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Project Summary

We study multiscale effects arising at interfaces or surfaces of bulk materials. Nontrivial interactions between bulk and interface phenomena will include diffusion and reaction of chemical species, elastic deformations, delamination, or dry friction. The emphasis is on the analysis of evolutionary models with interaction between competing effects, each having its own spatial or temporal scale, like the interface thickness, the length scale of structures in bulk/interface, or diffusion lengths. Using weak convergence methods such as G-convergence, homogenization, or multiscale convergence we aim to provide thermodynamically consistent root models for processes on many scales, to develop new analytical methodologies that allow us to derive and justify station­ary and evolutionary scaling limits, including hybrid models connecting small-scale models and effective models in different parts of the physical domain, to apply these methodologies to particular applications motivated by the CRC.

The general philosophy of this project is to consider structured equations such as gradi­ent systems or Hamiltonian systems with dissipation, namely GENERIC systems (Gen­eral Equations for Non-Equilibrium Reversible Irreversible Coupling, cf. [Ött05]), and to pass to scaling limits within these structures. The aim is to obtain in this way not only the information about the convergence of solutions but also of other relevant quantities as for example energies, entropies, or dissipation potentials.

The general strategies will be developed on the basis of well chosen concrete exam­ples that are motivated by the topics of the CRC. In particular, we plan to study scaling limits for bulk-interface reaction-diffusion systems coupled with elasticity, friction mod­els, or delamination. Thereby, we will investigate both, PDE-based models with many internal scales as well as discrete or atomistic models of elastic double strings with het­erogeneous interatomic potentials. From these root models the relevant scales shall be identi.ed by dimensional analysis. New methodologies will be developed in order to pass to the scaling limits in these complex evolutionary systems. The structure of the models as generalized gradient flow systems or as GENERIC systems will help to iden­tify suitable notions of convergence for evolutionary systems. In general, the separate convergence of energy functionals and dissipation potentials will not be sufficient in order to guarantee that solutions of the corresponding systems converge to solutions of the system defined by the limit functionals.
We expect that the research program of this project will also provide valuable infor­mation to identify effective quantities that can be simulated ef.ciently and that allow for the construction of numerical approximation schemes that are thermodynamically consistent and robust with respect to the multiple scales.

 

Link to the C05 Homepage at WIAS


Project Details

  • Status:  In Progress
  • Head(s):  Prof. Mielke
  • Project member(s):  Martin Heida,
  • Participating Institution(s):  Weierstrass Institute Berlin (WIAS)
  • Area:  C: Bridging the micro-macro scale range
  • Positions available?: 
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