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The objective of this paper is to make a review on recent advancements of the modified finite point method, named MFPM hereafter. This MFPM method is developed for solving general partial differential equations. Benchmark examples of employing this method to solve Laplace, Poisson, convection-diffusion, Helmholtz, mild-slope, and extended mild-slope equations are verified and then illustrated in fluid flow problems. Application of MFPM to numerical generation of orthogonal grids, which is governed by Laplace equation, is also demonstrated.

In recent decades, a new type of numerical methods for solving partial differential equations (PDEs) without meshes has been developed. These methods are called meshless or meshfree methods, which refer to the idea that information of connectivity of each mesh or element is no longer required as the input data in this new development. It saves a lot of manpower in the tedious preprocessing stage in preparing numerical computations. However, the information of the discretized nodes, such as the position vectors, is still a necessary requirement.

The idea of meshless methods can be traced back to the smoothed particle hydrodynamics (SPH) for modeling the astrophysical phenomenon [

Meshless methods are categorized mainly into two parts [

In the 1990s, a general collocation method to solve linear partial differential equations was proposed by adopting the radial basis functions (RBFs) [

Apart from the combination of RBFs, one could use a polynomial to approximate the solution of a PDE and seek the coefficients of all the monomials by applying the collocation technique. However, polynomials could not be used in a global solution form such as what is usually did in RBF collocation methods. This is because the values of high degree terms could become extremely large at points far from the origin. In most polynomial collocation methods, polynomials are just applied to approximate the solution piece-wisely around discretized nodes. The moving least square (MLS) or weighted least square (WLS) approaches are always accompanied by the polynomial approximation to transform and combine the locally approximated solutions into a global solution form. In each local solution, basis functions are the monomials and their factors are just the coefficients. The least square (LS) approach with Taylor series expansion was developed for the finite difference with arbitrary meshes [

In the global solution form, the basis functions are called the shape functions and their factors are the values of the solution itself. This localizing approach is somewhat like localized RBF collocation methods [

Like most strong-form methods, collocation methods with local polynomial approximation are reported to be unstable [

Though sometimes unstable, the simplicity renders the strong-form methods worthy of further investigations. Studies on improving the collocation techniques and the construction of local clouds, which means the selection of neighboring nodes involved in the local approximation, are important issues for increasing the numerical stability [

Besides methods mentioned above, one could also identify a few literatures about meshless methods employing the reproducing kernel approximation [

Adopting the advantage of localization, a robust local polynomial collocation method, which is similar to some finite point methods in which polynomials are localized by placing their origins at collocation points [

The local polynomial collocation method proposed in [

Taking a general 2-D linear second order PDE as an example, the descriptions of this method and the conventional method are given as follows.

When solving a general 2-D linear, second order PDE is

Illustration of the nodal index relationship between local and global domains.

Since the modified finite point method is originated from the conventional finite point method, the collocation method in the conventional finite point method will be presented briefly for completeness in this section.

The coefficients

It should be noted that this approximation is only valid in the vicinity of

When employing a conventional collocation method to solve a PDE, what one needs to do is just use the approximations of the solution and its partial derivatives to formulate a global matrix system and solve it. In the process of collocation, because it is anticipated to find local approximations of the solution and its partial derivatives very close to the relevant exact values which are not known a priori, the approximations listed in (

In the conventional method, when the collocation point is on the boundary, the satisfactory of the governing equation is not used. When the collocation point is on the boundary where two or more conditions are given, only one of them is chosen for the conventional collocation. The partial derivatives are thus inaccurately estimated as a result of lacking the complete satisfaction of all equations. Inspired by [

Assembling the local approximations into a global matrix system, one also gets a global matrix equation as (

where

Both in the conventional finite point method and in the modified one, there are several parameters to be chosen. They are the shape parameter

In this section, the following problems of different form of partial differential equations are chosen as benchmark problems to verify the accuracy of MFPM.

A potential uniform flow passing around a single cylinder in an infinite 2-D domain is chosen as the benchmark. The radius of the cylinder is

Due to symmetry, only the upper left quadrant is modeled. The values of

Boundary conditions for a problem of uniform flow passing a circular cylinder.

Two types of nodal arrangement for a potential uniform flow passing a circular cylinder.

The profile of computed flow speed (

Computed flow speed profile along the body surface in the problem of a uniform flow passing a circular cylinder by two different nodal arrangements.

Consider the following 2-D Poison equation:

The exact solution of this Poisson equation can be found as

The verification of the MFPM to this problem was demonstrated in [

Comparisons of the numerical results (a) with the exact solutions (b) for the benchmark problem of Poisson equation. From top to bottom, they are the value of the solution and partial derivatives in

Consider the following steady-state convection-diffusion equation [

A strong convective case of

Nodal arrangement for the problem of the steady-state convection-diffusion equation.

Results of problem of the steady-state convection-diffusion equation.

In previous three benchmark problems, unknowns of real-valued governing equations are dealt with. There are physical problems governed by equation with complex-valued unknowns. It is intended to choose free-surface water waves problems to demonstrate applicability of MFPM to such problems.

In the constant water depth, the motion of progressive, monochromatic free-surface waves can be reduced to a 2-D problem governed by Helmholtz equation

In the problem of monochromatic waves diffracted by a vertical cylinder in a constant water depth, shown as in Figure

A sketch of progressive monochromatic waves scattered by a vertical cylinder in a constant water depth.

For linear progressive, monochromatic waves, the total velocity potential of wave motion equals the linear combination of incident wave and scattered waves, that is,

The analytical solution [

A case of water depth of 10 m and incident waves with period ^{−1}, is chosen for demonstration. The radius of the cylinder is chosen as 30 m while the open outer boundary is located at

Nodal arrangement for MFPM computations of wave scattering.

Contours of equal-amplitude of waves scattered by a cylinder in constant water depth,

Contours of equal-phase of waves scattered by a cylinder on constant water depth,

Contours of magnitude of velocity of waves scattered by a cylinder on constant water depth,

Comparisons of MFPM results and analytical solutions of amplitude, phase, and magnitude of velocity around at

After the correctness and applicability of MFPM are verified in the previous section, applications of MFPM to several problems are further demonstrated in this section.

Though meshless numerical methods are becoming more and more popular in recent decades, grid-based numerical methods, which have been fully developed, are still indispensable. Transformation of coordinates from Cartesian system (

One application of MFPM is to employ it to solve the Laplace equations of numerically generate 2-D orthogonal grids. Based on Cauchy-Riemann conditions, the orthogonal transformation from the physical domain (

with the boundary conditions

where

The relations between the partial derivatives are listed below:

where

A sketch of the transformation is illustrated in Figure

Sketch of a transformation between two orthogonal coordinates from the physical domain to the transformed domain.

Conventionally the four corners of the irregular domain have to be right angles and the irregular domain is denominated as a “hyper-rectangle” [

The domain of an irregular area with zero-degree internal corners.

Arrangement of the collocation points in the up-right quarter of the irregular area with corners with zero-degree angle.

There is no exact solution for this problem, but one can still verify the correctness of the transformation by checking the orthogonality. Because the orthogonality is guaranteed mathematically, the following relative error criteria are used to show the correctness of the results:

The subscript “

Choosing

Orthogonal grid lines in an irregular area with internal corners of zero-degree angle,

Orthogonal grid lines around a zero-degree internal corner,

Various numerical methods have been developed to study the combined water-wave refraction and diffraction problems in the last decade. Notably, Tsay and Liu [

Alternatively, the mild-slope equation derived by Berkhoff [

Following the procedures that Berkhoff [

Generally, the treatment of the boundary far from the cylinder is the same as in (

In this section, the MFPM is implemented for the solution of the mild-slope equation for combined refraction and diffraction when monochromatic waves encountered with scatters, due to change of water depth or artificial structures. In present numerical model, the velocity of fluid particles at the grid points can be found easily as part of the coefficients resulted from the coefficients of collocation at each computational node. To verify present numerical model, analytical solutions of a benchmark problem in shallow water waves scattered by a circular island on a paraboloidal shoal by Homma [

As shown in Figure

Definition of a circular island on a paraboloidal shoal.

Analytical solutions of propagating shallow water wave responses around this circular island for

Differentiating (

A case for a circular island with radius,

Contours of wave amplitude lines around a circular island on a paraboloidal shoal,

Contours of wave phase lines around a circular island on a paraboloidal shoal,

Contours of magnitude of velocity around a circular island on a paraboloidal shoal,

Numerical and analytic results of amplitude, phase, and magnitude of velocity around at

Although present meshless methods have advantages over element- or mesh-based numerical methods to avoid tedious preprocessing of the mesh connectivity information in most the numerical computations, there are problems in engineering and science consisting of different regions in its physical behaviors divided by a very thin barrier or crack, such as a breakwater in water-wave diffraction, a crack in thin plates, and a cutoff in groundwater seepage. In present MFPM, the local collocation is performed by searching some closest nodes near the base point. When physical behaviors in separated regions by a thin barrier, it requires to identify the nodes searched by relative distance are not from the different regions with different physical behaviors. Therefore, it is required to divide the whole domain into subdomains of different physical behaviors. It is referred to as regional connectivity to complement the searching procedures of the local nodes for appropriate collocations [

To study resonance phenomena in a harbor basin, Lee [

However, there exist two drawbacks of these kinds of numerical models. First, a tedious preprocessing procedure is required to provide the connectivity information of the computational nodes and meshes in FEM. Secondly, numerical models with linear shape functions fall short of providing velocity fields of the water-wave motions.

For water-wave combined refraction and diffraction problems with abrupt change of geometry in computational domains, such as existence of breakwater(s), a concept of subdomains of regional connectivity has been established to guarantee application of local collocation that is performed within appropriate subdomains in applications of MFPM. The governing equation and boundary conditions can be referred to as the same of (

Protruding breakwaters or sharp geometrical corners exist in most harbors and may behave like a branch-cut line to divide the water area into subdomains with different characters such as up-wave and down-wave sides, respectively. In the collocation method of MFPM, the approximate polynomial function is established by identifying mostly and closely nearby computational grids in relative distances to a base point. When a protruding breakwater exists, special care must be taken to avoid identifying grid points from different subdomains in this searching process to evaluate coefficients of local polynomials in collocation approximation. Therefore, the computational domain is divided into subdomains and regional connectivity is specified for further local polynomial approximation at each node. In diffraction of water-wave problems, a computational domain can be divided into three subdomains which are the region of ocean, the entrance region of the harbor, and the basin region, shown as in Figure

Regional connectivity of subdomains for closest nodal searching.

A circular harbor with protruding breakwaters into the ocean side is considered, as shown in Figure

Definition of a circular harbor with protruding breakwaters.

and

In this problem, verifications of numerical calculations will be presented by comparisons with analytical results of Mei & Petroni [

A case of this application with incident wave direction

Response curves at

Response curve at

Contours of wave amplitude of a circular harbor of radius,

Velocity vectors of a circular harbor of radius,

For waves with very long periods, such as tidal waves, the Coriolis effects cannot be ignored. Based on the mild-slope equation, an extended mild-slope equation has been formulated to take the effects of Coriolis effects into account, when wave period becomes very long or the domain of concerns is large [

For real value of

Pedlosky [

In this section, Coriolis effects on long waves propagating in an infinite channel are investigated. The channel has two parallel side vertical walls of width

An infinite channel of width

According to Pedlosky [

For numerical computations of MFPM, both radiation condition at the far-field open boundary and no-flux condition on the solid boundaries need to be modified accordingly. Based on the wave modulation of a vertical distribution with

Assuming that the flow velocities associated with wave action,

From (

Then from the no-flux condition,

For practical purposes, outgoing propagation of scattered waves satisfy the Sommerfeld [

In order to illustrate the Coriolis effects of waves propagating in a channel with width ^{−1} (equivalently at latitude = 40°) have been taken. The wave in the channel is propagating from the left to right. The result of analytical solution is shown as in Figure

The results of analytical solutions: (a) distribution of wave amplitude; (b) distribution of wave velocity.

For numerical computations, the incident wave amplitude is chosen to be

The results of numerical model: (a) distribution of wave amplitude; (b) distribution of wave velocity.

Comparisons of amplitude and velocity at

For several decades, free-surface water wave problems have been studied as potential-flow problems governed by the Laplace equation with nonlinear free surface boundary conditions. If the motions of the free-surface are periodic, the problems can be treated as frequency-domain problems as illustrated above. But if one wants to obtain more detail in the transient process of the wave motion, time-domain water wave problems are recommended to be considered.

Most of these studies have been carried out with boundary-element methods (BEM), subjected to a mixed Eulerian-Lagrangian (MEL) time marching approach [

For solving this time dependent problem, the time-domain firstly has to be discretized. At each time step, the Laplace equation needs to be solved once to obtain the velocity potential for the entire domain thus to get the velocity. Boundary conditions are updated by the motion of the solid boundary and prediction from the free-surface boundary, which involves nonlinear terms. One could use the a mixed Eulerian-Lagrangian (MEL) time marching approach in which the 4th order Runge-Kutta time integration is used, or choose a simpler but satisfactory time marching scheme proposed in [

In the case of strong nonlinearity, the free-surface particle trajectory

The above equation is applicable after the velocity potential for the entire domain is solved. Note that there is no need to solve the Laplace equation again because there is barely difference between the free-surface velocity potential at

The first application of the MFPM to transient water wave problem is the simulation of the solitary wave generation by a piston-type wave maker [

Figure

Snapshots of fluid particles during wave generating process, for the case of water depth with 20 cm and input wave height of 7.4 cm.

In [

The formulae to determine the values of

The numerical results show that by applying the 9th order solitary wave solution [

Evolution of the free surface wave for the case of

A numerical model for simulating the free surface waves of the liquid sloshing in a swaying tank was developed in [

Layout of the sloshing experiment of Liu and Lin (2008).

Discretizing a wave length with at least 20 segments, the initial nodal spacing on the free surface is chosen as 5.18 cm. The collocation points are initially distributed as a hexagonal close packing array so that the most compact nodal arrangement can be achieved. Therefore, the vertical nodal spacing on the side walls is 3 cm. Totally, there are 127 collocation points. The time step chosen in the simulation is 1/80 of the swaying period. The initial nodal distribution is shown in Figure

Initial nodal distribution of the numerical model of the liquid sloshing in a swaying tank.

Figure

Comparison of the numerical results with the experimental data.

Snapshots of traced fluid particles in the time interval of

Recent advancement on the modified finite point method is reviewed in this paper. A normalized parameter in the Gaussian distribution of the weighting factor is presented with recommendation of relevant parameters for better numerical results, regardless of the size of the computational domain. A concept of regional connectivity is introduced and demonstrated for computational domains consisting of different physical behaviors in subdomains. This concept will be needed for similar problems such as a crack in a thin plate and a cutoff wall in groundwater seepage.

Benchmark examples of employing this method to solve Laplace, Poisson, convection-diffusion, Helmholtz, mild-slope, and extended mild-slope equations are verified. This method could be employed to solve governing equations with real-valued unknowns as well as complex ones. Application of MFPM to numerical generation of orthogonal grids is illustrated to overcome difficulty of other methods. Problems of free-surface water waves represented in frequency- and time-domain are demonstrated for the applicability of MFPM. The derivatives of the potentials can be easily obtained as part of the solutions after the local collocations are achieved. This advantage of MFPM will provide great potentials in numerical simulations for physical problems when accurate derivatives of the solution functions are of particular interest.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the National Science Council of Taiwan for the financial support through Projects nos. 102-2221-E-002-155 and NSC 103-2911-I-006-302 in these years for developing this MFPM method.