Goal-oriented error control for phase-field fracture coupled to multiphysics problems

In many applications in engineering, physics or also medicine, the focus is on the evaluation of goal functionals. Such quantities of interest are for example the accurate computation of displacements or stresses in solid mechanics, drag or lift forces in fluid dynamics, or the crack width and length in fracture mechanics. In this project, the main aim is on goal functional evaluations in fracture mechanics and its coupling to multiphysics and optimization. The basis fracture model is based on a phase-field technique, which is realized with the finite element method for carrying out computer simulations. This approach has been gaining increased attraction since the crack path is not required to be resolved by the finite element mesh. Consequently, the model formulation and code development for two and three dimensional simulations are relatively easy to achieve. However, a shortcoming is a diffusive zone around the crack path, which influences the accuracy of crack resolution. Moreover, this zone depends on a model regularization parameter eps, which depends itself on the spatial discretization parameter h of the finite element discretization. Therefore, the goal functional evaluation does always depend on these two parameters. Numerical analysis and corresponding simulations of the previously mentioned quantities of interest with the dual-weighted residual method for a posteriori error estimation constitute the first goal of this project. This part is already ambitious but successful findings can be expected. Two extensions including goal functional evaluations of the basic phase-field fracture approach towards multiphysics applications and optimization compose the innovative and challenging part of this project. Essentially, this is due to the fact that mathematical and numerical modeling (even without goal functionals) is only partially present to date and has to be accomplished first. A success of the entire program would unlock important and promising research fields. With respect to multiphysics problems these are for example applications in computational medicine (e.g., aortic dissections or rupture of plaque) or fractures in porous media or fracture networks in geology. With regard to optimization, important applications are optimal control problems (e.g., control of the crack path) and parameter estimation problems in which unknown model and material parameters can be estimated.

In many technical and scientific applications, computer simulations of multi-physics processes is an important tool for understanding and, above all, optimizing these processes. This class of problems includes many applications from solid and fluid mechanics such as the interaction of solids with fluids (FSI = fluid-structure interaction), and propagation of cracks (PFF = phase-field fracture), their interplay and interaction with other physical processes such as e.g., heat conduction and heat transport. The mathematical modelling of these processes yields coupled non-linear systems of unsteady partial differential equations, which can only be solved numerically. Fast numerical methods for computer simulations of FSI and PFF problems and their implementation on modern parallel computers were one of the two main research topics of the project. The development, implementation and testing of matrix-free, monolithic multigrid methods as an essential building block for the construction of numerical methods for solving the non-linear systems of equations with many millions of unknowns arising during the discretization was one of the main results of the project. The second research focus was devoted to the evaluation of the reliability and the improvement of the efficiency of the numerical methods by a posteriori error estimations and their use for the adaptive control of the discretization. Here, the main focus was on so-called goal-oriented error estimation. Such objective functionals result from technical applications with quantities derived from the solution. In many cases an accurate computation of the solution is only necessary where it affects the quantities under consideration. The main result here is the development of a new abstract-level theory on a posteriori error estimators that simultaneously consider multiple goals. Practically, one then has the possibility to adapt the discretization in such a way that these targets are computed as accurately as possible with the least numerical effort. Together with international cooperation partners in Germany, we have extended this technique to optimal control of non-linear partial differential equations.