<i>hp</i>‐Version finite elements for geometrically non‐linear problems

Krause, R.; Mücke, Roland; Rank, Ernst

doi:10.1002/cnm.1640111103

Cited by 15 publications

(4 citation statements)

References 8 publications

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“…Accuracy and convergence of the FCM versions will be assessed by

e_{r} MathClass-rel= \sqrt{\frac{MathClass-rel| U_{normalex} MathClass-bin- U_{FCM} MathClass-rel|}{U_{ex}}} MathClass-bin\times 100 % MathClass-punc,

denoting the relative error in terms of the total strain energy U of Equation . For fictitious domain problems, U ex denotes the exact strain energy of the original problem defined only over the physical domain Ω phys , corresponding to the first picture of Figure .…”

Section: The Concept Of the Hp‐d Finite Cell Methods In One Dimensionmentioning

confidence: 99%

The hp‐d‐adaptive finite cell method for geometrically nonlinear problems of solid mechanics

Schillinger

Düster

Rank

2011

Numerical Meth Engineering

103

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SUMMARYThe finite cell method (FCM) combines the fictitious domain approach with the p-version of the finite element method and adaptive integration. For problems of linear elasticity, it offers high convergence rates and simple mesh generation, irrespective of the geometric complexity involved. This article presents the integration of the FCM into the framework of nonlinear finite element technology. However, the penalty parameter of the fictitious domain is restricted to a few orders of magnitude in order to maintain local uniqueness of the deformation map. As a consequence of the weak penalization, nonlinear strain measures provoke excessive stress oscillations in the cells cut by geometric boundaries, leading to a low algebraic rate of convergence. Therefore, the FCM approach is complemented by a local overlay of linear hierarchical basis functions in the sense of the hp-d method, which synergetically uses the h-adaptivity of the integration scheme. Numerical experiments show that the hp-d overlay effectively reduces oscillations and permits stronger penalization of the fictitious domain by stabilizing the deformation map. The hp-d-adaptive FCM is thus able to restore high convergence rates for the geometrically nonlinear case, while preserving the easy meshing property of the original FCM. Accuracy and performance of the present scheme are demonstrated by several benchmark problems in one, two, and three dimensions and the nonlinear simulation of a complex foam sample.

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“…Accuracy and convergence of the FCM versions will be assessed by

e_{r} MathClass-rel= \sqrt{\frac{MathClass-rel| U_{normalex} MathClass-bin- U_{FCM} MathClass-rel|}{U_{ex}}} MathClass-bin\times 100 % MathClass-punc,

Section: The Concept Of the Hp‐d Finite Cell Methods In One Dimensionmentioning

confidence: 99%

The hp‐d‐adaptive finite cell method for geometrically nonlinear problems of solid mechanics

Schillinger

Düster

Rank

2011

Numerical Meth Engineering

103

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“…More details on the Fig. 6 Iterative computation of the contact interface point using the hp-method of the FEM hp-version can be found in [1,3,[7][8][9]29,32,33]. A flowchart of the applied hp-method is given in Fig.…”

Section: Hp-femmentioning

confidence: 99%

A comparison of the h-, p-, hp-, and rp-version of the FEM for the solution of the 2D Hertzian contact problem

Franke

Düster

Nübel³

et al. 2010

Comput Mech

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A comparison of the h-, p-, hp-, and r p-version of the FEM for the solution of the 2D Hertzian contact problemDavid Franke · A. Düster · V. Nübel · E. Rank Abstract Even after more than three decades of intensive research in the field of contact mechanics, there is still a great need for improving the numerical methods. Recent investigations on the 2D Hertzian contact problem clearly showed significantly varying results for different finite element versions. In this paper we will compare the analytical solution of the 2D Hertzian contact problem with the numerical results of the classical h-version with uniform and locally refined meshes, as well as with the p-, the hp-, and the r p-version of the FEM. The penalty method is used to incorporate the contact constraints.

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“…The poor numerical performance of standard p-FCM in the nonlinear case is further demonstrated by the relative error in strain energy [11] (e r ) E =…”

Section: Standard P-fcm: Strong Oscillations In the Stress Solutionmentioning

confidence: 99%

The finite cell method for geometrically nonlinear problems of solid mechanics

Schillinger

Kollmannsberger

Mundani

et al. 2010

IOP Conf. Ser.: Mater. Sci. Eng.

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The Finite Cell Method (FCM), which combines the fictitious domain concept with high-order p-FEM, permits the effective solution of problems with very complex geometry, since it circumvents the computationally expensive mesh generation and guarantees exponential convergence rates for smooth problems. The present contribution deals with the coupling of the FCM approach, which has been applied so far only to linear elasticity, with established nonlinear finite element technology. First, it is shown that the standard p-FEM based FCM converges poorly in a nonlinear formulation, since the presence of discontinuities leads to oscillatory solution fields. It is then demonstrated that the essential ideas of FCM, i.e. exponential convergence at virtually no meshing cost, can be achieved in the geometrically nonlinear setting, if high-order Legendre shape functions are replaced by a hierarchically enriched B-spline patch.

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hp‐Version finite elements for geometrically non‐linear problems

Cited by 15 publications

References 8 publications

The hp‐d‐adaptive finite cell method for geometrically nonlinear problems of solid mechanics

The hp‐d‐adaptive finite cell method for geometrically nonlinear problems of solid mechanics

A comparison of the h-, p-, hp-, and rp-version of the FEM for the solution of the 2D Hertzian contact problem

The finite cell method for geometrically nonlinear problems of solid mechanics

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