A phase-field model for fractures in incompressible solids

Within this work, we develop a phase-field description for simulating fractures in incompressible materials. Standard formulations are subject to volume-locking when the solid is (nearly) incompressible. We propose an approach that builds on a mixed form of the displacement equation with two unknowns: a displacement field and a hydro-static pressure variable. Corresponding function spaces have to be chosen properly. On the discrete level, stable Taylor-Hood elements are employed for the displacement-pressure system. Two additional variables describe the phase-field solution and the crack irreversibility constraint. Therefore, the final system contains four variables: displacements, pressure, phase-field, and a Lagrange multiplier. The resulting discrete system is nonlinear and solved monolithically with a Newton-type method. Our proposed model is demonstrated by means of several numerical studies based on two numerical tests. First, different finite element choices are compared in order to investigate the influence of higher-order elements in the proposed settings. Further, numerical results including spatial mesh refinement studies and variations in Poisson's ratio approaching the incompressible limit, are presented

Schur‐type preconditioning of a phase‐field fracture model in mixed form

In the context of phase-field modeling of fractures in incompressible materials, a mixed form of the elasticity equation can overcome possible volume locking effects. The drawback is that a coupled variational inequality system with three unknowns (displacements, pressure and phase-field) has to be solved, which increases the overall workload. Efficient preconditioning at this point is an indispensable tool. In this work, a problem-specific iterative solver is proposed leveraging the saddle-point structure of the displacement and pressure variable. A Schur-type preconditioner is developed to avoid ill-conditioning of the phase-field fracture problem. Finally, we show numerical results of a pressure-driven benchmark which to confirm the robustness of the solver

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 Studies of Different Mixed Phase-Field Fracture Models for Simulating Crack Propagation in Punctured EPDM Strips

We consider a monolithic phase-field description for fractures in nearly incompressible materials, i.e., carbon black filled ethylene propylene diene monomer rubber (EPDM). A quasi-static phasefield fracture problem is formulated in mixed form based on three different energy functionals (AT2, AT1 and Wu’s model) combined with two different stress splitting approaches (according to Miehe and Amor). It leads to six different phase-field fracture formulations in mixed form. The coupled variational inequality systems are solved in a quasi-monolithic manner with the help of a primal-dual active set method handling the inequality constraint. Further, adaptive mesh refinement is used to get a sharper crack zone. Numerical results based on the six different problem setups are validated on crack propagation experiments of punctured EPDM strips with five different test configurations. As a quantity of interest, the crack paths of experiments and numerical computations are discussed

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

Numerical Methods for Variational Phase-Field Fracture Problems

In these lectures notes, the variational phase-field approach for modeling fracture propagation is edited for the usage in classes and summer schools. Basic modeling is briefly reviewed first. The main emphasis is on the design of numerical methods. All algorithmic and theoretical advancements are illustrated with many examples and numerical tests.DFG/SPP1748/392587580/E

How To Overcome Confirmation Bias in Semi-Supervised Image Classification By Active Learning

Do we need active learning? The rise of strong deep semi-supervised methods raises doubt about the usability of active learning in limited labeled data settings. This is caused by results showing that combining semi-supervised learning (SSL) methods with a random selection for labeling can outperform existing active learning (AL) techniques. However, these results are obtained from experiments on well-established benchmark datasets that can overestimate the external validity. However, the literature lacks sufficient research on the performance of active semi-supervised learning methods in realistic data scenarios, leaving a notable gap in our understanding. Therefore we present three data challenges common in real-world applications: between-class imbalance, within-class imbalance, and between-class similarity. These challenges can hurt SSL performance due to confirmation bias. We conduct experiments with SSL and AL on simulated data challenges and find that random sampling does not mitigate confirmation bias and, in some cases, leads to worse performance than supervised learning. In contrast, we demonstrate that AL can overcome confirmation bias in SSL in these realistic settings. Our results provide insights into the potential of combining active and semi-supervised learning in the presence of common real-world challenges, which is a promising direction for robust methods when learning with limited labeled data in real-world applications.Comment: Accepted @ ECML PKDD 2023. This is the author's version of the work. The definitive Version of Record will be published in the Proceedings of ECML PKDD 202