meshfree explicit local radial basis function collocation ... · meshfree explicit local radial...
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ELSEVIER
An Intemational Journal Available online at www.scieneedirect.com computers &
.c,=.c= ~__~°..~¢T. mathematics with applications
Computers and Mathematics with Applications 51 (2006) 1269-1282 www.elsevier.com/locate/camwa
M e s h f r e e Expl i c i t Local Radia l Bas i s F u n c t i o n C o l l o c a t i o n M e t h o d for Di f fus ion P r o b l e m s
B . S A R L E R A N D R . V E R T N I K
L a b o r a t o r y for Mul t iphase Processes Nova Gorica Polytechnic , Vipavska 13
SI-5000, Nova Corica, S]ovenia
<bozidar. sarler><robert, vertnik>~p-ng, s i
A b s t r a c t This paper formulates a simple explicit local version of the classical meshless radial basis function collocation (Kansa) method. The formulation copes with the diffusion equation, ap- plicable in the solution of a broad spectrum of scientific and engineering problems. The method is structured on multiquadrics radial basis functions. Instead of global, the collocation is made locally over a set of overlapping domains of influence and the time-stepping is performed in an explicit way. Only small systems of linear equations with the dimension of the number of nodes included in the domain of influence have to be solved for each node. The computational effort thus grows roughly linearly with the number of the nodes. The developed approach thus overcomes the principal large- scale problem bottleneck of the original Kansa method. Two test cases are elaborated. The first is the boundary value problem (NAFEMS test) associated with the steady temperature field with simultaneous involvement of the Dirichlet, Neumann and Robin boundary conditions on a rectangle. The second is the initial value problem, associated with the Dirichlet jump problem on a square. Tile accuracy of the method is assessed in terms of the average and maximum errors with respect to the density of nodes, number of nodes in the domain of influence, multiquadrics free parameter, and timestep length on uniform and nonuniform node arrangements. The developed meshless method outperforms the classical finite difference method in terms of accuracy in all situations except imme- diately after the Dirichlet jump where the approximation properties appear similar. @ 2006 Elsevier Ltd. All rights reserved.
1. I N T R O D U C T I O N
The problems in science and engineering are usually reduced to a set of coupled partial differential equations. It is not easy to obtain their analytical solution, particularly in nonlinear and complex- shaped cases, and discrete approximate methods have to be employed accordingly. The finite- difference method (FDM), the finite-element method (FEM), and the boundary-element method (BEM) are most widely used among them at the present. Despite the powerful features of these methods, there are often substantial difficulties in their application to realistic, geometrically complex, three-dimensional transient situations with moving and/or deforming boundaries. A common complication in the mentioned methods is the need to create a polygonisation, either in the domain and/or on its boundary. This type of (re)meshing is often the most time-consuming part of the solution process and is far from being fully automated.
The authors acknowledge the support of the Slovenian Ministry for High Education, Science and Technology in the framework of the Grant J2-640-1540-04.
0898-1221/06/$ - see front matter @ 2006 Elsevier Ltd. All rights reserved. Typeset by AA.¢S- ~ doi: 10.1016/j.camwa.2006.04.013
1270 B. NARLEI{ AND R. VERTNIK
In recent years, a new class of methods is in development which does not require polygonisation but uses only a set of nodes to approximate the solution. The rapid development of these types of meshfree (meshless, polygon-free) methods and their classification is elaborated in the very recent monographs [1-4]. A broad class of meshfree methods in development today is based on radial basis functions (RBFs) [5]. The RBF collocation method or Kansa method [6] is the simplest of them. This method has been further upgraded to symmetric collocation [7,8], to modified colloca- tion [9] and to indirect collocation [10]. The method has been already used in a broad spectrum of computational fluid dynamics problems [11] such as the solution of Navier-Stokes equations [12] or porous media flow [13] and the solution of solid-liquid phase change problems [14]. In con- trast to advantages over mesh generation, all the listed methods unfortunately fail to perform for large problems, because they produce fully populated matrices, sensitive to the choice of the free parameters in RBFs. Sparse matrices can be generated by the introduction of the compactly supported RBFs, and the accuracy of such an approach can be improved by the multilevel tech- nique [15]. One of the possibilities for mitigating the large fully populated matrix problem is to employ the domain decomposition [16]. However, the domain decomposition reintroduces some sort of meshing which is not attractive. The concept of local collocation in the context of an RBF-based solution of the Poisson equation has been introduced in [17,18]. For interpolation of the function value in a certain node the authors use only data in the (neighbouring) nodes that fall into the domain of influence of this node. The procedure results in a matrix that is of the same size as the matrix in the original Kansa method, however it is sparse. Circular domains of influence have been used in [17] and stencil-shaped domains in [18]. In [17], the one-dimensional and two-dimensional Poisson equation has been solved by using multiquadries and inverse mul- tiquadrics RBFs with a detailed analysis of the influence of the free parameter on the results. In [18], a class of linear and nonlinear elasticity problems have been solved with a fixed free parameter. The differential quadrature method that calculates the derivatives of a function by a weighted linear sum of functional values at its neighbouring nodes has been structured with the RBFs in [19]. Despite the local properties, the matrix still has a similar form as in [17,18]. This paper formulates a simple meshfree solution procedure for solving the diffusion equation which overcomes even the solution of the large sparse matrices.
2. G O V E R N I N G E Q U A T I O N S
Let us limit our discussion to solution of the heat diffusion equation, defined on a fixed domain gt
with boundary F
pc~--~T = V. (kVT), (I)
with p, c, k, T, t standing for density, specific heat, thermal conductivity, temperature, and time. The material properties p, c, k may depend on the position and temperature, i.e., the problem might be inhomogeneous and nonlinear. The solution of the governing equation for the temperature at the final time to + At is sought, where to represents the initial time and At the positive time increment. The solution is constructed by the initial and boundary conditions that follow. The initial value of the temperature T(p, t) at a point with position vector p and time to
is defined through the known function To
T(p, t0 )=T0(p) ; p C f ~ + r . (2)
The boundary F is divided into not necessarily connected parts F = F > U F N U I "R with Dirichlet, Neumann, and Robin type boundary conditions, respectively. At the boundary point p with normal nr and time to _< t _< to + At, these boundary conditions are defined through known
Meshfree Explicit Local Radial Basis Function 1271
functions TD, TN T~, R Wl~ref
T = TD; p C F D, (3)
--~-0 T = TN; p C F N, (4) 0np
0 On--~T = TF R (T - T~.ef) ; p E F R. (5)
3. S O L U T I O N P R O C E D U R E
The representa t ion of t empera tu re over a set of t N arb i t ra r i ly spaced nodes lP,,; n = 1, 2,
. . . , i N is made in the following way:
,K
T(p) ~ E l~k(p)lak, (6) k--i
where l~k stands for the shape functions, lak for the coefficients of the shape functions, and iK
represents the number of the shape functions. The left lower index on entries of expression (6)
represents the domain of influence (subdomain) lw on which the coefficients lak are determined.
The domains of influence lw can in general be overlapping or nonoverlapping. Each of the domains
of influence lw includes iN nodes of which zNa can in general be in the domain and tNr on the
boundary, i.e., iN = lNa +INF. The domain of influence of the node IP is defined with the
nodes having the nearest iN - 1 distances to the node tP. The five node zN = 5 and nine node
supports IN = 9 are used in this paper. The coefficients can be calculated fl'om the subdomain
nodes in two distinct ways. The first way is collocation (interpolation) and the second way is
approximation by the least squares method. Only the simpler collocation version for calculation
of the coefficients is considered in this text. Let us assume the known function values iTn in the
nodes tP,~ of the subdomain lw. The collocation implies
zN
T(lp~) = E l~k(IPn)lak. (7) k--1
For the coefficients to be computable , the number of the shape functions has to match the
number of the collocation points zK = zN, and the collocation ma t r ix has to be nonsingulm.
The sys tem of equations (7) can be wr i t ten in a matr ix-vector no ta t ion
l ~ l a -- lT; ~Ok~ = l~k(IPn), 1T,, = T( Ipn) . (8)
The coefficients l a can be computed by invert ing sys tem (8)
ta -- 10-liT. (9)
By taking into account the expressions for the calculation of the coefficients ic~, the collocation
representation of temperature T(p) on subdomain lw can be expressed as
zN zN
T(p) Z l'Tn k=l n = l
Let us in t roduce a two-dimensional Car tes ian coordinate sys tem with base vectors i~; g = x, y and coordinates pC; q = x, y, i.e., p - ixpx + i~py. The first par t ia l spa t ia l derivatives of T (p ) on
subdomain lw can be expressed as
0 iN ~ iN - - - -1 ( I I ) ape T(p) "~ E ape l~#k(P) E z%bkn tTn; ; = x,y.
k=l n = l
1272 B NARLER AND R. VERTNIK
The second partial spatial derivatives of T(p) on subdomain lw can be expressed as
02 zN 02 zN _ _ --1 ( 1 2 ) Op~p T(p) ~ ~-~ O~p l~k(p) ~-~l%nlTn; ~,~= z,p.
k=l n= l
Radial basis functions [5], such as multiquadrics, can be used for the shape functions
= [lrk(p ) + l~bk (p) 2 c21r~] 1/2 ; lr~ = (p -/Pk)" (P -lPk), (13)
where c represents the dimensionless shape parameter. The scaling parameter lr~ is set to the maximum nodal distance in the domain of influence
rm(lp,~); m , n = 1 ,2 , . . . ,tN. ir~ = max 2 (14)
The explicit values of the involved first and second derivatives of ~k(P) are
a l~k(P) P~ - tPk~ (15) - - z
opx (lr~ + ~lrg) ~/~'
O l~k(P) Py - lPky (16) apv (lr~ + dlr2) 1/2'
02 (PY - lPkY)2 + c2lr~ (17)
oP~ l ~ ( p ) = (l,~ + dl,'~o) ~/~ '
0~2~lCk(p) = (Px -- lPkx) 2 + c2t r2 (18)
(l,~ + cOo~) ~/~ '
0 2 0 2 (P~ -lPk~)(Pu --zPkv) - - l C k (P) = - - 1 0 k (P) = (19) op~;~ Op~;~ (l,'~ + ch~) ~/~
What follows elaborates the solution of the heat diffusion equation (1), subject to the initial condition (2), and boundary conditions (3)-(5). The diffusion equation can be transformed into the following expression by taking into account the explicit time discretization:
aT pocoT - pocoTo = V . (k0VT0). (20) p c - ~ ~ At
The unknown function value Tl in domain node Pl can be calculated as
T~Tol + a t V . (ko~VTol) = Tol + A--L-t [Wo, . VTo, + kol" V2Tol] • (21) Pol col Pot cot
The explicit calculation of the above expression in 2D is
T l = T0 l + - -
At lCk(Pl) E l~b[,llk" + p01c01
n : ]
A~k0l [~-~ 02 IN l~knl n AFPoICo1 Ln=l O-~p2~ tCk(Pl) n=lE - -1 T
At /~k(P/) E l~b2l]Cn l~k(Pl) E - 1 T • l ~ ) k n l On PolCOl Lk=l n= l [k=l (Ypz n=l
~'~=1 LN ~2 IN
U --1
n= l Upy n=l
(22)
where formulas (11) and (12) have been employed. The complete solution procedure follows the below-defined Steps 1-4.
Meshfree Explici t Local Radia l Basis Function 1273
STEP 1. First, the initial conditions are set in the domain and boundary nodes and the required derivatives are calculated from the known nodal values.
STEP 2. Equation (22) is used to calculate the new values of the variable iT, at time to + At in the domain nodes.
STEP 3. What follows in Steps 3 and 4 defines variable zT,, at time to + At in the Dirichlet, Neumann, and Robin boundary nodes. For this purpose, in Step 3, the coefficients ~a have to be determined from the new values in the domain and from the information on the boundary conditions. Let us introduce domain, Diriehlet, Neumann, and Robin boundary indicators for this purpose. These indicators are defined as
1, p n E ~ , TD = 1, T f l , ,= 0, p~ ¢ f~, { 0 ,
TFN 1, Pn c F N, R { 1, 0, P n ~ F N, T r n = 0,
Pn C F D,
p . ~ F D,
Pn C F R,
P~ ~ F R.
(23)
The coefficients za are calculated from the system of linear equations
zN zN D E/Ta.~lVk(,p.~)la~ + E ITr'~lCe(tPn)'C~k
k = l k = l
tN N 0 zN R 0
k=l k=l
= l T a ~ l T n + tTrnlT,~ + tTr~zT~ + ITr~lT{'n t~k(~P~)zak -- I T ~ f ~ • \ k = l
(24)
System (24) can be written in a compact form
l t~loz = l b , (25)
with the following system matrix entries:
D IO~k = ITanl¢k(tPT,) + ITrnl~bk(zPn)
+ ~ T F ~ b ~ z¢~(~p~) + ~TF,, ~¢~(~p~) - ~T[% ~ ~ ( ~ p ~ ) , k = i
(26)
and with the following explicit form of the augmented right-hand side vector:
Ibn t T f t n T n D D N N "wR , ~ R , T R = l l F ~ . l ' l " F n l ~ F ref n ' (27)
STEP 4. The unknown boundary values are set from equation (7).
The steady-state is achieved when the criterion
max ITn -- To,~l _~ Tste (2s)
is satisfied in all computational nodes Pn; n = I, 2,..., N. The parameter Tste is defined as the
steady-state convergence margin. In case the steady-state criterion is achieved or the time of
calculation exceeds the foreseen time of interest, the calculation is stopped.
1274 B. SARLER AND R. VERTNIK
4. N U M E R I C A L E X A M P L E S
4.1. First Test: B o u n d a r y V a l u e P r o b l e m
The problem is posed on a two-dimensional rec tangular domain t2 : p~- < p~ < p+, p~- <
p < y < p+, and bounda ry F 2 : p~ p~, p , _< py <_ p+, F + : p~ = p+, p~- _< py _< p+,
< + = P + , P2 < < + F ~ : py = p y , p~ <_ p~ _ p ~ , F + : py - P~ - Px with p~- = 0m, p+ = 0.6m, p~ = 0m, p+ = 1.0 m. The mater ia l proper t ies are p = 7850 k g / m a, c = 460 J / k g K , and k = 52 W / m K . The
bounda ry condit ions are on the south bounda ry F~- of the Dirichlet type with T D = 100°C, on
the east and nor th boundar ies F + and F + of the Robin type with T r R -- - h / k , h = 750 W / m 2 K ,
TrRr~f = 0°C, and on the west bounda ry F x of the Neumann type with T N = 0 ( W / m 2 ) / ( W / m K ) . The analyt ica l solution of the test is
f i cos/3 .x [/3~ cosh/3. (p+ - py ) + h s i n h ~ . (p+ - py ) ]
T~n~(p~,py , t) = 2 h T ~ cosfl~p + [(fl~ + h 2~ ..+ h] [fl~ cosh fl~p + + h s i n h f l , ~ p + 1 '
w i t h / ~ represent ing the posit ive roots of the equation
(29)
9tan [9 ( . t - -px)] = h (30)
This solution represents the N A F E M S benchmark test No. 10 [20]. The solution is in refer-
ence [20] given in terms of analyt ical value for t empera tu re TNAFEMS = 18.25°C at PNAFEMS x = 0.6 m, PNAFEMS y = 0.2 iI1. The rounded eight digit accmate analyt ica l solution used in this paper
is TNAFEMS = 18.253756°C. The a n a l ~ i c a l solution has been calculated also in all computa t iona l nodes in order to be able to calculate the error measures tha t follow. The max imum absolute t empera tu re error Tm~x and the average absolute t empera tu re error T~vg of the numerical solution at t ime t are defined as
T m ~ × = m a x l T ( p n , t ) - T ~ n ~ ( p , ~ , t ) l ; n = 1 , 2 , . . . , N , (31)
N 1 T ~ v ~ = E - ~ l T ( p ~ , t ) - T ~ n ~ ( p n , t ) l ; n = l , 2 . . . . ,N , (32)
n = l
where T and T~,~ s tand for numerical and analyt ical solution, N represents the to ta l number of
all nodes p~ of which first NF nodes correspond to the bounda ry and the remaining N a to the
domain. The node with the max imum t empera tu re error is denoted as Pm~×. The chosen error measures have been made compat ib le wi th the studies [21,22].
4.2. C o m p u t a t i o n a l Parameters and Discuss ion of the First Test Case
The calculat ions are performed on three uniform node ar rangements 13 × 21 (N = 269, N r = 60,
N~ = 209), 31 x 51 (N = 1577, N r = 156, N~ = 1421), 61 × 101 (N = 6157, N r = 316,
N~ = 5841), and on the nonuniform node ar rangement 61 x 101 (N = 6157, N r = 316, Nf~ = 5841). A schematic of the uniform node ar rangement 31 × 51 is shown in Figure la . A randomly
displaced nonuniform node ar rangement is generated from the uniform node ar rangement through
t ransformat ion
P ~ (nonuniform) = P ~ (uniform) at- Crandom57"minPn~ (uniform); q = X, y, (33)
where Cr~ndom represents a r andom number - 1 _< Crandor a _~ +1, 5 represents a displacement
factor (in this work fixed to 0.25), and rmin represents the min imum dis tance between the two nodes in uniform node arrangement . The bounda ry nodes are displaced only in the direct ion
perpendicular to the bounda ry normal. A schematic of the r andomly displaced nonuniform node
Meshfree Explicit Local Radial Basis Punction 1275
a r r a n g e m e n t 31 x 51 is shown in F igu re l b . The s t e a d y s t a t e is a p p r o a c h e d by a t r ans i en t ca l cu la t ion using a fixed t ime - s t ep A t = l s in regu la r node a r r a n g e m e n t s and A t = 0.5 s in
i r regu la r node a r r a n g e men t . The s t e a d y s t a t e c r i te r ion used in all c o m p u t a t i o n s is T~t~ = 10 -6 °C.
Tables 1-3 show accu racy of the so lu t ion as a funct ion of (nonsca led) m u l t i q u a d r i c s free p a r a m - e ter c for di f ferent node a r r a n g e m e n t s and f ive-noded d o m a i n of influence. One can observe the
i m p r o v e m e n t of the accu racy wi th h igher values of c and denser nodes . One can also observe t h a t
in case the va lue of c is f ixed far f rom the o p t i m a l value (i.e., i t t akes values 1 or 2) the m e t h o d does not converge wi th denser nodes . The so lu t ion above c - 8 shows on ly s l ight i m p r o v e m e n t
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(a) 31 x 51 uniform node arrangement. (b) 31 x 51 nonuniform randomly displaced node arrangement
Figure 1.
5"
1
~0.i 0
j
0.01
0.001
] 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i .................................... ~ . . . . . . . . . . .
: i
0 0 . 0 1 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 6
A h [ m ]
F i g u r e 2 . C o n v e r g e n c e p l o t o f t h e m a x i m u m e r r o r ( u p p e r c u r v e ) a n d a v e r a g e e r r o r
( l o w e r c u r v e ) a s a f u n c t i o n o f t h e m i n i m u m n o d e d i s t a n c e f o r t h e f i v e - n o d e d s u p p o r t
a n d t w o m u l t i q u a d r i c s p a r a m e t e r s . S o l i d l i n e ~ 3 1 x 5 1 n o d e a r r a n g e m e n t , d a s h e d
l i n e - - 6 1 x 1 0 1 n o d e a r r a n g e m e n t , m a x i m u m e r r o r I I , a v e r a g e e r r o r ¢ .
1276 B. SARLER AND R. VERTNIK
100
10
0.1
0.01
0.001
"i I . . . . \ - i - • . . . . . . .
0.2 0.4 0.6 0.8 1 1.2 1.4
c.r0 [ml
Figure 3. Convergence plot of the maximum error (upper curves) and average er- ror ( lower curves) a~ a func t ion of the m u l t i q u a d r i c s p a r a m e t e r s for t h e f ive-noded s u p p o r t a n d two node a r r a n g e m e n t s . Solid l i n e - - 3 1 x 51 node a r r a n g e m e n t , d a s h e d line---61 x 101 node a r r a n g e m e n t , m a x i m u m error - i , ave rage e r r o r - - 0 .
1.00E+00
1.00E-01
1.00E-02
1.00E-03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.00E-04
100E05 t 1.00E-06 - - , i i
0.001 0.01 0.1 1
t[s]
F igu re 4. A v e r a g e and m a x i m u m errors as a func t ion of t i m e for t h e second t e s t case ca l cu l a t ed w i t h t he F D M a n d the deve loped m e t h o d . Solid l i n e - - d e v e l o p e d m e t h o d , d a s h e d l i n e ~ F D M , m a x i m u m e r r o r - - m , ave rage e r r o r - - 0 .
in maximum error; however the average error can still be significantly improved when changing
the parameter c from 8 to 16. The solution with node arrangement 13 x 21 and c = 32 did not
converge. Table 4 shows the same type of information as Table 3, however with the nine-noded
domain of influence. In this case the solution with c = 16 and c = 32 did not converge. This
result is consistent with the fact tha t more nodes are used in mult iquadrics collocation methods
the more the free parameter is restricted to smaller values. Comparison of Tables 3 and 4 shows
that bet ter results can be achieved with smaller domain of influence. At the fixed parameter
c = 8 the average error is smaller with the smaller domain of influence and the maximum error
is smaller with the larger domain of influence.
Next, the calculations are performed on the nonuniform randomly displaced node arrangement.
Here, the results did not converge with the five-noded domain of influence. Comparison of
Tables 5 and 4 shows the expected degradation of the accuracy with the node arrangement
randomisation. Tables 6 and 7 show accuracy of the developed meshfree method in the NAFEMS
reference point as compared with the classical FDM. The accuracy of the meshfree method with
the uniform 61 x 101 node arrangement and five-noded domain of influence is almost two orders
of magnitude higher than with the FDM method structured in the same gridpoints. The error is increased in case of nonuniform node arrangement. However, also in nonuniform case, the error
M e s h f r e e Exp l i c i t L o c a l R a d i a l Bas i s F u n c t i o n
T a b l e 1. A c c u r a c y o f t h e s o l u t i o n as a f u n c t i o n of i n u l t i q u a d r i c s free p a r a m e t e r c in t e r m s of a v e r a g e a n d m a x i m u m e r r o r s a n d t h e p o s i t i o n of m a x i m u m e r r o r for 13 x 21 n o d e a r r a n g e m e n t a n d f l ve -noded d o m a i n o f inf luence .
e A7:, ,g [°c]
2
4
8
16
32
fTmax [°C] p . . . . . [m] p m a x y [m]
12 .8558 29 .9918 0 .0000 0 .3000
2 .6133 5 .5789 0 .0000 0 .3500
0 .3114 3 .4979 0 .6000 0 .0500
0 .0891 3 .1707 0 .6000 0 .0500
0 .0719 3 .0920 0 .6000 0 .0500
d iv
T a b l e 2. A c c u r a c y of t h e s o l u t i o n as a f u n c t i o n of m u l t i q u a d r i c s fi'ee p a r a m e t e r c in t e r m s of a v e r a g e a n d m a x i m u m e r r o r s a n d t h e p o s i t i o n of m a x i m u m e r r o r for 31 x 51 n o d e a r r a n g e m e n t a n d f i ve -noded d o m a i n of inf luence .
AT~v+ [°c]
1 23 .0263
9 .8407
1 .1828
0 .1087
0 .0261
0 .0196
2
4
8
16
32
A T m a x [°O] p . . . . [In]
57 .0299 0 .0000
21 .5167 0 .0000
2 .9379 0 .6000
2 .5910 0 .6000
2 .5172 0 .6000
2 .4996 0 .6000
pmax~ [m]
0 .2000
0 .3200
0 .0200
0 .2000
0 .2000
0 .2000
T a b l e 3. A c c u r a c y of t h e s o l u t i o n as a f u n c t i o n of m u l t i q u a d r i c s free p a r a m e t e r c in t e r m s of a v e r a g e a n d m a x i m u m e r r o r s a n d t h e p o s i t i o n o f m a x i m u m e r r o r for 61 x 101
n o d e a r r a n g e m e n t a n d f i ve -noded d o m a i n o f inf luence .
c ATavg [°C] Pm~x~ [m] ATm~x r ~1 Lo~] p .... [m]
72.9256 0 .0000
42 .4449 0 .0000
8 .3054 0 .0000
1 .7675 0 .6000
1 .7042 0 .6000
1 .6903 0 .6000
27 .8036 0 .1300
18 .5475 0 .2500
3 .9476 0 .3500
0 .3210 0 .0100
0 .0314 0 .0100
0 .0107 0 .0100
2
4
8
16
32
T a b l e 4. Accuracy of t h e s o l u t i o n as a f u n c t i o n of m u l t i q u a d r i c s free p a r a m e t e r c in t e r m s of a v e r a g e a n d m a x i m u m e r r o r a n d t h e p o s i t i o n o f m a x i m u m e r r o r for 61 x 101
n o d e a r r a n g e m e n t a n d n i n e - n o d e d d o m a i n of inf luence .
Pm.x~ [m] ATma× [°C] p . . . . . [m]
50 .4141 0 .0000
15 .0366 0 .0000
1.4331 0 .0000
0 .8284 0 .6000
c ATavg [°C]
2
4
8
16
32
21 .3924 0 .2200
7 .0673 0 .3300
0 .6918 0 .3700
0 .0704 0 .0100
d iv
d iv
1277
1278 B. SARLER AND R. VERTNIK
T a b l e 5. A c c u r a c y o f t h e s o l u t i o n as a f u n c t i o n of m u l t i q u a d r i c s free p a r a m e t e r c in t e r m s of a v e r a g e a n d m a x i m u m e r r o r a n d t h e p o s i t i o n o f m a x i m u m e r r o r for 61 x 101 n o n u n i f o r m n o d e a r r a n g e m e n t a n d n i n e - n o d e d d o m a i n of inf luence .
c fT~vg [°c I
2
4
5
16
32
A T m a x [o@] p . . . . [m] Pmax~ [m]
22.5573 53 .1913 0 .0000 0 .2113
9 .3926 20 .0383 0 .0000 0 .3191
1 .2326 2 .5587 0 .0000 0 .3897
0 .5637 1 .1684 0 .0000 0 .3897
d iv
d iv
T a b l e 6. A c c u r a c y of t h e s o l u t i o n in N A F E M S r e f e r e n c e p o i n t PNAFEMS as a f u n c t i o n g r id d e n s i t y a t f ixed m u l t i q u a d r i c s free p a r a m e t e r for d i f fe ren t n o d e a r r a n g e m e n t s a n d five- a n d n i n e - n o d e d d o m a i n s of inf luence .
G r i d S u p p o r t
13 x 21 5
31 x 51 5
61 x 101 5
61 x 101 (un i fo rm) 9
61 x 101 ( n o n u n i f o r m ) 9
c T [°C] T - TNAFEMS [ ° e l
16 18 .3613 0 .1075
32 18 .2855 0 .0317
32 18 .2594 0 .0056
8 18 .2512 - -0 .0026
5 18 .0395 0 .2143
T a b l e 7. A c c u r a c y of t h e c l a s s i ca l F D M s o l u t i o n in N A F E M S re f e r ence p o i n t as a
f u n c t i o n of r e g u l a r g r i d dens i ty .
G r i d
61 x 101
121 x 201
241 x 401
T [°el
17.9827
18 .1074
18 .1754
T - TNAFEMS [°C]
- -0 .2711
- -0 .1464
- -0 .0783
in the reference point is only 0.2143°C (i.e., in the permille range) compared to the character is t ic
problem t empera tu re difference of 100°C.
4.3. S e c o n d Test: Ini t ia l Value P r o b l e m
The geomet ry of the problem is formally posed on a similar region as the first tes t case; however
the region is square with Px = 0m, p+ = 1.0m, py = 0m, p+ = 1.0m. The mate r i a l proper t ies
are set to unit values p = 1 k g / m 3, c = 1 J / k g K , k = 1 W / m K . Bounda ry condit ions on the
east Fx + and nor th boundar ies F + are of the Diriehlet type with T D = 0°C, and on the west F~-
and south boundar ies F~ are of tile Neumann type with T N = 0 ( W / m 2 ) / W / m K ) .
The init ial condit ions are To = 1°C. The analyt ical solution [23] of the test is
T~n~(p~,pu, t) = T~n~(p~, t)T~n~(pu, t)
with
<~,,~,(p~, t) = _4 ~ - 1 n 7~ 2 n + 1
k(2n + 1)%r2t - - exp cos [(2~ + 1)~(p~ - -2-(p~ -~-~ 1) ] ; -- x, y. (32)
M e s h f r e e E x p l i c i t L o c a l R a d i a l B a s i s F u n c t i o n 1279
4 . 4 . C o m p u t a t i o n a l P a r a m e t e r s a n d D i s c u s s i o n o f t h e S e c o n d T e s t C a s e
T h e c a l c u l a t i o n s a r e p e r f o r m e d o n f o u r d i f f e r e n t u n i f o r m n o d e a r r a n g e m e n t s 1 1 x 11 ( N = 1 1 7 ,
N r = 36, N a = 81), 21 x 21 ( N = 437, N r = 76, N a = 361), 41 x 41 ( N = 1677, N r = 156,
N a = 1521), 101 × 101 ( N = 10197, N r = 396, Nf~ = 9801), and f ive-noded d o m a i n of influence,
respect ively . T h e t imes t eps A t = 10 . 4 s and A t = 10 . 5 s are used in t h e ca lcu la t ions wi th 41 x 41
node a r r angemen t . T h e accuracy of the m e t h o d is assessed in t e r m s of t he m a x i m u m and average
errors a t t imes t = 0.001 s, t = 0.01s, t = 0.1s, and t = 1.0s.
Tables 8-10 show accuracy of t he so lu t ion as a func t ion of mu l t i quad r i c s free p a r a m e t e r c for
different grid a r r a n g e m e n t s and f ive-noded suppor t . One can observe t h e i m p r o v e m e n t of t he
T a b l e 8. A c c u r a c y o f t h e s o l u t i o n a s a f u n c t i o n o f n m l t i q u a d r i c s f r e e p a r a m e t e r c
a t t i m e s t = 0 . 0 0 1 s , t = 0 . 0 1 s , t = 0 . 1 s , a n d t = 1 . 0 s in t e r m s o f a v e r a g e e r r o r ,
m a x i m u m e r r o r , a n d t h e p o s i t i o n o f t h e m a x i m u m e r r o r for 11 x 11 n o d e a r r a n g e m e n t
w i t h f i v e - n o d e d d o m a i n o f i n f l u e n c e a n d A t = 10 . 4 s.
t Is] c ATavg [o c]
0 .001 8 1 . 1 8 4 E - 2
0 .001 16 1 . 1 8 2 E - 2
0 .001 32 1 . 1 8 1 E - 2
0 .01 8 4 . 8 6 2 E - 3
0 .01 16 4 . 7 6 7 E - 3
0 .01 32 4 . 7 4 6 E - 3
0 .1 8 1 . 2 4 2 E - 3
0.1 16 7 . 1 3 9 E - 4
0.1 32 6 . 2 3 9 E - 4
1.0 8 6 . 9 9 7 E - 5
1.0 16 9 . 0 6 8 E - 6
1.0 32 4 . 6 8 1 E - 6
i r m a x [°C] p . . . . [In] P m a x 9 [m]
1 . 2 5 9 E - 1 0 .900 0 . 9 0 0
1 . 2 5 5 E - 1 0 .900 0 . 9 0 0
1 . 2 5 4 E - 1 0 . 9 0 0 0 . 9 0 0
2 . 2 8 6 E - 2 0 . 7 0 0 0 . 7 0 0
2 . 2 4 5 E - 2 0 .700 0 . 7 0 0
2 . 2 3 5 E - 2 0 . 7 0 0 0 . 7 0 0
3 . 4 7 1 E - 3 0 . 2 0 0 0 . 2 0 0
2 . 5 9 2 E - 3 0 .200 0 . 2 0 0
2 . 4 0 3 E - 3 0 . 2 0 0 0 . 2 0 0
1 . 7 9 2 E - 4 0 . 1 0 0 0 . 1 0 0
2 . 8 1 8 E - 5 0 .100 0 . 1 0 0
1 . 1 2 0 E - 5 0 . 3 0 0 0 . 0 0 0
T a b l e 9. A c c u r a c y of t h e s o l u t i o n a s a f u n c t i o n o f m u l t i q u a d r i c s f r e e p a r a m e t e r c
a t t i m e s t = 0 . 0 0 1 s , t = 0 . 0 1 s , t = 0 . 1 s , a n d t = 1 . 0 s in t e r m s o f a v e r a g e e r r o r ,
m a x i m u m e r r o r , a n d t h e p o s i t i o n o f t h e m a x i m u m e r r o r f o r 21 x 21 n o d e a r r a n g e m e n t
w i t h f i v e - n o d e d d o m a i n o f i n f l u e n c e a n d A t = 10 . 4 s.
t Is]
0.001
0.001
0.001
0.01
0.01
0.01
0.1
0.1
0.1
1.0
1.0
1.0
C
8
16
32
8
16
32
8
16
32
8
16
32
A T a v g [°C] A T . . . . . [ °C] p . . . . [m] P m a x y [m]
4 . 5 9 3 E - 3 4 . 4 1 3 E - 2 0 .900 0 .900
4 . 5 2 7 E -- 3 4 . 3 7 7 E - 2 0 .900 0 . 9 0 0
4 . 5 1 2 E - 3 4 . 3 6 8 E - 2 0 .900 0 . 9 0 0
1 . 5 7 4 E - 3 7 . 1 2 1 E - 3 0 . 7 5 0 0 . 7 5 0
1 . 3 2 9 E - 3 6 . 4 2 1 E - 3 0 . 7 5 0 0 . 7 5 0
1 . 2 8 2 E - 3 6 . 2 5 5 E - 3 0 .750 0 . 7 5 0
1 . 2 4 1 E - 3 2 . 3 8 7 E - 3 0 . 2 0 0 0 . 2 0 0
4 . 3 4 4 E - 4 1 . 0 3 2 E - 3 0 .150 0 . 1 5 0
2 . 8 6 8 E 4 8 . 1 5 4 E - 4 0 .150 0 . 1 5 0
1 . 0 7 1 E - 4 2 . 6 9 5 E - 4 0 . 0 5 0 0 . 0 5 0
2 . 0 4 6 E - 5 5 . 2 6 2 E - 5 0 . 0 5 0 0 . 0 5 0
4 . 7 3 0 E - 6 1 . 3 3 7 E - 5 0 . 0 5 0 0 . 0 5 0
1280 B. SARLER AND R. VERTNIK
T a b l e 10. A c c u r a c y of t h e s o l u t i o n as a f u n c t i o n of m u l t i q u a d r i c s f ree p a r a m e t e r c
a t t i m e s t = 0 . 0 0 1 s , t = 0 ,01s , t = 0 . 1 s , a n d t = 1 .0 s in t e r m s of a v e r a g e e r ro r , m a x i m u m e r ro r , a n d t h e p o s i t i o n of t h e m a x i m u m e r r o r for 41 x 41 n o d e a r r a n g e m e n t w i t h f i ve -noded d o m a i n of in f luence a n d A t - - 10 - 4 s.
t Is] e ZXT~v~ [°C]
0.001 8 1 .798E - 3
0 .001 16 1 .699E - 3
0 .001 32 1 .679E - 3
0.01 8 1 .033E - 3
0.01 16 6 . 0 9 2 E - 4
0.01 32 5 . 4 1 5 E - 4
0.1 8 2 . 0 3 6 E - 3
0.1 16 3 . 8 1 5 E - 4
0.1 32 1 .815E - 4
1.0 8 2 .113E - 4
1.0 16 3 . 1 1 5 E - 5
1.0 32 9 . 1 6 0 E - 6
AZmax [ °C] p . . . . [m]
2 . 4 0 5 E - 2 0 .950
2 .319E - 2 0 .950
2 . 2 9 8 E - 2 0 .950
3 . 6 7 8 E - 3 0 .825
2 .647E - 3 0 .825
2 .418E - 3 0 .825
3 .761E - 3 0 .025
6 . 5 4 8 E - 4 0 .400
3 . 2 0 0 E - 4 0 .450
5 . 2 7 7 E - 4 0 .025
7 , 8 1 0 E - 5 0 .025
2 . 3 3 2 E - 5 0 .025
pmox~ [.1]
0 .950
0 .950
0 .950
0 .825
0 .825
0 .825
0 .025
0 .400
0 .450
0 .025
0 .025
0 .025
T a b l e 11. A c c u r a c y of t h e s o l u t i o n as a f u n c t i o n of m u l t i q u a d r i c s free p a r a m e t e r c a t t i m e s t = 0 . 0 0 1 s , t = 0 .01s , t = 0 . 1 s , a n d t = 1 .0 s in t e r m s of a v e r a g e e r ro r , m a x i m u m er ror , a n d t h e p o s i t i o n of t h e m a x i m u m e r r o r for 41 x 41 n o d e a r r a n g e m e n t w i t h f i ve -noded d o m a i n of in f luence a n d A t = 10 - 5 s.
t [ s ] c
0 .001 8
0 .001 16
0 .001 32
0.01 8
0.01 16
0.01 32
0.1 8
0.1 16
0.1 32
1.0 8
1.0 16
1.0 32
A T a v g [°C] A T . . . . . [°C] P . . . . [ml p . . . . . y [m]
1 .239E - 3 1 .571E - 2 0 .925 0 .925
1 .179E 3 1 . 5 0 2 E - 2 0 .925 0 .925
1 .168E - 3 1 .485E - 2 0 .925 0 .925
8 . 0 9 2 E - 4 2 .641E - 3 0 .250 0 .250
3 . 8 2 8 E - 4 1 .772E - 3 0 .275 0 .275
3 . 3 3 2 E - 4 1 .623E - 3 0 .275 0 .275
1 .969E - 3 3 . 8 6 4 E - 3 0 .975 0 .975
3 . 1 5 5 E - 4 6 . 3 2 8 E - 4 0 .900 0 .900
1 .098E - 4 3 .037E - 4 0 .925 0 .925
2 .065E - 4 5 . 1 5 2 E 4 0 .025 0 .025
2 . 6 1 9 E - 5 6 . 5 3 9 E - 5 0 .025 0 .025
4 . 1 9 1 E - 6 1 .058E - 5 0 .025 0 .025
a c c u r a c y w i t h h i g h e r v a l u e s o f c a n d d e n s e r n o d e a r r a n g e m e n t . C o m p a r i s o n o f T a b l e s 1 0 a n d 11
s h o w s e x p e c t e d c o n v e r g e n c e p r o p e r t i e s o f t h e m e t h o d , t h a t b e t t e r r e s u l t s c a n b e a c h i e v e d w i t h
r e d u c t i o n o f t h e t i m e s t e p f r o m A t = 1 0 - 4 s t o A t = 1 0 - 5 s . I n f o r m a t i o n i n T a b l e s 1 2 a n d 1 3
s h o w s s i m i l a r a c c u r a c y o f t h e d e v e l o p e d m e s h l e s s m e t h o d a s c o m p a r e d w i t h t h e c l a s s i c a l FDM a t
s h o r t e r t i m e s t = 0 . 0 0 1 s a n d t = 0 . 0 1 s i m m e d i a t e l y a f t e r t h e a b r u p t b o u n d a r y c o n d i t i o n s j u m p ,
a n d a n o r d e r o f m a g n i t u d e b e t t e r a c c u r a c y a t l o n g e r t r a n s i e n t t i m e s t = 0 . 1 s a n d t = 1 . 0 s . T h e
m e t h o d is u n s t a b l e w i t h t h e t i m e s t e p A t = 1 0 - 3 s b e c a u s e o f t h e e x p l i c i t a p p r o a c h .
B o t h t e s t c a s e s s h o w t h a t t h e a c c m - a c y o f t h e m e t h o d m o n o t o n i c a l l y i n c r e a s e s w i t h l a r g e r
v a l u e o f c . B e c a u s e o f t h e l a c k o f t h e s u i t a b l e t h e o r y f o r d e t e r m i n i n g t h e p r o p e r v a l u e o f t h e f l e e
p a r a m e t e r o n e c a n c o n f i d e n t l y u s e t h e h i g h e s t v a l u e s o f c w h i c h g i v e c o n v e r g e n c e .
Meshfree Explicit Local Radial Basis Function
Table 12. Accuracy of the solution at times t = 0.001s, t = 0.01s, t = 0.1s, and t = 1.0s in terms of average and maximum errors and the position of maximum error for 101 x 101 node arrangement, five-noded domain of influence, fixed multiquadrics free parameter c = 32 and At = 10 -5 s.
1281
t Is] c
0.001 32
0.01 32
0.1 32
1.0 32
ATavg [°C] AT ..... [°C] P . . . . [m] P . . . . y [m]
2.352E - 4 2.809E - 3 0.940 0.940
9.371E - 5 3.523E - 4 0.800 0.800
9.243E -- 5 1.582E -- 4 0.270 0.270
8.324E - 6 2.066E -- 5 0.010 0.010
Table 13. Accuracy of the FDM solution at times t = 0.001s, t = 0.0Is, t = 0.1s, and 1 = 1.0s in terms of average and maximum errors and the position of maximum error for 101 x 101 regular grid and At = 10 -5 s.
t Is]
0.001
0.01
0.1
1.0
At [s] AT~vg [°el ATm~x [cC] p . . . . [m] pmaxy [m]
1.0E - 5 1.368E - 4 1.273E - 3 0.925 0.950
1.0E - 5 3.722E - 4 1.298E - 3 0.020 0.890
1.0E - 5 3.363E - 4 2.786E - 3 0.000 0.010
1.0E - 5 2.029E - 4 5.649E - 4 0.000 0.010
5 . C O N C L U S I O N S
Thi s p a p e r r e p r e s e n t s a new (very) s imple mesh f ree f o r m u l a t i o n for so lv ing a wide range of
diffusion p rob lems . T h e numer i ca l t e s t s show m u c h h igher a ccu racy of t h e d e v e l o p e d m e t h o d
as c o m p a r e d w i t h t h e classical F D M . T h e only e x c e p t i o n o b s e rv ed is t h e so lu t ion at sho r t t imes
i m m e d i a t e l y af ter t he Dir ichle t j u m p where s imi lar numer i ca l a p p r o x i m a t i o n p r o p e r t i e s are ob-
served. T h e t i m e - m a r c h i n g is p e r f o r m e d in a s imple expl ic i t way. T h e govern ing eq u a t i o n is
solved in i ts s t r o n g form. No po lygon i s a t i on and i n t e g r a t i o n s are needed . T h e d e v e l o p e d m e t h o d
is a lmos t i n d e p e n d e n t of t h e p r o b l e m d imens ion . T h e c o m p l i c a t e d g e o m e t r y can easi ly be coped
wi th . T h e m e t h o d a p p e a r s efficient, because it does no t requ i re a so lu t ion of a large s y s t e m of
e q u a t i o n s like t he or ig inal K a n s a m e t h o d . I n s t ead , smal l s y s t e m s of l inear e q u a t i o n s have to be
solved in each t i m e s t e p for each n o d e a n d a s soc i a t ed d o m a i n of inf luence, p r o b a b l y r e p r e s e n t i n g
the m o s t n a t u r a l a n d a u t o m a t i c d o m a i n decompos i t i on . Th is f ea tu re of t h e deve loped m e t h o d
r ep re sen t s i ts p r inc ipa l difference f rom the o t h e r r e l a t ed local a p p r o a c h e s , whe re t he r e su l t an t
m a t r i x is large and spa r se [15,17-19]. T h e m e t h o d is s imple to l ea rn a n d s imple to code. T h e
m e t h o d can cope w i t h very large p r o b l e m s s ince the c o m p u t a t i o n a l effor t g rows a p p r o x i m a t e l y
l inear w i t h t h e n u m b e r of t h e nodes . T h e d e v e l o p m e n t s in th i s p a p e r can be s t r a i g h t f o r w a r d l y
e x t e n d e d to tackle o t h e r t ypes of pa r t i a l d i f ferent ia l equa t ions . O u r ongo ing resea rch is focused
on the e x t e n s i o n of t h e m e t h o d to impl ic i t t i m e - m a r c h i n g w h i c h m i g h t o v e rco me t h e i n h e ren t
s t ab i l i t y p r o b l e m of t h e expl ic i t approach .
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