Modified mixed least‐squares finite element formulations for small and finite strain plasticity

Igelbüscher, Maximilian; Schwarz, Alexander; Steeger, Karl; Schröder, Jörg

doi:10.1002/nme.5951

Cited by 6 publications

(4 citation statements)

References 55 publications

(80 reference statements)

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“…In the following, we will briefly describe the governing equations of the underlying material model which is based on the J 2 flow theory of elastoplasticity for finite strains accounting for nonlinear isotropic hardening. Please refer to Wriggers and Hudobivnik, Korelc and Stupkiewicz, Simo and Miehe, and Igelbüscher et al for a more detailed overview of the theory.…”

Section: Nonlinear Numerical Examplesmentioning

confidence: 99%

Spline‐ and hp‐basis functions of higher differentiability in the finite cell method

et al. 2019

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In this paper, the use of hp‐basis functions with higher differentiability properties is discussed in the context of the finite cell method and numerical simulations on complex geometries. For this purpose, Ck hp‐basis functions based on classical B‐splines and a new approach for the construction of C1 hp‐basis functions with minimal local support are introduced. Both approaches allow for hanging nodes, whereas the new C1 approach also includes varying polynomial degrees. The properties of the hp‐basis functions are studied in several numerical experiments, in which a linear elastic problem with some singularities is discretized with adaptive refinements. Furthermore, the application of the Ck hp‐basis functions based on B‐splines is investigated in the context of nonlinear material models, namely hyperelasticity and elastoplasicity with finite strains.

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Section: Nonlinear Numerical Examplesmentioning

confidence: 99%

Spline‐ and hp‐basis functions of higher differentiability in the finite cell method

et al. 2019

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“…Dekker et al [6] proposed an extended finite element model (XFEM) in an adaptive environment to capture the fatigue crack propagation and crack growth retardation under mixed-mode loading and overloading. A stress-displacement mixed least-squares finite element formulation was proposed by Igelbüscher et al [7] for elasticplastic material behaviour. Lee et al [8] analysed hyperelastic bodies with compressible and nearly incompressible neo-Hookean behaviour using the smoothed finite element method (S-FEM).…”

Section: Introductionmentioning

confidence: 99%

Techniques for Mesh Independent Displacement Recovery in Elastic Finite Element Solutions

Ahmed

2021

TFAMENA

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In this study, techniques for mesh dependent and independent displacement recovery for an a posteriori error estimation are presented. The error recovery of the field variable is made by fitting a higher order polynomial to the displacement over a mesh independent patch (support domain) using the moving least square (MLS) interpolation procedure. The mesh dependent recovery procedure is based on the recovery of the displacement over an element patch that consists of all elements surrounding the element under consideration using the least square (LS) interpolation procedure. The two-dimensional benchmark examples are analysed using linear and quadratic triangular elements to demonstrate the effectiveness and reliability of error estimations. Global and elemental errors of a finite element solution in the energy and L 2 norms are calculated directly from the post-processed displacement. The quality of error estimation obtained using the mesh independent displacement recovery technique in terms of convergence properties, effectivity and adaptive meshes under different error norms has been compared with that of the mesh dependent displacement recovery using the MLS interpolation and least square (LS) interpolation procedures. The performance of an adaptive scheme based on a mesh independent error estimator is compared with the adaptive scheme based on a mesh dependent error estimator. The numerical results show that the finite element analysis based on mesh independent recovery is very effective in converging to a predefined accuracy in a solution with a significantly smaller number of degrees of freedom.

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“…Furthermore, a modified first variation of the LS functional is introduced, which leads to an improvement of the performance of low order elements. An extension of the functional by adding an additional redundant residual for the fulfillment of stress symmetry is done in Schwarz et al [24,25] and Igelbüscher et al [26] The main aspect of this contribution is the investigation of the fulfillment of the balance of angular momentum, especially for coarse and moderate mesh densities. However, after some algebraic manipulations the balance of angular momentum is represented by two terms, the symmetry of the stress tensor = T and the cross product of the balance of linear momentum with the related distance to a fixed reference point (x − x 0 ) × (div + f ), where f denotes the body force.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A mixed least‐squares finite element formulation with explicit consideration of the balance of moment of momentum, a numerical study

2019

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Important conditions in structural analysis are the fulfillment of the balance of linear momentum (vanishing resultant forces) and the balance of angular momentum (vanishing resultant moment), which is not a priori satisfied for arbitrary element formulations. In this contribution, we analyze a mixed least-squares (LS) finite element formulation for linear elasticity with explicit consideration of the balance of angular momentum. The considered stress-displacement ( − u) formulation is based on the squared L 2 ()-norm minimization of the residuals of a first-order system of differential equations. The formulation is constructed by means of two residuals, that is, the balance of linear momentum and the constitutive equation. Motivated by the crucial point of weighting factors within LS formulations, a scale independent formulation is constructed. The displacement approximation is performed by standard Lagrange polynomials and the stress approximation with Raviart-Thomas functions. The latter ansatz functions do not a priori fulfill the symmetry of the Cauchy stress tensor. Therefore, a redundant residual, the balance of angular momentum ((x − x 0 ) × (div + f ) + axl[ − T ]), is introduced and the results are discussed from the engineering point of view, especially for coarse mesh discretizations.However, this formulation shows an improvement compared to standard LS − u formulations, which is considered here in a numerical study. K E Y W O R D Sbalance of moment of momentum, linear elasticity, mixed least-squares finite element method 1 The consideration of finite element formulations using standard displacement methods are well established in structural analysis of engineering problems within the frameworks of small and finite deformations. Unfortunately, the applicability of standard displacement formulations is limited by certain constraints, which can lead to locking behavior and unreliable results in the displacement and stress fields, see for example, Babuška and Suri. [1] An alternative to the primal finite element method are mixed variational principles where for example, displacements and stresses are approximated directly. One major advantage of mixed methods is the robustness in the presence of certain limiting situations, for example, incompressible or nearly This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.

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Modified mixed least‐squares finite element formulations for small and finite strain plasticity

Cited by 6 publications

References 55 publications

Spline‐ and hp‐basis functions of higher differentiability in the finite cell method

Spline‐ and hp‐basis functions of higher differentiability in the finite cell method

Techniques for Mesh Independent Displacement Recovery in Elastic Finite Element Solutions

A mixed least‐squares finite element formulation with explicit consideration of the balance of moment of momentum, a numerical study

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