Simulation of thermal field in mass concrete structures with cooling pipes by the localized radial basis function collocation method

International Journal of Heat and Mass Transfer 129 (2019) 449–459

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Review

Simulation of thermal field in mass concrete structures with cooling pipes by the localized radial basis function collocation method Yongxing Hong, Ji Lin ⇑, Wen Chen ** State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, International Center for Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing 211100, China

a r t i c l e

i n f o

Article history: Received 29 March 2018 Received in revised form 31 July 2018 Accepted 9 September 2018

Keywords: Mass concrete structure Cooling system Localized radial basis function collocation method Multiple-scale technique

a b s t r a c t During the construction of mass concrete structures, the embedded water pipe cooling system is regarded as a standard tool to control temperature. Prediction of thermal field in the structures plays an essential role in the design of cooling systems. In this study, the localized radial basis function collocation method is presented to simulate the thermal field in concrete structures with the water pipe cooling system. In the proposed scheme, the multiquadric radial basis function is used for the spatial discretization where a novel technique is applied for the determination of shape parameters. Because of the localized strategy, the sparse system can be formed to save storage and reduce computational cost. Five examples including a problem with five water cooling pipes and a three-dimensional problem with one cooling pipe are experimented. From the numerical results, it is evidently that the present method is feasible and has attractive advantages in the simulation of thermal fields in practical concrete structures with cooling systems. Ó 2018 Elsevier Ltd. All rights reserved.

Contents 1. 2. 3.

4.

5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Scheme for temporal discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Localized radial basis function collocation method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Formulation of the LRBFCM for considered problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Multiple-scale technique in MQ-RBF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical experiments and discussions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conflict of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

449 451 451 451 451 452 452 453 453 453 456 457 458 458 458 458

1. Introduction ⇑ Corresponding author. ⇑⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (J. Lin), chenwen@ hhu.edu.cn (W. Chen). https://doi.org/10.1016/j.ijheatmasstransfer.2018.09.037 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

Thermal stress is one of the most essential factors on crack germination in mass concrete structures, such as dams, grounds, buildings, bridges, etc. During the construction of concrete structures, a large amount of hydration heat is released which is not

450

Y. Hong et al. / International Journal of Heat and Mass Transfer 129 (2019) 449–459

easy to be dissipated especially in the early stage due to the poor heat transfer within the concrete [1,2]. Thus, the inner temperature of the structure rises sharply. On the contrary, the temperature on the surface decreases quickly due to the heat released into the environment. Consequently, the thermal difference between interior and exterior is generated and the tensile stress is produced on the concrete surface, which can lead to cracks and structural collapse. Therefore, reducing the inner temperature and increasing the thermal conductivity are both effective tools for the improvement of quality of the concrete structures. Despite of putting extra materials such as sand [3], bone [4], rubber [5], and fly ash [6] into the concrete, the water pipe cooling system [7] is regarded as a popular technology to release heat in mass concrete. By adopting the pipes, the heat of the concrete is absorbed by the water flow. The thermal capacity of the cooling system is upgraded and the hydration heat is decreased [8]. There are actually some flaws existing in this technology [9] even numerous of applications and experiments are conducted, such as the temperature of concrete around the embedded pipes should be well controlled, otherwise it will decrease shapely which results in high thermal gradient; apertures and interactive compression between concrete and pipe are unavoidable, where the structure is prone to the beginning of cracks; the distance between pipes has significant influence on temperature control and thermal gradient, but it is difficult to be determined; the surface temperature is not easy to satisfy the desired temperature requirements. Hence accurate prediction of the temperature distribution plays an important role in the design of cooling systems including the temperature and the velocity of running water, the arrangement of pipes, and the thermal retardation on the surface. Several models have been proposed to predict the thermal distribution [10–13]. One of the effective models is the equivalent equation of heat conduction in a concrete structure with a cooling water pipe proposed by Zhu [14] as described in the following equation

@T @h @/ ¼ ar2 T þ þ ðT c0  T w0 Þ ; @s @s @s

ð1Þ

where T is the unknown thermal field to determine, a is the thermal diffusivity of the considered concrete, T c0 is the pouring temperature of the concrete, T w0 is the initial temperature of cooling water, / is a known function associated with heat flux from concrete to water, and h is the adiabatic temperature raise, which can be obtained by experiments. Subsequently, this model has also been extended to the cases with embedded water pipe cooling systems [15,16]. Numerical techniques have witnessed a research boom in recent years. It is known to all that the finite element method (FEM) is one of the most widely used methods in science and engineering. There is no doubt that the FEM can be applied to solve the considered problem [17]. But the potential problem for the FEM simulation is the gap in size between the mass concrete structure and the tiny pipes. It means that a large quantity of small elements should be used around the pipes, otherwise the accuracy of the FEM is doubtful [18]. Furthermore, large-scale ratio problems make it worse, which is caused by the large difference in size between generated elements [19]. To avoid this problem, the practical shape and size of the pipes might be simply neglected whereby the equivalent temperature can be obtained, but the thermal field is remain unsolved [20]. For describing the thermal field, the composite element method (CEM) can be introduced to generate the long and tiny pipes [21]. By using strip elements, the pipes can be better depicted with small number of elements. Furthermore, the boundary element method (BEM) is proposed in combination with the FEM [22] for such problems. By generating elements only on the physical boundary, the dimension of the problem can be reduced by one, hence less number of elements

is needed. In addition, it can also take the advantage of the strip elements to reduce the number of elements. The virtual element method (VEM) can also be used for reducing elements [23]. Furthermore, the time-dependent fundamental solutions can be introduced to the BEM to efficiently handle the temporal term for the simulation of the heat transfer problems [24–28]. Instead of the mesh-based methods, a fast meshless method based on radial basis functions (RBF) is proposed in the paper. The localized radial basis function collocation method (LRBFCM) can be considered as an improved form of radial basis function collocation method (RBFCM), which is first proposed by Kansa [29] named as Kansa’s method. Unlike Kansa’s method, the LRBFCM divides the interest domain into several sub-domains. In each sub-domain, one center point is used to form a localized lowrank system with several neighboring points around it. Whilst, the k  d tree algorithm is employed to search the neighboring points of each center node, hence the low-rank matrix can be quickly formulated [30]. By reformulating all localized systems, a sparse system of equations is constructed [31]. As it is known to all that sparse data is by nature more easily to be compressed and thus requires significantly less storage. The LRBFCM has been developed for heat transfer and fluid flow problems by Kassab and Sarler [32–34]. Furthermore, the LRBFCM is widely used in solving partial differential equations (PDEs) [35–38]. The accuracy and stability of this approach are reported, whilst, the potential of the proposed method could be found in above-mentioned references and references therein. One of the fundamental problems of the RBF-based methods is the determination of the shape parameters in the RBFs such as the multiquatric (MQ) [39] and the Gaussian RBF [40]. Though the LRBFCM has reduced sensitivity of the shape parameters, it could still lead to doubtful results [41]. There are many attempts to choose good shape parameters [42,43]. However, most of them are only suitable for global RBF-based methods. As for the localized scheme, the leave-one-out cross validation (LOOCV) algorithm has been experimented by Lin [44]. Recently Maviric and Sarler [45] also reported an approach of using condition number of the collocation matrix to determine the shape parameter. Some good results have been reported in solving largescale realistic multi-physics engineering problems [46]. In this paper, the multiple-scale technique (MST) proposed by Liu [47] is used for the determination of shape parameters. The condition number of matrix system by the LRBFCM can be further reduced. Because in this technique the norm of all the rows or columns of an equilibrated matrix are the same, the shape parameters can be easily and quickly determined especially for the localized scheme. Although the selection of the shape parameter also depends on the fields that are supposed to be interpolated [48], the MST could accommodate the characteristics of the fields by an automatic function. Consequently, the LRBFCM combined with the MST is employed to predict the thermal field in mass concrete structures with water pipe cooling system.

Fig. 1. Water pipe cooling model.

451

Y. Hong et al. / International Journal of Heat and Mass Transfer 129 (2019) 449–459

The layout of this paper is as follows: Section 2 describes the mathematical model. In Section 3, the LRBFCM combined with the MST is proposed. Numerical experiments and discussions are presented in Section 4. Conclusions and remarks are provided in Section 5. 2. Mathematical model A typical water pipe cooling problem under the cuboid domain with a tube across the center of the domain is considered as shown in Fig. 1. It is assumed that the concrete is made up of the isotropic and homogeneous material. The water flow is laminar and the influence of the metal pipe on the heat conduction is quite little which can be neglected. All thermal properties are treated as constants. In this problem, the heat transport of the concrete is governed by Eq. (1) with the initial condition

Tðx; 0Þ ¼ T c0 ;

x 2 X;

ð2Þ

T n ¼ Tðx; sn Þ;

@hnþ1 @h nþ1 @/nþ1 @/ nþ1 ¼ ðs Þ; and ¼ ðs Þ; @s @s @s @s

ð9Þ

where sn ¼ n  Ds and n ¼ 0; 1; . . . ; S with a uniform time step size Ds ¼ T =S, T is the final time, T c0 ; T w0 ; h, and / are known functions. Rearranging Eq. (8) by moving the parts including T nþ1 to the left hand side and the others to the right hade side yields

ð1  aDs  r2 ÞT nþ1 ¼ g nþ1 ;

ð10Þ

where

g nþ1 ¼ T n þ Ds

@hnþ1 @/nþ1 þ DsðT c0  T w0 Þ : @s @s

ð11Þ

Once the boundary and initial conditions are given, the present LRBFCM can be applied to obtain the unknown function T nþ1 in Eq. (10) which will be described in the following subsection. 3.2. Localized radial basis function collocation method

under the following boundary conditions: 1. the Dirichlet boundary condition

x 2 C1 ;

Tðx; sÞ ¼ T b ðx; sÞ;

ð3Þ

2. the Neumann boundary conditions, which includes the adiabatic condition

@Tðx; sÞ ¼ 0; @n

x 2 C3 ;

ð4Þ

Kc

@Tðx; sÞ ¼ ba ðT a  Tðx; sÞÞ; @n

Kc

@Tðx; sÞ ¼ bw ðT w  Tðx; sÞÞ; @n

x 2 C2 ;

ð5Þ

x 2 Cp ;

ð6Þ

x 2 Cp ;

ð7Þ

where T pc is a known function for the approximation of the temperature on the concrete part contact with the water pipe. Consequently, the final model of this problem is formed by Eqs. (1)–(5) and (7).

3.1. Scheme for temporal discretization In this section, the Implicit-Euler scheme is introduced to discretize the temporal term in Eq. (1) as follows

T

nþ1

T @h ¼ ar2 T nþ1 þ @s Ds

with



asj U xs ; ysj ; xs ; ysj 2 Xs ;

ð12Þ

LT^ ðxs Þ ¼



Ns X



asj LU xs ; ysj ; xs ; ysj 2 Xs ;

ð13Þ

j¼1

where L denotes the differential operator, T^ ðxs Þ and LT^ ðxs Þ denote the numerical solutions of T and LT at any point xs in the subn o domain Xs , ysj are the selected neighboring points with the center in this sub-domain, Ns is the number of MQ-RBFs used in the subn o   domain, asj are unknown coefficients, U xs ; ysj is the MQ-RBF,   and LU xs ; ysj is the derivative of the MQ-RBF. The MQ-RBF is defined as





U xs ; ysj ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2    s x  ysj  þ c2 ;

ð14Þ

    where c is called shape parameter and xs  ysj  is the distance n o between xs and ysj . To solve asj , at least Ns number of conditions should be given. According to Eq. (12), by setting the collocation points to be the same as the neighboring points with the notation n o n o xsj ¼ ysj in each sub-domains, a symmetrical low-rank matrix can be formed as follows

T^ s ¼ Us as ;

ð15Þ

h i0 n o   ^ s ¼ T^ s ; T^ s ; . . . ; T^ s , as ¼ as ; as ; . . . ; as 0 , and Us ¼ Us where T 1 2 Ns 1 2 Ns ij    with the notations T^ si ¼ T^ xsi and Usij ¼ U xsi ; xsj . The matrix Us is

3. Numerical scheme

n



Ns X

and

where K c is the heat conductivity of concrete, ba or bw denotes the heat convection coefficient from concrete to the air or the water pipe, T b is the prescribed temperature of the boundary parts, T a or T w is the temperature of the air or the water flow, T c0 is the initial temperature of concrete, C1 ; C2 ; C3 , and Cp are the boundaries of the concrete exposing to the wooden box, air, insulation panel, and the water pipe, respectively. In this paper, we assume that the pipe temperature is a constant. It corresponds to the case that the water flow rate is very large such that any heated water is instantly taken away and replaced by new water. Furthermore, the heat convection from concrete to the water pipe Eq. (6) is presented by the Dirichlet boundary condition as follows:

nþ1

T^ ðxs Þ ¼

j¼1

and the heat convection conditions

Tðx; sÞ ¼ T pc ðx; sÞ;

In the LRBFCM, the interest domain is divided into several subdomains by endowing one center in each sub-domain and finding several neighboring points of the center. In each sub-domain, the Kansa method with linear combination of multiquadric radial basis functions (MQ-RBFs) is applied to approximate T and its derivative as follows

nþ1

@/ þ ðT c0  T w0 Þ @s

;

ð8Þ

positive definite as a result the coefficient a is unique and can be obtained

as ¼ ½Us 1 T^ s : Thus, the derivative of 1 LT^ si ¼ LUsi ½Us  T^ s ;

ð16Þ Tðxsi Þ

can be approximated as

ð17Þ

452

Y. Hong et al. / International Journal of Heat and Mass Transfer 129 (2019) 449–459

where LT^ si is the approximation of LTðxsi Þ, and LUsi denotes n oNs    with the notations LT^ si ¼ LT^ xsi and LUsij ¼ LU xsi ; xsj . LUsij j¼1

Then by combining with the expressions of Eq. (17) in each subdomains, a linear system of the localized formulation can be reformulated as follows Ncp X ^ k Þ; M mk Tðx

^ mÞ ¼ LTðx

ð18Þ

k¼1

where Ncp is the total number of collocation points and Mmk can be calculated by the following expressions



M mk ¼

0; 1

xk R Xm ;

ð19Þ

m LUm xk 2 Xm ; i ½U  ;

where k is the signed number of xm j and m is the signed number of the center xm i in the sub-domain Xm . Since most of the elements in N

cp are zero, a sparse system can be obtained. fMmk gk¼1

The numerical model of the problems can be described by the governing Eq. (1) as

x 2 X;

ð20Þ

with the initial and boundary conditions as described by Eqs. (2)– (5) and (7). Substituting Eq. (18) into Eq. (10), with the notation of L ¼ r2 , yields

^ nþ1 ¼ g nþ1 ; T^ nþ1 m  aDs  M m T m

xm 2 X;

ð21Þ

 where T^ nþ1 denotes the numerical solution of T xm ; snþ1 ; g nþ1 m m ; Mm , h i0 Ncp nþ1 ^ nþ1 nþ1 nþ1 ^ ^ and T denote gðxm ; s Þ, fMmk gk¼1 , and T 1 ; T 2 ; . . . ; T^ nþ1 . Ncp Here is the Kronecker delta function



dmk ¼

0;

m ¼ k;

ð22Þ

1; m – k:

Based on dmk , a unit vector Em is formed

Em ¼

Ncp : fdmk gk¼1

m ¼ Np1 ;

xm 2 X;

m ¼ Np2 ;

xm 2 C3 ;

m ¼ Np3 ;

xm 2 C2 ;

m ¼ Np4 ;

xm 2 C1 [ Cp ;

ð27Þ

where

8 Np1 > > > N p3 > > : Np4

¼ 1; . . . ; Ni ; ¼ Ni þ 1; . . . ; Ni þ Nb3 ;

ð28Þ

¼ Ni þ Nb3 þ 1; . . . ; Ni þ Nb3 þ Nb2 ; ¼ Ni þ Nb3 þ Nb2 þ 1; . . . ; Ncp ;

Ni denotes the number of the inner collocation points, Nb1 =Nb2 =Nb3 /Nb4 denotes the number of the collocation points on the C3 /C2 /C1 /Cp , and the total number of the collocation points is Ncp ¼ Ni þ Nb1 þ Nb2 þ Nb3 þ Nb4 . Once the initial and boundary conditions are given, the numerical solutions of T at any time history ^ nþ1 can be obtained by solving the sparse systems formulated by T (27).

3.3. Formulation of the LRBFCM for considered problems

@Tðx; sÞ @hðsÞ @/ðsÞ ¼ ar2 Tðx; sÞ þ þ ðT c0  T w0 Þ ; @s @s @s

8 ðEm  aDs  M m ÞT^ nþ1 ¼ g nþ1 > m ; > > > < M T^ ¼ 0; m > ðM m þ b ÞT^ ¼ b T a ; > a a > > : ^ nþ1 ^ nþ1 ¼ Tm ; Em T

3.4. Multiple-scale technique in MQ-RBF As described in [47], we can acquire that the matrix formed by MQ-RBF is symmetrical and the norm of the matrix can be easily solved, because of the appropriate attributes of the MQ-RBF. According to the definition of Eq. (14), the norm of each row of the formed MQ-RBF matrix (dr ) can be obtained

dr ¼

ð29Þ

Then, the norm of each column of the formed MQ-RBF matrix (dc ) can be the same as dr due to the symmetry of the matrix. In the LRBFCM, it can be found that the norm of the sparse system on the Dirichlet boundary (in Eq. (27)) is equal to 1, yielding dr ¼ dc ¼ 1. So, we have

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 XNs  s s 2 x  x þ N  c   s j i ¼ 1; j¼1 i and

ci ¼ ð23Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 XNs  s s 2 x  x þ N  c   s j i: j¼1 i

ð30Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  2 XNs  1 s s : 1 x  x  j j¼1 i Ns

ð31Þ

Then Eq. (21) can be rewritten as follows

ðEm  aDs  M m ÞT^ nþ1 ¼ g nþ1 m ;

xm 2 X:

ð24Þ

The same procedure can be adapted to obtain the expression on Neumann boundary parts Eqs. (4) and (5). With the notation of L ¼ @=@n, we have

M m T^ ¼ 0;

xm 2 C3 ;

ð25Þ

1

inner on Γ

0.8

on Γ1

2

on hole 0.6

and

ðM m þ ba ÞT^ ¼ ba T a ;

xm 2 C2 :

ð26Þ

It must be noted here that, to obtain more accurate solution on the Neumann boundary parts (C2 and C3 ), we may need more points matching in the sub-domain of the Neumann boundary parts, which is called Neumann sub-domain (NS). In the NS, the number of the points used is Nsn , which might be different from Ns . The effect of Nsn is discussed in Section 4. By using the LRBFCM to solve Eqs. (2)–(5) and (7), a set of sparse systems can be obtained as follows

0.4 0.2 0 0

0.5

1

1.5

Fig. 2. The profile of computational domain and distribution of collocation points with Ncp ¼ 253, including 171 inner points ðNi Þ, 59 points on C1 (Nb1 ), 20 points on C2 (Nb2 ), and 3 points on the hole (Nb3 ), for Example 1.

453

Y. Hong et al. / International Journal of Heat and Mass Transfer 129 (2019) 449–459

Thus, the shape parameter in each sub-domain can be easily achieved correspondingly. Eq. (31) is the function for the selection of the shape parameters in the proposed scheme. In Eq. (31), we can find that the shape parameter is different in each subdomains with different interpolated fields, which contributes good performance to the MST. From the above statements, we can find that the LRBFCM combined with the MST can achieve the shape parameters very fast, since just a few number of elements in each row or column are used. 4. Numerical experiments and discussions In this section, the accuracy and efficiency of the present method are tested by several examples including two- and threedimensional heat conduction problems in concrete structure with the water pipe cooling system. The numerical results obtained by the proposed LRBFCM are compared with the analytical solutions or those obtained by the FEM using COMSOL5.2a. The LRBFCM is tested on MATLAB R2014a. The accuracy of the obtained results is represented by the relative root mean square error (RMS) as

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u u 1 XNt Tðx Þ  Tðx ^ tÞ t ^ ¼t ; RMSðTÞ t¼1 Nt Tðxt Þ

ð32Þ

^ t Þ are the analytical and numerical solutions where Tðxt Þ and Tðx obtained by the proposed LRBFCM at the test point xt , and Nt is the total number of test points in the interest domain X. 4.1. Example 1 In the first example, the validity of the LRBFCM is tested in an Amoeba-like multiply connected domain using irregular distributions of the collocation points. The profile of the computational domain and the used points are as shown in Fig. 2, where C1 /C2 denote the Dirichlet/Neumann boundary parts and the total number of collocation points (Ncp ) is 253, including 171 inner points ðNi Þ, 59 points on C1 (Nb1 ), 20 points on C2 (Nb2 ), and 3 points on the hole (Nb3 ). Without heat source, the governing equation is as follows [49]

@T ¼ 0:5r2 T; @s

ð33Þ

under the initial condition

Tðx1 ; x2 ; 0Þ ¼ sinðx1 Þ sinðx2 Þ þ y cosðx1 Þ þ x sinðx2 Þ;

ð34Þ

In the LRBFCM, there are some free parameters, such as time step size (Ds), number of the points in a sub-domain (Ns and Nsn ), total number of the points (Ncp ), and shape parameters in MQRBF (c). First of all, Table 1 lists the RMS versus Nsn at s ¼ 0:5 obtained by the proposed LRBFCM, the LOOCV, and the coptimal , where the coptimal is obtained by trying a list of c. From this table, it can be seen that the RMS obtained by all of the methods decreases as the number of the points Nsn in NS increases. It is illustrated that the Nsn should be large enough in order to achieve good and stable results. Thus, we set the Nsn ¼ 21 in this subsection. The applicability of Nsn ¼ 21 is also studied in the other twodimensional examples. Compared with the LOOCV, the results obtained by the present method seem to be more accurate, which is the same to those obtained by the coptimal . This table also lists the RMS obtained by the LOOCV coupled with the MST (L-MST), where the MST is applied in the NS. We can find that the accuracy of the LMST is improved compared with the LOOCV. It indicates that the MST is more suitable in solving the problems with Neumann boundary parts. To investigate the convergence of the LRBFCM, Table 2 presents the RMS versus the time (s) with respect to different Ns . As shown in this table, we can find that the RMS decreases as Ns increases. For Ns ¼ 9, the obtained results are quite accurate with RMS around 105 . Good convergence of the proposed LRBFCM can also be observed. In Table 3, the RMS versus the time (s) with respect to different Ds is listed. From this table, it can be found that the RMS decreases as Ds declines. The accuracy of the LRBFCM con-

verges to a level around 105 , when Ds is smaller than 0:025. To further show the robustness of the proposed scheme, Fig. 3 displays the comparison of the RMS versus Ncp at s ¼ 1 obtained by the LRBFCM and the coptimal . The collocation points are always irregularly distributed in the domain with respect to different Ncp . As shown in Fig. 3, the RMS decreases fast with Ncp increases. Although the accuracy of the MST is slightly lower and the coptimal when Ncp is small, it turns to be as high as that obtained by the

coptimal with RMS less than 104 . Moreover, we study the accuracy of the proposed scheme in solving the problems with more chaotic arrangements of nodes. Table 4 presents the RMS obtained by the proposed LRBFCM with respect to the different arrangements of irregular collocation points, as shown in Fig. 4(a)–(d). We can see that, although the RMS increases from P(a) to P(d), the LRBFCM keeps accurate. The obtained RMS is still less than 102 with the chaotic collocation points in Fig. 4 (d). Despite of the complex domain and the irregular distribution of collocation points, all of the above results demonstrate that the present LRBFCM is of high accuracy and stability.

and boundary conditions

Tðx1 ; x2 ; sÞ ¼ T b ;

ðx1 ; x2 Þ 2 C1 or on hole;

ð35Þ

@Tðx1 ; x2 ; sÞ ; ¼q @n

ðx1 ; x2 Þ 2 C2 ;

ð36Þ

 are known functions. The analytical solution is as where T b and q follows

Tðx1 ; x2 ; sÞ ¼ sinðx1 Þ sinðx2 Þes þ ðy cosðx1 Þ þ x sinðx2 ÞÞe0:5s :

Table 1 Comparison of the RMS versus Nsn at

ð37Þ

4.2. Example 2 This example covers the thermal field in the simplified model with a water cooling pipe by the proposed LRBFCM. The domain is considered as a cross-section or a simplified model of a concrete cuboid, X ¼ fðx1 ; x2 Þj0 < ðx1 ; x2 Þ < 3 mg, as shown in Fig. 5 (left). The pipe (with radius r ¼ 0:025 m) is located at, the center of the structure ð1:5; 1:5Þ. In this figure (left), C1 ; C2 ; C3 , and Cp denote the boundaries of the structure contact with the wooden box, air, insulation panel, and the water pipe, correspondingly. Fig. 5 (right)

s ¼ 0:5 obtained by the proposed LRBFCM, the LOOCV, and the coptimal , with Ns ¼ 9 and Ds ¼ 0:02, for Example 1.

Nsn

9

13

16

21

24

LOOCV MST coptimal L-MST

2:0e  3 9:5e  5 9:5e  5 1:0e  4

6:6e  4 7:1e  5 6:9e  5 1:5e  4

4:6e  4 4:9e  5 4:8e  5 1:0e  4

4:3e  4 4:7e  5 4:7e  5 1:0e  4

5:3e  4 4:2e  5 4:2e  5 8:3e  5

454

Y. Hong et al. / International Journal of Heat and Mass Transfer 129 (2019) 449–459

Table 2 The RMS obtained by the proposed LRBFCM with respect to different Ns , where Nsn ¼ 21 and Ds ¼ 0:02, for Example 1. Ns

s ¼ 0:1

s ¼ 0:5

s ¼ 1:0

s ¼ 10

5 9 11

4:9e  3 4:5e  5 3:9e  5

5:6e  3 4:7e  5 3:8e  5

5:5e  3 5:0e  5 3:6e  5

5:0e  3 9:1e  5 5:6e  5

Table 3 The RMS obtained by the proposed LRBFCM with respect to different Ds, where Ns ¼ 9 and Nsn ¼ 21, for Example 1.

Ds

s ¼ 0:2

s ¼ 0:8

s ¼ 1:8

s ¼ 6:0

0.050 0.040 0.025 0.020

2:7e  4 1:8e  4 6:2e  5 4:6e  5

2:5e  4 1:6e  4 5:0e  5 4:9e  5

1:9e  4 1:1e  4 4:2e  5 5:9e  5

9:2e  5 4:9e  5 6:4e  5 8:7e  5

Here the coefficients for this problem can be calculated as k  2:09; AE ¼ 9; RE  1:692. Substituting the coefficients into Eq. (1), we have

−1

10

MST coptimal

−2

RMS

10

@T ¼ 0:1r2 T þ 10:26e0:27s  0:3e0:02s : @s

−3

Then, we consider the following initial condition

10

Tðx1 ; x2 ; 0Þ ¼ 30; −4

10

0

ðx1 ; x2 Þ 2 X;

ð41Þ

and boundary conditions

Tðx1 ; x2 ; sÞ ¼ 15 þ 15e0:18s ;

−5

10

ð40Þ

500

1000

1500

2000

Ncp

Kc

Fig. 3. The RMS versus Ncp at s ¼ 1 obtained by the proposed LRBFCM and the coptimal , where Ns ¼ 9, Nsn ¼ 21, and Ds ¼ 0:02, for Example 1.

ðx1 ; x2 Þ 2 C1 ;

@Tðx1 ; x2 ; sÞ ¼ ba ðT a  Tðx1 ; x2 ; sÞÞ; @n

@Tðx1 ; x2 ; sÞ ¼ 0; @n

ð42Þ

ðx1 ; x2 Þ 2 C2 ;

ðx1 ; x2 Þ 2 C3 ;

ð43Þ

ð44Þ

gives the profile of the collocation points distributed regularly with Ncp ¼ 262 . Three points are always put on Cp and three test points, point 1 (0.1, 0.1), point 2 (1, 1), and points 3 (1.5, 1.4), are selected to investigate the validity of the proposed scheme. The governing equation of this problem is Eq. (1), where T c0 ¼ 30  C and T w0 ¼ 15  C. The boundary conditions are given by Eqs. (3)–(5) and (7). Parameters of the applied concrete including heat conductivity (K c ), density (qc ), heat capacity (C c ), and adiabatic temperature raise (h) are assumed to be as K c ¼ 2:625 W=ðm  KÞ, qc ¼ 2400 kg=m3 , C c ¼ 945 J=ðkg  CÞ, and h ¼ 38ð1  e0:27s Þ  C=day. Then, the thermal conductivity (a) can be calculated, a ¼ K c =ðqc  C c Þ  0:1 m2 =day. According to [15,16], / can be simulated by the following relational expression 0

ka s A

/¼e

E

;

ð38Þ R2E

2

where k ¼ 2:09  1:35n þ 0:32n ; AE ¼ p is the area and RE denotes the radius of equivalent round of examine cross-section with each pipe, and

a0 ¼

lnð100Þ a: lnðRE =rÞ

ð39Þ

Tðx1 ; x2 ; sÞ ¼ T w0 þ 5=ð75s þ 1Þ þ 10e0:2s ;

ðx1 ; x2 Þ 2 Cp ;

ð45Þ

where the surrounding air temperature is supposed to be T a ¼ 18  C and the coefficient of heat convection from concrete to the air is ba ¼ 5:62 W=ðm2  CÞ [17]. By the LRBFCM, numerical results for this example are obtained with Ns ¼ 9 and Nsn ¼ 21. Table 5 lists the comparison between the numerical solutions obtained by the proposed LRBFCM with different time step size and those obtained by the FEM on some test points, where the FEM contains 4177 degrees of freedom with more than 2076 degrees used to simulate the structure surround the tiny pipe. From this table, it can be observed that numerical solutions obtained by the LRBFCM get closer to the compared (FEM) solutions when more collocation points (Ncp ) are applied. For Ncp ¼ 322 , the numerical solutions are quite consistent with the compared solutions and the obtained RMS is lower than 102 . It can be also found that, for both Ds ¼ 0:02 and 0:005, the obtained RMSs are almost the same with respect to the same Ncp . It is suggested that, for Ds ¼ 0:02, the LRBFCM has achieved numerical convergence. Since the sparse system is applied, the computational time of the LRBFCM is less than 10 s, for Ds ¼ 0:02. To reflect the

Table 4 The RMS obtained by the proposed LRBFCM with respect to the different arrangements of irregular collocation points, where Ns ¼ 9, Nsn ¼ 21, and Ds ¼ 0:02, for Example 1. Points

s ¼ 0:3

s ¼ 0:6

s ¼ 2:0

s ¼ 8:0

P(a) P(b) P(c) P(d)

8:1e  5 1:5e  4 2:8e  4 9:3e  4

8:1e  5 1:5e  4 3:4e  4 9:3e  4

8:3e  5 1:8e  4 4:9e  4 5:7e  4

8:9e  5 2:1e  4 6:7e  4 1:5e  3

455

Y. Hong et al. / International Journal of Heat and Mass Transfer 129 (2019) 449–459

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 0

0.5

1

1.5

0 0

0.5

(a) 1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 0

0.5

1

1.5

1

1.5

(b)

1

1.5

0 0

0.5

(c)

(d)

Fig. 4. The profile of increasing non-uniform arrangements of collocation points, for Example 1.

inner 3

Γ2

Γ3

Γ1

Γp

test

2.5

x2

2 1.5 1 0.5 0

0

0.5

1

1.5

x1

2

2.5

3

Fig. 5. Left: cross-section of water pipe cooling model, right: regular distribution of collocation points (Ncp ¼ 262 ), for Example 2.

function of the cooling pipe, we take off the cooling pipe. Then the boundary condition of C1 is replaced by the following function

Tðx1 ; x2 ; sÞ ¼ 15 þ 15e0:08s ;

ðx1 ; x2 Þ 2 C1 :

ð46Þ

For comparison, we also calculate the results of the thermal field without pipe. Fig. 6 displays the comparison of the numerical solutions of Max ðTÞ versus time (s) with a pipe or no pipe obtained by the proposed LRBFCM and the FEM. We can observed that the numerical solutions obtained by the LRBFCM are quite consistent

with those by the FEM. The accuracy of the proposed scheme can be obviously concluded. From this figure, we can see the maximum temperature occurs at s ¼ 4:7 (day) with a pipe about one day earlier than the case without pipe. The value of the maximum temperature decreases, from 54 to 51 by embedding a pipe. At the same time, we achieve the maximum difference of the temperature, DT ¼ Max ðTÞ  Min ðTÞ. Without pipe, the Max ðDTÞ is 29:9. When a pipe is embedded, the Max ðDTÞ increases to 32:4, which does not meet engineering requirements.

456

Y. Hong et al. / International Journal of Heat and Mass Transfer 129 (2019) 449–459

Table 5 Comparison between the numerical solutions obtained by the proposed LRBFCM and those obtained by the FEM on the test points, for Example 2. Ncp

Point 1

Ds; 0:02 262

s¼5 25:3788

s¼1 38:2205

s¼3 45:1218

s ¼ 21 15:9472

RMS 2:0e  2

s ¼ 24 2:0e  0 s

322

25:4060

38:2981

45:4948

16:4682

3:6e  3

3:7e  0 s

402

25:4203

38:3182

45:5933

16:6075

1:2e  3

7:4e  0 s

FEM

25:4485

38:3535

45:6070

16:5741

–

1:0e þ 2 s

2

25:4279

38:3478

45:6507

16:6069

1:2e  3

3:1e þ 1 s

322

25:4136

38:3278

45:5521

16:4676

3:4e  3

1:6e þ 1 s

262

25:3864

38:2508

45:1787

15:9467

1:9e  2

8:9e  0 s

Ds; 0:005

s¼5

s¼1

s¼3

s ¼ 14

RMS

s ¼ 24

40

Point 2

60

no pipe

LRBF FEM

40

@T ¼ 0:1r2 T þ 9e0:25s  1:335e0:089s : @s

51

30

50.5

20

50

0

Compute time

collocated on each pipes and four test points are listed as point 1 (0.1, 0.1), point 2 (1.0, 1.0), points 3 (1.2, 1.2), and points 4 (1.4, 1.4). The governing equation is given by Eq. (1) with a ¼ 0:1 m2 =day and h ¼ 36ð1  e0:25s Þ  C=day. Similar to Example 2, the coefficients for the considered problem can be calculated as k  2:09; AE ¼ 2:400; RE  0:874. Substituting the above coefficients and functions into Eq. (1), we have

a pipe

50

Max (T)

Point 3

ð47Þ

We consider the following initial condition

3

4

5

Tðx1 ; x2 ; 0Þ ¼ 30;

5

10

τ

15

20

25

Fig. 6. The numerical solutions of Max ðTÞ versus time (s) with a pipe or no pipe obtained by the proposed LRBFCM, with Ds ¼ 0:02 (day) and Ncp ¼ 322 , and by the FEM, for Example 2.

ðx1 ; x2 Þ 2 X;

ð48Þ

and boundary conditions

Tðx1 ; x2 ; sÞ ¼ 15 þ 15e0:24s ;

ðx1 ; x2 Þ 2 C1 ;

ð49Þ

Tðx1 ; x2 ; sÞ ¼ T w0 þ 3=ð150s þ 1Þ þ 12e0:25s ;

ðx1 ; x2 Þ 2 Cp ;

ð50Þ

where T w0 ¼ 15; T a ¼ 18; ba ¼ 5:62, and the boundary conditions on

C2 and C3 are the same as those described in the last example. For Ns ¼ 9; Nsn ¼ 21, Ds ¼ 0:02, and Ncp ¼ 282 , Fig. 8 presents the

4.3. Example 3 In this example, we further investigate the thermal field in the concrete cuboid with five embedded pipes. The efficiency of the LRBFCM is reported in the meantime. The centers of these pipes with radii r ¼ 0:025 m are located at (0.75, 0.75), (2.25, 0.75), (1.50, 1.50), (2.25, 2.25), and (0.75, 2.25). The computational domain with regular distribution of collocation points is similar to the last example, as displayed in Fig. 7. Three points are always

numerical solutions versus s obtained by the proposed LRBFCM and by the FEM at the test points. From the left of this figure, a consistence between the numerical solutions and the compared solutions can be obviously discovered. More details about the relative difference of the numerical solutions are displayed in the right of this figure. We can see that the relative difference is less than 102 , which is accurate enough for engineering cases. Furthermore, the LRBFCM only spends 3.2 s for solving this problem. All the

inner

3

Γ2

Γ3

Γ1

Γp

test

2.5

x2

2 1.5 1 0.5 0

0

0.5

1

1.5

x1

2

Fig. 7. The computational domain and the profile of collocation points with Ncp ¼ 282 , for Example 3.

2.5

3

457

Y. Hong et al. / International Journal of Heat and Mass Transfer 129 (2019) 449–459 −2

40

10

LRBF FEM

30 37.6 37.4 37.2 37 36.8 36.6

25 32.8 20 32.6 32.4 1.5 2 15 0 5

2

Relative difference

point 1 point 2 point 3 point 4

35

−3

10

−4

10

point 1 point 2

−5

point 3 point 4

10

3 −6

10

10

15

20

0

5

10

15

τ

20

Fig. 8. Comparison of the numerical solutions obtained by the proposed LRBFCM and the FEM, where the FEM contains 10813 degrees of freedom, for Example 3.

Γ2

inner

inner

on pipe

Γ

Γ

2

1

3

Γ

Γ

1

p

Γ

1

1

0.8

0.8

Γ

1

0.6

x2

x3

0.6

0.4

0.4

0.2 Γ1

Γ1

0 0

1 0.5

0.2

0.5

x1

1 0

0

x2

Γ3

0

0.2

0.4

x1

0.6

0.8

1

Fig. 9. The profile of collocation points with Ncp ¼ 13R , for Example 4.

0.6 LRBF FEM Relative difference

0.4

0.5

1

0.58

0.3 0.57

−2

10 0.2

−3

10

0.1 0 0

−4

2

4

τ

6

8

10 10

Relative difference

Max (T)/T

h

0.5

Fig. 10. The numerical solutions of Max ðTÞ=T h versus time (s) obtained by the proposed LRBFCM, with Ds ¼ 0:02 (day) and Ncp ¼ 13R , and by the FEM, where the FEM contains 51846 degrees of freedom, for Example 4.

above results demonstrate the accuracy and efficiency of the present LRBFCM in this problem. To further view the influence of the cooling pipe, we also achieve the maximum value and the maximum difference of the

temperature. We find that the maximum value of T is 42:4  C, which is about 10  C lower than that in the case without or just with one pipe. Meanwhile, the maximum difference of the temperature between the inner and surface parts is about 22:6  C. According to engineering experience, this difference (< 25  C) is safe enough to avoid the crack risks. It comes to the following conclusions that on one hand, the arrangement of cooling water pipes given in this case is truly helpful for the temperature control and the thermal crack prevention; on the other hand, the proposed LRBFCM is accurate and efficient in simulating the thermal field in the concrete structures with a cooling water pipe system.

4.4. Example 4 This example is designed to investigate the potential of the proposed LRBFCM to solve the three-dimensional heat conduction problems. The interesting domain is a cubic, X ¼ fðx1 ; x2 ; x3 Þj0 < x1 ; x2 ; x3 < 1 mg, with a tiny water pipe (r ¼ 0:025 m) across the center from its top surface to the bottom surface, as shown in Fig. 9. In this figure, the profile of collocation points is also displayed. The left of this figure displays the inner points and the points on the pipe, where the number of the inner points is 2184 (133  13) and the points on the pipe is 3  13.

458

Y. Hong et al. / International Journal of Heat and Mass Transfer 129 (2019) 449–459

Table 6 Comparison between the numerical solutions obtained by the proposed LRBFCM and those obtained by the FEM on the test points, for Example 4. Ncp

Point 1

Ds; 0:02 11R

s ¼ 4:9 21:5528

s ¼ 0:7 39:7924

s ¼ 2:7 31:3260

s ¼ 7:8 19:6375

RMS 3:0e  3

s ¼ 10 6:9e  0 s

Point 2

Point 3

13R

21:5905

39:8776

31:3694

19:6365

1:9e  3

1:2e þ 1 s

15R

21:6038

39:9241

31:4193

19:6415

1:8e  3

2:2e þ 1 s

FEM

21:5767

40:0212

31:3515

19:6488

–

8:0e þ 2 s

R

21:6100

40:0215

31:4181

19:6409

1:3e  3

8:8e þ 1 s

13R

21:5967

39:9743

31:3682

19:6359

8:5e  4

5:0e þ 1 s

11R

21:5591

39:8888

31:3248

19:6369

1:8e  3

2:6e þ 1 s

Ds; 0:005

s ¼ 4:9

s ¼ 0:7

s ¼ 2:7

s ¼ 7:8

RMS

s ¼ 10

15

The right of this figure presents the vertical view of this physical domain and the collocation points distributed regularly in the domain, where the number of the points on the top/bottom surface (C2 =C3 ) is 168 (132  1) and the points on each side surface (C1 ) is 169 (132 ). With the notation N R ¼ N 3 þ 6N 2 þ 2N  2, we have Ncp ¼ 13R in this figure. The governing equation of the target heat conduction problem is also considered as Eq. (1), where a ¼ 0:1 m2 =day and h ¼ 138ð1  e0:3s Þ  C=day. The coefficients for the target problem can be calculated as k  2:09; AE ¼ 1:000; RE  0:564. Then, we have

@T ¼ 0:1r2 T þ 41:4e0:3s  3:645e0:243s : @s

ð51Þ

In this example, the initial and boundary conditions conditions are given as follows

Tðx1 ; x2 ; x3 ; 0Þ ¼ 30;

ðx1 ; x2 ; x3 Þ 2 X;

Tðx1 ; x2 ; x3 ; sÞ ¼ 15 þ 15e0:20s ; Kc

Compute time

ðx1 ; x2 ; x3 Þ 2 C1 ;

@Tðx1 ; x2 ; x3 ; sÞ ¼ ba ðT a  Tðx1 ; x2 ; x3 ; sÞÞ; @n 2 C2 ;

@Tðx1 ; x2 ; x3 ; sÞ ¼ 0; @n

ð52Þ

ðx1 ; x2 ; x3 Þ 2 C3 ;

ð53Þ

ðx1 ; x2 ; x3 Þ ð54Þ ð55Þ

Tðx1 ; x2 ; x3 ; sÞ ¼ T w0 þ 15e0:22s þ x3 ð15e0:17s  15e0:22s Þ; ðx1 ; x2 ; x3 Þ 2 Cp ; ð56Þ

where T w0 ¼ 15; T a ¼ 15, and ba ¼ 5:62. For this three dimensional problem, we apply Ns ¼ 9 and Nsn ¼ 24, where the Nsn is a bit more than that appears in twodimensional problems. To begin with, numerical solutions at three test points, point 1 ð0:23; 0:68; 1Þ, point 2 ð0:5; 0:32; 0:5Þ, and point 3 ð0:77; 0:23; 0Þ, are investigated as listed in Table 6. It can be seen that numerical solutions obtained by the LRBFCM are quite consistent with the compared (FEM) solutions. The more collocation points (Ncp ) are used, the smaller RMS is obtained, for Ds ¼ 0:02. For Ds ¼ 0:005, more accurate results can be achieved. It also takes more computational time, when the larger Ncp or the smaller length of time step is used. From this table, we can discover the efficiency of the LRBFCM. Even for this three dimensional problem, just 6.9 s is needed to achieve a high level of the accuracy about 103 . To inspect the MaxðTÞ, Fig. 10 plots the MaxðTÞ=T h versus time s, where T h ¼ 70 denotes the dangerous temperature level of the concrete. From this figure, high accuracy of the proposed LRBFCM can be obviously observed. The MaxðTÞ keeps no more than 60 per cent of T h , which indicates the concrete is safety in the situa-

tion. Furthermore, the maximum DT is about 13  C, just 50 per cent of 25  C, which is also safety enough. 5. Conclusions This paper presented the localized radial basis function collocation method for the simulation of the thermal field in mass concrete structure with water pipe cooling system. In the proposed LRBFCM, the sparse matrices are introduced to accelerate the solution process and the multiple-scale technique is used for fast determination of the shape parameters in MQ-RBFs. The validity of the algorithm is illustrated through numerical investigations in solving several examples including a three-dimensional problem. In all examples, the Neumann boundary conditions are taken into account. We have found that the results in the sub-domain of Neumann boundary parts should be simulated by using more points, Nsn > Ns . Usually, the Nsn should be more than 20 for twodimensional problems and a bit more for three-dimensional problems. Compared with the analytical results or the results obtained by the FEM, all tested examples demonstrate that the proposed LRBFCM could provide accurate predictions. The convergence, stability and efficiency of the present scheme also are presented in these examples. The obtained numerical results indicate that the proposed LRBFCM has great potential for the design of the water pipe cooling system. Conflict of interest Authors declare that there is no con flict of interest. Acknowledgement The author thanks the editor and anonymous reviewers for their constructive comments on the manuscript. The work in this paper is supported by the Fundamental Research Funds for the Central Universities (No. 2018B16714), the National Natural Science Foundation of China (Nos. 11702083, 11572111), the Natural Science Foundation of Jiangsu Province (No. BK20150795), and the State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) (No. MCMS-0218G01). References [1] N. Shi, J. Ouyang, R. Zhang, et al., Experimental study on early-age crack of mass concrete under the controlled temperature history, Adv. Mater. Sci. Eng. 2014 (4) (2015) 1–10. [2] Y.H. Kwak, J. Walewski, D. Sleeper, et al., What can we learn from the Hoover Dam project that influenced modern project management, Int. J. Proj. Manag. 32 (2) (2014) 256–264. [3] B. Bhardwaj, P. Kumar, Waste foundry sand in concrete: a review, Constr. Build. Mater. 156 (2017) 661–674.

Y. Hong et al. / International Journal of Heat and Mass Transfer 129 (2019) 449–459 [4] K. Rahal, Mechanical properties of concrete with recycled coarse aggregate, Build. Environ. 42 (1) (2007) 407–415. [5] X. Wang, J. Xia, O. Nanayakkara, et al., Properties of high-performance cementitious composites containing recycled rubber crumb, Constr. Build. Mater. 156 (2017) 1127–1136. [6] T. Hemalatha, A. Ramaswamy, A review on fly ash characteristics–towards promoting high volume utilization in developing sustainable concrete, J. Clean Prod. 147 (2017) 546–559. [7] N.A. Abdulla, Concrete filled PVC tube: a review, Constr. Build. Mater. 156 (2017) 321–329. [8] P. Lin, Q. Li, P. Jia, A real-time temperature data transmission approach for intelligent cooling control of mass concrete, Math. Probl. Eng. 2014 (8) (2014) 1–10. [9] Y. Qiu, G. Zhang, Stress and damage in concrete induced by pipe cooling at mesoscopic scale, Adv. Mater. Sci. Eng. 9 (2) (2017), https://doi.org/10.1177/ 1687814017690509. [10] K.K. Jin, K.H. Kim, J.K. Yang, Thermal analysis of hydration heat in concrete structures with pipe-cooling system, Comput. Struct. 79 (2) (2001) 163–171. [11] J.K. Yang, Y. Lee, Heat transfer coefficient in flow convection of pipe-cooling system in massive concrete, J. Adv. Concr. Technol. 9 (1) (2011) 103–114. [12] X.H. Liu, Y. Duan, W. Zhou, et al., Modeling the piped water cooling of a concrete dam using the heat-fluid coupling method, J. Eng. Mech. 139 (9) (2013) 1278–1289. [13] Y. Hong, W. Chen, J. Lin, et al., Thermal field in water pipe cooling concrete hydrostructures simulated with singular boundary method, Water Sci. Eng. 10 (2) (2017) 107–114. [14] B.F. Zhu, Equivalent equation of heat conduction in mass concrete considering the effect of pipe cooling, J. Hydraul. Eng. (1991). [15] B.F. Zhu, The equivalent heat conduction equation of pipe cooling in mass concrete considering influence of external temperature, J. Hydraul. Eng. 34 (3) (2003) 49–54. [16] Y.X. Ma, X.G. Chen, The application of equivalent heat conduction equation in cooling temperature field calculation of mass concrete water pipe, Concrete (2007). [17] P.R. Singh, D.C. Rai, Effect of piped water cooling on thermal stress in mass concrete at early ages, J. Eng. Mech. 144 (3) (2018) 04017183, https://doi.org/ 10.1061/10.106(ASCE)EM.1943-7889.0001418. [18] S.H. Chen, P.F. Su, I. Shahrour, Composite element algorithm for the thermal analysis of mass concrete: simulation of lift joint, Finite Elem. Anal. Des. 47 (5) (2011) 536–542. [19] J. Cheng, T.C. Li, X. Liu, et al., A 3D discrete FEM iterative algorithm for solving the water pipe cooling problems of massive concrete structures, Int. J. Numer. Anal. Methods 40 (4) (2016) 487–508. [20] X. Liu, C. Zhang, X. Chang, et al., Precise simulation analysis of the thermal field in mass concrete with a pipe water cooling system, Appl. Therm. Eng. 78 (2015) 449–459. [21] R. Zhong, G.P. Hou, S. Qiang, An improved composite element method for the simulation of temperature field in massive concrete with embedded cooling pipe, Appl. Therm. Eng. 124 (2017) 1409–1417. [22] A.J. Deeks, J.P. Wolf, A virtual work derivation of the scaled boundary finiteelement method for elastostatics, Comput. Mech. 28 (6) (2002) 489–504. [23] L.B.D. Veiga, F. Brezzi, L.D. Marini, et al., Virtual element method for general second-order elliptic problems on polygonal meshes, J. Bus. Ethics 26 (4) (2016) 75–89. [24] F. Wang, W. Chen, A. Tadeu, C. Correia, Singular boundary method for transient convection-diffusion problems with time-dependent fundamental solution, Int. J. Heat Mass Transf. 114 (2017) 1126–1134. [25] J. Lin, C.Z. Zhang, L.L. Sun, J. Lu, Simulation of seismic wave scattering by embedded cavities in an elastic half-plane using the novel singular boundary method, Adv. Appl. Math. Mech. 10 (2018) 322–342. [26] J.P. Li, W. Chen, A modified singular boundary method for three-dimensional high frequency acoustic wave problems, Appl. Math. Model 54 (2018) 189– 201.

459

[27] Z.J. Fu, Q. Xi, W. Chen, A.H.D. Cheng, A boundary-type meshless solver for transient heat conduction analysis of slender functionally graded materials with exponential variations, Comp. Math. Appl. 76 (2018) 760–773. [28] J. Lin, S.Y. Reutskiy, J. Lu, A novel meshless method for fully nonlinear advection-diffusion-reaction problems to model transfer in anisotropic media, Appl. Math. Comput. 339 (2018) 459–476. [29] E.J. Kansa, Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics, I. Surface approximation and partial derivative estimates, Comput. Math. Appl. 19 (8–9) (1990) 127–145. [30] G. Yao, Z. Yu, A localized meshless approach for modeling spatial-temporal calcium dynamics in ventricular myocytes, Int. J. Numer. Meth. Biomed. 28 (2) (2012) 187–204. [31] J. Lin, C.S. Chen, C.S. Liu, J. Lu, Fast simulation of multi-dimensional wave problems by the sparse scheme of the method of fundamental solutions, Comput. Math. Appl. 72 (3) (2016) 555–567. [32] B. Sarler, R. Vertnik, Meshfree local radial basis function collocation method for diffusion problems, Comput. Math. Appl. 51 (8) (2006) 1269–1282. [33] R. Vertnik, B. Sarler, Meshless local radial basis function collocation method for convective-diffusive solid-liquid phase change problems, Int. J. Numer. Method H 16 (5) (2006) 617–640. [34] E. Divo, A.J. Kassab, An efficient localized radial basis function meshless method for fluid flow and conjugate heat transfer, J. Heat Transf. 129 (2) (2007) 124–136. [35] M. Li, W. Chen, C.S. Chen, The localized RBFs collocation methods for solving high dimensional PDEs, Eng. Anal. Bound. Elem. 37 (10) (2013) 1300–1304. [36] Siraj-ul-Islam, R. Vertnik, B. Sarler, Local radial basis function collocation method along with explicit time stepping for hyperbolic partial differential equations, Appl. Numer. Math. 67 (4) (2013) 136–151. [37] A. Safdari-Vaighani, A. Heryudono, E. Larsson, A radial basis function partition of unity collocation method for convection-diffusion equations arising in financial applications, J. Sci. Comput. 64 (2) (2015) 341–367. [38] M. Hamaidi, A. Naji, A. Charafi, Space-time localized radial basis function collocation method for solving parabolic and hyperbolic equations, Eng. Anal. Bound. Elem. 67 (2016) 152–163. [39] E.J. Kansa, P. Holoborodko, On the ill-conditioned nature of C-infinity RBF strong collocation, Eng. Anal. Bound. Elem. 78 (2017) 26–30. [40] A.R. Lamichhane, C.S. Chen, The closed-form particular solutions for Laplace and biharmonic operators using Gaussian function, Appl. Math. Lett. 46 (2015) 50–56. [41] A.H.-D. Cheng, M.A. Golberg, E.J. Kansa, et al., Exponential convergence and H– c multiquadric collocation method for partial differential equations, Numer. Meth. Part Differ. Equ. 19 (5) (2010) 571–594. [42] G.E. Fasshauer, J.G. Zhang, On choosing optimal shape parameters for RBF approximation, Numer. Algorithms 45 (1–4) (2007) 345–368. [43] W. Chen, Y. Hong, J. Lin, The sample solution approach for determination of the optimal shape parameter in the Multiquadric function of the Kansa method, Comp. Math. Appl. 75 (8) (2018) 2942–2954. [44] J. Lin, Y. Hong, L.H. Kuo, et al., Numerical simulation of 3D nonlinear Schrödinger equations by using the localized method of approximate particular solutions, Eng. Anal. Bound. Elem. 78 (2017) 20–25. [45] B. Maviric, B. Sarler, Application of the RBF collocation method to transient coupled thermoelasticity, Int. J. Numer. Method H 27 (5) (2017) 1064–1077. [46] V. Hatic, B. Mavric, N. Kosnik, B. Sarler, Simulation of direct chill casting under the influence of a low-frequency electromagnetic field, Appl. Math. Model. 54 (2018) 170–188. [47] C.S. Liu, W. Chen, Z. Fu, A multiple-scale MQ-RBF for solving the inverse Cauchy problems in arbitrary plane domain, Eng. Anal. Bound. Elem. 68 (2016) 11–16. [48] M. Harris, A.J. Kassab, E. Divo, Application of an RBF blending interpolation method to problems with shocks, Comput. Assisted Meth. Eng. Sci. 22 (2017) 229–241. [49] Y. Hong, J. Lin, W. Chen, A typical backward substitution method for the simulation of Helmholtz problems in arbitrary 2D domains, Eng. Ana. Bound. Elem. 93 (2018) 167–176.