1,078 research outputs found
Complementary geometric formulations for electrostatics
No full textThe simultaneous use of a pair of complementary discrete formulations for electrostatic boundary value problems (BVPs) allows to accurately compute electromagnetic quantities, such as capacitance or electrostatic force with a minimum computational effort. In fact, the two formulations provide the upper and lower bounds for these quantities and their averages result quite close to the exact solution even for extremely coarse meshes. Despite the potential benefit to the many three-dimensional large-scale applications, taking advantage of this feature is not exploited in practice due to theoretical difficulties in the potential design. The aim of this paper is to fill this gap by rigorously introducing a pair of three-dimensional complementary geometric formulations to solve electrostatic BVPs on domains covered by conformal polyhedral meshes. In particular, an original formulation based on a vector potential is introduced by using cohomology theory with integer coefficients. It is shown how the so-called thick links are needed, which are representatives of the second cohomology group generators of the dielectric region. Two easy-to-implement graph-theoretic algorithms to automatically find such a basis with optimal computational complexity are described. Some benchmark problems are presented to show how the simultaneous use of both formulations yields to a sensible computational advantage. Therefore, solvers based on complementary formulations should be embedded in the next-generation of electromagnetic Computer-Aided Engineering (CAE) softwares
A novel approach for solving three dimensional eddy current problems in fusion devices
No full textWe present a technique to efficiently solve 3D eddy current problems in fusion devices, whose structuresexhibit a geometrical symmetry. The proposed approach is based on the exploitation of symmetry viaharmonic analysis and it is suitable to treat also problems where the field sources (plasma or externalcoils) do not share the domain symmetry. A simple test case is presented to describe the methodology.Then, a more complicated geometry is considered, which represents a realistic vacuum vessel of an ITERlike fusion device (double layer structure with several portholes). The results are discussed for bothaxisymmetric and non-axisymmetric excitations in frequency domain
Lean cohomology computation for electromagnetic modeling
Full text linkSolving eddy current problems formulated by using a magnetic scalar potential in the insulator requires a topological pre-processing to find the so-called first cohomology basis of the insulating region, which may be very time-consuming for challenging industrially driven problems. The physics-inspired Dłotko-Specogna (DS) algorithm was shown to be superior to alternatives in performing such a topological pre-processing. Yet, the DS algorithm is particularly fast when it produces as output not a regular cohomology basis but a so-called lazy one, which contains the regular one but it keeps also some additional redundant elements. Having a regular basis may be advantageous over the lazy basis if a technique to produce it would take about the same time as the computation of a lazy basis. In the literature, such a technique is missing. This paper covers this gap by introducing modifications to the DS algorithm to compute a regular basis of the first cohomology group in practically the same time as the generation of a lazy cohomology basis. The speedup of this modified DS algorithm with respect to the best alternative reaches more than two orders of magnitudes on challenging benchmark problems. This demonstrates the potential impact of the proposed contribution in the low-frequency computational electromagnetics community and beyond. © 2017 IEEE
Fast Frequency and Material Properties Sweeps for Quasi-Static Problems
No full textWe introduce a novel technique that speeds up the computation of a frequency sweep or some parametric change of material properties - assumed uniform over the entire domain - around a nominal value in electroquasi-static problem or magnetoquasi-static problem. In place of using the usual practice of solving the complex systems arising at each frequency and at each material parameter value independently, our technique requires only one factorization of a real, symmetric, and positive definite matrix. The solution at each frequency and each value of material parameter is, then, found with a few back-substitutions only. The obtained speedup is sensible and the implementation is straightforward, showing the usefulness of the proposed technique in practical applications. © 1965-2012 IEEE
Optimal cohomology generators for 2d eddy-current problems in linear time
No full textThe aim of this paper is to present an automatic and efficient algorithm to find cohomology generators suitable for 2d eddycurrent problems formulated by means of complementary formulations. The algorithm is general, straightforward to implement, exhibits a linear worst-case computational complexity and produces optimal representatives of generators. By optimal we mean the representatives that minimize in practical cases the fill-in of the system of equations matrix and guarantee that the current flowing in each conductor is in one-to-one correspondence with a generator. As a numerical example, the complementary formulations are used to compute the frequency-dependent per-unit-length impedance in integrated circuits
Lean Complementarity for Poisson Problems
No full textWe introduce a novel technique - lean complementarity - that attempts to eliminate any waste of computational resources occurring during the pursuing of complementarity. First, contrarily to the widely used practice of solving the problem two times with a pair of complementary or complementary-dual formulations, lean complementarity requires just one solution with the computationally cheap formulation based on the scalar potential. This result is enabled by a novel and explicit flux equilibration technique that produces tight bounds and is computationally inexpensive, because no system has to be solved. Second, the systems arising during the adaptive mesh refinement procedure are solved inexactly on purpose, by stopping the iterations of the iterative solver when the algebraic error gets negligible with respect to the discretization error. The discretization error is bounded with complementarity, whereas the algebraic error is computed very accurately with a novel and cheap technique. © 1965-2012 IEEE
Computation of stationary 3D halo currents in fusion devices with accuracy control
No full textThis paper addresses the calculation of the resistive distribution of halo currents in three-dimensional structures of large magnetic confinement fusion machines. A Neumann electrokinetic problem is solved on a geometry so complicated that complementarity is used to monitor the discretization error. An irrotational electric field is obtained by a geometric formulation based on the electric scalar potential, whereas three geometric formulations are compared to obtain a solenoidal current density: a formulation based on the electric vector potential and two geometric formulations inspired from mixed and mixed-hybrid Finite Elements. The electric vector potential formulation is usually considered impractical since an enormous computing power is wasted by the topological pre-processing it requires. To solve this challenging problem, we present novel algorithms based on lazy cohomology generators that enable to save orders of magnitude computational time with respect to all other state-of-the-art solutions proposed in literature. Believing that our results are useful in other fields of scientific computing, the proposed algorithm is presented as a detailed pseudocode in such a way that it can be easily implemented
Numerical determination of upper and lower bounds of the transmembrane potential with complementarity
No full textOne stroke complementarity for poisson-like problems
No full textTaking electrokinetics as a paradigm problem for the sake of simplicity, complementarity originates when an irrotational electric field and a solenoidal current density satisfying boundary conditions are in hand. We first compare three formulations to obtain a solenoidal current density, both in terms of pure computational advantage and in the ability to pursue symmetric energy bounds with respect to the standard electric scalar potential formulation. For these formulations, we devise post-processing techniques that promise to provide bilateral bounds in one stroke, hence requiring the solution of just one linear system
Diagonal discrete hodge operators for simplicial meshes using the signed dual complex
No full textWe present a technique to extend the geometric construction of diagonal discrete Hodge operators to arbitrary triangular and tetrahedral boundary conforming Delaunay meshes in the frequent case of piecewise uniform and isotropic material parameters. The technique is based on the novel concept of signed dual complex that originates from a physical argument. In particular, it is shown how the positive definiteness of the mass matrix obtained with the signed dual complex is easily ensured for all boundary conforming Delaunay meshes without requiring—as expected by the common knowledge—that each circumcenter has to lie inside the corresponding element. Eliminating this requirement, whose fulfillment presents otherwise formidable practical difficulties, enables
one to easily obtain efficient, consistent, and stable schemes
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