A Particle-Partition of Unity Method Part V: Boundary Conditions

Griebel, Michael; Schweitzer, Marc Alexander

doi:10.1007/978-3-642-55627-2_27

Cited by 92 publications

(110 citation statements)

References 28 publications

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“…The use of Nitsche's technique in the imposition of interpatch-continuity also represents a common and well explored alternative [5,44,[47][48][49][50][51][52]. While such option is clearly superior to the use of simpler approaches, it is less efficient in implementation and computational costs, in the sense that it requires modifying the variational form as well as evaluating boundary integrals at the boundaries of interest.…”

Section: Analysis-related Enhancement Of Geometrical Datamentioning

confidence: 99%

Realization of CAD-integrated shell simulation based on isogeometric B-Rep analysis

Teschemacher

Bauer

Oberbichler

et al. 2018

Adv. Model. and Simul. in Eng. Sci.

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An entire design-through-analysis workflow solution for isogeometric B-Rep analysis (IBRA), including both the interface to existing CADs and the analysis procedure, is presented. Possible approaches are elaborated for the full scope of structural analysis solvers ranging from low to high isogeometric simulation fidelity. This is based on a systematic investigation of solver designs suitable for IBRA. A theoretically ideal IBRA solver has all CAD capabilities and information accessible at any point, however, realistic scenarios typically do not allow this level of information. Even a classical FE solver can be included in the CAD-integrated workflow, which is achieved by a newly proposed meshless approach. This simple solution eases the implementation of the solver backend. The interface to the CAD is modularized by defining a database, which provides IO capabilities on the base of a standardized data exchange format. Such database is designed to store not only geometrical quantities but also all the numerical information needed to realize the computations. This feature allows its use also in codes which do not provide full isogeometric geometrical handling capabilities. The rough geometry information for computation is enhanced with the boundary topology information which implies trimming and coupling of NURBS-based entities. This direct use of multi-patch trimmed CAD geometries follows the principle of embedding objects into a background parametrization. Consequently, redefinition and meshing of geometry is avoided. Several examples from illustrative cases to industrial problems are provided to demonstrate the application of the proposed approach and to explain in detail the proposed exchange formats.

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Section: Analysis-related Enhancement Of Geometrical Datamentioning

confidence: 99%

Realization of CAD-integrated shell simulation based on isogeometric B-Rep analysis

Teschemacher

Bauer

Oberbichler

et al. 2018

Adv. Model. and Simul. in Eng. Sci.

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“…In fact, the last term in (24) is required to ensure coercivity of the bilinear form a(·, ·) provided that β is large enough. The parameter β must be larger than a minimum value that can be assessed as γ/ℓ, where ℓ is a characteristic size of the discretization (say H or h for the discretizations mentioned here), and γ is a constant that can be obtained as the maximum eigenvalue of some eigenvalue problem [31]. In practice that means that β must increase when the discretization gets finer.…”

Section: Dirichlet Boundary Conditionsmentioning

confidence: 99%

“…are derived taking into account equation (31). The same behavior is assumed to hold in the new discretization…”

Section: Adaptive Strategymentioning

confidence: 99%

Bounds for quantities of interest and adaptivity in the element‐free Galerkin method

Vidal

Mariné

Dı́ez

et al. 2008

Numerical Meth Engineering

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1 of estimators ensuring bounds for the error. Classical residual type estimators, which provide upper bounds of the error, require flux-equilibration procedures (hybrid-flux techniques) to properly set boundary conditions for local problems [10, 2]. Flux-equilibration requires a domain decomposition, which is natural in finite elements but not in mesh-free methods. Moreover, it is performed by a complex algorithm, strongly dependent on the finite element type and requiring a data structure that is not natural in a standard finite element code.The idea of using flux-free estimators, based on the partition-of-the-unity concept and using local subdomains different than elements, has been already proposed in [11,12,13,14] for finite elements. The main advantage of the flux-free approach is the simplicity in the implementation. Obviously, this is especially important in the 3D case.From the mesh-free point of view, another advantage is the fact that the local subdomains where the error equation is solved are the support of the functions characterizing the partition of unity. This is a concept that also exists in mesh-free methods and thus the extension is possible. Moreover, boundary conditions of the local problems are natural and the usual data structure of a code is directly employed.In the last few years, some research has been devoted to develop error estimation procedures for mesh-free methods. To the authors knowledge implicit residual based estimators have not been proposed for mesh-free methods. However, in [19] a strategy to assess the energy norm of the error for the Generalized Finite Element Method (GFEM) is presented and the possible generalizations to mesh-free methods are also devised. Implicit residual error estimators are now standard in finite elements because they are mathematically sound, more accurate and allow computing upper and lower bounds for energy norms as well as for functional outputs. In this paper the implicit residual-type flux-free error estimator proposed in [14], which is comparable in effectivity to the standard hybrid-flux estimators, is extended to the Element Free Galerkin Method. In the present work, the difficulties found when extending the implicit residual methods to mesh-free are related with the fact that the refined discretization do not induce nested functional spaces. This is specially important for the error representation. Moreover, the bounds for the specific quantity of interest must be in this case carefully computed. In reference [19] these difficulties are not encountered because the error estimator is performed in terms of the energy norm and in the GFEM the refined subspaces are nested.Mesh-free methods are especially well suited for adaptivity. Enriching the mesh-free discretization is straightforward because the new particles are added without any connectivity restriction. Here, a refinement scheme based on inserting particles following a quadtree structure in a background rectangular grid is proposed. The refinement criterion uses the information provided...

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“…Symmetric variants of Nitsche methods are accurate and robust, but their performance crucially depends on appropriate estimates of the stabilization parameters involved [40,44,45]. If estimates are too large, the method degrades to a penalty method, which adversely influences consistency, accuracy and robustness.…”

Section: Introductionmentioning

confidence: 99%

A parameter-free variational coupling approach for trimmed isogeometric thin shells

2016

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The non-symmetric variant of Nitsche's method was recently applied successfully for variationally enforcing boundary and interface conditions in non-boundary-fitted discretizations. In contrast to its symmetric variant, it does not require stabilization terms and therefore does not depend on the appropriate estimation of stabilization parameters. In this paper, we further consolidate the non-symmetric Nitsche approach by establishing its application in isogeometric thin shell analysis, where variational coupling techniques are of particular interest for enforcing interface conditions along trimming curves. To this end, we extend its variational formulation within Kirchhoff-Love shell theory, combine it with the finite cell method, and apply the resulting framework to a range of representative shell problems based on trimmed NURBS surfaces. We demonstrate that the non-symmetric variant applied in this context is stable and can lead to the same accuracy in terms of displacements and stresses as its symmetric counterpart. Based on our numerical evidence, the non-symmetric Nitsche method is a viable parameter-free alternative to the symmetric variant in elastostatic shell analysis.

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A Particle-Partition of Unity Method Part V: Boundary Conditions

Cited by 92 publications

References 28 publications

Realization of CAD-integrated shell simulation based on isogeometric B-Rep analysis

Realization of CAD-integrated shell simulation based on isogeometric B-Rep analysis

Bounds for quantities of interest and adaptivity in the element‐free Galerkin method

A parameter-free variational coupling approach for trimmed isogeometric thin shells

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