Fundamental solutions for the collocation method in three-dimensional elastostatics

A method referred to as the fundamental collocation method is applied to traction problems of three-dimensional linear isotropic elastostatics. In the method the governing equations are satisfied exactly using fundamental solutions corresponding to concentrated forces, while the boundary conditions are satisfied approximately using an overdeterminate collocation technique. Numerical results are given for three sample problems. The paper is concluded with a critical discussion of the merits of the method.     

A mass conserving multi-domain spectral collocation method for the Stokes problem

A muli-domain spectral collocation method is developed for the approximation of the Stokes problem in two dimensions. Compatible velocity and pressure approximations are constructed to ensure that the discrete problem is well-posed. The collocation scheme possesses the property that the continuity equation is satisfied at all the collocation points. In addition, for problems defined in domains which are rectangularly decomposable, the scheme yields velocity approximations that are globally divergence-free.     

An adaptive method of fundamental solutions for solving the Laplace equation

In this paper, we propose a residual-type adaptive method of fundamental solutions (AMFS) for solving the two-dimensional Laplace equation. An error estimator is defined only on the boundary of the domain. Initial distributions of source points and collocation points are determined by using approaches proposed in Chen et al. (2006). The adding, removing, and stopping strategies are designed so that the required accuracy can be satisfied within finite steps. Numerical experiments reveal that AMFS improves the accuracy of the MFS approximation obtained from uniformly distributed sources and collocation points, which makes the MFS more practical for non-harmonic and non-smooth boundary conditions. Moreover, it is shown that the error estimator becomes equidistributed after an adaptive iteration. A detailed comparison between AMFS and MFS using uniformly distributed points is also presented for each numerical example.     

Radial basis function meshless method for the steady incompressible Navier–Stokes equations

A meshless collocation method based on radial basis functions is proposed for solving the steady incompressible Navier–Stokes equations. This method has the capability of solving the governing equations using scattered nodes in the domain. We use the streamfunction formulation, and a trust-region method for solving the nonlinear problem. The no-slip boundary conditions are satisfied using a ghost node strategy. The efficiency of this method is demonstrated by solving three model problems: the driven cavity flows in square and rectangular domains and flow over a backward-facing step. The results obtained are in good agreement with benchmark solutions.     

The fundamental collocation method applied to the nonlinear poisson equation in two dimensions

The fundamental collocation method is adapted to the nonlinear Poisson equation in two dimensions with mixed boundary conditions of the Dirichlet and Neumann type. The technique is an iterative collocation procedure which requires a representation of the boundary of a finite region by N points and of the interior by M points. The order of the problem as determined by the dimensions of the collocation matrices is N × N for each iteration. The method also employs an adjustable parameter S which can be used to check for stability. The accuracy and efficiency are shown to be quite good on three example problems, two of which are for heat-transfer and non-Newtonian laminar flow. Suggestions for improving the method are made.     

Moving least-squares approximations for linearly-solvable stochastic optimal control problems

Nonlinear stochastic optimal control problems are fundamental in control theory.A general class of such problems can be reduced to computing the principal eigenfunction of a linear operator.Here,we describe a new method for finding this eigenfunction using a moving least-squares function approximation.We use efficient iterative solvers that do not require matrix factorization,thereby allowing us to handle large numbers of basis functions.The bases are evaluated at collocation states that change over iterations of the algorithm,so as to provide higher resolution at the regions of state space that are visited more often.The shape of the bases is automatically defined given the collocation states,in a way that avoids gaps in the coverage.Numerical results on test problems are provided.     

A Hermite‐Lobatto Pseudospectral Method for Optimal Control

A pseudospectral (PS) method based on Hermite interpolation and collocation at the Legendre‐Gauss‐Lobatto (LGL) points is presented for direct trajectory optimization and costate estimation of optimal control problems. A major characteristic of this method is that the state is approximated by the Hermite interpolation instead of the commonly used Lagrange interpolation. The derivatives of the state and its approximation at the terminal time are set to match up by using a Hermite interpolation. Since the terminal state derivative is determined from the dynamic, the state approximation can automatically satisfy the dynamic at the terminal time. When collocating the dynamic at the LGL points, the collocation equation for the terminal point can be omitted because it is constantly satisfied. By this approach, the proposed method avoids the issue of the Legendre PS method where the discrete state variables are over‐constrained by the collocation equations, hence achieving the same level of solution accuracy as the Gauss PS method and the Radau PS method, while retaining the ability to explicitly generate the control solution at the endpoints. A mapping relationship between the Karush‐Kuhn‐Tucker multipliers of the nonlinear programming problem and the costate of the optimal control problem is developed for this method. The numerical example illustrates that the use of the Hermite interpolation as described leads to the ability to produce both highly accurate primal and dual solutions for optimal control problems.     

Boundary element analysis of linear viscoelastic problems using realistic relaxation functions

This paper presents an efficient method for the stress analysis of realistic viscoelastic solids by the time-domain boundary element method. The fundamental solutions and stress kernels are obtained using the elastic-viscoelastic correspondence principle. Since it is inconvenient to obtain the Laplace transform of the relaxation functions of realistic viscoelastic solids, the method of collocation has been employed and the relaxation function has been expanded in a sum of exponentials. Numerical results of example problems show the effectiveness and applicability of the proposed method.     

A symmetric variational finite element-boundary integral equation coupling method

This paper is concerned with the development of a mixed variational principle for coupling finite element and boundary integral methods in interface problems, using the generalized Poisson's equation as a prototype situation. One of its primary objectives is to compare the performance of fully variational procedures with methods that use collocation for the treatment of boundary integral equations. A distinctive feature of the new variational principle is that the discretized algebraic equations for the coupled problem are automatically symmetric since they are all derived from a single functional. In addition, the condition that the flux remain continuous across interfaces is satisfied naturally. In discretizing the problem within inhomogeneous or loaded regions, domain finite elements are used to approximate the field variable. On the other hand, only boundary elements are used for regions where the medium is homogeneous and free of external agents. The corresponding integral equations are discretized both by fully variational and by collocation techniques. Results of numerical experiments indicate that the accuracy of the fully variational procedure is significantly greater than that of collocation for the complete interface problem, especially for complex disturbances, at little additional computational cost. This suggests that fully variational procedures may be preferable to collocation, not only in dealing with interface problems, but even for solving integral equations by themselves.     

Collocation for initial value problems based on hermite interpolation

This paper proposes a technique to approximate the solutions of nonlinear initial value problems with Hermite interpolation polynomials, using collocation in a special way. Neverthless the method is shown to be equivalent with a classical collocation method involving a larger collocation system. The order of the method is investigated bringing out some connection with a class of hybrid multistep formulas. Sufficient conditions are given guarenteeing convergence of iterative method for the solution of the nonlinear collocation system. Work supported by the italian Ministero della Pubblica Istruzione.     

Genetic Programming Approaches for Solving Elliptic Partial Differential Equations

In this paper, we propose a technique based on genetic programming (GP) for meshfree solution of elliptic partial differential equations. We employ the least-squares collocation principle to define an appropriate objective function, which is optimized using GP. Two approaches are presented for the repair of the symbolic expression for the field variables evolved by the GP algorithm to ensure that the governing equations as well as the boundary conditions are satisfied. In the case of problems defined on geometrically simple domains, we augment the solution evolved by GP with additional terms, such that the boundary conditions are satisfied by construction. To satisfy the boundary conditions for geometrically irregular domains, we combine the GP model with a radial basis function network. We improve the computational efficiency and accuracy of both techniques with gradient boosting, a technique originally developed by the machine learning community. Numerical studies are presented for operator problems on regular and irregular boundaries to illustrate the performance of the proposed algorithms.      

Numerical investigation on convergence of boundary knot method in the analysis of homogeneous Helmholtz, modified Helmholtz, and convection-diffusion problems

This paper concerns a numerical study of convergence properties of the boundary knot method (BKM) applied to the solution of 2D and 3D homogeneous Helmholtz, modified Helmholtz, and convection-diffusion problems. The BKM is a new boundary-type, meshfree radial function basis collocation technique. The method differentiates from the method of fundamental solutions (MFS) in that it does not need the controversial artificial boundary outside physical domain due to the use of non-singular general solutions instead of the singular fundamental solutions. The BKM is also generally applicable to a variety of inhomogeneous problems in conjunction with the dual reciprocity method (DRM). Therefore, when applied to inhomogeneous problems, the error of the DRM confounds the BKM accuracy in approximation of homogeneous solution, while the latter essentially distinguishes the BKM, MFS, and boundary element method. In order to avoid the interference of the DRM, this study focuses on the investigation of the convergence property of the BKM for homogeneous problems. The given numerical experiments reveal rapid convergence, high accuracy and efficiency, mathematical simplicity of the BKM.     

求解最优控制问题的Chebyshev-Gauss伪谱法

提出了一种求解最优控制问题的Chebyshev-Gauss伪谱法, 配点选择为Chebyshev-Gauss点. 通过比较非线性规划问题的Kaursh-Kuhn-Tucker条件和伪谱离散化的最优性条件, 导出了协态和Lagrange乘子的估计公式. 在状态逼近中, 采用了重心Lagrange插值公式, 并提出了一种简单有效的计算状态伪谱微分矩阵的方法. 该法的独特优势是具有良好的数值稳定性和计算效率. 仿真结果表明, 该法能够高精度地求解带有约束的复杂最优控制问题.     

The method of fundamental solutions for oscillatory and porous buoyant flows

Accurate solutions of oscillatory Stokes flows in convection and convective flows in porous media are studied using the method of fundamental solutions (MFS). In the solution procedure, the flows are represented by a series of fundamental solutions where the intensities of these sources are determined by the collocation on the boundary data. The fundamental solutions are derived by transforming the governing equation into the product of harmonic and Helmholtz-type operators, which can be classified into three types depending on the oscillatory frequencies of temperature field. All the velocities, the pressure, and the stresses corresponding to the fundamental solutions are expressed explicitly in tensor forms for all the three cases. Three numerical examples were carried out to validate the proposed fundamental solutions and numerical schemes. Then, the method was also applied to study exterior flows around a sphere. In these studies, we derived the MFS formulas of drag forces. Numerical results were compared accurately with the analytical solutions, indicating the ability of the MFS for obtaining accurate solutions for problems with smooth boundary data. This study can also be treated as a preliminary research for nonlinear convective thermal flows if the particular solutions of the operators can be supplied, which are currently under investigations.     

Parallelizing spectral deferred corrections across the method

In this paper we present two strategies to enable “parallelization across the method” for spectral deferred corrections (SDC). Using standard low-order time-stepping methods in an iterative fashion, SDC can be seen as preconditioned Picard iteration for the collocation problem. Typically, a serial Gauß–Seidel-like preconditioner is used, computing updates for each collocation node one by one. The goal of this paper is to show how this process can be parallelized, so that all collocation nodes are updated simultaneously. The first strategy aims at finding parallel preconditioners for the Picard iteration and we test three choices using four different test problems. For the second strategy we diagonalize the quadrature matrix of the collocation problem directly. In order to integrate non-linear problems we employ simplified and inexact Newton methods. Here, we estimate the speed of convergence depending on the time-step size and verify our results using a non-linear diffusion problem.

     

Adaptive Techniques for Spline Collocation

We integrate optimal quadratic and cubic spline collocation methods for second-order two-point boundary value problems with adaptive grid techniques, and grid size and error estimators. Some adaptive grid techniques are based on the construction of a mapping function that maps uniform to non-uniform points, placed appropriately to minimize a certain norm of the error. One adaptive grid technique for cubic spline collocation is mapping-free and resembles the technique used in COLSYS (COLNEW) [2], [4]. Numerical results on a variety of problems, including problems with boundary or interior layers, and singular perturbation problems indicate that, for most problems, the cubic spline collocation method requires less computational effort for the same error tolerance, and has equally reliable error estimators, when compared to Hermite piecewise cubic collocation. Comparison results with quadratic spline collocation are also presented.     

Mixed operator problems using least squares finite element collocation

A finite element method based on least squares collocation on an element is formulated for problems of mixed type. The least squares method is developed using an incomplete quintic (C¹) element and overdetermined element collocation equations. Numerical experiments are conducted for the classical Tricomi equation, and the accuracy of computed solutions is examined at points in elliptic and hyperbolic subdomains.     

Solution of some boundary value problems in applied mechanics by the collocation least square method

The conventional simple but crude method of collocation is greatly improved by a least square augmentation. Simplicity in application and good accuracy of the proposed collocation least square scheme is demonstrated by the solution of some complex problems in applied mechanics.Example solutions of such problems include the linear and nonlinear analyses of isotropic plates, orthotropic plates and plates on elastic foundations. Numerical and graphical results are presented and compared with existing solutions. For the problems considered herein, the present method proves to be much less laborious than other numerical methods frequently employed by previous investigators.     

An effective pseudospectral method for constraint dynamic optimisation problems with characteristic times

Dynamic optimisation problem with characteristic times, widely existing in many areas, is one of the frontiers and hotspots of dynamic optimisation researches. This paper considers a class of dynamic optimisation problems with constraints that depend on the interior points either fixed or variable, where a novel direct pseudospectral method using Legendre–Gauss (LG) collocation points for solving these problems is presented. The formula for the state at the terminal time of each subdomain is derived, which results in a linear combination of the state at the LG points in the subdomains so as to avoid the complex nonlinear integral. The sensitivities of the state at the collocation points with respect to the variable characteristic times are derived to improve the efficiency of the method. Three well-known characteristic time dynamic optimisation problems are solved and compared in detail among the reported literature methods. The research results show the effectiveness of the proposed method.     

Reduced Collocation Methods: Reduced Basis Methods in the Collocation Framework

In this paper, we present the first reduced basis method well-suited for the collocation framework. Two fundamentally different algorithms are presented: the so-called Least Squares Reduced Collocation Method (LSRCM) and Empirical Reduced Collocation Method (ERCM). This work provides a reduced basis strategy to practitioners who prefer a collocation, rather than Galerkin, approach. Furthermore, the empirical reduced collocation method eliminates a potentially costly online procedure that is needed for non-affine problems with Galerkin approach. Numerical results demonstrate the high efficiency and accuracy of the reduced collocation methods, which match or exceed that of the traditional reduced basis method in the Galerkin framework.