Finite element approaches to the PTC thermistor problem

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Abstract

In this paper, subdomain collocation (SC) and Petrov–Galerkin (PG) methods based on spline finite elements have been applied to the one-dimensional positive temperature coefficient (PTC) thermistor problem with a temperature dependent ramp function electrical conductivity to obtain the predicted temperature distributions and the locations of the free boundaries in the steady-state case. The numerical results obtained by the present methods have been compared with the exact solution and also those obtained by earlier authors. A Fourier stability analysis of each method is investigated.

Introduction

In this paper we consider an initial boundary value problem of coupled non-linear partial differential equations which consists of a heat-flow equation with the joule heating as a source and a current flow equation with a temperature dependent electrical conductivity. This problem is a thermo-electric problem called as a PTC thermistor problem which is mathematically modeled as follows:

Heat-flow problem:Ut=Uxx+ασ(φx)2,0<x<1,t>0,Ux(0,t)=0,t>0,Ux(1,t)+βU(1,t)=0,t>0,U(x,0)=0,0⩽x⩽1.

Current-flow problem:(σφx)x=0,0<x<1,t>0,φ(0,t)=0,t>0,φ(1,t)=1,t>0,φ(x,0)=x,0⩽x⩽1,where U(x,t) is the temperature, φ(x,t) is the electrical potential, β is a surface heat transfer coefficient, α is the ratio of electrical conductivity and σ is a temperature dependent electrical conductivity which decreases as the temperature increases.

The existence, uniqueness and behaviour of solutions to this problem were given in [1], [2], [3], [4], [5], [6]. Numerical solutions of the PTC thermistor problem have been undertaken by employing various forms of the finite element [7], [8], [9], finite difference [10], [11], [12], [13] and heat balance integral methods [14], [15].

In this paper, SC and PG methods have been applied to the PTC thermistor problem with an electrical conductivity proposed in the next section to obtain its approximate steady-state solutions. The numerical results obtained by both methods are compared with exact ones and also earlier work.

Section snippets

Electrical conductivities for the problem

Many authors [7], [11], [15] considered the above given thermistor problem with a step function electrical conductivity of the formσ(U)=1,0⩽U⩽1,δ∼10−5,U>1.

In a postscript, Howison [16] makes the point that a step function is a crude representation for σ(U). Recently, Kutluay and Wood [12] introduced a slightly more realistic model for σ(U) of the formσ(U)=1,0⩽U⩽1,1+(1−U)(1−δ),1<U<2,δ∼10−5,U⩾2,which is mathematically equivalent to a ramp function. They have applied a standard explicit finite

Warm/cold solution

We obtain the exact steady-state solution U(x) to this phase by using a standard analytical method asU(x)=2c1coshα(1−δ)s1x+1(1−δ)s1+1,wherec1=βeα(1−δ)s11(δ−1)s1−1β−α(1−δ)s1+β+α(1−δ)s1e2α(1−δ)s1.Inserting U(s1)=1 into Eq. (13) leads to a non-linear equation for the free interface s1:2c1(1−δ)s1coshs1α(1−δ)s1+1=0,which can be solved numerically. If s1 becomes equal to 1, Eq. (14) givesβ=α(1−δ)tanhα(1−δ)(1−δ)and so the problem enters a warm phase. Since 0<s1⩽1 in this phase, we obtain an

The finite element solutions

The region [0,1] is partitioned into N finite elements of equal length h by the knots xi such that 0=x0<x1<⋯<xN−1<xN=1. Over the element [xm,xm+1], an approximation to the exact solution U(x,t) in terms of quadratic B-splines Qm with the required properties can be given as [17]UN=Qm−1δm−1+Qmδm+Qm+1δm+1,where the δm are time dependent element parameters to be determined from the boundary and weighted residual conditions. In terms of a local coordinate system ξ given by ξ=xxm (0⩽ξh) the base

Numerical results and conclusion

In order to obtain the steady-state temperature distribution U(x,t) and the locations of the free boundaries s1 and s2, respectively, in the warm/cold and hot/warm phases constructed in the previous sections we used criteria1kmaxU0new−U0old,UNnew−UNold⩽0.5×10−8with k=0.01 for both the SC and PG solutions.

In order to show how good the numerical solutions of the problem in comparison with the exact ones we used the error norm L2 defined byL2=1N+1m=0NU(xm)−U(xm)21/2.

To compare our results

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