Solving convection dominant problems by standard Galerkin finite element methods, with an arbitrary choice of the finite element space, inevitably produces spurious oscillations. For example, to compute fluid flows with high Reynolds number described by the Navier-Stokes equations, a numerical stabilization need to be added. Several stabilized finite element methods where developed during the course of the last 40 years. The Galerkin Least Squares method also named as Petrov-Galerkin Method adds a weighted least squares term to the equations. Using an additional artificial viscosity based on the residual, results in the Streamline Upwind Petrov-Galerkin (SUPG) Method, sometimes also named as Streamline Diffusion Method (SD). The drawback is that the stabilization can result in a decrease of the accuracy and efficiency. The challenge is to find a consistent formulation, which can be implemented as an efficient algorithm which gives solutions with high accuracy.

• Model: Two Phase Incompressible Flow
• Method: Level Set Method with Stabilized Finite Element Method
• Stabilization: Residual Viscosity

We model waves using a two phase flow model that involves an interface between phases. One technique to solve interface propagation problems is the Level Set Method. The Method uses an implicit function with specific properties to represent the interface. The evolution of the interface is determined by solving the Level Set equation. The implicit function can lose its required property in the process of solving the Level Set equation, in that case, it needs to be reinitialized. We study various numerical methods to solve the Level Set equation within the finite element setting in order to find an efficient and accurate method. The figure in this sections shows the Zalesaks Problem. A slotted disk is rotated around the center of the domain. The initial and final interface are compared after a number of periods to evaluate the accurate representation of the interface.

Collaborator: Aurélien Larcher

• Model: Variable Density Incompressible Flow
• Method: Stabilized Finite Element Method
• Stabilization: Residual Viscosity

Tsunamis can damage or destroy bridges which an cause delays in emergency response, complicate recovery, and adversely impact the local economy. To offer design alternatives for the construction and or reinforcement on bridges, this research project studies the impact of tsunamis on bridges using numerical methods. Simulation techniques are being developed to model the complex phenomena associated with tsunami related fluid-air-structure interactions because experiments have limitations such as imperfect boundary conditions or scaling issues and can be impractical to perform in many circumstances.The numerical simulations provide qualitative results that can be used to improve bridges along the coast or to propose design alternatives.
We approach the tsunami-structure interaction problem in the computational setting by modeling air and water as an incompressible variable density fluid.This allows us to take the entrapped air into account which plays a key role in the interaction dynamics. The model equations are solved using the open source software dolfin-hpc and Licorne which are software components of the FeniCS project.
The turbulent nature of the flow is taken into account by the implicit large eddy simulation (LES), where the unresolved scales are represented by the dissipation of the numerical scheme. The computational framework can utilize massively parallel architectures to achieve the required model resolutions.
The figure in this section shows pictures of an experiment conducted by a research group in Japan and early results of a simulation performed by us.
Collaborator: Michael Motley

Adaptive algorithms in finite element method are used to improve the accuracy of a computed quantity. Doing an uniform refinement is a possible approach, and it can increase the accuracy, but is inefficient, since not all areas of the domain contributing to the error in the same amount. To do an efficient refinement, the areas which contribute most to the error need to be identified. This means that it is necessary to add an error indicator. An a posteriori error estimation, extracted from the computed solution, as well as the given data, is usually used. To add sensitivity to local error sources, duality techniques are employed.

• Model: Regularized 13-Moment
• Method: Finite Element Method

A gas contains particles in the order of 1016 per cubic millimeter. Due to the many collisions between particles, the gas behaves as a continuum. The condition of a gas can be described by the Knudsen Number which is defined by Kn = λ₀/L where λ₀ is the mean free path length and L is the macroscopic length scale. In situations where the typical length scale L is much larger than the mean free path length λ₀ the flow is described well through the Navier- Stokes and Fourier equations. These equations are valid for Kn = 0.01, which is called the Hydrodynamic Regime. Gases outside the Hydrodynamic Regime are called rarefied gases. In this region the Navier-Stokes Fourier equations fail, and have to be replaced by a set of more refined equations. A good example is the Boltzmann equation which describes the gas on a microscopic level that refers to the translation and collision of particles. The Boltzmann equation is the main equation in the kinetic theory of gases. A new theory developed by Struchtrup and Torrilhon (2003) describe the rarefied gas by the Regularized 13-Moment-Equations short R13. The state of the gas is characterized by 13 quantities containing density, velocity, pressure, temperature, stress tensor and heat flux. Since the R13 equations can be written in conservational form, it is suitable to use Finite Element Methods to solve them.
Collaborator: Manuel Torrilhon

C++ code maintained and developed mainly by Aurélien Larcher and Niclas Jansson

• Fully distributed unstructured mesh
• Load balancing
• Read and write with MPI I/O
• Mesh partitioning with ParMetis
• Linear algebra back-end PETSc or JANPACK
• Post-processing

C++ and Python code maintained and developed by Aurélien Larcher and myself, containing

• Solvers
• Convergence tests
• Applications

The software is installed and run on various computer systems. We follow the following plan:

• Development and testing: Local machines with few number of cores
• Parallel testing and convergence studies: Servers with around 64 cores
• Large applications: Supercomputer systems with several thousand of cores