On 8/25/21 11:53 AM, Anton Ermakov wrote:
I I¯ A 0 ¯I *eigenvalue + I¯ 0 B ¯ I I * eigenvector =0
I I_ 0 0 _I I_ C D _I I
¯ ¯
where eigenvector consists of a vector displacement and a scalar
pressure perturbation:
eigenvector = transpose([U P])
It seems from the literature (e.g., Wang & Bathe 1997) that it is
standard to rewrite this system to eliminate the pressure variable:
eigenvalue * A * U - B * D^-1 * C * U = 0
or
eigenvalue * A * U + M * U = 0
with M = - B * D^-1 * C and P = -D^-1 * C * U
You can do that, but I believe that it's also possible to work with the
original system where one of the two matrices is only semidefinite.
What I am having trouble with is how to efficiently represent matrix
products and inverses that go into matrix M. All the matrices (A, B, C,
D) are sparse, but due to the inverse, it seems that M will be a full
matrix and it might be unaffordable to store it in the memory. There was
a discussion on a similar topic here:
https://groups.google.com/g/dealii/c/5P_fv0zg7jg/m/CPYcYrbsDQAJ
It seems that LinearOperator was used in that discussion to manipulate
the matrix products and inverses, which seemed like an approach to
follow if you use ARPACK eigensolvers. However, I am using SLEPc and it
seems the inputs to SLEPc solvers would have to be objects of class
PETScWrappers::MatrixBase
<https://nam10.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.dealii.org%2Fcurrent%2Fdoxygen%2Fdeal.II%2FclassPETScWrappers_1_1MatrixBase.html&data=04%7C01%7CWolfgang.Bangerth%40colostate.edu%7Ce376b299e56c44690ada08d967f13673%7Cafb58802ff7a4bb1ab21367ff2ecfc8b%7C0%7C0%7C637655108014800275%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=QQLBoH27O1sZ6GMXDatmucrxAye8edSeN3XNqAybONI%3D&reserved=0>.
So, my question is how to prepare such a matrix M involving matrix
products and matrix inverse efficiently and represent it as
PETScWrappers::MatrixBase
That might not easily be possible with our PETSc interfaces.
PETSc has the concept of MatShell, which is in essence what deal.II's
LinearOperator is: A class that implements a matrix-vector operation,
without giving access to individual matrix elements. You could try
whether you can implement something derived from MatrixBase that is a
MatShell operation in the same way as the existing classes derived from
MatrixBase model other types of PETSc matrices.
I don't know how difficult that would be, though.
Best
W.
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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