TY - JOUR

T1 - A modified combined active-set Newton method for solving phase-field fracture into the monolithic limit

AU - Kolditz, Leon

AU - Mang, Katrin

AU - Wick, Thomas

N1 - Funding Information: All authors thank Viktor Kosin (Université Paris-Saclay) for fruitful discussions on the iteration on the extrapolation part and Johannes Lankeit (Leibniz University Hannover) for giving valuable advice and comments on solution spaces in connection with equivalence proofs within this work. Moreover, the authors thank Sebastian Bohlmann for the Scientific Computing environment at IfAM. The present work has been partially carried out within the DFG Collaborative Research Center (CRC) 1463 “Integrated design and operation methodology for offshore megastructures”, which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 434502799, SFB 1463. Moreover, the authors thank the (anonymous) reviewers for their questions that helped to improve the manuscript.

PY - 2023/9/1

Y1 - 2023/9/1

N2 - 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 from Heister and Wick (2020). 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.

AB - 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 from Heister and Wick (2020). 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.

KW - Adaptive finite elements

KW - Complementarity system

KW - Modified Newton's method

KW - Monolithic scheme

KW - Phase-field fracture

KW - Primal–dual active set

UR - http://www.scopus.com/inward/record.url?scp=85162861062&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2306.03501

DO - 10.48550/arXiv.2306.03501

M3 - Article

AN - SCOPUS:85162861062

VL - 414

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

M1 - 116170

ER -