Complex system matrices

[supreet]

In the "Solving eigenvalue problems" chapter of the Solver manual, it is mentioned that the keyword "Eigen System Complex" can set to true when the system matrices are complex.

How can complex system matrices be created? Because for linear elasticity problems, density, which forms the mass matrix, must be real, as per solver. And for acoustic problems, damping coefficient, which forms the damping matrix, must be real, as per solver.

Are there any tutorials that make use of this keyword?

Thanks.

[Rich_B]

Hello,

Searching in elmerfem\fem\tests for the phrase 'Eigen System Complex' returns four results. One result is for 'ShoeboxFsiEigen2D', which includes a note:

This is test case for strongly coupled eigenmode solution of coupled wave equation and elasticity equation.

Maybe this will be relevant for your case?

Rich.

[kevinarden]

Complex eigenvalues are required to account for damping in a linear eigenvalue solution.
Working from ElmerGUI Tutorial Number 10, Case 1 is no damping, Only real eigen values are computed
Case 2, add damping to the material, but still only real eigenvalues since the complex solver is
not switched on, damping is ignored. Case 3 damping with the complex eigen solver turned on, eigen values are now complex, contains real and imaginary values.

Case1 No Damping

case1.sif

(1.58 KiB) Downloaded 28 times

EigenSolve: Computed 10 Eigen Values
EigenSolve: 1: 1.884402E+01 0.000000E+00
EigenSolve: 2: 8.075503E+01 0.000000E+00
EigenSolve: 3: 8.076752E+01 0.000000E+00
EigenSolve: 4: 2.122355E+02 0.000000E+00

Case2 Damping defined in material but complex not turned on

case2.sif

(1.71 KiB) Downloaded 33 times

EigenSolve: Computed 10 Eigen Values
EigenSolve: 1: 1.884402E+01 0.000000E+00
EigenSolve: 2: 8.075503E+01 0.000000E+00
EigenSolve: 3: 8.076752E+01 0.000000E+00
EigenSolve: 4: 2.122355E+02 0.000000E+00

Cse3 Damping included complex eigen solution turned on

case3.sif

(1.75 KiB) Downloaded 35 times

Eigenvalues have real and imaginary components
EigenSolveComplex: EIGEN SYSTEM SOLUTION COMPLETE:
EigenSolveComplex:
EigenSolveComplex: Number of converged Ritz values is: 10
EigenSolveComplex: Computed Eigen Values:
EigenSolveComplex: 1 (2.35779681324004681E-005,-9.55825719520030495E-006)
EigenSolveComplex: 2 (2.20256595972161914E-005,1.31875981601833986E-005)
EigenSolveComplex: 3 (-1.02390128222069986E-005,-2.59052059504491354E-005)
EigenSolveComplex: 4 (-1.30324599576908262E-005,2.48602021200610741E-005)

[supreet]

Hi Kevin,

The manual says to set the keyword "Eigen System Damped" to true when we add damping term to the equation. I did this for a toy problem whose analytical eigenvalues are known, and got the right result.

Further, it also says not to set the keyword "Eigen System Complex" to true when damping is considered. It specifically says to include this keyword when the system matrices are complex.

[supreet]

Hi Rich,

I'll check these tutorials and get back to you.

Thanks.

[raback]

Hi

When you have a time-dependent PDE you can go into harmonic one by making some ansatz with time dependence exp(i*w*t). When this is derived by time you get i*w out. So basically d/dt -> i*w. Hence d^2/dt^2 -> -w^2. This means that even if you have a real valued damping matrix it results to complex eigen matrix. The 2nd order inertial term is always real.

-Peter