Convergence of eigenvalue solution in conforming plate bending finite elements

Lynn, Paul P.; Dhillon, Balaur S.

doi:10.1002/nme.1620040208

Cited by 8 publications

(2 citation statements)

References 7 publications

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“…Consider, for instance, the element with linear bending strains and constant tensional strains. Its rate of convergence is nearly 0(h 4 and the ratio r/t is eliminated from it. This element will converge to the inextensional solution without the ratio r/t appearing in its condition number.…”

Section: Condition Of Stiffness Matrixmentioning

confidence: 97%

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Shape functions and the accuracy of arch finite elements.

Fried

1973

AIAA Journal

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If the estimation error has zero mean and has Gaussian distribution it can be shown that Eq. (A27) reduces to [./P-] (A28)The variance of g is the mean square minus the squared mean Var[0] =h r p-h + rr[Jp-Jp-]/2 (A29) The numerator in Eq. (A21) becomes -tr[Jp-]/2)] = P~h Thus, the optimum gain vector isThe covariance of the estimation error after incorporating the measurement was given in Eq. (A9). Recall b = 0. Using Eqs.(A19, 20, 29, 30) the covariance is P + -(/-kh r )p-(/-kh r ) r + k(r + rr[Jp-Jp-]/2)k r (A32)Note if the nonlinearity J is zero, then Eq. (A31) and Eq. (A32) are equivalent to the extended (linearized) Kalman measurement incorporation equations.To summarize, the measurement incorporation equations arewhere g 0 , h, and J were defined in Eqs. (A23-25). These equations are equivalent to those proposed previously byThe rate of convergence of the finite element method is determined by the ability of the approximate strains to assume arbitrary polynomial states. A systematic way to generate shape functions assuring a certain accuracy is to start, then, with assumed polynomial strains and integrate for the displacements. In curved structures, this leads to a coupled system of shape functions for the normal and tangential displacements, which are not necessarily polynomials. Some further simplifications reduce these to coupled polynomial shape functions which might prove more efficient than uncoupled shape functions assumed independently for the different displacements. The spectral condition number of the global stiffness matrix for the curved beam increases with (r/t) 2 -the ratio of radius of curvature to thickness squared. The extensional energy portion of the elastic energy is proportional to (t/r) 2 . For a coarse mesh this might be smaller than the discretization error, not warranting the introduction of the exact small t/r into the extensional energy. By relating r/t to the number of elements so as to balance the extensional energy and the energy discretization error, r/t is eliminated from the extensional energy and consequently from the condition number. The resulting element converges with its full rate to the inextensional solution but without undue ill-conditioning in the stiffness matrix.

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Section: Condition Of Stiffness Matrixmentioning

confidence: 97%

“…(2) and (3) an energy norm ||e,/c|| defined by (4) If £ and K are approximate strains such that \\s.K\\ < oo, derived from continuous u, w and dw/ds, then the energy error in these strains is measured in e-S, K- (5) where Et was assumed, for the sake of simplicity, to be equal to 1. Equation (5) can be written also in the form…”

Section: Variational Principlementioning

confidence: 99%

Shape functions and the accuracy of arch finite elements.

Fried

1973

AIAA Journal

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Observations regarding assumed‐stress hybrid plate elements

Cook

Ladkany

1974

Numerical Meth Engineering

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SUMMARYEdge displacement of assumed-stress hybrid plate elements are commonly chosen as either linear, quadratic or cubic functions of the edge co-ordinate. The present paper examines the effect of this choice on the acceptability, accuracy, and possible zero energy modes of elements based on a linear stress field. It is concluded that triangular elements may be preferable to quadrilaterals, and that linear edge displacements are not desirable.

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Least squares finite element analysis of laminar boundary layer flows

Lynn

1974

Numerical Meth Engineering

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SUMMARYBased on the least squares error criterion, a class of finite element is formulated for the numerical analysis of steady state viscous boundary layer Row problems. The method is essentially a discrete element-wise minimization of square and weighted residuals which arise from the attempts in approximately satisfying boundary layer equations. An iterative linearization scheme is developed to circumvent the mathematical difficulties posed by the non-linear boundary layer equations. It results in a process of successive least squares minimizations of residual errors arising from satisfying a set of linear differential equations. A mathematical justification for the method is presented. A major feature of the method lies in the linearization approach which renders nonlinear differential equations amenable to linear least squares finite element analysis. Another important feature rests on the proposed finite element formulation which preserves the symmetric nature of finite element matrix equations through the use of the least squares error criterion. Numerical examples of viscous Row along a flat plate are presented to demonstrate the applicability of the method as &ell as to illuminate discussions on the theoretical aspects of the method. I NTRO DUCT1 0 N As a major extension of the finite element method, a considerable amount of interest has been recently directed to the numerical analysis of viscous flow problems. Without presenting a practical method, Oden' has given a theoretical analogue of Navier-Stokes equations. Due to the fact that there exists no variational functions describing Navier-Stokes equations,2 several alternative approaches such as Galerkin's weighted residual method and pseudo-variational principle have been used to formulate proper finite elements. The former approach is applied by Baker334 and the latter is developed by Olson.5 Because of the non-self-adjoint character of Navier-Stokes and Prandtl's boundary layer equations, the above formulations fail to preserve the symmetric.property of the finite element matrix equations which is one of the important practical features of matrix methods.In this paper, the least squares error criterion is employed, in an element-wise minimization of differential equation residuals,6 to construct finite elements for two-dimensional boundary layer analyses. Consequently, the positive definite symmetric nature of the element stiffness matrix has been retained.6 To overcome the mathematical difficulties presented by the non-linear Navier-Stokes or the boundary layer equations, there are, in general, two basic iterative numerical approaches. One is to linearize initially the original non-linear differential equations and then solve the problem by a process of successive linear finite element analyses. The other approach is to formulate directly non-linear finite element equations and then numerically solve the resulting system of non-linear algebraic equations by the Newton-Raphson m e t h~d .~ The iterative * Associate Professor.

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Convergence of eigenvalue solution in conforming plate bending finite elements

Cited by 8 publications

References 7 publications

Shape functions and the accuracy of arch finite elements.

Shape functions and the accuracy of arch finite elements.

Observations regarding assumed‐stress hybrid plate elements

Least squares finite element analysis of laminar boundary layer flows

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