Point Collocation Method Research Articles

Axisymmetric postbuckling of moderately thick laminated annular plate using FSDT

This work presents axisymmetric postbuckling response of polar orthotropic laminated moderately thick circular and annular plates subjected to uniformly distributed inplane radial compressive load at the outer edge. First-order shear deformation theory (FSDT) is used in terms of deflection w, rotation ψ of the normal to the midplane and the stress function Φ. These field variables are expanded in polynomials and orthogonal point collocation method is used to discretise the governing equations. The buckling load is obtained by solving a linear eigenvalue problem. Postbuckling loads for different deflections are obtained by solving nonlinear equations by Newton–Raphson method. Clamped and simply supported circular and annular plates with free hole or reinforced by a rigid ring are considered. The results are compared with the classical lamination theory to assess the dependence of the effect of shear deformation on boundary conditions, ratio of Young's moduli, lamination scheme, number of layers, the radius-to-thickness ratio and annular ratio. The response of unsymmetrically laminated plates under inplane radial load is also presented.

Meshless analysis of piezoelectric devices

The sensor and actuator properties of piezoelectric materials make them well suited for applications in a variety of microelectromechanical systems (MEMS). Simulating the response of piezoelectric devices requires solving coupled electrical and mechanical partial differential equations. In this paper, we have implemented a meshless point collocation method (PCM) to solve the governing equations. Interpolation functions are constructed from a reproducing kernel approximation, and the governing equations are discretized using a collocation approach. PCM is implemented using either a relaxation algorithm or a fully-coupled algorithm. Comparisons between the two algorithms are given. To demonstrate the performance of PCM, the behavior of two static single-layer problems and a piezoelectric bimorph have been modeled. The bimorph analysis is extended to model a prototype MEMS device.

Geometrically nonlinear axisymmetric analysis of thick laminated annular plate using FSDT

This paper presents geometrically nonlinear, axisymmetric, static and transient, moderately large deflection response of a laminated, composite, moderately thick annular plate subjected to uniformly distributed ring loads. The layers are considered to be polar orthotropic. Rotational inertia is included but the inplane inertia is neglected. First-order shear deformation theory is used in terms of the deflection w, rotation ψ of the midplane and the stress function Φ. The governing equations are solved by the orthogonal point collocation method and Newmark-β scheme. The effect of the radius-to-thickness ratio of the plate on the deflection and stress response is studied for symmetrically and unsymmetrically laminated plates.

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.

Stability of Galerkin and collocation time domain boundary element methods as applied to the scalar wave equation

The Galerkin method for time domain boundary element analysis is presented in this paper and used for the first time as an alternative procedure to improve the stability of BEM formulations applied to the scalar wave equation. The paper includes a short review of the classical time-stepping BEM formulation and the modifications required to implement a Galerkin approach. No mathematical proof is given here, concerning the stability improvement of the Galerkin formulation over the point collocation method. However, strong numerical evidence of such stability improvement is provided by the second example of this paper where the Galerkin approach has shown to be stable for rather small time steps.

A galerkin boundary element method for solving the generalized Helmholtz decomposition

The objective of this work is to more accuretely calculate velocities, particularly on bondaries, by using the Galerkin form of the generalized Helmholtz decomposition. This work describes advantages of using the generalized Helmholtz decomposition over the other formulations and shows the accuracy and convergence advantages of Galerkin numerical methods over point collocation methods. Solutions based on Galerkin and point collocation methods are compared for two-dimensional problems. The comparaison shows that Galerkin solutions are significantly more accurate and less oscillatory. Accordingly, the Galerkin method is recommended for use with vortex methods.

Spline Kantorovich method and analysis of general slab bridge deck

In this paper, the spline Kantorovich method is developed and applied to the analysis and design of bridge decks. First, the bridge deck is mapped into a unit square in the Xi - eta plane. The governing partial differential equation of the plate is reduced to the ordinary differential equation in the longitudinal direction of the bridge by the routine Kantorovich method. Spline point collocation method is then used to solve the derived ordinary differential equation to obtain the displacements and internal forces of the bridge deck. Mindlin plate theory is incorporated into the differential equation and, as a result, the effect of shear deformation of the plate is also considered. Possible shear locking is avoided by the reduced integration technique. Numerical examples show that the proposed new numerical model is versatile, efficient, and reliable.Key words: Kantorovich method, spline function, partial differential equations, ordinary differential equations, point collocation method, bridge deck.

Time weighting in time domain BEM

The time-weighted time domain boundary element method (TWTDBEM) is presented as an alternative approach to improve the stability of the traditional time domain BEM (TTDBEM) formulation for dynamics. A forward time prediction method is used, which does not cause significant errors as long as the time step is not too large. Numerical experiments carried out show that the time weighting method, presented for the first time here, is stable for applications in which the TTDBEM approach is not, even when only a small number of time weighting discrete points is employed. The use of only two weighting points for each time step leads to a substantial improvement in the stability of the TWTDBEM over the TTDBEM at the expense of a computational cost increase which is not significant. In applications where the time marching process is likely to present instabilities, the time weighting method should be preferred rather than the standard point collocation method.

Simplified Model for Hollow Fine Fiber Modules with Radial Dispersion

There is enough experimental evidence indicating that the reciprocal of the rejection of hollow fine fiber modules is linearly related to the reciprocal of the product flow rate. We show that this relationship can be obtained from a rigorous mathematical model that accounts for radial dispersion using approximations based on the application of one point collocation method.

Nonlinear static and dynamic analysis for laminated, annular, spherical caps of moderate thickness

Based on Timoshenko-Mindlin kinematic hypothesis, the shallow shell theory is extended to include the transverse shear deformation for the nonlinear axisymmetric dynamic analysis of the symmetric cross-ply shallow spherical shell. Using the orthogonal point collocation method and the Newmark scheme, an iterative solution is formulated. The numerical results for the nonlinear static and dynamic responses and dynamic buckling of these shallow spherical shells with circular holes under uniformly distributed static or dynamic normal impact loads are presented and compared with available data.

Subject index of volume 112

For appropriate management of available groundwater, the flow behavior in the porous media has to be analyzed. The complex problem of groundwater flow can be studied by solving the governing equations analytically or by using numerical methods. As the analytical solutions are available only for simple idealized cases, numerical methods such as finite difference method (FDM) and finite element method (FEM) are generally used for field problems. Meshfree (MFree) method is an alternative numerical approach to solve complex groundwater problems in simple manner. MFree method eliminates the drawback of meshing and remeshing as in FDM and FEM which can translate to substantial cost and time savings in modeling. In this paper, a model using MFree point collocation method (PCM) with multi-quadric radial basis function (MQ-RBF) is proposed for 2D groundwater flow simulation. The accuracy of the developed model is verified with available analytical solution in literature. The developed model is applied initially for a hypothetical problem and further for a field problem to compute head distribution. The PCM model results for the hypothetical problem are compared with FEM simulations while that of field problem are compared with boundary element based model results. The PCM model results are found to be satisfactory showing the applicability of the present approach.

On Trajectory Planning for Flexible Robots

A computational technique, based on radial basis functions, for deriving trajectories for a single link flexible robot arm, is presented in this paper. Using an approximation based on radial basis functions and point collocation method, the trajectory planning problem is transformed into a linear least-squares problem. This technique is easy to implement and is computationally efficient. Furthermore, the trajectories obtained do not produce any residual vibration at the end of the move. A computer simulation based on a finite-element model indicates the effectiveness of the proposed technique.

Nonlinear dynamic response and dynamic buckling of shallow spherical shells with circular hole

In this paper, the nonlinear equations of motion for shallow spherical shells with axisymmetric deformation including transverse shear are derived. The nonlinear static and dynamic response and dynamic buckling of shallow spherical shells with circular hole on elastically restrained edge are investigated. By using the orthogonal point collocation method for space and Newmark-β scheme for time, the displacement functions are separated and the nonlinear differential equations are replaced by linear algebraic equations to seek solutions. The numerical results are presented for different cases and compared with available data.

An approximate analytical solution for the performance of reverse osmosis plants

Recent developments in membrane technology and appropriate construction material made reverse osmosis plant attractive for large desalting capacity. Reserve osmosis showed a growing demand specially in sea water desalting. It can be considered the optimum process in areas where sufficient electric power is available at low cost. Due to these reasons, mathematical modeling of reverse osmosis plants became an important task in the design procedure. In this paper, the partial differential equations representing the material and momentum balances inside a hollow fine fiber reverse osmosis model are discretized by the method of orthogonal collocation. The approximate analytical solution is obtained by applying the one point collocation method in the radial and axial direction. This leads to simple expressions for the recovery and product concentration. The obtained expressions are compared by the more exact results obtained by using higher order collocation method. The applicability of these results to the design and operation of reserve osmosis plants will be disucssed.

Non-linear vibration of orthotropic thin rectangular plates on elastic foundations

This study deals with the large amplitude free vibrations of rectilinearly orthotropic thin rectangular plates resting on elastic foundations. The von Kármán-type of governing equations are employed. The displacements are assumed to be harmonic in time and are expanded space-wise in terms of polynomial basis functions which satisfy the boundary conditions exactly. The time co-ordinate is eliminated by the Kantorovich averaging method. The orthogonal point collocation method is used to obtain the discretised equations. The non-linear eigenvalue problem is solved by an iterative method. Numerical results are presented for the linear frequency of the fundamental mode and for the ratio of the non-linear frequency to the linear frequency. Plates with opposite edges clamped or simply supported with immovable in-plane conditions and resting on Winkler, Pasternak and non-linear Winkler foundations are considered. The effects of foundation parameters, material parameters, edge conditions and the aspect ratio on the non-linear vibration behaviour have been investigated.

Nonlinear static analysis of rectangular plates on elastic foundations by the orthogonal point collocation method

The conventional interior point collocation method is greatly improved if the collocation points are located at the zeros of orthogonal polynomials. In this paper, the simplicity and good accuracy of this orthogonal point collocation method is demonstrated for the solution of some geometrically nonlinear problems of moderately large deflections of rectangular plates subjected to static loads. Von Kármán-type governing equations are employed. Clamped and simply supported plates with immovable inplane conditions at the edges are considered. Plates which are simply supported at two opposite edges and clamped at the other two edges are also analyzed. Winkler and Pasternak models of elastic foundation are used. The present results agree quite well with the available ones. New results are presented for orthotropic plates resting on Winkler and Pasternak foundations.

Nonlinear forced vibration of orthotropic thin rectangular plates

This study deals with the large amplitude forced vibrations of specially orthotropic thin rectangular plates. Geometric nonlinearity is included in accordance with von Kármán type large deflection plate theory. The transverse displacement is assumed to be harmonic at the same frequency as that of the uniformly distributed harmonic force. The displacements are expanded space-wise in terms of polynomial basis functions which satisfy the boundary conditions. Time is eliminated by averaging over the time-period of the harmonic force. The orthogonal point collocation method is used to obtain the discretized equations. These nonlinear equations are solved by an iterative procedure. Plates with all edges clamped or simply-supported and immovable inplane are considered. The relations of amplitude and frequency ratio for rectangular plates made of various orthotropic materials are presented.

Large deflection axisymmetric analysis of orthotropic annular plates on elastic foundations

Large deflection axisymmetric response analyses of cylindrically orthotropic thin annular plates, resting on annular elastic foundations and subjected to uniformly distributed loads are presented. Static and step function loads applied to clamped and simply supported annular plates are considered. The natural boundary conditions employed at the hole are consistent with those obtained from Hamilton's principle. Von Kármán type governing equations are solved using the orthogonal point collocation method in conjunction with the Newmark-β scheme. The maximum response to step loads obtained from static analysis is shown to agree very well with that obtained from a transient analysis. Thus one static analysis is sufficient to obtain the maximum response to a whole set of step loads, instead of separate transient solutions for each load. New results are presented to study the effects of the foundation parameters on the plate response.

Nonlinear transient response of orthotropic thin rectangular plates on elastic foundations

This paper presents geometrically nonlinear transient analysis of rectilinearly orthotropic thin rectangular plates resting on Winkler and Pasternak foundations for uniformly distributed step function and sinusoidal pulse loadings. The orthogonal point collocation method in the space domain and Newmark-β scheme in the time domain are employed. Immovable clamped and simply-supported plates are analysed. An approximate method is used to predict the maximum dynamic response to step loads from the results for static loads and is found to yield sufficiently accurate results.

Nonlinear Axisymmetric Static and Transient Response of Orthotropic Annular Shallow Spherical Caps

ABSTRACT This investigation deals with the nonlinear axisymmetric static and transient response of orthotropic annular shallow spherical caps with a free edge hole and with a rigid central mass, subjected to uniformly distributed load and a central load. The dynamic analogue of Marguerre equations, in terms of normal displacement w and stress function Ψ, are employed. An orthogonal point collocation method is used for spatial discretization and a Newmark-β scheme is used for time integration. Static load, step function load, and sinusoidal pulse load are considered. The influence of annular ratio on the response is studied for isotropic (β = 1) and orthotropic (β = 3) clamped caps, with a rise to thickness ratio of 1.5.