HIGH ACCURACY FINITE VOLUME ELEMENT METHOD FOR TWO-POINT BOUNDARY VALUE PROBLEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference methods. It is proved that the method has optimal order error estimate O(h3) in H1 norm. Finally, two examples show that the method is effective.

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.

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.

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.

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.

Elastic-plastic analytical solution for centric crack loaded by two pairs of point shear forces in finite plate

The near crack line analysis method was used to investigate a centric crack loaded by two pairs of point shear forces in a finite plate, and the analytical solution was obtained. The solution includes the unit normal vector of the elastic-plastic boundary near the crack line, the elastic-plastic stress fields near the crack line, the variations of the length of the plastic zone along the crack line with an external load, and the bearing capacity of a finite plate with a centric crack loaded by two pairs of point shear forces. The results are sufficiently precise near the crack line because the assumptions of small scale yielding theory have not been made and no other assumptions are taken.

Shape optimization of axisymmetric solids with the finite cell method using a fixed grid

In this work, a design procedure extending the B-spline based finite cell method into shape optimization is developed for axisymmetric solids involving the centrifugal force effect. We first replace the traditional conforming mesh in the finite element method with structured cells that are fixed during the whole design process with a view to avoid the sophisticated re-meshing and eventual mesh distortion.Then, B-spline shape functions are further implemented to yield a high-order continuity field along the cell boundary in stress analysis. By means of the implicit description of the shape boundary, stress sensitivity is analytically derived with respect to shape design variables. Finally, we illustrate the efficiency and accuracy of the proposed protocol by several numerical test cases as well as a whole design procedure carried out on an aeronautic turbine disk.

An improved local radial point interpolation method for transient heat conduction analysis

The smoothing thin plate spline （STPS） interpolation using the penalty function method according to the optimization theory is presented to deal with transient heat conduction problems. The smooth conditions of the shape functions and derivatives can be satisfied so that the distortions hardly occur. Local weak forms are developed using the weighted residual method locally from the partial differential equations of the transient heat conduction. Here the Heaviside step function is used as the test function in each sub-domain to avoid the need for a domain integral. Essential boundary conditions can be implemented like the finite element method （FEM） as the shape functions possess the Kronecker delta property. The traditional two-point difference method is selected for the time discretization scheme. Three selected numerical examples are presented in this paper to demonstrate the availability and accuracy of the present approach comparing with the traditional thin plate spline （TPS） radial basis functions.

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.

ASYMPTOTIC ERROR EXPANSIONS OF QUADRATIC SPLINE COLLOCATION SOLUTIONS FOR TWO-POINT BOUNDARY VALUE PROBLEMS

In this paper, we consider the following problem:The quadratic spline collocation, with uniform mesh and the mid-knot points are taken as the collocation points for this problem is considered. With some assumptions, we have proved that the solution of the quadratic spline collocation for the nonlinear problem can be written as a series expansions in integer powers of the mesh-size parameter. This gives us a construction method for using Richardson’s extrapolation. When we have a set of approximate solution with different mesh-size parameter a solution with high accuracy can he obtained by Richardson’s extrapolation.

Polynomial-time interior-point algorithm based on a local self-concordant finite barrier function

The choice of self-concordant functions is the key to efficient algorithms for linear and quadratic convex optimizations, which provide a method with polynomial-time iterations to solve linear and quadratic convex optimization problems. The parameters of a self-concordant barrier function can be used to compute the complexity bound of the proposed algorithm. In this paper, it is proved that the finite barrier function is a local self-concordant barrier function. By deriving the local values of parameters of this barrier function, the desired complexity bound of an interior-point algorithm based on this local self-concordant function for linear optimization problem is obtained. The bound matches the best known bound for small-update methods.

Mathematical analysis of EEP method for one-dimensional finite element postprocessing

For a class of two-point boundary value problems, by virtue of onedimensional projection interpolation, it is proved that the nodal recovery derivative obtained by Yuan＇s element energy projection （EEP） method has the accuracy O（h^min{2k,k＋4}） The theoretical analysis coincides the reported numerical results.

二次B-Spline时域基函数的TDFEM的应用

提出了时域有限元法(TDFEM)的一种新的基函数—二次B-spline时域基函数.首先简述了时域有限元法的原理和基本公式;然后提出了新型的基于B-spline函数的条件稳定和无条件稳定的时域有限元法方案,并应用于三维电磁辐射问题.通过典型的算例对这两种方案的精度、运算时间进行了比较,证实了基于二次B-spline函数的时域有限元法的有效性.通过稳定性理论分析得出该算法的精确稳定性,并且通过数值计算的结果得到验证.

IMAGE SUPER-RESOLUTION RECONSTRUCTION BY HUBER REGULARIZATION AND TAILORED FINITE POINT METHOD

In this paper,we propose using the tailored finite point method(TFPM)to solve the resulting parabolic or elliptic equations when minimizing the Huber regularization based image super-resolution model using the augmented Lagrangian method(ALM).The Hu-ber regularization based image super-resolution model can ameliorate the staircase for restored images.TFPM employs the method of weighted residuals with collocation tech-nique,which helps get more accurate approximate solutions to the equations and reserve more details in restored images.We compare the new schemes with the Marquina-Osher model,the image super-resolution convolutional neural network(SRCNN)and the classical interpolation methods:bilinear interpolation,nearest-neighbor interpolation and bicubic interpolation.Numerical experiments are presented to demonstrate that with the new schemes the quality of the super-resolution images has been improved.Besides these,the existence of the minimizer of the Huber regularization based image super-resolution model and the convergence of the proposed algorithm are also established in this paper.

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.

基于Coons曲面与B-spline曲线的有限元网格自动生成

在概述了有限元网格自动生成的基础上,介绍了一种能对复杂曲边区域作有限元网格生成的技术。这种技术以Coons曲面作映射函数,以B-spline曲线作拟合区域边界,充分发挥ICG的交互能力,利用B-spline曲线的形状控制,对多种复杂曲边区域作网格自动生成。用该技术研制的网格生成系统在交互式人机界面支持下,能快速输入几何信息,实现图形交互编辑,边界贴合良好,内部网格点理想。

Finite Element Modeling and Design of Rectangular Patch Antenna with Different Feeding Techniques

The finite element modeling of three dimensional structures is important for researchers especially in the field of antennas and other domains of electromagnetic waves. This paper presents a finite element calculations and numerical analysis for the microstrip patch antennas. In this paper, two different designs have been modelled and analyzed and both designs are based on the rectangular patches. The feeding point of one design is inside the patch while the other design contains feeding point outside the patch is T shaped. The computational analysis showed some interesting results for radiation pattern and far field domain. For these designs, the characteristic impedance taken is 50 Ω and the operating frequency domain is 1.4 to 1.7 GHz. The microstrip patch antennas are encapsulated in the inert spherical atmosphere of 20 mm thickness containing air inside it.

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.

Arc-length technique for nonlinear finite element analysis

Nonlinear solution of reinforced concrete structures, particularly complete load-deflection response, requires tracing of the equilibrium path and proper treatment of the limit and bifurcation points. In this regard, ordinary solution techniques lead to instability near the limit points and also have problems in case of snap-through and snap-back. Thus they fail to predict the complete load-displacement response. The arc-length method serves the purpose well in principle, received wide acceptance in finite element analysis, and has been used extensively. However modifications to the basic idea are vital to meet the particular needs of the analysis. This paper reviews some of the recent developments of the method in the last two decades, with particular emphasis on nonlinear finite element analysis of reinforced concrete structures.

Numerical solution of Poisson equation with wavelet bases of Hermite cubic splines on the interval

A new wavelet-based finite element method is proposed for solving the Poisson equation. The wavelet bases of Hermite cubic splines on the interval are employed as the multi-scale interpolation basis in the finite element analysis. The lifting scheme of the wavelet-based finite element method is discussed in detail. For the orthogonal characteristics of the wavelet bases with respect to the given inner product, the corresponding multi-scale finite element equation can be decoupled across scales, totally or partially, and suited for nesting approximation. Numerical examples indicate that the proposed method has the higher efficiency and precision in solving the Poisson equation.