Traditional Collocation Method Research Articles

A new type of radial basis functions for problems governed by partial differential equations.

The aim of this paper is to introduce a novel category of radial basis functions that incorporate smoothing techniques. Initially, we employ the power augmented and shape parameter schemes to create the radial basis functions. Subsequently, we apply the newly-constructed radial basis functions using the traditional collocation method and singular values decomposition algorithm to solve the corresponding linear system equations. Finally, we analyze several pairs of radial basis functions in depth to address physical problems linked to thermal science that are governed by partial differential equations. The numerical results demonstrate that the radial basis functions constructed using the power augmented and shape parameter schemes exhibit remarkable performance.

Superconvergence points of integer and fractional derivatives of special Hermite interpolations and its applications in solving FDEs

In this paper, we study the theory of convergence and superconvergence for integer and fractional derivatives of the one-point and two-point Hermite interpolations. When considering the integer-order derivatives, exponential decay of the error is proved, and superconvergence points are located, at which the convergence rates areO(N−2) andO(N−1.5) better than the global rates for the one-point and two-point interpolations, respectively. HereNrepresents the degree of the interpolation polynomial. It is proved that theαth fractional derivative of (u−uN), withk<α<k+1, is bounded by its (k+1) th derivative. Furthermore, the corresponding superconvergence points are predicted for fractional derivatives, and an eigenvalue method is proposed to calculate the superconvergence points for the Riemann–Liouville derivatives. In the application of the knowledge of superconvergence points to solve FDEs, we discover that a modified collocation method makes numerical solutions much more accurate than the traditional collocation method.

Free element collocation method: A new method combining advantages of finite element and mesh free methods

In this paper, a new numerical method, named as the Free Element Collocation Method (FECM), is proposed for solving general engineering problems governed by the second order partial differential equations (PDEs). The method belongs to the group of the collocation method, but the spatial partial derivatives of physical quantities are computed based on the isoparametric elements as used in FEM. The key point of the method is that the isoparametric elements used can be freely formed by the nodes around the collocation node. To achieve a narrow bandwidth of the final system of equations, elements with a central node are recommended. For this purpose, a new 21-node quadratic element for 3D problems is constructed for the first time. Attributed to the use of isoparametric elements, FECM can result in more stable results than the traditional collocation method. In addition, the elements can be freely formed by local nodes, FECM has the advantage of mesh-free methods to fit complicated geometries of engineering problems. A number of numerical examples of 2D and 3D thermal and mechanical problems are given to demonstrate the correctness and efficiency of the proposed method.

Element differential method for solving transient heat conduction problems

In this paper, a new numerical method, Element Differential Method (EDM), is developed for solving transient heat conduction problems with variable conductivity. The key point of this method is based on the direct differentiation of shape functions of isoparametric elements used to evaluate the geometry and physical variables. A new collocation method is proposed for establishing the system of equations, in which the governing differential equation is collocated at nodes inside elements, and the flux equilibrium equation is collocated at interface nodes between elements and outer surface nodes of the problem. Attributed to the use of the Lagrange elements that can guarantee the variation of physical variables consistent through all elemental nodes, EDM has higher stability than the traditional collocation method. The other main characteristics of EDM are that no variational principle or a control volume are required to set up the system of equations and no integrals are included to form the coefficients of the system. Based on the implicit backward differentiation scheme, an unconditionally stable and non-oscillatory time marching solution scheme is developed for solving the time-dependent system equations. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

Hermite Radial Basis Collocation Method for Unsaturated Soil Water Movement Equation

Due to the nonlinear diffusion term, it is hard to use the collocation method to solve the unsaturated soil water movement equation directly. In this paper, a nonmesh Hermite collocation method with radial basis functions was proposed to solve the nonlinear unsaturated soil water movement equation with the Neumann boundary condition. By preprocessing the nonlinear diffusion term and using the Hermite radial basis function to deal with the Neumann boundary, the phenomenon that the collocation method cannot be used directly is avoided. The numerical results of unsaturated soil moisture movement with Neumann boundary conditions on the regular and nonregular regions show that the new method improved the accuracy significantly, which can be used to solve the low precision problem for the traditional collocation method when simulating the Neumann boundary condition problem. Moreover, the effectiveness and reliability of the algorithm are proved by the one-dimensional and two-dimensional engineering problem of soil water infiltration in arid area. It can be applied to engineering problems.

Element differential method and its application in thermal‐mechanical problems

SummaryIn this paper, a new numerical method, element differential method (EDM), is proposed for solving general thermal‐mechanical problems. The key point of the method is the direct differentiation of the shape functions of Lagrange isoparametric elements used to characterize the geometry and physical variables. A set of analytical expressions for computing the first‐ and second‐order partial derivatives of the shape functions with respect to global coordinates are derived. Based on these expressions, a new collocation method is proposed for establishing the system of equations, in which the equilibrium equations are collocated at nodes inside elements, and the traction equilibrium equations are collocated at interface nodes between elements and outer surface nodes of the problem. Attributed to the use of the Lagrange elements that can guarantee the variation of physical variables consistent through all elemental nodes, EDM has higher stability than the traditional collocation method. The other main features of EDM are that no mathematical or mechanical principles are required to set up the system of equations and no integrals are involved to form the coefficients of the system. A number of numerical examples of 2‐ and 3‐dimensional problems are given to demonstrate the correctness and efficiency of the proposed method.

A fast collocation method for a static bond-based linear peridynamic model

We develop a fast collocation method for a steady-state bond-based linear peridynamic model in two space dimensions. The method reduces the computational work from O(N2) per Krylov subspace iteration in a traditional collocation method to O(NlogN) and the memory requirement from O(N2) in a collocation method to O(N), where N is the number of unknowns in the discrete system. Furthermore, the method reduces the computational work of evaluating and assembling the stiffness matrix, which often constitutes a major portion of the overall computational work, from O(N2) in a collocation method to O(N). The significant reduction of the CPU times and storage in the fast collocation method is achieved by carefully exploring the structure of the stiffness matrix of the collocation method without any lossy compression involved. In other words, the fast method is evaluated in an equivalent but more efficient manner. Therefore, the fast method generates identical numerical solutions as the traditional collocation method does and naturally inherits the stability and convergence properties that were already proved for a traditional collocation method. Numerical results are presented to show the utility of the fast method.

Método de elementos de borde jerárquico basado en el árbol de Barnes-Hut aplicado a flujo reptante exterior

In this work, a hierarchical variant of a boundary element method and its use in Stokes flow around three-dimensional rigid bodies in steady regime is presented. The proposal is based on the descending hierarchical low-order and self-adaptive algorithm of Barnes-Hut, and it is used in conjunction with an indirect boundary integral formulation of second class, whose source term is a function of the undisturbed velocity. The solution field is the double layer surface density, which is modified in order to complete the eigenvalue spectrum of the integral operator. In this way, the rigid modes are eliminated and both a non-zero force and a non-null torque on the body could be calculated. The elements are low order flat triangles, and an iterative solution by generalized minimal residual (GMRES) is used. Numerical examples include cases with analytical solutions, bodies with edges and vertices, or with intricate shapes. The main advantage of the presented technique is the possibility of considering a greater number of degrees of freedom regarding traditional collocation methods, due to the decreased demand of main memory and the reduction in the computation times.

A new radial basis function for Helmholtz problems

In this paper, a new radial basis function (RBF) is proposed to solve Helmholtz problems in the traditional collocation method. Since the matrix equation arising from the RBF interpolation is ill-conditioned, a regularized singular value decomposition method is used to obtain a more accurate solution. Numerical examples of both direct and inverse problems are presented to demonstrate the effectiveness and applicability of the proposed RBF versus the traditional multiquadric RBF.

Analysis of a moving collocation method for one-dimensional partial differential equations

A moving collocation method has been shown to be very efficient for the adaptive solution of second- and fourth-order time-dependent partial differential equations and forms the basis for the two robust codes MOVCOL and MOVCOL4. In this paper, the relations between the method and the traditional collocation and finite volume methods are investigated. It is shown that the moving collocation method inherits desirable properties of both methods: the ease of implementation and high-order convergence of the traditional collocation method and the mass conservation of the finite volume method. Convergence of the method in the maximum norm is proven for general linear two-point boundary value problems. Numerical results are given to demonstrate the convergence order of the method.

Direct Trajectory Optimization by a Chebyshev Pseudospectral Method

We present a Chebyshev pseudospectral method for directly solving a generic Bolza optimal control problem with state and control constraints. This method employs Nth-degree Lagrange polynomial approximations for the stateand control variables with the values of these variables at the Chebyshev-Gauss-Lobatto (CGL) points as the expansion coefficients. This process yields a nonlinear programming problem (NLP) with the state and control values at the CGL points as unknown NLP parameters. Numerical examples demonstrate that this method yields more accurate results than those obtained from the traditional collocation methods.

A Spectral Patching Method for Direct Trajectory Optimization

A new direct method for trajectory optimization is presented in this paper. This method is based on using global orthogonal polynomials instead of the traditional spline-integration methods to discretize the dynamical constraints. The states and controls are approximated by interpolating polynomials with unknown coefficients as the values of the functions at the collocation points. The collocation points are the Legendre-Gauss-Lobatto points which provide the smallest error in interpolation in the least-squares sense. They also provide an expression for the Lagrange interpolating polynomials used as trial functions in terms of Legendre polynomials. In order to achieve a greater freedom in the distribution of the collocation points, we present a patching method wherein the time domain may be arbitrarily divided into a few subintervals. The state equations over these subintervals are discretized by the spectral collocation method. Then, the solutions on these subintervals are “patched” by a continuity condition. In this patching method, by judiciously combining the ideas of the traditional collocation methods with global spectral methods, we achieve advantages of both methods while using neither. Results for the simplified Moon-landing problem (selected in honor of Professor Battin) show that the spectral patching method works well for just two subintervals.

A collocation method for convection dominated flows

AbstractA collocation method based on multiple regions with moving boundaries placed in a flow field in which convection effects dominate, is proposed. By making the moving boundaries of the regions coincide with moving sharp fronts present in the solution of convection dominated problems, and thereby allowing higher concentration of meshes to be placed about the fronts, the proposed method is able to achieve very high accuracy. By having a moving mesh, the Peclet number characterizing the flow field depends upon velocity relative to a moving mesh in a region. Consequently by choosing proper velocities of the moving boundaries, the value of this Peclet number can be made as small as desired. The traditional collocation method based on centred discretization, when applied to each region in the field, produces oscillation free solutions even when the values of Peclet number based on absolute velocity are extremely large. In view of these characteristics the method appears to be an excellent candidate for the solution of any two‐phase flow problem containing sharp fronts.