The electrical conductivity, , can be isotropic, orthotropic, or fully anisotropic (see “Electrical conductivity,” Section 20.6.1). Ohm's law assumes that the electrical conductivity is independent of the electrical field, . The coupled thermal-electrical problem is nonlinear when the electrical conductivity depends on temperature.

A steady-state analysis has no intrinsic physically meaningful time scale. Nevertheless, you can assign a “time” scale to the analysis step, which is often convenient for output identification and for specifying prescribed temperatures, electrical potential, and fluxes (heat flux and current density) with varying magnitudes. Thus, when steady-state analysis is chosen, you specify a “time” increment and a “time” period for the step; ABAQUS/Standard then increments through the step accordingly.

Any fluxes or boundary condition changes to be applied during a steady-state step should be given using appropriate amplitude references to specify their “time” variations (“Amplitude curves,” Section 27.1.2). If fluxes and boundary conditions are specified for the step without amplitude references, they are assumed to change linearly with “time” during the step—from their magnitudes at the end of the previous step (or zero, if this is the beginning of the analysis) to their newly specified magnitudes at the end of this step (see “Procedures: overview,” Section 6.1.1).

Time integration in the transient heat transfer problem is done with the same backward Euler method used in uncoupled heat transfer analysis. This method is unconditionally stable for linear problems. You can specify the time increments directly, or ABAQUS can select them automatically based on a user-prescribed maximum nodal temperature change in an increment. Automatic time incrementation is generally preferred.

*COUPLED THERMAL-ELECTRICAL, DELTMX=

*COUPLED THERMAL-ELECTRICAL

Spurious oscillations due to small time increments

In transient heat transfer analysis with second-order elements there is a relationship between the minimum usable time increment and the element size. A simple guideline is

where is the time increment, is the density, c is the specific heat, k is the thermal conductivity, and is a typical element dimension (such as the length of a side of an element). If time increments smaller than this value are used in a mesh of second-order elements, spurious oscillations can appear in the solution, in particular in the vicinity of boundaries with rapid temperature changes. These oscillations are nonphysical and may cause problems if temperature-dependent material properties are present. In transient analyses using first-order elements the heat capacity terms are lumped, which eliminates such oscillations but can lead to locally inaccurate solutions for small time increments. If smaller time increments are required, a finer mesh should be used in regions where the temperature changes rapidly.

There is no upper limit on the time increment size (the integration procedure is unconditionally stable) unless nonlinearities cause convergence problems.

By default, a transient analysis will end when the specified time period has been completed. Alternatively, you can specify that the analysis should continue until steady-state conditions are reached. Steady state is defined by the temperature change rate; when the temperature changes at a rate that is less than the user-specified rate (given as part of the step definition), the analysis terminates.

*COUPLED THERMAL-ELECTRICAL, END=PERIOD (default)

*COUPLED THERMAL-ELECTRICAL, END=SS

An exact implementation of Newton's method involves a nonsymmetric Jacobian matrix as is illustrated in the following matrix representation of the coupled equations:

Solving this system of equations requires the use of the unsymmetric matrix storage and solution scheme. Furthermore, the electrical and thermal equations must be solved simultaneously. The method provides quadratic convergence when the solution estimate is within the radius of convergence of the algorithm. The exact implementation is used by default.

Some problems require a fully coupled analysis in the sense that the electrical and thermal solutions evolve simultaneously, but with a weak coupling between the two solutions. In other words, the components in the off-diagonal submatrices are small compared to the components in the diagonal submatrices . For these problems a less costly solution may be obtained by setting the off-diagonal submatrices to zero, so that we obtain an approximate set of equations:

As a result of this approximation the electrical and thermal equations can be solved separately, with fewer equations to consider in each subproblem. The savings due to this approximation, measured as solver time per iteration, will be of the order of a factor of two, with similar significant savings in solver storage of the factored stiffness matrix. Further, in situations without strong thermal loading due to cavity radiation, the subproblems may be fully symmetric or approximated as symmetric, so that the less costly symmetric storage and solution scheme can be used. The solver time savings for a symmetric solution is an additional factor of two. Unless you explicitly select the unsymmetric solver for the step (“Procedures: overview,” Section 6.1.1), the symmetric solver will be used with this separated technique.

This modified form of Newton's method does not affect solution accuracy since the fully coupled effect is considered through the residual vector at each increment in time. However, the rate of convergence is no longer quadratic and depends strongly on the magnitude of the coupling effect, so more iterations are generally needed to achieve equilibrium than with the exact implementation of Newton's method. When the coupling is significant, the convergence rate becomes very slow and may prohibit the attainment of a solution. In such cases the exact implementation of Newton's method is required. In cases where it is possible to use this approximation, the convergence in an increment will depend strongly on the quality of the first guess to the incremental solution, which you can control by selecting the extrapolation method used for the step (see “Procedures: overview,” Section 6.1.1).

*SOLUTION TECHNIQUE, TYPE=SEPARATED