Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions

Holst, Michael; Larson, Mats G.; Målqvist, Axel; Söderlund, Robert

doi:10.1007/s10543-010-0287-z

Cited by 12 publications

(11 citation statements)

References 11 publications

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“…Theoretical analyses for the nonlinear thermistor system have been done extensively, see, e.g., [3,5,9,19,32,33]. Numerical methods and analysis for the time-dependent nonlinear thermistor system (1.1)-(1.4) can be found in [2,4,12,31,34,35].…”

Section: Introductionmentioning

confidence: 99%

Unconditional Optimal Error Estimates of BDF–Galerkin FEMs for Nonlinear Thermistor Equations

Gao

2015

J Sci Comput

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In this paper we study linearized backward differential formula (BDF) type schemes with Galerkin finite element approximations for the time-dependent nonlinear thermistor equations. Optimal L 2 error estimates for the proposed schemes are proved unconditionally. The proof consists of two steps. First, the boundedness of the numerical solution in certain strong norms is obtained by a temporal-spatial error splitting argument. Second, a traditional approach is used to provide an optimal L 2 error estimate for r -th order FEMs (r ≥ 1). Numerical experiments in both two and three dimensional spaces are conducted to confirm our theoretical analysis and show the high order accuracy and unconditional stability (convergence) of the linearized BDF-Galerkin FEMs.

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Section: Introductionmentioning

confidence: 99%

Unconditional Optimal Error Estimates of BDF–Galerkin FEMs for Nonlinear Thermistor Equations

Gao

2015

J Sci Comput

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“…An application of Grönwall's lemma then shows that the left-hand side is bounded by Ck 2 . Using Equation (16) From these preliminary bounds, we may deduce the desired regularity of Θ n and Φ n and then test (17) with −∆e n θ to acquire e n θ 2…”

Section: Error Analysismentioning

confidence: 99%

Finite element convergence analysis for the thermoviscoelastic Joule heating problem

Målqvist

Stillfjord

2017

Bit Numer Math

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We consider a system of equations that model the temperature, electric potential and deformation of a thermoviscoelastic body. A typical application is a thermistor; an electrical component that can be used e.g. as a surge protector, temperature sensor or for very precise positioning. We introduce a full discretization based on standard finite elements in space and a semi-implicit Euler-type method in time. For this method we prove optimal convergence orders, i.e. second-order in space and firstorder in time. The theoretical results are verified by several numerical experiments in two and three dimensions.

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“…(1) In the analysis of similar coupled problems in the field of Joule heating, the difficulties with the right-hand side data not having sufficient regularity in order to ensure uniqueness of the weak solution are also encounted. By following a similar approach to [32,33] and imposing boundedness on the solution to weak form of the electrostatics problem we can (22) thus ensuring uniqueness of the solution to (24).…”

Section: Electrostrictionmentioning

confidence: 99%

“…In the case of linearised electrostriction, the weak forms of electrostatics (32) and linearised elasticity (using either the displacement (33) or the mixed formulation (34)) are coupled in the sense that the deformation dependent permittivity tensor OEOE r .u/ becomes a function of the small strain tensor OEOE".u/, and the Cauchy stress tensor OEOE Q becomes a function of the electric field E . For materials with a weak electrostrictive coupling, Knops [3] has proposed a simplified mathematical model in which the permittivity tensor is assumed to be invariant to the linearised strain tensor, so that the electrostatic and linearised elasticity problems decouple resulting in OEOE r .u/ D r I.…”

Section: Coupled Strategymentioning

confidence: 99%

A coupled hp‐finite element scheme for the solution of two‐dimensional electrostrictive materials

Gil

Ledger

2012

Numerical Meth Engineering

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As part of the ongoing research within the field of computational analysis for the coupled electro-magnetomechanical response of smart materials, the problem of linearised electrostriction is revisited and analysed for the first time using the framework of hp-finite elements. The governing equations modelling the physics of the dielectric are suitably modified by introducing a new total Cauchy stress tensor (A. Dorfmann and R.W. Ogden. Nonlinear electroelasticity. Acta Mechanica, 174:167-183, 2005), which includes the electrostrictive effect and a staggered partitioned scheme for the numerical solution of the coupling phenomena. With the purpose of benchmarking numerical results, the problem of an infinite electrostrictive plate with a circular/elliptical dielectric insert is revisited. The presented analytical solution is based on the theoretical framework for two-dimensional electrostriction proposed by Knops (R.J. Knops. Twodimensional electrostriction. Quarterly Journal of Mechanics and Applied Mathematics, 16:377-388, 1963) and uses classical techniques of complex variable analysis. Our presentation, to the best of our knowledge, provides the first correct closed form expression for the solution to the infinite electrostrictive plate with a circular/elliptical dielectric insert, correcting the errors made in previous presentations of this problem. We use this analytical solution to assess the accuracy, efficiency and robustness of the hp-formulation in the case of nearly incompressible electrostrictive materials. 1159where Maxwell's equations, which govern electromagnetic phenomena, reduce to the electrostatic model in which the electric field can be described by the gradient of a scalar potential, and the Navier-Stokes equations, which describe the mechanical-fluid behaviour, reduce to the equations of linearised elasticity in the case of small strains. The coupling being through nonlinear constitutive laws describing the electrostrictive materials.Under the assumption of very small strains, so that the effect of the electric field on the deformation of the dielectric can be ignored, Knops [3] has presented a theoretical framework for analysing two-dimensional problems in electrostriction. With the use of classical techniques of complex variable analysis, this framework has been subsequently used to obtain analytical solutions to a number of benchmark problems including an infinite electrostrictive plate with a rigid circular/elliptical dielectric insert [3][4][5][6]. Clearly, this type of analytical solution can only be obtained for simple geometries and therefore has only limited practical application. Nevertheless, such solutions are extremely useful for benchmarking computation procedures for electrostriction. We therefore revisit the classical solution of an infinite electrostrictive plate with a rigid circular/elliptical dielectric insert and correct the mistakes found in previous presentations and present, to the best of our knowledge, the first correct solution to this problem.Our computational ap...

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Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions

Cited by 12 publications

References 11 publications

Unconditional Optimal Error Estimates of BDF–Galerkin FEMs for Nonlinear Thermistor Equations

Unconditional Optimal Error Estimates of BDF–Galerkin FEMs for Nonlinear Thermistor Equations

Finite element convergence analysis for the thermoviscoelastic Joule heating problem

A coupled hp‐finite element scheme for the solution of two‐dimensional electrostrictive materials

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