Feature Preserving Parameterization for Quadrilateral Mesh Generation Based on Ricci Flow and Cross Field

We propose a newmethod to generate surface quadrilateralmesh by calculating a globally defined parameterization with feature constraints.In the field of quadrilateral generation with features,the cross field methods are wellknown because of their superior performance in feature preservation.The methods based on metrics are popular due to their sound theoretical basis,especially the Ricci flow algorithm.The cross field methods’major part,the Poisson equation,is challenging to solve in three dimensions directly.When it comes to cases with a large number of elements,the computational costs are expensive while the methods based on metrics are on the contrary.In addition,an appropriate initial value plays a positive role in the solution of the Poisson equation,and this initial value can be obtained from the Ricci flow algorithm.So we combine the methods based on metric with the cross field methods.We use the discrete dynamic Ricci flow algorithm to generate an initial value for the Poisson equation,which speeds up the solution of the equation and ensures the convergence of the computation.Numerical experiments show that our method is effective in generating a quadrilateral mesh for models with features,and the quality of the quadrilateral mesh is reliable.

New Ranking of Generalized Quadrilateral Shape Fuzzy Number Using Centroid Technique

The output of the fuzzy set is reduced by one for the defuzzification procedure.It is employed to provide a comprehensible outcome from a fuzzy inference process.This page provides further information about the defuzzifica-tion approach for quadrilateral fuzzy numbers,which may be used to convert them into discrete values.Defuzzification demonstrates how useful fuzzy ranking systems can be.Our major purpose is to develop a new ranking method for gen-eralized quadrilateral fuzzy numbers.The primary objective of the research is to provide a novel approach to the accurate evaluation of various kinds of fuzzy inte-gers.Fuzzy ranking properties are examined.Using the counterexamples of Lee and Chen demonstrates the fallacy of the ranking technique.So,a new approach has been developed for dealing with fuzzy risk analysis,risk management,indus-trial engineering and optimization,medicine,and artificial intelligence problems:the generalized quadrilateral form fuzzy number utilizing centroid methodology.As you can see,the aforementioned scenarios are all amenable to the solution pro-vided by the generalized quadrilateral shape fuzzy number utilizing centroid methodology.It’s laid out in a straightforward manner that’s easy to grasp for everyone.The rating method is explained in detail,along with numerical exam-ples to illustrate it.Last but not least,stability evaluations clarify why the Gener-alized quadrilateral shape fuzzy number obtained by the centroid methodology outperforms other ranking methods.

Application of the quadrilateral area coordinate method:a new element for laminated composite plate bending problems

Recently, some new quadrilateral finite elements were successfully developed by the Quadrilateral Area Coordinate （QAC） method. Compared with those traditional models using isoparametric coordinates, these new models are less sensitive to mesh distortion. In this paper, a new displacement-based, 4-node 20-DOF （5-DOF per node） quadrilateral bending element based on the first-order shear deformation theory for analysis of arbitrary laminated composite plates is presented. Its bending part is based on the element AC-MQ4, a recent-developed high-performance Mindlin-Reissner plate element formulated by QAC method and the generalized conforming condition method; and its in-plane displacement fields are interpolated by bilinear shape functions in isoparametric coordinates. Furthermore, the hybrid post-rocessing procedure, which was firstly proposed by the authors, is employed again to improve the stress solutions, especially for the transverse shear stresses. The resulting element, denoted as AC-MQ4-LC, exhibits excellent performance in all linear static and dynamic numerical examples. It demonstrates again that the QAC method, the generalized conforming condition method, and the hybrid post-processing procedure are efficient tools for developing simple, effective and reliable finite element models.

Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The fiat shell element is constructed by combining a quadrilateral area co- ordinate method （QAC） based membrane element AGQ6- II, and a Timoshenko beam function （TBF） method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory （FSDT）, for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical re- sults show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.

Vibration of quadrilateral embedded multilayered graphene sheets based on nonlocal continuum models using the Galerkin method

Free vibration analysis of quadrilateral multilayered graphene sheets（MLGS） embedded in polymer matrix is carried out employing nonlocal continuum mechanics.The principle of virtual work is employed to derive the equations of motion.The Galerkin method in conjunction with the natural coordinates of the nanoplate is used as a basis for the analysis.The dependence of small scale effect on thickness,elastic modulus,polymer matrix stiffness and interaction coefficient between two adjacent sheets is illustrated.The non-dimensional natural frequencies of skew,rhombic,trapezoidal and rectangular MLGS are obtained with various geometrical parameters and mode numbers taken into account,and for each case the effects of the small length scale are investigated.

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.

A new method of quality improvement for quadrilateral mesh based on small polygon reconnection

In this paper, a new method of topological cleanup for quadrilateral mesh is presented. The method first selects a patch of mesh around an irregular node. It then seeks the best connection of the selected patch according to its irregular valence using a new topological operation： small polygon reconnection （SPR）. By replacing the original patch with an optimal one that has less irregular valence, mesh quality can be improved. Three applications based on the proposed approach are enumerated： （1） improving the quality of a quadrilateral mesh, （2） converting a triangular mesh to a quadrilateral one, and （3） adapting a triangle generator to a quadrilateral one. The presented method is highly effective in all three applications.

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.

Quadrilateral 2D linked-interpolation finite elements for micropolar continuum

Quadrilateral finite elements for linear micropolar continuum theory are developed using linked interpolation.In order to satisfy convergence criteria,the newly presented finite elements are modified using the Petrov-Galerkin method in which different interpolation is used for the test and trial functions.The elements are tested through four numerical examples consisting of a set of patch tests,a cantilever beam in pure bending and a stress concentration problem and compared with the analytical solution and quadrilateral micropolar finite elements with standard Lagrangian interpolation.In the higher-order patch test,the performance of the first-order element is significantly improved.However,since the problems analysed are already describable with quadratic polynomials,the enhancement due to linked interpolation for higher-order elements could not be highlighted.All the presented elements also faithfully reproduce the micropolar effects in the stress concentration analysis,but the enhancement here is negligible with respect to standard Lagrangian elements,since the higher-order polynomials in this example are not needed.

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.

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.

Automated quadrilateral mesh generation for digital image structures

With the development of advanced imaging technology, digital images are widely used. This paper proposes an automatic quadrilateral mesh generation algorithm for multi-colour imaged structures. It takes an original arbitrary digital image as an input for automatic quadrilateral mesh generation, this includes removing the noise, extracting and smoothing the boundary geometries between different colours, and automatic all-quad mesh generation with the above boundaries as constraints. An application example is provided to demonstrate the usefulness and effectiveness of the proposed approach.

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.

Quadrilateral plate fractures of the acetabulum:Classification,approach,implant therapy and related research progress

The quadrilateral plate(QP)is an essential structure of the inner wall of the acetabulum,an important weight-bearing joint of the human body,which is often involved in acetabular fractures.The operative exposure,reduction and fixation of QP fractures have always been the difficulties in orthopedics due to the special morphological structure and anatomical features of the QP.Fortunately,there have been many effective methods and instruments developed for QP exposure,reduction and fixation by virtue of the combined efforts of numerous orthopedists.At the same time,each method presents with its own advantages and disadvantages,resulting in different prognoses.It is necessary to have a thorough understanding of the anatomy,radiology and fixation techniques of the QP in terms of patient prognosis optimization.In this paper,the anatomical features,definition and classification of QP,operative approach selection,implant internal fixation methods and efficacy were reviewed.

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.

C1-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations

This paper is devoted to Professor Benyu Guo's open question on the C1-conforming quadrilateral spectral element method for fourth-order equations which has been endeavored for years. Starting with generalized Jacobi polynomials on the reference square, we construct the C1-conforming basis functions using the bilinear mapping from the reference square onto each quadrilateral element which fall into three categories-interior modes, edge modes, and vertex modes. In contrast to the triangular element, compulsively compensatory requirements on the global C1-continuity should be imposed for edge and vertex mode basis functions such that their normal derivatives on each common edge are reduced from rational functions to polynomials, which depend on only parameters of the common edge. It is amazing that the C1-conforming basis functions on each quadrilateral element contain polynomials in primitive variables, the completeness is then guaranteed and further confirmed by the numerical results on the Petrov-Galerkin spectral method for the non-homogeneous boundary value problem of fourth-order equations on an arbitrary quadrilateral. Finally, a C1-conforming quadrilateral spectral element method is proposed for the biharmonic eigenvalue problem, and numerical experiments demonstrate the effectiveness and efficiency of our spectral element method.

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 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.

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.