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An Efficient Treatment of the Laplacian in a Gradient‐Enhanced Damage Model
Author(s) -
Schwarz Stephan,
Hackl Klaus,
Junker Philipp
Publication year - 2019
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201900125
Subject(s) - regularization (linguistics) , laplace operator , finite element method , mathematics , boundary value problem , viscoplasticity , operator (biology) , mathematical optimization , mathematical analysis , computer science , physics , constitutive equation , biochemistry , chemistry , repressor , artificial intelligence , transcription factor , gene , thermodynamics
Abstract As is well‐known, softening effects that are characteristic for damage models are accompanied by ill‐posed boundary value problems. This arises from non‐convex and non‐coercive energies and results in mesh‐dependent finite element results. For that reason, regularization strategies, which somehow take into account the non‐local behavior, have to be applied in order to prevent ill‐posedness and to achieve mesh‐independence. Hereto, most commonly gradient‐enhanced formulations [1, 2] are considered, but also integral‐type [3,4] and viscous [5,6] regularization are well‐known. Gradient‐enhanced damage models such as [1,2], to what group our new model [7] basically belongs, come along with a field function acting on the non‐local level. Two variational equations are resulting and, however, usually the number of nodal unknowns is increased and consequently the numerical effort is increased as well. In contrast, we present an improved algorithm for brittle damage [7] combining finite element and meshless methods resulting in a quick update of the field function. Thereby, an efficient evaluation of the Laplace operator is applied as introduced in [8]. In the end, the numerical effort of the novel approach is almost comparable with an elastic problem while maintaining well‐posedness and therefore mesh‐independent results.