Sep 23, 2014
Benjamin Müller, University of Duisburg-Essen, Essen, Germany Joint work with Gerhard Starke, Jörg Schröder and Alexander Schwarz The finite element method is an important tool for the simulation of elasticity problems in solid mechanics. It is well known that the linear elastic theory does not cover arising real life problems. Physically more realistic models lead to nonlinear partial differential equations. In this talk we present least squares finite element methods based on the momentum balance and nonlinear constitutive equations for hyperelastic materials. Our approach is motivated by a well-studied least squares formulation for linear elasticity. The idea in this approach is to invert the given stress-strain relation such that it is possible to consider fully incompressible materials. Our aim is to generalize this idea to an approach which takes nonlinear kinematics and nonlinear stress-strain relations into account. General least squares formulations for isotropic homogeneous frame-indifferent hyperelastic materials based on inverse stress-strain relations will be derived. For the special case of a Neo-Hooke material we can consider, similar to linear elasticity, the full incompressible case. A detailed analysis for the nonlinear and the linearized problem will be provided. It will be shown under strong regularity assumptions that the (nonlinear) least squares functional is an efficient and reliable a-posteriori error estimator. A further novelty of our approach, in comparison to other discretization methods, is that next to the displacement u the full first Piola- Kirchhoff stress tensor P is considered and both are approximated simultaneously. At the end of the talk we will illustrate the performance of our method in some numerical experiments using second to lowest order Raviart-Thomas elements for the stress tensor, continuous piecewise quadratic elements for the displacement vector and adaptive refinement.
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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