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Solver Robustness in Joule Heating Model with Field and Temperature Dependent Conductivity
Posted Mar 6, 2013, 5:13 a.m. EST 1 Reply
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Hi there,
I am trying to solve a Joule heating problem in 2D axial symmetry. The challenge in this system is the temperature "T" and electric field "E" dependence of the conductivity "sigma" of one of the materials. It follows:
sigma(E,T) ~ p / ( 1 - c1 * exp(E / E0) )
p(T) ~ T^(3/2) * exp( - c2 / T )
c1, c2 and E0 are constants. At the critical field "E0", the conductivity has a singularity. Thus above c1 * exp(E / E0) > 1, the equation is not well defined anymore. Therefore, I introduced a cut off conductivity "sigma0":
sigma0(T) ~ T^(3/2)
The total conductivity becomes now:
min( (c1 * exp(E / E0) < 1) * sigma(E,T) + (c1 * exp(E / E0) > 1) * sigma0(T) , sigma0(T) )
Without the temperature dependence the system is solvable with a fully coupled solver using the double dogleg nonlinear method. However, it is not very robust. For some input voltages I have to switch from the direct solver (MUMPS) to an iterative solver (FGMRES) and play with the relaxation factor of the incomplete LU preconditioner.
It seems that the convergence problem arises from the steepness of the field dependence before the critical field. When I use an expansion of the field dependence 1/(1-x) --> 1+x+x^2+x^3+... up to the 5th order I can still solve the problem very robust after that it becomes more and more unstable.
Including the temperature dependence, Comsol solves the problem for low input voltages (resulting in almost no heating). In this case I use the automatic highly nonlinear (Newton) setting for the nonlinear method. As soon as the temperature rise is significant the system does not converge anymore.
I would like to get help now with respect to the robustness. Is there any way, change of settings or change of implementation of the equations, that will make this problem solvable for Comsol for most of the input voltages?
Many thanks in advance,
Daniel Krebs
I am trying to solve a Joule heating problem in 2D axial symmetry. The challenge in this system is the temperature "T" and electric field "E" dependence of the conductivity "sigma" of one of the materials. It follows:
sigma(E,T) ~ p / ( 1 - c1 * exp(E / E0) )
p(T) ~ T^(3/2) * exp( - c2 / T )
c1, c2 and E0 are constants. At the critical field "E0", the conductivity has a singularity. Thus above c1 * exp(E / E0) > 1, the equation is not well defined anymore. Therefore, I introduced a cut off conductivity "sigma0":
sigma0(T) ~ T^(3/2)
The total conductivity becomes now:
min( (c1 * exp(E / E0) < 1) * sigma(E,T) + (c1 * exp(E / E0) > 1) * sigma0(T) , sigma0(T) )
Without the temperature dependence the system is solvable with a fully coupled solver using the double dogleg nonlinear method. However, it is not very robust. For some input voltages I have to switch from the direct solver (MUMPS) to an iterative solver (FGMRES) and play with the relaxation factor of the incomplete LU preconditioner.
It seems that the convergence problem arises from the steepness of the field dependence before the critical field. When I use an expansion of the field dependence 1/(1-x) --> 1+x+x^2+x^3+... up to the 5th order I can still solve the problem very robust after that it becomes more and more unstable.
Including the temperature dependence, Comsol solves the problem for low input voltages (resulting in almost no heating). In this case I use the automatic highly nonlinear (Newton) setting for the nonlinear method. As soon as the temperature rise is significant the system does not converge anymore.
I would like to get help now with respect to the robustness. Is there any way, change of settings or change of implementation of the equations, that will make this problem solvable for Comsol for most of the input voltages?
Many thanks in advance,
Daniel Krebs
1 Reply Last Post Mar 6, 2013, 6:34 a.m. EST
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Posted:
1 decade ago
Dear Daniel,
Your intention to avoid the singularity is in generla very good.
However, note that a function min(x,y) is not differentiable. The solvers however, especially for nonlinear
problems, depend on the fact that functions can be differentiated ( because a Newton approach is used).
Hence it would be better to multiply your function with a step function (of finite smoothness).
sigma = sigma_ugly-step1(T)*(sigma_ugly-sigma_safe)
step1 will be equal to one above the threshold and thus constrain sigma to sigma_safe.
sigma_ugly and sigma_safe can both be expressions.
The key is that you can control the smoothness of step1.
Further questions on that I prefer to answer in your support case.
Sven
Your intention to avoid the singularity is in generla very good.
However, note that a function min(x,y) is not differentiable. The solvers however, especially for nonlinear
problems, depend on the fact that functions can be differentiated ( because a Newton approach is used).
Hence it would be better to multiply your function with a step function (of finite smoothness).
sigma = sigma_ugly-step1(T)*(sigma_ugly-sigma_safe)
step1 will be equal to one above the threshold and thus constrain sigma to sigma_safe.
sigma_ugly and sigma_safe can both be expressions.
The key is that you can control the smoothness of step1.
Further questions on that I prefer to answer in your support case.
Sven