Bachelor / Master Thesis
Multigrid with Higher-order Prolongation and Restriction in waLBerla
for elektrokinetic flow applications
In lab-on-a-chip (LoC) systems, samples consisting of fluid and particles are transported,
manipulated and analyzed in structures of length scales from 100 µm to several nm.
The mechanism of choice for microfluidic manipulation and actuation is electrokinetics,
since devices are cheaper and easier to control and manufacture compared to other mechanisms.
Electrokinetics includes electro-osmosis and electrophoresis.
Electro-osmosis is the flow of a dilute electrolyte solution caused by an applied electric field.
Electrophoresis is the movement of charged particles relative to a fluid in an electric field.
Typical applications are electrokinetic pumps, traps or filters for driving, fixing or
separating bioparticles depending on their charge.
Electrokinetic phenomena can be simulated by coupling a fluid simulation (considering fluid-particle interaction)
to a solver for a partial differential equation (PDE) that describes the electric field.
Such PDEs are the Poisson-Boltzmann or Guoy-Chapman equation.
(widely applicable Lattice Boltzmann from Erlangen) software framework provides a powerful
simulation environment for multi-physics applications.
As illustrated in the movie above, we can simulate the behavior of charged particles in a fluid.
The electric field is then modeled by PDEs (in the simplest case Poisson's equation) that are then
discretized to an LSE with finite differences.
We already integrated a multigrid solver for Poisson's equation into the waLBerla framework so that we can
model basic electrokinetic effects. This solver as well as other linear solvers are encapsulated in a
The main drawback of the current multigrid method is that convergence cannot be guaranteed for more complex equations,
such as the Poisson-Boltzmann or Guoy-Chapman equation.
This results from the low-order restriction and prolongation operators that are implemented at the moment.
The goal of this thesis is to implement higher-order grid transfer operators into the waLBerla module
(considering the performance of the implementation) and to evaluate the convergence properties of the extended algorithm.
Approaches can be found in . Besides the Galerkin-coarsening approach that is used at the moment, also
a collocation-coarse approximation scheme shall be discussed since it can preserve the seven-point stencil .
Finally, simulations with the schemes solving the Poisson-Boltzmann or Guoy-Chapman equation should be performed.
The physical correctness of the simulations should be verified.
- Background in numerics
- C++ programming skills
- Interest in simulation and modelling of physical phenomena
 M. Mohr and R. Wienands, "Cell-centred Multigrid Revisited", Computing and Visualization in Science, 7:129-140 (2004).
 R. Wienands and I. Yavneh, Collocation Coarse Approximation in Multigrid. SIAM Journal on Scientific Computing, 31:3643-3660 (2009).