A modified combined active-set Newton method for solving phase-field fracture into the monolithic limit

In this work, we examine a numerical phase-field fracture framework in which the crack irreversibility constraint is treated with a primal-dual active set method and a linearization is used in the degradation function to enhance the numerical stability. The first goal is to carefully derive from a complementarity system our primal-dual active set formulation, which has been used in the literature in numerous studies, but for phase-field fracture without its detailed mathematical derivation yet. Based on the latter, we formulate a modified combined active-set Newton approach that significantly reduces the computational cost in comparison to comparable prior algorithms for quasi-monolithic settings. For many practical problems, Newton converges fast, but active set needs many iterations, for which three different efficiency improvements are suggested in this paper. Afterwards, we design an iteration on the linearization in order to iterate the problem to the monolithic limit. Our new algorithms are implemented in the programming framework pfm-cracks [T. Heister, T. Wick; pfm-cracks: A parallel-adaptive framework for phase-field fracture propagation, Software Impacts, Vol. 6 (2020), 100045]. In the numerical examples, we conduct performance studies and investigate efficiency enhancements. The main emphasis is on the cost complexity by keeping the accuracy of numerical solutions and goal functionals. Our algorithmic suggestions are substantiated with the help of several benchmarks in two and three spatial dimensions. Therein, predictor-corrector adaptivity and parallel performance studies are explored as well.Comment: 49 pages, 45 figures, 9 table

Numerical Methods for Algorithmic Systems and Neural Networks

These lecture notes are devoted to numerical concepts and solution of algorithmic systems and neural networks. The course is divided into four parts: traditional AI (artificial intelligence), deep learning in neural networks, applications to (and with) differential equations, and project work. Throughout this course an emphasis is on mathematical ingredients from which several are rigorously proven. In the project work, the participants usually form groups and work together on a given problem to train themselves on mathematical modeling, design of algorithms, implementation, and analysis and intepretation of the simulation results

Modeling, Discretization, Optimization, and Simulation of Phase-Field Fracture Problems

This course is devoted to phase-field fracture methods. Four different sessions are centered around modeling, discretizations, solvers, adaptivity, optimization, simulations and current developments. The key focus is on research work and teaching materials concerned with the accurate, efficient and robust numerical modeling. These include relationships of model, discretization, and material parameters and their influence on discretizations and the nonlinear (Newton-type methods) and linear numerical solution. One application of such high-fidelity forward models is in optimal control, where a cost functional is minimized by controlling Neumann boundary conditions. Therein, as a side-project (which is itself novel), space-time phase-field fracture models have been developed and rigorously mathematically proved. Emphasis in the entire course is on a fruitful mixture of theory, algorithmic concepts and exercises. Besides these lecture notes, further materials are available, such as for instance the open-source libraries pfm-cracks and DOpElib. The prerequisites are lectures in continuum mechanics, introduction to numerical methods, finite elements, and numerical methods for ODEs and PDEs. In addition, functional analysis (FA) and theory of PDEs is helpful, but for most parts not necessarily mandatory. Discussions with many colleagues in our research work and funding from the German Research Foundation within the Priority Program 1962 (DFG SPP 1962) within the subproject Optimizing Fracture Propagation using a Phase-Field Approach with the project number 314067056 (D. Khimin, T. Wick), and support of the French-German University (V. Kosin) through the French-German Doctoral college ``Sophisticated Numerical and Testing Approaches" (CDFA-DFDK 19-04) is gratefully acknowledged