Data Structures

In this chapter the basic data structures are summarized and shortly described that are used in package ModiaBase.

1. Bi-partite Graph

The bi-partite Graph G of a DAE system defines the functional dependency of the equations $e_i$ from time-varying variables $v_j$. G is also a sparse representation of the incidence matrix of the DAE system. Example:

# Bi-partite graph of low pass filter
G = Vector{Int}[ [1,2,4], # equation 1 depends on variables 1,2,4
                 [1,7],   # equation 2 depends on variables 1,7
                 [3,4],
                 [3,7],
                 [6,7],
                 [2] ]

In ModiaBase, only potential states and unknown variables are described with G. Dependency on parameters (= constant quantities) is not shown.

The number of variables is usually larger as the number of equations, because both the potential states and their derivatives are stored in G.

2. Assignment Vector

The assignment vector assign defines for all equations and variables the unique variable $v_j$ that is solved from equation $e_i$. Example:

assign = [2,6,3,1,0,5,4]  # Variable 1 is solved from equation 2
                          # Variable 2 is solved from equation 6
                          #  ...
                          # Variable 5 is not solved from any equation

The inverted assignment vector invAssign defines the unique equation $e_i$ that is solved for the unique variable $v_j$. This vector can be directly computed from vector assign. Example:

(invAssign, unAssigned) = invertAssign(assign)

# invAssign = [4,1,3,7,6,2,0]   # Equation 1 is solved for variable 4
                                # Equation 2 is solved for variable 1
                                # ...
                                # Equation 7 is not solved for any variable

3. Block Lower Triangular Form

The Block Lower Triangular form blt of an equation system describes the sorted set of equations, in order to solve for the unknown variables. With vector invAssign (see subsection 2 above) the information is provided for which variable the respective equation is solved.

Example:

blt = [ [6],        # Solve first equation 6
        [3,4,2,1],  # Afterwards solve equations 3,4,2,1 (they form an algebraic loop)
        [5] ]       # Finally solve equation 5

invAssign = [4, 1, 3, 7, 6, 2, 0]

The meaning is:

Equation 6 is solved for variable 2.
Equations 3,4,2,1 are solved simultaneously for variables 3, 7, 1, 4.
Equation 5 is solved for variable 6.

4. Variable Association Vector

The derivative relationship between variables is described with the variable association vector Avar and its inverted vector invAvar:

\[\begin{aligned} Avar_j &= \left\{ \begin{array}{rl} k & \text{\textbf{if}}~ \dot{v}_j \equiv v_k \\ 0 & \text{\textbf{if}}~ v_j \text{ is not a differentiated variable} \end{array}\right. \\ invAvar_j &= \left\{ \begin{array}{rl} k & \text{\textbf{if}}~ v_j \equiv \dot{v}_k \\ 0 & \text{\textbf{if}}~ v_j \text{ is not a differentiated variable} \end{array}\right. \end{aligned}\]

Example:

The following derivative relationships between variables v1,v2,v3,v4,v5

   1. v1
   2. v2 = der(v1)
   3. v3 = der(v2)
   4. v4
   5. v5 = der(v4)

are expressed by the following variable association vector and its inverted form:

   Avar    = [2,3,0,5,0]   # The derivative of variable 1 is variable 2
                           # The derivative of variable 2 is variable 3
                           #  ...

   invAvar = [0,1,2,0,4]   # Variable 1 is not a derivative
                           # Variable 2 is the derivative of variable 1
                           #   ...
                           # Variable 5 is the derivative of variable 4

5. Equation Association Vector

The derivative relationship between equations is described with the equation association vector Bequ and its inverted vector invBequ:

\[\begin{aligned} Bequ_i &= \left\{ \begin{array}{rl} k & \text{\textbf{if}}~ \dot{e}_i \equiv e_k \\ 0 & \text{\textbf{if}}~ \dot{e}_i \text{ does not exist} \end{array}\right. \\ invBequ_i &= \left\{ \begin{array}{rl} k & \text{\textbf{if}}~ e_i \equiv \dot{e}_k \\ 0 & \text{\textbf{if}}~ _i \text{ is not a differentiated equation} \end{array}\right. \end{aligned}\]

Example:

The following derivative relationships between equations e1,e2,e3,e4,e5

   1. e1
   2. e2 = der(e1)
   3. e3 = der(e2)
   4. e4
   5. e5 = der(e4)

are expressed by the following equation association vector and its inverted form:

   Bvar    = [2,3,0,5,0]   # The derivative of equation 1 is equation 2
                           # The derivative of equation 2 is equation 3
                           #  ...

   invBvar = [0,1,2,0,4]   # Equation 1 is not a differentiated equation
                           # Equation 2 is the derivative of equation 1
                           #   ...
                           # Equation 5 is the derivative of equation 4