A Local Meshless Method for Approximating 3D Wind Fields

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A Local Meshless Method for Approximating 3D Wind Fields DARRELL W. PEPPER AND JIAJIA WATERS Nevada Center for Advanced Computational Methods, University of Nevada, Las Vegas, Nevada (Manuscript received and in final form 5 August 2015) ABSTRACT An efficient, mesh-free numerical method has been developed for creating 3D wind fields using data from meteorological towers. Node points are placed within a region of interest, generally based upon topological features. Since meshless methods do not require connective mesh generation, storage is greatly reduced, permitting implementation of the code using MATLAB on a personal computer. Utilizing locally collocated nodes and radial basis functions, a 3D wind can be quickly created that satisfies mass consistency. The meshless method yields close approximations to results obtained with mesh-dependent finite-difference, finite-volume, and finite-element techniques.

1. Introduction An accurate depiction of 3D wind fields, especially in complex terrain, is crucial when conducting atmospheric studies. Unfortunately, the lack of adequate meteorological data makes this particularly difficult. Extracting realistic estimates of 3D wind from sparse meteorological data requires the use of numerical modeling—typically employing procedures based on extensive numerical meshing involving finite-difference, finite-volume, and finite-element methods (FDM, FVM, and FEM, respectively). Localized mesh adaptation has been shown to help alleviate global meshing requirements and improve accuracy (Pepper and Wang 2009). Meshless methods are becoming popular, principally due to their abilities to deal with complex geometrical problems with inhomogeneous or variable properties, the use of general-purpose algorithms, and the lack of a mesh requirement. There has been increasing research interest for applying meshless (or mesh free) methods to obtain approximate solutions of differential equations (see Atluri and Zhu 1998; Balachandran et al. 2008; Liu 2003; Li and Mulay 2013). In a meshless method, there is no need to compute mesh elements induced by the selected nodes in the domain; the meshless method uses a group of nodes to derive a

Corresponding author address: Darrell W. Pepper, Department of Mechanical Engineering, University of Nevada, Las Vegas, 4505 Maryland Parkway, Las Vegas, NV 89154-4027. E-mail: [email protected] DOI: 10.1175/JAMC-D-15-0246.1 Ó 2016 American Meteorological Society

solution without the need for a connectivity relationship. In many situations where explicit mesh generation is difficult, the meshless method is emerging as an alternative to mesh-based methods. Selection of node placement in mesh-free applications is addressed in Atluri and Zhu (1998), Choi and Kim (1999), and Gewali and Pepper (2010). Numerically creating mass-consistent 3D wind fields has been employed for many years and has been especially effective in modeling atmospheric dispersion. Many of these early models are discussed in Pepper and Wang (2009). In all of these studies, the construction of a mesh in order to simulate the physical domain requires time and effort and subsequently dictates the storage requirements of the model. This study used a meshless approach to generate a 3D wind field over the Nevada National Security Site (NNSS), previously known as the Nevada Test Site, located northwest of the city of Las Vegas, Nevada. Topological elevations were obtained from Digital Elevation Map (DEM) data, developed by the U.S. Geological Survey. Initial tests of the meshless method were conducted using several well-known benchmark test cases, and results were compared using FreeFEM (a free finite-element code; Hecht 2012). This is described in more detail in Waters and Pepper (2015). In this study, we apply the meshless method to establish 3D wind fields over the NNSS, and the results are compared with an h-adaptive finite-element model, previously described in Pepper and Wang (2009). When compared with using mesh-dependent schemes,

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FIG. 1. Structured meshed domain vs meshless domain with random spacing.

the meshless method is computationally more efficient and much easier to implement.

2. Mass-consistent wind fields The continuity equation for an incompressible flow can be written as ›u ›y ›w 1 1 5 0. ›x ›y ›z

(1)

A Sasaki variational technique (Sasaki 1958) is used to establish a Poisson equation for a set of Lagrange multipliers l(x, y, z) utilizing observed velocity values (u0, y 0, and w0 obtained from tower data), and incorporating Gauss moduli a to account for vertical versus horizontal influence. The resulting Euler– Lagrange equation for l(x, y, z) is expressed as 

2 2 a1 › l ›y0 ›w0 ›2 l ›2 l 2 ›u0 , 1 1 5 22a 1 1 1 ›x2 ›y2 a2 ›z2 ›x ›y ›z (2) where u0, y 0, and w0 are observed velocity values in the x, y, and z directions, and ai are the Gauss precision moduli, where a2i [ 1/(2s2i ) (si are observation tower errors; i.e., deviations of the observed field from the desired adjusted field). The Gauss precision moduli are assumed identical in the horizontal (x, y) plane. As discussed in Sherman (1978), these moduli are important in determining nondivergent wind fields over irregular terrain, with (a1/a2)2 being proportional to the magnitude of the expected (w/u)2. As previously discussed in Pepper and Wang (2009), minimum residual divergence occurs at about (a1/a2)2 5 0.01, with a1 (horizontal adjustment) and a2 (vertical adjustment) set to 0.01 and 0.1, respectively. Solving for l, the velocities are then adjusted to satisfy continuity, keeping the velocities obtained at the tower

locations essentially fixed. The velocities are adjusted once the Lagrange multipliers are obtained, that is, u 5 u0 1

1 ›l , 2a21 ›x

y 5 y0 1

1 ›l , 2a21 ›y

w 5 w0 1

1 ›l . 2a22 ›z

(3) and

(4) (5)

The measured tower velocities are generally updated every 15 min (averaged), and a new 3D wind field is generated again. Equations (3)–(5) are updated only once per cycle as new tower data is obtained. Nonzero adjustment of the velocity normal to the boundary implies mass entering or leaving the volume. The boundary condition l 5 0 is appropriate for open or ‘‘flow-through’’ boundaries. When ›l/›n 5 0 on the boundary, the adjusted values of the normal velocity are the same as the observed value, and is used for closed or ‘‘no-flow-through’’ boundaries. The resulting mass consistent wind field is realistic. Sherman (1978) and Dickerson (1978) showed that the technique was within a factor of 2 approximately 50% of the time and within an order of magnitude about 90% of the time.

3. Meshless approach Traditionally, PDEs are solved using FDM, FVM, and FEM techniques. These methods require that a mesh be created of the problem (or computational) domain. However, the meshless method does not require this process. Figure 1 shows structured, uniform nodal spacing that is common to finite-difference methods and a randomly spaced nodal array; while a nonuniform mesh can be created for the randomly

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spaced nodes, nodal connectivity is not required in the meshless method (see Fasshauer 2007, 2002). The roots of the development of meshless methods began in the 1970s (Franke 1982). However, they have become more widely recognized within the last 15 or so years. They have attracted notice for their ease in implementation, relative to the more traditional FDM, FVM, and FEM techniques, which rely on a mesh consisting of interconnected node points to calculate values of interest. The strengths of a nonstructured mesh become apparent when the problem involves complex geometries or time-varying boundaries, but it can still be troublesome to generate. Although a variety of meshless method implementations exist, most have one property in common: they do not require nodal connectivity (i.e., a formal mesh; there are some that claim to be meshless, but still rely on a mesh foundation). A more extensive explanation of the meshless method is provided in Liu (2003). To begin, the problem domain is populated with nodes scattered across the domain and boundary, as shown in Fig. 1. Generally, the nodes do not need to be distributed in a uniform manner, and can be denser in regions of greater interest. Shape functions are used to relate the influence of each node to the other nodes in the domain. This is particularly necessary when the nodes are unevenly distributed. Shape functions generally are referred to as the support domain for the node of interest, and can have weighted influence. Figure 2 shows an irregular shaped domain with a circular support region surrounding a node (note that this is done for each node located within the problem domain). A field variable [e.g., l(x)] is interpolated using the displacements at its nodes within the support domain. The shape functions can be used to write a PDE in nodal matrix form, and global matrices assembled for the entire problem domain. All that remains is to solve the matrices to obtain PDE solutions. Various categories of meshless methods include smoothed particle hydrodynamics, reproducing kernel particle, meshless Petrov–Galerkin, local radial point interpolation, finite point, and finite differences with arbitrary irregular grids. Each method has benefits and drawbacks. In this study, radial basis functions (RBFs) were used as this did not require special consideration in terms of nodal placement.

4. Radial basis functions One of the simplest implementations of a meshless technique is to use the RBFmethod, in which a basis

FIG. 2. Node placement and circle of influence.

function relates the influence of surrounding nodes to the node of interest. The nodes closest to the node of interest have the greatest influence. Nodes that are increasingly farther away have decreasing influence. The distance d between the radial positions r is defined as follows: di 5 [(r 2 ri )2 ]1/2 .

(6)

Table 1 shows different basis functions f that have been proposed. The most commonly used basis function is the multiquadratics (MQ), proposed by Hardy (1971) with an exponent of b 5 10.5. The MQ form, shown in Table 1, is a general form with the exponent b as another parameter to be optimized. If the exponent is set TABLE 1. Radial basis functions.

Item

Name

Expression

Shape parameters

1 2 3 4

MQ Gaussian TPS Logarithmic RBF

fj (d) 5 (d2i 1 c2j )b fj (d) 5 exp(2Cj d2j ) fj (d) 5 dhj fj (di ) 5 dhi log(di )

c, b C h h

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at 20.5, this gives the inverse MQ form. There are two shape parameters to be tuned using the MQ form: c and b. The shape parameter c varies with position and is discussed later. The Gaussian RBF gives an exponential function of the distance. The shape parameter in this RBF controls the decay rate. The thin-plate spline (TPS) is a special case of the MQ, having only one shape parameter to optimize rather than two. Additionally, the logarithmic RBF can be used with only one shape parameter to optimize. The MQ form is widely used in constructing approximate solutions to PDEs and was selected as the preferred approach for this study. Upon substitution of the distance formula, the basis function f is defined as fj 5 [(r 2 rj )2 1 c2j ]b , j 5 1, 2, . . . , N ,

(7)

where N is the total number of nodes. Upon defining the problem domain, the set of Lagrange multipliers l(x) can be expressed as N

l(x) 5

å fj (x)lj ,

(8)

j51

where lj is the Lagrange coefficient obtained at each point. These coefficients must be determined for each continuous function value. To solve an elliptic PDE, a linear operator (e.g., L [ =2 ) is applied to the interior domain V of the continuous function. For example, N

Ll(x) 5

å Lfj (x)lj .

(9)

j51

From this relation, the linear operator (the PDE) is applied to the basis function. Equation (2) requires calculating the first and second derivatives of the basis functions. Additionally, PDEs are subject to boundary conditions; this is taken into account by applying a boundary operator (B [ Dirichlet or Neumann condition on the boundary G), as

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TABLE 2. Shape parameter formulations. Reference

Shape parameter c

Hardy (1971) Franke (1982) Fasshauer (2002) Kansa (2005)

c 5 0:815d pffiffiffiffi c 5 1:25/ pffiffiffiffi N c 5 2/ N c2j 5 C1 [1 1 C2 (21) j ]

Most of the shape parameter formulations rely on the number of nodes N and distance d, where N d 5 (1/NI )åi51 di , with di being the distance between the ith data point and its nearest neighbor. Some formulations show a changing shape parameter based on the node position. The shape parameter can depend on many factors, including the number and distribution of nodes, the basis function, and computer precision (see Roque and Ferreira 2009). The formulation used in this study is based on the expression used by Kansa (2005), as listed in Table 2. Different constants for the shape parameter can be used for the interior V and the boundary G. For the interior, the parameter C1 is defined as an initial shape parameter for the domain. Parameter C2 is used to determine the amplitudes of C1 depending on whether j is even or odd. The constant C2 varies from 0.25 to 0.33. For the boundary, the constant C2 varies between 0.49 and 0.55.

b. Basis function Equation (2) can be expressed as
2 2 a1 › l ›2 l ›2 l 1 1 5 f (x) , ›x2 ›y2 a2 ›z2

x 2 V,

(11)

where x [ (x, y, z) with
f (x) 5 22a21

›u0 ›y0 ›w0 1 1 ›x ›y ›z

 (12)

at all interior points, and l(x) 5 g(x) ,

x 2 G,

(13)

N

Bl(x) 5

å Bfj (x)lj .

(10)

where g(x) denotes the divergence of the observed velocity values at the boundaries G. At this point, we introduce the MQ form of the basis functions for fj (x):

The shape parameter c is important for determining the accuracy of PDE solutions when applying radial basis functions. Various expressions for determining values of the shape parameter are listed in Table 2.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fj (x) 5 rj2 1 c2 5 (x 2 xj )2 1 (y 2 yj )2 1 (z 2 zj )2 1 c2 .

j51

a. Shape parameter

(14) Likewise, the derivatives can be expressed as

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›fj

x 2 xj 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ›x rj2 1 c2

›2 fj ›y2

5

›fj

y 2 yj 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ›y rj2 1 c2

(x 2 xj )2 1 (z 2 zj )2 1 c2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 3 rj2 1 c2

and

›fj

z 2 zj 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ›z rj2 1 c2 ›2 fj ›z2

Substituting Eq. (8) into Eqs. (11) and (13), one obtains "

N

å

›2 fj (xi ) ›x2

j51

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›2 fj (xi ) ›y2

1

#
2 2 a1 › fj (xi ) a2

›z2

lj (xi ) 5 f (xi ) ,

i 5 1, 2, . . . , NI , and (16) N

å fj (xi )lj 5 g(xi ),

j51

i 5 NI11 , NI12 , . . . , N ,

(17)

which produces an N 3 N linear system of equations for the unknown, lj.

5. Local RBF approach There are essentially two approaches to using RBFbased meshless methods: global and local collocation. In the more general global approach, as described in section 4, the collocation is made globally over the whole domain (total number of node points), where the size of the discretization matrices scales as the number of the nodes in the domain N. In the localized method, a local collocation is defined over each node point, ultimately creating a set of overlapping domains of influence (see Fig. 2). When using the local method, small systems of linear equations are solved for each node. In this case, the amount of computational work grows with the number of nodes. Although the localized approach is slightly more expensive when computing on serial processors, the method permits easy parallelization. In this study we use the local approach. When the nodes are nonuniformly spaced or the domain is irregularly shaped (Fig. 2), matrices formed from the global RBFs meshless method can become ill conditioned; that is, the condition numbers can become very large (see the appendix). When there are many points in the domain, the global method can lead to large computational times and storage demands (since a dense, fully populated matrix must be solved directly). Although some methods such as domain decomposition and appropriate preconditioning have been used to address some of these global meshless method issues, the choice of the shape parameters in the RBFs is sensitive to the distance between nodes and the variables in the

5

(y 2 yj )2 1 (z 2 zj )2 1 c2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 , 3 ›x2 rj2 1 c2

›2 f j

(x 2 xj )2 1 (y 2 yj )2 1 c2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 3 rj2 1 c2

(15)

equations. The global RBF meshless method is best suited for square and small domains. Pepper et al. (2014b) discuss the use of the global RBF approach for generating 3D wind fields. Most of the problematic issues arising from global RBFs meshless method can be mitigated by using the localized RBF method. In addition, preconditioning methods are not required. More details regarding the use of localized RBF methods is discussed in Waters and Pepper (2015). The localized RBF method collocates a small number of points on a domain V, essentially dividing the entire domain into much smaller subdomains. To approximate l(x) of Eq. (11) over the domain V a collocation on the subdomain, Vj 5 P m j,k51 , j 5 1, 2, . . . , N, will be considered. Here N denotes all points on the domain V and Pj is the center point, while the index k goes from 1 to m, and m is the number of nearest points from the point Pj. Figure 3 shows an example of a local domain of influence containing 15 points (i.e., m 5 15); the dark point is the center node point. We can approximate l(x) in the following form for every point Pj 5 (xj ) 2 Vj : m

l(xi ) 5

å fk (xi )lk,j ,

k51

xi 2 Vj ,

(18)

where lk,j are the coefficients of RBFs for Vj. The RBFs fk are the shape functions. Substituting Eq. (18) into Eq. (11), an m 3 m linear algebraic system is obtained for each Vj defined by interior point Pj m

"

å

k51

#
2 2 ›2 fk (xi ) ›2 fk (xi ) a1 › fk (xi ) lk,j (xi ) 5 f (xi ), 1 1 ›x2 ›y2 ›z2 a2

i 5 1, 2, . . . , m, xi 2 Vj (19) subject to the boundary conditions m

å fk (xi )lk,j (xi ) 5 g(xi ) ,

k51

i 2 NI11 , NI12 , . . . , N, xi 2 Vj . (20)

Equations (19) and (20) form an m 3 m linear system for the unknowns flk,j gm k51 for each Vj defined by

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FIG. 3. Localized nine-point stencil.

interior points Pj. Note that Eq. (19) applies to interior points (xi , i 5 1, 2, . . . , NI ) and Eq. (20) is for boundary points in Vj.

6. Simulation of the Nevada National Security Site Mass-consistent wind fields generated by the local meshless method were compared with results obtained from an h-adaptive finite-element model. More detailed information regarding the NNSS datasets, topography, wind tower locations, and adaptive finite-element technique are given in Pepper and Wang (2009). Model results were compared with the 10- and 50-m levels, as shown in Figs. 4 and 5. The additional velocity vectors that appear in the FEM solution occur as a result of the local refinement h associated with the adaptive process. Various test cases were run using a larger array of interior nodes. However, utilizing only 240 nodes in the meshless method yielded nearly matching values obtained in the FEM method with refined local meshing, ultimately employing over 12 500 nodes. Even with such a coarse density of nodes, the meshless results were still close in values and patterns when compared with the high-resolution finite-element model. While both numerical methods produce realistic 3D wind

fields, the computational work associated with the meshless approach is significantly less. Further, the meshless simulation used MATLAB on a personalcomputer platform whereas the h-adaptive FEM ran on a supercomputer and was written in FORTRAN.

7. Conclusions A meshless method was used to create massconsistent 3D wind fields utilizing a Sasaki variational approach and sparse data from meteorological towers. Wind fields were simulated from the Nevada National Security Site. Model results obtained with the meshless method were in good agreement with results obtained from an h-adaptive finite-element model. Meshless methods are cost efficient. To capture rapidly changing flow features accurately, conventional numerical approaches typically require localized fine meshing with large-scale calculations. The meshless technique requires fewer points to formulate a problem domain, and can give qualitatively similar results to mesh-based methods while requiring less computational time. The results of the meshless method do not appear to be sensitive to the location of the nodes. Using a meshless method, arbitrary points can be

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FIG. 4. (a) Meshless results for 10-m level, and (b) h-adaptive FEM results.

added or extracted to existing problems easily without resolving the entire system. The application of localized meshless methods for meteorological simulations appears promising. Such methods are particularly effective for large or irregularly shaped domains and appear advantageous over the use of the more common global meshless methods. Although

the localized meshless method is sensitive to the distribution of nodes in the influence domain, it is more flexible and stable than global meshless methods. The ability to select the number of points for collocation in the localized meshless technique results in a unique method that can be significantly more efficient but with comparable accuracy when compared with mesh-based methods.

FIG. 5. (a) Meshless results for 50-m level, and (b) h-adaptive FEM results.

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FIG. A1. Temperature distribution over L 5 1 with Dirichlet boundary conditions, T0 5 15 and TL 5 25 (from Pepper et al. 2014a).

APPENDIX

fj (x) 5 [r2 (x, xj ) 1 c2 ]1/2 ,

A 1D Meshless Example

where r(x, xj ) is the radial (Euclidean) distance from the expansion point xj to any point x, and c is a shape parameter that controls the flatness of the RBF. We use a global expansion for the 1D temperature as

Consider the one-dimensional problem in which temperature is governed in a region with the following boundary conditions (Pepper et al. 2014a):

e T(x) 5

2

d T(x) 1 T(x) 1 x 5 0, x 2 [0, L], dx2 T(0) 5 T0 , and T(L) 5 TL .

(A4)

N

å fj (x, xj )aj ,

(A5)

j51

(A1)

The exact solution to this problem is   TL 1 L 2 T0 cos(L) sin(x) 2 x T(x) 5 T0 cos(x) 1 sin(L) (A2)

where aj is the expansion coefficient at point j. The coefficients of the expansion have no physical meaning in contrast to mesh-based schemes. The shape factor is c 5 102, with r normalized with respect to a point spacing, and Dx taken to be constant using N 5 6 points discretization. High levels of accuracy can be obtained by controlling the shape parameter to provide a nearly flat profile for functions that do not feature discontinuities.

with the exact derivative of the temperature given by   TL 1 L 2 T0 cos(L) q(x) 5 2T0 sin(x) 1 cos(x) 2 1. sin(L) (A3) This temperature profile is shown in Fig. A1 for values of T0 5 15 and TL 5 25. Meshless methods stem from spectral and pseudospectral techniques where global Chebycheff and Legendre polynomial expansion on regular point distributions are used in the method of weighted residuals along with domain decomposition. The application of global and local RBF expansions on arbitrary point distributions produces the meshless method. In this example problem, the Hardy multiquadric RBF is used, that is,

FIG. A2. MATLAB output; circles denote exact values.

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TABLE A1. Comparison of the error in computing the temperature at the interior points i 5 2, 3, . . . , N 2 1 (from Pepper et al. 2014a).

i

FDM (1024)

FVM (1024)

FEM (1024)

BEM (1028)

Meshless (1025)

2 3 4 5

3.437 4.653 4.375 2.833

1.699 2.299 2.162 1.4

3.398 4.6 4.325 2.801

5.166 2.091 2.924 7.915

29.493 29.39 29.405 29.538

N

d2 fj (x, xj )

j51

dx2

å aj

N

d2 fj (x, xj )

j51

dx2

å aj

å aj fj (xi ,

j51

f1 (xN21 , x1 )

f2 (x1 , x2 ) L2 (x2 , x2 )

(A7)

N

å aj fj (x1 ,

j51

xj ) 5 T0

for

i 5 1 and

xj ) 5 TL

for

i5N.

N

å aj fj (xN ,

j51

(A8)

Defining the operator .

(A6)

Introducing the RBF expansion for the temperature and its second derivative into the governing equation, and collocating at the interior points, we get f3 (x1 , x3 ) L3 (x2 , x3 )

f4 (x1 , x4 ) L4 (x2 , x4 )

... ...

L4 (x3 , x4 ) ... .. . L4 (xN21 , x4 ) . . .

f2 (xN21 , x2 )

f4 (xN21 , x4 )

.

L2 (x3 , x2 ) L3 (x3 , x3 ) .. .. . . L2 (xN21 , x2 ) L3 (xN21 , x3 ) f3 (xN21 , x3 )

or in compact form [C]fag5fdg. These equations are readily solved by direct methods to yield the expansion coefficients a. Once these are found the temperature can be evaluated anywhere using the RBF expansion. The following code, written in MATLAB, used MQ as the RBFs with six nodes that were equally positioned within the domain L. This program uses the global RBF and is written to illustrate the ease of programming and implementation. Even though the matrix condition number is high, an accurate solution can still be obtained. The approximate solution is shown in Fig. A2, and the errors obtained using the FDM, FVM, FEM, boundary-element method (BEM), and meshless method are shown in Table A1. % MESHLESS METHOD USING MULTIQUADRIC RBF %number of grid points N 5 6; %mesh spacing dx 5 1/(N-1); T0 5 15; TL 5 25; %x location of the ith grid point

d2 fj (x, xj ) dx2

1 fj (xi , xj ) ,

(A9)

the meshless method can be assembled into the matrix set of equations:

..

f1 (x1 , x1 ) 6 L 6 1 (x2 , x1 ) 6 6 L (x , x ) 6 1 3 1 6 .. 6 6 . 6 6 L (x , x ) 4 1 N21 1

for

At the boundaries, we collocate the RBF expansion to impose the boundary conditions

Lj (x, xj ) 5

2

xj ) 5 2xi

i 5 2, 3, . . . , N 2 1:

We solve the problem in strong form, with the second derivative of the temperature defined as e d2 T(x) 5 dx2

N

1

...

9 8 9 38 a1 > fN (x1 , xN ) > T0 > > > > > > > > 7> > > > > > > LN (x1 , xN ) 7> a2 > 2x2 > > > > > > > > > > > > 7> > > > > > > > > LN (x2 , xN ) 7 a 2x < < = 3 3 = 7 7 .. 7> .. > 5 > .. > . > 7> . . > > > . > > > > > > 7> > > > > > > > > 7 > > > > a 2x LN (xN21 , xN ) 5> > > N21 > > N21 > > > > > > > : ; : ; aN fN (xN , xN ) TL

(A10)

for i 5 1:N xx(i) 5 (i-1)*dx; end %Define multiquadric interpolant and shape factor c1 c1 5 1000; n1 5 1; chi 5 @ (y,x) (11((y-x)^2/(c1*dx^2)))^(n1(3/2)); %Second derivative of the RBF ddchi5 @ (y,x) (3*((x-y)/20)^2/(4*((x-y)^ 2/40 1 1)^(5/2))-(1/(40*((x-y)^2/40 1 1)^ (3/2)))); %Build meshless matrix equations C 5 zeros(N,N); b 5 zeros(N,1); for i 5 2:N-1 for k 5 1:N C(i,k)5ddchi(xx(i),xx(k))1chi(xx(i), xx(k)); b(i,1)5-xx(i); C(il,k)5chi(xx(il),xx(k));

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C(1,k)5chi(xx(1),xx(k)); end end T0 5 15; TL525; b(1,1)5T0; b(N)5TL; %echo matrix C fprintf('The Condition Number is %d\n', cond(C)) %Solve for the coefficients a5C\b; %use meshless expansion to find the solution at interior points TMEM5zeros(N,1); for i 5 1:N for j 5 1:N TMEM(i,1)5TMEM(i,1)1a(j,1)*chi(xx(i), xx(j)); end end TMEM %Compare with exact solution TE5[15;18.72608;21.69763;23.78822; 24.90653;25]; plot(xx(1:6),TMEM,'or'); hold on; plot(xx(1:6),TE); C5 1.0000 0.9995 0.9980 0.9955 0.9921 0.9877 0.9746 0.9750 0.9746 0.9735 0.9715 0.9688 0.9735 0.9746 0.9750 0.9746 0.9735 0.9715 0.9715 0.9735 0.9746 0.9750 0.9746 0.9735 0.9688 0.9715 0.9735 0.9746 0.9750 0.9746 0.9877 0.9921 0.9955 0.9980 0.9995 1.0000 The Condition Number is 6.145000e110 TMEM 5 15.0000 18.7262 21.6977 23.7883 24.9066 25.0000

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