Vectorial Field: Offline

Online phase of the Empirical Interpolation Method applied to Vector fields

Preparation

The structure of the case study folder is the following (in this example Folder3 and Folder4 are the test case folders)

>> ./Study_case
	>> /Folder_1  			
	>> /Folder_2
	>> /Folder_3  			
	>> /Folder_4		
	>> /EIM_(fieldName)
		>> /0		
		>> /constant	        		
		>> /system
			controlDict
			blockMeshDict
			...
			EIMsolverDict  <--- Dictionary needed for the input parameters
		>> /(fieldName)_EIM_Offline_files

The EIMsolverDict must be put inside ./Study_case/EIM_(fieldName)/system/

An example of EIMsolverDict can be found in application/EIM/ScalarEIM_Online, which requires the following entries:

Online_parameters
{
	field      U;			<---- VectorField on which EIM is performed 
	mfNumber   20;			<---- number of EIM magic functions to use
	foldersList  ( 
			"Folder_3" 
			"Folder_4") ;	<---- List of folder names containig the snapshots to be reconstructed
}

Usage

Inside ./Study_case/EIM_(fieldName) launch

VectorialEIM_Online

To include folder “0” use

VectorialEIM_Online -withZero

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

VectorialEIM_Online -region <regionName>

Results

The interpolant and the residual field, defined as

\[r_M = \left| \mathbf{u}-\mathcal{I}_M[\mathbf{u}]\,\right|\]

are stored in the correspondent snapshot folders

>> ./Study_case
	>> /Folder_1  		  		
	>> /Folder_2
	>> /Folder_3
		>> /0
			(fieldName)EIMInterpolant  <---(fieldName) EIM interpolant obtained with mfNumber basis
			(fieldName)EIMresidual     <---(fieldName) EIM residual obtained with mfNumber basis
		>> /1	
			(fieldName)EIMInterpolant
			(fieldName)EIMresidual
		>>  ...			
				
	>> /Folder_4
		>> /0
			(fieldName)EIMInterpolant 
			(fieldName)EIMresidual
		>> /1	
			(fieldName)EIMInterpolant
			(fieldName)EIMresidual
		>> /...		
			
	>> /EIM_(fieldName)		
		>> /0		        				
		>> /system			
		>> /constant
		>> /(fieldName)_EIM_Offline_files
		>> /(fieldName)_EIM_Online_files
			maximum_L2_relative_error.txt <---- max L2 absolute error as a function of basis number
			average_L2_relative_error.txt <---- max L2 realtive error as a function of basis number

The absolute and relative error are computed as

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

recalling that the norms are defined as

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