Scalar Field: Offline

Offline phase of the Generalised Empirical Interpolation Method applied to scalar fields

Preparation

The structure of the case study folder is the following

>> ./Study_case
	>> /Folder_1  		
		>> /0
		>> ...
		>> /constant
		>> /system
			controlDict
			blockMeshDict
			...
			GEIMsolverDict  <--- Dictionary needed for the input parameters					
	>> /Folder_2

The GEIMsolverDict must be put inside ./Study_case/Folder_1/system/

An example of GEIMsolverDict can be found in application/GEIM/ScalarGEIM_Offline, which requires the following entries:

{
	field      T;			<---- ScalarField on which GEIM is performed
	error      0.001;		<---- relative L_2 error desidered
	MaxSensorsNumber 20;		<---- Max number of Sensors
	SensorsVariance 0.0001;		<---- variance of the Gaussian function describing the sensors
	foldersList (
			"Folder_1"
			"Folder_2") ;	<---- List of folder names containig the snapshots
	SensorsPositions ( 
			(0.1 0.5 0.3)
			(0.6 0.8 0.9)
			...
			(0.25 0.41 0.9) 
			 )		<---- List of admissible sensor locations
} 

If ‘SensorsPositions’ is missing, as many sensors as cell centers will be created. However, for big mashes this option may be very RAM consuming !!! It has been implemented a way to tacke these problem, that samples every n cell (the maximum size can be changed according to the available capabilities).

Usage

Inside ./Study_case/Folder_1 launch

ScalarGEIM_Offline

To include folder “0” use

ScalarGEIM_Offline -withZero

To perform on a specified region (for multi-region cases) use

ScalarGEIM_Offline -region <regionName>

Results

The Magic Functions, Magic Sensors and all the other .txt files are saved in a separate folder called GEIM_(fieldName)s(SensorsVariance), which has the classical OpenFOAM structure.

>> ./Study_case
	>> /Folder_1  				 		
	>> /Folder_2		
	>> /GEIM_T_s_0.0001		
		>> /0
			TGEIMMagicFunction0
			TGEIMMagicFunction1
			TGEIMMagicFunction2
			...
			TMagicSensor0
			TMagicSensor1
			TMagicSensor2
			...						
		>> /system			
		>> /constant
		>> /T_GEIM_Offline_files
			Lebesgue_constant.txt
			max_relative_L2_error.txt
			max_absolute_L2_error.txt
			betaCoefficients.txt			<---- file needed in the indirect reconstruction (required for the mapping procedure)
			matrixB.txt				<---- file needed in the indirect reconstruction (required for the mapping procedure)
			GEIM_Offline_Coeffs_avarage_values.txt	<---- file needed for 'ScalarTR-GEIM'
			GEIM_Offline_Coeffs_std.txt		<---- file needed for 'ScalarTR-GEIM'

The absolute and relative error are computed as

\[E_M = || T-\mathcal{I}_M[T]||_{L^2}\qquad \epsilon_M = \frac{|| T-\mathcal{I}_M[T]||_{L^\infty}}{||T||_{L^2}}\]

recalling that the norms are defined as

\[|| T ||_{L^2(\Omega)}^2 =\int_\Omega T^2\, d\Omega\]