Adaptive computations on conforming quadtree meshes

Tabarraei, Alireza; Sukumar, N.

doi:10.1016/j.finel.2004.08.002

Cited by 107 publications

(85 citation statements)

References 34 publications

(25 reference statements)

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“…One of the simplest is using quad tree structures (in 2D) and has been used in the context of finite volume method in [14][15][16], for FEM in [17], and with the meshfree element-free Galerkin method [18], where level-1 quad-tree structures are employed even though this is not explicitly mentioned in the paper. The level-1 (or 1-irregular) grid refinement has been originally introduced in [1].…”

Section: Literature Reviewmentioning

confidence: 99%

Convergence, adaptive refinement, and scaling in 1D peridynamics

Bobaru

Yang

Alves

et al. 2008

Numerical Meth Engineering

449

331

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SUMMARYWe introduce here adaptive refinement algorithms for the non-local method peridynamics, which was proposed in (J. Mech. Phys. Solids 2000; 48:175-209) as a reformulation of classical elasticity for discontinuities and long-range forces. We use scaling of the micromodulus and horizon and discuss the particular features of adaptivity in peridynamics for which multiscale modeling and grid refinement are closely connected. We discuss three types of numerical convergence for peridynamics and obtain uniform convergence to the classical solutions of static and dynamic elasticity problems in 1D in the limit of the horizon going to zero. Continuous micromoduli lead to optimal rates of convergence independent of the grid used, while discontinuous micromoduli produce optimal rates of convergence only for uniform grids. Examples for static and dynamic elasticity problems in 1D are shown. The relative error for the static and dynamic solutions obtained using adaptive refinement are significantly lower than those obtained using uniform refinement, for the same number of nodes.

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Section: Literature Reviewmentioning

confidence: 99%

Convergence, adaptive refinement, and scaling in 1D peridynamics

Bobaru

Yang

Alves

et al. 2008

Numerical Meth Engineering

449

331

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“…This problem is solved in Reference [9] using the exact solution to estimate the error. In this paper, the problem is solved using the residual-based error estimator.…”

Section: Examplementioning

confidence: 99%

“…Special techniques have been used to construct conforming approximations over quadtree meshes: constraining hanging nodes to corner nodes [1], adding temporary elements to construct a compatible mesh [2,3], Lagrange multipliers and penalty or Nitsche's method to impose constraints [4,5], using hierarchical enrichment [6,7] or B-splines [8], and natural neighbor basis functions [9,10]. In this paper the method developed in Reference [9] is employed to resolve the problem associated with the presence of hanging nodes. This technique is based on the polygonal finite element method introduced in References [11,12].…”

Section: Introductionmentioning

confidence: 99%

Adaptive computations using material forces and residual-based error estimators on quadtree meshes

Tabarraei

Sukumar

2007

Computer Methods in Applied Mechanics and Engineering

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Quadtree is a hierarchical data structure that is well-suited for h-adaptive mesh refinement. Due to the presence of hanging nodes, classical shape functions are non-conforming on quadtree meshes. In this paper, we use natural neighbor basis functions to construct conforming interpolants on quadtree meshes. To this end, the recently proposed construction of polygonal basis functions is adapted to quadtree elements. A fast technique for calculating stiffness matrix on quadtree meshes is introduced. Residual-based error estimators and material force technique are used to estimate the error on quadtree meshes. The performance of the adaptive technique is demonstrated through the solution of linear and nonlinear boundary-value problems.

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“…Tabarraei and Sukumar [11] demonstrated that, for the special case of quadtree meshes, for the Poisson equation and for the elasticity problem, the stiffness matrix of a subelement is the same as the stiffness matrix of the parent. Suzuki and Tabata [20] also showed the importance of reusing previous calculations studying the structure of the finite element mass and stiffness matrices of congruent subdomains (each of them being an image of a reference subdomain by an affine transformation, see [20] for further details).…”

Section: Hierarchical Properties Between Geometrically Similar Elementsmentioning

confidence: 99%

“…The use of a structured element splitting, as in the quadtree and octree methods [8,11] is rather efficient as it allows improving the storage of the mesh related information. However, other splitting techniques, that could be termed non-structured element splitting techniques, are more flexible as they can be applied on any mesh.…”

Section: Introductionmentioning

confidence: 99%

A hierarchical h adaptivity methodology based on element subdivision

Ródenas

Albelda

Tur

et al. 2017

revuin

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Este artículo puede compartirse bajo la licencia CC BY-ND 4.0 y se referencia usando el siguiente formato: J.J. Ródenas, J.Albelda, M. Tur, F.J. Fuenmayor, "A hierarchical h adaptivity methodology based on element subdivision", UIS Ingenierías, vol. 16, no. ABSTRACTThis paper presents a hierarchical h adaptive methodology for Finite Element Analysis based on the hierarchical relations between parent and child elements that come out if these elements are geometrically similar. Under this similarity condition the terms involved in the evaluation of element stiffness matrices of parent and child elements are related by a constant which is a function of the element sizes ratio (scaling factor). These relations have been the basis for the development of a hierarchical h adaptivity methodology based on element subdivision and the use of multipoint-constraints to ensure C 0 continuity. The use of a hierarchical data structure significantly reduces the amount of calculations required for the mesh refinement, the evaluation of the global stiffness matrix, element stresses and element error estimation. The data structure also produces a natural reordering of the global stiffness matrix that improves the behaviour of the Cholesky factorization. KEYWORDS:Adaptive Modelling, Hierarchical properties, Mesh Enrichment, Mesh Generation. RESUMENEn este artículo se presenta una metodología h adaptativa para el Análisis por Elementos Finitos basada en las relaciones jerárquicas entre elementos padre e hijo que surgen si estos elementos son geométricamente similares. Bajo esta condición de similitud, los términos resultantes de la evaluación de las matrices de rigidez de elementos padre e hijo están relacionados por una constante que es una función de la relación de tamaños de elemento (factor de escala). Estas relaciones han sido la base para el desarrollo de una metodología jerárquica h adaptativa basada en la subdivisión de elementos y el uso de restricciones multipunto para asegurar la continuidad C 0 . El uso de una estructura de datos jerárquica reduce significativamente la cantidad de cálculos requeridos para el refinamiento de la malla, la evaluación de la matriz de rigidez global, las tensiones de los elementos y la estimación del error del elemento. La estructura de datos también produce un reordenamiento natural de la matriz de rigidez global que mejora el comportamiento de la factorización de Cholesky. PALABRAS CLAVE:Modelado Adaptativo, Propiedades jerárquicas, Enriquecimiento de malla, Mallado.

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Adaptive computations on conforming quadtree meshes

Cited by 107 publications

References 34 publications

Convergence, adaptive refinement, and scaling in 1D peridynamics

Convergence, adaptive refinement, and scaling in 1D peridynamics

Adaptive computations using material forces and residual-based error estimators on quadtree meshes

A hierarchical h adaptivity methodology based on element subdivision

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