1. Helmholtz equation
FEM can be used to approximately solve the Helmholtz equation. However, the solution is rapidly oscillating and a sufficiently fine mesh is required to capture the oscillation. In this project, a solver for the Helmholtz equation is implemented and the dependency between mesh size and frequency of the oscillation required before the method converges is studied numerically.
2. Finite element method in 3D
In this project, a finite element solver is implemented in three spatial dimensions. The aim is to compare the added computational cost arising from 2D→3D transition.
3. Axisymmetric reduction
Many engineering objects are axially symmetric. In this project, one studies the use of symmetry to reduce the computational cost.
4. The multigrid method
One challenge of the 2D→3D transition is the increased cost of solving the linear system related to finite element method. Very efficient iterative solution methods have been developed for this purpose. In this project, one example of such methods in two dimensional setting is investigated.
5. Contact problem
Especially in solving problems related to mechanical engineering, the partial differential equation is coupled with constrants for the solution. In this project, one studies solution of such problems by using the Poisson equation as a model problem.
6. Multi-scale finite element method
The topic of this problem is solution of the heat equation with oscillating material data. When the material data oscillates rapidly, one has to use mesh that has smaller elements in comparison to the length of the oscillation. In this project, a special multi-scale finite element method that allows one to use larger elements is studied.
7. Solution of non-linear PDEs
PDEs arising in applications are very often non-linear. In this project, one investigates application of finite element method with Newton-iteration to solution of non-linear PDEs.
8. Electrical impedance tomography
In this project a non-constant conductivity of a 2D domain is solved based on boundary electric potential (voltage) measurements with a collection of current injection patterns. The inverse boundary value problem is solved by formulating an energy minimization problem with regularization.
9. Rotating subdomain
In this project, one considers a domain with a rotating part. One way to deal with rotation in finite element simulation is to divide the domain into two sudomains and to generate a finite element mesh separately for them. Continuity of the solution is forced by aproppriately modifying the bilinear form.
10. Own topic
Feel free to propose your own topic to Antti and Topi!
