Application of localized meshless methods to 2D shallow water equation problems

This study aims to apply the meshless local radial-basis-function differential quadrature (LRBFDQ) method to solve the shallow water equations (SWE). This localized approach is developed from the differential quadrature (DQ) method by employing the radial-basis functions (RBFs) as the trial functions. Comparing with global-type meshless methods, the present method is more appropriate to large-scale problems with complex shapes. Moreover the drawbacks rising from the poor selection of shape parameter and also the full resultant matrix with high condition number are reduced. For real hydraulic-engineering applications located in irregular domains, the LRBFDQ method is very suitable to solve these kinds of shallow-water problems. In this work, the numerical models are applied to simulate three typical 2D SWE problems: (1) a tidal-wave propagation, (2) a dam-break problem and (3) an inverse engineering problem: the numerical analysis of the inflow discharge of the Yuanshantze Flood Diversion (YFD) project in Taiwan. As a result, the adopted meshless method not only shows its algorithm superiority over other mesh-dependent numerical schemes, but also brings more efficiency than several conventional mesh or meshless methods. The application of YFD project also delivers its applicability of this meshless scheme to solve real-world engineering projects.     

自适应网格方法在Stokes问题形状最优控制中的应用

为了求解流体力学中的形状最优控制问题,本文提出了一种与最优化准则方法相耦合的自适应网格方法.优化的目标是使得流体流动的能量耗散达到最小,状态方程是Stokes问题.本算法可以在减少计算量的情况下,保证流体的界面达到较高的分辨率.最优化算法采用的是非常稳定的经典最优化准则方法,自适应网格的指示函数是通过材料分布的信息得到的.虽然本文只是考虑了Stokes问题,但所得算法可以用来解决很广泛的一类流体动力学中的形状或拓扑最优化问题.     

An h-p- multigrid method for finite element analysis

The finite element method generates solutions to partial differential equations by minimizing a strain energy based functional. Strain energy based techniques for adaptive mesh refinements are not always effective, however. The adaptive refinement technique proposed in this paper uses strain energy but also incorporates advantages from the h- and p- finite element methods, the multigrid method and a Delaunay based mesh generation method. The refinement technique converged rapidly and was numerically efficient when applied to determining stress concentrations around the circular hole of a thick plate under tension.     

A meshless method of lines for the numerical solution of KdV equation using radial basis functions

In this paper, a meshless method of lines (MOL) is presented for the numerical solution of the Korteweg–de Vries (KdV) equation. This novel method has an advantage over the traditional method of lines which approximates the spatial derivatives using finite difference method (FDM) or finite element method (FEM), because it does not need the mesh in the domain, and it approximates the solution using the radial basis functions (RBFs) on a set of node scattered in problem domain. A comparison among some RBFs is made in numerical examples. Numerical examples demonstrate the accuracy and easy implementation of this novel method and it is an efficient method for the nonlinear time-dependent partial differential equations (PDEs).     

Meshless methods based on collocation with radial basis functions

Meshless methods based on collocation with radial basis functions (RBFs) are investigated in detail in this paper. Both globally supported and compactly supported radial basis functions are used with collocation to solve partial differential equations (PDEs). Using RBFs as a meshless collocation method to solve PDEs possesses some advantages. It is a truly mesh-free method, and is space dimension independent. Furthermore, in the context of scattered data interpolation it is known that some radial basis functions have spectral convergence orders. This study shows that the accuracy of derivatives of interpolating functions are usually very poor on boundary of domain when a direct collocation method is used, therefore it will result in significant error in solving a PDE with Neumann boundary conditions. Based on this fact, a Hermite type collocation method is proposed in this paper, in which both PDEs and prescribed traction boundary conditions are imposed on prescribed traction boundary. Numerical studies shows that the Hermite type collocation method improve the accuracy significantly. Received 31 January 2000     

A multi‐level adaptive mesh refinement method for level set simulations of multiphase flow on unstructured meshes

An adaptive mesh refinement (AMR) technique is proposed for level set simulations of incompressible multiphase flows. The present AMR technique is implemented for two‐dimensional/three‐dimensional unstructured meshes and extended to multi‐level refinement. Smooth variation of the element size is guaranteed near the interface region with the use of multi‐level refinement. A Courant–Friedrich–Lewy condition for zone adaption frequency is newly introduced to obtain a mass‐conservative solution of incompressible multiphase flows. Finite elements around the interface are dynamically refined using the classical element subdivision method. Accordingly, finite element method is employed to solve the problems governed by the incompressible Navier–Stokes equations, using the level set method for dynamically updated meshes. The accuracy of the adaptive solutions is found to be comparable with that of non‐adaptive solutions only if a similar mesh resolution near the interface is provided. Because of the substantial reduction in the total number of nodes, the adaptive simulations with two‐level refinement used to solve the incompressible Navier–Stokes equations with a free surface are about four times faster than the non‐adaptive ones. Further, the overhead of the present AMR procedure is found to be very small, as compared with the total CPU time for an adaptive simulation. Copyright © 2016 John Wiley & Sons, Ltd.     

Development of RBF-DQ method for derivative approximation and its application to simulate natural convection in concentric annuli

 The radial basis functions (RBFs) have been proven to have excellent properties for interpolation problems, which can be considered as an efficient scheme for function approximation. In this paper, we will explore another type of approximation problem, that is, the derivative approximation, by the RBFs. A new approach, which is based on the differential quadrature (DQ) approximation for the derivative with RBFs as test functions, is proposed to approximate the first, second, and third order derivatives of a function. The performance of three commonly-used RBFs for some typical expressions of derivatives as well as the computation of one-dimensional Burgers equation are studied. Furthermore, the proposed method is applied to simulate natural convection in a concentric annulus by solving Navier–Stokes equations. The obtained results are compared well with exact data or benchmark solutions. Received: 27 June 2001 / Accepted: 29 July 2002     

How to adaptively resolve evolutionary singularities in differential equations with symmetry

Many time-dependent partial differential equations have solutions which evolve to have features with small length scales. Examples are blow-up singularities and interfaces. To compute such features accurately it is essential to use some form of adaptive method which resolves fine length and time scales without being prohibitively expensive to implement. In this paper we will describe an r-adaptive method (based on moving mesh partial differential equations) which moves mesh points into regions where the solution is developing singular behaviour. The method exploits natural symmetries which are often present in partial differential equations describing physical phenomena. These symmetries give an insight into the scalings (of solution, space and time) associated with a developing singularity, and guide the adaptive procedure. In this paper the theory behind these methods will be developed and then applied to a number of physical problems which have (blow-up type) singularities linked to symmetries of the underlying PDEs. The paper is meant to be a practical guide towards solving such problems adaptively and contains an example of a Matlab code for resolving the singular behaviour of the semi-linear heat equation.     

Adaptive moving mesh methods for simulating one‐dimensional groundwater problems with sharp moving fronts

Accurate modelling of groundwater flow and transport with sharp moving fronts often involves high computational cost, when a fixed/uniform mesh is used. In this paper, we investigate the modelling of groundwater problems using a particular adaptive mesh method called the moving mesh partial differential equation approach. With this approach, the mesh is dynamically relocated through a partial differential equation to capture the evolving sharp fronts with a relatively small number of grid points. The mesh movement and physical system modelling are realized by solving the mesh movement and physical partial differential equations alternately. The method is applied to the modelling of a range of groundwater problems, including advection dominated chemical transport and reaction, non‐linear infiltration in soil, and the coupling of density dependent flow and transport. Numerical results demonstrate that sharp moving fronts can be accurately and efficiently captured by the moving mesh approach. Also addressed are important implementation strategies, e.g. the construction of the monitor function based on the interpolation error, control of mesh concentration, and two‐layer mesh movement. Copyright © 2002 John Wiley & Sons, Ltd.     

A two-dimensional adaptive nodes technique in irregular regions applied to meshless-type methods

We propose a two-dimensional (2D) adaptive nodes technique for irregular regions. The method is based on equi-distribution principle and dimension reduction. The mesh generation is carried out by first producing some adaptive nodes in a rectangle based on equi-distribution along the coordinate axes and then transforming the generated nodes to the physical domain. Since the produced mesh is applied to the meshless-type methods, the connectivity of the points is not used and only the grid points are important, though the grid lines are utilized in the adapting process. The performance of the adaptive points is examined by considering a collocation meshless method which is based on interpolation in terms of a set of radial basis functions. A generalized thin plate spline with sufficient smoothness is used as a basis function and the arc-length is employed as a monitor in the equi-distribution process. Some experimental results will be presented to illustrate the effectiveness of the proposed method.     

Discrete differential operators on irregular nodes (DDIN)

A previous research made an integral mathematical contribution for obtaining local function interpolation using neighboring nodal values of the solution function. Subsequent researchers developed mesh‐free methods for Finite Element Method (FEM). This principle can also be used to obtain discrete differential operators on irregular nodes. They may be successfully applied to Finite Difference method, Moving Particle Semi‐implicit (MPS) method and Random Collocation Method (RCM). In this paper, we obtain discrete differential operators on irregular nodes and successfully apply them to solve differential equations using the RCM. We also discuss mathematical aspects of the MPS method. Copyright © 2011 John Wiley & Sons, Ltd.     

Implicit boundary method for finite element analysis using non‐conforming mesh or grid

In the finite element method (FEM), a mesh is used for representing the geometry of the analysis and for representing the test and trial functions by piece‐wise interpolation. Recently, analysis techniques that use structured grids have been developed to avoid the need for a conforming mesh. The boundaries of the analysis domain are represented using implicit equations while a structured grid is used to interpolate functions. Such a method for analysis using structured grids is presented here in which the analysis domain is constructed by Boolean combination of step functions. Implicit equations of the boundary are used in the construction of trial and test functions such that essential boundary conditions are guaranteed to be satisfied. Furthermore, these functions are constructed such that internal elements, through which no boundary passes, have the same stiffness matrix. This approach has been applied to solve linear elastostatic problems and the results are compared with analytical and finite element analysis solutions to show that the method gives solutions that are similar to the FEM in quality but is less computationally expensive for dense mesh/grid and avoids the need for a conforming mesh. Copyright © 2007 John Wiley & Sons, Ltd.     

Dynamic analysis of laminated composite plates traversed by a moving mass based on a first-order theory

The dynamic response of angle-ply laminated composite plates traversed by a moving mass or a moving force is investigated. For this purpose, a finite element method based on the first-order shear deformation theory is used. Stationary and adaptive mesh techniques have been applied as two different meshing schemes. The adaptive mesh strategy is then used to avoid off-nodal position of moving mass. In this manner, the finite element mesh is continuously adapted to follow and comply with the path of moving mass. A Newmark direct integration method is employed to solve the equations of motion. Parametric study is directed to find out how different parameters like mass of the moving object as well as the type of the angle-ply laminated composite plates affect the dynamic response. Numerical results show the significant effects of the stacking order on the dynamic responses of the composite structures under a moving mass. It is found that although [30/−60/−60/30] lamination shows the highest maximum vertical deflection but [−45/45/45/−45] lamination has the highest value of the dynamic amplification factor. The dynamic amplification factor for different stacking orders and mass velocities is less than 1.25.     

Dual boundary‐element method: Simple error estimator and adaptivity

This paper is concerned with the effective numerical implementation of the adaptive dual boundary‐element method (DBEM), for two‐dimensional potential problems. Two boundary integral equations, which are the potential and the flux equations, are applied for collocation along regular and degenerate boundaries, leading always to a single‐region analysis. Taking advantage on the use of non‐conforming parametric boundary‐elements, the method introduces a simple error estimator, based on the discontinuity of the solution across the boundaries between adjacent elements and implements the p, h and mixed versions of the adaptive mesh refinement. Examples of several geometries, which include degenerate boundaries, are analyzed with this new formulation to solve regular and singular problems. The accuracy and efficiency of the implementation described herein make this a reliable formulation of the adaptive DBEM. Copyright © 2011 John Wiley & Sons, Ltd.     

The method of fundamental solutions for nonlinear elliptic problems

A meshless method was presented, which couples the method of fundamental solutions (MFS) with radial basis functions (RBFs) and the analog equation method (AEM), to solve nonlinear problems. In this method, the AEM is used to convert the nonlinear governing equation into a corresponding linear inhomogeneous equation, so that a simpler fundamental solution can be employed. Then, the RBFs and the MFS are, respectively, used to construct the expressions of particular and homogeneous solution parts of the substitute equation, from which the approximate solution of the original problem and its derivatives involved in the governing equation are represented via the unknown coefficients. After satisfying all equations of the original problem at collocation points, a nonlinear system of equations can be obtained to determine all unknowns. Some numerical tests illustrate the efficiency of the method proposed.     

Refined h-adaptive finite element procedure for large deformation geotechnical problems

Adaptive finite element procedures automatically refine, coarsen, or relocate elements in a finite element mesh to obtain a solution with a specified accuracy. Although a significant amount of research has been devoted to adaptive finite element analysis, this method has not been widely applied to nonlinear geotechnical problems due to their complexity. In this paper, the h-adaptive finite element technique is employed to solve some complex geotechnical problems involving material nonlinearity and large deformations. The key components of h-adaptivity including robust mesh generation algorithms, error estimators and remapping procedures are discussed. This paper includes a brief literature review as well as formulation and implementation details of the h-adaptive technique. Finally, the method is used to solve some classical geotechnical problems and results are provided to illustrate the performance of the method.     

Solution of non‐linear boundary integral equations in complex geometries with auxiliary integral subtraction

The boundary integral equation that results from the application of the reciprocity theorem to non‐linear or non‐homogeneous differential equations generally contains a domain integral. While methods exist for the meshless evaluation of these integrals, mesh‐based domain integration is generally more accurate and can be performed more quickly with the application of fast multipole methods. However, polygonalization of complex multiply‐connected geometries can become a costly task, especially in three‐dimensional analyses. In this paper, a method that allows a mesh‐based integration in complex domains, while retaining a simple mesh structure, is described. Although the technique is intended for the numerical solution of more complex differential equations, such as the Navier–Stokes equations, for simplicity the method is applied to the solution of a Poisson equation, in domains of varying complexity. It is shown that the error introduced by the auxiliary domain subtraction method is comparable to the discretization error, while the complexity of the mesh is significantly reduced. The behaviour of the error in the boundary solution observed with the application of the new method is analogous to the behaviour observed with conventional cell‐based domain integration. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical simulation of two-dimensional combustion using mesh-free methods

The purpose of this research was to develop tools for numerical simulations of flame propagation with mesh-free radial basis functions (RBFs). Mesh-free methods offer many distinct advantages over traditional finite difference, finite element, and finite volume methods. Traditional Lagrangian methods with significant swirl require mesh stiffeners and periodic remeshing to avoid excessive mesh distortion; such codes often require user interaction to repair the meshes before the simulation can proceed again.A propagating flame of infinite extent is simulated as a collection of normalized cells with periodic boundary conditions. Rather than capturing the flame front, it is tracked as a discontinuity. The flame front is approximated as a product of a Heaviside function in the normal propagation direction and a piece-wise continuous function represented by RBFs in the tangential direction. The cells are subdivided into the burned and unburned sub-domains approximated by two-dimensional periodic RBFs that are constrained to be strictly conservative. The underlying steady flow is vortical with an input turbulent intensity. The governing equations are rotationally and translationally transformed to produce exact differentials that are integrated exactly in time.In the present paper, the previous results of Aldredge who used a finite-difference level-set method were compared. The physical behavior was remarkably similar, whereas the finite-difference level-set method required 14 h of CPU time, the RBF approach required only 120 CPU seconds on a desktop computer for the case with the largest turbulent intensity. Although there are no other papers that tried to duplicate the original results of Aldredge, the results that are reported here are consistent with the physics observed in other experimental and numerical investigations.     

A meshfree interpolation method for the numerical solution of the coupled nonlinear partial differential equations

This paper formulates a simple classical radial basis functions (RBFs) collocation (Kansa) method for the numerical solution of the coupled Korteweg-de Vries (KdV) equations, coupled Burgers’ equations, and quasi-nonlinear hyperbolic equations. Contrary to the mesh oriented methods such as the finite-difference and finite element methods, the new technique does not require mesh to discretize the problem domain, and a set of scattered nodes provided by initial data is required for realization of solution of the problem. Accuracy of the method is assessed in terms of the error norms $L_2,L_{\infty }$, number of nodes in the domain of influence, time step length, parameter free and parameter dependent RBFs. Numerical experiments are performed to demonstrate the accuracy and robustness of the method for the three classes of partial differential equations (PDEs).     

On approximate cardinal preconditioning methods for solving PDEs with radial basis functions

The approximate cardinal basis function (ACBF) preconditioning technique has been used to solve partial differential equations (PDEs) with radial basis functions (RBFs). In [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press], a preconditioning scheme that is based upon constructing the least-squares approximate cardinal basis function from linear combinations of the RBF–PDE matrix elements has shown very attractive numerical results. This preconditioning technique is sufficiently general that it can be easily applied to many differential operators.In this paper, we review the ACBF preconditioning techniques previously used for interpolation problems and investigate a class of preconditioners based on the one proposed in [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press] when a cardinality condition is enforced on different subsets. We numerically compare the ACBF preconditioners on several numerical examples of Poisson's, modified Helmholtz and Helmholtz equations, as well as a diffusion equation and discuss their performance.