The Triangular Equilibrium Element in the Solution of Plate Bending Problems

Morley, L. S. D.

doi:10.1017/s0001925900004546

Cited by 325 publications

(163 citation statements)

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“…Although both methods work well in approximating the viscosity solution of (1)-(2), the resulting nonlinear algebraic system is very large and costly to compute. The main reason to use the Morley finite element to approximate (6)- (8) is that it has the least number of degrees of freedom on each element for fourth order problems, as its basis functions consist of only quadratic polynomials [24,26,27].…”

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confidence: 99%

A nonconforming Morley finite element method for the fully nonlinear Monge-Ampère equation

Neilan

2010

Numer. Math.

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In this paper, we study finite element approximations of the viscosity solution of the fully nonlinear Monge-Ampère equation, det(D 2 u) = f (> 0) using the well-known nonconforming Morley element. Our approach is based on the vanishing moment method, which was recently proposed as a constructive way to approximate fully nonlinear second order equations by the author and Feng (J Sci Comput 38(1):74-98, 2009). The vanishing moment method approximates the MongeAmpère equation by the fourth order quasilinear equation − ∆ 2 u + det(D 2 u ) = f with appropriate boundary conditions. We develop a finite element scheme using the n-dimensional Morley element introduced in Wang and Xu (Numer Math 103: 155-169, 2006) to approximate the regularized fourth order problem in two and three dimensions, and then derive optimal order error estimates.

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A nonconforming Morley finite element method for the fully nonlinear Monge-Ampère equation

Neilan

2010

Numer. Math.

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“…The P 1 nonconforming finite element [3], different from the Courant element, is a bivariate linear weak spline. The well-known Morley element [3][4][5][6], which is very useful in the study of thin plane bending, is a bivariate quadratic Project was supported by the National Natural Science Foundation of China (No.60533060). weak spline.…”

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confidence: 99%

On the Dimension of Bivariate Weak Spline Space Over Regular Rectilinear Partition

Lang

Wang

2009

Results. Math.

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In this paper, we mainly study the dimensions of bivariate weak spline spaces W µ k (I1∆) (k ≥ 2µ+1) and W 1 2 (I * 1 ∆) by using the smoothing cofactor-conformality method, where I1∆ and I * 1 ∆ are regular rectilinear partitions with appointed point sets. Some future works relative to bivariate weak splines are also listed at the end of this paper. Mathematics Subject Classification (2000). Primary 65D07; Secondary 41A15.

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“…As a consequence, nonconforming elements are a widely adopted choice. A well known finite element for the Kirchhoff problem is the Morley element which uses just second order piecewise polynomial functions (see for example [9,14]). …”

Section: Introductionmentioning

confidence: 99%