A set of pathological tests to validate new finite elements

Rao, K. Mallikarjuna; Shrinivasa, U.

doi:10.1007/bf02703459

Cited by 14 publications

(6 citation statements)

References 71 publications

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“…A three-dimensional patch test [27] is considered. The patch of elements shown in Figure 1 is tested under constant and linear fields [28]. Deflections at the tip of load application and stresses at the root, normalized with the exact solution, are presented in Table I for the case of linear strain patch test.…”

Section: Patch Testmentioning

confidence: 99%

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A 20‐node hexahedron element with enhanced distortion tolerance

Ooi

Rajendran

Yeo

2004

Numerical Meth Engineering

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SUMMARYThis paper presents a distortion resistant 20-node hexahedron element that employs two different sets of shape functions for the trial and test functions. The formulation seeks to satisfy the continuity and completeness requirements by exploiting the intrinsic properties of these two sets of shape functions. Several test problems are used to assess the performance of the element under various mesh distortions. The ability of the proposed as well as the classical 20-node element to maintain solution accuracy under severe mesh distortions has been studied. The proposed element exhibits a very high tolerance to mesh distortions. In particular, for problems involving linear and quadratic displacement fields, the element is capable of reproducing exact solution under all admissible geometrical distortions of the mesh. For test problems involving higher-order displacement fields, the performance of the present element is in general better than that of the classical element.

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Section: Patch Testmentioning

confidence: 99%

“…The cantilever beam [28] as shown in Figure 15 is used to test the locking effect when the Poisson ratio approaches 0.5 simulating material incompressibility. The test includes the effects of trapezoidal-shaped elements.…”

Section: Volumetric Lockingmentioning

confidence: 99%

A 20‐node hexahedron element with enhanced distortion tolerance

Ooi

Rajendran

Yeo

2004

Numerical Meth Engineering

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“…Much attention is given to testing new numerical methods for locking [39,40,56,63]. Numerous so-called pathological tests for other bottlenecks in numerical methods are collected in [56].…”

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confidence: 99%

Mixed functionals in the theory of nonlinearly elastic shells

Maksimyuk

Chernyshenko

2004

Int Appl Mech

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Results obtained on the basis of linearized functionals in the theory of nonlinearly elastic composite shells are analyzed and generalized. The Kirchhoff-Love and Timoshenko hypotheses are used. Possible membrane or shear locking is taken into account. New approaches are proposed to improve the convergence of numerical solution for new classes of nonlinear problems for thin and nonthin shells with a curvilinear (circular, elliptical) hole. The stress-strain state of shells is analyzed using different versions of shell theory. The influence of the nonlinear properties and orthotropy of composite materials on the stress distribution in structural members is studied.Introduction. At a certain stage of development of numerical mesh-type methods for shells, so-called locking effects became to manifest themselves as slowed convergence. A similar effect called volumetric or dilatational locking is widely known in relation to weakly compressible bodies [36,38,41]. One of the first locking effects associated with shells was shear locking [61] mainly manifested in attempts to apply shear finite-element models to thin-walled structures [4,41]. Under certain conditions, even the classical model based on the Kirchhoff-Love hypotheses may lead to membrane locking [4,41,61,63]. Further increase in the degrees of freedom of the finite-element method (FEM), beginning with the consideration of reduction and ending with a three-dimensional model, revealed new types of locking: thickness or Poisson's thickness locking [41,62,67] and trapezoidal or curvature thickness locking [67].Originally, locking was overcome using sophisticated mathematical methods such as reduced integration and the like. These methods happened to have a number of shortcomings: they are not general enough to encompass all types of locking, geometrical parameters of shells, and properties of materials, they may produce false solutions, and they are limited to the FEM and involve finite-element models of high order (see [35,36,43] and references therein). These shortcomings are apparently because of the heuristic nature of such methods. This motivated researchers to develop meshfree, or meshless, methods [45,52,54] and mesh methods other than the FEM [2, 6, 7], approaches based on mixed variational principles [42,57,58], and finite-element models of low order [55,60].In the variational difference method (VDM) in the form of the finite-difference method (FDM), locking effects are less significant [2]. The authors of [2] point out that an attempt to satisfy, as accurately as possible, the additional conditions of functionals with the minimum number of varied parameters in the FEM adversely affects the accuracy of the stationarity conditions, while the reasonable inaccuracy of the former, as in the FDM and mixed FEM, produces more accurate trade-off schemes.Different types of locking and modern methods used to overcome them are summarized in Table 1.In the table, the methods are named as is customary in the English-language literature, since their Russian names are ...

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“…Though the stability theorems for the Cholesky method are proved for positive definite matrices, experience suggests that the method also works in the case being considered. Numerous examples of solving nonlinear, linear, so-called pathological [57] test problems are given in [52]. Noteworthy is the positive experience of [8,18] in using Lagrangian multipliers to implement the kinematic hypotheses of the theory of thin shells in solving linear and nonlinear problems by the VDM.…”

Section: Methodical Aspects Of Numerical Solution Of Nonlinear Problemsmentioning

confidence: 99%

Using mesh-based methods to solve nonlinear problems of statics for thin shells

2009

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The paper outlines a numerical procedure for solving physically and geometrically nonlinear problems of statics for thin shells based on three mesh-based methods: finite-difference, variational difference, and finite-element methods. The methodological, algorithmic, and analytical aspects of implementing the Kirchhoff-Love hypotheses are analyzed. The algorithmic approach employs Lagrangian multipliers. The advantages and disadvantages of these methods are evaluated Keywords: finite-difference method, variational difference method, finite-element method, Kirchhoff-Love hypotheses, Lagrangian multipliers, nonlinear elastic composite, elastoplastic stateIntroduction. Strength analysis of structural members such as shells of complex geometry is mainly conducted on a computer using numerical mesh-based methods, including the finite-difference method (FDM), the variational difference method (VDM), and the finite-element method (FEM). To derive the basic equations of the theory of thin shells, it is sufficient to use the classical model based on the Kirchhoff-Love hypotheses. The Kirchhoff-Love kinematics is implemented differently in these three methods.The system of governing equations in the FDM is known to be derived from the equilibrium equations for an element of the shell, while the Kirchhoff-Love constraints are incorporated analytically, by substituting the corresponding relations into the formulas describing the distribution of displacements throughout the thickness of the shell. This way causes no complications. Contrastingly, the system of governing equations in the VDM and FEM is derived from the extremum condition for some functional. An attempt to implement Kirchhoff-Love constraints analytically gives rise to higher-than-first-order derivatives of displacements in the functionals. Note that the complications in the VDM are only restricted to more awkward equations and the necessity of deriving them because the VDM imposes no special requirements upon the smoothness of the integrands and functions. In the FEM, however, certain requirements are placed upon the smoothness of the coordinate functions, and the construction of an element is complicated if the integrands include higher-order derivatives.These circumstances led to a search for alternative ways of incorporating the Kirchhoff-Love constraints such as those that would not produce derivatives of higher than the first order in the integrands. Noteworthy are two groups of approaches that employ Lagrangian multipliers and penal functions. Applying them requires certain correctness [17,52] associated with the variation order and additional conditions. However, the use of Lagrangian multipliers to implement the Kirchhoff-Love hypotheses increases the number of varied functions and requires much computing. This is obviously why the Kirchhoff-Love constraints were first (1967) used [46] in the FEM for a plate by specifying them at discrete points of an element (discrete Kirchhoff element). Later, this method was extended to thin shells [45,70] and is wi...

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A set of pathological tests to validate new finite elements

Cited by 14 publications

References 71 publications

A 20‐node hexahedron element with enhanced distortion tolerance

A 20‐node hexahedron element with enhanced distortion tolerance

Mixed functionals in the theory of nonlinearly elastic shells

Using mesh-based methods to solve nonlinear problems of statics for thin shells

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