A novel four-node quadrilateral element with continuous nodal stress

Formulation and numerical evaluation of a novel four-node quadrilateral element with continuous nodal stress（Q4-CNS）are presented.Q4-CNS can be regarded as an improved hybrid FE-meshless four-node quadrilateral element（FE-LSPIM QUAD4）, which is a hybrid FE-meshless method.Derivatives of Q4-CNS are continuous at nodes, so the continuous nodal stress can be obtained without any smoothing operation.It is found that,compared with the standard four-node quadrilateral element（QUAD4）,Q4- CNS can achieve significantly better accuracy and higher convergence rate.It is also found that Q4-CNS exhibits high tolerance to mesh distortion.Moreover,since derivatives of Q4-CNS shape functions are continuous at nodes,Q4-CNS is potentially useful for the problem of bending plate and shell models.

On the Error Estimate of h-Convergence in Quadrilateral Elements

To provide a iheoreiical basis for h-type .finire elemenl analysis with quadrilateralelements, in the present paper, the h-convergence of quadrilateral elements is established. whose related lemmas and theorems being presented, and therefore, theerror estimate problems are investigated.

A NEW ELEMENT FOR THIN PLATE OF BENDING WITH CURVILINEAR BOUNDARY—CURVILINEAR BOUNDARY QUADRILATERAL ELEMENT

This paper presents a curvilinear boundary quadrilateral element for the problem of thin plate of bending with curvilinear boundary. A coordinate transformation of two dimensions is performed in the calculation of FEM. The introduction of an additional stiffness matrix based on the generalized variational principles results in high accuracy and less computation time. The numerical results agree with the analytical solution very well.

A 17-node quadrilateral spline finite element using the triangular area coordinates

Isopaxametric quadrilateral elements are widely used in the finite element method. However, they have a disadvantage of accuracy loss when elements are distorted. Spline functions have properties of simpleness and conformality. A 17onode quadrilateral element has been developed using the bivaxiate quaxtic spline interpolation basis and the triangular area coordinates, which can exactly model the quartic displacement fields. Some appropriate examples are employed to illustrate that the element possesses high precision and is insensitive to mesh distortions.

INCOMPATIBLE CURVED QUADRILATERAL PLATE BENDING ELEMENT WITH 12-DEGREES OF FREEDOM

This paper presents a new curved quadrilateral plate element with 12-degree freedom by the exact element method[1]. The method can be used to arbitrary non-positive and positive definite partial differential equations without variation principle. Using this method, the compatibility conditions between element can be treated very easily, if displacements and stress resultants are continuous at nodes between elements. The displacements and stress resultants obtained by the present method can converge to exact solution and have the second order convergence speed. Numerical examples are given at the end of this paper, which show the excellent precision and efficiency of the new element.

A cubic quadrilateral spline element with concave shapes

Basic requirement for applying isoparametric element is that the element has to be convex and no violent distortion is allowed. In this paper, a cubic quadrilateral spline element with 12 nodes has been developed using the triangular area coordinates and the B-net method, which can exactly model the cubic field for quadrilateral element with both convex and concave shapes. Neither mapping nor coordinate transformation is required and the spline element can obtain high accuracy solutions and insensitive to mesh distortions.

Development of quadrilateral spline thin plate elements using the B-net method

The quadrilateral discrete Kirchhoff thin plate bending element DKQ is based on the isoparametric element Q8, however, the accuracy of the isoparametric quadrilateral elements will drop significantly due to mesh distortions. In a previous work, we constructed an 8-node quadrilateral spline element L8 using the triangular area coordinates and the B- net method, which can be insensitive to mesh distortions and possess the second order completeness in the Cartesian co- ordinates. In this paper, a thin plate spline element is devel- oped based on the spline element L8 and the refined tech- nique. Numerical examples show that the present element indeed possesses higher accuracy than the DKQ element for distorted meshes.

THE ADAPTIVE FINITE ELEMENT REMESHING IN LARGE DEFORMATION OF METAL FORMING

The adaptive remeshing technique for quadrilateral elements consists of modules thetrigger of remeshing, the new mesh generation, adaptive refinement and interpolationof field variables. The new adaptive mesh genemtion is the key problem. First, acoarse mesh is created by using 'loop algorithm'. Subsequent local mesh adaptiverefinement is performed based on effective strain. Finally, a typical example of upset-ting is given to test efficient of techniques, from which it is verified that the remeshingalgorithm developed here exhibits good performance and has high accuracy.

Vibration analysis of two-dimensional structures using micropolar elements

Based on the micropolar theory(MPT),a two-dimensional(2 D)element is proposed to describe the free vibration response of structures.In the context of the MPT,a 2 D formulation is developed within the ABAQUS finite element software.The user-defined element(UEL)subroutine is used to implement a micropolar element.The micropolar effects on the vibration behavior of 2 D structures with arbitrary shapes are studied.The effect of micro-inertia becomes dominant,and by considering the micropolar effects,the frequencies decrease.Also,there is a considerable discrepancy between the predicted micropolar and classical frequencies at small scales,and this difference decreases when the side length-to-length scale ratio becomes large.

A Modified Paving Approach and Automatic Mesh Grading Method to Quadrilateral Mesh Generation

A modified paving technique for automatic generation of all-quadrilateral mesh fromarbitrary 2-D geometry is presented. The generated mesh elementS are nearly square andperpendicular to boundaries. Aner the nodes and elementS formation is completed. a fully automaticgrading method is applied to increase the accuracy and reliability of engineering analysis. In thispaper, we mainly describe the theory of mathematical algorithm and present some examples ofautomatically generated mesh.

Developing Serpent-Type Wave Generators to Create Solitary Wave Simulations with BEM

Developing serpent-type wave generators to generate solitary waves in a 3D-basin was investigated in this study. Based on the Lagrangian description with time-marching procedures and finite differences of the time derivative, a 3D multiple directional wave basin with multidirectional piston wave generators was developed to simulate ocean waves by using BEM with quadrilateral elements, and to simulate wave-caused problems with fully nonlinear water surface conditions. The simulations of perpendicular solitary waves were conducted in the first instance to verify this scheme. Furthermore, the comparison of the waveform variations confirms that the estimation of 3D solitary waves is a feasible scheme.

A class of nonconforming quadrilateral finite elements for incompressible flow

This paper focuses on the low-order nonconforming rectangular and quadrilateral finite elements approximation of incompressible flow.Beyond the previous research works,we propose a general strategy to construct the basis functions.Under several specific constraints,the optimal error estimates are obtained,i.e.,the first order accuracy of the velocities in H1-norm and the pressure in L2-norm,as well as the second order accuracy of the velocities in L2-norm.Besides,we clarify the differences between rectangular and quadrilateral finite element approximation.In addition,we give several examples to verify the validity of our error estimates.

C^1 conforming quadrilateral finite elements with complete secondorder derivatives on vertices and its application to Kirchhoff plates

The classical problem of the construction of C^1 conforming single-patch quadrilateral finite elements has been solved in this investigation by using the blending function interpolation method.In order to achieve the C^1 conformity on the interfaces of quadrilateral elements,complete second-order derivatives are used at the element vertices,and the information of geometrical mapping is also considered into the construction of shape functions.It is found that the shape functions and the polynomial spaces of the present elements vary with element shapes.However,the developed quadrilateral elements are at least third order for general quadrilateral shapes and fifth order for rectangular shapes.Therefore,very fast convergence can be achieved.A promising feature of the present elements is that they can be used in cooperation with those high-precision rectangular and triangular elements.Since the present elements are over conforming on element vertices,an approach for handling problems of material discontinuity is also proposed.Numerical examples of Kirchhoff plates are employed to demonstrate the computational performance of the present elements.

NONCONFORMING QUADRILATERAL ROTATED ELEMENT FOR REISSNER-MINDLIN PLATE

In this paper, we extend two rectangular elements for Reissner-Mindlin plate [9] to the quadrilateral case. Optimal H and L error bounds independent of the plate hickness are derived under a mild assumption on the mesh partition.

MATHEMATICAL ANALYSIS FOR QUADRILATERAL ROTATED Q_1 ELEMENT Ⅲ: THE EFFECT OF NUMERICAL INTEGRATION

This is the third part of the paper for the rotated Q1 nonconforming element on quadrilateral meshes for general second order elliptic problems. Some optimal numerical formulas are presented and analyzed. The novelty is that it includes a formula with only two sampling points which excludes even a Q1 unisolvent set. It is the optimal numerical integration formula over a quadrilateral mesh with least sampling points up to now.

MATHEMATICAL ANALYSIS FOR QUADRILATERAL ROTATED Q_1 ELEMENT Ⅱ: POINCARE INEQUALITY AND TRACE INEQUALITY

This is the second part of the paper for the mathematical study of nonconforming rotated Q1 element (NRQ1 hereafter) on arbitrary quadrilateral meshes. Some Poincare Inequalities are proved without assuming the quasi-uniformity of the mesh subdivision. A discrete trace inequality is also proved.

Some Numerical Quadrature Schemes of a Non-conforming Quadrilateral Finite Element

Numerical quadrature schemes of a non-conforming finite element method for general second order elliptic problems in two dimensional （2-D） and three dimensional （3-D） space are discussed in this paper. We present and analyze some optimal numerical quadrature schemes. One of the schemes contains only three sampling points, which greatly improves the efficiency of numerical computations. The optimal error estimates are derived by using some traditional approaches and techniques. Lastly, some numerical results are provided to verify our theoretical analysis.

GENERALIZED DIFFERENCE METHODS ON ARBITRARYQUADRILATERAL NETWORKS

This paper considers the generalized difference methods on arbitrary networks for Poisson equations. Convergence order estimates are proved based on some a priori estimates. A supporting numerical example is provided.

On the Full C_(1)-Q_(k) Finite Element Spaces on Rectangles and Cuboids

We study the extensions of the Bogner-Fox-Schmit element to the whole family of Q_(k) continuously differentiable finite elements on rectangular grids,for all k≥3,in 2D and 3D.We show that the newly defined C_(1) spaces are maximal in the sense that they contain all C_(1)-Q_(k) functions of piecewise polynomials.We give examples of other extensions of C_(1)-Q_(k) elements.The result is consistent with the Strang’s conjecture(restricted to the quadrilateral grids in 2D and 3D).Some numerical results are provided on the family of C_(1) elements solving the biharmonic equation.