Exact integration formulas for the finite volume element method on simplicial meshes

Voitovich, Tatiana; Vandewalle, Stefan

doi:10.1002/num.20210

Cited by 7 publications

(4 citation statements)

References 16 publications

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“…The correspondence of notations between those found in [17] and those shown in Figure 3b will be as follows: Sfalse˜1⇔Sfalse˜34, Sfalse˜2⇔Sfalse˜14, Sfalse˜3⇔Sfalse˜42, Sfalse˜4⇔Sfalse˜23, Sfalse˜5⇔Sfalse˜13, Sfalse˜6⇔Sfalse˜12. Then the integral (14) is expressed as:wfalse˜ρCpv3=ρCpVfalse→e·nfalse→3Sfalse˜3432(13T1+5T2+13T3+5T4)…”

Section: The Constitutive Convective Matrix In the CMmentioning

confidence: 99%

“…Applying the formulas of the exact integration (22), Equation (21) is transformed into (23), see [17]:∫Sfalse˜kλδds={left1118|Sfalse˜k| δ=k736|Sfalse˜k| δ≠k, wfalse˜conv1=heffsans-serifΔ3(1118T1+736T2+736T3)−heffsans-serifΔTam3, where Δ is the area of each triangle of the contour. Analogously it is done with the other two dual surfaces, leaving the final Equation (24).…”

Section: The Constitutive Convective Matrix In the CMmentioning

confidence: 99%

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Thermal Analysis of a Magnetic Brake Using Infrared Techniques and 3D Cell Method with a New Convective Constitutive Matrix

Monzón-Verona

González-Domínguez

García-Alonso

et al. 2019

Sensors

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In this work we analyse the temperature distribution in a conductor disk in transitory regime. The disk is in motion in a stationary magnetic field generated by a permanent magnet and so, the electric currents induced inside it generate heat. The system acts as a magnetic brake and is analysed using infrared sensor techniques. In addition, for the simulation and analysis of the magnetic brake, a new thermal convective matrix for the 3D Cell Method (CM) is proposed. The results of the simulation have been verified by comparing the numerical results with those obtained by the Finite Element Method (FEM) and with experimental data obtained by infrared technology. The difference between the experimental results obtained by infrared sensors and those obtained in the simulations is less than 0.0459%.

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Section: The Constitutive Convective Matrix In the CMmentioning

confidence: 99%

Section: The Constitutive Convective Matrix In the CMmentioning

confidence: 99%

Thermal Analysis of a Magnetic Brake Using Infrared Techniques and 3D Cell Method with a New Convective Constitutive Matrix

Monzón-Verona

González-Domínguez

García-Alonso

et al. 2019

Sensors

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“…where k = 1, 2, 3. For a detailed evaluation of the integrals and their values for the relevant cases we refer to [9].…”

Section: Exact Integration Of Polynomials Using Barycentric Coordinatesmentioning

confidence: 99%

Barycentric Interpolation and Exact Integration Formulas for the Finite Volume Element Method

Voitovich¹,

Vandewalle²

2008

AIP Conference Proceedings

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This contribution concerns with the construction of a simple and effective technology for the problem of exact integration of interpolation polynomials arising while discretizing partial differential equations by the finite volume element method on simplicial meshes. It is based on the element-wise representation of the local shape functions through barycentric coordinates (barycentric interpolation) and the introducing of classes of integration formulas for the exact integration of generic monomials of barycentric coordinates over the geometrical shapes defined by a barycentric dual mesh. We discuss especially a related problem of the approximation of the diffusion operators with spatially varying diffusion tensors, resulting in asymmetric stiffness matrices. Numerical examples are presented that illustrate the validity of the technology.

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“…Conforming polygonal finite elements [1][2][3][4] and finite elements on convex polyhedra [5][6][7] require the integration of nonpolynomial basis functions. The integration of polynomials on irregular polytopes arises in the non-conforming variable-element-topology finite element method [8,9], discontinuous Galerkin finite elements [10], finite volume element method [11] and mimetic finite difference schemes [12][13][14]. In partition-of-unity methods such as the extended finite element method (X-FEM) [15,16], discontinuous functions are integrated to form the stiffness matrix of elements that are cut by a crack or an interface.…”

mentioning

confidence: 99%

Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons

2010

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We construct efficient quadratures for the integration of polynomials over irregular convex polygons and polyhedrons based on moment fitting equations. The quadrature construction scheme involves the integration of monomial basis functions, which is performed using homogeneous quadratures with minimal number of integration points, and the solution of a small linear system of equations. The construction of homogeneous quadratures is based on Lasserre's method for the integration of homogeneous functions over convex polytopes. We also construct quadratures for the integration of discontinuous functions without the need to partition the domain into triangles or tetrahedrons. Several examples in two and three dimensions are presented that demonstrate the accuracy and versatility of the proposed method.Keywords Numerical integration · Lasserre's method · Euler's homogeneous function theorem · Irregular polygons and polyhedrons · Homogeneous and nonhomogeneous functions · Strong and weak discontinuities · Polygonal finite elements · Extended finite element method

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Exact integration formulas for the finite volume element method on simplicial meshes

Cited by 7 publications

References 16 publications

Thermal Analysis of a Magnetic Brake Using Infrared Techniques and 3D Cell Method with a New Convective Constitutive Matrix

Thermal Analysis of a Magnetic Brake Using Infrared Techniques and 3D Cell Method with a New Convective Constitutive Matrix

Barycentric Interpolation and Exact Integration Formulas for the Finite Volume Element Method

Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons

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