Model order reduction in computational solid mechanics
Model order reduction techniques decrease the complexity of mathematical models in numerical simulations. They play a key role in dealing with parametrized systems that require fast and frequent model evaluation. This course provides an introduction to model order reduction with a focus on applications in computational solid mechanics. After successful completion of the first part, students know the foundations of parametrized partial differential equations and understand the challenges associated with their finite element approximation. They know the mathematical basis of different reduced order methods, including their specific advantages, and are able to decide in what scenario which method should be applied. After completion of the second part, students are able to bridge the gap between multiscale models in solid mechanics, discretization based on the finite element method, and model order reduction. They are able to implement different model order reduction techniques for linear problems and are able to critically assess their results in terms of accuracy and efficiency. After completing the third part, students understand limitations of model order reduction techniques and know about open questions and challenges related to current research.
- Motivation of reduced order modeling (many-query, real-time, high-dimensional scenarios).
- Traditional engineering approaches: static condensation, modal decomposition.
- Foundations of parametrized partial differential equations.
- Proper orthogonal decomposition, snapshots, offline/online strategies.
- Reduced basis methods, Galerkin projection and orthonormalization, sampling strategies.
- Computational homogenization of heterogeneous materials.
- Generalized multiscale finite element methods.
- Stability, system conditioning, empirical interpolation methods.
The course is accompanied by a computer lab, where illustrative model problems are implemented in Matlab.
This course requires a solid knowledge on the finite element method and continuum mechanics.