FEM: Local lumped mass matrices for elements on no slip boundaries.

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SUMMARY

This discussion focuses on the implementation of the Lumped Mass Matrix in a SUPG FEM code for flow problems, specifically using P2P1 tetrahedral elements. It highlights the 60% loss of mass at the local level for elements on no slip boundaries due to the presence of velocity nodes that represent known values. The consensus is that additional mass should not be lumped at unknown nodes, as the lumped mass matrix inherently conserves the total mass of the fluid in the computational domain, despite the boundary conditions affecting the global system of equations.

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  • Concept of Lumped Mass Matrix in computational fluid dynamics
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Researchers and engineers working with computational fluid dynamics, particularly those implementing SUPG methods and analyzing the effects of boundary conditions in finite element simulations.

Geaux
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I am trying to debug a SUPG FEM code for flow problems (based on Brooks Hughes 1982) and have a question about the Lumped Mass Matrix. My understanding of the Lumped Mass Matrix is that for any given element the mass of that element is simply distributed evenly over the number of nodes.

In my case I am using P2P1 tetrahedral elements which, in the case of an element on the no slip boundary, results in a 60% loss of mass at the local level because of the 6 velocity nodes (3 vertex and 3 edge nodes) that would lie on the no slip surface. The reason for this is that nodes on the boundary represent known velocity values and are not required in the global system of equations. This ulimately results in the sum of my diagonal entries in the global Lumped Mass Matrix not being equal to the mass of fluid in my system.

My question is: For an element on the no slip boundary, should additional mass be lumped at those unknown nodes such that the total mass of the fluid in my computational domain is equal to the sum of diagonal entries in the global Lumped Mass Matrix?
 
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I'd say "no" and think it's ok as is. Think the lumped mass matrix in principle does contain the whole mass of the fluid, even though the boundary conditions lead to a different matrix being actually processed within the solution. As long as mass is conserved on the global scale, i.e. it's at least lumped somewhere irrespective of what the mass actually "does" in the analysis.
 
Thanks for those comments as the more I think about it I certainly agree with what you have said.

Every node in the flow domain has a discrete equation associated with it, and will therefore have an associated entry in the global lumped mass matrix. If that node happens to be on a dirichlet boundary then the equaion is not needed and the subsequent mass 'lumped' at that node is no longer required.

I think this is just a convoluted way of re-stating what you have said. Thanks again.
 

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