Collaboration and funding
The research is done in collaboration with NASA Langley Research Center, Stanford University and Southern Methodist University in USA, the Council for Scientific and Industrial Research (CSIR) in South Africa, the Swedish Defence Research Agency (FOI) and the Swedish Meteorological and Hydrological Institute (SMHI) in Sweden.
The base funding is from Linköping University while the external research is funded by Swedish Research Council, SMHI and the European Union FP7 projects Industrialisation of High-Order Methods - A Top-Down Approach (IDIHOM) and Uncertainty Management for Robust Industrial Design in Aeronautics (UMRIDA).
Our ambition
To develop mathematically based and provable convergent methods for solving time-dependent partial differential equations governing physical processes.
Main activities:
High Order Finite Difference Methods (FDM)
We have developed summation-by-parts operators and penalty techniques for boundary and interface conditions. We have applied FDM to demanding problems in fluid mechanics and wave propagation.
High speed separated flow including shocklets
Collaborator: Flow Research Unit, University of the Witwatersrand
Earthquake simulation related to Japananese tsunam
Collaborator: Department of Geophysics, Stanford University
Contact: Jan Nordström
We have extended the FDM to handle partial differential equations posed on time varying deforming domains. As an application, sound propagation by the linearized Euler equations in a time-dependent domain is considered.
A time varying deforming domain
The rate of dilatation
Contact: Jan Nordström
We have developed a method for constructing finite differences adjusted to periodic wave propagation problems. The resulting stencils minimise the wave speed error that arises in the discrete solution.
Comparison of pulse propagation between the new Remez stencils and other stencils in the literature
Pressure profile of propagating Euler vortex
Contact: Jan Nordström
Finite Volume Methods (FVM)
We have reused the framework for the high order finite difference methods and derived summation-by-parts operators for first and second derivatives as well as suitable artificial dissipation operators.
Unstructured mesh around high lift configuration
Collaborator: FOI, Swedish Defence Research Agency
Flow in the nose region of a Naca0012 wing profile
Collaborator: FOI, Swedish Defence Research Agency
Contact: Jan Nordström
Numerical Coupling
We combine the advantages of FDM and FVM in an efficient and stable way. This numerical technique has been used in both aerodynamics (the Euler equations) and wave propagation (elastic wave equations) problems.
Unstructured/structured mesh around rods
Collaborator: CTR, Center for Turbulence Research at Stanford University
Unstructured/structured mesh around Landers fault
Collaborator: Department of Geophysics, Stanford University
Contact: Jan Nordström
Multi Physics Coupling
We develop well-posed and stable numerical coupling procedures for multiple equations sets. Lately we have considered the Navier-Stokes and the heat equation, fluid structure interaction and the elastic wave equation with friction laws.
FSI, low accuracy misses eigenfrequency
Collaborator: Department of Information Technology, Uppsala University
Conjugate heat transfer between air and gold
Collaborator: Department of Information Technology, Uppsala University
Contact: Jan Nordström
Boundary Conditions
We analyze existing techniques and derive new formulations. Lately we have focused on solid wall boundary conditions for the Navier-Stokes equations.
Navier-Stokes solution close to solid boundary
Collaborator: Department of Information Technology, Uppsala University
Weak enforcement of solid wall boundary conditions
Collaborator: FOI, Swedish Defence Research Agency
Contact: Jan Nordström
SAT terms are used at the boundaries to implement the boundary conditions. However, if data is available, additional SAT terms can be applied in the interior of the domain. We refer to this technique as multiple penalty technique (MPT). By using this technique, we can reduce the errors in the computational domain and increase the rate of convergence.
An oscillating pulse at different time levels that we want to model. Upper left: t=0, upper right: t=0.25, lower left: t=0.75 lower right: t=1.15
The error as a function of time when using 80 grid points in each space direction
Contact: Jan Nordström
Unceartainty Quantification (UQ)
We take into account various kinds of uncertainties or stochastic variations related to aerodynamic problems. Typical examples include stochastic uncertainties in the geometry of a wing, the speed of the aircraft and the angle of attack.
Unceartainty in density for the Sod test case approximated using Haar-wavelets
Collaborator: Uncertainty Quantification (UQ) Laboratory at Stanford University
Unceartainty in density for the Sod test case, exact solution
Collaborator:
Uncertainty Quantification (UQ) Laboratory at Stanford University
Contact: Jan Nordström
Fluid Mechanics of Vortices
By using the energy method on the constant coefficient compressible Navier-Stokes equations we can show why vortices decay slowly in viscous flows.
Velocity field in a vortex
Collaborator: Department of Information Technology, Uppsala University
Energy deacy for different types of flowfields
Contact: Jan Nordström
Propagation of nerv signals in living tissue
By solving the cable equation in combination with the Hodgkin-Huxley's equations we can simulate the propagation of nerve signals in the denditric tree.
Nerves and somas
Collaborator: Department of Aeronautics and Astronautics, Stanford University
Sketch of a nerve connecting to a soma
Collaborator: Department of Aeronautics and Astronautics, Stanford University
Contact: Jan Nordström
Heat Conduction
In industrial applications one would sometimes estimate the temperature in inaccessible areas where the temperature can not be measured directly. You then place a measuring element inside the material and calculate the sought surface temperature from the measured data. Mathematically, it is described as a Cachy problem for the heat equation with data available along the line x = L and the solution at x = 0 is requested. The situation is illustrated in Figure 1. The problem is ill-posed when small measurement errors can cause large errors in the computed solution. Special regularization methods are used to minimize the effect of measurement errors of calculated solutions. Similar mathematical problems arise in applications where mathematical analysis is used to compensate for errors caused by, for example, the measurement devices.
Research is done within the methodological and mathematical analysis of both the application problems and numerical methods.
Figure 1: Illustration of lateral heat conduction. Measurements at x = L are used to calculate the temperature at the surface x = 0.
Contact: Fredrik Berntsson
