Non-Conventional Numerical Strategies in the Advanced Simulation of Materials and Processes

In this work we analyze the possibilities of applying model reduction in the advanced simulation of materials and processes. The use of such strategies allows impressive computing time savings in the numerical simulations of complex models without degrading the solution accuracy. For this purpose we apply proper generalized decompositions of multidimensional models that can be associated to usual models in computational mechanics. The proposed approach is based in introducing all the parameters of interest as new extra-coordinates in the model and then solving it allowing computing the solution in the space and time for any value of the selected parameters. This approach increases the dimensionality of the model, but by applying a separated representation of the unknown field the redoubtable curse of dimensionality can be circumvented

On the solution of the multidimensional Langer's equation using the proper generalized decomposition method for modeling phase transitions

The dynamics of phase transition in a binary mixture occurring during a quench is studied taking into account composition fluctuations by solving Langer's equation in a domain composed of a certain number of micro-domains. The resulting Langer's equation governing the evolution of the distribution function becomes multidimensional. Circumventing the curse of dimensionality the proper generalized decomposition is applied. The influence of the interaction parameter in the vicinity of the critical point is analyzed. First we address the case of a system composed of a single micro-domain in which phase transition occurs by a simple symmetry change. Next, we consider a system composed of two micro-domains in which phase transition occurs by phase separation, with special emphasis on the effect of the Landau free energy non-local term. Finally, some systems consisting of many micro-domains are considered

Non-Conventional Numerical Strategies in the Advanced Simulation of Materials and Processes

International audienceIn this work we analyze the possibilities of applying model reduction in the advanced simulation of materials and processes. The use of such strategies allows impressive computing time savings in the numerical simulations of complex models without degrading the solution accuracy. For this purpose we apply proper generalized decompositions of multidimensional models that can be associated to usual models in computational mechanics. The proposed approach is based in introducing all the parameters of interest as new extra-coordinates in the model and then solving it allowing computing the solution in the space and time for any value of the selected parameters. This approach increases the dimensionality of the model, but by applying a separated representation of the unknown field the redoubtable curse of dimensionality can be circumvented

Atoms, Molecules and Flows: Recent Advances and New Challenges in their Multi-Scale Numerical Modeling at the Beginning of the Third Millenium

International audienceNano-science and nano-technology as well as the fine description of the structure and mechanics of materials from the nanometric to the micrometric scales need descriptions ranging from the quantum mechanics to the kinetic theory descriptions characteristic of statistical mechanics. This paper explores the modelling at these scales and points out the main challenges related to the numerical solution of such models that sometimes are discrete but involve an extremely large number of particles (as in the case of molecular dynamics simulations or coarse-grained molecular dynamics) and other times are continuous but they are defined in highly multidimensional spaces leading to the well known curse of dimensionality issues

Recent Advances in Material Homogenization

Heterogeneous materials involve different length scales in their mechanical properties. Obviously a mechanical description taking into account all the microscopic details is impossible from a computational point of view except for parts of very small dimensions. The main aim of material homogenization is defining macroscopic homogeneous properties able to represent at the macroscopic scale the real material and allowing for ignoring the microscopic scale in the numerical representation