Prof. Dr.-Ing. Paul Steinmann
Friedrich-Alexander-University of Erlangen-Nuremberg
Lehrstuhl für Technische Mechanik

Prof. Dr. Ellen Kuhl
Stanford University
Department of Mechanical Engineering

Diffusion is of utmost scientific and technological importance in various fields of engineering, materials science, natural sciences and their intersections. Prominent examples are the modelling and simulation of solder joints, micro-structure evolution in advanced materials as used e.g. in modern and future turbine blades, mineral unmixing in geology, contaminant distribution in environmental systems, and drug transport and delivery in biological tissues. The expected output of this project is the clarification of the underlying micromechanical origin of a particular class of higher order continuum models of diffusion. This will be of great importance in various fields such as engineering, materials science and natural sciences from both the scientific and the technological view point. In particular the design and understanding in the area of novel and advanced materials will be strongly enhanced by the findings of this project.

Final report:

The project resulted in two publications [1,2] and an approved DFG-proposal [3].

In the [1,2] a natural element method (NEM) to treat higher-order spatial derivatives in the Cahn- Hilliard equation as a paradigm for non-classical diffusion is proposed. The Cahn-Hilliard equation is a fourth-order nonlinear partial differential equation that allows to model phase separation in binary mixtures. Standard classical C₀-continuous finite element solutions are not suitable because primal variational formulations of fourth order operators are well-defined and integrable only if the finite element basis functions are piecewise smooth and globally C₁-continuous. Thus, to ensure C₁-continuity, a natural-element-based spatial discretisation scheme has been developed. Numerical examples are presented to demonstrate the efficiency of the computational algorithm. Both, periodic Dirichlet and homogeneous Neumann boundary conditions are applied. Also constant and degenerate mobilities are considered. It is demonstrated that the use of C₁-continuous natural element shape functions enables the computation of topologically correct solutions on arbitrarily shaped domains.

[1] Rajagopal A., Fischer P., Kuhl E., Steinmann P., Natural Element Analysis of the Cahn-Hillard Phase-Field Model. Computational Mechanics 46, (2010), 471-494.

[2] Fischer P., Rajagopal A., Kuhl E., Steinmann P., Cahn-Hillard Generalized Diffusion Modeling using the Natural Element Method. In: Altenbach, Maugin, Erofeev, editors, Mechanics of Generalized Continua (2011) 326-336, Springer.

[3] STE 544-48: On the Formulation and the Micromechanical Origin of Non-Classical Models of Diffusion.