Nonconforming Quadrilateral Element Research Articles

Non-conforming and conforming five-node quadrilateral graded finite elements

This study evaluates the performance of conforming and non-conforming transition graded finite elements dealing with graded materials. Two-dimensional 5-node non-conforming quadrilateral element (Q5) is used to transition from one large 4-node linear quadrilateral to two small 4-node linear quadrilateral elements (Q4). In addition, a conforming Q5 element is used in between one Q4 and one 8-node quadratic elements (Q8). A single element test of Q5 graded element is done. Then, the accuracy and convergence study of Q5 graded element are examined. We found that the use of Q5 transition elements improved computational efficiency while maintaining the accuracy of the solutions.

Effect of numerical integration on a new rotated nonconforming quadrilateral element

A new nonparametric nonconforming quadrilateral finite element is used to approximate the general second-order elliptic problem in two dimensions. Some optimal numerical integration formulas are presented and analyzed. These formulas are derived on a reference quadrilateral which can be linearly mapped to a physical quadrilateral. The novelty of these formulas is that they only involve two quadrature nodes which excludes even a Q1 unisolvent set, and they are not required to be exact for all the shape functions. Numerical tests show that the presented quadrature rules can also be used coupled with other low-order elements.

A superconvergent nonconforming quadrilateral spline element for biharmonic equation using the B-net method

In this paper, we construct a new nonconforming quadrilateral element with 12 degrees of freedom to solve the biharmonic problems. $${\mathscr {T}}_{h}$$ is a triangulated quadrangulation of the domain $$\varOmega $$ . For a quadrilateral element $$Q_{T}$$ , the finite element space, which contains $${\mathbb {P}}_{3}(Q_{T})$$ , is a subspace of the bivariate spline space $${\mathbf {S}}_{3}^{1}(Q_{T})$$ . The degrees of freedom are chosen as the four point values, the four edge integrals of the shape functions and the edge integrals of their normal derivatives such that the weak continuity between elements can be satisfied. Accordingly, we explicitly establish 12 spline interpolation bases in the B-net form. Error estimates are given with optimal convergence order in both discrete $$H^{2}$$ and $$H^{1}$$ seminorms. The proposed element NCQS12 can get the superconvergence results with theoretical proof when $${\mathscr {T}}_{h}$$ is an uniform parallelogram mesh. Some degenerate meshes are considered subsequently. Finally, we do some numerical experiments to verify the theoretical analysis. Numerical results show that the proposed element performs well over the asymptotically regular parallelogram meshes, which is same as over the uniform parallelogram meshes.

A simple and robust quadrilateral non-conforming element with a special 5-point quadrature

Abstract A 4-node, 8-DOF non-conforming quadrilateral element with four internal modes, denoted as iQ8, is developed. It takes the four edges' midpoints of the linear element Q4 as the internal virtual nodes, whose shape functions are the same as those of the mid-side nodes of the quadratic element Q8. In order to deal with the rank deficiency of the new element stiffness matrix under 2 × 2 Gaussian quadrature, a new reduced 5-point integration scheme is constructed. By both the eigenvalue analysis of the element stiffness matrix and the example for the patch test, the weight of the central integration point is determined to a value close to 4, correspondingly the 5-point quadrature converges to 1-point quadrature. The non-central integration points are far away from the element domain, which is much different from conventional quadratures. Finally, some typical numerical examples are tested, and the results show that, except for only passing the weak patch test, the new element achieves the best performance of other 4-node quadrilateral elements in the aspects of anti-distortion, being free of trapezoidal locking, and exactness in pure bending.

A nonconforming quadrilateral finite element approximation to nonlinear schrödinger equation

In this article, a nonconforming quadrilateral element (named modified quasi-Wilson element) is applied to solve the nonlinear schrödinger equation (NLSE). On the basis of a special character of this element, that is, its consistency error is of order O(h3) for broken H1-norm on arbitrary quadrilateral meshes, which is two order higher than its interpolation error, the optimal order error estimate and superclose property are obtained. Moreover, the global superconvergence result is deduced with the help of interpolation postprocessing technique. Finally, some numerical results are provided to verify the theoretical analysis.

Stable cheapest nonconforming finite elements for the Stokes equations

We introduce two pairs of stable cheapest nonconforming finite element space pairs to approximate the Stokes equations. One pair has each component of its velocity field to be approximated by the P1 nonconforming quadrilateral element while the pressure field is approximated by the piecewise constant function with globally two-dimensional subspaces removed: one removed space is due to the integral mean-zero property and the other space consists of global checker-board patterns. The other pair consists of the velocity space as the P1 nonconforming quadrilateral element enriched by a globally one-dimensional macro bubble function space based on DSSY (Douglas–Santos–Sheen–Ye) nonconforming finite element space; the pressure field is approximated by the piecewise constant function with mean-zero space eliminated. We show that two element pairs satisfy the discrete inf–sup condition uniformly. And we investigate the relationship between them. Several numerical examples are shown to confirm the efficiency and reliability of the proposed methods.

A piecewiseP2-nonconforming quadrilateral finite element

We introduce a piecewise P 2 -nonconforming quadrilateral finite element. First, we decompose a convex quadrilateral into the union of four triangles divided by its diagonals. Then the finite element space is defined by the set of all piecewise P 2 -polynomials that are quadratic in each triangle and continuously differentiable on the quadrilateral. The degrees of freedom (DOFs) are defined by the eight values at the two Gauss points on each of the four edges plus the value at the intersection of the diagonals. Due to the existence of one linear relation among the above DOFs, it turns out the DOFs are eight. Global basis functions are defined in three types: vertex-wise, edge-wise, and element-wise types. The corresponding dimensions are counted for both Dirichlet and Neumann types of elliptic problems. For second-order elliptic problems and the Stokes problem, the local and global interpolation operators are defined. Also error estimates of optimal order are given in both broken energy and L 2 (Ω ) norms. The proposed element is also suitable to solve Stokes equations. The element is applied to approximate each component of velocity fields while the discontinuous P 1 -nonconforming quadrilateral element is adopted to approximate the pressure. An optimal error estimate in energy norm is derived. Numerical results are shown to confirm the optimality of the presented piecewise P 2 -nonconforming element on quadrilaterals.

A refined nonconforming quadrilateral element for couple stress/strain gradient elasticity

AbstractC0−1 patch test (Int. J. Numer. Meth. Engng 2004; 61:433–454) proposed by Soh and Chen is a reliable method to ensure convergence of nonconforming finite element for the couple stress/strain gradient elasticity. The C0−1 patch test function is a complete quadratic polynomial that satisfies the equilibrium equations. To pass the C0−1 patch test, the element displacement functions used to calculate strains must satisfy C0 continuity (or weak C0 continuity) and quadratic completeness. In this paper, a 24‐DOF (degrees of freedom) quadrilateral element (CQ12+RDKQ) for the couple stress/strain gradient elasticity is developed by combining the refined thin plate element RDKQ and the nonconforming element CQ12. The element RDKQ, which satisfies weak C1 continuity, is used to calculate strain gradients, whereas strains are computed by the element CQ12, which is established based on an extended variational functional and satisfies weak C0 continuity and quadratic completeness. Numerical examples show that the element (CQ12+RDKQ) passes the C0−1 patch test and it is also more efficient than the existing available triangular and quadrilateral elements in stress concentration problems with size effects. Copyright © 2010 John Wiley & Sons, Ltd.

TWO NONCONFORMING QUADRILATERAL ELEMENTS FOR THE REISSNER–MINDLIN PLATE

We construct two low order nonconforming quadrilateral elements for the Reissner–Mindlin plate. The first one consists of a modified nonconforming rotated Q1 element for one component of the rotation and the standard four-node isoparametric element for the other component as well as for the the approximation of the transverse displacement, a modified rotated Raviart–Thomas interpolation operator is employed as the shear reduction operator. The second differs from the first only in the approximation of the rotation, which employs the modified rotated Q1 element for both components of the rotation, and a jump term accounting the discontinuity of the rotation approximation is included in the variational formulation. Both elements give optimal error bounds uniform in the plate thickness with respect to the energy norm as well as the L2 norm.

P1-Nonconforming Quadrilateral Finite Element Methods for Second-Order Elliptic Problems

A P1 -nonconforming quadrilateral finite element is introduced for second-order elliptic problems in two dimensions. Unlike the usual quadrilateral nonconforming finite elements, which contain quadratic polynomials or polynomials of degree greater than 2, our element consists of only piecewise linear polynomials that are continuous at the midpoints of edges. One of the benefits of using our element is convenience in using rectangular or quadrilateral meshes with the least degrees of freedom among the nonconforming quadrilateral elements. An optimal rate of convergence is obtained. Also a nonparametric reference scheme is introduced in order to systematically compute stiffness and mass matrices on each quadrilateral. An extension of the P1 -nonconforming element to three dimensions is also given. Finally, several numerical results are reported to confirm the effective nature of the proposed new element.

A parallel iterative Galerkin method based on nonconforming quadrilateral elements for second-order partial differential equations

A parallel iterative Galerkin method based on domain decomposition technique with nonconforming quadrilateral finite elements will be analyzed for second-order elliptic equations subject to the Robin boundary condition. Optimal order error estimates are derived with respect to a broken H 1-norm and L 2-norm. Applications to time-dependent problems will be considered. Some numerical experiments supporting the theoretical results will be given. This paper is to extend the work in [J. Douglas Jr., J.E. Santos, D. Sheen, X. Ye, Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems, Mathematical Modelling and Numerical Analysis, RAIRO, Modél. Math. Anal. Numér. 33 (4) (1999) 747] to the non-self-adjoint case of second-order equations including the term b ·∇u . We suppose that uniformly ellipticity holds. Hence the arguments in (loc. cit.) may be applied, word for word. So some proofs will be omitted.

A streamline-diffusion method for nonconforming finite element approximations applied to convection-diffusion problems

We consider a nonconforming streamline-diffusion finite element method for solving convection-diffusion problems. The theoretical and numerical investigation for triangular and tetrahedral meshes recently given by John, Maubach and Tobiska has shown that the usual application of the SDFEM gives not a sufficient stabilization. Additional parameter dependent jump terms have been proposed which preserve the same order of convergence as in the conforming case. The error analysis has been essentially based on the existence of a conforming finite element subspace of the nonconforming space. Thus, the analysis can be applied for example to the Crouzeix/Raviart element but not to the nonconforming quadrilateral elements proposed by Rannacher and Turek. In this paper, parameter free new jump terms are developed which allow to handle both the triangular and the quadrilateral case. Numerical experiments support the theoretical predictions.

Size effect in beams with rounded-tip V-notch

Stresses and strains in a beam with a single-edge rounded-tip V-notch subjected to bending were analyzed by the method of finite elements by using nonconforming quadrilateral elements under the assumption that the material is linearly elastic. We studied the influence of the depth of the notcha, its radius of curvature ρ, the angle of the notch ω, and the height of the beamh on the stress intensity factorKt. It is established that the relative depth of the notch ζ=a/h causes practically no influence if the notch angle ω lies within the range 0°–135°. The influence of the notch rapidly weakens as the relative radius of curvature ρd=ρ/h increases from 0.01 to 1.0 and becomes insignificant for ρd ≤ 1. For small ρd (ρd ≤ 1), the quantityKt attains its maximum value provided that ζ lies within the range 0.1–0.3. As the height of the beam increases, the quantityKt first remains practically constant, then increases, and finally, ash infinitely increases, approaches a certain constant value.

Use of incompatible displacement modes in a finite element model to analyze the dynamic behavior of unreinforced masonry panels

A finite element model, where a non-conforming quadrilateral element is utilized, capable of analyzing the dynamic nonlinear behavior in a biaxial stress field of unreinforced masonry panels is presented. For the material, the linear elastic-plastic constitutive law is adopted. The formulation for the linear element and the extension for the linear elastic-plastic element are proposed. The solution is carried out by a direct step by step integration procedure in time domain, based on the Newmark method of the equilibrium equations, inclusive of inertial and damping actions, the latter evaluated using the Rayleigh hypothesis. The procedure was implemented in a computer program and verified by the analysis of an unreinforced masonry shear panel, the dynamic behavior of which was analyzed experimentally [1, 2]. The comparisons between the numeric results and laboratory test measurements show good agreement, proving the good performance of the non-conforming quadrilateral element also for time-dependent and markedly nonlinear analyses. In addition, the case of Parma Cathedral Bell-Tower subjected to a dynamic excitation available in literature, was analyzed using the proposed model. The same case was approached by a reliable finite element code [3], using quadratic serendipity elements and a more dense mesh than in the previous analysis. The results, in terms of kinematic parameters, stress and strain fields, etc. obtained by the two models, agree, proving that the use of a non-conforming quadrilateral element leads to analyses which are computationally economical and simple to use in input.