Global Equations
A default Global Equations node () is added to The Global ODEs and DAEs Interface. To add additional Global Equations nodes, either right-click and select it from the context menu or click Global Equations on the toolbar.
In any other physics interface, click the Show More Options button () and select Equation-Based Contributions in the Show More Options dialog box. Then right-click the physics interface and select Global>Global Equations to add a node directly, without needing to add a separate Global ODEs and DAEs interface.
Global Equations
The global equations that you can solve have the following form:
with the initial conditions u(t0) = u0 and ut(t0) = ut,0 (where the subscript t indicates the time derivative). Several equations can be added and the equation can be coupled.
The first time derivative of u is written ut, and the second time derivative of u is utt. With time derivatives, this equation is an ODE (ordinary differential equation). With no time derivatives, the equation is an algebraic equation or a transcendental equation. If some equations include time derivatives and others do not, the system is a DAE (differential-algebraic equation).
Initial conditions are necessary for ODEs and DAEs. For the DAEs, it is important to specify a set of initial conditions consistent with the algebraic part of the system. Otherwise, the solvers may be forced to modify the initial values to make them consistent, or they may fail.
In the Global Equations table, each row corresponds to a named equation; that is, it defines a single degree of freedom and one equation.
The selected row in the table of global equations may also be edited using the Name, f(u,ut,utt,t), Initial value (u_0), Initial value (u_t0), and Description fields underneath the table.
In each column enter as follows:
•
Enter the Name of the equation variable. This also defines time-derivative variables. If a variable is called u, its first and second time derivatives are ut and utt, respectively. These variables become available in all geometries. Therefore the names must be unique.
•
Use the f(u,ut,utt,t) column to specify the right-hand side of the equation that is to be set equal to zero.
The software then adds this global equation to the system of equations. When solving the model, the value of the dependent variable u is adapted in such a way that the associated global equation is satisfied. All dependent variables and their time derivatives can be used as well as any parameters, global variables, and coupling operators with a scalar output and global domain of definition in the f(u,ut,utt,t) column. The variables can be functions of the dependent variables in the global equations. Setting an equation for a dependent variable is optional. The default value of 0 means that the software does not add any additional condition to the model.
•
If the time derivative of a dependent variable appears somewhere in the model during a time-dependent solution, the dependent variable needs an initial condition. Models that contain second time derivatives also require an initial value for the first time derivatives of the dependent variables. Set these conditions in the third (Initial value (u_0)) and fourth (Initial value (u_t0)) columns.
•
Enter comments about the equation in the last column, Description.
•
To add another equation, make additional entries in the first empty row.
Move equation rows up and down using the Move Up () and Move Down () buttons.
To remove an equation, select some part of that equation’s row in the table and click the Delete button ().
Save the definitions of the global equations to a text file by clicking the Save to file button () and using the Save to File dialog box that appears. To load a text file with global equation definitions, use the Load from File button () and the Load from File dialog box that appears. Data must be separated by spaces or tabs (or be in a Microsoft Excel Workbook spreadsheet if the license includes LiveLink™ for Excel®).
Discretization
To display this section, click the Show More Options button () and select Advanced Physics Options in the Show More Options dialog box.
The Discretization section for Global Equations is used to specify the Value type (Real or Complex) of the variables. The Split complex variables in real and imaginary parts setting is activated in the Compile Equations node of any solver sequence. The default for the split complex variables setting is to be not active and in that case you do not need to specify the value type for global equations variables (the value type specified would be ignored in such cases). The value type (complex or real) for all the variables defined by this Global Equations node is selected in the Value type when using splitting of complex variables selection. The default value type is Complex.
Units
By default, the global equations are dimensionless, but units can be defined for the dependent variable and the source term (that is, the overall left and right side of the equation). The units for these quantities — in combination with the unit for time — fully define the units for all other terms in the equations. Select the units from a list of physical quantities or enter the unit directly.
Select the Dependent variable quantity that defines the unit for the dependent variable u. The default is Dimensionless (with 1 in the Unit column). Click the Select Dependent Variable Quantity button () to open the Physical Quantity dialog box to browse to find a physical quantity to use. You can also type a search string in the text field at the top of the dialog box and then click the Filter button () to filter the list of physical quantities. For example, type potential and click the Filter button to only list physical quantities that represent some kind of potential. Alternatively, click the Define Dependent Variable Unit button () to edit the unit directly in the Unit column, typing a unit to define the dependent variable quantity. The quantity column then contains Custom unit.
Select the Source term quantity that defines the unit for the source term f (the unit for the right — and left — side of the global equation). The default is Dimensionless (with 1 in the Unit column). Click the Select Source Term Quantity button () to open the Physical Quantity dialog box to browse to find a physical quantity to use. You can also type a search string in the text field at the top of the dialog box and then click the Filter button () to filter the list of physical quantities. For example, type potential and click the Filter button to only list physical quantities that represent some kind of potential. Alternatively, click the Define Source Term Unit button () to edit the unit directly in the Unit column, typing a unit (for example, W/m^3 or A/m^3) to define the dependent variable quantity. The quantity column then contains Custom unit.