Numerical Heat Transfer, Part B, 44: 575–599, 2003 Copyright # Taylor & Francis Inc. ISSN: 1040-7790 print/1521-0626 online DOI: 10.1080/10407790390231761
SOLVING THE STEFAN PROBLEM WITH THE RADIAL BASIS FUNCTION COLLOCATION METHOD I. Kovacˇevic´ and A. Poredosˇ Laboratory for Refrigeration, Faculty of Mechanical Engineering, University of Ljubljana, Ljubljana, Slovenia
B. Sˇarler Laboratory for Multiphase Processes, Nova Gorica Polytechnic, Nova Gorica, Slovenia In this paper the classical Stefan problem in its enthalpy formulations is solved with the radial basis function collocation method. The phase change is treated using the general source-based method of V.R. Voller & C.R. Swaminathan. Tests include numerical convergence studies for the two-dimensional Dirichlet jump problem with a solid-liquid phase change. The numerical results are compared with a semi-analytical solution given by K.A. Rathjen & L.M. Jiji, and with finite-volume method. The suitability of the radial basis function collocation method is demonstrated using a realistic engineering problem involving a charging tank for latent cool thermal storage.
1.
INTRODUCTION
Various aspects of science and technology are nowadays related to the prediction of the behavior of solid–liquid phase-change problems. The discrete approximate solution of the macroscopic transport phenomena in solid–liquid phase-change systems follows two major strategies [1]. The first one is called the twodomain approach, with the conservation equations written separately for the domain with the solid phase and for the domain with the liquid phase. The phase-change process is taken into account through interphase boundary conditions, moving of the interphase boundary, and changing of the domains occupied by the phases. The second strategy is called the one-domain approach, in which a single set of conservation equations is written for both the solid and the liquid phases. The phasechange process is taken into account through a jump in the transport quantity, which is built into transient, convective, diffusive, or source terms in a variety of possible combinations. The first approach requires the use of so-called moving-grid Received 17 January 2003; accepted 12 May 2003. I. Kovacˇevic´ thanks The Slovenian Science Foundation for providing a M.Sc. scholarship at the Faculty of Mechanical Engineering, University of Ljubljana, Slovenia. B. Sˇarler acknowledges the support of NATO grant PST.CLG.977633. Address correspondence to B. Sˇarler, Laboratory for Multiphase Processes, Nova Gorica Polytechnic, Vipavska 13, SI-5000 Nova Gorica, Slovenia. E-mail:
[email protected] 575
576
ˇ EVIC ´ ET AL. I. KOVAC
NOMENCLATURE A b c cp F(T) g h dh k L m M n N p q r T t Dt S Dx b G D E R B
coefficient matrix vector shape parameter specific heat liquid fraction temperature relationship volume fraction enthalpy difference between enthalpies thermal conductivity latent heat number of iterations per time step number of nodes normal vector number of radial basis functions vector position heat flux distance temperature time time step source term typical grid distance ration between shape parameter and typical grid distance boundary error convergence parameter density coefficient
w c O
boundary condition indicator radial basis function domain
Subscripts ana avg i l M max n ref rel s x,y G O 0
analytical value average value ith node liquid phase melting temperature maximum value nth radial function reference temperature relaxation solid phase Cartesian coordinates value on the boundary value in the domain initial condition
Superscripts D Dirichlet boundary condition j iteration level j jþ1 iteration level j þ 1 N Neumann boundary condition old value at the beginning of time interval 0 scaling constant þ maximum coordinate 7 minimum coordinate
(or front-tracking) numerical schemes in order to capture the moving boundary. The second approach requires the use of fixed-grid schemes that are able to cope with strong nonlinearities. In recent years a number of mesh-free methods have been developed to circumvent the problem of polygonization encountered in the classical numerical methods. In mesh-free methods the approximation is constructed entirely in terms of a set of gridpoints. A class of such methods is based on the collocation with radial basis functions. These functions were first intensively researched in the areas of multivariate data and function interpolation [2]. Kansa used them for a scattered data approximation [3], and afterwards for a solution of the partial different equations (PDEs) [4]. The method and its new variants are currently under intensive investigation by Chen ([5, 6]). The key point of the radial basis function collocation method or Kansa method (RBFCM or KM) is the approximation of the fields on the boundary and in the domain by a set of global approximation functions and subsequent representation of the partial derivatives by the partial derivatives of these global approximation functions. With these methods the discretization is represented only by gridpoints, in contrast to the finite-element method (FEM) and
SOLVING THE STEFAN PROBLEM WITH THE RBFCM
577
the finite-volume method (FVM), in which the approximate polygonization also needs to be generated, or the finite-difference method (FDM), in which the points are constrained to the coordinate lines. The main advantage of using the RBFCM for solving the PDEs is in its simplicity, applicability to different, PDEs, and effectiveness in dealing with arbitrary dimensions and complicated domains. The method has begun to be successfully applied in many scientific and engineering discipline. It was initially used in a heat transport context by Zerroukat et al. for diffusion problems [7], and later for advection-diffusion problems [8]. The method has also been applied to the solution of Navier-Stokes equations [9], the natural-convection problem [10], the steady natural-convection problem with a free boundary associated with the solid–liquid phase change [11], and natural convection in porous media [12].
2.
GOVERNING EQUATIONS
The enthalpy formulation for the Stefan problem [13] is represented using the mixture theory of Bennon and Incropera [14] by q ðRhÞ ¼ H ðkHTÞ qt
ð1Þ
where t is time, and R, h, k, and T stand for the density, enthalpy, thermal conductivity, and temperature of the mixture, respectively. The constitutive equations are defined as h ¼ gs h s þ g l h l
ð2Þ
where it is assumed that the density of both phases is equal: R ¼ Rs ¼ Rl . gP is the volume fraction and hP is the enthalpy of phase P. The subscripts P ¼ s; l are used for the solid and liquid phases, respectively. Only two phases are present in the system (without porosity), therefore gs ¼ 1 gl , and in the following g ¼ gl. The enthalpies of the solid and liquid phases are defined as hs ¼
Z
T
cps ðyÞdy
Tref
hl ¼
Z
T
cpl ðyÞdy þ L
ð3Þ
Tref
where cpP is the specific heat of phase P, L is the latent heat, and Tref is an arbitrary reference temperature. The conductivity and the specific heat of the mixture are k ¼ ð1 gÞks þ gkl
cp ¼ ð1 gÞcps þ gcpi
ð4Þ
Using Eqs. (2) and (3) the term qðRhÞ=qt can be expanded as q qT qg ðRhÞ ¼ Rcp þ Rdh qt qt qt
ð5Þ
where dh ¼
Z
T
ðcpl cps Þdy þ L Tref
ð6Þ
ˇ EVIC ´ ET AL. I. KOVAC
578
represents the difference between the liquid and solid enthalpies (hl 7 hs). Substitution of Eq. (5) into Eq. (1) gives the temperature formulation of Eq. (1), Rcp
qT ¼ H ðkHTÞ þ S qt
ð7Þ
where the source term is S ¼ Rdh
qg qt
ð8Þ
By choosing a reference temperature of Tref ¼ 0 and assuming the specific heats cp do not depend on temperature, Eq. (6) can be simplified as dh ¼ ðcpl cps ÞT þ L
ð9Þ
The solution Tðp; t0 þ DtÞ, where Dt represents a positive time increment, is obtained using the initial and boundary conditions that follow. The initial temperature Tðp; t0 Þ at a point with a vector position p and a time t0 is defined using the known function T0(p), Tðp; t0 Þ ¼ T0 ðpÞ pEO [ G ð10Þ where a two-dimensional domain O with a boundary G is considered. The boundary G is assumed to be divided into the not-necessarily-connected parts GD and GN with the Dirichlet and Neumann thermal boundary conditions, respectively: Tðp; tÞ ¼ TG ðp; tÞ;
p E GD
k
qTðp; tÞ ¼ qG ðp; tÞ; qnG
p E GN ; t0 < t t0 þ Dt ð11Þ
where nG is the normal on the boundary, and TG(p, t) and qG(p, t) are known functions. 2.1. The Voller-Swaminathan Scheme In 1991, Voller and Swaminathan [15] proposed a scheme for solving the nonlinear partial differential Eq. (7) in the context of the finite-volume method. In this article their method is put into RBFCM context. Equation (7) is presented in terms of two variables, T and g. It is assumed that the liquid fraction is a function of temperature alone, i.e., g ¼ FðTÞ
ð12Þ
The change of phase in a pure material takes place at the constant melting temperature TM, whereas in mixtures it is generally between the solidus temperature Ts and the liquidus temperature Tl. In this article only a pure material is considered. F(T) is approximated by using a narrow phase-change interval DT and a piecewise linear function of temperature: 8 0 T < Ts > < g ¼ ðT Ts Þ=ðTl Ts Þ Ts T Tl ð13Þ > : 1 T > Tl ;
SOLVING THE STEFAN PROBLEM WITH THE RBFCM
579
where Tl ¼ TM þ DT=2, and Ts ¼ TM DT=2. A fully implicit (backward Euler) time integration of Eq. (7) in matrix form is Rcp ðT T old Þ ¼ DtðkH2 T þ Hk HTÞ þ Rdhðgold gÞ
ð14Þ
where [ ]old is the value in an old time interval. The size of the vectors equals the number of nodes in the domain. The coefficients k, cp, and the source term in Eq. (14) represent a nonlinearity, and the solution inherently involves iterations. The iterative scheme can be written as Rcpj ðT jþ1 T old Þ ¼ Dtðk j H2 T jþ1 þ Hk j HT jþ1 Þ þ Rdh j ðgold g jþ1 Þ
ð15Þ
where the superscripts [ ] j and [ ] j+1 refer to the iteration levels j and j þ 1, respectively. In the following the superscript [ ] j is omitted for reasons of clarity. By using a truncated Taylor-series expansion for g j+1, one can write g jþ1 ¼ g þ
dF 1 ½F ðgÞ½T jþ1 F 1 ðgÞ dt
ð16Þ
By using this linearization, the source term can be written as S ¼ Sp T jþ1 þ Sc where Sp ¼ Rdh
dF 1 ½F ðgÞ dt
Sc ¼ Rdhðgold gÞ Sp F 1 ðgÞ
ð17Þ ð18Þ
The iterative scheme can now be written as ðRcp Sp ÞT jþ1 DtðkH2 T jþ1 þ Hk HT jþ1 Þ ¼ Rcp Told þ Sc
ð19Þ
The liquid fraction in the node i for the iteration level j þ 1 is obtained from an appropriate form of Eq. (16), i.e., ( dF=dT 0 < gi < 1 jþ1 jþ1 1 gi ¼ g i þ ½Ti F ðgi Þ ð20Þ cp =dh gi ¼ 1 or gi ¼ 0 If the node i is undergoing the phase change (in the mushy region), then Eq. (20) is the same as Eq. (16). If the node i is outside the mushy region, then dF=dT ¼ 0, but in order to allow the initiation of the phase change within a time step the slope is assumed to take the value of cp =dh [16]. Equation (20) is applied at every node, followed by an undershoot=overshoot correction that constrains the nodal liquid fractions to take values between 0 and 1, i.e., 8 >1 1 gjþ1 > i > < jþ1 jþ1 ð21Þ gi ¼ gi 0 gi jþ1 1 > > : 0 gi jþ1 < 0 The criterion for reaching convergence is defined in terms of the enthalpy of the mixture: hjþ1 h i i max ð22Þ Eh hi
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where Eh is a sufficiently small parameter. In general, the coefficients k and cp have to be relaxed: kjþ1 ¼ k þ Erel ½kðgjþ1 Þ k
cjþ1 ¼ cp þ Erel ½cp ðgjþ1 Þ cp p
ð23Þ
where the relaxation coefficient Erel is a value in the interval from 0 to 1.
3.
THE RBFCM OR KANSA METHOD
The temperature field is calculated in gridpoints pi; i ¼ 1; 2; . . . ; M; M ¼ MG þ MO . The first MG gridpoints are distributed on the boundary G and the last MO in the domain O. Let us continue the discussion in two-dimensional Cartesian coordinate system with base vectors i, j and coordinates x, y of point p: p ¼ xi þ yj
ð24Þ
The unknown fields are approximated by N global approximation functions, cn ðx; yÞ, and their coefficients Bn , Tðx; yÞ
N X
cn ðx; yÞBn
ð25Þ
n¼1
The indices i, n are introduced. They take values i ¼ 1,2,. . ., M, n ¼ 1,2,. . ., N if not stated otherwise. The approximation coefficients Bn can be calculated from collocation equations in points (xi, yi): Tðxi ; yi Þ Ti ¼
N X n¼1
cn ðxi ; yi ÞBn
N X
cin Bn
ð26Þ
n¼1
Since the number of functions N might be chosen to be greater than the number of collocation points M, the following augmentation equations are needed in order to determine all of the coefficient Bn : M X
cn ðxi ; yi ÞBi ¼ 0
n ¼ M þ 1; M þ 2; . . . ; N
ð27Þ
i¼1
The first and the second partial derivatives of function T and the first partial derivative of the mixture’s thermal conductivity k over coordinate x can be approximated as N X q q Tðx; yÞ cn ðx; yÞBn qx qx n¼1
N X q2 q2 Tðx; yÞ c ðx; yÞBn qx2 qx2 n n¼1
N X q q kðx; yÞ cn ðx; yÞBkn qx qx n¼1
ð28Þ
ð29Þ
SOLVING THE STEFAN PROBLEM WITH THE RBFCM
3.1.
581
The Radial Basis Functions
In this work, multiquadric (MQ) RBFs, cn ðrÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2n þ c2
ð30Þ
with c as a free parameter, the thin-plate spline (TPS) of order two, cn ðrÞ ¼ r4n logðrn Þ
ð31Þ
cn ðrÞ ¼ r6n logðrn Þ
ð32Þ
cn ðrÞ ¼ r3n
ð33Þ
order three,
and Duchon cubics,
are used. The distance rn between the points with coordinate (x, y) and the center of the RBFs (xn, yn) is r2n ¼ ðx xn Þ2 þ ðy yn Þ2
n ¼ 1; 2; . . . ; M
ð34Þ
MQ RBFs are used without augmentation, because no major improvement was obtained by adding the polynomial [17]. The augmentation polynomials for the TPS of order two are cMþ1 ðx; yÞ ¼ ðx x0 Þ2
cMþ2 ðx; yÞ ¼ ðx x0 Þðy y0 Þ
cMþ3 ðx; yÞ ¼ ðy y0 Þ2
cMþ4 ðx; yÞ ¼ ðx x0 Þ
cMþ5 ðx; yÞ ¼ ðy y0 Þ
ð35Þ
cMþ6 ðx; yÞ ¼ 1
and of order three are cMþ1 ðx; yÞ ¼ ðx x0 Þ3
cMþ2 ðx; yÞ ¼ ðx x0 Þ2 ðy y0 Þ
cMþ3 ðx; yÞ ¼ ðx x0 Þðy y0 Þ2
cMþ4 ðx; yÞ ¼ ðy y0 Þ3
cMþ5 ðx; yÞ ¼ ðx x0 Þ2
cMþ6 ðx; yÞ ¼ ðx x0 Þðy y0 Þ
cMþ7 ðx; yÞ ¼ ðy y0 Þ2
cMþ8 ðx; yÞ ¼ ðx x0 Þ
cMþ9 ðx; yÞ ¼ ðy y0 Þ
ð36Þ
cMþ10 ðx; yÞ ¼ 1
The scaling constants x0 and y0 are set to 1 x0 ¼ ðx þ xþ Þ 2
1 y0 ¼ ðy þ yþ Þ 2
ð37Þ
where x þ , y þ represent the maximum, and x 7 , y 7 the minimum, coordinates x, y, respectively, of the domain O. Among all the methods tested in Franke’s review article [2], Hardy’s MQ ranked as the best, followed by the TPSs. Hardy’s MQ
ˇ EVIC ´ ET AL. I. KOVAC
582
involves a free parameter that introduces additional degrees of freedom into the present discussion. Since the first-order TPSs involve a singularity in the second derivatives, the second-order and the third-order TPSs have been chosen as a possible simplest alternative.
3.2. The RBFCM Implementation of the Voller-Swaminathan Scheme The temperature field Tðx; y; t0 Þ of the initial condition T0 ðx; yÞ [Eq. (10)] is approximated with Eq. (26) and Eq. (27), i.e., N X
cin Bn ¼ T0 ðxi ; yi Þ
n¼1 M X
ð38Þ cin Bi ¼ 0
n ¼ M þ 1; M þ 2; . . . ; N
i¼1
The thermal boundary conditions Eq. (11) are presented with the RBFCM: ! N N N X X X qcin jþ1 qcin jþ1 D jþ1 N B þ nGyi B wi cin Bn þ wi nGxi qx n qy n n¼1 n¼1 n¼1 N ¼ wD i TGi þ wi qGi
i ¼ 1; 2; . . . ; MG
ð39Þ
where the Dirichlet and Neumann boundary-conditions indicator has been introduced: ( ( 1 pðx; yÞ 2 GD 1 pðx; yÞ 2 GN D N w ðx; yÞ ¼ w ðx; yÞ ¼ ð40Þ 0 pðx; yÞ 2 = GD 0 pðx; yÞ 2 = GN In the domain, Eq. (19) is presented with the RBFCM ðRcpi Spi Þ
N X
" cin Bjþ1 n
Dt ki
n¼1
þ
N X qc 2 in
n¼1
qx
Bkn Bjþ1 n
þ
N X qc 2 in
n¼1
¼ Rcpi Told i þ Sci
N N X q2 cin jþ1 X q2 cin jþ1 B þ B n 2 qx qy2 n n¼1 n¼1
qy
!
# Bkn Bjþ1 n
i ¼ MG þ 1; MG þ 2; . . . ; M
ð41Þ
If the number of functions N is greater than the number of colloaction points M, the following augmentation equations are needed in order to determine all of the coefficients Bjþ1 : M X i¼1
cin Bjþ1 ¼0 i
n ¼ M þ 1; M þ 2; . . . ; N
ð42Þ
SOLVING THE STEFAN PROBLEM WITH THE RBFCM
583
For MQ [Eq. (30)] and Duchon cubics RBFs [Eq. (33)], N ¼ M, and for TPSs of order two [Eq. (31)] and three [Eq. (32)], N ¼ M þ 6 and N ¼ M þ 10, respectively. The previous Eq. (39), (41), and (42) in matrix form are simply ABjþ1 ¼ b
ð43Þ
with explicit expressions for boundary conditions gridpoints qcin qcin N þ n Ain ¼ wD c þ w n Gxi Gyi in i i qx qy bi ¼
wD i TGi
þ
wN i qGi
i ¼ 1; 2; . . . ; MG ;
for domain gridpoints
q2 cin q2 cin Ain ¼ ðcpi Spi Þcin Dtki þ qx2 qy2 bi ¼ Rcpi Tiold þ Sci
ð44Þ
n ¼ 1; 2; . . . ; N
" # qcin 2 qcin 2 k þ Dt Bn qx qy
i ¼ MG þ 1; MG þ 2; . . . ; M;
ð45Þ
n ¼ 1; 2; . . . ; N
and for augmentation rows Ani ¼ cin
bn ¼ 0
n ¼ M þ 1; M þ 2; . . . ; N;
i ¼ 1; 2; . . . ; M
ð46Þ
The stability of the numerical solution is estimated by a condition number of the coefficient matrix A. The coefficient matrix A is calculated in every iteration and at every time step, and the maximum condition number of the matrix A is the highest value of the condition number during the computation. The condition number is computed for the Euclidean norm. The solution procedure in a single time step is as follows: 1. At the start of the time step the iterative procedure is initiated by setting g0 ¼ gold. 2. The values k, cp, Sp, Sc, and dh and the coefficients of Bk are calculated, and the system, Eq. (43), is solved to give the coefficients Bjþ1 , and with Eq. (25), the temperature field Tj+1. The enthalpy of the mixture is computed from Eq. (2). 3. An appropriate liquid fraction correction follows from Eq. (20). The correction is applied at every node and is followed by the overshoot=undershoot correction [Eq. (21)]. The procedure outlined in steps 2–3 is continued until convergence occurs. 4. BENCHMARK TESTS The numerical calculations are divided into two segments. In the first segment the RBFCM is used in the test rectangle where the analytical solution exists. The problem is computed by applying different RBFs and the classical FVM, and the obtained results are considered in relation to the analytical solution. In the second segment the RBFCM is used to calculate the process of freezing around tubes embedded in a water vessel.
ˇ EVIC ´ ET AL. I. KOVAC
584
4.1. Freezing in a Rectangular Corner The domain O is a rectangle with the Cartesian coordinates x < x < xþ and y < y < yþ , with x ¼ 0:0 ½m, xþ ¼ þ1:5 ½m, y ¼ 0:0 ½m, and yþ ¼ þ1:5 ½m. The test example corresponds to phase-change material with the constant unit material properties ks ¼ kl ¼ 1½W=m K, cps ¼ cpi ¼ 1½J=kg K, and Rs ¼ Rl ¼ 1½kg=m3 . The isothermal melting temperature TM ¼ 1½K is approximated in the calculations by a narrow phase-change interval of width DT. The linear variation of the liquid fraction g is used on the phase-change interval. Three different latent heat values are used: L ¼ 0.25 [J=kg], 0.50 [J=kg], and 1.00 [J=kg]. The initial condition is T0 ¼ 1.3 [K]. The boundary conditions on the west and south boundaries with coordinates x ¼ x and y ¼ y are of the Dirichlet type with TG ¼ 0½K, and the boundary conditions on the remaining east and north rectangular boundaries x ¼ xþ and y ¼ yþ are of the Neumann type with qG ¼ 0 [W=m2]. The reference solution for the test example is constructed using the quasi-analytical technique as proposed by Rathjen and Jiji in [18]. In this article the computation of the analytical solution as used by Sˇarler and Kuhn [19] is applied. The corresponding reference solution values are listed in Table 1. The defined test example is the same as used by Dalhuijsen and Segal in [20] for testing the FEM. The maximum absolute temperature error Dmax and the average absolute temperature error Davg of the numerical solution at time t are defined as
Dmax ðtÞ ¼ maxjTðxi ; yi ; tÞ Tana ðxi ; yi ; tÞj
Davg ðtÞ ¼
ð47Þ
Mg 1 X jTðxi ; yi ; tÞ Tana ðxi ; yi ; tÞj Mg i¼1
ð48Þ
where T and Tana are the numerical and analytical solution, and Mg is the total number of gridpoints. Two different grids are used for computing the maximum and the average errors. The first grid, denoted Type A, includes gridpoints, where either centers of the RBFs or centers of finite volumes are located. The second grid, denoted Type B, is equidistant 1006100 grid, with a total of Mg ¼ 10; 201 points. This grid serves for the global error assessment only, not for computational purposes. The RBFCM solutions are interpolated from the Type A grids to the Type B grid by using the formula Eq. (25). Three time-step lengths of 0.010, 0.005, and 0.001 [s] are used. All the accuracy comparisons are performed at time 0.1 [s]. The criterion for convergence is Eh ¼ 104 , and the relaxation coefficient is Erel ¼ 1.
Table 1. Rectangular corner freezing. Parameters of the reference analytical solution used in the comparisons. The nomenclature is identical to the nomenclature used by Rathjen and Jiji [18]. L [ J=kg] l b x0 x1 c m
0.25
0.50
1.00
0.7076615274E þ 00 2.122984582E þ 00 0.8957660560E þ 00 1.041029971E þ 00 0.1595087143E þ 00 4.795854597E þ 00
0.6232488949E þ 00 2.492995580E þ 00 0.7833238284E þ 00 0.9112988963E þ 00 0.4900333819E 7 01 4.196262251E þ 00
0.5202617454E þ 00 2.601308727E þ 00 0.6495522355E þ 00 0.7563678399E þ 00 0.1310427777E 7 01 3.666133944E þ 00
SOLVING THE STEFAN PROBLEM WITH THE RBFCM
585
4.1.1. 15615 Grid. The grid schematic for the RBFCM is presented in Figure 1a, and for the reference solution with the FVM in Figure 1b. The only difference between these two grids is at the corners, as can be seen from Figures 1a and 1b. The 15615 grid includes a total number of M ¼ ð15 þ 1Þ2 ¼ 256 nodes, with MG ¼ 4615 ¼ 60 being on the boundary, and MO ¼ (15 7 1)2 ¼ 196 in the domain. Sensitivity to shape parameter in the multiquadrics. First, MQ RBFs with the shape parameter c are used [Eq. (30)]. The choice of the optimum value for the shape parameter is still an unsolved problem, and the optimum value is found using numerical experiments. The shape parameter is considered to be constant with time and the same for all gridpoints in the test rectangle. Some authors claims that the shape parameter is related to the typical grid distance (Dx in Figure 1b), c ¼ b Dx [21]. Other researches did not find any relation, and claim simply that the optimum shape parameter is problem-dependent [22]. Very recently, Wang and Liu [23] analyzed the use of the extended multiquadric form cn ¼ ðr2n þ c21 Þc2 with two shape parameter c1 and c2 . The authors concluded that by proper fixing of both parameters to c1 ¼ 1:42, c2 ¼ 1:03, the solution becomes independent of node density, node distribution, and problem. The first attempted value for the shape parameter in our calculations is chosen to be the same as the distance between neighboring nodes (b ¼ 1). With increasing shape parameter the value of the maximum and the average errors Dmax and Davg becomes smaller. Also, with increasing value of the shape parameter the condition number of the coefficient matrix grows and the system of equations becomes ill-conditioned. The best numerical solution is obtained when the value of the shape parameter c ¼ 0.40 [m] (b ¼ 4.00) is used. With a further increase in the value of c, the solution diverges. Table 2 shows good agreement between the maximum and the average absolute errors calculated using grids of Type A 15615 and Type B.
Figure 1. Grid Type A 15615. Left: RBFCM grid (Figure 1a). Right: FVM mesh (Figure 1b).
ˇ EVIC ´ ET AL. I. KOVAC
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Table 2. Rectangular corner freezing. Sensitivity to the shape parameter in MQ RBFs in terms of the number of iterations per time step, maximum condition number of coefficient matrix, maximum and average errors in grid Types A 15 6 15 and B. MQ
No. of iter.
Dt [s]
c [m]
b
Stability max [cond(A)]
0.010 0.010 0.010 0.010 0.005 0.005 0.005 0.005 0.001 0.001 0.001 0.001
0.10 0.20 0.30 0.40 0.10 0.20 0.30 0.40 0.10 0.20 0.30 0.40
1.00 2.00 3.00 4.00 1.00 2.00 3.00 4.00 1.00 2.00 3.00 4.00
1.9711E þ 06 1.4850E þ 07 2.8351E þ 08 5.7001E þ 09 2.1234E þ 06 1.6143E þ 07 2.9586E þ 08 7.1497E þ 09 1.9952E þ 06 1.8084E þ 07 4.4916E þ 08 2.5383E þ 10
Type A 15 6 15
Type B
mmax
mavg
Dmax [K]
Davg [K]
Dmax [K]
Davg [K]
4 4 4 5 4 4 5 5 4 4 4 4
3.70 3.60 3.60 3.70 3.40 3.45 3.35 3.25 2.61 2.59 2.58 2.52
0.1273 0.0888 0.0631 0.0457 0.1211 0.0877 0.0518 0.0311 0.1160 0.0892 0.0394 0.0275
0.0267 0.0172 0.0121 0.0102 0.0245 0.0146 0.0088 0.0064 0.0251 0.0137 0.0071 0.0045
0.1279 0.0937 0.0656 0.0462 0.1232 0.0954 0.0532 0.0311 0.1204 0.1000 0.0393 0.0329
0.0259 0.0169 0.0122 0.0105 0.0236 0.0141 0.0088 0.0066 0.0244 0.0132 0.0070 0.0046
Comparison of the RBFCM with the FVM. Table 3 shows number of iterations per time step, maximum condition number of the coefficient matrix, and maximum and average errors computed in gridpoints of Type A 15615 and Type B using different RBFs and three time-step lengths. The computations were performed by MQ [Eq. (30)], TPS of order two [Eq. (31)] and three [Eq. (32)] with augmentations, and Duchon cubics [Eq. (33)]. With a good choice of the shape parameter, the maximum and average errors have similar values to those obtained with the
Table 3. Rectangular corner freezing. Comparison of the RBFCM involving different RBFs with the FVM, in terms of the number of iterations per time-step, maximum condition number of the coefficient matrix, maximum and average errors in grid Types A 15 6 15 and B. No. of iter. Method
Dt [s]
Stability max [cond(A)]
MQ b ¼ 4.00 r4 log(r) r6 log(r) r3 FVM MQ b ¼ 4.00 r4 log(r) r6 log(r) r3 FVM MQ b ¼ 4.00 r4 log(r) r6 log(r) r3 FVM
0.010 0.010 0.010 0.010 0.010 0.005 0.005 0.005 0.005 0.005 0.001 0.001 0.001 0.001 0.001
5.7001E þ 09 1.2320E þ 08 8.6091E þ 09 8.7167E þ 07 2.2003E þ 01 7.1497E þ 09 1.2938E þ 08 9.0951E þ 09 9.4276E þ 07 1.7705E þ 01 2.5383E þ 10 1.2575E þ 08 9.3804E þ 09 7.5512E þ 07 1.4343E þ 01
Type A 15 6 15
Type B
mmax
mavg
Dmax [K]
Davg [K]
Dmax [K]
Davg [K]
5 4 5 4 4 5 5 4 4 4 4 4 4 4 3
3.70 3.70 3.60 3.90 3.60 3.25 3.35 3.25 3.40 3.00 2.52 2.52 2.54 2.58 2.40
0.0457 0.0427 0.0334 0.0784 0.0319 0.0311 0.0291 0.0202 0.0766 0.0252 0.0275 0.0189 0.0173 0.0618 0.0171
0.0102 0.0112 0.0102 0.0131 0.0113 0.0064 0.0066 0.0056 0.0090 0.0069 0.0045 0.0038 0.0028 0.0064 0.0039
0.0462 0.0473 0.0340 0.0978 — 0.0311 0.0296 0.0258 0.0947 — 0.0329 0.0335 0.0323 0.0770 —
0.0105 0.0117 0.0108 0.0128 — 0.0066 0.0070 0.0062 0.0085 — 0.0046 0.0041 0.0032 0.0061 —
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FVM. MQ RBFs with shape parameter c ¼ 0.40 [m] (b ¼ 4.00) are the best for timestep length Dt ¼ 0.010 [s]. The results in Table 3 show that the average errors computed in gridpoints by using TPS RBFs are smaller than those obtained with the classical FVM for all three time-step lengths. The best results for two time-step lengths, Dt ¼ 0.001 [s] and Dt ¼ 0.005 [s], are obtained by using a TPS of the order three. The largest errors are encountered when using Duchon cubic RBFs. The condition numbers of the coefficient matrix are the highest for MQ and RBFs of order three. The difference is that in MQ RBFs the value of the condition number can be tuned by the shape parameter, which is not possible in TPS. Generally, the stability of the numerical solution calculated with the FVM is very good. The RBFCM converges a little more slowly with shorter time steps than the FVM. The number of iterations is approximately equal for all the RBFs used. The maximum error computed in gridpoints of Type A is smaller than for gridpoints of Type B, as expected. What is more important, however, is that the average errors computed in gridpoints of Type A and Type B are approximately the same. The precision of the FVM is calculated in gridpoints of Type A, which seems to be the only reasonable choice. Position of the interphase boundary. The comparison of the positions of the interphase boundaries calculated with different RBFs and the FVM is presented in Figure 2. The interphase boundary are computed like isotherms with TM ¼ 1[K].
Figure 2. Rectangular corner freezing. A comparison of the RBFCM and FVM interphase boundaries calculated with grid Type A, 15615.
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Two solid–liquid boundaries with latent heats L ¼ 0.25 [J=kg] and L ¼ 1.00 [J=kg] after time t ¼ 0.1 [s] of solidification are presented. The solid–liquid boundaries are calculated numerically using MQ RBFs with a shape parameter c ¼ 0.40 [m] (b ¼ 4.00), TPS of order two, and the FVM. The isotherms are calculated from the matrices in which the values of the temperature fields at the centers of the RBFs are contained. The numerical calculations of the position of interphase boundary with latent heat L ¼ 0.25 [J=kg] are accurate for all three methods. When the latent heat is L ¼ 1.00 [J=kg], all the numerical calculations of the position are inaccurate. The RBFCM for both functions gives approximately the same results. 4.1.2. 30630 Grid. The same test is repeated with a 30630 grid. The grid schematic for the RBFCM is presented in Figure 3a, and for the reference solution with the FVM in Figure 3b. The 30630 grid includes a total number of M ¼ ð30 þ 1Þ2 ¼ 961 nodes, with MG ¼ 4 30 ¼ 120 being on the boundary, and MO ¼ ð30 1Þ2 ¼ 841 in the domain. Sensitivity to the shape parameter in MQ RBFs. The MQ RBFs were first used with the shape parameter c ¼ Dx (b ¼ 1.00). With an increase in the shape parameter, the accuracy of the results improves, as experienced in Section 4.1.1. Using MQ RBFs with the shape parameter c ¼ 0.20 [m] (b ¼ 4.00) and time-step length Dt ¼ 0.001 [s], the solution diverges. Instead of this value, a slightly smaller value of shape parameter, c ¼ 0.18 [m] (b ¼ 3.60) is used, and results in the best performance (Table 4). The shape parameter is tuned so that the coefficient matrix is invertible and the system of equations is not ill-conditioned. The use of a 30630 grid needs a much larger number of iterations than a 15615 grid. The difference between the errors calculated using a Type A 30630 grid and a Type B grid is smaller than 3%. Comparison of the RBFCM with the FVM. Table 5 shows the maximum and average errors computed with different RBFs using a 30630 grid, as well as with the
Figure 3. Grid Type A, 30630. Left: RBFCM grid (Figure 3a). Right: FVM mesh (Figure 3b).
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Table 4. Rectangular corner freezing. Sensitivity to the shape parameter in MQ RBFs in terms of the number of iterations per time step, maximum condition number of coefficient matrix, maximum and average errors in grid Types A 30 6 30 and B. MQ
No. of iter.
Dt [s]
c[m]
b
Stability max [cond(A)]
0.010 0.010 0.010 0.010 0.005 0.005 0.005 0.005 0.001 0.001 0.001 0.001
0.05 0.10 0.15 0.20 0.05 0.10 0.15 0.20 0.05 0.10 0.15 0.18
1.00 2.00 3.00 4.00 1.00 2.00 3.00 4.00 1.00 2.00 3.00 3.60
3.4610E þ 07 1.8828E þ 08 5.2622E þ 09 2.1854E þ 11 2.6131E þ 07 1.4819E þ 08 5.0037E þ 09 1.9997E þ 11 2.4614E þ 07 1.6096E þ 08 4.5541E þ 09 4.2405E þ 10
Type A 30 6 30
Type B
mmax
mavg
Dmax [K]
Davg [K]
Dmax[K]
Davg[K]
6 6 6 6 5 5 5 5 4 4 4 4
4.90 4.60 4.70 4.60 3.95 3.90 3.90 4.00 3.12 3.15 3.11 3.12
0.1035 0.0596 0.0333 0.0317 0.0995 0.0444 0.0212 0.0157 0.0973 0.0395 0.0208 0.0159
0.0205 0.0133 0.0101 0.0092 0.0174 0.0100 0.0065 0.0050 0.0154 0.0075 0.0040 0.0029
0.1035 0.0599 0.0333 0.0317 0.1005 0.0444 0.0215 0.0182 0.1000 0.0408 0.0215 0.0167
0.0203 0.0134 0.0103 0.0095 0.0171 0.0099 0.0065 0.0051 0.0151 0.0074 0.0039 0.0029
FVM. For all RBFCMs and the FVM, the accuracies are better and the condition numbers are higher with the 30630 grid than with the 15615 grid. The solution computed with the TPS of order three is not convergent, because the system of equations is ill-conditioned. The maximum errors are smaller with all the RBFCMs than with the FVM. The best solution is calculated using the MQ RBFs with two time-step lengths: Dt ¼ 0.010 [s] and Dt ¼ 0.005 [s]. The TPS, but now with order two, has the lowest average error for the time-step length Dt ¼ 0.001 [s]. The big advantage of the RBFCM is the possibility for a straightforward approximation of the solution at the points which do not belong to the grid during the computation. In contrast,
Table 5. Rectangular corner freezing. Comparison of the RBFCM involving different RBFs with the FVM, in terms of the number of iterations per time step, maximum condition number of coefficient matrix, maximum and average errors in grid Types A 30 6 30 and B. No. of iter. Method
Dt [s]
Stability max [cond(A)]
MQ b ¼ 4.00 r4 log(r) r3 FVM MQ b ¼ 4.00 r4 log(r) r3 FVM MQ b ¼ 3.60 r4 log(r) r3 FVM
0.010 0.010 0.010 0.010 0.005 0.005 0.005 0.005 0.001 0.001 0.001 0.001
2.1854E þ 11 1.3870E þ 10 6.8932E þ 09 5.4950E þ 01 1.9997E þ 11 1.1949E þ 10 6.0983E þ 09 3.1379E þ 01 4.2405E þ 10 1.4049E þ 10 5.0521E þ 09 1.6880E þ 01
Type A 30 6 30
Type B
mmax
mavg
Dmax [K]
Davg [K]
Dmax [K]
Davg [K]
6 6 6 6 5 5 5 5 4 4 4 4
4.60 4.50 4.80 4.40 4.00 4.00 3.95 3.85 3.13 3.09 3.07 2.99
0.0317 0.0319 0.0311 0.0321 0.0157 0.0159 0.0182 0.0190 0.0159 0.0138 0.0137 0.0162
0.0092 0.0099 0.0102 0.0097 0.0050 0.0052 0.0055 0.0052 0.0029 0.0016 0.0019 0.0018
0.0317 0.0319 0.0311 — 0.0182 0.0176 0.0182 — 0.0167 0.0156 0.0147 —
0.0095 0.0102 0.0105 — 0.0051 0.0053 0.0056 — 0.0029 0.0016 0.0019 —
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only the FVM straightforwardly allows for a linear, inaccurate interpolation. With the RBFCM the first and the second derivatives can be calculated, which is not possible in an easy way with the FVM. Position of the interphase boundary. The interphase boundaries computed with the 30630 grid are shown in Figure 4. A comparison between the MQ RBFCM and the FVM is presented. The numerical calculation is now much better than when the 15615 grid is used. For latent heat L ¼ 0.25 [J=kg], the value of the shape parameter c ¼ 0.18 [m] (b ¼ 3.60) is used, and when we used the same value of c with the latent heat L ¼ 1.00 [J=kg], the solution diverges. Because of this, in the example with latent heat L ¼ 1.00 [J=kg], the value of the shape parameter c ¼ 0.15 [m] (b ¼ 3.00) is used. For both values of latent heat, numerical calculations of the interphase boundaries with two different numerical methods show good agreement with the solid–liquid interphases computed with the analytical solution. 4.1.3. Latent heat convergence study. The influence of the latent heat on the accuracy of the results, maximum condition number of the coefficient matrix, and number of iterations per time step is shown in Table 6. The accuracy of the results is reduced slightly with increasing latent heat, which is a common feature of the one-domain methods. Generally, with an increase in the value of the latent heat, the
Figure 4. Rectangular corner freezing. A comparison of the RBFCM and FVM interphase boundaries calculated with grid Type A, 30630.
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SOLVING THE STEFAN PROBLEM WITH THE RBFCM
Table 6. Rectangular corner freezing. The influence of the latent heat in terms of the number of iterations, maximum condition number of the coefficient matrix, maximum and average errors in grid Types A 15 6 15 and B. Dt ¼ 0.001 [s], DT=2 ¼ 0.01 [K]. No. of iter. Method
Stability max [cond(A)]
mmax
mavg
Type A 15 6 15
Type B
Dmax [K]
Davg [K]
Dmax [K]
Davg [K]
0.0289 0.0301 0.0232 0.0359 0.0243
0.0041 0.0046 0.0035 0.0045 0.0053
0.0341 0.0317 0.0326 0.0367 —
0.0047 0.0045 0.0039 0.0043 —
0.0768 0.0749 0.0708 0.1043 0.0851
0.0167 0.0142 0.0141 0.0152 0.0109
0.0837 0.0938 0.0819 0.1144 —
0.0180 0.0143 0.0150 0.0146 —
L ¼ 0.50 [J=kg] MQ b ¼ 4.00 r4 log(r) r6 log(r) r3 FVM
4.9464E þ 10 2.5426E þ 08 1.9528E þ 10 1.4869E þ 08 2.7056E þ 01
4 4 4 4 3
2.44 2.46 2.42 2.47 2.37
L ¼ 1.00 [J=kg] MQ b ¼ 4.00 r4 log(r) r6 log(r) r3 FVM
9.6711E þ 10 5.0687E þ 08 3.8168E þ 10 2.9359E þ 08 5.2473E þ 01
4 4 4 4 3
2.36 2.36 2.32 2.43 2.26
errors become higher. Exceptions to this are the MQ and Duchon cubic RBFs with latent heat L ¼ 0.50 [J=kg]. The difference between the maximum errors computed by the Type A and Type B grids grows with increase in the value of the latent heat. The value of the latent heat does not influence the difference between average errors. The condition number of the coefficient matrix increases approximately linearly with the increase of the latent heat. The influence of the latent heat is higher for the RBFCM than for the FVM. The accuracy of the FVM is lowest at L ¼ 0.50 [J=kg], but the FVM gives the best results with the value of the latent heat equal to L ¼ 1.00 [J=kg]. The number of iterations decreases slightly with increase of the latent heat. Table 7. Rectangular corner freezing. The influence of the artificial phase-change interval in terms of the number of iterations, maximum condition number of the coefficient matrix, maximum and average errors in grid Types A 15 6 15 and B. L ¼ 0.25 [J=kg], Dt ¼ 0.001 [s]. No. of iter.
Type A 15 6 15
Type B
Stability max [cond(A)]
mmax
mavg
Dmax [K]
Davg [K]
Dmax [K]
Davg [K]
MQ b ¼ 4.00 r4 log(r) r6 log(r) r3 FVM
2.3293E þ 11 1.1474E þ 09 — 6.9295E þ 08 1.2879E þ 02
DT=2 ¼ 0.001 [K] 4 4 — 4 4
2.52 2.55 — 2.63 2.44
0.0273 0.0198 — 0.0638 0.0221
0.0046 0.0041 — 0.0066 0.0041
0.0354 0.0356 — 0.0787 —
0.0047 0.0043 — 0.0063 —
MQ b ¼ 4.00 r4 log(r) r6 log(r) r3 FVM
6.3410E þ 09 5.2387E þ 07 5.0302E þ 09 1.9698E þ 07 3.0104E þ 00
DT=2 ¼ 0.100 [K] 4 4 4 4 3
2.52 2.48 2.47 2.52 2.42
0.0160 0.0133 0.0142 0.0327 0.0108
0.0045 0.0032 0.0027 0.0050 0.0026
0.0164 0.0154 0.0148 0.0367 —
0.0045 0.0033 0.0029 0.0051 —
Method
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Figure 5. Thermal storage application. Dimensions of the cross section of the vessel (in mm) and modeled region (shaded).
4.1.4. Artificial phase-change interval convergence study. The influence of the width of the phase-change interval is presented in Table 7. The accuracy of the results is better with the wider phase-change interval DT=2 ¼ 0.100 [K] for all the methods. The stability of the RBFCMs is very sensitive to the value of the phase-change interval. The condition number of the coefficient matrix increases very much with the decrease of the phase-change interval. Therefore, the TPS of order three is not convergent when the half-value of the phase-change interval DT=2 ¼ 0.001 [K] is used. The width of the artificial phase-change interval does not influence the number of iterations.
4.2. An Engineering Application of Thermal Storage In recent decades, studying cool thermal storage has become important for decreasing the electricity demand of air conditioners. The goal of thermal storage is to accumulate cool thermal energy when the electricity tariff is lower, and use it during times of peak thermal demand, which usually coincide with a high electricity
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593
Figure 6. Thermal storage application. Layout of grid Type I and reference point A.
tariff [24]. Latent heat is normally used in such systems because of the larger effective volumetric capacity. In this example, the claimed advantages of the mesh-free RBFCM are demonstrated using an engineering example. We have simulated the process of charging the tank for latent cool thermal storage. The dimensions of the cross section of such a vessel are presented in Figure 5. The problem is considered to
Figure 7. Thermal storage application. Layout of grid Type II and reference point A.
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possess symmetry, and only a small region, around half of the tube, is modeled (shaded in Figure 5). The geometry is complex, and obtaining a solution to this problem by standard methods such as the FEM and the FVM is not trivial. Because of this, the mesh-free RBFCM is a very attractive alternative for simulating these type of problems. The initial temperature of the water is T0 ¼ 10 [ C]. It is assumed that the tank is isolated, i.e, described by a Neumann-type boundary condition with qG ¼ 0 ½W=m2 . The boundary conditions on the tube are of the Dirichlet type, with TG ¼ 10 ½ C. Constant but different values of the specific heat and the thermal conductivity are used for the phase. The specific heat of water and ice are cpi ¼ 4.186 [kJ=kg K] and cps ¼ 1.762 [kJ=kg K], and the thermal conductivity of water and ice are kl ¼ 0.556 [W=m K] and k s ¼ 2.220 [W=m K]. The densities of water and ice are considered to be the same, R ¼ 1,000 [kg=m3]. The value of the latent heat is 332 [kJ=kg], and the artificial phase-change interval is set to DT=2 ¼ 0.1[ C]. The change of the thermal conductivity in the artificial phase-change interval is assumed to be a linear function of the liquid fraction. Again, the criterion for convergence is Eh ¼ 104 , and the relaxation coefficient is Erel ¼ 1. The computations are conducted with two grids. Type I (see Figure 6, Dx ¼ 4.3 [mm]) includes 307 gridpoints, 76 on the boundaries and 231 in the domain. Type II (Figure 7, Dx=2.5 [mm]) includes 1,199 gridpoints, 150 on the boundaries and 1,049 in the domain. The Type I and
Figure 8. Thermal storage application. Temperature history of reference point A as a function of multiquadrics shape parameter for grid Type I with Dt ¼ 120 s.
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595
Figure 9. Thermal storage application. Temperature history of reference point A as a function of multiquadrics shape parameter for grid Type II with Dt ¼ 120 s.
Type II gridpoints are represented by and 6, respectively. The gridpoints do not change positions in the grid during the computation. The RBFCM allows for simple increase or decrease of the number of gridpoints in a grid without difficulty. The value of the latent heat (L ¼ 332 [kJ=kg]) is high, and for a reasonably accurate calculation of the temperature field of the distance between the neighboring gridpoints needs to be sufficiently small. MQ RBFs are used in the simulation of described engineering system. The value of the shape parameter is once again determined by several numerical experiments. The shape parameters c ¼ 4.30 [mm] (b ¼ 1.00), c ¼ 6.45 [mm] (b ¼ 1.50), and c ¼ 8.60 [mm] (b ¼ 2.00) are used for grid Type I. The shape parameters c ¼ 1.00 [mm] (b ¼ 0.46), c ¼ 1.50 [mm] (b ¼ 0.70), and c ¼ 2.15 [mm] (b ¼ 1.00) are used for grid Type II. The presentation of the results is given in Figures 8, 9, 10 and 11. The changes of temperature at gridpoint A (see Figures 6 and 7) during the first hour of solidification are plotted in Figure 8 for the coarser grid and in Figure 9 for the finer grid. The time-step length is Dt ¼ 120 [s] or 2 min. The sensitivity of the results on the shape parameter decreases with the grid refinement. The use of other RBFs that do not involve the shape parameter fails in this engineering example due to the ill-conditioned system matrix. One can observe fairly good agreement between the results obtained with coarser and finer grids. The oscillations are typical for one-domain computational procedures [25]. The interphase boundaries after 1 and 2 h of charging the tank are presented in Figure 10.
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Figure 10. Thermal storage application. The interphase boundaries after 1 and 2 h of freezing as a function of grid Type I with b ¼ 1.5 and grid Type II with b ¼ 1.0.
The interphase boundary of the phase change calculated with grid Type I is ahead of the interphase boundary calculated with grid Type II, a consequence of the less dense grid. For the same reason, the interphase boundary calculated with grid Type II is smoother than the interphase boundary calculated with grid Type I. The change of temperature at gridpoint A, calculated with three time-step lengths, is presented in Figure 11 for grid Type I with c ¼ 6.45 [mm] (b ¼ 1.50). The temperature histories corresponding to all three time-step lengths used are in good agreement even for the coarse grid used.
5.
CONCLUSIONS
The RBFCM was successfully developed for Stefan problems. The use of different radial basis functions in the numerical implementation of the RBFCM for solid–liquid phase change processes is demonstrated. When MQ RBFs are used, the value of the shape parameter has a big influence on the accuracy. With an increase in the value of the shape parameter, the maximum and average errors begin to decrease. The growth of the shape parameter is confined because the value of the condition number of the coefficient matrix increases and the system of equations becomes
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597
Figure 11. Thermal storage application. Temperature history of reference point A as a function of timestep length. Grid Type I, b ¼ 1.5.
ill-conditioned. The influence of the shape parameter on the results decreases with the grid spacing. The best results are obtained with MQ RBFs and longer time-step lengths. TPS RBFs give the best accuracy for smaller time-step lengths. The difference between the average errors calculated by grids Type A and Type B are approximately equal. By increasing the number of gridpoints in Type A, the difference between the errors becomes smaller. The accuracy of the position of the interphase boundary is influenced by the value of the latent heat. For a large value of latent heat, the distance between neighboring gridpoints needs to be sufficiently small, as explained by Crank [25]. The RBFCM represented is very simple to upgrade to three dimensions and implement numerically in complex geometry. The latter is demonstrated in the latent heat thermal storage application. The highest errors in the RBFCM are on the boundaries with Neumann boundary conditions. Zhang et al. [22] proposed a collocation method of a Hermitetype interpolation in order to upgrade the RBFCM or the direct Kansa method and subsequently alleviate the mentioned inaccuracies. Another appealing feature [23] represents the use of the two-parametric MQs. Both issues are left to be elaborated for Stefan problems in the continuation of the present work.
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