Model: Transformation – MOSAICmodeling

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Optional element

There are situations in which you have formulated a differential-algebraic equation system, but now you need a purely algebraic equation system without having to re-write all algebraic and differential equations.

Purpose

The Transformation is used to translate differential equations within equation systems into algebraic equations. A possible example is orthogonal collocation on finite elements.

Explanation of the editor

A transformation can be applied by setting up an equation system that contains the rule for how this discretization scheme works. For example, the system below represents the discretization scheme of orthogonal collocation on finite elements. Here, the differential term is translated into a summation term that incorporates the derivates of the Lagrange basis polynomials. The second term represents the continuity of the state variables between finite elements:

$\frac{d y}{d z}= \frac{1}{\Delta z_{fe}}\sum_{ip=0}^{Nip} \frac{dL_{ip,cp}}{d \tau} y_{cp=ip,fe}$

and

$y_{cp=3,fe-1} = y_{cp=0,fe}$ .

The transformation editor is shown in Figure 1. On the left side, the discretization scheme is loaded. The super notation is the notation of your original equation system. An explanation of the fields in the transformation tab is given in Table 1.

Figure 1: The transformation editor allows you to specify a transformation that is then applied on your equation system to transform differential equations into algebraic ones.

Table 1: Elements of the transformation editor.

Element	Explanation
File	Filename of your transformation once you have saved or loaded it
Description	A description of the current transformation
Keywords	Optional keywords for your transformation
Usages	Model elements, i.e., equations or functions, in which the transformation is used
Set Notations	Select the equation system that contains the discretization scheme as well as the super notation, i.e., the notation of the equation system you would like to transform
Variable Matching	Match the independent variables in the discretization and the original model. In the example above, $z$ is the indepent variable of the discretization, $t$ could be the independent variable in your system in case it is a dynamic system
Index Matching	Match indices that are relevant for your transformation. Therefore, your super notation must contain all indices used in the transformation scheme, but they can be different. For example, if you prefer the index $fez$ in your transformed equation system for the finite elements, you can add this index to the super notation and match it accordingly in this tab
Variable Predetermination	Pre-select which variables will be treated as differential (= state) variables
Variable Bounds	Select boundary conditions for certain variables