A collocation solution for Burgers' equation using cubic B-spline finite elements

The non-linear Burgers' equation is solved numerically by a B-spline finite element method. The approach used is based on collocation of cubic B-splines over finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range. A linear stability analysis shows that a numerical scheme based on a Crank-Nicolson approximation in time is unconditionally stable. Three standard problems are used to validate the algorithm. Comparisons are made with published numeric and analytic solutions. The proposed method performs well.     

求解对流扩散方程的混合三次B-样条配点法

通过对三次B-样条和三次三角B-样条基函数引入权因子[ω],给出了对流扩散方程的混合三次B-样条配点法。对对流扩散方程空间离散采用混合三次B-样条配点法和时间离散采用向前有限差分,引入参数[θ],建立差分格式。对差分格式的稳定性进行分析,得到稳定性条件。数值实验表明所构造方法的有效性,并且适当调整权因子[ω]和参数[θ]的值,可提高计算的精度。     

Modal Hermite spectral collocation method for solving multi-dimensional hyperbolic telegraph equations

The present research is contemplated proposing a numerical solution of multi-dimensional hyperbolic telegraph equations with appropriate initial time and boundary space conditions. The truncated Hermite series with unknown coefficients are used for approximating the solution in both of the spatial and temporal variables. The basic idea for discretizing the considered one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) telegraph equations is based on the collocation method together with the Hermite operational matrices of derivatives. The resulted systems of linear algebraic equations are solved by some efficient methods such as LU factorization. The solution of the algebraic system contains the coefficients of the truncated Hermite series. Numerical experiments are provided to illustrate the accuracy and efficiency of the presented numerical scheme. Comparisons of numerical results associated to the proposed method with some of the existing numerical methods confirm that the method is accurate and fast experimentally.     

Hermite三次样条插值的车型机器人路径规划研究

针对车型机器人在移动过程中的路障规避和寻找最优路径的问题,提出了一种基于Hermite三次样条的基线平滑路径,作为移动机器人穿越复杂环境的可行路径,并给出了相应的迭代优化算法。该算法在ODE仿真环境下进行了测试,其效果令人满意。     

An offset spline approximation for plane cubic splines

A plane cubic spline segment is given. We want to approximate its offset line by another cubic spline segment. Therefore we take the known curvatures and tangents at the end-points of the offset line and calculate the corresponding spline. Finally examples are given.     

A nonlinear solution scheme for multistory, multibay plane frames

If a plane frame experiences bending deformations in the fundamental state or is in a post-buckled secondary state, then a precise determination of the load-deflection relationships requires consideration of member rotations. This leads to highly nonlinear equations: and for multistory, multibay frames with many members, the solution is difficult to obtain. This paper presents a systematic procedure for handling the geometrically nonlinear equations, leading to an exact engineers' beam theory solution for rectangular plane frames with linearly elastic material. A procedure for locating the critical load is also discussed.     

Orthogonal spline collocation scheme for the multi-term time-fractional diffusion equation

A novel numerical technique is considered for the solution of a multi-term time-fractional diffusion equation. The orthogonal spline collocation method is used for in space, and a finite difference method in time. The stability and convergence are provided. The numerical experiments for one and two dimensional problems support our theoretical analysis.     

A finite element method for plane strain deformations of incompressible solids

A finite element method for incompressible deformation is formulated from the virtual work equation based on deviatoric quantities. Particular emphasis is given to nonlinear material behavior. The incompressibility constraint is imposed on the admissible displacement field by direct elimination of nodal displacements. The resulting stiffness matrix is symmetric and positive definite. Once a convergent solution for the displacement is obtained, the hydrostatic stress is determined from the principle of virtual work. A variety of illustrative examples are presented. The efficiency, economy and limitations of the method are discussed.     

Stability and convergence of the two parameter cubic spline collocation method for delay differential equations

In this paper, we propose the cubic spline collocation method with two parameters for solving delay differential equations (DDEs). Some results of the local truncation error and the convergence of the spline collocation method are given. We also obtain some results of the linear stability and the nonlinear stability of the method for DDEs. In particular, we design an algorithm to obtain the ranges of the two parameters α,β which are necessary for the P-stability of the collocation method. Some illustrative examples successfully verify our theoretical results.     

A numerical solution scheme for softening problems involving total strain control

Nonlinear structural problems in which material softening is present constitute a severe challenge to solution algorithms. Most of the existing computational codes are based on using an incremental procedure with iterations and suitable constraints. Among the constraints are load control, direct or indirect displacement control, and arc-length control involving a combination of load and displacement parameters. For many problems in which softening and localization occur, these algorithms fail at some point in the post-peak regime. In an attempt to remedy this problem, an alternative constraint condition is proposed whereby a combination of the total strain components is prescribed at the most critical point in the body. The critical point is defined to be that point where a suitable measure of strain is a maximum. Numerical solutions for both plane strain and plane stress problems are given to illustrate the ability of the procedure to capture post-peak responses of structures governed by materially nonlinear behavior.     

Galerkin collocation for an improved boundary element method for a plane mixed boundary value problem

We present the numerical implementation of a boundary element approximation for the plane mixed boundary value problem of the Laplacian. The performed Galerkin procedure is based on the direct boundary integral method. Its accuracy is improved by using appropriate singularity functions as additional test and trial functions besides quadratic splines. We analize the consistency error of the used numerical integrations and present asymptotic error estimates for the fully discretized numerical scheme which are of the same optimal orders as the Galerkin errors.     

一种水轮机应变检测仪的设计

针对水轮发电机组在运行不久后就出现叶片裂纹的情况,分析了水轮机叶片裂纹产生的原因,通过设计水轮机应变检测仪能够对水轮机叶片维护及提高水轮机的使用寿命具有直接指导作用,具体介绍了应变测量原理,最后,给出了应变检测仪的设计思路和系统的软件、硬件组成。经过对水轮机叶片的现场监测,应变检测仪达到了设计要求。     

A finite element procedure for plane strain metal flow within specified plastic boundaries

In this paper a new set of governing equations are developed for analyzing two-dimensional plane strain plasticity problems. Two sets of stress components are introduced—one set satisfying the equilibrium equations and the second ensuring that the yield criterion is satisfied. The differences between these two sets of stress components constitute the approximation errors. Using the method of least squares a simple triangular finite element model is developed by means of which the differences between the two sets of stress components are minimized.The formulation and the element developed are verified by analyzing three metal forming examples. Results computed with the proposed method correlate well with the exact solution of one example, and with the classical slip-line fields solutions of the other two examples.     

A plane stress to plane strain functional technique to compute stress-intensity factors in three-dimensional problems

A functional technique to calculate stress-intensity factors for cracks intersecting or approaching free surfaces is presented. Results are compared to previous finite elements and alternating solutions. The comparison shows that use of a full plane strain formula could be questionable in three-dimensional problems where cracks approach or intersect free surfaces.     

A boundary collocation method for cracked plates

A boundary collocation method is developed for analyzing cracked thin plates. Complex stress functions which satisfy the equilibrium equations of an infinite domain having a single crack are first derived. As the functions have also satisfied the stress singularity at the crack tips, it is only necessary to enforce the stress functions to satisfy the boundary conditions along the edges of the plates and the surfaces of the cracks, if there is more than one crack. This is achieved by the collocation least square approach. The unknown coefficients of the stress functions having been determined, the stress intensity factors can then be computed according to the related formulae. Examples of rectangular and circular plates with a different number of cracks and under different loadings are used to demonstrate the accuracy, versatility and advantages of the method.     

An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method

In this paper we propose a collocation method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value problems. The proposed approach is based on a Hermite function collocation (HFC) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The new method reduces the solution of a problem to the solution of a system of algebraic equations. Hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable.     

An integration scheme for electromagnetic scattering using plane wave edge elements

Finite element techniques for the simulation of electromagnetic wave propagation are, like all conventional element based approaches for wave problems, limited by the ability of the polynomial basis to capture the sinusoidal nature of the solution. The Partition of Unity Method (PUM) has recently been applied successfully, in finite and boundary element algorithms, to wave propagation. In this paper, we apply the PUM approach to the edge finite elements in the solution of Maxwell’s equations. The electric field is expanded in a set of plane waves, the amplitudes of which become the unknowns, allowing each element to span a region containing multiple wavelengths. However, it is well known that, with PUM enrichment, the burden of computation shifts from the solver to the evaluation of oscillatory integrals during matrix assembly. A full electromagnetic scattering problem is not simulated or solved in this paper. This paper is an addition to the work of Ledger and concentrates on efficient methods of evaluating the oscillatory integrals that arise. A semi-analytical scheme of the Filon character is presented.     

A practical chebyshev collocation method for differential equations

This paper describes a method for solving ordinary and partial differential equations in Chebyshev series. The main feature of the method, which is based on the collocation principle, (Lanczos [8]) is that it solves the problem of differentiating a Chebyshev series directly by the use of a stable recurrence relation. As a practical consequence the method is very simple and can easily be coded into a general-purpose program for solving some differential equations.