RepresentativeMassAlgorithm

How to distribute representative masses to MESHFREE points

Motivation

Inherently, MESHFREE points do not carry mass as they are information carriers only. That gives a lot of freedom regarding adaptive refinement and much more. That also means that MESHFREE is not inherently mass conservative, but we provide strategies for conserving mass that act locally.

Basic Idea

Within MESHFREE, mass can only be produced at inflow boundaries or DropletSource items. Mass can only be reduced at outflow boundary elements or by EVENT statements. Besides that, mass cannot be generated. That means that the total mass can be determined analytically by measuring the mass being brought into and out of the system. The idea is to distribute mass packages among the MESHFREE points. The sum of all masses shall represent the analytical mass to be in the system. Mass can be re-distributed among points, which will become necessary in regions of local refinement/coarsening.

Parameters

The RepresentativeMassAlgorithm is triggered by common_variables parameter RepresentativeMass_iData .

Algorithm

First, please see DefinitionRepresentativeMass (%ind_mi_rep%) and DefinitionRepresentativeDensity (%ind_r_rep%). The algorithm is sketched here:

algorithm DeletedOrInactivePoints is launched after point deletion in order to project back the representative masses of the vanishing points onto the active part of the boundary

After the end of point cloud organization

launch DeletedOrInactivePoints for all recently deactivated boundary points onto the remaining active part of boundary
launch NewPoints for all new points or newly activated boundary points
launch FlowPenetrationBoundaries in order to update representative mass at inflow, outflow, or permeable walls
iterate Smoothing for a given number of iteration loops (to be defined in RepresentativeMass_iData )
finally, launch Repair in order to remove negative masses in active points, that might have occurred during the previous steps

Relevant Indices

%ind_mi_rep% , %ind_r_rep% , %ind_Vi% , %ind_BNDfree_defect%, %ind_cluster%

Postprocessing - Options for mass measurement with MESHFREE

Mass measurements using the representative mass index %ind_mi_rep% can be done via two distinct ways.

Volume-Based Integration Using Control Volumes

This is based on the InDom() functionality.

INTEGRATION($MassRepControlVolume$) = ( %SUMMATION_INT%, [InDom("cube") * Y%ind_mi_rep%], $MAT$, %INTEGRATION_Header%, "RepMassCalc_InDom" )

Here, "cube" denotes the control volume, which is defined using %IDENT_BlindAndEmpty%. This method is straightforward and effective for simple geometries. However, in case of complex geometries, the control volume construction may not be feasible.

Flux-Based Measurement Across a Control Surface

This is based on %INTEGRATION_FLUX_TIME%.

INTEGRATION($MassRepFluxPlane$) = ( %INTEGRATION_FLUX_TIME%, [Y%ind_mi_rep%/ Y%ind_Vi%], $FluxPlane$, %INTEGRATION_Header%, "RepMassCalc_Flux" )

$FluxPlane$ represents the control surface across which the representative mass flux is evaluated. This flux-based method mitigates potential inaccuracies arising from volume integration, particularly in regions with isolated or irregular free surface points. Furthermore, this measurement can become unreliable near wall-adjacent regions, especially when no-slip boundary conditions are used. Note: In regions where the flow is well-resolved, both the volume-based and flux-based methods are expected to yield approximately the same results for the representative mass calculations.

List of members:
Smoothing	smooth the representative masses in order to obtain smooth representative density
DefinitionRepresentativeMass	define the representative mass
NewPoints	newly created points acquiring mass from existing points
DeletedOrInactivePoints	deleted or inactive points giving away their mass to active points
FlowPenetrationBoundaries	adapt mass of flow-penetrated boundaries
Repair	fix negative of too small representative masses, obey upper hard limit as well
DefinitionRepresentativeDensity	define the representative density