Stability and vibration analyses of carbon nanotube-reinforced composite beams with elastic boundary conditions: Chebyshev collocation method

This article aims to investigate stability and vibration behavior of carbon nanotube-reinforced composite beams supported by classical and nonclassical boundary conditions. To include significant effects of shear deformation and rotary inertia, Timoshenko beam theory is used to formulate the coupled equations of motion governing buckling and vibration analyses of the beams. An effective mathematical technique, namely Chebyshev collocation method, is employed to solve the coupled equations of motion for determining critical buckling loads and natural frequencies of the beams with different boundary conditions. The accuracy and reliability of the proposed mathematical models are verified numerically by comparing with the existing results in the literature for the cases of classical boundary conditions. New results of critical buckling loads and natural frequencies of the beams with nonclassical boundary conditions including translational and rotational springs are presented and discussed in detail associated with many important parametric studies.     

A Chebyshev collocation multidomain method to solve the Reissner–Mindlin equations for the transient response of an anisotropic plate subjected to impact

The transient response of an anisotropic rectangular plate subjected to impact is described through a Chebyshev collocation multidomain discretization of the Reissner–Mindlin plate equations. The trapezoidal rule is used for time-integration. The spatial collocation derivative operators are represented by matrices, and the subdomains are patched by natural and essential conditions. At each time level the resulting governing matrix equation is reduced by two consecutive block Gaussian eliminations, so that an equation for the variables at the subdomain corners has to be solved. Back-substitution gives the variables at all other collocation points. The time history as represented by computed contour plots has been compared with analytical results and with photos produced by holographic interferometry. The agreements are satisfactory. © 1997 John Wiley & Sons, Ltd.     

A Chebyshev collocation based sequential matrix exponential method for the generalized density evolution equation

In perspective of global approximation, this study presents a numerical method for the generalized density evolution equation (GDEE) based on spectral collocation. A sequential matrix exponential solution based on the Chebyshev collocation points is derived in consideration of the coefficient or velocity term of GDEE being constant in each time step, then the numerical procedure could be successively implemented without implicit or explicit difference schemes. The results of three numerical examples indicate that the solutions in terms of the sequential matrix exponential method for GDEE have good agreement with the analytical results or Monte Carlo simulations. For sufficiently smooth cases, there need no more than one hundred representative points to achieve a satisfied solution by the proposed method, whereas for the case in presence of severe discontinuity a few more sampling points are required to keep numerical stability and accuracy.     

Natural frequencies and critical loads of beams and columns with damaged boundaries using Chebyshev polynomials

The effects of damaged boundaries on natural frequencies and critical loads of beams and columns of variable cross section with conservative and non-conservative loads are investigated. The shifted Chebyshev polynomials are used to solve the one-dimensional transverse vibration problem, in which the ordinary differential equation is reduced to an algebraic eigenvalue problem. The advantages of this method are that it is easily employed in a symbolic form and that the number of polynomials may be adjusted to attain convergence. In the present study, the damaged boundary is modeled by linear translational and torsional springs, and the effects of the damage severity on the natural frequencies are studied. It is shown that as the amount of damage increases the natural frequencies decrease at rates which vary with the mode number. The method is applied to the instability problems of both uniform and uniformly tapered beams with and without follower forces, and the results for the undamaged cases show agreement when compared with results available in the literature. Convergence studies are carried out to determine the number of Chebyshev polynomials that should be used in the proposed method.     

A fast collocation method for an inverse boundary value problem

In this paper, we present an implementation of a fast multiscale collocation method for boundary integral equations of the second kind, and its application to solving an inverse boundary value problem of recovering a coefficient function from a boundary measurement. We illustrate by numerical examples the insensitive nature of the map from the coefficient to measurement, and design and test a Gauss–Newton iteration algorithm for obtaining the best estimate of the unknown coefficient from the given measurement based on a least‐squares formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

A gradient reproducing kernel collocation method for boundary value problems

The earlier work in the development of direct strong form collocation methods, such as the reproducing kernel collocation method (RKCM), addressed the domain integration issue in the Galerkin type meshfree method, such as the reproducing kernel particle method, but with increased computational complexity because of taking higher order derivatives of the approximation functions and the need for using a large number of collocation points for optimal convergence. In this work, we intend to address the computational complexity in RKCM while achieving optimal convergence by introducing a gradient reproduction kernel approximation. The proposed gradient RKCM reduces the order of differentiation to the first order for solving second‐order PDEs with strong form collocation. We also show that, different from the typical strong form collocation method where a significantly large number of collocation points than the number of source points is needed for optimal convergence, the same number of collocation points and source points can be used in gradient RKCM. We also show that the same order of convergence rates in the primary unknown and its first‐order derivative is achieved, owing to the imposition of gradient reproducing conditions. The numerical examples are given to verify the analytical prediction. Copyright © 2012 John Wiley & Sons, Ltd.     

Subdomain radial basis collocation method for heterogeneous media

Strong form collocation in conjunction with radial basis approximation functions offer implementation simplicity and exponential convergence in solving partial differential equations. However, the smoothness and nonlocality of radial basis functions pose considerable difficulties in solving problems with local features and heterogeneity. In this work, we propose a simple subdomain strong form collocation method, in which the approximation in each subdomain is constructed separately. Proper interface conditions are then imposed on the interface. Under the subdomain strong form collocation construction, it is shown that both Neumann and Dirichlet boundary conditions should be imposed on the interface to achieve the optimum convergence. Error analysis and numerical tests consistently confirm the need to impose the optimal interface conditions. The performance of the proposed methods in dealing with heterogeneous media is also validated. Copyright © 2009 John Wiley & Sons, Ltd.     

A robust local polynomial collocation method

In this paper, a robust local polynomial collocation method is presented. Based on collocation, this method is rather simple and straightforward. The present method is developed in a way that the governing equation is satisfied on boundaries as well as boundary conditions. This requirement makes the present method more accurate and robust than conventional collocation methods, especially in estimating the partial derivatives of the solution near the boundary. Studies about the sensitivity of the shape parameter and the local supporting range in the moving least square approach and the convergence of the nodal resolution are carried out by using some benchmark problems. This method is further verified by applying it to a steady‐state convection–diffusion problem. Finally, the present method is applied to calculate the velocity fields of two potential flow problems. More accurate numerical results are obtained.Copyright © 2012 John Wiley & Sons, Ltd.     

A mixed DOF collocation method for elastic problems in heterogeneous structure

In this paper, a collocation method with mixed degrees of freedom (DOFs) is proposed for heterogeneous structures. Local tractions of the outer and interface boundaries are introduced as DOFs in the mixed collocation scheme. Then, the equilibrium equations of all the nodes and the outer boundary conditions are discretized and assembled into the global stiffness matrix. A local force equilibrium equation for modeling the stress discontinuity through the interface is developed and added into the global stiffness matrix as well. With those contributions, a statically determined stiffness matrix is obtained. Numerical examples show that the present method is superior to the classical mixed collocation method in the heterogeneous structure because it improves the accuracy and the convergence and remains the efficiency. Besides, almost constant convergence rates of displacements and stresses are found in all the examples, even for three-dimensional problems.     

A point collocation method based on reproducing kernel approximations

A reproducing kernel particle method with built‐in multiresolution features in a very attractive meshfree method for numerical solution of partial differential equations. The design and implementation of a Galerkin‐based reproducing kernel particle method, however, faces several challenges such as the issue of nodal volumes and accurate and efficient implementation of boundary conditions. In this paper we present a point collocation method based on reproducing kernel approximations. We show that, in a point collocation approach, the assignment of nodal volumes and implementation of boundary conditions are not critical issues and points can be sprinkled randomly making the point collocation method a true meshless approach. The point collocation method based on reproducing kernel approximations, however, requires the calculation of higher‐order derivatives that would typically not be required in a Galerkin method, A correction function and reproducing conditions that enable consistency of the point collocation method are derived. The point collocation method is shown to be accurate for several one and two‐dimensional problems and the convergence rate of the point collocation method is addressed. Copyright © 2000 John Wiley & Sons, Ltd.     

Direct collocation method for identifying the initial conditions in the inverse wave problem using radial basis functions

A direct collocation method associated with explicit time integration using radial basis functions is proposed for identifying the initial conditions in the inverse problem of wave propagation. Optimum weights for the boundary conditions and additional condition are derived based on Lagrange’s multiplier method to achieve the prime convergence. Tikhonov regularization is introduced to improve the stability for the ill-posed system resulting from the noise, and the L-curve criterion is employed to select the optimum regularization parameter. No iteration scheme is required during the direct collocation computation which promotes the accuracy and stability for the solutions, while Galerkin-based methods demand the iteration procedure to deal with the inverse problems. High accuracy and good stability of the solution at very high noise level make this method a superior scheme for solving inverse problems.     

Direct solution of ill‐posed boundary value problems by radial basis function collocation method

Numerical solution of ill‐posed boundary value problems normally requires iterative procedures. In a typical solution, the ill‐posed problem is first converted to a well‐posed one by assuming the missing boundary values. The new problem is solved by a conventional numerical technique and the solution is checked against the unused data. The problem is solved iteratively using optimization schemes until convergence is achieved. The present paper offers a different procedure. Using the radial basis function collocation method, we demonstrate that the solution of certain ill‐posed problems can be accomplished without iteration. This method not only is efficient and accurate, but also circumvents the stability problem that can exist in the iterative method. Copyright © 2005 John Wiley & Sons, Ltd.     

A univariate Chebyshev polynomials method for structural systems with interval uncertainty

This paper presents an effective univariate Chebyshev polynomials method (UCM) for interval bounds estimation of uncertain structures with unknown-but-bounded parameters. The interpolation points required by the conventional collocation methods to generate the surrogate model are the tensor product of each one-dimensional (1D) interpolating point. Therefore, the computational cost is expensive for uncertain structures containing more interval parameters. To deal with this issue, the univariate decomposition is derived through the higher-order Taylor expansion. The structural system is decomposed into a sum of several univariate subsystems, where each subsystem only involves one uncertain parameter and replaces the other parameters with their midpoint value. Then the Chebyshev polynomials are utilized to fit the subsystems, in which the coefficients of these subsystems are confirmed only using the linear combination of 1D interpolation points. Next, a surrogate model of the actual structural system composed of explicit univariate Chebyshev functions is established. Finally, the extremum of each univariate function that is obtained by the scanning method is substituted into the surrogate model to determine the interval ranges of the uncertain structures. Numerical analysis is conducted to validate the accuracy and effectiveness of the proposed method.     

A numerical method for heat transfer problems using collocation and radial basis functions

Simple, mesh/grid free, numerical schemes for the solution of heat transfer problems are developed and validated. Unlike the mesh or grid-based methods, these schemes use well-distributed quasi-random collocation points and approximate the solution using radial basis functions. The schemes work in a similar fashion as finite differences but with random points instead of a regular grid system. This allows the computation of problems with complex-shaped boundaries in higher dimensions with no extra difficulty. © 1998 John Wiley & Sons, Ltd.     

A stochastic collocation method for large classes of mechanical problems with uncertain parameters

The paper presents a self-contained and didactic approach to the stochastic collocation method. The method relies on the Lagrange polynomials and the Gauss quadrature rule. It is presented for large classes of mechanical problems, i.e. static problems, dynamic problems and spectral problems. After a general presentation of each of them, examples and results are provided. Numerical results show the high rate of convergence of the proposed method.     

A collocation meshless method based on local optimal point interpolation

This paper deals with the use of the local optimal point interpolating (LOPI) formula in solving partial differential equations (PDEs) with a collocation method. LOPI is an interpolating formula constructed by localization of optimal point interpolation formulas that reproduces polynomials and verifies the delta Kronecker property. This scheme results in a truly meshless method that produces high quality output and accurate solutions. Copyright © 2003 John Wiley & Sons, Ltd.     

A complex variable boundary collocation method for plane elastic problems

 Lagrange interpolation is extended to the complex plane in this paper. It turns out to be composed of two parts: polynomial and rational interpolations of an analytical function. Based on Lagrange interpolation in the complex plane, a complex variable boundary collocation approach is constructed. The method is truly meshless and singularity free. It features high accuracy obtained by use of a small number of nodes as well as dimensionality advantage, that is, a two-dimensional problem is reduced to a one-dimensional one. The method is applied to two-dimensional problems in the theory of plane elasticity. Numerical examples are in very good agreement with analytical ones. The method is easy to be implemented and capable to be able to give the stress states at any point within the solution domain. Received: 20 August 2002 / Accepted: 31 January 2003     

Meshfree point collocation method for elasticity and crack problems

A generalized diffuse derivative approximation is combined with a point collocation scheme for solid mechanics problems. The derivatives are obtained from a local approximation so their evaluation is computationally very efficient. This meshfree point collocation method has other advantages: it does not require special treatment for essential boundary condition nor the time‐consuming integration of a weak form. Neither the connectivity of the mesh nor differentiability of the weight function is necessary. The accuracy of the solutions is exceptional and generally exceeds that of element‐free Galerkin method with linear basis. The performance and robustness are demonstrated by several numerical examples, including crack problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Weighted radial basis collocation method for boundary value problems

This work introduces the weighted radial basis collocation method for boundary value problems. We first show that the employment of least‐squares functional with quadrature rules constitutes an approximation of the direct collocation method. Standard radial basis collocation method, however, yields a larger solution error near boundaries. The residuals in the least‐squares functional associated with domain and boundary can be better balanced if the boundary collocation equations are properly weighted. The error analysis shows unbalanced errors between domain, Neumann boundary, and Dirichlet boundary least‐squares terms. A weighted least‐squares functional and the corresponding weighted radial basis collocation method are then proposed for correction of unbalanced errors. It is shown that the proposed method with properly selected weights significantly enhances the numerical solution accuracy and convergence rates. Copyright © 2006 John Wiley & Sons, Ltd.     

Plane elasticity problems by barycentric rational interpolation collocation method and a regular domain method

Barycentric rational interpolation collocation method (BRICM) for solving plane elasticity problems with high accuracy is presented. The plane elasticity problems on a circular or rectangular domain can be solved directly by BRICM. Embedded the irregular domain into a regular (circular or rectangular) domain, the governing equations of plane elasticity on regular domain are discretized by the differentiation matrices based on barycentric rational interpolation to form a system of algebraic equations. Discrete boundary conditions are obtained using barycentric rational interpolation. The irregular boundary conditions are imposed by the additional method to form an over-constraint linear system of algebraic equations. Numerical experiments are presented to illustrate the efficiency and high computing precision of proposed method.