Hallo John, many thanks for your answer.
Yes, I agree, The condition I wrote is true only for low order elements. For serendipity, and maybe also second order, elements it is not true. For second order, some basis functions have zero integral. Actually the difference that I have founds more related to "where" you apply a quadrature that results in a lumped mass matrix. You can write the integral over a generic element and use a "generic lumping", (i.e. a quadrature rule that gives a diagonal mass matrix on this configuration) or map to a reference element and use a "reference lumping" there. If the map between the two configurations is not linear, as in the case of a generic hexahedral/quadrialteral elements, the two approaches result in different quadrature rules. What I have also observed, but not proven and maybe it was only by chance, is that the "algebraic lumping" (i.e. summing the entradiagonal element to the diagonal one) coincides with the approach "generic lumping" if the the tridiagonal term does not appear in the map between the two configurations. If this term is present, the three approaches may give three "different" lumping methods. Best regards Marco On Tue, Jun 4, 2019 at 4:24 PM John Peterson <jwpeter...@gmail.com> wrote: > > > On Wed, May 22, 2019 at 10:03 AM John Peterson <jwpeter...@gmail.com> > wrote: > >> >> >> On Wed, May 22, 2019 at 3:30 AM marco <ing....@gmail.com> wrote: >> >>> Mass lumping is defined by using trapezoidal quadrature rule. >>> In this way, the weights of a trapezoidal rule should be the integral >>> over >>> the element of each basis function. >>> >> > I looked into this a bit more carefully over the weekend, and while it may > be technically true, I don't think it necessarily results in a "good" > quadrature rule for every element, at least not for the "serendipity > elements" (Quad8, Hex20). > > If you compute the element integral of one of the vertex basis functions > of the Hex20, you get -1. The edge basis functions integrate to 4/3. > Interestingly, such a rule is exact for quadratics, but using negative > weights in a nodal quadrature rule seems very problematic to me. For > example, it would give you negative diagonal entries for the mass matrix > rows associated with the vertex basis functions... > > > I noticed that this is not always true for hexahedral elements. Am I >>> missing something or could there be a bug? >>> >> > That being said, I think I have convinced myself that using the tensor > product of 1D QSimspson rules on a Hex20 or Quad8 a) does not result in a > nodal quadrature for such an element, and b) produces a non-diagonal mass > matrix. It's definitely possible to "fix" this issue, but it's not clear to > me what the best choice of weights should be. One choice, which is exact > for quadratics, is the one with negative vertex weights mentioned above. > Another choice of weights would be (vertex=1/7, edge=4/7), but this is > chosen somewhat arbitrarily to match the 4:1 ratio of edge:vertex weights > in Simpson's rule, and is only exact for linears. It may be possible to > optimize the choice so that e.g. the mass matrix diagonals resulting from > this quadrature are as close as possible to the "true" mass matrix > diagonals (or their lumped equivalents) > > -- > John > _______________________________________________ Libmesh-users mailing list Libmesh-users@lists.sourceforge.net https://lists.sourceforge.net/lists/listinfo/libmesh-users
