- Email: [email protected]
Numerical solution of 3D non-stationary heat conduction problems using the Finite Pointset Method
International Journal of Heat and Mass Transfer 87 (2015) 104–110
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical solution of 3D non-stationary heat conduction problems using the Finite Pointset Method Edgar O. Reséndiz-Flores a,⇑, Irma D. García-Calvillo b a Division of Postgraduate Studies and Research, Department of Metal-Mechanical Engineering, The Technological Institute of Saltillo, Blvd. V. Carranza 2400 Col. Tecnológico, C.P. 25280 Saltillo, Coahuila, Mexico b Research Center on Applied Mathematics, Autonomous University of Coahuila, Campus Saltillo, Building S, C.P. 25000 Saltillo, Coahuila, Mexico
a r t i c l e
i n f o
Article history: Received 23 October 2014 Received in revised form 26 March 2015 Accepted 26 March 2015 Available online 11 April 2015
a b s t r a c t This paper presents the application of the Finite Pointset Method for the numerical solution of three dimensional and non-stationary heat conduction problems. The strong formulation of the parabolic partial differential equation is directly used instead of the corresponding weak form. Moreover, a numerical comparison between the Finite Pointset Method and the corresponding analytical solutions is reported. Ó 2015 Elsevier Ltd. All rights reserved.
Keywords: Three dimensional heat conduction Non-stationary heat conduction Meshless method Finite Pointset Method
1. Introduction During the past 30 years different type of meshless methods for solving partial differential equations have been developed with the objective of eliminating part of the difficulties arising when mesh-based methods are used. The starting point of these methods was the smooth particle hydrodynamics (SPH) method proposed by Lucy [1] and Gingold and Monaghan [2] in the area of astrophysics and applied later in other research fields. For a good overview on meshless methods we refer to [3] and references therein. In the field of heat transfer by conduction different meshless methods have been already applied. Among these scientific works some of the recent publications, to the authors knowledge, are the following works: Liu et al. [4] uses a meshless weighted least squares method for heat conduction. Chen et al. [5] applied a corrective SPH method to solve unsteady heat conduction problems. Cheng and Liew [6,7] applied the reproducing kernel particle method (RKPM) for two and three-dimensional unsteady heat conduction problems, respectively. We refer to [7] for a concise and
⇑ Corresponding author. Tel./fax: +52 (844)438 9500. E-mail addresses: [email protected] (E.O. Reséndiz-Flores), irma.garcia @uadec.edu.mx (I.D. García-Calvillo). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.03.084 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
recent overview on meshless methods on the field of heat conduction problems. As an alternative to the solution for the unsteady heat conduction equation we propose in this work the application of a slightly different version of the finite point method developed by Oñate [8]. This method is called the Finite Pointset Method (FPM) and to the authors knowledge this version of the method has been developed by Kuhnert [9] in the Fraunhofer-Institut für Techno- und Wirtschaftsmathematik, in Kaiserslautern, Germany, and it has been already applied in the fields of fluid mechanics [10–13] and radiative heat transfer problems [14]. Later on, a very close version of the FPM of Kuhnert has been also developed by Cheng and Liu, [15] in the field of fluid mechanics. The presented work in this article is an extension to three dimensions of the work already published by the authors in [16] and to the authors knowledge this is the first time that three dimensional numerical results regarding heat conduction problems using the Finite Pointset Method have been reported. In order to get some insight into FPM performance we compare the numerical solution to some specific problems whose analytical solutions are known. The structure of the paper is as follows: Section 2 shortly describe the basic ideas behind FPM. Section 3 presents some issues regarding the numerical implementation of FPM. The test examples are presented in Section 4 with the corresponding results. Finally some conclusions are given in last section.
105
E.O. Reséndiz-Flores, I.D. García-Calvillo / International Journal of Heat and Mass Transfer 87 (2015) 104–110
2. The FPM method
~f ðxÞ ¼
In this section we describe the main ideas of the FPM method proposed by [9]. The FPM is a member of the family of the least square (LS) methods and it is closely related to the finite point method by Oñate et al. [8,17]. Although they are very similar they are not identical. The main difference is that finite point method of Oñate uses polynomial basis and the FPM method uses Taylor series which allow to compute, by an LS approach, the function and its derivatives values that naturally appear as unknown coefficients in the series. The method is based on the so-called moving least squares procedure which is shortly described next, following [14]: Let X be a given domain with boundary @ X and suppose that the set of points x1 ; x2 ; . . . ; xn is distributed with corresponding function values f ðx1 Þ; f ðx2 Þ; . . . ; f ðxn Þ. The problem is to find an approximate value of f at some arbitrary location f ðxÞ. Thus, the following procedure can be applied: Define the approximation to f ðxÞ as
~f ðxÞ ¼
m X pk ðxÞbk ðxÞ ¼ pt ðxÞbðxÞ
ð1Þ
k¼1
m X Þbk ðxÞ ¼ pt ðx ÞbðxÞ pk ðx
ð2Þ
k¼1
where pk ðxÞ denotes a set of linear independent functions, in particular, they can be linear monomials. Now, minimize the quadratic form
J ¼
n X wðx; xj Þe2j
ð3Þ
j¼1
¼
n m X X wðx; xj Þ pk ðxj Þbk ðxÞ f ðxj Þ j¼1
!2 ð4Þ
k¼1
in order to get the optimal coefficients
b ¼ A1 Bf ¼ ðBPÞ1 ðPt WÞf
ð5Þ
where
p1 ðx1 Þ
p2 ðx1 Þ
pm ðx1 Þ
3
6 p ðx2 Þ 6 1 P¼6 .. 6 4 .
p2 ðx2 Þ .. .
.. .
pm ðx2 Þ .. .
7 7 7 7 5
2
6 6 W¼6 6 4
wðx; x1 Þ 0 .. . 0
0
If the base functions pi ðxÞ are defined as follows:
pt ¼ ½1; Dxj ; Dyj ; Dzj ; . . .;
ð11Þ
where Dxj ¼ xj x; Dyj ¼ yj y and Dzj ¼ zj z for j ¼ 1; . . . ; np , the following equivalent representation is obtained
~f ðxÞ ’
m X pk ðxÞbk ðxÞ ¼ f ðxj Þ þ rf ðxj Þ Dxj þ
ð12Þ
k¼1
which implies that under this representation the new vector of unknown coefficients becomes
bðxÞ ¼ ½f ðxÞ; @ x f ðxÞ; @ y f ðxÞ; @ z f ðxÞ; . . .
t
ð13Þ
In this way we automatically get the values of the function and its derivatives at point x. We refer to [10] for a more explicit presentation of the FPM method applied to the Poisson equation. 3. Numerical implementation
0
ð6Þ
3
f ¼ ½f ðx1 Þ; f ðx2 Þ; . . . ; f ðxnp Þt ¼ ½f 1 ; f 2 ; . . . ; f np t
@T kDT ¼ Q_ ; @t
in X
ð14Þ
with the following boundary conditions
T ¼ T on C1 on C2 n k rT ¼ q
ð15Þ ð16Þ
n krT ¼ hc ðT T 1 Þ on C3
ð17Þ
and initial condition
Tjt¼0 ¼ T 0
ð18Þ
where X denotes de domain of interest, @ X ¼ C1 [ C2 [ C3 is the boundary. T; Q_ ; k; c; q and n denotes the temperature, source of heat generation per unit volume, thermal conductivity, specific heat of the material, density of the material and the unit outward normal to the boundary, respectively.
3.1.1. FPM form for the heat equation In the FPM representation for the unsteady three dimensional heat equation, the matrices we need to compute by each particle in X take the following form: If xi 2 X, then 0
7 wðx; x2 Þ 0 7 7 .. .. .. 7 5 . . . 0 wðx; xnp Þ
bðxÞ ¼ ½b1 ðxÞ; b2 ðxÞ; . . . ; bm ðxÞ
qc
3.1. FPM discretization
p1 ðxnp Þ p2 ðxnp Þ pm ðxnp Þ 2
ð10Þ
k¼1
Along this section we will describe some issues regarding the numerical implementation for FPM applied to two dimensional problems of the following form
whose local version reads
~f ðx; x Þ ¼
m X pk ðxÞbk ðxÞ ¼ pt ðxÞAðxÞ1 BðxÞf ¼ UðxÞf
ð7Þ
t
1 Dx1 Dy1 Dz1 B B 1 Dx Dy Dz 2 2 B 2 B .. .. . . P¼B B .. .. . . B B @ 1 Dxn Dyn Dzn 2 0 0 0
1 ð D x1 Þ 2 2 1 ð D x2 Þ 2 2
.. . 1 ðDxn Þ2 2
Dt
Dx1 Dy1 Dx1 Dz1 Dx2 Dy2 Dx2 Dz2 .. . Dxn Dyn Dxn Dzn 0 0
1 ðDy1 Þ2 2 1 ðDy2 Þ2 2
Dy1 Dz1
1 ðDyn Þ2 2
Dyn Dzn 0
Dt
np denotes the number of neighbor points xj of x and wðx; xj Þ denotes a weight function with compact support. Moreover, different weight functions have been used in the literature and the most common functions are the cubic spline and the Gaussian functions, being the last one chosen to be used in this paper. Once b is known the function approximation at point x reads
C C C C 2C 1 ðDzn Þ A 2 Dt
ð19Þ
ð8Þ ð9Þ
Dy2 Dz2
1
1 ðD z1 Þ2 2 C 1 ðD z2 Þ2 C C 2
2 6 6 6 6 W ¼6 6 6 4
wðx x1 Þ 0 .. . 0 0
0
0
wðx x2 Þ .. .. . .
0 .. .
0 0
3
7 7 7 7 7 7 7 wðx xn Þ 5 1
ð20Þ
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E.O. Reséndiz-Flores, I.D. García-Calvillo / International Journal of Heat and Mass Transfer 87 (2015) 104–110 1.2
FPM Analytical Solution
t= 0.1
t f ¼ T s;lþ1 ðx1 Þ; T s;lþ1 ðx2 Þ; . . . ; T s;lþ1 ðxnp Þ; 2DtQ_ þ 2T s;l ðxÞ þ DtDT s;l ð21Þ
1
bðxÞ ¼ T s;lþ1 ðxÞ; @ x T s;lþ1 ðxÞ; @ y T s;lþ1 ðxÞ; @ z T s;lþ1 ðxÞ; @ xx T s;lþ1 ðxÞ;
0.8
t @ xy T s;lþ1 ðxÞ; @ xz T s;lþ1 ðxÞ; @ yy T s;lþ1 ðxÞ; @ yz T s;lþ1 ðxÞ; @ zz T s;lþ1 ðxÞ
t= 0.3
0.6
ð22Þ 0.4
where s and l denotes the iteration and time counters, respectively. Moreover, since the FPM method is an iterative method over each particle in X, therefore the stopping criteria used in the implemented algorithm is a relative error of the following form
t= 0.5 t= 0.7
0.2
t= 1
0
0
0.5
1
1.5
2
2.5
3
Fig. 1. Exact and approximating solution corresponding to Example 1 at position x ¼ 0:9921; y ¼ 0:1653 and t ¼ 0:1; 0:3; 0:5; 0:7; 1.
Pnp
i¼1 jT
sþ1;l
Pnp
ðxi Þ T s;l ðxi Þj
i¼1 jT
sþ1;l
ðxi Þj
ð23Þ
where the solution at each time step is obtained once we reach convergence, i.e., T s;lþ1 ðxi Þ ¼ T sþ1 ðxi Þ as i ¼ 1; . . . ; np .
s ! 1 and this holds for
Fig. 2. Sliced 3D view for the approximating solution corresponding to Example 1 at t ¼ 0:1; 0:3; 0:5; 0:7; 1.
E.O. Reséndiz-Flores, I.D. García-Calvillo / International Journal of Heat and Mass Transfer 87 (2015) 104–110 0.14
3.2. Shape function
FPM Analytical Solution
t= 0.1
0.12
Regarding the shape function, the following Gaussian type function is implemented:
t= 0.3
0.1
(
t= 0.5
0.08
wðx xi Þ ¼
t= 0.7
0.06
107
eckxxi k
2
=h2
0
; if
kxxi k h
6 1;
else:
ð24Þ
t= 1
0.04
4. Numerical examples
0.02
The Finite Pointset Method is implemented in a threedimensional non-stationary setting in order to solve four particular examples taken from [7,6] which are described below. These particular examples are of interest since their analytic solutions are known, thus the approximate numerical solution can be compared with respect to the exact one.
0
0
0.5
1
1.5
2
2.5
3
Fig. 3. Exact and approximating solution corresponding to Example 2 at position x ¼ 0:3158; y ¼ 0:1053 and t ¼ 0:1; 0:3; 0:5; 0:7; 1.
Fig. 4. Sliced 3D view for the approximating solution corresponding to Example 2 at t ¼ 0:1; 0:3; 0:5; 0:7; 1.
108
E.O. Reséndiz-Flores, I.D. García-Calvillo / International Journal of Heat and Mass Transfer 87 (2015) 104–110 11
Example 1. Homogeneous Dirichlet conditions.
FPM Analytical Solution
10
t= 1
Consider the following problem
9
@T @ 2 T @ 2 T @ 2 T ¼ 2þ 2þ 2; @t @x @y @z
8 t= 0.7
7
0 6 x; y; z 6 p;
6
t= 0.5
with boundary conditions
5
t= 0.3
Tj@X ¼ 0
t= 0.1
4
and initial condition
3
Tðx; y; z; 0Þ ¼ 10sinðxÞsinðyÞsinðzÞ
2 1
0
The corresponding analytical solution reads 0
0.5
1
1.5
2
2.5
3
Tðx; y; z; tÞ ¼ 10sinðxÞsinðyÞsinðzÞe3t :
Fig. 5. Exact and approximating solution corresponding to Example 3 at position x ¼ 0:3158; y ¼ 0:1053 and t ¼ 0:1; 0:3; 0:5; 0:7; 1.
Fig. 6. Sliced 3D view for the approximating solution corresponding to Example 3 at t ¼ 0:1; 0:3; 0:5; 0:7; 1.
109
E.O. Reséndiz-Flores, I.D. García-Calvillo / International Journal of Heat and Mass Transfer 87 (2015) 104–110 110
Example 3. Non-homogeneous Dirichlet boundary conditions t= 0.2
100
Consider the following problem
t= 0.1
90
1 1 1 @T @ 2 T @ 2 T @ 2 T þ þ ¼ 2þ 2þ 2 a2 b2 c2 @t @x @y @z
80 70
t= 0.05
60
0 6 x 6 a;
50
20 10
z
Tða; y; z; tÞ ¼ etþ1þbþc
y
x
z
Tðx; b; z; tÞ ¼ etþaþ1þc
t= 0.01
0
0.2
0.4
0.6
0.8
x
y
Tðx; y; c; tÞ ¼ etþaþbþ1
Example 2. Homogeneous Dirichlet conditions.
1
1 1 1 @T @ 2 T @ 2 T @ 2 T þ þ ¼ 2þ 2þ 2 a2 b2 c2 @t @x @y @z 0 6 z 6 c;
x
Tðx; y; 0; tÞ ¼ etþaþb ;
z
z
y
and initial condition x
y
z
Tðx; y; z; 0Þ ¼ eaþbþc
The corresponding analytical solution reads x
y
z
Tðx; y; z; tÞ ¼ etþaþbþc
Consider the following problem
0 6 y 6 b;
x
Tðx; 0; z; tÞ ¼ etþaþc ;
FPM Analytical Solution
Fig. 7. Exact and approximating solution corresponding to Example 4 along the radial coordinate ðrÞ and t ¼ 0:01; 0:05; 0:1; 0:2.
0 6 x 6 a;
0
y
Tð0; y; z; tÞ ¼ etþbþc ;
30
p2
0 6 z 6 c;
with boundary conditions
40
0
0 6 y 6 b;
0
with boundary conditions
Example 4. Homogeneous Neumann boundary conditions Consider the following problem which involves a cylindrical region, the cylinder has initial temperature zero, the top and the bottom surfaces are insulated and the wall temperature is kept constant
1 @T @ 2 T @ 2 T @ 2 T ¼ 2þ 2þ 2 K @t @x @y @z
Tj@ X ¼ 0 and initial condition
px py pz Tðx; y; z; 0Þ ¼ 10sin sin sin a b c The corresponding analytical solution reads
px py pz Tðx; y; z; tÞ ¼ et sin sin sin a b c
with boundary conditions
Tðr; tÞ ¼ 100;
r ¼ 1;
ðr denotes the radial coordinateÞ
@Tðx; y; 2; tÞ @Tðx; y; 0; tÞ ¼ ¼0 @z @z
Fig. 8. Sliced 3D view for the approximating solution corresponding to Example 4 at t ¼ 0:01; 0:05; 0:1; 0:2.
110
E.O. Reséndiz-Flores, I.D. García-Calvillo / International Journal of Heat and Mass Transfer 87 (2015) 104–110
and initial condition
Tðx; y; z; 0Þ ¼ 0
background mesh arrangement for Gaussian points is not needed, therefore it is a feasible and much less involved method for real problems solving in a three dimensional setting.
The corresponding analytical solution reads
Tðr; tÞ ¼ T 1
! 1 2X J ðr an Þ 2 eK an t 0 a n¼1 an J1 ðaan Þ
where T is the surface temperature, a is the radius of cylinder, K and an , for n ¼ 1; 2; . . ., are the roots of J0 ðaaÞ ¼ 0. In order to numerically solve the presented problems the following parameters have been used: N ¼ 20; dt ¼ 103 ; c ¼ 6 for all examples, h ¼ 0:3927 for the first example while h ¼ 0:2625 and h ¼ 0:375 for Examples 2 and 3, respectively. The constants defining the domains are taken as a ¼ 1; b ¼ 2; c ¼ 3 in these examples. Finally we choose K ¼ 5 and h ¼ 0:25 in Example 4. The behaviour of the numerical solutions are sketched in Figs. 1, 3, 5 and 7, where the exact and approximating solutions are plotted for each example at different time instances. Since the problems are three dimensional the displayed graphics correspond to a two dimensional slice at the positions x ¼ 0:9921 and y ¼ 0:1653 in the first example while x ¼ 0:3158 and y ¼ 0:1053 are fixed for Examples 2 and 3. Finally a cut along the radial coordinate r has been considered in the last example. Moreover, a sliced 3D view is reported in Figs. 2, 4, 6 and 8 corresponding to Examples 1–4, respectively. The approximating solution obtained by FPM is in a good agreement with the exact solution. Moreover, a good advantage of the FPM is that there is no need to keep a regular particle distribution in order to obtain good numerical approximating solutions not even have to use Gaussian points like other meshless methods. 5. Conclusions We have successfully implemented, for the first time and to the authors knowledge, the discussed version of the FPM method for unsteady three dimensional heat transfer problems. Based on the numerical performance we can conclude that FPM is other meshless method that can be used to successfully solve more complex energy transfer problems. The biggest advantage of FPM over other meshless methods is that the implementation of boundary conditions is very simple and it is a real full meshfree method, since it does not need to compute any numerical quadratures, thus a
Conflict of interest None declared. References [1] L.B. Lucy, A numerical approach to the testing of the fission hypothesis, Astron. J. 82 (1977) 1013–1024. [2] R.A. Gingold, J.J. Monaghan, Smoothed particle hydrodynamics: theory and applications to non-spherical stars, Month. Notices R. Astron. Soc. 181 (1977) 375–389. [3] V.P. Nguyen, T. Rabczuk, S. Bordas, M. Duflot, Meshless methods: a review and computer implementation aspects, Math. Comput. Simul. 79 (2008) 763–813. [4] Y. Liu, X. Zhang, M.W. Lu, A meshless method based on least-squares approach for steady-and unsteady-state heat conduction problems, Numer. Heat Transfer, Part B 47 (2005) 257–275. [5] J.K. Chen, J.E. Beraun, T.C. Carney, A corrective smoothed particle method for boundary value problems in heat conduction, Int. J. Numer. Methods Eng. 46 (1999) 231–252. [6] R. Cheng, K.M. Liew, The reproducing kernel particle method for twodimensional unsteady heat conduction problems, Comput. Mech. 45 (2009) 1–10. [7] R. Cheng, K.M. Liew, A meshless analysis of three-dimensional transient heat conduction problems, Eng. Anal. Bound. Elem. 36 (2012) 203–210. [8] E. Oñate, S. Idelsohn, O. Zienkiewics, R. Taylor, A finite point method in computational mechanics. applications to convective transport and fluid flow, Int. J. Numer. Methods Eng. 39 (1996) 3839–3866. [9] J. Kuhnert, General smoothed particle hydrodynamics (Ph.D. thesis), Technische Universität Kaiserslautern, 1999. [10] S. Tiwari, J. Kuhnert, Grid free method for solving the poisson equation, Berichte des Fraunhofer ITWM 25. [11] S. Tiwari, J. Kuhnert, Finite pointset method based on the projection method for simulations of the incompressible Navier–Stokes equations, Springer LNCSE: Meshfree methods for Partial Differential Equations, 26. [12] S. Tiwari, S. Antonov, D. Hietel, J. Kuhnert, F. Olawsky, R. Wegener, A meshfree method for simulations of interactions between fluids and flexible structures, Lect. Notes Comput. Sci. Eng. 57 (2006) 249–264. [13] S. Tiwari, J. Kuhnert, Modeling of two-phase flows with surface tension by finite pointset method (FPM), J. Comput. Appl. Math. 203 (2007) 376–386. [14] A. Wawren´czuk, J. Kuhnert, N. Siedow, FPM computations of glass cooling with radiation, Comput. Methods Appl. Mech. Eng. 196 (2007) 4656–4671. [15] M. Cheng, G.R. Liu, A novel finite point method for flow simulation, Int. J. Numer. Methods Fluids 39 (2002) 1161–1178. [16] E.O. Reséndiz-Flores, I.D. García-Calvillo, Application of the finite pointset method to non-stationary heat conduction problems, Int. J. Heat Mass Transfer 71 (2014) 720–723. [17] E. Oñate, S. Idelsohn, O.C. Zienkiewics, R.L. Taylor, S. Sacco, A stabilized finite point method for analysis of fluid mechanics problems, Comput. Methods Appl. Mech. Eng. 139 (1996) 315–346.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical solution of 3D non-stationary heat conduction problems using the Finite Pointset Method Edgar O. Reséndiz-Flores a,⇑, Irma D. García-Calvillo b a Division of Postgraduate Studies and Research, Department of Metal-Mechanical Engineering, The Technological Institute of Saltillo, Blvd. V. Carranza 2400 Col. Tecnológico, C.P. 25280 Saltillo, Coahuila, Mexico b Research Center on Applied Mathematics, Autonomous University of Coahuila, Campus Saltillo, Building S, C.P. 25000 Saltillo, Coahuila, Mexico
a r t i c l e
i n f o
Article history: Received 23 October 2014 Received in revised form 26 March 2015 Accepted 26 March 2015 Available online 11 April 2015
a b s t r a c t This paper presents the application of the Finite Pointset Method for the numerical solution of three dimensional and non-stationary heat conduction problems. The strong formulation of the parabolic partial differential equation is directly used instead of the corresponding weak form. Moreover, a numerical comparison between the Finite Pointset Method and the corresponding analytical solutions is reported. Ó 2015 Elsevier Ltd. All rights reserved.
Keywords: Three dimensional heat conduction Non-stationary heat conduction Meshless method Finite Pointset Method
1. Introduction During the past 30 years different type of meshless methods for solving partial differential equations have been developed with the objective of eliminating part of the difficulties arising when mesh-based methods are used. The starting point of these methods was the smooth particle hydrodynamics (SPH) method proposed by Lucy [1] and Gingold and Monaghan [2] in the area of astrophysics and applied later in other research fields. For a good overview on meshless methods we refer to [3] and references therein. In the field of heat transfer by conduction different meshless methods have been already applied. Among these scientific works some of the recent publications, to the authors knowledge, are the following works: Liu et al. [4] uses a meshless weighted least squares method for heat conduction. Chen et al. [5] applied a corrective SPH method to solve unsteady heat conduction problems. Cheng and Liew [6,7] applied the reproducing kernel particle method (RKPM) for two and three-dimensional unsteady heat conduction problems, respectively. We refer to [7] for a concise and
⇑ Corresponding author. Tel./fax: +52 (844)438 9500. E-mail addresses: [email protected] (E.O. Reséndiz-Flores), irma.garcia @uadec.edu.mx (I.D. García-Calvillo). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.03.084 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
recent overview on meshless methods on the field of heat conduction problems. As an alternative to the solution for the unsteady heat conduction equation we propose in this work the application of a slightly different version of the finite point method developed by Oñate [8]. This method is called the Finite Pointset Method (FPM) and to the authors knowledge this version of the method has been developed by Kuhnert [9] in the Fraunhofer-Institut für Techno- und Wirtschaftsmathematik, in Kaiserslautern, Germany, and it has been already applied in the fields of fluid mechanics [10–13] and radiative heat transfer problems [14]. Later on, a very close version of the FPM of Kuhnert has been also developed by Cheng and Liu, [15] in the field of fluid mechanics. The presented work in this article is an extension to three dimensions of the work already published by the authors in [16] and to the authors knowledge this is the first time that three dimensional numerical results regarding heat conduction problems using the Finite Pointset Method have been reported. In order to get some insight into FPM performance we compare the numerical solution to some specific problems whose analytical solutions are known. The structure of the paper is as follows: Section 2 shortly describe the basic ideas behind FPM. Section 3 presents some issues regarding the numerical implementation of FPM. The test examples are presented in Section 4 with the corresponding results. Finally some conclusions are given in last section.
105
E.O. Reséndiz-Flores, I.D. García-Calvillo / International Journal of Heat and Mass Transfer 87 (2015) 104–110
2. The FPM method
~f ðxÞ ¼
In this section we describe the main ideas of the FPM method proposed by [9]. The FPM is a member of the family of the least square (LS) methods and it is closely related to the finite point method by Oñate et al. [8,17]. Although they are very similar they are not identical. The main difference is that finite point method of Oñate uses polynomial basis and the FPM method uses Taylor series which allow to compute, by an LS approach, the function and its derivatives values that naturally appear as unknown coefficients in the series. The method is based on the so-called moving least squares procedure which is shortly described next, following [14]: Let X be a given domain with boundary @ X and suppose that the set of points x1 ; x2 ; . . . ; xn is distributed with corresponding function values f ðx1 Þ; f ðx2 Þ; . . . ; f ðxn Þ. The problem is to find an approximate value of f at some arbitrary location f ðxÞ. Thus, the following procedure can be applied: Define the approximation to f ðxÞ as
~f ðxÞ ¼
m X pk ðxÞbk ðxÞ ¼ pt ðxÞbðxÞ
ð1Þ
k¼1
m X Þbk ðxÞ ¼ pt ðx ÞbðxÞ pk ðx
ð2Þ
k¼1
where pk ðxÞ denotes a set of linear independent functions, in particular, they can be linear monomials. Now, minimize the quadratic form
J ¼
n X wðx; xj Þe2j
ð3Þ
j¼1
¼
n m X X wðx; xj Þ pk ðxj Þbk ðxÞ f ðxj Þ j¼1
!2 ð4Þ
k¼1
in order to get the optimal coefficients
b ¼ A1 Bf ¼ ðBPÞ1 ðPt WÞf
ð5Þ
where
p1 ðx1 Þ
p2 ðx1 Þ
pm ðx1 Þ
3
6 p ðx2 Þ 6 1 P¼6 .. 6 4 .
p2 ðx2 Þ .. .
.. .
pm ðx2 Þ .. .
7 7 7 7 5
2
6 6 W¼6 6 4
wðx; x1 Þ 0 .. . 0
0
If the base functions pi ðxÞ are defined as follows:
pt ¼ ½1; Dxj ; Dyj ; Dzj ; . . .;
ð11Þ
where Dxj ¼ xj x; Dyj ¼ yj y and Dzj ¼ zj z for j ¼ 1; . . . ; np , the following equivalent representation is obtained
~f ðxÞ ’
m X pk ðxÞbk ðxÞ ¼ f ðxj Þ þ rf ðxj Þ Dxj þ
ð12Þ
k¼1
which implies that under this representation the new vector of unknown coefficients becomes
bðxÞ ¼ ½f ðxÞ; @ x f ðxÞ; @ y f ðxÞ; @ z f ðxÞ; . . .
t
ð13Þ
In this way we automatically get the values of the function and its derivatives at point x. We refer to [10] for a more explicit presentation of the FPM method applied to the Poisson equation. 3. Numerical implementation
0
ð6Þ
3
f ¼ ½f ðx1 Þ; f ðx2 Þ; . . . ; f ðxnp Þt ¼ ½f 1 ; f 2 ; . . . ; f np t
@T kDT ¼ Q_ ; @t
in X
ð14Þ
with the following boundary conditions
T ¼ T on C1 on C2 n k rT ¼ q
ð15Þ ð16Þ
n krT ¼ hc ðT T 1 Þ on C3
ð17Þ
and initial condition
Tjt¼0 ¼ T 0
ð18Þ
where X denotes de domain of interest, @ X ¼ C1 [ C2 [ C3 is the boundary. T; Q_ ; k; c; q and n denotes the temperature, source of heat generation per unit volume, thermal conductivity, specific heat of the material, density of the material and the unit outward normal to the boundary, respectively.
3.1.1. FPM form for the heat equation In the FPM representation for the unsteady three dimensional heat equation, the matrices we need to compute by each particle in X take the following form: If xi 2 X, then 0
7 wðx; x2 Þ 0 7 7 .. .. .. 7 5 . . . 0 wðx; xnp Þ
bðxÞ ¼ ½b1 ðxÞ; b2 ðxÞ; . . . ; bm ðxÞ
qc
3.1. FPM discretization
p1 ðxnp Þ p2 ðxnp Þ pm ðxnp Þ 2
ð10Þ
k¼1
Along this section we will describe some issues regarding the numerical implementation for FPM applied to two dimensional problems of the following form
whose local version reads
~f ðx; x Þ ¼
m X pk ðxÞbk ðxÞ ¼ pt ðxÞAðxÞ1 BðxÞf ¼ UðxÞf
ð7Þ
t
1 Dx1 Dy1 Dz1 B B 1 Dx Dy Dz 2 2 B 2 B .. .. . . P¼B B .. .. . . B B @ 1 Dxn Dyn Dzn 2 0 0 0
1 ð D x1 Þ 2 2 1 ð D x2 Þ 2 2
.. . 1 ðDxn Þ2 2
Dt
Dx1 Dy1 Dx1 Dz1 Dx2 Dy2 Dx2 Dz2 .. . Dxn Dyn Dxn Dzn 0 0
1 ðDy1 Þ2 2 1 ðDy2 Þ2 2
Dy1 Dz1
1 ðDyn Þ2 2
Dyn Dzn 0
Dt
np denotes the number of neighbor points xj of x and wðx; xj Þ denotes a weight function with compact support. Moreover, different weight functions have been used in the literature and the most common functions are the cubic spline and the Gaussian functions, being the last one chosen to be used in this paper. Once b is known the function approximation at point x reads
C C C C 2C 1 ðDzn Þ A 2 Dt
ð19Þ
ð8Þ ð9Þ
Dy2 Dz2
1
1 ðD z1 Þ2 2 C 1 ðD z2 Þ2 C C 2
2 6 6 6 6 W ¼6 6 6 4
wðx x1 Þ 0 .. . 0 0
0
0
wðx x2 Þ .. .. . .
0 .. .
0 0
3
7 7 7 7 7 7 7 wðx xn Þ 5 1
ð20Þ
106
E.O. Reséndiz-Flores, I.D. García-Calvillo / International Journal of Heat and Mass Transfer 87 (2015) 104–110 1.2
FPM Analytical Solution
t= 0.1
t f ¼ T s;lþ1 ðx1 Þ; T s;lþ1 ðx2 Þ; . . . ; T s;lþ1 ðxnp Þ; 2DtQ_ þ 2T s;l ðxÞ þ DtDT s;l ð21Þ
1
bðxÞ ¼ T s;lþ1 ðxÞ; @ x T s;lþ1 ðxÞ; @ y T s;lþ1 ðxÞ; @ z T s;lþ1 ðxÞ; @ xx T s;lþ1 ðxÞ;
0.8
t @ xy T s;lþ1 ðxÞ; @ xz T s;lþ1 ðxÞ; @ yy T s;lþ1 ðxÞ; @ yz T s;lþ1 ðxÞ; @ zz T s;lþ1 ðxÞ
t= 0.3
0.6
ð22Þ 0.4
where s and l denotes the iteration and time counters, respectively. Moreover, since the FPM method is an iterative method over each particle in X, therefore the stopping criteria used in the implemented algorithm is a relative error of the following form
t= 0.5 t= 0.7
0.2
t= 1
0
0
0.5
1
1.5
2
2.5
3
Fig. 1. Exact and approximating solution corresponding to Example 1 at position x ¼ 0:9921; y ¼ 0:1653 and t ¼ 0:1; 0:3; 0:5; 0:7; 1.
Pnp
i¼1 jT
sþ1;l
Pnp
ðxi Þ T s;l ðxi Þj
i¼1 jT
sþ1;l
ðxi Þj
ð23Þ
where the solution at each time step is obtained once we reach convergence, i.e., T s;lþ1 ðxi Þ ¼ T sþ1 ðxi Þ as i ¼ 1; . . . ; np .
s ! 1 and this holds for
Fig. 2. Sliced 3D view for the approximating solution corresponding to Example 1 at t ¼ 0:1; 0:3; 0:5; 0:7; 1.
E.O. Reséndiz-Flores, I.D. García-Calvillo / International Journal of Heat and Mass Transfer 87 (2015) 104–110 0.14
3.2. Shape function
FPM Analytical Solution
t= 0.1
0.12
Regarding the shape function, the following Gaussian type function is implemented:
t= 0.3
0.1
(
t= 0.5
0.08
wðx xi Þ ¼
t= 0.7
0.06
107
eckxxi k
2
=h2
0
; if
kxxi k h
6 1;
else:
ð24Þ
t= 1
0.04
4. Numerical examples
0.02
The Finite Pointset Method is implemented in a threedimensional non-stationary setting in order to solve four particular examples taken from [7,6] which are described below. These particular examples are of interest since their analytic solutions are known, thus the approximate numerical solution can be compared with respect to the exact one.
0
0
0.5
1
1.5
2
2.5
3
Fig. 3. Exact and approximating solution corresponding to Example 2 at position x ¼ 0:3158; y ¼ 0:1053 and t ¼ 0:1; 0:3; 0:5; 0:7; 1.
Fig. 4. Sliced 3D view for the approximating solution corresponding to Example 2 at t ¼ 0:1; 0:3; 0:5; 0:7; 1.
108
E.O. Reséndiz-Flores, I.D. García-Calvillo / International Journal of Heat and Mass Transfer 87 (2015) 104–110 11
Example 1. Homogeneous Dirichlet conditions.
FPM Analytical Solution
10
t= 1
Consider the following problem
9
@T @ 2 T @ 2 T @ 2 T ¼ 2þ 2þ 2; @t @x @y @z
8 t= 0.7
7
0 6 x; y; z 6 p;
6
t= 0.5
with boundary conditions
5
t= 0.3
Tj@X ¼ 0
t= 0.1
4
and initial condition
3
Tðx; y; z; 0Þ ¼ 10sinðxÞsinðyÞsinðzÞ
2 1
0
The corresponding analytical solution reads 0
0.5
1
1.5
2
2.5
3
Tðx; y; z; tÞ ¼ 10sinðxÞsinðyÞsinðzÞe3t :
Fig. 5. Exact and approximating solution corresponding to Example 3 at position x ¼ 0:3158; y ¼ 0:1053 and t ¼ 0:1; 0:3; 0:5; 0:7; 1.
Fig. 6. Sliced 3D view for the approximating solution corresponding to Example 3 at t ¼ 0:1; 0:3; 0:5; 0:7; 1.
109
E.O. Reséndiz-Flores, I.D. García-Calvillo / International Journal of Heat and Mass Transfer 87 (2015) 104–110 110
Example 3. Non-homogeneous Dirichlet boundary conditions t= 0.2
100
Consider the following problem
t= 0.1
90
1 1 1 @T @ 2 T @ 2 T @ 2 T þ þ ¼ 2þ 2þ 2 a2 b2 c2 @t @x @y @z
80 70
t= 0.05
60
0 6 x 6 a;
50
20 10
z
Tða; y; z; tÞ ¼ etþ1þbþc
y
x
z
Tðx; b; z; tÞ ¼ etþaþ1þc
t= 0.01
0
0.2
0.4
0.6
0.8
x
y
Tðx; y; c; tÞ ¼ etþaþbþ1
Example 2. Homogeneous Dirichlet conditions.
1
1 1 1 @T @ 2 T @ 2 T @ 2 T þ þ ¼ 2þ 2þ 2 a2 b2 c2 @t @x @y @z 0 6 z 6 c;
x
Tðx; y; 0; tÞ ¼ etþaþb ;
z
z
y
and initial condition x
y
z
Tðx; y; z; 0Þ ¼ eaþbþc
The corresponding analytical solution reads x
y
z
Tðx; y; z; tÞ ¼ etþaþbþc
Consider the following problem
0 6 y 6 b;
x
Tðx; 0; z; tÞ ¼ etþaþc ;
FPM Analytical Solution
Fig. 7. Exact and approximating solution corresponding to Example 4 along the radial coordinate ðrÞ and t ¼ 0:01; 0:05; 0:1; 0:2.
0 6 x 6 a;
0
y
Tð0; y; z; tÞ ¼ etþbþc ;
30
p2
0 6 z 6 c;
with boundary conditions
40
0
0 6 y 6 b;
0
with boundary conditions
Example 4. Homogeneous Neumann boundary conditions Consider the following problem which involves a cylindrical region, the cylinder has initial temperature zero, the top and the bottom surfaces are insulated and the wall temperature is kept constant
1 @T @ 2 T @ 2 T @ 2 T ¼ 2þ 2þ 2 K @t @x @y @z
Tj@ X ¼ 0 and initial condition
px py pz Tðx; y; z; 0Þ ¼ 10sin sin sin a b c The corresponding analytical solution reads
px py pz Tðx; y; z; tÞ ¼ et sin sin sin a b c
with boundary conditions
Tðr; tÞ ¼ 100;
r ¼ 1;
ðr denotes the radial coordinateÞ
@Tðx; y; 2; tÞ @Tðx; y; 0; tÞ ¼ ¼0 @z @z
Fig. 8. Sliced 3D view for the approximating solution corresponding to Example 4 at t ¼ 0:01; 0:05; 0:1; 0:2.
110
E.O. Reséndiz-Flores, I.D. García-Calvillo / International Journal of Heat and Mass Transfer 87 (2015) 104–110
and initial condition
Tðx; y; z; 0Þ ¼ 0
background mesh arrangement for Gaussian points is not needed, therefore it is a feasible and much less involved method for real problems solving in a three dimensional setting.
The corresponding analytical solution reads
Tðr; tÞ ¼ T 1
! 1 2X J ðr an Þ 2 eK an t 0 a n¼1 an J1 ðaan Þ
where T is the surface temperature, a is the radius of cylinder, K and an , for n ¼ 1; 2; . . ., are the roots of J0 ðaaÞ ¼ 0. In order to numerically solve the presented problems the following parameters have been used: N ¼ 20; dt ¼ 103 ; c ¼ 6 for all examples, h ¼ 0:3927 for the first example while h ¼ 0:2625 and h ¼ 0:375 for Examples 2 and 3, respectively. The constants defining the domains are taken as a ¼ 1; b ¼ 2; c ¼ 3 in these examples. Finally we choose K ¼ 5 and h ¼ 0:25 in Example 4. The behaviour of the numerical solutions are sketched in Figs. 1, 3, 5 and 7, where the exact and approximating solutions are plotted for each example at different time instances. Since the problems are three dimensional the displayed graphics correspond to a two dimensional slice at the positions x ¼ 0:9921 and y ¼ 0:1653 in the first example while x ¼ 0:3158 and y ¼ 0:1053 are fixed for Examples 2 and 3. Finally a cut along the radial coordinate r has been considered in the last example. Moreover, a sliced 3D view is reported in Figs. 2, 4, 6 and 8 corresponding to Examples 1–4, respectively. The approximating solution obtained by FPM is in a good agreement with the exact solution. Moreover, a good advantage of the FPM is that there is no need to keep a regular particle distribution in order to obtain good numerical approximating solutions not even have to use Gaussian points like other meshless methods. 5. Conclusions We have successfully implemented, for the first time and to the authors knowledge, the discussed version of the FPM method for unsteady three dimensional heat transfer problems. Based on the numerical performance we can conclude that FPM is other meshless method that can be used to successfully solve more complex energy transfer problems. The biggest advantage of FPM over other meshless methods is that the implementation of boundary conditions is very simple and it is a real full meshfree method, since it does not need to compute any numerical quadratures, thus a
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