EigenValue Problem using dealii

[kai huang]

Dear all,

   I am working on some problems involving eigenvalues and eigenvectors. 

   I am having some issues when assembling the stiffness matrix. Generally, there are three methods to handle Dirichlet boundary conditions, such as the penalty method (multiplying by a large number); setting the diagonal elements to 1 and the remaining elements to zero; or removing the corresponding degrees of freedom from the stiffness matrix. However, the penalty method (multiplying by a large number) can cause the matrix to become ill-conditioned. Setting the diagonal elements to 1 and the remaining elements to zero can affect the computation of lower-order eigenvalues, leading to eigenvalues of 1 that might not be the desired eigenvalues. The best method for solving eigenvalue problems is to remove the corresponding degrees of freedom from the stiffness matrix, as this not only saves computational effort but also avoids the aforementioned issues. However, in sparse matrices, this method is not easy to handle. Is there a function in deal.II that can manage this?

    Additionally, I would like to ask if isogeometric analysis can be done in deal.II?

    Thank you for your great kindness and generosity.

Best

Huang

[Wolfgang Bangerth]

On 6/20/24 22:09, kai huang wrote:
>    I am working on some problems involving eigenvalues and eigenvectors.
>    I am having some issues when assembling the stiffness matrix. Generally,
> there are three methods to handle Dirichlet boundary conditions, such as the
> penalty method (multiplying by a large number); setting the diagonal elements
> to 1 and the remaining elements to zero; or removing the corresponding degrees
> of freedom from the stiffness matrix. However, the penalty method (multiplying
> by a large number) can cause the matrix to become ill-conditioned. Setting the
> diagonal elements to 1 and the remaining elements to zero can affect the
> computation of lower-order eigenvalues, leading to eigenvalues of 1 that might
> not be the desired eigenvalues. The best method for solving eigenvalue
> problems is to remove the corresponding degrees of freedom from the stiffness
> matrix, as this not only saves computational effort but also avoids the
> aforementioned issues. However, in sparse matrices, this method is not easy to
> handle. Is there a function in deal.II that can manage this?

You are in luck, we thought of this when we wrote step-36 :-) Take a look at
this section:
https://dealii.org/developer/doxygen/deal.II/step_36.html#step_36-EigenvaluesandDirichletboundaryconditions

>     Additionally, I would like to ask if isogeometric analysis can be done in
> deal.II?

This depends on how exactly you define "isogeometric". We can use arbitrary
geometries (in particular NURBS-based geometries) and there are tutorial
programs about that as well (step 54, for example).

Best
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/