Boundary integration over linear polyhedra

Cattani, Carlo; Paoluzzi, Alberto

doi:10.1016/0010-4485(90)90007-y

Cited by 50 publications

(39 citation statements)

References 12 publications

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“…Lien and Kajaya [6] presented an outline of a closed form formula for volume integration for a linear tetrahedron and suggested that volume integration over a linear polyhedron can be obtained by simple means of disjoint decomposition technique. Cattani and Paoluzzi [7,8] have obtained finite integration over plane polygons and space polyhedra via surface and volume integration methods based on Green's and Gauss's Divergence theorems. In a recent paper, Bernardini [9] has presented explicit formulas and algorithms over a n-dimensional solid by using decomposition representation and boundary representation.…”

Section: Introductionmentioning

confidence: 99%

“…In a recent paper, Bernardini [9] has presented explicit formulas and algorithms over a n-dimensional solid by using decomposition representation and boundary representation. In recent works, Rathod and Govinda Rao [ 10,l l] addressed these problems, with an aim of giving more efficient and explicit algorithms than the previous works of Cattani and Paoluzzi [7,8] which made reference to combined use of well-known Taylor series expansion and Leibniz's theorem on differentation to obtain finite integration formulas for the integration of monomials over plane polygons and space polyhedra. Integration of a triple product, viz.…”

Section: Introductionmentioning

confidence: 99%

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Boundary integration of polynomials over an arbitrary linear tetrahedron in Euclidean three-dimensional space

Rathod¹,

Hiremath²

1998

Computer Methods in Applied Mechanics and Engineering

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This paper is concerned with explicit integration formulas and algorithms for computing integrals of trivariate polynomials over an arbitrary linear tetrahedron in Euclidean. three-dimensional space. This basic three-dimensional integral governing the problem 1s transformed to surface integrals by use of the divergence theorem. The resulting two-dimensional integrals are then transformed into convenient and computationally efficient line integrals. These algorithms and explicit finite integration formulas are followed by an application--example for which we have explained the detailed computational scheme. The numerical result thus found is in complete agreement with previous works. Further. It 1s shown that the present algorithms are much simpler and more economical as well. in terms of arithmetic operations. The symbolic finite integration formulas presented in this paper may lead to an easy incorporation of geometric properties of solid objects, for example, the centre of mass, moment of inertia, etc. required in the engineering design process as well as several applications of numerical analysis where integration is required, for example in the finite element and boundary integral equation methods. Rathod, S.V Hiremath I Comput. Methods Appl. Mech. Engrg. 153 (1998)

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Boundary integration of polynomials over an arbitrary linear tetrahedron in Euclidean three-dimensional space

Rathod¹,

Hiremath²

1998

Computer Methods in Applied Mechanics and Engineering

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“…Lien and Kajiya [6] presented an outline of a closed formula of volume integration for a tetrahedron and suggested that volume integration for a linear polyhedron can be obtained by decomposing it into a set of solid tetrahedrons. Cattani and Paoluzzi [2,3] gave a symbolic solution to both surface and volume integration of trivariate polynomials in K 3 by using a triangulation of the solid based on the concept proposed by Timmer and Stern [8]. In a recent paper, Bernardini [1] presented the evaluation of integrals over n-dimensional linear polyhedra which are based on methods proposed earlier by Timmer and Stern [8] and Lien and Kajiya [6].…”

Section: Introductionmentioning

confidence: 99%

“…In the present paper, we have developed closed-form integration formulas which mainly follow the concepts proposed by Timmer and Stern [8] and Cattani and Paoluzzi [2,3]. In the derivation of these formulas, our approach differs from that of Cattani and Paoluzzi in that we have transformed the surface integral in three-space to a double integral over a polygon in the jcv-plane via the use of the equation of a plane spanning the three co-ordinates of a triangle in three-space.…”

Section: Introductionmentioning

confidence: 99%

“…The double integral in plane over a polygon can then be computed by either transforming it into unit triangles in two-parameter spaces or by means of line integrals along the edges of the two-dimensional polygons. We have also addressed these problems, with an aim to give more efficient algorithms than previous work of Cattani and Paoluzzi [2,3]. In these derivations, we have made reference to the well known Gauss's Divergence theorem, Taylor series expansion and Leibnitz theorem on differentiation to produce the present form of analytical integration formulas.…”

Section: Introductionmentioning

confidence: 99%

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Integration of polynomials over linear polyhedra in Euclidean three-dimensional space

Rathod

Rao

1995

Computer Methods in Applied Mechanics and Engineering

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This paper concerns with analytical integration of trivariate polynomials over linear polyhedra in Euclidean three-dimensional space. The volume integration of trivariate polynomials over linear polyhedra is computed as sum of surface integrals in R 3 on application of the well known Gauss's divergence theorem and by using triangulation of the linear polyhedral boundary. The surface integrals in R 3 over an arbitrary triangle are connected to surface integrals of bivariate polynomials in R 2 . The surface integrals in R 2 over a simple polygon or over an arbitrary triangle are computed by two different approaches. The first algorithm is obtained by transforming the surface integrals in R 2 into a sum of line integrals in a one-parameter space, while the second algorithm is obtained by transforming the surface integrals in R 2 over an arbitrary triangle into a parametric double integral over a unit triangle. It is shown that the volume integration of trivariate polynomials over linear polyhedra can be obtained as a sum of surface integrals of bivariate polynomials in R 2 . The computation of surface integrals is proposed in the beginning of this paper and these are contained in Lemmas 1-6. These algorithms (Lemmas 1-6) and the theorem on volume integration are then followed by an example for which the detailed computational scheme has been explained. The symbolic integration formulas presented in this paper may lead to an easy and systematic incorporation of global properties of solid objects, for example, the volume, centre of mass, moments of inertia etc., required in engineering design processes.

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Symbolic integration of polynomial functions over a linear polyhedron in Euclidean three-dimensional space

Rathod

Rao²,

Hiremath

1996

Commun. Numer. Meth. Engng.

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SUMMARYThe paper concerns analytical integration of polynomial functions over linear polyhedra in threedimensional space. To the authors' knowledge this is a first presentation of the analytical integration of monomials over a tetrahedral solid in 3D space. A linear polyhedron can be obtained by decomposing it into a set of solid tetrahedrons, but the division of a linear polyhedral solid in 3D space into tetrahedra sometimes presents difficulties of visualization and could easily lead to errors in nodal numbering, etc We have taken this into account and also the linearity property of integration to derive a symbolic integration formula for linear hexahedra in 3D space. We have also used yet another fact that a hexahedron could be built up in two, and only two, distinct ways from five tetrahedral shaped elements These symbolic integration formulas are then followed by an illustrative numerical example for a rectangular prism element, which clearly verifies the formulas derived for the tetrahedron and hexahedron elements.

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Boundary integration over linear polyhedra

Cited by 50 publications

References 12 publications

Boundary integration of polynomials over an arbitrary linear tetrahedron in Euclidean three-dimensional space

Boundary integration of polynomials over an arbitrary linear tetrahedron in Euclidean three-dimensional space

Integration of polynomials over linear polyhedra in Euclidean three-dimensional space

Symbolic integration of polynomial functions over a linear polyhedron in Euclidean three-dimensional space

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