Point Collocation Method Research Articles

Nonlinear Dynamic Response and Active Control of Piezoelastic Laminated Shallow Spherical Shells with Damage

Based on Talreja’s damage model with tensor valued internal state variables and geometric nonlinear theory, the constitutive relations for a moderately thick shallow spherical shell with damage are derived. The distribution of electric potential along the thickness direction in the piezoelectric layer is simulated by a sinusoidal function, and accordingly the dynamic analytical model for the cross-ply laminated moderately thick piezoelectric shallow spherical shell is established. Using the negative velocity feedback control algorithm, an analytical model for active vibration control of the piezoelectric laminated moderately thick piezoelectric shallow spherical shell is built when the damage effect is considered. And the solutions to the whole problem are obtained with synthetical utilization of the orthogonal collocation point method and the Newark method. In numerical examples, the effects of damage, piezoelectric effect, and the structure’s geometrical parameters on the dynamic response and vibration control of the piezoelastic laminated shallow spherical shells with damage are investigated.

The finite point method for the p-Laplace equation

In this paper, the finite point method (FPM) is presented for solving the 2D, nonlinear, elliptic p-Laplace or p-harmonic equation. The FPM is a truly meshfree technique based on the combination of the moving least squares approximation on a cloud of points with the point collocation method to discretize the governing equation. The lack of dependence on a mesh or integration procedure is an important feature, which makes the FPM simple, efficient and applicable to solve nonlinear problems. Applications are demonstrated through illustrative examples.

Free vibration of symmetric angle-ply laminated cylindrical shells of variable thickness

In this paper, free vibrations of symmetric angle-ply laminated cylindrical shells are presented under different thickness variations. The thickness variations of the shells are linear, exponential, and sinusoidal. Spline function techniques are adopted to approximate the displacement functions, and the point collocation method is used to obtain the generalized eigenvalue problem with appropriate boundary conditions. The effects of the geometric parameters, thickness parameters, and material parameters on the frequencies of symmetric angle-ply laminated cylindrical shells with two kinds of material properties under different boundary conditions are analyzed.

2차원 채널 내의 수직 평판을 지나는 스톡스 유동에 대한 연구

A two-dimensional Stokes flow past a vertical plate in a channel is analyzed. The vertical plate is located at the center of the channel, and plane Poiseuille flow exists far upstream and downstream of the vertical plate. The Stokes approximation is used, and the flow is investigated analytically using the method of eigenfunction expansion and the point collocation method. From the analysis, the stream function and pressure distribution are obtained, and the pressure and shear stress distributions on the plate and channel wall are calculated. The additional pressure drop induced by the vertical plate and the force exerted on it are calculated as functions of the length of the vertical plate. For a typical length of the vertical plate, the streamline pattern and pressure distribution are shown. In addition, numerical analysis of laminar flow with a small Reynolds number is carried out to analyze the effect of a small Reynolds number on the flow pattern.

A meshless collocation method for the plane problems of functionally graded material beams and plates using the DRK interpolation

A meshless collocation method is developed for the static analysis of plane problems of functionally graded (FG) elastic beams and plates under transverse mechanical loads using the differential reproducing kernel (DRK) interpolation, in which the DRK interpolant is constructed by the randomly distributed nodes. A point collocation method based on this DRK interpolation is developed for the plane stress and strain problems of homogeneous and FG elastic beams and plates. It is shown that the present DRK interpolation-based collocation method is indeed a truly meshless approach with excellent accuracy and has a fast convergence rate.

Upwinding meshfree point collocation method for steady MHD flow with arbitrary orientation of applied magnetic field at high Hartmann numbers

In this paper, a meshfree point collocation method, with an upwinding scheme, is presented to obtain the numerical solutions of the coupled equations in velocity and magnetic field for the fully developed magnetohydrodynamic (MHD) flow through an insulated straight duct of rectangular section. The moving least-square (MLS) approximation is employed to construct the shape functions in conjunction with the framework of the point collocation method. Computations have been carried out for different applied magnetic field orientations and a wide range of values of Hartmann number from 5 to 10 6. As the adaptive upwinding local support domain is introduced in the meshless point collocation method, numerical results show that the method may compute MHD problems not only at low and moderate values but also at high values of the Hartmann number with high accuracy and good convergence.

Simulation of groundwater flow in unconfined aquifer using meshfree point collocation method

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.

MESHLESS POINT COLLOCATION METHOD FOR 1D AND 2D GROUNDWATER FLOW SIMULATION

ABSTRACT For appropriate management of available groundwater, the flow behavior in the porous media has to be analyzed. Meshfree (MFree) method is an alternative numerical approach to commonly used Finite Difference Method (FDM) and Finite Element Method (FEM) to solve more complex engineering problems by simple and accurate manner. MFree method eliminates the drawback of meshing and remeshing as in FDM and FEM. Since there is no need to create a mesh, the time one would spend on conventional mesh generation can be saved. This can translate to substantial cost and time savings in modelling and simulation. In this paper, a meshless method called as Polynomial Point Collocation method (PPCM) with radial basis function has been developed for the groundwater flow simulation in porous media in one and two dimensions. The developed model has been applied for computing head distribution in a hypothetical confined aquifer having different boundary conditions and source, and sink terms. The developed model is tested with analytical and FEM solutions available in literature and found to be satisfactory.

Nonlinear dynamic response for functionally graded shallow spherical shell under low velocity impact in thermal environment

Based on Giannakopoulos’s 2-D functionally graded material (FGM) contact model, a modified contact model is put forward to deal with impact problem of the functionally graded shallow spherical shell in thermal environment. The FGM shallow spherical shell, having temperature dependent material property, is subjected to a temperature field uniform over the shell surface but varying along the thickness direction due to steady-state heat conduction. The displacement field and geometrical relations of the FGM shallow spherical shell are established on the basis of Timoshenko– Midlin theory. And the nonlinear motion equations of the FGM shallow spherical shell under low velocity impact in thermal environment are founded in terms of displacement variable functions. Using the orthogonal collocation point method and the Newmark method to discretize the unknown variable functions in space and in time domain, the whole problem is solved by the iterative method. In numerical examples, the contact force and nonlinear dynamic response of the FGM shallow spherical shell under low velocity impact are investigated and effects of temperature field, material and geometrical parameters on contact force and dynamic response of the FGM shallow spherical shell are discussed.

Meshless global radial point collocation method for three-dimensional partial differential equations

This study deals with the numerical solution of three-dimensional partial differential equations by the meshless global radial point collocation method based on various radial basis functions. First, second, third, and fourth-order three-dimensional partial differential equations are considered. The effect of shape parameters of various radial basis functions on the numerical accuracy is studied. The effect of grid pattern on accuracy is also studied by several numerical examples.

Spline Subdomain Approach to Thermal Conduction Problem for Composite with Rectangular Shape Particle

In this paper the spline subdomain approach is applied to the 2D simulations of the temperature distributions for composites containing a single rectangular particle with an interfacial thermal resistance at the interface between the particle and matrix. The bicubic B-splines are used to construct the trial functions for the approximations of the potential fields of composites. Applying the weighted residual point collocation method inside each subdomain and also on the boundaries between different subdomains, a system of linear algebraic equations is set up to determine the unknowns of the trial functions. The temperature distributions both inside the rectangular particle and along the interfaces under different interfacial contact conditions can be simulated approximately. Numerical results which are compared with the available solutions obtained by FEM method illustrate the accuracy and suitability of the present approach for steady-state conduction. Even in the adjacent areas of corners in the rectangular particle, the simulation results are also satisfactory.

A meshless local radial point collocation method for free vibration analysis of laminated composite plates

In this paper, a meshless local radial point collocation method based on multiquadric radial basis function is proposed to analyze the free vibration of laminated composite plates. This method approximates the governing equations based on first-order shear deformation theory using the nodes in the support domain of any data center. Natural frequencies of the laminated composite plates with various boundary conditions, side-to-thickness ratios, and material properties are computed by present method. The choice of shape parameter, effect of dimensionless sizes of the support domain on accuracy, convergence characteristics are studied by several numerical examples. The results are compared with available published results which demonstrate the accuracy and efficiency of present method.

A nth-order meshless generalization of Reddy’s third-order shear deformation theory for the free vibration on laminated composite plates

Abstract In this paper, a nth-order shear deformation theory is proposed to analyze the free vibration of laminated composite plates. The present nth-order shear deformation theory satisfies the zero transverse shear stress boundary conditions on the top and bottom surface of the plate. Reddy’s third-order theory can be considered as a special case of present nth-order theory (n = 3). Natural frequencies of the laminated composite plates with various boundary conditions, side-to-thickness ratios, material properties are computed by present nth-order theory and a meshless radial point collocation method based on the thin plate spline radial basis function. The results are compared with available published results which demonstrate the accuracy and efficiency of present nth-order theory.

Localized meshless point collocation method for time-dependent magnetohydrodynamics flow through pipes under a variety of wall conductivity conditions

In this article a numerical solution of the time dependent, coupled system equations of magnetohydrodynamics (MHD) flow is obtained, using the strong-form local meshless point collocation (LMPC) method. The approximation of the field variables is obtained with the moving least squares (MLS) approximation. Regular and irregular nodal distributions are used. Thus, a numerical solver is developed for the unsteady coupled MHD problems, using the collocation formulation, for regular and irregular cross sections, as are the rectangular, triangular and circular. Arbitrary wall conductivity conditions are applied when a uniform magnetic field is imposed at characteristic directions relative to the flow one. Velocity and induced magnetic field across the section have been evaluated at various time intervals for several Hartmann numbers (up to 105) and wall conductivities. The numerical results of the strong-form MPC method are compared with those obtained using two weak-form meshless methods, that is, the local boundary integral equation (LBIE) meshless method and the meshless local Petrov–Galerkin (MLPG) method, and with the analytical solutions, where they are available. Furthermore, the accuracy of the method is assessed in terms of the error norms L 2 and L ∞, the number of nodes in the domain of influence and the time step length depicting the convergence rate of the method. Run time results are also presented demonstrating the efficiency and the applicability of the method for real world problems.

Numerical Solution of Non-Isothermal Fluid Flows Using Local Radial Basis Functions (LRBF) Interpolation and a Velocity-Correction Method

Meshfree point collocation method (MPCM) is developed, solving the velocity-vorticity formulation of Navier-Stokes equations, for two-dimensional, steady state incompressible viscous flow problems in the presence of heat transfer. Particular emphasis is placed on the application of the velocity-correction method, ensuring the continuity equation. The Gaussian Radial Basis Functions (GRBF) interpolation is employed to construct the shape functions in conjunction with the framework of the point collocation method. The cases of forced, natural and mixed convection in a 2D rectangular enclosure are examined. The accuracy and the sta- bility of the proposed scheme are demonstrated through three representative, well known and established benchmark problems. Results are presented for high values of the characteristics non-dimensional numbers of the flow, that is, the Reynolds, the Rayleigh and the Richardson number.

Extended Meshfree Point Collocation Method for Electromagnetic Problems With Layered Singularity

A novel meshfree point collocation method for solving the electromagnetic potential problems with layered singularity is presented. The approximation based on the moving least squares technique is enriched by Heaviside step function, scissor function and wedge function. It effectively models discontinuities in electromagnetic potential and its gradient fields. Direct discretization of the partial differential equations yields the system of equations so that the computational speed is very fast compared to the Galerkin formulation. Solving the system provides the nodal solutions and jump strengths on the singular layer at the same time. Numerical results show that the developed method is highly efficient and reliable in solving electrostatic problems with charge layer or insulating layer.

The postbuckling analysis of laminated circular plate with elliptic delamination

Based on the Von Karman plate theory, considering the effect of transverse shear deformation, and using the method of the dissociated three regions, the postbuckling governing equations for the axisymmetric laminated circular plates with elliptical delamination are derived. By using the orthogonal point collocation method, the governing equations, boundary conditions and continuity conditions are transformed into a group of nonlinear algebraically equation and the equations are solved with the alternative method. In the numerical examples, the effects of various elliptical in shape, delamination depth and different material properties on buckling and postbuckling of the laminated circular plates are discussed and the numerical results are compared with available data.

Meshfree Point Collocation Schemes for 2D Steady State Incompressible Navier-Stokes Equations in Velocity-Vorticity Formulation for High Values of Reynolds Number

A meshfree point collocation method has been developed for the velocity- vorticity formulation of two-dimensional, steady state incompressible Navier-Stokes equations. Particular emphasis was placed on the application of the velocity-correc- tion method, ensuring the continuity equation. The Moving Least Squares (MLS) approximation is employed for the construction of the shape functions, in conjunc- tion with the general framework of the point collocation method. Computations are obtained for regular and irregular nodal distributions, stressing the positivity con- ditions that make the matrix of the system stable and convergent. The accuracy and the stability of the proposed scheme are demonstrated through two representative, well-known, and established benchmark problems. The numerical scheme was also applied to a case with irregular geometry for marginally high Reynolds numbers.

A meshless collocation method based on the differential reproducing kernel interpolation

A differential reproducing kernel (DRK) interpolation-based collocation method is developed for solving partial differential equations governing a certain physical problem. The novelty of this method is that we construct a set of differential reproducing conditions to determine the shape functions of derivatives of the DRK interpolation function, without directly differentiating the DRK interpolation function. In addition, the shape function of the DRK interpolation function at each sampling node is separated into a primitive function processing Kronecker delta properties and an enrichment function constituting reproducing conditions, so that the nodal interpolation properties are satisfied. A point collocation method based on the present DRK interpolation is developed for the analysis of one-dimensional bar problems, two-dimensional potential problems, and plane problems of elastic solids. It is shown that the present DRK interpolation-based collocation method is indeed a truly meshless approach, with excellent accuracy and fast convergence rate.

Free Vibration of Layered Circular Cylindrical Shells of Variable Thickness Using Spline Function Approximation

Free vibration of layered circular cylindrical shells of variable thickness is studied using spline function approximation by applying a point collocation method. The shell is made up of uniform layers of isotropic or specially orthotropic materials. The equations of motions in longitudinal, circumferential and transverse displacement components, are derived using extension of Love's first approximation theory. The coupled differential equations are solved using Bickley-type splines of suitable order, which are cubic and quintic, by applying the point collocation method. This results in the generalized eigenvalue problem by combining the suitable boundary conditions. The effect of frequency parameters and the corresponding mode shapes of vibration are studied with different thickness variation coefficients, and other parameters. The thickness variations are assumed to be linear, exponential, and sinusoidal along the axial direction. The results are given graphically and comparisons are made with those results obtained using finite element method.