Local mesh refinement with finite elements for elliptic problems

Gregory, John; Fishelov, Dalia; Schiff, B.; Whiteman, J. R.

doi:10.1016/0021-9991(78)90114-6

Cited by 13 publications

(5 citation statements)

References 13 publications

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“…The transition to a coarser mesh with increasing distance from the superelement was achieved by using triangular elements, with linear trial functions, and five-point rectangular elements [18]. The transition to a coarser mesh with increasing distance from the superelement was achieved by using triangular elements, with linear trial functions, and five-point rectangular elements [18].…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In other words, we set / qj = Z a,,,f,,,(x2, Yj), (2) m=l where qj is the vector of the nodal values at the mesh point (xj, yj), for each mesh point of Din. In other words, we set / qj = Z a,,,f,,,(x2, Yj), (2) m=l where qj is the vector of the nodal values at the mesh point (xj, yj), for each mesh point of Din.…”

Section: The Constrained Finite Elementmentioning

confidence: 99%

“…In one, exemplified by [1][2][3], the usual elements are used throughout the domain, but the mesh is refined locally in the neighborhood of the singularity. The other approach, exemplified by [4][5][6][7][8][9][10][11][12][13], takes into account the known form of the solution near the singular point by including singular functions in the finite element trial space.…”

Section: Introductionmentioning

confidence: 99%

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Stress intensity factors for two-dimensional crack problems using constrained finite elements

Ogen

Schiff

1985

Int J Fract

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Two-dimensional crack problems, in common with other elliptic problems containing a boundary singularity, may be solved efficiently with the aid of a constrained finite element. The singularity is surrounded by a superelement containing a refined mesh whose interior nodal values are constrained to agree with the first few terms of the known expansion for the solution. The superelement conforms with linear or bilinear elements, and may thus be included in standard finite dement programs. The calculation yields the expansion coefficients directly, and the method has been applied to determine stress intensity factors for a variety of two-dimensional configurations, including mixed-mode. The results are in excellent agreement with those obtained by other methods.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: The Constrained Finite Elementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stress intensity factors for two-dimensional crack problems using constrained finite elements

Ogen

Schiff

1985

Int J Fract

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“…In fracture mechanics, numerical methods are an invaluable tool to compute fracture parameters that are associated with the fracture and failure of cracked-bodies. Bernal and Whiteman [42] used finite difference approximations, and Gregory et al [43] used local mesh refinement with modified C 1 bicubic interpolants to solve the two-dimensional biharmonic problem of an edge-cracked plate under uniaxial tension. Let w be the Airy stress function in two-dimensional elasticity.…”

Section: A Airy Stress Functionmentioning

confidence: 99%

C1natural neighbor interpolant for partial differential equations

Sukumar

Moran

1999

Numer. Methods Partial Differential Eq.

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Natural neighbor coordinates [20] are optimum weighted-average measures for an irregular arrangement of nodes in R n .[26] used the notion of Bézier simplices in natural neighbor coordinates Φ to propose a C 1 interpolant. The C 1 interpolant has quadratic precision in Ω ⊂ R 2 , and reduces to a cubic polynomial between adjacent nodes on the boundary ∂Ω. We present the C 1 formulation and propose a computational methodology for its numerical implementation (Natural Element Method) for the solution of partial differential equations (PDEs). The approach involves the transformation of the original Bernstein basis functions B 3 i (Φ) to new shape functions Ψ(Φ), such that the shape functions ψ3I−2(Φ), ψ3I−1(Φ), and ψ3I (Φ) for node I are directly associated with the three nodal degrees of freedom wI , θI x , and θI y , respectively. The C 1 shape functions interpolate to nodal function and nodal gradient values, which renders the interpolant amenable to application in a Galerkin scheme for the solution of fourth-order elliptic PDEs. Results for the biharmonic equation with Dirichlet boundary conditions are presented. The generalized eigenproblem is studied to establish the ellipticity of the discrete biharmonic operator, and consequently the stability of the numerical method.

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“…In [, Section 3], an optimal convergence rate for the LDG method mentioned above was achieved. In , the authors treat the biharmonic problem using a conformal bicubic Hermite polynomial. Using interpolation theory it may be shown that if the solution is smooth enough then the error is bounded by Ch 4 , where h is the mesh size.…”

Section: Introductionmentioning

confidence: 99%

Time evolution of discrete fourth‐order elliptic operators

Ben‐Artzi

Croisille

Fishelov

2019

Numerical Methods Partial

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The evolution equationA discrete parabolic methodology is developed, based on a discrete elliptic (fourth-order) calculus. The main ingredient of this calculus is a discrete biharmonic operator (DBO). In the general case, it is shown that the approximate solutions converge to the continuous one. An "almost optimal" convergence result (O(h 4 − )) is established in the case of constant coefficients, in particular in the pure biharmonic case. Several numerical test cases are presented that not only corroborate the theoretical accuracy result, but also demonstrate high-order accuracy of the method in nonlinear cases. The nonlinear equations include the well-studied Kuramoto-Sivashinsky equation. Numerical solutions for this equation are shown to approximate remarkably well the exact solutions. The numerical examples demonstrate the great improvement achieved by using the DBO instead of the standard (five-point) discrete bilaplacian.

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Local mesh refinement with finite elements for elliptic problems

Cited by 13 publications

References 13 publications

Stress intensity factors for two-dimensional crack problems using constrained finite elements

Stress intensity factors for two-dimensional crack problems using constrained finite elements

C1natural neighbor interpolant for partial differential equations

Time evolution of discrete fourth‐order elliptic operators

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