Sep 22, 2014
Hiromichi Itou, Tokyo University of Science, Tokyo, Japan Theory of partial differential equations has been thoroughly developed mainly in smooth domains. In non-smooth domains such as polyhedral or cracked domains, difficulties appear because domains have singular points. Then, it is important to analyze the precise behavior of the solution of the corresponding boundary value problems near singular points. And they have possibility of application in various fields of science and engineering such as Fracture problems, Inverse problems (nondestructive evaluation) and so on. The aim of this talk is to introduce some convergent expansion formulae of solutions of two dimensional linearized elasticity equation (called Navier’s equation) around singular points such as a crack tip and a tip of a rigid line inclusion, and furthermore, to clarify the relation between the order of singularities in series expansions and boundary conditions. Some formulae themselves may be known in engineering, however, the explicit formulae with rigorous convergent proof seem non-trivial. The derivation is based on complex analysis, especially, analytic continuation, Goursat-Kolosov-Muskhelishvili stress functions, Lekhnitskii formalism, Riemann-Hilbert problem, etc.
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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