Moderate-degree tetrahedral quadrature formulas

Keast, P.

doi:10.1016/0045-7825(86)90059-9

Cited by 166 publications

(71 citation statements)

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“…Numerical integration was performed using a fifth-order, 15-point volume Gauss quadrature. 47 The equations were solved iteratively using GMRES with diagonal preconditioning, 48 based on a square root scaling introduced in Ref. 49.…”

Section: B Direct-fe Methodsmentioning

confidence: 99%

Permeability calculations in three-dimensional isotropic and oriented fiber networks

et al. 2008

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Hydraulic permeabilities of fiber networks are of interest for many applications and have been studied extensively. There is little work, however, on permeability calculations in three-dimensional random networks. Computational power is now sufficient to calculate permeabilities directly by constructing artificial fiber networks and simulating flow through them. Even with today's high-performance computers, however, such an approach would be infeasible for large simulations. It is therefore necessary to develop a correlation based on fiber volume fraction, radius, and orientation, preferably by incorporating previous studies on isotropic or structured networks. In this work, the direct calculations were performed, using the finite element method, on networks with varying degrees of orientation, and combinations of results for flows parallel and perpendicular to a single fiber or an array thereof, using a volume-averaging theory, were compared to the detailed analysis. The detailed model agreed well with existing analytical solutions for square arrays of fibers up to fiber volume fractions of 46% for parallel flow and 33% for transverse flow. Permeability calculations were then performed for isotropic and oriented fiber networks within the fiber volume fraction range of 0.3%-15%. When drag coefficients for spatially periodic arrays were used, the results of the volume-averaging method agreed well with the direct finite element calculations. On the contrary, the use of drag coefficients for isolated fibers overpredicted the permeability for the volume fraction range that was employed. We concluded that a weighted combination of drag coefficients for spatially periodic arrays of fibers could be used as a good approximation for fiber networks, which further implies that the effect of the fiber volume fraction and orientation on the permeability of fiber networks are more important than the effect of local network structure.

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Section: B Direct-fe Methodsmentioning

confidence: 99%

Permeability calculations in three-dimensional isotropic and oriented fiber networks

et al. 2008

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“…Estas reglas de integración son clásicas y fáciles de encontrar en la bibliografía, por ejemplo en [33] y posteriormente en [34]. Para aprovechar el método de colapso de Stroud en un tetraedro seguimos la siguiente regla: la arista donde se colapsa el hexaedro se hace coincidir con la arista que comparte el tetraedro con el frente de grieta, y el nodo donde se colapsan más nodos del hexaedro se hace coincidir con uno de los vértices que coinciden con el frente de la grieta.…”

Section: Integración Basada En El Método Clásico De Stroudunclassified

“…La integración de los tetraedros que contienen singularidad se realiza mediante un hexaedro colapsado de 64 puntos, siguiendo las reglas clásicas preentadas por [33][34] o mediante una regla de 42 puntos según el esquema de [25]. Solo se analizan los casos en que la solución presenta KI o KII, lo que permite estudiar el comportamiento de las reglas de integración adaptadas al extremo de grieta.…”

Section: Verificación Numéricaunclassified

Evaluación de la rigidez a flexión de puentes de viga y losa en concreto presforzado a partir de pruebas de carga. Caso de estudio: Puente La Parroquia vía La Renta - San Vicente de Chucurí

Chávez

Nova

Jaime³

2017

revuin

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Este artículo puede compartirse bajo la licencia CC BY-ND 4.0 (https://creativecommons.org/licenses/by-nd/4.0/) V.F. González-Albuixech, Giner y J.E. Tarancón. "Comparación de esquemas de integración 3D para elementos enriquecidos en XFEM", UIS Ingenierias, vol. 15 RESUMENEl XFEM es una técnica desarrollada para la simulación numérica de problemas relacionados con la mecánica de la fractura. Este método tiene diversas ventajas, pero también aparecen ciertas cuestiones que deben ser abordadas con cuidado. La integración numérica de los elementos enriquecidos es uno de esos puntos. En este trabajo se han comparado dos técnicas posibles para realizar dicha integración, una clásica y una desarrollada especificamente para este tipo de elementos. Las diferencias en el resultado no son destacables, pero no así en la implementación. Por tanto, se recomienda el uso de la técnica clásica.Palabras Clave: elementos enriquecidos, integración, mecánica de la fractura, XFEM. ABSTRACTThe XFEM is a technique developed for the numerical simulation of fracture mechanics problems. The method shows several advantages. But some questions arise that should be handled with some care. Numerical integration of the enriched elements is one of these issues. In this work we have compared two avalaible techniques for its integration. One is a classic integration scheme and another is a scheme specifically developed for this kind of elements. The differences on the results are minimal, but not in their implementation difficulty. Hence, the classical integration is recommended.Keywords: enriched elements, fracture mechanics, integration, XFEM. INTRODUCCIÓNEl estudio analítico del comportamiento mecánico de componentes, incluyendo el análisis del efecto de la presencia de grietas mediante el uso de la Mecánica de la Fractura Elástico Lineal (MFEL), solo puede realizarse analíticamente en casos sencillos. Los casos más complicados tienen que abordarse inevitablemente mediante técnicas numéricas que permiten obtener una solución aproximada del problema.Una de las técnicas numéricas más consolidada y usada, tanto en ámbitos científicos como técnicos, es el método de los elementos finitos (MEF). No obstante, el MEF presenta algunos inconvenientes para la resolución de problemas donde existen singularidades, de los cuales la existencia de grietas es un caso particular. La razón es que la aproximación del MEF se basa en un desarrollo en un espacio polinómico, que puede utilizarse para estudiar, con elevada precisión, problemas cuya solución presenta un campo continuo y suave, pero presenta poca eficacia al emplearse en la descripción de comportamientos singulares.En particular, la solución correspondiente a la existencia de la fisura presenta una discontinuidad y una región con comportamiento singular, que no se puede describir fácilmente mediante funciones polinómicas. Para conseguir una precisión adecuada en este tipo de problemas, resulta obligatorio el uso de un refinamiento local de la malla en la región donde domina la singularidad, lo que con...

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“…Here, the test function space is the set of δ u's that are piecewise linear and continuous and vanish on Γ D . The integral in (3) is evaluated with a quadrature rule; we have used a 6-point formula having degree 4 precision from [7] for our quadrature in 2D and a 15-point formula having degree 5 precision from [8] for 3D. It suffices to solve (3) for the dm choices of δ u that compose the standard basis for the test function space.…”

Section: The Problem Under Considerationmentioning

confidence: 99%

A robust solution procedure for hyperelastic solids with large boundary deformation

Shontz

Vavasis

2011

Engineering with Computers

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Compressible Mooney-Rivlin theory has been used to model hyperelastic solids, such as rubber and porous polymers, and more recently for the modeling of soft tissues for biomedical tissues, undergoing large elastic deformations. We propose a solution procedure for Lagrangian finite element discretization of a static nonlinear compressible Mooney-Rivlin hyperelastic solid. We consider the case in which the boundary condition is a large prescribed deformation, so that mesh tangling becomes an obstacle for straightforward algorithms. Our solution procedure involves a largely geometric procedure to untangle the mesh: solution of a sequence of linear systems to obtain initial guesses for interior nodal positions for which no element is inverted. * Much of the work of the first author was performed while a graduate student in the Center for Applied Mathematics at Cornell University. The author's work is supported by a fellowship from the National Physical Science Consortium (with support from Sandia National Laboratories, and Cornell University); it is also supported in part by NSF grants ACI-0085969, CNS-0720749, and NSF CAREER Award OCI-1054459. † The work of the second author was supported in part by NSF grant ACI-0085969, a Discovery grant from NSERC (Canada), and a grant from the U. After the mesh is untangled, we take Newton iterations to converge to a mechanical equilibrium. The Newton iterations are safeguarded by a line search similar to one used in optimization. Our computational results indicate that the algorithm is up to 70 times faster than a straightforward Newton continuation procedure and is also more robust (i.e., able to tolerate much larger deformations). For a few extremely large deformations, the deformed mesh could only be computed through the use of an expensive Newton continuation method while using a tight convergence tolerance and taking very small steps.Keywords solids · elasticity · nonlinear solver · large deformation · moving mesh The problem under considerationWe consider the problem of solving for the deformed shape of a hyperelastic solid body under static loads. The continuum mechanical model under consideration has the following description [1]. Let B 0 ⊂ R d be an undeformed solid body whose boundary is ∂ B 0 . Here d, the space dimension, is 2 or 3. Assume boundary conditions (either displacement or traction, i.e., Dirichlet or Neumann) are given as follows. The boundary ∂ B 0 is partitioned into two subsets Γ D and Γ N . A function φ 0 :Γ D → R d specifies new-position (Dirichlet) boundary conditions. A second function t 0 :Γ N → R d specifies traction (Neumann) boundary conditions. Everything in this paper extends to the more general case that some coordinate entries are Neumann while others are Dirichlet at certain boundary points, but we limit the discussion to the special case that each boundary point is Dirichlet or Neumann in all d coordinates in order to simplify notation. Finally, the model requires a specification of the model's body forces, that is, a function b...

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Moderate-degree tetrahedral quadrature formulas

Cited by 166 publications

References 4 publications

Permeability calculations in three-dimensional isotropic and oriented fiber networks

Permeability calculations in three-dimensional isotropic and oriented fiber networks

Evaluación de la rigidez a flexión de puentes de viga y losa en concreto presforzado a partir de pruebas de carga. Caso de estudio: Puente La Parroquia vía La Renta - San Vicente de Chucurí

A robust solution procedure for hyperelastic solids with large boundary deformation

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