Fully coupled thermal-electrical-structural analysis

Assigning a “time” scale to the analysis

In steady-state cases you should assign an arbitrary “time” scale to the step: you specify a “time” period and “time” incrementation parameters. This time scale is convenient for changing loads and boundary conditions through the step and for obtaining solutions to highly nonlinear (but steady-state) cases; however, for the latter purpose, transient analysis often provides a natural way of coping with the nonlinearity.

Accounting for frictional slip heat generation

Frictional slip heat generation is normally neglected in the steady-state case. However, it can still be accounted for if user subroutine FRIC provides the incremental frictional dissipation through the variable SFD. If frictional heat generation is present, the heat flux into the two contact surfaces depends on the slip rate of the surfaces. The “time” scale in this case cannot be described as arbitrary, and a transient analysis should be performed.

Automatic incrementation controlled by a maximum allowable temperature change

The time increments can be selected automatically based on a user-prescribed maximum allowable nodal temperature change in an increment, $\mathrm{\Delta }{\theta }_{max}$. Abaqus/Standard will restrict the time increments to ensure that this value is not exceeded at any node (except nodes with boundary conditions) during any increment of the analysis (see Time integration accuracy in transient problems).

COUPLED TEMPERATURE-DISPLACEMENT, ELECTRICAL, DELTMX=$\mathrm{\Delta }{\theta }_{max}$

Step module: Create Step: General: Coupled thermal-electrical-structural: Basic: Response: Transient; Incrementation: Type: Automatic: Max. allowable temperature change per increment: $\mathrm{\Delta }{\theta }_{max}$

Fixed incrementation

If you do not specify $\mathrm{\Delta }{\theta }_{max}$, fixed time increments equal to the user-specified initial time increment, $\mathrm{\Delta }{t}_0$, will be used throughout the analysis, except when the explicit creep integration scheme is used. In this case Abaqus/Standard might decrease the time increment if the stability limit is exceeded.

COUPLED TEMPERATURE-DISPLACEMENT, ELECTRICAL
$\mathrm{\Delta }{t}_0$

Step module: Create Step: General: Coupled thermal-electrical-structural: Basic: Response: Transient; Incrementation: Type: Fixed: Increment size: $\mathrm{\Delta }{t}_0$

Spurious oscillations due to small time increments

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

where $\mathrm{\Delta }t$ is the time increment, $\rho$ is the density, c is the specific heat, k is the thermal conductivity, and $\mathrm{\Delta }\mathrm{\ell }$ 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.

Automatic incrementation controlled by the creep response

The accuracy of the integration of time-dependent (creep) material behavior is governed by the user-specified accuracy tolerance parameter, $tolerance\ge \left(\dot{\overline{\epsilon }}^{cr}{}_{t+\mathrm{\Delta }t}-\dot{\overline{\epsilon }}^{cr}{}_t\right)\mathrm{\Delta }t$. This parameter is used to prescribe the maximum strain rate change allowed at any point during an increment, as described in Rate-dependent plasticity: creep and swelling. The accuracy tolerance parameter can be specified together with the maximum allowable nodal temperature change in an increment, $\mathrm{\Delta }{\theta }_{max}$ (described above); however, specifying the accuracy tolerance parameter activates automatic incrementation even if $\mathrm{\Delta }{\theta }_{max}$ is not specified.

COUPLED TEMPERATURE-DISPLACEMENT, ELECTRICAL, DELTMX=$\mathrm{\Delta }{\theta }_{max}$, CETOL=tolerance

Step module: Create Step: General: Coupled thermal-electrical-structural: Basic: Response: Transient, toggle on Include creep/swelling/viscoelastic behavior; Incrementation: Type: Automatic: Max. allowable temperature change per increment: $\mathrm{\Delta }{\theta }_{max}$, Creep/swelling/viscoelastic strain error tolerance: tolerance

Selecting explicit creep integration

Nonlinear creep problems (Rate-dependent plasticity: creep and swelling) that exhibit no other nonlinearities can be solved efficiently by forward-difference integration of the inelastic strains if the inelastic strain increments are smaller than the elastic strains. This explicit method is efficient computationally because, unlike implicit methods, iteration is not required as long as no other nonlinearities are present. Although this method is only conditionally stable, the numerical stability limit of the explicit operator is in many cases sufficiently large to allow the solution to be developed in a reasonable number of time increments.

For most coupled thermal-electrical-structural analyses, however, the unconditional stability of the backward difference operator (implicit method) is desirable. In such cases the implicit integration scheme may be invoked automatically by Abaqus/Standard.

Explicit integration can be less expensive computationally and simplifies implementation of user-defined creep laws in user subroutine CREEP; you can restrict Abaqus/Standard to using this method for creep problems (with or without geometric nonlinearity included). See Rate-dependent plasticity: creep and swelling for further details.

COUPLED TEMPERATURE-DISPLACEMENT, ELECTRICAL, CETOL=tolerance, CREEP=EXPLICIT 

Step module: Create Step: General: Coupled thermal-electrical-structural: Basic: Response: Transient, toggle on Include creep/swelling/viscoelastic behavior; Incrementation: Creep/swelling/viscoelastic strain error tolerance: tolerance, Creep/swelling/viscoelastic integration: Explicit

Excluding creep and viscoelastic response

You can specify that no creep or viscoelastic response will occur during a step even if creep or viscoelastic material properties have been defined.

COUPLED TEMPERATURE-DISPLACEMENT, ELECTRICAL, DELTMX=$\mathrm{\Delta }{\theta }_{max}$, CREEP=NONE 

Step module: Create Step: General: Coupled thermal-electrical-structural: Basic: Response: Transient, toggle off Include creep/swelling/viscoelastic behavior

Unstable problems

Some types of analyses may develop local instabilities, such as surface wrinkling, material instability, or local buckling. In such cases it may not be possible to obtain a quasi-static solution, even with the aid of automatic incrementation. Abaqus/Standard offers a method of stabilizing this class of problems by applying damping throughout the model in such a way that the viscous forces introduced are sufficiently large to prevent instantaneous buckling or collapse but small enough not to affect the behavior significantly while the problem is stable. The available automatic stabilization schemes are described in detail in Automatic stabilization of unstable problems.