Use of material forces in adaptive finite element methods

Mueller, Ralf; Groß, Dietmar; Maugin, Gérard A.

doi:10.1007/s00466-003-0543-z

Cited by 80 publications

(51 citation statements)

References 15 publications

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“…In this section, the concept of material forces and a finite element-based technique to evaluate material forces is reviewed. For a detailed discussion on material forces, we point to References [14][15][16][17] and to see its application within finite elements the interested reader can refer to References [18,41,42]. …”

Section: Materials Forcesmentioning

confidence: 99%

“…Residual-based error estimators measure the error by evaluating the local residual of the differential equation on the element domain and evaluating the flux jump across the element boundaries [13]. Material forces are associated with the Eshelby stress tensor [14][15][16][17], and have been recently used as an error indicator in finite element analysis [18,19].…”

Section: Introductionmentioning

confidence: 99%

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Adaptive computations using material forces and residual-based error estimators on quadtree meshes

Tabarraei

Sukumar

2007

Computer Methods in Applied Mechanics and Engineering

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Quadtree is a hierarchical data structure that is well-suited for h-adaptive mesh refinement. Due to the presence of hanging nodes, classical shape functions are non-conforming on quadtree meshes. In this paper, we use natural neighbor basis functions to construct conforming interpolants on quadtree meshes. To this end, the recently proposed construction of polygonal basis functions is adapted to quadtree elements. A fast technique for calculating stiffness matrix on quadtree meshes is introduced. Residual-based error estimators and material force technique are used to estimate the error on quadtree meshes. The performance of the adaptive technique is demonstrated through the solution of linear and nonlinear boundary-value problems.

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Section: Materials Forcesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Adaptive computations using material forces and residual-based error estimators on quadtree meshes

Tabarraei

Sukumar

2007

Computer Methods in Applied Mechanics and Engineering

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“…The Material Force Method has some roots in the contribution [3] which showed that the gradient of the discrete potential energy with respect to the material node point positions can be expressed in terms of the Eshelby stress tensor and which already denoted the components of the gradient associated with the nodes as discrete configurational forces. In [19][20][21] the application of the Material Force Method has been investigated, e.g., in the context of adaptivity, fracture mechanics, and inhomogeneities.…”

Section: Materials Force Methodsmentioning

confidence: 99%

“…Many recent impressive developments are in the application of concepts from configurational mechanics to computational mechanics, see, e.g., [17][18][19][20][21][22]31]. Our own contributions to the field are, e.g., [1,2,4,6,14,[23][24][25][26][27][28][29][30]32].…”

Section: Introductionmentioning

confidence: 99%

Secret and joy of configurational mechanics: From foundations in continuum mechanics to applications in computational mechanics

Steinmann¹,

Scherer

Denzer

2009

Z Angew Math Mech

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Key words Configurational mechanics, computational mechanics. This manuscript contains the material presented in a plenary lecture given at the GAMM annual meeting 2007 in ZurichOver the recent years configurational mechanics has developed into a very active and successful topic both in continuum mechanics as well as in computational mechanics. On the continuum mechanics side the basic idea is to consider energy variations that go along with changes of the material configuration. Configurational forces are then energetically dual to these configurational changes. Configurational forces take the interpretation as being the driving forces in the kinetics of defects; like e.g., cracks, inclusions, phase boundaries, dislocations and the like. On the computational side it turns out that a discretisation scheme brings in artificial, discrete configurational forces that indicate in a certain sense the quality, e.g., of a finite-element mesh. This information can then be used to optimize the nodal material positions. Surprisingly, even driven by energetical arguments, it turns out that a finite element mesh optimized with respect to discrete configurational forces also renders superior results in terms of classical error measures. The manuscript will span the field from the underlying theoretical foundations over the algorithmic challenges to various computational applications.

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“…One of them is the so called r-adaptive FE mesh refinement strategy which was introduced by Braun (1997) [1]. According to definition the configurational force [4] should be zero in the interior of the material when the material is assumed to be homogeneous and isotropic [6]. However, in case of finite element computation this requirement won't be fulfilled and nodal configurational forces appear on interior nodes too.…”

Section: Introductionmentioning

confidence: 99%

Configurational-force-based finite element mesh refinement for elastic-plastic problems

Hénap¹

2012

Per. Pol. Mech. Eng.

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In this paper the computation of configurational forces in case of elastic-ideally plastic material will be examined. Numerical computation of the error in configurational forces will also be introduced in elastic and plastic domain. It will be shown that the so-called r-adaptive mesh refinement procedure [1] is also applicable for elastic-plastic problems as well as the configurational force driven h-adaptive scheme. In some special examples configurational forces are computable in analytic way. This is useful to compare the solution with numerical results, therefore validating the finite element procedure. Two plane problems will be considered where analytical solutions are known. The first one is the thick walled tube model loaded by internal pressure. Second one is an artificial problem where the displacement field assumed to be known in every point of the domain considered. According to the papers Krieg [5] and Szabó [8] analytical solution is obtainable for stress and strain distributions, if the time derivative of the strain is constant. R− and h−adaptive procedures will demonstrated on these two examples.Keywords configurational force · plasticity · r-adaptivity Acknowledgement The work reported in the paper has been developed in the framework of the project "Talent care and cultivation in the scientific workshops of BME" project. This project is supported by the grant

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Use of material forces in adaptive finite element methods

Cited by 80 publications

References 15 publications

Adaptive computations using material forces and residual-based error estimators on quadtree meshes

Adaptive computations using material forces and residual-based error estimators on quadtree meshes

Secret and joy of configurational mechanics: From foundations in continuum mechanics to applications in computational mechanics

Configurational-force-based finite element mesh refinement for elastic-plastic problems

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