Nonconforming Element Research Articles

A new quadrilateral 5-node non-conforming membrane element with drilling DOF

In this paper, a new quadrilateral 5-node non-conforming membrane element with drilling degrees of freedom is presented. The main advantage of these elements is the relatively small number of integration points to evaluate a stiffness matrix comparing to the existing transition membrane elements (CLM elements). Moreover, the presented elements pass the patch test by virtue of the Direct Modification Method incorporated into the element formulation. The presented 5-node elements are proved to be very efficient when used in the local mesh refinement for the in-plane structures which have stress concentrations. And some numerical studies also show the good performance of the new element developed in this study.

Wilson non‐conforming element in numerical manifold method

AbstractNumerical manifold method (NMM) by Shi tries to unify the solution of problems in both continuous and discontinuous domains. Presently, NMM is commonly adopted on finite triangular and quadrilateral covers and the shape functions used in classical finite element analysis are also commonly adopted in manifold analysis. In this paper, the Wilson non‐conforming element is used in quadrilateral covers. The corresponding matrices of equilibrium equations are derived and a cantilever slab is considered. The results from the use of non‐conforming element in NMM are in good agreement with theoretical solution as compared with the classical Q4 isoparametric element. Copyright © 2002 John Wiley & Sons, Ltd.

Three-dimensional stable nonconforming parallelepiped elements for the Stokes problem

Stability result is obtained for the approximation of the stationary Stokes problem with nonconforming elements proposed by Jim Douglas Jr. et al. [Math. Model. Numer. Anal. 33 (4) (1999) 747] for the velocity with conforming bubble functions and discontinuous piecewise linear for the pressure on parallelepiped elements. Optimal order H 1 and L 2 error estimates are derived.

STABLE LOW ORDER NONCONFORMING QUADRILATERAL FINITE ELEMENTS FOR THE STOKES PROBLEM

Stability result is obtained for the approximation of the stationary Stokes problem with nonconforming elements proposed by Douglas et al [1] for the velocity and discontinuous piecewise constants for the pressure on qudrilateral elements. Optimal order <TEX>$H^1$</TEX>and <TEX>$L^2$</TEX>error estimates are derived.

Covolume method for new nonconforming rectangle element for the Stokes problem

We analyze a covolume method based on the new nonconforming element introduced by Douglas et al. [1]. We show the H 1 optimal order convergence of the scheme for Stokes problem and study the hybrid domain decomposition procedure for this covolume scheme. The numerical experiment shows that the covolume scheme is somewhat better than finite element scheme in the computation of pressure.

Uniform convergence of Galerkin‐multigrid methods for nonconforming finite elements for second‐order problems with less than full elliptic regularity

AbstractIn this article we prove uniform convergence estimates for the recently developed Galerkin‐multigrid methods for nonconforming finite elements for second‐order problems with less than full elliptic regularity. These multigrid methods are defined in terms of the “Galerkin approach,” where quadratic forms over coarse grids are constructed using the quadratic form on the finest grid and iterated coarse‐to‐fine intergrid transfer operators. Previously, uniform estimates were obtained for problems with full elliptic regularity, whereas these estimates are derived with less than full elliptic regularity here. Applications to the nonconforming P1, rotated Q1, and Wilson finite elements are analyzed. The result applies to the mixed method based on finite elements that are equivalent to these nonconforming elements. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 203–217, 2002; DOI 10.1002/num.10004

High Accuracy Analysis of the Adini's Nonconforming Element

In this paper, we present the asymptotic error expansions of the Adini's nonconforming finite element for solving the biharmonic equation with the Dirchlet boundary value condition. Based on these expansions, we have obtained a series of sharp error estimates and extrapolations.

Spectral Element Methods for Transitional Flows in Complex Geometries

We describe the development and implementation of an efficient spectral element code for simulating transitional flows in complex three-dimensional domains. Critical to this effort is the use of geometrically nonconforming elements that allow localized refinement in regions of interest, coupled with a stabilized high-order time-split formulation of the semi-discrete Navier–Stokes equations. Simulations of transition in a model of an arteriovenous graft illustrate the potential of this approach in biomechanical applications.

On V-cycle multigrid implementation for plate elements

In this paper an optimal V-cycle multigrid method is presented for the nonconforming element approximation of the plate bending problem does not need full elliptic regularity. The uniform convergence rate independent of mesh sizes and levels is established. Some numerical tests are given to illustrate the efficiency of the method.

Variational principles for non‐conforming finite element methods

AbstractVariational principles with relaxed inter‐element continuity requirement for non‐conforming element methods in linear and non‐linear analyses are developed. Based on the principles, any non‐conforming element displacement can be used directly to derive the explicit expressions of non‐conforming displacement function, which can ensure the passage of the patch test C for the requirement of convergence Copyright © 2001 John Wiley &amp; Sons, Ltd.

Uniform convergence and Schwarz method for the mortar projection nonconforming and mixed element methods for nonselfadjoint and indefinite problems

In this paper, two mortar versions of the so-called projection nonconforming and the mixed element methods are proposed, respectively, for nonselfadjoint and indefinite second-order elliptic problems. It is proven that the mortar mixed element method is equivalent to the mortar projection nonconforming element method. Based on this equivalence, the existence, uniqueness, and uniform convergence of the solution for mortar mixed element method are shown only under minimal regularity assumption. Meanwhile, the optimal error estimate is obtained under certain regularity assumption. Furthermore, an additive Schwarz preconditioning method is proposed for solving the discrete problem and the nearly optimal convergence rate for the preconditioned GMRES method is proven under minimal regularity assumption. Finally, the practical implementation of the method is adderssed and numerical experiments are presented.

On mortar-type Morley element method for plate bending problem

The purpose of this paper is to consider a mortar element method with locally nonconforming elements for solving plate bending problem. On the subdomains, Morley nonconforming elements are used with different mesh triangulations as desire; at the interfaces two mortar conditions, one for the solution itself in pointwise way, the other for the normal derivative of the solution in the projection way, are given to keep the global convergence of this method. After some detailed analysis, an error estimate in energy norm is obtained, which is optimal in approximation order and only requires the regularity just like that for Morley element method (Arnold and Brezzi (1985), Shi (1990)), namely, u∈H 3(Ω)∩H 2 0(Ω),f∈L 2(Ω) .

Finite volume method based on the Crouzeix-Raviart element for the Stokes equation

The author provides a new discretization method-the finite volume method(FVM). For the Stokes equation the velocity space is approximated by the nonconforming linear element based on the dual partition and the pressure by the piecewise constant based on the primal triangulation. Under the suitable smoothness of the solution, the optimal convergence rateO(h) is obtained, whereh denotes the parameter of the space discretization.

Multigrid for the mortar element method for P1 nonconforming element

In this paper, a multigrid algorithm is presented for the mortar element method for P1 nonconforming element. Based on the theory developed by Bramble, Pasciak, Xu in [5], we prove that the W-cycle multigrid is optimal, i.e. the convergence rate is independent of the mesh size and mesh level. Meanwhile, a variable V-cycle multigrid preconditioner is constructed, which results in a preconditioned system with uniformly bounded condition number.

A direct modification method for strains due to non-conforming modes

This paper addresses an efficient modification method that eliminates the undesirable effects of strains due to various non-conforming modes so that the non-conforming element can pass the patch test unconditionally. The scheme is incorporated in the element formulation to establish new types of non-conforming hexahedral elements designated as NHx and NVHx for the regular element and variable node element, respectively. Non-conforming displacement modes are selectively added to the ordinary (conforming) element displacement assumptions to improve the bending behavior of the distorted solid element. To verify the validation of proposed direct modification method and the improvement of element behavior, several numerical tests are carried out. Test results show that the proposed method is effective and its applications to non-conforming solid elements guarantee for the element to pass the patch test.

Stability analysis of cylindrical shells using refined non‐conforming rectangular cylindrical shell elements

AbstractThe accuracy of stability analysis depends on the accuracy of both the element stiffness matrix and geometry stiffness matrix. Therefore, when carrying out the stability analysis of thin cylindrical shells using the finite element methods will require, firstly, a refined non‐conforming rectangular curved cylindrical shell element RCSR4 is proposed according to the refined non‐conforming FE method, in which both the C1 and C0 weak continuity conditions are satisfied and as a result, can ensure the convergence of computation. At the same time, a refined geometrical stiffness matrix is introduced to replace the standard consistent geometrical stiffness matrix. Simple expressions of the refined constant strain matrices with adjustable constants are introduced with respect to the weak continuity conditions. Numerical examples are presented to show that the present method can indeed improve the performance and the accuracy in stability analysis. Copyright © 2001 John Wiley &amp; Sons, Ltd.

An efficient rectangular plate element

A new 12-parameter rectangular plate element is presented by use of the double set parameter method. The error in the energy norm is of order O ( h 2 ), one order higher than the commonly used Adini nonconforming element.

On the Necessity and Sufficiency of the Patch Test for Convergence of Nonconforming Finite Elements

This paper proposes a weak patch test that improves both the patch test and the generalized patch test. Under a weak superapproximation assumption, which can be obtained from certain approximation and weak continuity property, it is shown that the weak patch test is equivalent to the generalized patch test. Thus if a nonconforming finite element passes the patch test and satisfies some approximation and weak continuity property, its convergence is then guaranteed. Furthermore, the consistency term of such a nonconforming element can be proved to be of order O(h). It is shown that if a nonconforming element does not pass the patch test and a condition relative to subdividing a domain by the element is true, then one can find a family of triangulations for which the element is divergent.

A New Anisoparametric Out-of-Plane Deformable Curved Beam Element

It is known that the reduced integration, modified shape function, anisoparametric and non-conforming element can reduce the error induced by stiffness locking phenomenon in the finite element analysis. In this study, we propose new anisoparametric curved beam element. The new element based on reduced minimization theory is composed of different shape functions in each displacement field. By the substitution of this modified shape function, the unmatched coefficient that cause stiffness locking in the constraint energy is eliminated. To confirm the availability of this new model, we performed numerical tests for a simple model. As a result of numerical test, the undulate stress patterns are disappeared in static analysis, and displacements and stresses are close to exact solution. Not only in the static analysis but also in the eigen analysis of free vibrated curved beam model, this element shows successful convergent results.

Multigrid for the Wilson Mortar Element Method

AbstractIn this paper, a mortar finite element method is proposed for the Wilson nonconforming element. Multigrid method is used to solve the resulting discrete system.