The phase-field method is an non-local approach which allows the computation of the dynamical evolution of an interface. This technique is applicable wherever several phases are separated by a diffuse interface. The basic idea is to differentiate different phases by means of an order parameter. The variation of the systems free energy with respect to this order parameter then reveals the time evolution of the interface. Phase-field models of fracture are conceptually similar to models of continuum damage mechanics, whereby the order parameter weights the damage (or integrity) of the structure. Generally, a phase-field model may be considered as a gradient-type material model with a free energy function that is composed of classical bulk energy and a gradient-type regularized surface energy. Phase-field approaches to fracture offer important new perspectives towards the computational modelling of complex crack topologies. Diffusive crack zones avoid the modelling of discontinuities and can be implemented in a straightforward manner by coupled multi-field finite element solvers. Structured meshes are able to account for complex 3D crack patterns, which removes the need for costly remeshing and the precise tracking of the discontinuities. This simplifies tremendously the arising implementation issues in terms of mesh handling, adaptivity, and solution methods. In fact, structured meshes simplify the application of efficient solution methods as multi-grid methods, as the construction of the multilevel hierarchy is straightforward.
An additional advantage of the diffusive crack zone is that it allows for adaptive techniques based on locally structured meshes. As a recent development, adaptive techniques for parallel computing have been developed which combine the advantages of structured meshes, such as implementational simplicity and a low memory footprint, with the flexibility of adaptive methods. These new methods have been tested within the context of cardiac electrophysiology, where moving sharp gradients, i.e. the gradient of the activation potential, have to be resolved accurately. As this fits perfectly the needs for the adaptive resolution of the diffusive crack zone, in the framework of our phase-field approach, this lightweight adaptivity technique will allow us to combine the advantages of structured meshes with a highly accurate resolution of the crack topologies. As for shape functions we will follow a less traditional approach using Non-uniform rational B-splines (NURBS) instead of conventional Lagrangian shape functions, which is expected to provide higher accuracy and flexibility. NURBS are motivated as basis functions to link the field of Computer Aided Design (CAD) with Galerkin- based discretisation. However, in addition to their advantages with handling complex geometries, the higher regularity of NURBS is also highly beneficial in fracture mechanics. Among other, the stress field at the boundary is smoother, they prevent possible notch effects of C0 elements and they provide enhanced stability for higher order ansatz functions as can be shown using simple eigenproblem benchmarks. Moreover, NURBS based shape functions provide efficient global p- and k-refinement capabilities, reducing the numerical costs for large-scale, three-dimensional problems.
Concerning the temporal discretisation we want to avoid issues with non-conservation of important invariants of non-linear elastodynamics, such as total energy and total angular momentum, as they arise within implicit time-stepping schemes such as the Newmark algorithm. Nowadays, it is well accepted that energy-momentum conserving time-stepping schemes and energy decaying variants thereof provide enhanced stability for applications in non-linear structural dynamics and thermoelastodynamics.
In this project we plan to develop the described new simulation framework along driven fracturing, typically referred to as pneumatic and hydraulic fracturing, or “fracking”. Hydraulic fracturing as a complex application will serve as an ideally challenging problem for developing, testing, and improving our new approaches and methods. This joint project will establish a common basis for the development of non-conventional finite element analysis in space and time for phase-field simulation of fracture undergoing finite deformations. The composition of the research group from different fields of material and computer science is inspired by actual initiatives in the Computer Science and Engineering (CSE) community, and combines complementarity expertise in engineering, modelling, computational science, and high performance computing.
Images: C. Hesch, K. Weinberg; Int. J. Numer. Meth. Engng. 99(1097), 2002
This joint project will be carried out in collaboration with Prof. Dr. Rolf Krause and Dr. Roger A. Müller, Università della Svizzera Italiana, Lugano, Institute of Coputational Science, Prof. Dr. Kerstin Weinberg, Universität Siegen, Chair for Solid Mechanics and Dr. Christian Hesch, Karlsruher Institut für Technologie, Institute of Mechanics.This is a joint project with Universität Duisburg Essen and is supported by the German Research Foundation.
Prof. Dr. Rolf Krause; ; PI; ICS Institute of Computational Science