On the relation of Gamma-convergence parameters for pressure-driven quasi-static phase-field fracture

Kolditz, Leon; Mang, Katrin

doi:10.1016/j.exco.2022.100047

Cited by 7 publications

(3 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For a decoupled linearized system, such estimates are shown in [54, Section 5.5]. A numerical error analysis for this test on a good choice of κ can be found in [34].…”

Section: 3mentioning

confidence: 99%

Robust preconditioning for a mixed formulation of phase-field fracture problems

Heister¹,

Mang²,

Wick³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

In this work, we consider fracture propagation in nearly incompressible and (fully) incompressible materials using a phase-field formulation. We use a mixed form of the elasticity equation to overcome volume locking effects and develop a robust, nonlinear and linear solver scheme and preconditioner for the resulting system. The coupled variational inequality system, which is solved monolithically, consists of three unknowns: displacements, pressure, and phase-field. Nonlinearities due to coupling, constitutive laws, and crack irreversibility are solved using a combined Newton algorithm for the nonlinearities in the partial differential equation and employing a primal-dual active set strategy for the crack irreverrsibility constraint. The linear system in each Newton step is solved iteratively with a flexible generalized minimal residual method (GMRES). The key contribution of this work is the development of a problem-specific preconditioner that leverages the saddle-point structure of the displacement and pressure variable. Four numerical examples in pure solids and pressure-driven fractures are conducted on uniformly and locally refined meshes to investigate the robustness of the solver concerning the Poisson ratio as well as the discretization and regularization parameters.

show abstract

“…For a decoupled linearized system, such estimates are shown in [54, Section 5.5]. A numerical error analysis for this test on a good choice of κ can be found in [34].…”

Section: 3mentioning

confidence: 99%

Robust preconditioning for a mixed formulation of phase-field fracture problems

Heister¹,

Mang²,

Wick³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Carbotti et al [22] proved the local minimality of half spaces in Carnot groups and provided a lower bound of Γ-lim inf of the recalled energy in the form of horizontal parameters. Kolditz and Mang [23] studied the numerical solutions of quasi-static phase-field fracture problems based upon the Γconvergence of parameters. For a comprehensive study on Γ-convergence, we refer to [24].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Analysis of Low Energy Extremals with Γ-Convergence in Variable Exponent Lebesgue Spaces

et al. 2022

View full text Add to dashboard Cite

In many physical models, internal energy will run out without external energy sources. Therefore, finding optimal energy sources and studying their behavior are essential issues. In this article, we study the following variational problem: Gϵ*=sup∫ΩG(u)ϵq(x)dx:∥∇u∥Lp(.)≤ϵ,u=0on∂Ω, with the help of Γ-convergence, where G:R→R is upper semicontinuous, non zero in the L1 sense, 0≤G(u)≤c|u|q(.), Ω is a bounded open subset of Rn, n≥3, 1<p(.)<n, and p(.)≤q(.)≤p*(.). For special choices of G, we can study Bernoulli’s free-boundary and plasma problems in variable exponent Lebesgue spaces.

show abstract

“…Therein, the overall goals are computational investigations of the performance of the reduced Newton algorithm (NLP), the linear conjugate gradient (CG) method, and convergence of the residuals, cost functionals, tracking parts, Tikhonov parts and optimal controls. We recall upfront that such investigations are even challenging for forward phasefield fracture problems due to the interaction of model, discretization and material parameters (see [10,30] and closely related work on image segmentation [9], and the prior seminal work on Gamma convergence [3,4,13]), and possibly also penalization parameters for treating the crack irreversibility constraint [45]. All of them have an impact on mathematical wellposedness [11] (and references cited therein), and on numerical approximations and nonlinear and linear solution algorithms [11,12,47,45,19,17].…”

Section: Introductionmentioning

confidence: 99%

Space-time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems

Denis¹,

Steinbach²,

Wick³

2022

Preprint

View full text Add to dashboard Cite

The purpose of this work are computational demonstations for a newly developed space-time phase-field fracture optimal control framework. The optimal control solution algorithm is a Newton algorithm, which is obtained with the reduced approach by eliminating the state constraint. Due to the crack irreversibility constraint, a rate-independent problem arises, which is treated by penalization and for which we utilize a space-time approach. Therein, we deal with the state, adjoint, tangent, and adjoint Hessian equations. Our fully discretized space-time optimization algorithm is presented and extensively tested with various numerical experiments.

show abstract

On the relation of Gamma-convergence parameters for pressure-driven quasi-static phase-field fracture

Cited by 7 publications

References 16 publications

Robust preconditioning for a mixed formulation of phase-field fracture problems

Robust preconditioning for a mixed formulation of phase-field fracture problems

Asymptotic Analysis of Low Energy Extremals with Γ-Convergence in Variable Exponent Lebesgue Spaces

Space-time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems

Contact Info

Product

Resources

About