Use of Functions IV – Definition by Interface and Empty Body Functions

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Functions defined by interfaces can also be called “empty body functions”, because only inputs and outputs and no function bodies are specified within MOSAICmodeling. They can be used as placeholders in equation systems. The output and input variables must be defined via an interface, but the mathematical expression will not be specified at that point. In this way, the equation system is independent of the functional correlation used to calculate the output variable. Similar to functions with defined bodies, several variable inputs are possible while only one output variable is calculated. Both input and output variables can be scalars or vectors. Hence, more than one functional dependence can be expressed by the use of one empty body function.

Model description

In this exercise the use of functions defined by interfaces is practiced. The following equation must hold:

$\begin{equation*}Z = \sum_{i=1}^4 z_i.\end{equation*}$

The components $z_i$ of the vector $z$ are calculated via the following mathematical operations:

$\begin{align*}&z_1 = x + y \\[2ex]&z_2 = x - y \\[2ex]&z_3 = \frac{x}{y} \\[2ex]&z_4 = x \cdot y. \\[2ex]\end{align*}$

This shall be achieved with an empty body function. The mathemical expressions shall thus be added later, in this case in Matlab. The function’s output variable $z$ is thus a vector. In the simulation, $y$ shall be determined iteratively for a given $x$ and $Z$ .

Workflow

Notation of equation system

Set up a notation with the following base names and indices:

Base names

$x$ , constant
$y$ , iteration variable
$Z$ , sum of all components of $z_i$
$z$ , vector

Indices

$i$ , index for components of $z$ 1…NI

The resulting notation has ID 182679.

Equations

Now formulate the model equation defined above. The resulting equation has ID 182680.

Definition of interface

To create the interface, do the following in the Interface tab:

Load the notation
Add an informative description of the interface
Add fields for $x$ , $y$ , and $z_i$ by clicking on “+ Field Name”
Select the dimensions of these three variables – Scalar for $x$ and $y$ , and Vector for $z_i$
Select “In” for $x$ and $y$ , and “Out” for $z_i$ in the In/Out column
Go to the Pattern Assistance tab and select Function in General within the Type of Interface
Save the interface

This interface is available in MOSAICmodeling with ID 182681.

Definition of function

To create the empty body function, we take the following steps in the Function tab:

Load the notation of this example
Add an informative description of the function
Make sure that Load interface is selected in the Interface & Body Settings
Load the interface you created
Save the function

The function has ID 182682.

Equation system

Go to the tab Equation System and add your equation in the same way as in the previous examples. Then go to the Functions tab within the equation system and take the following steps:

Add the function by clicking on the “Add” button at the bottom
Add an application by clicking on “Add Application”
Add $z_i$ as Applied Naming for the output and $x$ as well as $y$ as Applied Naming for the input
Confirm that and confirm again to close the popup windows. You should now see one application of the function within the equation system

The resulting equation system has ID 182683.

Evaluation / Simulation

Next, we can move to the “Simulation” section and load the equation system we just created. Then, we take the following steps:

Add a description for your simulation
In the tab on the Generic System, go to Functions and check whether MOSAICmodeling confirms one usage of the function in one application
Go to Indexing and enter 4 as Max Value for NI; then click on Confirm Index Data
Go to the Specifications tab and assign $Z$ and $x$ as design values
The variables $z_1$ through $z_4$ should be classified as calculated values

Initialization, filling out the the empty body, and results

To initialize and specify the model, take the following steps:

Initialize this example with the design values and initial guesses given in Table 1
Save the variable specification
Save the simulation
Go to the Evaluation tab and generate the code for Matlab
Copy the generated code into Matlab and search for the following code:

function[std_z_i] = fun_182682__03_function_emptybody(std_x,std_y)
	std_z_i =  --- NO BODY DEFINED: ADD YOUR CODE HERE.--- ;
end

Replace the body of the function with the following code:

function[std_z_i] = fun_182682__03_function_emptybody(std_x,std_y)
    std_z_i(1) = std_x + std_y;
    std_z_i(2) = std_x - std_y;
    std_z_i(3) = std_x / std_y;
    std_z_i(4) = std_x * std_y;
end

This simulation is available with ID 182684 with variable specification 182685. The solution for the iteration variable is also given in Table 1.

Name	Description	Value / Initial guess	Solution
$Z$	Parameter determining the sum of $z_i$	18.0
$x$	Parameter	4.0
$z_1$	First component of $z$		6.0
$z_2$	Second component of $z$		2.0
$z_3$	Third component of $z$		2.0
$z_4$	Fouth component of $z$		8.0
$y$	Iteration variable	10.0	2.0