High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles

Saye, Robert I.

doi:10.1137/140966290

Cited by 138 publications

(146 citation statements)

References 32 publications

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“…On the one hand, both require special attention towards the numerical integration of elements cut by boundaries. In this context, a series of papers have recently highlighted the importance of geometrically faithful quadrature in embedded domain methods (see, for example, other studies [37][38][39][40][41][42][43] and the references therein).…”

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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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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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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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“…For an extensive overview on quadrature methods for level sets respectively cut cells, we refer to our original work on HMF [5]. A rather new, alternative approach to quadrature on cut cells was proposed by Saye [27], who also demonstrated an XDG method for a Poisson problem with a jump at the interface.…”

Section: Motivation and Objectivesmentioning

confidence: 99%

Extended discontinuous Galerkin methods for two‐phase flows: the spatial discretization

Kummer

2016

Numerical Meth Engineering

102

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This work discusses a discontinuous Galerkin (DG) discretization for two-phase flows. The fluid interface is represented by a level set, and the DG approximation space is adapted such that jumps and kinks in pressure and velocity fields can be approximated sharply. This adaption of the original DG space, which can be performed 'on-the-fly' for arbitrary interface shapes, is referred to as extended discontinuous Galerkin. By combining this ansatz with a special quadrature technique, one can regain spectral convergence properties for low-regularity solutions, which is demonstrated by numerical examples. This work focuses on the aspects of spatial discretization, and special emphasis is devoted on how to overcome problems related to quadrature, small cut cells, and condition number of linear systems. Temporal discretization will be discussed in future works.interface were investigated by Babuska [2] as well as by Barrett and Elliott [3]. For the mathematical properties, that is, convergence rate and spectral properties of the matrix, the technique of interface representation should have no effect. However, especially for moving interfaces, one typically wants to avoid repetitive remeshing, leading to so-called extended finite element methods (XFEM). A very prominent work in this field, although in the domain of solid mechanics, was presented by Moës et al. [4].The present work is based on the discontinuous Galerkin (DG) method. In cells that are cut by the fluid interface, the standard DG space is extended in order to provide separate degrees of freedom (DOF) for both phases. One difficulty is that one has to integrate accurately over an interface that is only known implicitly and can have an almost arbitrary shape. This is solved by a special procedure for numerical integration, the so-called hierarchical-moment-fitting (HMF) [5] that eliminates the need for reconstructing the interface. On this basis, we demonstrate a DG method for which we can experimentally show a convergence order of approximately k C 1 for velocity and k for pressure, where k denotes the total polynomial degree of the velocity approximation. Many names for this idea seem at hand, for example, cut-cell DG or unfitted DG [6]. In reminiscence of XFEM [4], we prefer the name extended discontinuous Galerkin (XDG). While the idea seems simple at first glance, it introduces a bunch of technical difficulties, some of which are addressed in this work.This work represents the second paper in a series of publications on the road to a fully transient, 3D XDG-multiphase solver. Within the first work [5], the HMF procedure was developed. Because an XDG method like the one proposed in this work can only be 'h kC1 -accurate' if the quadrature also is sufficiently accurate, this is a necessary prerequisite for this work. For the actual work, one simplification we make is to consider only polynomial level set functions in C 1 . /. Such a choice allows us to evaluate the curvature of the fluid interface, which is required to describe the pressure jump due to sur...

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“…We have implemented the Galerkin method for (26) for the linear case j(r, ψ(r, z)) = j(r, z) in CONCEPTS [9,27,34] (www.concepts.math.ethz.ch), using finite dimensional hp-FEM spaces V h,p for the approximations ψ h,p of ψ. CONCEPTS provides tensor product basis functions based on integrated Legendre polynomials on the reference square. We make use of CONCEPTS' ability to resolve arbitrary curved boundaries with known parametrization exactly in the mesh by transfinite interpolation techniques.…”

Section: 2mentioning

confidence: 99%

“…A non-exhaustive list of references about this topic is [7,22,10] and the many references therein. We also mention the more recent [26], [17] and [14] that address the issue of integrals over implicitly defined hypersurfaces as important ingredient of fictitious domain methods, unfitted finite element methods and numerical methods of partial differential equations on surfaces.…”

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confidence: 99%

A High Order Method for the Approximation of Integrals Over Implicitly Defined Hypersurfaces

Drescher¹,

Heumann²,

Schmidt³

2017

SIAM J. Numer. Anal.

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Abstract. We introduce a novel method to compute approximations of integrals over implicitly defined hypersurfaces. The new method is based on a weak formulation in L 2 (0, 1), that uses the coarea formula to circumvent an explicit integration over the hypersurfaces. As such it is possible to use standard quadrature rules in the spirit of hp/spectral finite element methods, and the expensive computation of explicit hypersurface parametrizations is avoided. We derive error estimates showing that high order convergence can be achieved provided the integrand and the hypersurface defining function are sufficiently smooth. The theoretical results are supplemented by numerical experiments including an application for plasma modeling in nuclear fusion.

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High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles

Cited by 138 publications

References 32 publications

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

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

Extended discontinuous Galerkin methods for two‐phase flows: the spatial discretization

A High Order Method for the Approximation of Integrals Over Implicitly Defined Hypersurfaces

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