4.3.1. Equation Specification

This chapter documents the EQ line commands within the current version of Aria. EQ line commands add equations and independent variables to Aria’s specification of the equation set to be solved for within each region. The equation is also associated with a field variable here that becomes part of the solution vector for Aria. EQ commands occur in Aria’s input file within Region blocks.

Note

For more information on the equations aria solves, refer to the theory section. A comprehensive list of EQ line commands can be found in the command reference.

The EQ command line adds an equation to be solved for on a particular MESHPART. The format is as follows

EQ [equation] FOR [DOF] on [MESHPART] using [INTERP] with TERM0 ... TERMn

Equation is the string identifier for the individual equations listed in General Naming Convention. Namely the material phase, species, or LS phase can be specified in the Equation argument e.g.

# Equation for solid phase voltage
EQ current for voltage in solid_phase on block_1 ...

# Equation for Li+ in liquid phase
EQ species for species of Li+ in liquid_phase on block_1 = ...

The DOF keyword specifies the independent unknown that is solved for in order to satisfy the equation. Normally, it is a strict function of the equation keyword. In other words, the temperature is the only valid DOF entry if the energy equation is being solved. MESHPART is usually the name of an active element block in the finite element model. Unfortunately, if an equation is to be solved on the entire finite element model, this means that there must be multiple EQ keywords for each element block defined in the mesh. Alternative an assembly or meshgroup can be used (see Assemblies and Mesh Groups).

INTERP defines the finite element interpolation to be used. Currently the valid entries for this keyword are P0, P1, Q1, Q2 and Q2S with standard volumes, e.g. hexahedron, tetrahedron, wedge, pyramid, quadrilateral, triangle and bar. These keywords imply the interpolation and polynomial order for the solution field where P denotes (polynomial) element interpolation and Q denotes the nodal interpolation - Q1: linear - Q2: quadratic - Q2S: near-quadratic , serendipity

We note that no syntactical distinction is made between element topology (i.e. triangle and quadrilateral or tetrahedron and hexahedron), that is, the syntax indicates only the level of interpolation. The association of element topology with the equation is handled internal to the code and allowing the specification of interpolation independent of the topology enables the use of superparametric elements (i.e. quadratic geometry with linear interpolation).

Parametric coordinates of shell and bar elements differ from those of conventional bulk volume elements (e.g. hexahedron and tetrahedron) since their meshed geometric description lacks a thickness or area dimension. This missing dimension (SHELL THICKNESS and BAR AREA) can be provided for each material independent of the meshed discretization. Hence when defining equations on shell elements the INTERP keywords previously mentioned are paired (e.g. Q1P0) according to interpolation in the shell plane and interpolation through the shell thickness. For example Q1P0 represents linear nodal interpolation in the plane of the shell and constant interpolation through the shell thickness. Similarly for bar elements an INTERP specification of Q1P0 represents linear nodal interpolation along the length of the bar and constant interpolation in the bar section. In sets of coupled equations, arbitrary combinations of interpolations are sometimes not permitted for solution variables appearing in coupling terms of an equation as interpolation must be consistent in any given equation.

TERM_n refer to the broad categories for the terms in a general advection-diffusion continuity equation. Each term in the equation must be explicitly turned on for it to appear in the conservation equation. Admissible values of TERM are MASS, LUMPED_MASS, ADV, DIFF, SRC, and XFER. While MASS, LUMPED_MASS, ADV, DIFF, SRC terms imply the activation of mathematical operators appearing in an equation, XFER implies that a solution field values for this equation are being provided externally so the equation can still be thought of as belonging to the overall equation set.

Provisions are made for supplying multiple equation contributions for DIFF. These contributions are usually activated via settings in the Aria Materials command block (see Material Properties). Multiple contributions can also be defined for SRC by adding additional source command lines (see Volumetric Sources). In order to accommodate different definitions of velocity for ADV the user can request the velocity to be provided from another equation, from transfer or from other available velocity models. See also Advection Velocity.

All equations are assumed to be formulated in Cartesian coordinates. For two-dimensional problems, axisymmetric formulations are available as well as the Cartesian forms. Invocation of the axisymmetric option is described in the Model Definition chapter in Finite element model.