Scipedia: Computational Electromagnetics

An immersed boundary hierarchical B-spline method for flexoelectricity

Jesús Sánchez Pinedo — Tue, 25 Aug 2020 10:36:57 +0200

This paper develops a computational framework with unfitted meshes to solve linear piezoelectricity and flexoelectricity electromechanical boundary value problems including strain gradient elasticity at infinitesimal strains. The high-order nature of the coupled PDE system is addressed by a sufficiently smooth hierarchical B-spline approximation on a background Cartesian mesh. The domain of interest is embedded into the background mesh and discretized in an unfitted fashion. The immersed boundary approach allows us to use B-splines on arbitrary domain shapes, regardless of their geometrical complexity, and could be directly extended, for instance, to shape and topology optimization. The domain boundary is represented by NURBS, and exactly integrated by means of the NEFEM mapping. Local adaptivity is achieved by hierarchical refinement of B-spline basis, which are efficiently evaluated and integrated thanks to their piecewise polynomial definition. Nitsche’s formulation is derived to weakly enforce essential boundary conditions, accounting also for the non-local conditions on the non-smooth portions of the domain boundary (i.e. edges in 3D or corners in 2D) arising from Mindlin’s strain gradient elasticity theory. Boundary conditions modeling sensing electrodes are formulated and enforced following the same approach. Optimal error convergence rates are reported using high-order B-spline approximations. The method is verified against available analytical solutions and well-known benchmarks from the literature.

Electromagnetic axisymmetric finite elements based on a gauged four-potential variational principle

María Jesús Samper — Mon, 17 Aug 2020 10:25:47 +0200

Electromagnetic finite elements are derived based on a variational principle that uses the electromagnetic four-potential as a primary variable. The Lorentz gage normalization is incorporated as a constraint condition through a Lagrange multiplier field to construct elements suitable for downstream coupling with mechanical and thermal finite elements for the analysis of high-temperature superconductor devices with aerospace applications. The main advantages are: jump discontinuities on interfaces are naturally handled; no a priori approximations are invoked; and the number of degrees of freedom per node remains modest as the problem dimensionality increases. The new elements are tested on two magnetostatic axisymmetric problems. The results are in excellent agreement with analytical solutions and previous solutions for the 1D problem of a conducting infinite wire, in which case the multiplier field has no effect. For materials of widely different permeability, jump conditions are naturally accommodated by the present formulation.

Electromagnetic finite elements based on a four-potential variational principle

María Jesús Samper — Fri, 14 Aug 2020 13:20:43 +0200

We derive electromagnetic finite elements based on a variational principle that uses the electromagnetic four-potential as primary variable. This choice is used to construct elements suitable for downstream coupling with mechanical and thermal finite elements for the analysis of electromagnetic/mechanical systems that involve superconductors. The main advantages of the four-potential as a basis for finite element formulation are: the number of degrees of freedom per node remains modest as the problem dimensionality increases, jump discontinuities on interfaces are naturally accomodated, and statics as well as dynamics may be treated without any a priori approximations. The new elements are tested on an axisymmetric problem under steady-state forcing conditions. The results are in excellent agreement with analytical solutions.

Superconducting axisymmetric finite elements based on a gauged potential variational principle—II. Solution and numerical results

María Jesús Samper — Fri, 14 Aug 2020 09:29:18 +0200

Part II of a two part paper discusses an incremental-iterative nonlinear solution technique for solving the nonlinear finite element equations of the superconducting state of a superconductor. The untreated equations are highly ill-conditioned and are impossible to solve within the typical 16-place double precision supplied by most computers. A combination of matrix scaling and mesh grading techniques is used to reduce the condition number of the tangent stiffness matrix and increase the accuracy of the current carrying boundary layer representation. Numerical results for a one-dimensional model of a time-independent superconductor treated by the Ginzburg-Landau model are presented and discussed. The computed solutions clearly display the Meissner effect of magnetic field expulsion from the central region of the superconductor. These results are compared to the physics of a low-viscosity fluid problem. From this analogy, a physical argument is advanced about the macroscopic behavior of superconductors.

Superconducting axisymmetric finite elements based on a gauged potential variational principle—I. Formulation

María Jesús Samper — Thu, 13 Aug 2020 14:43:09 +0200

The present work is part of a research program for the numerical simulation of electromagnetic (EM) fields within conventional Ginzburg-Landau (GL) superconductors. The final goal of this research is to formulate, develop and validate finite element (FE) models that can accurately capture electromagnetic, thermal and material phase changes in a superconductor. The formulations presented here are for a time-independent Ginzburg-Landau superconductor and are derived from a potential-based variational principle.

In Part I of this paper, we develop an appropriate variational formulation of time-independent superconductivity for the general three-dimensional case and specialize it to the one-dimensional case. Also developed are expressions for the material-dependent parameters α and β of GL theory and their dependence upon the temperature T. The one-dimensional formulation is then discretized for finite element purposes and the first variation of these equations is obtained. The resultant Euler equations contain nonlinear terms in the primary variables. To solve these equations, an incremental-iterative solution method is used. Expressions for the internal force vector, external force vector, loading vector and tangent stiffness matrix are therefore developed for use with the solution procedure.

Electromagnetics via the Taylor-Galerkin Finite Element Method on Unstructured Grids

María Jesús Samper — Thu, 09 Jul 2020 11:40:00 +0200

Traditional techniques for computing electromagnetic solutions in the time domain rely on finite differences. These so-called finite-difference time-domain (FDTD) methods are usually defined only on regular lattices of points and can be too restrictive for geometrically demanding problems. Great geometric flexibility can be achieved by abandoning the regular latticework of sample points and adopting an unstructured grid. An unstructured grid allows one to place the grid points anywhere one chooses, so that curved boundaries can be fit with ease and local regions in which the field gradients are steep can be selectively resolved with a fine mesh. In this paper we present a technique for solving Maxwell's equations on an unstructured grid based on the Taylor-Galerkin finite-element method. We present several numerical examples which reveal the fundamental accuracy and adaptability of the method. Although our examples are in two dimensions, the techniques and results generalize readily to 3D.

Finite element simulation of a turbulent MHD system: comparison to a pseudo-spectral simulation

María Jesús Samper — Wed, 08 Jul 2020 10:36:50 +0200

A finite element MHD algorithm is used to simulate a two-dimensional, viscous and resistive turbulent model, namely the Orszag-Tang vortex. The results are compared to a pseudo-spectral simulation of the same system reported by Dahlburg and Picone (Phys. Fluids B 1 (1989) 2153). The agreement of results from both methods supports the contention that the finite element method can appropriately simulate systems exhibiting turbulence, thus enabling the use of realistic geometries and boundary conditions, as well as adaptive refinement on simulations of turbulent systems. A short discussion on the behavior of ▿·B is presented. An inverse correlation between spatial resolution and the magnitude of ▿·B was found. The relevance of our findings to Adaptive Mesh Refinement is briefly discussed.

Simulation of the magnetosphere with a new three dimensional MHD code and adaptive mesh refinement: Preliminary results

María Jesús Samper — Tue, 07 Jul 2020 16:25:22 +0200

We present the first results from a new unstructured mesh three dimensional finite element MHD code which uses dynamic solution-adaptive mesh refinement in a manner similar to our two dimensional finite element MHD code /31/. The problem being considered here is the interaction of the solar wind with the earth's magnetosphere, using a three-dimensional Cartesian approximation. Our results strongly indicate that such adaptive mesh techniques have the ability to resolve structures in the three dimensional MHD flow field that would otherwise be possible only with orders of magnitude greater cost and that are most likely beyond the capability of present supercomputers.

Electromagnetic scattering calculations using a finite—element solver for the Maxwell equations

María Jesús Samper — Mon, 06 Jul 2020 11:44:32 +0200

We describe a pair of finite-element codes which use unstructured meshes to solve the time-dependent Maxwell equations, with particular emphasis on their application to electromagnetic scattering problems. A two-step, flux-corrected transport scheme is used to effect the time integration, while the spatial structure of the field is determined by a Galerkin procedure. The basis functions are piecewise-linear on three-noded triangles in two dimensions and four-noded tetrahedra in three. For the periodic scattering problems with which we are presently concerned, adaptive remeshing is a convenient and powerful method for improving the quality of the solutions. Results for the analytically tractable case of scattering by a perfectly conducting circular cylinder are used to illustrate the performance of the codes.

Iterative solution applied to the Helmholtz equation: Complex deflation on unstructured grids

María Jesús Samper — Wed, 01 Jul 2020 09:58:38 +0200

Extensions of deflation techniques developed for the Poisson and Navier equations (Aubry et al., 2008; Mut et al., 2010; Löhner et al., 2011; Aubry et al., 2011) [1], [2], [3], [4] are presented for the Helmholtz equation. Numerous difficulties arise compared to the previous case. After discretization, the matrix is now indefinite without Sommerfeld boundary conditions, or complex with them. It is generally symmetric complex but not Hermitian, discarding optimal short recurrences from an iterative solver viewpoint (Saad, 2003) [5]. Furthermore, the kernel of the operator in an infinite space typically does not belong to the discrete space. The choice of the deflation space is discussed, as well as the relationship between dispersion error and solver convergence. Similarly to the symmetric definite positive (SPD) case, subdomain deflation accelerates convergence if the low frequency eigenmodes are well described. However, the analytic eigenvectors are well represented only if the dispersion error is low. CPU savings are therefore restricted to a low to mid frequency regime compared to the mesh size, which could be still relevant from an application viewpoint, given the ease of implementation.

An outdoor Test Reference Environment for double skin applications of Building Integrated PhotoVoltaic Systems

María Jesús Samper — Tue, 05 May 2020 14:01:14 +0200

This article presents and discusses an outdoor Test Reference Environment (TRE) for double skin applications of Building Integrated PhotoVoltaic (BIPV) Systems.

From the experience gained during the past 20 years in several EC research projects, an experimental tested design for a common Test Reference Environment is proposed. This outdoor test set-up allows the assessment of experimental data for electrical and thermal performance evaluation of photovoltaic systems integrated as double skin applications in the building envelope. The specific design of the Test Reference Environment makes it possible to study in a harmonised way through electrical and thermal energy flow analysis, the impact of different materials for PV modules and construction design of building envelopes. The energy balance for BIPV double skin applications is presented as well.

The experimental data has been used for validation of modelling work by several academic groups which has resulted in an improved knowledge on the heat transfer, in particular the convective heat exchange coefficient for the specific double skin boundary conditions.

Development of a dynamic model for natural ventilated photovoltaic components and of a data driven approach to validate and identify the model parameters

María Jesús Samper — Tue, 05 May 2020 11:56:33 +0200

In the development of dynamic models for the energy performance evaluation of building integrated natural ventilated PV components there are still many open questions regarding the uncertainty of the estimated parameters of the models. Traditionally, the dynamic models for these complex components are derived from the heat transfer balance equations, and the unknown heat transfer coefficients (convection and radiation), the solar properties of the materials or the pressure coefficients for the air mass flow rate balance, are assigned based on literature or on manufacturer prescriptions. However, there is a lack of systematic methods able to validate the simulation outputs with the measured data, taking into consideration the uncertainty of the parameters and their effect over the results. This research is focused on the development of a dynamic simulation model for a PV ventilated component, and on the application of a data-driven iterative approach to identify the unknown parameters, to evaluate their influence in the simulation outputs and finally, to determine the deviations of the simulations outputs against the measured data. During the identification process, 43 unknown parameters are detected and 13 of them are categorized as strong parameters. The implemented data driven approach is able to achieve high goodness of fit with the measured data and it is recommended to analyses which aim at evaluating the influence of some component parameters or the thermal and electrical energy produced by these natural ventilated PV components.

Comparing time-series clustering approaches for individual electrical load patterns

María Jesús Samper — Tue, 05 May 2020 10:41:36 +0200

This work positions the task of grouping electricity load time series among the vast field of clustering, and highlights corresponding research issues. A selection of the most performant time-series clustering approaches from the signal processing community are compared on the same dataset, composed by domestic electricity load profiles from Spain. The cross-correlation-based distance of Paparrizos and Gravano (2015) is shown to provide the best trade-off between clustering accuracy and CPU times

Using matrix factorisation for the prediction of electrical quantities

María Jesús Samper — Tue, 05 May 2020 10:24:38 +0200

The prediction task is attracting more and more attention among the power system community. Accurate predictions of electrical quantities up to a few hours ahead (e.g. renewable production, electrical load etc.) are for instance crucial for distribution system operators to operate their network in the presence of a high share of renewables, or for energy producers to maximise their profits by optimising their portfolio management. In the literature, statistical approaches are usually proposed to predict electrical quantities. In the present study, the authors present a novel method based on matrix factorisation. The authors' approach is inspired by the literature on data mining and knowledge discovery and the methodologies involved in recommender systems. The idea is to transpose the problem of predicting ratings in a recommender system to a problem of forecasting electrical quantities in a power system. Preliminary results on a real wind speed dataset tend to show that the matrix factorisation model provides similar results than auto regressive integrated models in terms of accuracy (MAE and RMSE). The authors' approach is nevertheless highly scalable and can deal with noisy data (e.g. missing data).

EMPOWERING, a smart Big Data framework for sustainable electricity suppliers

María Jesús Samper — Tue, 05 May 2020 09:46:15 +0200

This paper presents the EMPOWERING project, a Big Data environment aimed at helping domestic customers to save electricity by managing their consumption positively. This is achieved by improving the information received about energy bills and offering online tools. The main contributions of EMPOWERING are the creation of a novel workflow in the electricity utility sector regarding the implementation of data analytics for their customers and the fast implementation of data-mining techniques in massive datasets within a Big Data platform to achieve scalability. The results obtained show that EMPOWERING can be of use for customers of electrical suppliers by changing their energy habits to decrease consumption and so increase environmental sustainability.

Open tools for electromagnetic simulation programs

María Jesús Samper — Thu, 16 Apr 2020 11:43:57 +0200

Purpose

The aim of the paper is to propose three computer tools to create electromagnetic simulation programs: GiD, Kratos and EMANT.

Design/methodology/approach

The paper presents a review of numerical methods for solving electromagnetic problems and presentation of the main features of GiD, Kratos and EMANT.

Findings

The paper provides information about three computer tools to create electromagnetic simulation packages: GiD (geometrical modeling, data input, visualisation of results), Kratos (C++ library) and EMANT (finite element software for solving Maxwell equations).

Research limitations/implications

The proposed platforms are in development and future work should be done to validate the codes for expecific problems and to provide extensive manual and tutorial information.

Practical implications

The tools could be easily learnt by different user profiles: from end‐users interested in simulation programs to developers of simulation packages.

Originality/value

This paper offers an integrated vision of open and easily customisable tools for the demands of different users profiles.

Scalable solvers for complex electromagnetics problems

María Jesús Samper — Tue, 14 Apr 2020 11:25:32 +0200

In this work, we present scalable balancing domain decomposition by constraints methods for linear systems arising from arbitrary order edge finite element discretizations of multi-material and heterogeneous 3D problems. In order to enforce the continuity across subdomains of the method, we use a partition of the interface objects (edges and faces) into sub-objects determined by the variation of the physical coefficients of the problem. For multi-material problems, a constant coefficient condition is enough to define this sub-partition of the objects. For arbitrarily heterogeneous problems, a relaxed version of the method is defined, where we only require that the maximal contrast of the physical coefficient in each object is smaller than a predefined threshold. Besides, the addition of perturbation terms to the preconditioner is empirically shown to be effective in order to deal with the case where the two coefficients of the model problem jump simultaneously across the interface. The new method, in contrast to existing approaches for problems in curl-conforming spaces does not require spectral information whilst providing robustness with regard to coefficient jumps and heterogeneous materials. A detailed set of numerical experiments, which includes the application of the preconditioner to 3D realistic cases, shows excellent weak scalability properties of the implementation of the proposed algorithms.

An approach to verification and validation of MHD codes for fusion applications

María Jesús Samper — Wed, 01 Apr 2020 17:25:23 +0200

We propose a new activity on verification and validation (V&V) of MHD codes presently employed by the fusion community as a predictive capability tool for liquid metal cooling applications, such as liquid metal blankets. The important steps in the development of MHD codes starting from the 1970s are outlined first and then basic MHD codes, which are currently in use by designers of liquid breeder blankets, are reviewed. A benchmark database of five problems has been proposed to cover a wide range of MHD flows from laminar fully developed to turbulent flows, which are of interest for fusion applications: (A) 2D fully developed laminar steady MHD flow, (B) 3D laminar, steady developing MHD flow in a non-uniform magnetic field, (C) quasi-two-dimensional MHD turbulent flow, (D) 3D turbulent MHD flow, and (E) MHD flow with heat transfer (buoyant convection). Finally, we introduce important details of the proposed activities, such as basic V&V rules and schedule. The main goal of the present paper is to help in establishing an efficient V&V framework and to initiate benchmarking among interested parties. The comparison results computed by the codes against analytical solutions and trusted experimental and numerical data as well as code-to-code comparisons will be presented and analyzed in companion paper/papers

Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

María Jesús Samper — Wed, 01 Apr 2020 16:41:34 +0200

The thermally coupled incompressible inductionless magnetohydrodynamics (MHD) problem models the flow of an electrically charged fluid under the influence of an external electromagnetic field with thermal coupling. This system of partial differential equations is strongly coupled and highly nonlinear for real cases of interest. Therefore, fully implicit time integration schemes are very desirable in order to capture the different physical scales of the problem at hand. However, solving the multiphysics linear systems of equations resulting from such algorithms is a very challenging task which requires efficient and scalable preconditioners. In this work, a new family of recursive block LU preconditioners is designed and tested for solving the thermally coupled inductionless MHD equations. These preconditioners are obtained after splitting the fully coupled matrix into one-physics problems for every variable (velocity, pressure, current density, electric potential and temperature) that can be optimally solved, e.g., using preconditioned domain decomposition algorithms. The main idea is to arrange the original matrix into an (arbitrary) 2 x 2 block matrix, and consider an LU preconditioner obtained by approximating the corresponding Schur complement. For every one of the diagonal blocks in the LU preconditioner, if it involves more than one type of unknowns, we proceed the same way in a recursive fashion. This approach is stated in an abstract way, and can be straightforwardly applied to other multiphysics problems. Further, we precisely explain a flexible and general software design for the code implementation of this type of preconditioners.

Unconditionally stable operator splitting algorithms for the incompressible magnetohydrodynamics (MHD) system discretized by a stabilized finite element formulation based on projections

María Jesús Samper — Tue, 31 Mar 2020 13:02:49 +0200

In this article, we propose different splitting procedures for the transient incompressible magnetohydrodynamics (MHD) system that are unconditionally stable. We consider two levels of splitting, on one side we perform the segregation of the fluid pressure and magnetic pseudo‐pressure from the vectorial fields computation. At the second level, the fluid velocity and induction fields are also decoupled. This way, we transform a fully coupled indefinite multi‐physics system into a set of smaller definite ones, clearly reducing the CPU cost. With regard to the finite element approximation, we stick to an unconditionally convergent stabilized finite element formulation because it introduces convection stabilization, allows to circumvent inf‐sup conditions (clearly simplifying implementation issues), and is able to capture non‐smooth solutions of the magnetic subproblem. However, residual‐based finite element formulations are not suitable for segregation, because they lose the skew‐symmetry of the off‐diagonal blocks. Therefore, in this work, we have proposed a novel term‐by‐term stabilization of the MHD system based on projections that is still unconditionally convergent.