I³MS Markert Seminar
Prof. Dr. Bernd Markert - On Coupled Problems and Multi-X Modeling
General Mechanics, RWTH Aachen University
The theoretical description and numerical solution of volume-coupled problems is enjoying an enduring interest, as they are relevant to many science and engineering applications. On the macroscopic scale of interest, one is commonly concerned with a strong interaction of multiple field quantities over the entire domain of a continuum body. In fact, the interaction is inherent to the material and modeled by a coupled set of partial differential equations (PDE) and not only restricted to the domain boundaries as in surface-coupled problems such as fluid-structure interaction (FSI).
By their very nature, volumetrically or materially coupled problems make high demands on the theoretical modeling as the multi-field character of the interacting physical, chemical, biological etc. phenomena is often nonlinear and hard to retrace. In this regard, extended and multi-field continuum theories provide a suitable framework, where additional field quantities or higher-order terms are augmented to capture the specific properties arising from the internal coupling effects. Typical examples are multi-phase and mixture theories, phase-field and gradient models or coupled mean-field approaches of electro- and thermodynamics in combination with suitable constitutive relations.
When it comes to the numerical solution of coupled problems, the immense power of modern computing systems together with highly sophisticated numerical methods in principle enables us to solve large-scale initial-boundary-value problems. However, despite some very specialized codes tailored for very specific problems, the majority of software tools available is not anywhere near to exploit the praised new hardware features in combination with modern numerics. But this is a prerequisite for the solution of huge coupled multivariate PDE systems, which after the spatial discretization may comprise millions of evolution equations acting on different time scales along with algebraic equations representing possible constraints of the problem. In view of stability, flexibility and scalability, efficient time-stepping and solution strategies, regardless of whether proceeding from a monolithic or partitioned approach, remain the key ingredients to fast computation of coupled systems.
In this context, the present contribution aims at giving an elucidating insight into the possibilities offered by advanced continuum approaches as well as the macroscopic description of coupled multi-physics, multiphase or multi-field (for short multi-x) nature of many materials. Thereby, special attention will be drawn to the modeling and simulation of interfacial problems by use of non-local diffusion approaches (phase-field models), the multi-phase description of saturated porous media as well as some computational multiphysics aspects in the multi-scale modeling of spider silk, see e.g. [1,2,3].
 Markert, B. (ed.): Advances in Extended and Multifield Theories for Continua, Lecture Notes in Applied and Computational Mechanics, vol. 59. Springer, Berlin 2011.
 Markert, B.: A survey of selected coupled multifield problems in computational mechanics. Journal of Coupled Systems and Multiscale Dynamics 1 (2013), 22–48.
 Cetinkaya, M., Xiao, S., Markert, B., Stacklies, W. & Gräter, F.: Silk fiber mechanics from multiscale force distribution analysis. Biophysical Journal 100 (2011), 1298–1305.