25. Září 2014
Thomas Richter, Universität Heidelberg, Germany Eulerian formulations for fluid-structure interaction problems have the benefit, that large deformation and contact is not a limit. In contrast to the Arbitrary Lagrangian Eulerian formulation, no artificial coordinate transformation is involved. Instead, all calculations are carried out on a fixed Eulerian background mesh. This casts the method into the frame of front-capturing techniques. The interface between fluid and solid will cut through mesh elements. Furthermore, the interface is not fixed, but it will move over time. We will show, that the motion of the interface through the domain will give rise to severe numerical problems, if we aim at getting high order convergence in space and time. First, as an interface problem, the spatial discretization must take care of discontinuities at the interface. This can be achieved by a modification of the finite elements spaces (e.g. XFEM or fitted finite elements). Second, the motion of the interface from time-step to time-step leads to a deterioration of temporal accuracy. Even though similar problems appear for all Eulerian models with moving interfaces (like levelset-formulations of multiphase flows), this problem is still insufficiently discussed in literature. In this talk, we focus on the temporal discretization. We will present a high order spatial and temporal discretization scheme for Eulerian models with sharp and moving interfaces. The discretization is based on a space-time Galerkin approach that considers both the spatial and temporal motion of the domains.
MOdelling REvisited + MOdel REduction Modeling, analysis and computing in nonlinear PDEs. September 21-26, 2014, Chateau Liblice, Czech Republic. Recently developed implicit constitutive theory allows one to describe nonlinear response of complex materials in complicated processes and to model phenomena in both fluid and solid mechanics that have hitherto remained unexplained. The theory also provides a thermodynamically consistent framework for technologically important but so far only ad hoc engineering models without sound footing. The overall goal of the project is to develop accurate, efficient and robust numerical methods that allow one to perform large-scale simulations for the models arising from the new theoretical framework. A natural part of the goal is rigorous mathematical analysis of the models. The nonstandard structure of constitutive relations arising from the new framework requires the reconsideration of many existing approaches in the mathematical theory of partial differential equations and the development of new ones. In particular, basic notions such as the concept of the solution and its well-posedness need to be reconsidered. Further, the complexity of the constitutive relations calls for rigorous investigation on model reduction - the identification of simplified models that capture the chosen (practically relevant) information about the behaviour of the system and disregard irrelevant information. Reliable numerical simulations require the derivation of sharp a posteriori error estimates to control all possible sources of errors, including rarely studied but important algebraic errors. We believe that in solving difficult problems in mathematical modelling the individual aspects discussed above - physics, mathematical analysis and numerical analysis - are so closely interrelated that no breakthrough can be achieved without emphasising the holistic approach as the main principle. Our vision is to follow this principle: the entire process of modelling of complex materials will be revisited in an innovative manner.
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