Symmetric quadrature rules for simplexes based on sphere close packed lattice arrangements

Williams, David M.; Shunn, Lee; Jameson, A.

doi:10.1016/j.cam.2014.01.007

Cited by 58 publications

(40 citation statements)

References 16 publications

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“…However, the cubatures generated by this rule have negative weights and the number of integration points for higher orders are not optimal. Over the past decade, only a few contributions have appeared that have produced cubature rules on a tetrahedron, with integration rules of order p =15 being the maximum that is currently available . The presence of only a handful of studies in this important topic in computational mathematics is due to the fact that with increasing p in 3D, constructing cubature rules is a notoriously difficult problem.…”

Section: Introductionmentioning

confidence: 99%

High‐order cubature rules for tetrahedra

Jaśkowiec

Sukumar

2020

Numerical Meth Engineering

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In this paper, we construct new high-order numerical integration schemes for tetrahedra, with positive weights and integration points that are in the interior of the domain. The construction of cubature rules is a challenging problem, which requires the solution of strongly nonlinear algebraic (moment) equations with side conditions given by affine inequality constraints. We present a robust algorithm based on a sequence of three modified Newton procedures to solve the constrained minimization problem. In the literature, numerical integration rules for the tetrahedron are available up to order p = 15. We obtain integration rules for the tetrahedron from p = 2 to p = 20, which are computed using multiprecision arithmetic. For p ≤ 15, our approach provides integration rules that have the same or fewer number of integration points than existing rules; for p = 16 to p = 20, our rules are new. Numerical tests are presented that verify the polynomial-precision of the cubature rules. Convergence studies are performed for the integration of exponential, rational, weakly singular and trigonometric test functions over tetrahedra with flat and curved faces. In all tests, improvements in accuracy is realized as p is increased, though in some cases nonmonotonic convergence is observed. K E Y W O R D Sleast squares, numerical integration, polynomial basis, positive weights, strictly interior integration points, tetrahedral domain 1 2418

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Section: Introductionmentioning

confidence: 99%

High‐order cubature rules for tetrahedra

Jaśkowiec

Sukumar

2020

Numerical Meth Engineering

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“…This is perhaps the most studied cubature domain, with a correspondingly large body of literature a selection of which is presented here. While rules of degree up to 20, thus covering most cases of practical interest, were progressively developed by 1985 [1,4,5,6], this is still an active field [7,8,9,10,11,12,13,14,15,16]. This happens for two distinct reasons, the first being that different applications require different properties of the cubature rules; the previously cited work for example focuses only on fully symmetric rules (which are also the easier to determine), while only a few works consider rotationally symmetric [17,12] or asymmetric [18,19] rules.…”

Section: Introductionmentioning

confidence: 99%

New fully symmetric and rotationally symmetric cubature rules on the triangle using minimal orthonormal bases

Papanicolopulos

2016

Journal of Computational and Applied Mathematics

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Cubature rules on the triangle have been extensively studied, as they are of great practical interest in numerical analysis. In most cases, the process by which new rules are obtained does not preclude the existence of similar rules with better characteristics. There is therefore clear interest in searching for better cubature rules.Here we present a number of new cubature rules on the triangle, exhibiting full or rotational symmetry, that improve on those available in the literature either in terms of number of points or in terms of quality. These rules were obtained by determining and implementing minimal orthonormal polynomial bases that can express the symmetries of the cubature rules. As shown in specific benchmark examples, this results in significantly better performance of the employed algorithm.

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“…Here, an integration degree of 5 with 15 integration points or a degree of 7 with 84 integration is used (Williams et al, 2014). The stresses at the integration points int are calculated from the nodal results by using the element shape functions, while the volumes associated to the integration points are calculated from the element volumes considering the weights of the integration points.…”

Section: A Modified Failure Criterion and Its Implementation In A Relmentioning

confidence: 99%

A Stochastic Reliability Model for Application in a Multidisciplinary Optimization of a Low Pressure Turbine Blade Made of Titanium Aluminide

Dresbach

Becker

Reh

et al. 2016

Lat. Am. j. solids struct.

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Currently, there are a lot of research activities dealing with gamma titanium aluminide (-TiAl) alloys as new materials for low pressure turbine (LPT) blades. Even though the scatter in mechanical properties of such intermetallic alloys is more distinctive as in conventional metallic alloys, stochastic investigations on -TiAl alloys are very rare. For this reason, we analyzed the scatter in static and dynamic mechanical properties of the cast alloy Ti48Al-2Cr-2Nb. It was found that this alloy shows a size effect in strength which is less pronounced than the size effect of brittle materials. A weakest-link approach is enhanced for describing a scalable size effect under multiaxial stress states and implemented in a post processing tool for reliability analysis of real components. The presented approach is a first applicable reliability model for semi-brittle materials. The developed reliability tool was integrated into a multidisciplinary optimization of the geometry of a LPT blade. Some processes of the optimization were distributed in a wide area network, so that specialized tools for each discipline could be employed. The optimization results show that it is possible to increase the aerodynamic efficiency and the structural mechanics reliability at the same time, while ensuring the blade can be manufactured in an investment casting process.

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Symmetric quadrature rules for simplexes based on sphere close packed lattice arrangements

Cited by 58 publications

References 16 publications

High‐order cubature rules for tetrahedra

High‐order cubature rules for tetrahedra

New fully symmetric and rotationally symmetric cubature rules on the triangle using minimal orthonormal bases

A Stochastic Reliability Model for Application in a Multidisciplinary Optimization of a Low Pressure Turbine Blade Made of Titanium Aluminide

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