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NTFA-enabled goal-oriented adaptive space-time finite elements for micro-heterogeneous elastoplasticity problems. (English) Zbl 1507.74079

Summary: In this work, we establish a goal-oriented space-time finite element method for a class of dissipative heterogeneous materials. Those materials are modeled on both micro- and macroscale, with a scale transition of volume averaging type satisfying the Hill-Mandel condition. A nonuniform transformation field analysis is performed on the microscopic inelastic strain fields for a model reduction. Reduced variables are deduced from a space-time decomposition of those inelastic strain fields. Closed-form constitutive relations are derived from some dissipative considerations, thus resulting into a reduced order homogenization problem. The resulting model error is sufficiently small for the considered class of materials, thus leaving the discretization error of the finite element method as a main error source. For ease of error estimate, we rewrite the reduced order problem in a multifield formulation. Based on duality techniques, a backward-in-time dual problem is derived from a Lagrange method, rendering error representations of a user-defined quantity of interest. Combining a patch recovery technique, a computable error estimator is developed to quantify both spatial and temporal discretization errors. By means of a localization technique, local error estimators are used to drive a greedy adaptive refinement algorithm in space and time. The effectiveness of the resulting algorithm is confirmed by several numerical examples w.r.t. a prototype model.

MSC:

74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics

Software:

VCFEM-HOMO; FEAPpv
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Full Text: DOI

References:

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