[fredroy]

This was 99.99% AI generated 🥴 as I am not really an expert on mass.
But I felt that it was quite serious to make the AI do it 🤖
But is will a good starting point for those who know

and I am a bit amazed that the references itselves stored a blow-up 🙃

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The non-lumped mass matrix only stores vertex and edge masses. On hexahedra it used rhoV * (1/20 per vertex, 1/40 per edge), whose element eigenvalues are rhoV/40 * {5, 3, 1, -1}: the checkerboard mode has negative mass, so any hexahedral MeshMatrixMass was indefinite (e.g. 42 negative eigenvalues out of 576 in TetrahedronHyperelasticityFEMForceField_invertedTets.scn). On quads, rhoA * (1/8, 1/16) was singular (eigenvalues rhoA/16 * {4, 2, 2, 0}).

Start from the consistent mass matrices (trilinear hexa: rhoV/216 * (8, 4, 2, 1), bilinear quad: rhoA/36 * (4, 2, 1)) and add the couplings that cannot be stored (face and body diagonals, quad diagonals) to the vertex mass:

• hexa: rhoV/216 * (15 vertex, 4 edge), eigenvalues rhoV/216 * {27, 19, 11, 3}
• quad: rhoA/36 * (5 vertex, 2 edge), eigenvalues rhoA/36 * {9, 5, 5, 1} Total mass and row sums (the lumped masses, also used for the weight) are unchanged; the lumping coefficients are updated accordingly (27/15 and 9/5).

Re-enable MeshMatrixMass_test.cpp, which was dropped from the CMake list in #5920, and add tests checking that the assembled matrix is positive definite, sums to totalMass and has the lumped masses as row sums, including after topological changes.

Example scenes:

Mass spectrum:
Inverted-tets scene: it had 42 negative eigenvalues out of 576 (smallest −4.2e-4); it now has none (smallest +1.2e-4).
Other hexahedral scenes: the remaining non-lumped ones were also indefinite on master and are now positive definite: TetrahedronHyperelasticityFEMForceField.scn, EulerExplicitIntegrationScheme.scn, MatrixLinearSystem.scn and rigidifiedSectionsInBeam.scn.
Quad scene: VisualMesh.scn was singular and is now positive definite.
Tetra and triangle scenes: unchanged.
Explicit scene: EulerExplicitIntegrationScheme.scn blew up on master (displacement 524 at t=0.03 s). With the fix it falls freely as expected (0.0045, about ½gt²).
Implicit scenes: trajectories change by less than 0.1%.

Regression references:

EulerExplicitIntegrationScheme.scn (3000 steps), whose stored reference is the blow-up.

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[with-all-tests]

[NOPE-depends-on https://github.com/https://github.com/sofa-framework/Regression/pull/145]
(to fix when the new regression system is merged)

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Reviewers will merge this pull-request only if

• it builds with SUCCESS for all platforms on the CI.
• it does not generate new warnings.
• it does not generate new unit test failures.
• it does not generate new scene test failures.
• it does not break API compatibility.
• it is more than 1 week old (or has fast-merge label).

[alxbilger]

I don't understand how the coupling terms were perceived by Claude in this PR. They are still missing with this PR right?
Note that FEMMass should give the exact mass matrix, including the coupling terms. I don't knowe exactly how it compares to the this PR.

[alxbilger]

For the records, this PR fixes diffusion on hexas in #6431. Without it, it explodes.