Numerical integration of discontinuities on arbitrary domains based on moment fitting

Joulaian, Meysam; Hubrich, Simeon; Düster, Alexander

doi:10.1007/s00466-016-1273-3

Cited by 97 publications

(74 citation statements)

References 55 publications

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“…where f j (x) are the basis functions, x i the integration points, and w i the corresponding weights [3,4]. The integrals of the right-hand side are also referred to as the moments.…”

Section: Moment Fittingmentioning

confidence: 99%

“…The integrals of the right-hand side are also referred to as the moments. Since the system is non-linear in terms of the location of the integration points, in [3,4] the position of the integration points is fixed a priori to simplify the solution process. In doing so, it is ensured that the number of integration points is equal or higher than the number of the basis functions (n ≥ m) so that the moment fitting equation system results in a linear equation system that is regular or underdetermined.…”

Section: Moment Fittingmentioning

confidence: 99%

“…The corresponding weights are then determined by computing the minimum norm solution of the resulting linear least square problem. In [3,4], it is shown that using this approach the number of integration points could be reduced significantly as compared to the adaptive integration method based on a subdivision of the broken cells that is commonly used in the context of the FCM. In addition it is shown that the resulting quadrature rules perform very well for the problems under consideration.…”

Section: Moment Fittingmentioning

confidence: 99%

“…Since the standard Gauss quadrature rules do not perform well for discontinuities, special treatment during the numerical integration process of the cut cells is required. To this end, recently, a new integration method has been introduced in the context of the FCM [3,4]. In this method individual quadrature rules for every broken cell are generated by solving the moment fitting equation system.…”

Section: Introductionmentioning

confidence: 99%

“…In this method individual quadrature rules for every broken cell are generated by solving the moment fitting equation system. In [3,4] to simplify the procedure it is proposed to select the location of the integration points in advance transforming the non-linear moment fitting equation system into a linear one. Consequently, the location and the number of the integration points are not optimal.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Efficient numerical integration of arbitrarily broken cells using the moment fitting approach

Hubrich

Joulaian

Stolfo³

et al. 2016

Proc Appl Math and Mech

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The finite cell method is based on a fictitious domain approach, providing a simple and fast mesh generation of structures with complex geometries. However, this simplification leads to intersected cells where the standard Gauss quadrature does not perform well. To perform the numerical integration of these cells, we use the moment fitting approach that generates an individual quadrature rule for every broken cell. In this paper, we will perform a non-linear optimization approach to find the optimal position and number of the integration points. The findings show that the proposed method leads to efficient quadrature rules that require less integration points than other existing integration methods.

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“…where f j (x) are the basis functions, x i the integration points, and w i the corresponding weights [3,4]. The integrals of the right-hand side are also referred to as the moments.…”

Section: Moment Fittingmentioning

confidence: 99%

Section: Moment Fittingmentioning

confidence: 99%

Section: Moment Fittingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Efficient numerical integration of arbitrarily broken cells using the moment fitting approach

Hubrich

Joulaian

Stolfo³

et al. 2016

Proc Appl Math and Mech

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Thep‐Version of the Finite Element and Finite Cell Methods

Düster

Rank

Szabó

2017

Encyclopedia of Computational Mechanics Second Edition

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In the first part of this chapter, the basic algorithmic structure and performance characteristics of the p ‐version of the finite element method (FEM) are surveyed with reference to elliptic problems in solid mechanics. For this class of problems, the theoretical basis of the p ‐version is fully established, and a very substantial amount of engineering experience covering linear and nonlinear applications is available. It is shown that p ‐extensions on properly designed meshes make realization of exponential rates of convergence in practical computations possible and provide for the estimation and control of relative errors in terms of any quantity of interest. In the second part, the p ‐version of the FEM is extended to a high‐order fictitious domain approach, the finite cell method (FCM). While the FCM inherits the advantages of the p ‐version with respect to accuracy and robustness, it relieves analysts from the necessity of generating finite element meshes. Thus, it strongly supports the analysis of problems with highly complicated geometry, for which meshing with finite elements would be very difficult. Given the growing demand for verified numerical solutions, the p ‐version of the finite element and FCMs are expected to play an increasingly important role.

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2022

IGA: Non‐conforming Coupling and Shape Optimization of Complex Multipatch Structures

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Numerical integration of discontinuities on arbitrary domains based on moment fitting

Cited by 97 publications

References 55 publications

Efficient numerical integration of arbitrarily broken cells using the moment fitting approach

Efficient numerical integration of arbitrarily broken cells using the moment fitting approach

Thep‐Version of the Finite Element and Finite Cell Methods

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