For first-order elements, one way to obtain a lumped diagonal mass matrix is to sum up all the rows of the original mass matrix and assign such sum to the diagonal. However, this does not work for second-order element as there are some nodes that have negative contributions and a simple sum may result in negative values for the diagonal.
Does anybody know an efficient way of obtaining a diagional mass matrix when the domain is composed of second-order elements (particularlty tet10s)?
Hi! I noticed that you read my paper on local mass matrix inversion. That paper provides one way to answer your question. Let me know if you have any further questions. Also, there have been some more recent papers on new nodal basis functions for triangles that could probably be extended to tetrahedra. I can try to dig them up if you are interested.
-Mike
To my knowledge, the most common approaches for mass lumping in finite elements are:
1. Row-summing (as you mentioned). It's very straightforward but as you said can lead to an indefinite mass matrix, depending on the type of element.
2. 'HRZ'-lumping, as described in Hinton, E., Rock, T., & Zienkiewicz, O. C. (1976). A note on mass lumping and related processes in the finite element method. Earthquake Engineering & Structural Dynamics, 4(3), 245–249. It avoids negative entries in the mass matrix and works well for many cases.
3. 'Spectral lumping': Placing the nodes in the same position as the integration points. This approach will directly lead to a diagonal mass matrix without computing the consistent mass matrix first. This works for arbitrary element order with high accuracy and is probably the most popular approach, particularly for quadrilateral elements. It can be used on triangular elements too. You will find hundreds of publications if you search for keywords such as e.g. 'high order spectral elements', 'Gauss-Lobatto quadrature'
You have to be aware of the fact when using 'Spectral lumping' you slightly under integrate the system matrices. For case where distorted elements are within the mesh one should actually use p+3 Integration points in each direction and therefore, I would always recommend HRZ-lumping. The technique is very easy to implement and takes the mass distribution within the element into account (this is very beneficial for fictitious domain methods as spectral lumping does not work for non-geometry conforming meshes). Furthermore, as pointed out by Hauke it ensures that all components are positive and therefore, the results will converge in the limit. So far I have not notice any drawbacks of the HRz lumping scheme.
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