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

Summary We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase‐field approximations of sharp domains. Leveraging the properties of the phase‐field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase‐field solutions of the Allen‐Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase‐field in the diffuse boundary region and a uniform mesh for the representation of the physics‐based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty‐type methods. In the context of imaging‐based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, ie, the interface width of the phase‐field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body.

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“…We note that there exists a nonsymmetric penalty‐free variant of Nitsche's method that does not require stabilization. ()…”

Section: Variational Formulation and Discretizationmentioning

confidence: 99%

The diffuse Nitsche method: Dirichlet constraints on phase‐field boundaries

Nguyen

Stoter

Ruess

et al. 2017

Numerical Meth Engineering

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“…where γ 1 and γ 2 are penalty parameters. It is straightforward to consider alternative constrain enforcement techniques, like the Nitsche method [33,34] or Lagrange multipliers [31,35]. As usual, the discrete finite element equilibrium equations are derived by first writing the potential (41) as a sum over the set of reference elements { e } using the Jacobian |∂X/∂η|.…”

Section: Thin-shell Formulation and Discretisationmentioning

confidence: 99%

Manifold-based isogeometric analysis basis functions with prescribed sharp features

Zhang

Cirak

2020

Computer Methods in Applied Mechanics and Engineering

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We introduce manifold-based basis functions for isogeometric analysis of surfaces with arbitrary smoothness, prescribed C 0 continuous creases and boundaries. The utility of the manifold-based surface construction techniques in isogeometric analysis was demonstrated in Majeed and Cirak (CMAME, 2017). The respective basis functions are derived by combining differential-geometric manifold techniques with conformal parametrisations and the partition of unity method. The connectivity of a given unstructured quadrilateral control mesh in R 3 is used to define a set of overlapping charts. Each vertex with its attached elements is assigned a corresponding conformally parametrised planar chart domain in R 2 so that a quadrilateral element is present on four different charts. On the collection of unconnected chart domains, the partition of unity method is used for approximation. The transition functions required for navigating between the chart domains are composed out of conformal maps. The necessary smooth partition of unity, or blending, functions for the charts are assembled from tensor-product B-spline pieces and require in contrast to earlier constructions no normalisation. Creases are introduced across user tagged edges of the control mesh. Planar chart domains that include creased edges or are adjacent to the domain boundary require special local polynomial approximants in the partition of unity method. Three different types of chart domain geometries are necessary to consider boundaries and arbitrary number and arrangement of creases. The new chart domain geometries are chosen so that it becomes trivial to establish local polynomial approximants that are always C 0 continuous across the tagged edges. The derived non-rational manifold-based basis functions correspond to the vertices of the mesh and may have an arbitrary number of creases and prescribed smoothness. This makes them particularly well suited for isogeometric analysis of Kirchhoff-Love thin shells with kinks, which require C 1 continuous basis functions that are C 0 continuous across the kinks. We demonstrate the convergence and utility of the new basis functions with linear and nonlinear beam, plate and shell examples.

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

Cited by 57 publications

References 82 publications

The diffuse Nitsche method: Dirichlet constraints on phase‐field boundaries

The diffuse Nitsche method: Dirichlet constraints on phase‐field boundaries

Manifold-based isogeometric analysis basis functions with prescribed sharp features

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

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