of arbitrary order of approximation. They bear similarities with discontinuous Galerkin and especially with Brezzi's virtual element method (VEM), but they turn out to be more efficient. At the first order, they are akin to Finite Integration Technique (FIT)like discretizations, whereas the link with FIT for high order elements is still unclear. Given that the solution to most practical electrostatic problems contain strong singularities, we developed a residual based error estimator for automatic mesh adaptivity. The solution of full Maxwell problems is an ongoing work.
Category: CE SeminarTechnische Universität Darmstadt
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