A meshless numerical approach for the analysis of earthing systems in electrical installations

[Abstract] In the last three decades some numerical formulations have been developed for solving potential problems in electrical engineering applications. In the particular case of the grounding analysis area, in recent years we have developed a general numerical approach based on the Boundary Element Method for homogeneous and isotropic soil models, which has been succesfully applied to the analysis of large grounding systems. This numerical approach has been recently extended for the study of earthing grids embedded in stratified soils, which enables to solve some frequent practical cases, such as the two-layered soil models. Nevertheless, boundary element approaches imply a considerable computational effort when applied to the grounding analysis buried in more stratified soils or completely heterogeneous. This difficulty of the extremely high cost also arises which the use of standard numerical techniques (Finite Differences or Finite Elements) which require the discretization of the whole domain: the ground. Since early nineties, several numerical methods where meshes are unnecessary ("meshless methods") have been proposed in several engineering applications. In this paper, we briefly review some of these meshless techniques, and propose the use of a Moving Least Square methodology with a point collocation scheme for solving problems in electrical engineering. Furthermore, the use of enrichment procedure in these meshless formulations is explored to improve results and decrease the computational cost required.Ministerio de Educación y Cultura; 1FD97-010

Multi-field variational formulations and mixed finite element approximations for electrostatics and magnetostatics

The final publication is available at link.springer.com[EN] In this paper we propose different multi-field variational formulations for electrostatics and magnetostatics, which can provide optimal discrete approximation of any particular vector field. The proposed formulations are constructed by appealing to mechanics point of view amenable to using general constitutive equations, which is quite different from electrostatics and magnetostatics formulations typical of physics and electrical engineering focusing on the corresponding global form suitable only for linear case. In particular, the formulations we propose can be combined with mixed discrete approximations that can ensure the continuity of tangential component of electric ormagnetic field and normal component of electric displacement and magnetic flux even for low order interpolations. The choice of this kind is quite different from currently favorite choice of high order finite element interpolations used for coupling electromagnetism with mechanics. The discrete approximation is based upon Whitney's interpolations representing the vector fields in terms of corresponding differential forms, with electric and magnetic fields as one-form and electric displacement and magnetic flux as two-form. The implementation of interpolations of this kind is made for 3D tetrahedron elements with non-standard approximation parameters defined not only at vertices (for zero-form), but at edges (for one-form) and at facets (for two-form). The results of several numerical simulations are presented to illustrate the performance of different formulations proposed herein.This work was supported jointly by Haut-deFrance Region (CR Picardie) (120-2015-RDISTRUCT-000010 and RDISTRUCT-000010) and EU funding (FEDER) for Chaire-deMecanique (120-2015-RDISTRUCTF-000010 and RDISTRUCTI000004). AI was also supported by IUF.Moreno-Navarro, P.; Ibrahimbegovic, A.; Ospina, A. (2020). Multi-field variational formulations and mixed finite element approximations for electrostatics and magnetostatics. Computational Mechanics. 65(1):41-59. https://doi.org/10.1007/s00466-019-01751-xS4159651Albanese R, Rubinacci G (1997) Finite element methods for the solution of 3d eddy current problems. In: Advances in imaging and electron physics, vol 102, pp 1–86. ElsevierAlotto P, Freschi F, Repetto M, Rosso C (2013) The cell method for electrical engineering and multiphysics problems: an introduction, vol 230. Springer, BerlinAngoshtari A, Shojaei MF, Yavari A (2017) Compatible-strain mixed finite element methods for 2d compressible nonlinear elasticity. Comput Methods Appl Mech Eng 313:596–631Arnold DN, Falk RS, Winther R (2006) Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15:1–155Balanis CA (1999) Advanced engineering electromagnetics. Wiley, HobokenBamberg P, Sternberg S (1991) A course in mathematics for students of physics, vol 2. Cambridge University Press, CambridgeBochev PB, Robinson AC (2002) Matching algorithms with physics: exact sequences of finite element spaces. Collected Lectures on Preservation of Stability Under Discretization, pp 145–166Bossavit A (1988) A rationale for ‘edge-elements’ in 3-d fields computations. IEEE Trans Magn 24(1):74–79Bossavit A (1988) Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism. IEE Proc 135(8):493–500Bossavit A (2010) Discrete magneto-elasticity: a geometrical approach. IEEE Trans Magn 46(8):3485–3491Desbrun M, Kanso E, Tong Y (2008) Discrete differential forms for computational modeling. In: Discrete differential geometry, pp 287–324. SpringerDular P, Geuzaine C (1997) Getdp: a general environment for the treatment of discrete problemsDular P, Hody J-Y, Nicolet A, Genon A, Legros W (1994) Mixed finite elements associated with a collection of tetrahedra, hexahedra and prisms. IEEE Trans Magn 30(5):2980–2983Felippa CA, Park KC, Farhat C (2001) Partitioned analysis of coupled mechanical systems. Comput Methods Appl Mech Eng 190(24–25):3247–3270Gil AJ, Ortigosa R (2016) A new framework for large strain electromechanics based on convex multi-variable strain energies: variational formulation and material characterisation. Comput Methods Appl Mech Eng 302:293–328Golias NA, Tsiboukis TD, Bossavit A (1994) Constitutive inconsistency: rigorous solution of maxwell equations based on a dual approach. IEEE Trans Magn 30(5):3586–3589Gradinaru V, Hiptmair R (1999) Whitney elements on pyramids. Electron Trans Numer Anal 8:154–168Hale HW (1961) A logic for identifying the trees of a graph. Trans Am Inst Electr Eng Part III Power Apparatus Syst 80(3):195–197Hammond P (2013) Electromagnetism for engineers: an introductory course. Elsevier, AmsterdamHammond P, Penman J (1976) Calculation of inductance and capacitance by means of dual energy principles. In: Proceedings of the Institution of Electrical Engineers, vol 123, pp 554–559. IETHammond P, Tsiboukis TD (1983) Dual finite-element calculations for static electric and magnetic fields. IEE Proc A (Phys Sci Measurement Instrum Manag Educ Rev) 130(3):105–111Ibrahimbegovic A (2009) Nonlinear solid mechanics: theoretical formulations and finite element solution methods, vol 160. Springer, DordrechtJackson JD (1999) Classical electrodynamics. Wiley, New YorkKameari A (1989) Three dimensional eddy current calculation using edge elements for magnetic vector potential. In: Applied electromagnetics in materials, pp 225–236. ElsevierKassiotis C, Ibrahimbegovic A, Niekamp R, Matthies HG (2011) Nonlinear fluid–structure interaction problem. Part I: implicit partitioned algorithm, nonlinear stability proof and validation examples. Comput Mech 47(3):305–323Kassiotis C, Ibrahimbegovic A, Niekamp R, Matthies HG (2011) Nonlinear fluid–structure interaction problem. Part II: space discretization, implementation aspects, nested parallelization and application examples. Comput Mech 47(3):335–357Keip Marc-André, Steinmann Paul, Schröder Jörg (2014) Two-scale computational homogenization of electro-elasticity at finite strains. Comput Methods Appl Mech Eng 278:62–79Ladevèze P, Pelle J-P (2005) Mastering calculations in linear and nonlinear mechanics, vol 171. Springer, New YorkMacneal RH, Harder Robert L (1985) A proposed standard set of problems to test finite element accuracy. Finite Elem Anal Des 1(1):3–20Manges J, Cendes Z (1996) Generation of tangential vector finite elements. Int Compumag Soc Newsl 3(1):4–10Mitchell BS (2004) An introduction to materials engineering and science for chemical and materials engineers. Wiley, HobokenMoreno-Navarro P, Ibrahimbegovic A, Pérez-Aparicio JL (2017) Plasticity coupled with thermo-electric fields: thermodynamics framework and finite element method computations. Comput Methods Appl Mech Eng 315:50–72Moreno-Navarro P, Ibrahimbegovic A, Pérez-Aparicio JL (2018) Linear elastic mechanical system interacting with coupled thermo-electro-magnetic fields. Coupled Syst Mech 7(1):5–25Nédélec J-C (1980) Mixed finite elements in $$\mathbb{R}^3$$. Numer Math 35(3):315–341Penman J, Fraser J (1982) Complementary and dual energy finite element principles in magnetostatics. IEEE Trans Magn 18(2):319–324Razek A (1995) Computation of 3d electrostatic local fields and forces using complementarity of dual formulations. Application for capacitance and torque calculations in micromotorsRen Z (1995) A 3d vector potential formulation using edge element for electrostatic field computation. IEEE Trans Magn 31(3):1520–1523Ren Z (2009) On the complementarity of dual formulations on dual meshes. IEEE Trans Magn 45(3):1284–1287Repetto M, Trevisan F (2004) Global formulation of 3d magnetostatics using flux and gauged potentials. Int J Numer Meth Eng 60(4):755–772Schröder J, Keip M-A (2012) Two-scale homogenization of electromechanically coupled boundary value problems. Comput Mech 50(2):229–244Stark S, Semenov AS, Balke H (2015) On the boundary conditions for the vector potential formulation in electrostatics. Int J Numer Meth Eng 102(11):1704–1732Taylor RL (2012) Feap–a finite element analysis program. Version 8.4 Theory ManualTonti E (2013) The mathematical structure of classical and relativistic physics. Springer, New YorkVogel F, Bustamante R, Steinmann P (2012) On some mixed variational principles in electro-elastostatics. Int J Nonlinear Mech 47(2):341–354Wang J-S, Ida N (1993) Curvilinear and higher order ‘edge’ finite elements in electromagnetic field computation. IEEE Trans Magn 29(2):1491–1494Webb JP, Forgahani B (1993) Hierarchal scalar and vector tetrahedra. IEEE Trans Magn 29(2):1495–1498Yioultsis TV, Tsiboukis TD (1996) The mystery and magic of Whitney elements—an insight in their properties and construction. ICS Newsl 3:1389–1392Zienkiewicz OC, Taylor RL (1977) The finite element method, vol 36. McGraw-Hill, Londo

Nonlinear Elastic Material Property Estimation of Lower Extremity Residual Limb Tissues

The interface stresses between the residual limb and prosthetic socket have been studied to investigate prosthetic fit. Finite-element models of the residual limb-prosthetic socket interface facilitate investigation of the mechanical interface and may serve as a potential tool for future prosthetic socket design. However, the success of such residual limb models to date has been limited, in large part due to inadequate material formulations used to approximate the mechanical behavior of residual limb soft tissues. Nonlinear finite-element analysis was used to simulate force-displacement data obtained during in vivo rate-controlled (1, 5, and 10 mm/s) cyclic indentation of the residual limb soft tissues of seven individuals with transtibial amputation. The finite-element models facilitated determination of an appropriate set of nonlinear elastic material coefficients for bulk soft tissue at discrete clinically relevant test locations. Axisymmetric finite-element models of the residual limb bulk soft tissue in the vicinity of the test location, the socket wall and the indentor tip were developed incorporating contact analysis, large displacement, and large strain, and the James-Green-Simpson nonlinear elastic material formulation. Model dimensions were based on medical imaging studies of the residual limbs. The material coefficients were selected such that the normalized sum of square error (NSSE) between the experimental and finite-element model indentor tip reaction force was minimized. A total of 95% of the experimental data were simulated using the James-Green-Simpson material formulation with an NSSE less than 5%. The respective James-Green-Simpson material coefficients varied with subject, test location, and indentation rate. Therefore, these coefficients cannot be readily extrapolated to other sites or individuals, or to the same site and individual some time after testing

Stochastic post-processing calculation of iron losses – application to a PMSM

To take account of the uncertainties introduced on the magnetic properties during the manufacturing process, the present work aims to focus on the stochastic modelling of iron losses in electrical machine stators. The investigated samples are composed of 28 slinky stators, coming from the same production chain. The stochastic modelling approach is first described. Thereafter, the Monte-Carlo sampling method is used to calculate, in post-processing, the iron loss density in a PMSM that is modelled by the finite element method. The interest of such an approach is emphasized by calculating the main statistical characteristics associated to the losses variability, which are Gaussian distributed for A and O formulations. The originality of the approach is due to the fact that the global influence of the manufacturing process (cutting, assembly, …) on magnetic properties of the considered samples is taken into account in the way of computing the iron losses.This work is supported by the program MEDEE (Nord Pas-de-Calais Region, France

Some Key Developments in Computational Electromagnetics and their Attribution

Key developments in computational electromagnetics are proposed. Historical highlights are summarized concentrating on the two main approaches of differential and integral methods. This is seen as timely as a retrospective analysis is needed to minimize duplication and to help settle questions of attribution

Evolution and Modern Approaches for Thermal Analysis of Electrical Machines

In this paper, the authors present an extended survey on the evolution and the modern approaches in the thermal analysis of electrical machines. The improvements and the new techniques proposed in the last decade are analyzed in depth and compared in order to highlight the qualities and defects of each. In particular, thermal analysis based on lumped-parameter thermal network, finite-element analysis, and computational fluid dynamics are considered in this paper. In addition, an overview of the problems linked to the thermal parameter determination and computation is proposed and discussed. Taking into account the aims of this paper, a detailed list of books and papers is reported in the references to help researchers interested in these topics

Multiscale Finite Element Modeling of Nonlinear Magnetoquasistatic Problems Using Magnetic Induction Conforming Formulations

In this paper we develop magnetic induction conforming multiscale formulations for magnetoquasistatic problems involving periodic materials. The formulations are derived using the periodic homogenization theory and applied within a heterogeneous multiscale approach. Therefore the fine-scale problem is replaced by a macroscale problem defined on a coarse mesh that covers the entire domain and many mesoscale problems defined on finely-meshed small areas around some points of interest of the macroscale mesh (e.g. numerical quadrature points). The exchange of information between these macro and meso problems is thoroughly explained in this paper. For the sake of validation, we consider a two-dimensional geometry of an idealized periodic soft magnetic composite.Comment: Paper accepted for publication in the SIAM MMS journa

Error estimation of a proper orthogonal decomposition reduced model of a permanent magnet synchronous machine

Model order reduction methods, like the proper orthogonal decomposition (POD), enable to reduce dramatically the size of a finite element (FE) model. The price to pay is a loss of accuracy compared with the original FE model that should be of course controlled. In this study, the authors apply an error estimator based on the verification of the constitutive relationship to compare the reduced model accuracy with the full model accuracy when POD is applied. This estimator is tested on an example of a permanent magnet synchronous machine.This work is supported by the IAP7/M2E2S (Belgium state) and MEDEE pole supported by the region council of Nord Pas de Calais (France) and the European Community