Tilings of the Sphere by Congruent Quadrilaterals I:Edge Combination a^(2)bc

Abstract Edge-to-edge tilings of the sphere by congruent a^(2)bc-quadrilaterals are classified as 3 classes:(1)A 1-parameter family of quadrilateral subdivisions of the octahedron with24 tiles,and a flip modification for one special parameter;(2)a 2-parameter family of 2-layer earth map tilings with 2n tiles for each n≥3;(3)a 3-layer earth map tiling with 8n tiles for each n≥2,and two flip modifications for each odd n.The authors also describe the moduli of parameterized tilings and provide the full geometric data for all tilings.

VERTEX-DISJOINT QUADRILATERALS IN BIPARTITE GRAPHS

H. Wang considered the minimum degrees condition that G has largevertex-disjoint cycles in bipartite graphs. Motivated by this, we consider the small vertex-disjointcycles in bipartite graphs in this paper. We prove the following result: Let m ≥ 3, n ≥ 2 and k≥ 1 be three integers. Let G = (V_1,V_2;E) be a bipartite graph with |V_1| = |V_2| = n ≥ 2k+1.

A NEW PSEUDOSPECTRAL METHOD ON QUADRILATERALS

In this paper, we investigate a new pseudospectral method for mixed boundary value problems defined on quadrilaterals. We introduce a new Legendre-Gauss type interpolation and establish the basic approximation results, which play important roles in pseudospectral method for partial differential equations defined on quadrilaterals. We propose pseudospee- tral method for two model problems and prove their spectral accuracy. Numerical results demonstrate their high efficiency. The approximation results developed in this paper are also applicable to other problems defined on complex domains.

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.

Existence and Uniqueness of Nested Regular Quadrilateral Central Configurations

Two cases of the nested configurations in R3 consisting of two regular quadrilaterals are discussed. One case of them do not form central configuration, the other case can be central configuration. In the second case the existence and uniqueness of the central configuration are studied. If the configuration is a central configuration, then all masses of outside layer are equivalent, similar to the masses of inside layer. At the same time the following relation between r(the ratio of the sizes) and mass ratio b = m/m must be satisfied in which the masses at outside layer are not less than the masses at inside layer, and the solution of this kind of central configuration is unique for the given ratio (6) of masses.

Bounds of Interpolation Error for Arbitrary Narrow Quadrilateral Quasi-Wilson Element

Based on the construction of reference e le ment and bilinear transformation, a quasi-Wilson element for arbitrary narrow q uadrilateral is presented. Using the interpolation Theorem for narrow quadrilate ral isoparametric finite element and related methods, the bounds of interpolatio n error for arbitrary narrow quadrilateral quasi-Wilson element are obtained in case when the condition ρ K/h K≥σ 0】0 is not satisfied, where h K is the diameter of the element K and ρ K is the diameter of an ins cribed circle in K. The interpolation error is O(h2 K) in the L2( K)-norm and O(h K) in the H1(K) -norm provided that the in terpolated function belongs to H2(K).

Positioning of grid points in wave front construction

In view of the relative positioning problem between non-regular quadrilateral grids and regular rectangle grid nodes in the wave front construction method, concrete realization problems with four grid positioning methods （vector cross product judgment, angle sum, intersection-point, and signs comparison algorithms） in wave front construction which are commonly used in computer graphics are compared and analyzed in this paper. Based on the stability analysis of the location method, the calculation examples show that the vector cross product judgment method is faster and more accurate than other methods in the realization of the relative positioning between non-regular quadrilateral grids and regular rectangle grid nodes in wave front construction. It provides precise grid point attribute values for the next steps of migration and demigration.

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.

Influence of anchorage length and pretension on the working resistance of rock bolt based on its tensile characteristics

In coal mining roadway support design,the working resistance of the rock bolt is the key factor affecting its maximum support load.Effective improvement of the working resistance is of great significance to roadway support.Based on the rock bolt’s tensile characteristics and the mining roadway surrounding rock deformation,a mechanical model for calculating the working resistance of the rock bolt was established and solved.Taking the mining roadway of the 17102(3)working face at the Panji No.3 Coal Mine of China as a research site,with a quadrilateral section roadway,the influence of pretension and anchorage length on the working resistance of high-strength and ordinary rock bolts in the middle and corner of the roadway is studied.The results show that when the bolt is in the elastic stage,increasing the pretension and anchorage length can effectively improve the working resistance.When the bolt is in the yield and strain-strengthening stages,increasing the pretension and anchorage length cannot effectively improve the working resistance.The influence of pretension and anchorage length on the ordinary and high-strength bolts is similar.The ordinary bolt’s working resistance is approximately 25 kN less than that of the high-strength bolt.When pretension and anchorage length are considered separately,the best pretensions of the high-strength bolt in the middle of the roadway side and the roadway corner are 41.55 and 104.26 kN,respectively,and the best anchorage lengths are 1.54 and 2.12 m,respectively.The best anchorage length of the ordinary bolt is the same as that of the high-strength bolt,and the best pretension for the ordinary bolt in the middle of the roadway side and at the roadway corner is 33.51 and 85.12 kN,respectively.The research results can provide a theoretical basis for supporting the design of quadrilateral mining roadways.

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.

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.

Application of the VOF method based on unstructured quadrilateral mesh

To simulate two-dimensional free-surface flows with complex boundaries directly and accurately, a novel VOF (Volume-of-fluid) method based on unstructured quadrilateral mesh is presented. Without introducing any complicated boundary treatment or artificial diffusion, this method treated curved boundaries directly by utilizing the inherent merit of unstructured mesh in fitting curves. The PLIC (Piecewise Linear Interface Calculation) method was adopted to obtain a second-order accurate linearized reconstruction approximation and the MLER (Modified Lagrangian-Eulerian Re-map) method was introduced to advect fluid volumes on unstructured mesh. Moreover, an analytical relation for the interface’s line constant vs. the volume clipped by the interface was developed so as to improve the method’s efficiency. To validate this method, a comprehensive series of large straining advection tests were performed. Numerical results provide convincing evidences for the method’s high volume conservative accuracy and second-order shape error convergence rate. Also, a dramatic improvement on computational accuracy over its unstructured triangular mesh counterpart is checked.

Rigid graph-based three-dimension localization algorithm for wireless sensor networks

This paper investigates the node localization problem for wireless sensor networks in three-dimension space. A distributed localization algorithm is presented based on the rigid graph. Before location, the communication radius is adaptively increasing to add the localizability. The localization process includes three steps: firstly, divide the whole globally rigid graph into several small rigid blocks; secondly, set up the local coordinate systems and transform them to global coordinate system; finally, use the quadrilateration iteration technology to locate the nodes in the wireless sensor network. This algorithm has the advantages of low energy consumption, low computational complexity as well as high expandability and high localizability. Moreover, it can achieve the unique and accurate localization. Finally, some simulations are provided to demonstrate the effectiveness of the proposed algorithm.

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.

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.

An Equivalent Definition of the Crossed Cube

For the crossed cube,an equivalent definition based on the quadrilateral is given,by which we obtain the main properties of this topological architecture,and propose a procedure to find a shortest path between any two vertices in it.

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.