Mat-1.3656 Seminar on numerical analysis and computational science

Monday, September 7, 2009, room U322 at 14.15, Eirola & Stenberg

Toni Lassila, TKK Mat

Reduced basis method for parametric PDEs in computational fluid and structural mechanics problems

Computational mechanics problems are often modelled using PDEs with one or more free parameters. These parameters can be either physical, such as material constants, or geometric, such as dimensions or even shape parameters. There is typically an engineering interest in computing outputs of the model for different parameter values for purposes of sensitivity analysis, optimization, shape design etc. The outputs are assumed to be linear functionals of the field solution. The objective is to compute efficiently and reliably the outputs of the model for many different parameter values in a real-time or repeated evaluation context.

Since solving repeatedly the full finite element problem turns out to be too costly in many cases, a reduced basis method was proposed in the 1980's to approximate the finite element solution. This method is based on the use of "snapshot" solutions of the PDE at well-chosen parameter points as global basis functions, followed by standard Galerkin projection. We introduce some basic concepts of reduced basis methods and their a posteriori error estimates. Some examples of reduced basis computations for computational mechanics problems in haemodynamics and aerodynamic design are presented.