Point Collocation Method Research Articles

New efficient algorithm for the isometric embedding of 2-surface metrics in three dimensional Euclidean space

We present a new numerical method for the isometric embedding of 2-geometries specified by their 2-metrics in three-dimensional Euclidean space. Our approach is to directly solve the fundamental embedding equation supplemented by six conditions that fix translations and rotations of the embedded surface. This set of equations is discretized by means of a pseudospectral collocation point method. The resulting nonlinear system of equations are then solved by a Newton–Raphson scheme. We explain our numerical algorithm in detail. By studying several examples we show that our method converges provided we start the Newton–Raphson scheme from a suitable initial guess. Our novel method is very efficient for smooth 2-metrics.

An efficient meshfree point collocation moving least squares method to solve the interface problems with nonhomogeneous jump conditions

We are going to study a simple and effective method for the numerical solution of the closed interface boundary value problem with both discontinuities in the solution and its derivatives. It uses a strong‐form meshfree method based on the moving least squares (MLS) approximation. In this method, for the solution of elliptic equation, the second‐order derivatives of the shape functions are needed in constructing the global stiffness matrix. It is well‐known that the calculation of full derivatives of the MLS approximation, especially in high dimensions, is quite costly. In the current work, we apply the diffuse derivatives using an efficient technique. In this technique, we calculate the higher‐order derivatives using the approximation of lower‐order derivatives, instead of calculating directly derivatives. This technique can improve the accuracy of meshfree point collocation method for interface problems with nonhomogeneous jump conditions and can efficiently estimate diffuse derivatives of second‐ and higher‐orders using only linear basis functions. To introduce the appropriate discontinuous shape functions in the vicinity of interface, we choose the visibility criterion method that modifies the support of weight function in MLS approximation and leads to an efficient computational procedure for the solution of closed interface problems. The proposed method is applied for elliptic and biharmonic interface problems. For the biharmonic equation, we use a mixed scheme, which replaces this equation by a coupled elliptic system. Also the application of the present method to elasticity equation with discontinuities in the coefficients across a closed interface has been provided. Representative numerical examples demonstrate the accuracy and robustness of the proposed methodology for the closed interface problems. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1031–1053, 2015

A boundary collocation meshfree method for the treatment of Poisson problems with complex morphologies

A new meshfree method based on a discrete transformation of Green's basis functions is introduced to simulate Poisson problems with complex morphologies. The proposed Green's Discrete Transformation Method (GDTM) uses source points that are located along a virtual boundary outside the problem domain to construct the basis functions needed to approximate the field. The optimal number of Green's functions source points and their relative distances with respect to the problem boundaries are evaluated to obtain the best approximation of the partition of unity condition. A discrete transformation technique together with the boundary point collocation method is employed to evaluate the unknown coefficients of the solution series via satisfying the problem boundary conditions. A comprehensive convergence study is presented to investigate the accuracy and convergence rate of the GDTM. We will also demonstrate the application of this meshfree method for simulating the conductive heat transfer in a heterogeneous materials system and the dissolved aluminum ions concentration in the electrolyte solution formed near a passive corrosion pit.

A multiple scale time domain collocation method for solving non-linear dynamical system

Recently, a simple time domain collocation (TDC) method was proposed by researchers including the first author of the present paper, and has been successfully applied to obtain the harmonic, subharmonic, and superharmonic responses of the nonlinear Duffing oscillator. The TDC method is based on the point-collocation method performing within an appropriate period of the periodic solution, wherein the approximate solution is assumed as a Fourier series. Upon using the TDC method, the ordinary differential equation is transformed into a system of non-linear algebraic equations (NAEs), which can be readily solved by an NAE solver. In this study, using the Duffing oscillator as the prototype, we develop a multiple scale time domain collocation (MSTDC) method, by introducing a series of optimal multiple scales to the Fourier series of the approximate solution, to alleviate the ill-posedness of the system of collocation-resulting NAEs due to the inclusion of very high order harmonics in the approximate solution. Besides the MSTDC method, a multiple scale differential transformation (MSDT) method is proposed by introducing the multiple scales to the classical differential transformation method. Finally, numerical experiments are carried out to verify the accuracy and efficiency of the MSTDC and the MSDT methods.

Nonlinear transient response of functionally graded shallow spherical shells subjected to mechanical load and unsteady temperature field

This paper is concerned with the transient deformation of functionally graded (FG) shallow spherical shells subjected to time-dependent thermomechanical load. Based on Timoshenko-Mindlin hypothesis and von Karman nonlinear theory, a set of nonlinear governing equations of motion for FG shallow spherical shells in regard to transverse shear deformation and all the inertia terms are established using Hamilton's principle. The collocation point method and Newmark-beta scheme in conjunction with the finite difference method are adopted to solve the governing equations of motion and the unsteady heat conduction equation numerically. In the numerical examples, the transient deflection and stresses of FG shallow spherical shells with various material properties under different loading conditions are presented.

MHD natural-convection flow in an inclined square enclosure filled with a micropolar-nanofluid

Abstract Transient, laminar, natural-convection flow of a micropolar-nanofluid (Al2O3/water) in the presence of a magnetic field in an inclined rectangular enclosure is considered. A meshless point collocation method utilizing a velocity-correction scheme has been developed. The governing equations in their velocity–vorticity formulation are solved numerically for various Rayleigh (Ra) and Hartman (Ha) numbers, different nanoparticles volume fractions (φ) and considering different inclination angles and magnetic field directions. The results show that, both, the strength and orientation of the magnetic field significantly affect the flow and temperature fields. For the cases considering herein, experimentally given forms of dynamic viscosity, thermal conductivity and electrical conductivity are utilized.

On the acoustic performance of rectangular splitter silencers in the presence of mean flow

Dissipative splitter silencers are often used to reduce the noise emitted in ventilation and gas turbine systems. It is well known that the acoustic performance of a splitter silencer changes under the influence of the convective effects of a mean gas flow and so in this article a theoretical model is developed to include the effects of mean flow. The theoretical model is based on a hybrid finite element method which enables the inclusion of bull nose fairings and a perforated screen separating the mean gas flow from a bulk reacting porous material. Predictions are compared against experimental measurements obtained both with and without mean flow. Good agreement between prediction and measurement is generally observed in the absence of mean flow, although it is seen that for silencers with a low percentage open area the silencer insertion loss is over predicted at higher frequencies. When mean flow is present, problems with the experimental methodology are observed at relatively modest mean flow velocities, and so comparison between prediction and experiment is limited to relatively low face velocities. However, experiment and theory both show that the insertion loss reduces at low frequencies when mean flow is in the direction of sound propagation, and at high frequencies the influence of mean flow is generally much smaller. Following additional theoretical investigations it is concluded that the influence of mean flow on splitter silencer performance should be accounted for at low frequencies when silencer airway velocities are greater than about 20m/s; however, at higher frequencies one may generally neglect the effect of mean flow, even at higher velocities. Predictions obtained using the hybrid method are also compared to a simplified point collocation approach and it is demonstrated that the computationally efficient point collocation method may be used to investigate the effects of mean flow in a splitter silencer without loss of accuracy.

The Calculation of the Dynamic Problem for the Thin Shell by Method of Convolution-Type Point Collocation

Dynamic analysis of elastic body for plate and shell is important and difficult to solute. The convolution-type weighted residuals method is a new method, which is applied to solve dynamic problems. And the method of convolution-type point collocation is applied to solve dynamic problems of thin shell. An effective approach to structural dynamic analysis of thin shell is provided.

Optimal scale polynomial interpolation technique for obtaining periodic solutions to the Duffing oscillator

A novel optimal scale polynomial interpolation method (OSPIM) is proposed to attack the Duffing oscillator. This method is based on the ideas of multi-scaling and equilibrated matrix, such that the condition number of the coefficient matrix of the polynomial interpolation is minimized. The OSPIM can eliminate the Runge phenomenon, which occurs in the conventional polynomial interpolation, and is well suited for solving nonlinear oscillatory systems. In addition, we further alleviate the ill-posedness of polynomial interpolation by proposing a half-order technique, with which one can use a \(m\)-order polynomial to interpolate as many as \(2m+1\) points. We then employ the half-order OSPIM, i.e., OSPIM(H), as a trial function in conjunction with the simple point-collocation method, to solve the nonlinear Duffing equation. Moreover, the differential transformation method is used for the first time to solve a forced Duffing oscillator to compare with the present method. Finally, illustrative examples verify the accuracy and efficiency of the present method.

The Calculation of the Dynamic Problem for the Beam with Arbitrary Boundary Condition by Method of Convolution-Type Point Collocation

The method of convolution-type point collocation to calculate the beam with arbitrary boundary conditions was presented by using series function in time domain and the mode shape function of the beam in space domain. And this method is applied to solve dynamic problems of beam with arbitray boundary condition. The calculation example showed this method was very simple, highly accurate and effective.

Multiobjective Groundwater Remediation Design Using a Coupled MFree Point Collocation Method and Particle Swarm Optimization

Pump and treat (PAT) is one of the commonly used techniques for groundwater remediation in which the contaminated groundwater is pumped, treated, and put back to the aquifer system or other sources. For an effective PAT system design, simulation-optimization (S/O) models are very useful; the simulation model helps in predicting the spatial and temporal variation of the contamination plume, and optimization models are used to minimize the cost of pumping. For large-scale groundwater flow and transport simulation, grid-based or mesh-based models such as the finite difference method (FDM) and the FEM have been found to be cumbersome and time consuming. Meshfree (MFree)-based numerical models that use a set of nodes scattered within the problem domain and on the boundaries of the domain regardless of the connectivity information between them are found to be very effective for large-scale field problems. In this study, the MFree point collocation method (PCM) is used for the groundwater flow and transport simulation, and an evolutionary algorithm based on particle swarm optimization (PSO) is proposed for optimization. The S/O model based on PCM and PSO with multiobjective (PCM-PSO-PAT-MO) strategies are developed and applied for remediation of an unconfined aquifer polluted by total dissolved solids (TDS) using PAT. The PCM-PSO–based models are found to be simple and effective in groundwater remediation design using PAT.

Nonlinear bending of size-dependent circular microplates based on the modified couple stress theory

The present study proposes a nonclassical Kirchhoff plate model for the axisymmetrically nonlinear bending analysis of circular microplates under uniformly distributed transverse loads. The governing differential equations are derived from the principle of minimum total potential energy based on the modified couple stress theory and von Karman geometrically nonlinear theory in terms of the deflection and radial membrane force, with only one material length scale parameter to capture the size-dependent behavior. The governing equations are firstly discretized to a set of nonlinear algebraic equations by the orthogonal collocation point method, and then solved numerically by the Newton–Raphson iteration method to obtain the size-dependent solutions for deflections and radial membrane forces. The influences of length scale parameter on the bending behaviors of microplates are discussed in detail for immovable clamped and simply supported edge conditions. The numerical results indicate that the microplates modeled by the modified couple stress theory causes more stiffness than modeled by the classical continuum plate theory, such that for plates with small thickness to material length scale ratio, the difference between the results of these two theories is significantly large, but it becomes decreasing or even diminishing with increasing thickness to length scale ratio.

Three-dimensional stress field analysis of rotating thick bidirectional functionally graded axisymmetric annular plates with nonuniform loads and elastic foundations

Based on the three-dimensional theory of elasticity, a comprehensive stress analysis is performed for the rotating bidirectional functionally graded thick axisymmetric circular/annular plates, for the first time. The plate may be subjected to arbitrary distributions of the transverse load and various mixed (Dirichlet-type and Neumann-type) edge conditions. Furthermore, the circular plate may be supported by a nonuniform elastic foundation or a rigid substrate. In contrast to the very limited works presented for the rotating functionally graded circular plates so far, the transverse flexibility and the transverse stress components are considered and studied in the present research. Since finite element and boundary element techniques, due to their integral natures, cannot adequately trace abrupt changes of the quantities, a second-order point collocation method with forward–backward schemes is adopted to solve the system of the governing and boundary conditions. Effects of the distributions of the various material properties (Poisson’s ratio, Young’s modulus, and mass density), angular velocity, foundation compliance, and edge conditions are evaluated. Results reveal that radially graded or transversely graded material properties significantly affect distribution and magnitude and location of the extrema of the stress components and the lateral deflections and orientation of the general neutral surface of the plate.

A POINT COLLOCATION SCHEME FOR THE STATIONARY INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

An efficient and stable point collocation scheme based on a meshfree method is studied for the stationary incompressible Navier-Stokes equations. We describe the diffuse derivatives associated with the moving least square method. Using these diffuse derivatives, we propose a point collocation method to fit in solving the Navier-Stokes equations which improves the stability of the direct point collocation scheme. The convergence of the numerical solution is investigated from numerical examples. The driven cavity ow and the backward facing step ow are implemented for the reliability of the scheme. Also, the viscous ow on complicated geometry is successfully calculated such as the ow past a circular cylinder in duct.

High Accurate Solutions of Nonlocal Elasticity for Sphere

The analysis of sphere nonlocal elasticity is carried out by using the improved point collocation method. The approach is based on the Eringen’s model and two and three dimension problems are transformed to one dimension problems considering the polar symmetry. One dimension second order differential equation in terms of radial displacement is derived with domain integral. Due to the excellent accuracy of the point collocation method to one dimension differential equation using the radial basis function interpolation, the numerical solutions can be used as benchmarks. This approach can be easily extended to dynamic nonlocal elasticity and plasticity for sphere.

A comparative study of point collocation based MeshFree and finite element methods for groundwater flow simulation

For the simulation of groundwater flow, numerical models based on either finite difference method or finite element method (FEM) are most commonly used. These methods require well-defined grids or mesh and also large efforts in pre-processing. But, recently developed MeshFree (MFree) method does not require a predefined grid or mesh and this translates into savings on both computational cost incurred and the time required. Several MFree methods have been proposed over the last decade, though some of them are not totally MFree in the truest sense. Some MFree methods, like those based on the point collocation method (PCM), do not have an associated mesh but others, such as those based on the Galerkin method, actually require an auxiliary mesh or cell structure. In this paper, a MFree method based on PCM is discussed and comparison of results obtained with the conventional FEM is presented for groundwater flow problems. Here, PCM- and FEM-based groundwater flow models for confined aquifer in two-dimensional are developed. The PCM model has been applied to hypothetical groundwater flow problems and results are compared with analytical solutions and FEM model results. MFree-based PCM model has been found to be more accurate than FEM results when compared with analytical solutions. The present study shows the effectiveness of PCM-based models for groundwater flow simulation.

A meshless scheme for incompressible fluid flow using a velocity–pressure correction method

A meshless point collocation method is proposed for the numerical solution of the steady state, incompressible Navier–Stokes (NS) equations in their primitive u–v–p formulation. The flow equations are solved in their strong form using either a collocated or a semi-staggered “grid” configuration. The developed numerical scheme approximates the unknown field functions using the Moving Least Squares approximation. A velocity, along with a pressure correction scheme is applied in the context of the meshless point collocation method. The proposed meshless point collocation (MPC) scheme has the following characteristics: (i) it is a truly meshless method, (ii) there is no need for pressure boundary conditions since no pressure constitutive equation is solved, (iii) it incorporates simplicity and accuracy, (iv) results can be obtained using collocated or semi-staggered “grids”, (v) there is no need for the usage of a curvilinear system of coordinates and (vi) it can solve steady and unsteady flows. The lid-driven cavity flow problem, for Reynolds numbers up to 5000, has been considered, by using both staggered and collocated grid configurations. Following, the Backward-Facing Step (BFS) flow problem was considered for Reynolds numbers up to 800 using a staggered grid. As a final example, the case of a laminar flow in a two-dimensional tube with an obstacle was examined.

Nonlinear transient response of fibre metal laminated shallow spherical shells with interfacial damage under unsteady temperature fields

This paper is concerned with the transient response of fibre metal laminated (FML) shallow spherical shells with interfacial damage subjected to the unsteady temperature field. An exact displacement field which satisfies the boundary conditions on the outer and inner surfaces and stress continuity conditions at interfaces is presented. The nonlinear governing equations of motion for FML shallow spherical shells including the transverse shear deformation are established using the Hamilton’s principle. The transient temperature is determined from the heat conduction equation by using the finite difference method. The collocation point method and Newmark method are adapted to solve the governing equations of motion numerically. The present model provides an effective method for nonlinear dynamic analysis of composites laminated structures with interfacial damage subjected to transient temperature fields. In the numerical examples, the transient temperature, thermal deformation and stresses of FML shallow spherical shells are presented. The transient temperature will increase from initial value with time to a steady state. The displacements and stresses of the shell increase with time and remain unchanged when the temperature is in a steady state. And the effects of interfacial damage on the mechanical behaviours of FML shallow spherical shells are also discussed.

An implicit meshless scheme for the solution of transient non-linear Poisson-type equations

Abstract A meshfree point collocation method is used for the numerical simulation of both transient and steady state non-linear Poisson-type partial differential equations. Particular emphasis is placed on the application of the linearization method with special attention to the lagging of coefficients method and the Newton linearization method. The localized form of the Moving Least Squares (MLS) approximation is employed for the construction of the shape functions, in conjunction with the general framework of the point collocation method. Computations are performed for regular nodal distributions, stressing the positivity conditions that make the resulting system stable and convergent. The accuracy and the stability of the proposed scheme are demonstrated through representative and well-established benchmark problems.

Interfacial damage analysis of shallow spherical shell with FGM coating under low velocity impact

Interfacial damage would seriously reduce the structures' property, especially structures with coating when under dynamic loading. In this paper, the interfacial damage of shallow spherical shell with functionally graded material (FGM) coating subjected to low velocity impact is studied. An interfacial damage analytical model is established based on the continuum theory-based interfacial damage constitutive relations. The A.E. Giannakopoulos's 2-D functionally graded material (FGM) contact model is applied to predict contact force. Motion equations for shallow spherical shell substrate and FGM coating are obtained by Reissner variation, respectively, and the dynamic analytical model is established by using interfacial connection relations to relate the motion equations for FGM coating and those for elastic shallow spherical substrate. The orthogonal collocation point method, the Newmark method and iterative method are used synthetically to solve the whole question. In numerical examples, the dynamic response of shallow spherical shell with FGM coating and contact force are obtained, and the effects of material and geometrical parameters of FGM coating on interfacial damage and contact force have been discussed.