ANALYSIS OF BENDING, VIBRATION AND STABILITY FOR THIN PLATE ON ELASTIC FOUNDATION BY THE MULTIVARIABLE SPLINE ELEMENT METHOD

In this paper, the bicubic splines in product form are used to construct the multi-field functions for bending moments, twisting moment and transverse displacement of the plate on elastic foundation. The multivariable spline element equations are derived, based on the mixed variational principle. The analysis and calculations of bending, vibration and stability of the plates on elastic foundation are presented in the paper. Because the field functions of plate on elastic foundation are assumed independently, the precision of the field variables of bending moments and displacement is high.

A 3D pyramid spline element

In this paper,a 13-node pyramid spline element is derived by using the tetrahedron volume coordinates and the B-net method,which achieves the second order completeness in Cartesian coordinates.Some appropriate examples were employed to evaluate the performance of the proposed element.The numerical results show that the spline element has much better performance compared with the isoparametric serendipity element Q20 and its degenerate pyramid element P13 especially when mesh is distorted,and it is comparable to the Lagrange element Q27.It has been demonstrated that the spline finite element method is an efficient tool for developing high accuracy elements.

Two 8-node quadrilateral spline elements by B-net method

Isoparametric quadrilateral elements are widely used in the finite element method, but the accuracy of the isoparametric quadrilateral elements will drop obviously deteriorate due to mesh distortions. Spline functions have some properties of simplicity and conformality. Two 8-node quadrilateral elements have been developed using the trian- gular area coordinates and the B-net method, which can ex- actly model the quadratic field for both convex and concave quadrangles. Some appropriate examples are employed to evaluate the performance of the proposed elements. The nu- merical results show that the two spline elements can obtain solutions which are highly accurate and insensitive to mesh distortions.

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.

Construction of n-sided polygonal spline element using area coordinates and B-net method

In general, triangular and quadrilateral elements are commonly applied in two-dimensional finite element methods. If they are used to compute polycrystalline materials, the cost of computation can be quite significant. Polygonal elements can do well in simulation of the materials behavior and provide greater flexibility for the meshing of complex geometries. Hence, the study on the polygonal element is a very useful and necessary part in the finite element method. In this paper, an n-sided polygonal element based on quadratic spline interpolant, denoted by PS2 element, is presented using the triangular area coordinates and the B-net method. The PS2 element is conforming and can exactly model the quadratic field. It is valid for both convex and non-convex polygonal element, and insensitive to mesh distortions. In addition, no mapping or coordinate transformation is required and thus no Jacobian matrix and its inverse are evaluated. Some appropriate examples are employed to evaluate the performance of the proposed element.

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.

A B-spline active contour model based on finite element method

A B-spline active contour model based on finite element method is presented, into which the advantages of a B-spline active contour attributing to its fewer parameters and its smoothness is built accompanied with reduced computational complexity and better numerical stability resulted from the finite element method. In this model, a cubic B-spline segment is taken as an element, and the finite element method is adopted to solve the energy minimization problem of the B-spline active contour, thus to implement image segmentation. Experiment results verify that this method is efficient for B-spline active contour, which attains stable, accurate and faster convergence.

INFINITE SPLINE BOUNDARY ELEMENT METHOD AND ITS APPLICATION TO STRUCTURE-FOUNDATION INTERACTION

In view of the infinity behaviors of 3-D Kelvin solution, we constructed an infinite spline boundary element which has fine precision in the analysis of the half space foundation subjected to uniform pressure on the circular domain. We also analysed a square plate resting on elastic half space foundation. The results indicate that this model not only fits for the coupled analysis of foundation and structures but also has the advantage of fewer degrees of freedom and fine precision.

THE CONSTRUCTION OF WAVELET-BASED TRUNCATED CONICAL SHELL ELEMENT USING B-SPLINE WAVELET ON THE INTERVAL

Based on B-spline wavelet on the interval （BSWI）, two classes of truncated conical shell elements were constructed to solve axisymmetric problems, i.e. BSWI thin truncated conical shell element and BSWI moderately thick truncated conical shell element with independent slopedeformation interpolation. In the construction of wavelet-based element, instead of traditional polynomial interpolation, the scaling functions of BSWI were employed to form the shape functions through the constructed elemental transformation matrix, and then construct BSWI element via the variational principle. Unlike the process of direct wavelets adding in the wavelet Galerkin method, the elemental displacement field represented by the coefficients of wavelets expansion was transformed into edges and internal modes via the constructed transformation matrix. BSWI element combines the accuracy of B-spline function approximation and various wavelet-based elements for structural analysis. Some static and dynamic numerical examples of conical shells were studied to demonstrate the present element with higher efficiency and precision than the traditional element.

Vibration Analysis of Beams by Spline Finite Element

In this paper,the spline finite element method is developed to investigate free vibration problems of beams.The cubic B-spline functions are used to construct the displacement field.The assembly of elements and the introduction of boundary conditions follow the standard finite element procedure.The results under various boundary conditions are compared with those obtained by the exact method and the finite difference method.It shows that the results are in excellent agreement with the analytical results and much more accurate than the results obtained by the finite difference method,especially for higher order modes.

A Finite Element Cable Model and Its Applications Based on the Cubic Spline Curve

For accurate prediction of the deformation of cable in the towed system, a new finite element model is presented that provides a representation of both the bending and torsional effects. In this paper, the cubic spline interpolation function is applied as the trial solution. By using a weighted residual approach, the discretized motion equations for the new finite element model are developed. The model is calculated with the computation program complier by Matlab. Several numerical examples are presented to illustrate the numerical schemes. The results of numerical simulation are stable and valid, and consistent with the mechanical properties of the cable. The model can be applied to kinematics analysis and the design of ocean cable, such as mooring lines, towing, and ROV umbilical cables.

ELASTIC ANALYSIS OF ORTHOTROPIC PLANE PROBLEMS BY THE SPLINE FICTITIOUS BOUNDARY ELEMENT METHOD

Non-singular fictitious boundary integral equations for orthotropic elastic plane problems were deduced according to boundary conditions by the techniques of singular-points-outside-domain. Then the unknown fictitious load functions along the fictitious boundary were expressed in terms of basic spline functions, and the boundary-segment-least-squares method was proposed to eliminate the boundary residues obtained. By the above steps, numerical solutions to the integral equations can be achieved. Numerical examples are given to show the accuracy and efficiency of the proposed method.

Stability Analysis of Spatial Cubic Spline Geometric Nonlinear Beam Element Considering the Second-Order Effect

To analyze the stability problem of spatial beam structure more accurately,a spatial cubic spline geometric nonlinear beam element was proposed considering the second-order effect.The deformation field was built with cubic spline function,and its curvature degree of freedom(DOF) was eliminated by static condensation method.Then we got the geometric nonlinear stiffness matrix of the new spatial two-node Euler-Bernoulli beam element.Several examples proved calculation accuracy of the critical load by meshing a bar to one element using the method of this paper was equivalent to mesh a bar to 3 or 4 traditional nonlinear beam elements.

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.

Study on spline wavelet finite-element method in multi-scale analysis for foundation

A new finite element method （FEM） of B-spline wavelet on the interval （BSWI） is proposed. Through analyzing the scaling functions of BSWI in one dimension, the basic formula for 2D FEM of BSWI is deduced. The 2D FEM of 7 nodes and 10 nodes are constructed based on the basic formula. Using these proposed elements, the multiscale numerical model for foundation subjected to harmonic periodic load, the foundation model excited by external and internal dynamic load are studied. The results show the pro- posed finite elements have higher precision than the tradi- tional elements with 4 nodes. The proposed finite elements can describe the propagation of stress waves well whenever the foundation model excited by extemal or intemal dynamic load. The proposed finite elements can be also used to con- nect the multi-scale elements. And the proposed finite elements also have high precision to make multi-scale analysis for structure.

B-Spline高阶元方法在三维水弹性力学中的应用

本文简要叙述了Ｂ－Ｓｐｌｉｎｅ样条函数的基本性质；将二元高阶Ｂ－Ｓｐｌｉｎｅ函数与Gａｌｅｒｋｉｎ方法相结合，形成了任意光滑空间曲面的二元高阶 Ｂ－ Ｓｐｌｉｎｅ函数簇解析表达方法； 讨论了 Ｂ－ Ｓｐｌｉｎｅ函数在三维水弹性力学边界积分方程中应用的一些数值计算问题。

浮动花键接触有限元模型及不对中接触特性研究

针对浮动花键进行有限元建模,分析不同扭矩、不同齿侧间隙等对航空花键接触特性的影响。结果表明:花键Mises应力值、齿面压力、整体变形都随着扭矩的增大,近似线性增大;随着不对中角度的增大,齿侧间隙大的花键的Mises应力增速快;齿侧间隙对花键的Mises应力及齿面压力的影响是非线性变化。

DYNAMIC RESPONSE OF ELASTIC RECTANGULAR PLATES BY SPLINE STATE VARIABLE METHOD

Application of spline element and state space method for analysis of dynamic response of elastic rectangular plates is presented. The spline element method is used for space domain and the state space method in control theory of system is used for time domain. A state variable recursive scheme is developed, then the dynamic response of structure can he calculated directly. Several numerical examples are given. The results which are presented to demonstrate the accuracy and efficiency of the present method are quite satisfactory.

电磁场B样条间断有限元方法

在计算电磁学领域,时域间断有限元算法(DGTD)一般基于模型空间的不规则网格划分和单元上高阶多项式插值计算。同样的插值阶数,二维空间四边形网格划分比三角形网格划分具有更少的自由度和更高的计算效率。然而,传统基于等参变换和多项式张量积插值的基函数空间在四边形单元上仅具有低阶完备性,且稳定性和精度受网格畸变影响较大。为此,提出了一种基于不规则四边形网格的高阶B样条插值DGTD方法,用于Maxwell方程的求解。文章采用的B样条基不仅具有高阶多项式空间的插值完备性,而且完全消除了单元内部自由度。此外,Maxwell方程离散系统的各系数矩阵还具有精确的解析形式。使用该方法分析腔体的本征模和楔形体的电磁散射,结果表明,相较于COMSOL软件最大允许时间步长提高2.5倍,计算所需未知量减少25%,证实了本文算法的高稳定性和高精度特点。

Bézier和B-spline曲线的绝对结点坐标列式有限元离散方法

CAD系统中广泛采用Bézier,B-spline描述复杂的几何形状,而传统有限元则采用完全不同的插值函数来描述分析模型,对几何形状的不同描述导致由CAD模型转换为有限元模型非常困难,且带来模型上的不一致。研究Bézier和B-spline曲线离散为绝对结点坐标列式(Absolute nodal coordinate formulation,ANCF)有限单元的方法,建立Bézier和B-spline曲线与ANCF索单元之间的线性转换关系,实现了两者之间的自动转换,从而为整合CAD和CAA系统提供一种方法,表明ANCF单元可完全涵盖Bézier曲线的特性,Bézier表达其实是ANCF表达的特例,采用ANCF单元可以精确表达Bézier和B-spline曲线。同时研究了B-spline曲线的连续性与ANCF单元结点之间的连续性关系,B-spline采用节点重复度控制所定义图形的连续性,ANCF单元因为采用位置和位置导数作为结点坐标可以自动保证C1连续,C2连续性要求实际上是在单元之间添加曲率相等的约束方程,从而实现消除一个位置导数坐标;对于3阶曲线,证明C3连续性要求下任意两个ANCF单元均可以合并为一个大的单元。进而得出在实际转换和分析中不必通过添加约束方程得到不同的连续性,可以直接由B-spline表达得到不同连续性要求的ANCF单元网格。以悬臂梁在冲击作用下的动态响应为例说明了不同的连续性要求对有限元分析的影响。