Joule heating arises when the energy dissipated by an electrical current flowing through a
conductor is converted into thermal energy. Abaqus/Standard provides a fully coupled thermal-electrical procedure for analyzing this type of problem:
the coupled thermal-electrical equations are solved simultaneously for both temperature and
electrical potential at the nodes.
The capability includes the analysis of the electrical problem, the thermal problem, and
the coupling between the two problems. Coupling arises from two sources:
temperature-dependent electrical conductivity or temperature-dependent electrical
resistivity, and
internal heat generation, which is a function of the electrical current density.
The thermal part of the problem can include heat conduction and heat storage (About Thermal Properties) as well as cavity
radiation effects (Cavity Radiation in Abaqus/Standard). Forced
convection caused by fluid flowing through the mesh is not considered.
The thermal-electrical equations are unsymmetric; therefore, the unsymmetric solver is
invoked automatically if you request coupled thermal-electrical analysis. For problems where
coupling between the thermal and electrical solutions is weak or where a pure electrical
conduction analysis is required for the entire model, the unsymmetric terms resulting from
the interfield coupling may be small or zero. In these problems you can invoke the less
costly symmetric storage and solution scheme by solving the thermal and electrical equations
separately. The separated technique uses the symmetric solver by default. The
thermal-electrical solution schemes are discussed below.
The electric field in a conducting material is governed by Maxwell's equation of
conservation of charge. Assuming steady-state direct current, the equation reduces to
where V is any control volume whose surface is
S, is the outward normal to S, is the electrical current density (current per unit area), and is the internal volumetric current source per unit volume.
The flow of electrical current is described by Ohm's law:
where
is the electrical field intensity, defined as the negative of the gradient of the
electrical potential ,
is the electrical potential,
)
is the electrical conductivity matrix,
is the temperature, and
are predefined field variables.
Using Ohm's law in the conservation equation, written in variational form, provides the
governing equation of the finite element model:
where is the current density entering the control volume across
S.
Defining the Electrical Conductivity
The electrical conductivity, , can be isotropic, orthotropic, or fully anisotropic (see Electrical Conductivity). 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. The (electrical) conduction properties of the
material may alternatively be described in terms of the electrical resistivity of the
material, as described below.
Defining the Electrical Resistivity
Alternatively, the electrical conduction properties of the material can be defined in
terms of the electrical resistivity, , which is the inverse of the electrical conductivity, . The electrical resistivity can be isotropic, orthotropic, or fully
anisotropic (see Electrical Resistivity). Ohm's law
assumes that the electrical resistivity is independent of the electrical field, . The coupled thermal-electrical problem is nonlinear when the electrical
resistivity depends on temperature.
Specifying the Amount of Thermal Energy Generated due to Electrical Current
Joule's law describes the rate of electrical energy, , dissipated by current flowing through a conductor as
The amount of this energy released as internal heat within the body is , where is an energy conversion factor. You specify in the material definition. It is assumed that all the electrical energy
is converted into heat () if you do not include the Joule heat fraction in the material
description. The fraction given can include a unit conversion factor, if required.
Property module: material editor: ThermalJoule Heat Fraction
Steady-State Analysis
Steady-state analysis provides the steady-state solution directly. Steady-state thermal
analysis means that the internal energy term (the specific heat term) in the governing heat
transfer equation is omitted. Only direct current is considered in the electrical problem,
and it is assumed that the system has negligible capacitance. (Electrical transient effects
are so rapid that they can be neglected.)
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” period and “time” incrementation parameters 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). 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 Defining an Analysis).
Transient Analysis
Alternatively, the thermal portion of the coupled thermal-electrical problem can be
considered transient. As in steady-state analysis, electrical transient effects are
neglected. See Uncoupled Heat Transfer Analysis for a more detailed description of
the heat transfer capability in Abaqus/Standard.
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.
Automatic Incrementation
The time increment size can be selected automatically based on a user-prescribed
maximum allowable nodal temperature change in an increment, . Abaqus/Standard will restrict the time increments to ensure that these values are not exceeded at any
node (except nodes with boundary conditions) during any increment of the analysis (see
Time Integration Accuracy in Transient Problems).
Step module: Create Step: General: Coupled thermal-electric: Basic: Response: Transient; Incrementation: Type: Automatic: Max. allowable temperature change per increment:
Fixed Incrementation
If you select fixed time incrementation and do not specify , fixed time increments equal to the user-specified initial time
increment, , will then be used throughout the analysis.
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.
Ending a Transient Analysis
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.
Input File Usage
Use the following option to end the analysis when the time period is reached:
Step module: Create Step: General: Coupled thermal-electric: Basic: Response: Transient; Incrementation: End step when temperature change is less than
Fully Coupled Solution Schemes
Abaqus/Standard offers an exact as well as an approximate implementation of Newton's method for coupled
thermal-electrical analysis.
Exact Implementation
An exact implementation of Newton's method involves a nonsymmetric Jacobian matrix as is
illustrated in the following matrix representation of the coupled equations:
where and are the respective corrections to the incremental electrical potential
and temperature, are submatrices of the fully coupled Jacobian matrix, and and are the electrical and thermal residual vectors, respectively.
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.
Approximate Implementation
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 (Defining an Analysis), 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 Defining an Analysis).
Input File Usage
Use the following option to specify a separated solution scheme:
Uncoupled Electric Conduction and Heat Transfer Analysis
The coupled thermal-electrical procedure can also be used to perform uncoupled electric
conduction analysis for the whole model or just part of the model (using coupled
thermal-electrical elements). Uncoupled electrical analysis is available by omitting the
thermal properties from the material description, in which case only the electric potential
degrees of freedom are activated in the element and all heat transfer effects are ignored.
If heat transfer effects are ignored in the entire model, you should invoke the separated
solution technique described above. Use of this technique will then invoke the symmetric
storage and solution scheme, which is an exact representation of a purely electrical
problem.
Similarly, coupled thermal-electrical elements can be used in an uncoupled heat transfer
analysis (Uncoupled Heat Transfer Analysis), in which case all electric conduction
effects are ignored. This feature is useful if a thermal-electrical analysis is followed by
a pure heat conduction analysis. A typical example is a welding process, where the electric
current is applied instantaneously, followed by a cooldown period during which no electrical
effects need to be considered. The symmetric solver is activated by default in an uncoupled
heat transfer analysis.
Cavity Radiation
Cavity radiation can be activated in a heat transfer step. This feature involves
interacting heat transfer between all of the facets of the cavity surface, dependent on the
facet temperatures, facet emissivities, and the geometric view factors between each facet
pair. When the thermal emissivity is a function of temperature or field variables, you can
specify the maximum allowable emissivity change during an increment in addition to the
maximum temperature change to control the time incrementation. See Cavity Radiation in Abaqus/Standard for more
information.
Input File Usage
Use the following option in the step definition to activate cavity radiation:
By default, the initial temperature of all nodes is zero. You can specify nonzero initial
temperatures or field variables (see Initial Conditions). Since only
steady-state electrical currents are considered, the initial value of the electrical
potential is not relevant.
Boundary Conditions
Boundary conditions can be used to prescribe the electrical potential, (degree of freedom 9), and the temperature, (degree of freedom 11), at the nodes. See Boundary Conditions.
Boundary conditions can be specified as functions of time by referring to amplitude curves
(see Amplitude Curves).
A boundary without any prescribed boundary conditions corresponds to an insulated surface.
Loads
Both thermal and electrical loads can be applied in a coupled thermal-electrical analysis.
Applying Thermal Loads
The following types of thermal loads can be prescribed in a coupled thermal-electrical
analysis, as described in Thermal Loads:
Concentrated heat fluxes.
Body fluxes and distributed surface fluxes.
Average-temperature radiation conditions.
Convective film conditions and radiation conditions.
Applying Electrical Loads
The following types of electrical loads can be prescribed, as described in Electromagnetic Loads:
Concentrated current.
Distributed surface current densities and body current densities.
Predefined Fields
You can define initial temperature fields in coupled thermal-electrical analyses; other
predefined temperature fields are not allowed. Instead, you should use boundary conditions
to specify temperatures, as described above.
You can specify other predefined field variables in a coupled thermal-electrical analysis.
These values affect only field-variable-dependent material properties, if any. See Predefined Fields.
Material Options
Both thermal and electrical properties are active in coupled thermal-electrical analyses.
If thermal properties are omitted, an uncoupled electrical analysis will be performed.
All mechanical behavior material models (such as elasticity and plasticity) are ignored in
a coupled thermal-electrical analysis.
Thermal Material Properties
For the heat transfer portion of the analysis, the thermal conductivity must be defined
(see Conductivity). The specific
heat must also be defined for transient heat transfer problems (see Specific Heat). If changes in
internal energy due to phase changes are important, latent heat can be defined (see Latent Heat). Thermal
expansion coefficients (Thermal Expansion) are not
meaningful in a coupled thermal-electrical analysis since deformation of the structure is
not considered. Internal heat generation can be specified (see Uncoupled Heat Transfer Analysis).
Electrical Material Properties
For the electrical portion of the analysis, either the electrical conductivity (see Electrical Conductivity) or the
electrical resistivity (see Electrical Resistivity) must be defined.
The electrical conductivity or the electrical resistivity can be a function of temperature
and user-defined field variables. The fraction of electrical energy dissipated as heat can
also be defined, as explained above.
Elements
The simultaneous solution in a coupled thermal-electrical analysis requires the use of
elements that have both temperature (degree of freedom 11) and electrical potential (degree
of freedom 9) as nodal variables. The finite element model can also include pure heat
transfer elements (so that a pure heat transfer analysis is provided for that part of the
model) and coupled thermal-electrical elements for which no thermal properties are given (so
that a pure electrical conduction solution is provided for that part of the model).
Heat flux per unit area generated by the electrical current.
SJDA
SJD multiplied by area.
SJDT
Time integrated SJD.
SJDTA
Time integrated SJDA.
WEIGHT
Heat distribution between interface surfaces, f.
Considerations for Steady-State Coupled Thermal-Electrical Analysis
In a steady-state coupled thermal-electrical analysis the electrical energy dissipated
due to flow of electrical current at an integration point (output variable
JENER) is computed using the following
relationship:
where denotes the electrical energy dissipated due to flow of electrical
current and is the current step time. In the above relationship it is assumed that
the rate of the electrical energy dissipation, , has a constant value in the step that is equal to the value currently
computed.
The output variable JENER and the
derived output variables ELJD and
ALLJD contain the values of electrical
energies dissipated in the current step only. Similarly, the contribution from the
electrical current flow to the output variable
ALLWK includes only the external work
performed in the current step.
Input File Template
HEADING
…
MATERIAL, NAME=mat1CONDUCTIVITYData lines to define thermal conductivityELECTRICAL CONDUCTIVITYData lines to define electrical conductivityJOULE HEAT FRACTIONData lines to define the fraction of electric energy released as heat
**
STEPCOUPLED THERMAL-ELECTRICALData line to define incrementation and steady stateBOUNDARYData lines to define boundary conditions on electrical potential and
temperature degrees of freedomCECURRENTData lines to define concentrated currentsDECURRENT and/or DSECURRENTData lines to define distributed current densitiesCFLUX and/or DFLUX and/or DSFLUXData lines to define thermal loadingFILM and/or SFILM and/or RADIATE and/or SRADIATEData lines to define convective film and radiation conditions
…
CONTACT PRINT or CONTACT FILEData lines to request output of surface interaction variablesEND STEP