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for Structure preserving discretizations for coupled problems in mechanics: overview and applications

=Ignacio Romero, Ph.D.=
===March 8, 2012===


"Structure preserving discretizations for coupled problems in mechanics:
overview and applications"

Ignacio Romero, Mechanical Engineering School, Technical University of Madrid

The development of numerical methods which preserve the mathematical structure of the equations whose solution they are meant to approximate has guided much of the efforts of Applied Mathematics and Mechanics in the last two decades. In Mechanics, most of the advances have taken place in the numerical solution of Hamiltonian problems, whose well-known mathematical structure has assisted the algorithmic design.

There exists, however, great interest in Mechanics and other fields to find structure preserving methods that can be applied to general classes of problems, far beyond conservative ones. In this talk, a family of methods will be presented designed to be applied in the numerical solution of a wide class of evolution problems, namely, those who derive from the principles of thermodynamics under particular modeling assumptions.  The most salient feature of such methods is that they exactly preserve the two laws of
thermodynamics, as well as symmetries in the equations.

In a way, the methods that will be presented can be understood as a generalization of the Energy-Momentum methods for Hamiltonian problems developed at Stanford during the 80s and 90s by Prof. Simo. Similarly to the Energy-Momentum methods, the ones described in the talk provide accurate qualitative descriptions of the dynamical behavior of the systems they approximate, as well as exhibiting remarkable numerical robustness.

In addition to describing the theory, examples will be given to illustrate the practical application of such methods to specific problems of interest. To showcase their general applicability, their implementation to solid mechanics and a phase field model for dendritic growth will be presented.

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