Numerical integration of discontinuous functions: moment fitting and smart octree

Hubrich, Simeon; Stolfo, Paolo Di; Kudela, László; Kollmannsberger, Stefan; Rank, Ernst; Schröder, Jörg; Düster, Alexander

doi:10.1007/s00466-017-1441-0

Cited by 65 publications

(52 citation statements)

References 60 publications

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“…A crucial point is accurate numerical integration of the cells cut by the boundary of the original domain which feature a discontinuity. As Gauss quadrature shows poor convergence for non-smooth functions a number of alternative integration schemes has been proposed in the context of FCM, see Hubrich et al [2017]. In this contribution the octree subdivision approach is used which was first combined with FCM in Düster et al [2008].…”

Section: The Finite Cell Methodsmentioning

confidence: 99%

A 3D benchmark problem for crack propagation in brittle fracture

Hug

Kollmannsberger

Yosibash

et al. 2020

Computer Methods in Applied Mechanics and Engineering

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We propose a full 3D benchmark problem for brittle fracture based on experiments as well as a validation in the context of phase-field models. The example consists of a series of four-point bending tests on graphite specimens with sharp V-notches at different inclination angles. This simple setup leads to a mixed mode (I + II + III) loading which results in complex yet stably reproducible crack surfaces. The proposed problem is well suited for benchmarking numerical methods for brittle fracture and allows for a quantitative comparison of failure loads and propagation paths as well as initiation angles and the fracture surface. For evaluation of the crack surfaces image-based 3D models of the fractured specimen are provided along with experimental and numerical results. In addition, measured failure loads and computed load-displacement curves are given. To demonstrate the applicability of the benchmark problem, we show that for a phase-field model based on the Finite Cell Method and multi-level hp-refinement the complex crack surface as well as the failure loads can be well reproduced.

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Section: The Finite Cell Methodsmentioning

confidence: 99%

A 3D benchmark problem for crack propagation in brittle fracture

Hug

Kollmannsberger

Yosibash

et al. 2020

Computer Methods in Applied Mechanics and Engineering

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“…This implies that Ω phys must have a mathematically valid description. Due to the discontinuity of α, the integrands in cut cells need to be computed by specially constructed quadrature rules, see, e.g., [35,38,63] for a recent overview of possible schemes. To perform a suitable integration, the domain is approximated by a space-tree TR int .…”

Section: Geometry Treatmentmentioning

confidence: 99%

Integrating CAD and numerical analysis: ‘Dirty geometry’ handling using the Finite Cell Method

Wassermann

Kollmannsberger

Yin

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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This paper proposes a computational methodology for the integration of Computer Aided Design (CAD) and the Finite Cell Method (FCM) for models with "dirty geometries". FCM, being a fictitious domain approach based on higher order finite elements, embeds the physical model into a fictitious domain, which can be discretized without having to take into account the boundary of the physical domain. The true geometry is captured by a precise numerical integration of elements cut by the boundary. Thus, an effective Point Membership Classification algorithm that determines the inside-outside state of an integration point with respect to the physical domain is a core operation in FCM. To treat also "dirty geometries", i.e. imprecise or flawed geometric models, a combination of a segment-triangle intersection algorithm and a flood fill algorithm being insensitive to most CAD model flaws is proposed to identify the affiliation of the integration points. The present method thus allows direct computations on geometrically and topologically flawed models. The potential and merit for practical applications of the proposed method is demonstrated by several numerical examples.

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“…To perform the numerical integration of broken cells more efficiently, recently, within the framework of the finite cell method [1] an integration method based on moment fitting was introduced [2,3]. Thereby, for every broken finite cell an individual quadrature rule is generated by solving the moment fitting equation system.…”

Section: Adaptive Integration Based On Moment Fittingmentioning

confidence: 99%

“…Thereby, for every broken finite cell an individual quadrature rule is generated by solving the moment fitting equation system. In [3], we showed that the moment fitting results in good conditioned quadrature rules when choosing the Gauss-Legendre points and using the Legendre polynomials as basis functions. However, this approach still requires the solution of the moment fitting equation system.…”

mentioning

confidence: 99%

Adaptive numerical integration of broken finite cells based on moment fitting applied to finite strain problems

Hubrich

Düster

2018

Proc Appl Math and Mech

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In the finite cell method (FCM) which is based on the fictitious domain approach the numerical integration of broken cells represents a major challenge. Commonly, an adaptive integration scheme is used which, usually, results in a large number of integration points and thus increases the numerical effort, especially for nonlinear applications. To reduce the number of integration points, we present an adaptive scheme which is based on moment fitting. Thereby, we introduce an approach based on Lagrange polynomials which avoids the necessity of solving the moment fitting equation system. We study the performance of this integration method considering two numerical examples for finite strain problems of the FCM. Adaptive integration based on moment fittingTo perform the numerical integration of broken cells more efficiently, recently, within the framework of the finite cell method [1] an integration method based on moment fitting was introduced [2, 3]. Thereby, for every broken finite cell an individual quadrature rule is generated by solving the moment fitting equation system. In [3], we showed that the moment fitting results in good conditioned quadrature rules when choosing the Gauss-Legendre points and using the Legendre polynomials as basis functions. However, this approach still requires the solution of the moment fitting equation system. In this contribution, we replace the Legendre basis by an equivalent Lagrange polynomial basis through the Gauss-Legendre points. For the onedimensional case, then, the moment fitting readsIn Eq.(1) f j define the moment fitting basis functions, l j (ξ ) the Lagrange polynomials, Ω phy is the physical domain, and the upper index GL indicates the Gauss-Legendre points. Following this approach, we avoid to solve the moment fitting equation system by taking advantage of the Kronecker delta property of the Lagrange polynomials f j (ξ GL i ) = δ ji , thus, the moment fitting weights can be computed on the fly as w i = Ω phy f j (ξ )dΩ. For the multidimensional case, the extension is straightforward applying the tensor product of the one-dimensional basis functions and points.For linear applications the presented moment fitting method works very well. However, for nonlinear applications this method performs less stable than the adaptive integration in which broken cells are subdivided applying spacetrees and performing a standard Gauss quadrature on subcell level. Due to this reason, we present an adaptive moment fitting approach. In this approach, we subdivide broken cells and subcells if their volume fraction of the physical domain is smaller than a predefined tolerance. Thus, the moment fitting is performed on cell or subcell level depending on the material distribution within the cells. Numerical examplesTo demonstrate the efficiency of the presented integration methods, we investigate two three-dimensional FCM examples for finite strain J 2 flow theory of plasticity [4]. In doing so, we compare the results of the moment fitting (MF) and the adaptive moment fitting (AMF) with th...

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Numerical integration of discontinuous functions: moment fitting and smart octree

Cited by 65 publications

References 60 publications

A 3D benchmark problem for crack propagation in brittle fracture

A 3D benchmark problem for crack propagation in brittle fracture

Integrating CAD and numerical analysis: ‘Dirty geometry’ handling using the Finite Cell Method

Adaptive numerical integration of broken finite cells based on moment fitting applied to finite strain problems

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