Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 56, 4, pp. 1123-1137, Warsaw 2018 DOI: 10.15632/jtam-pl.56.4.1123 AN EFFICIENT ANALYSIS OF STEADY-STATE HEAT CONDUCTION INVOLVING CURVED LINE/SURFACE HEAT SOURCES IN TWO/THREE-DIMENSIONAL ISOTROPIC MEDIA Mehrdad Mohammadi Department of Mechanical Engineering, Shiraz Branch, Islamic Azad University, Shiraz, Iran Mohammad R. Hematiyan Department of Mechanical Engineering, Shiraz University, Shiraz, Iran Yuichuin C. Shiah Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan, R.O.C. e-mail: ycshiah@mail.ncku.edu.tw In thispaper, anewformulationbasedonthemethodof fundamental solutions for two/three- -dimensional steady-state heat conduction problems involving internal curved line/surface heat sources is presented. Arbitrary shapes and non-uniform intensities of the curved heat sources can be modeled by an assemblage of several parts with quadratic variations. The presented mesh-free modeling does not require any internal points as in domain methods. Four numerical examples are studied to verify the validity and efficiency of the proposed method. Our analyses have shown that the presentedmesh-free formulation is very efficient in comparison with conventional boundary or domain solution techniques. Keywords: heat conduction, concentratedheat source, curvedheat source,mesh-freemethod 1. Introduction Heat conduction and thermoelasticity involving boundary or domain heat sources have been subject of many studies in the last years, and they are still active areas of researches (e.g. Ro- gowski, 2016; Hidayat et al., 2017). In real applications, it is quite often to have internal heat sources concentrated onpoints, lines or curvedpaths due to electrical heating or some other heat sources like laser beams. As examples, we canmention infrared heating, amethod of electric he- ating that is frequently used in themetallurgy and textile industries, laser beamheating/welding that is used in automotive and aerospace industries, and friction heating formaterial processing and joining.Despite several analytical solutions for simple problems involving concentrated heat sources (e.g. Chao andTan, 2000; Han andHasebe, 2002), practical problems with complicated conditions still need to resort to numerical tools. The accuracy analysis of the domain solution methods such as the finite element method (FEM) depends on themesh density especially near the concentrated heat source. As powerful alternative approaches, the boundarymethods such as the boundary element method (BEM) and themethod of fundamental solutions (MFS) only require boundary discretization. To date, the BEM has been effectively used to solve direct and inverse problems containing concentrated sources of heat generation. Le Niliot (1998) proposed a boundary element formu- lation for identification of the intensity of point heat sources in diffusive systems. In another work, Le Niliot and Lefèvre (2001) proposed a BEM to identify the location and strength of multiple point heat sources in a transient heat conduction problem. Karami and Hematiyan (2000a,b) proposed a formulation based on the BEM for direct and inverse analyses of heat 1124 M.Mohammadi et al. conduction problems containing concentrated sources of heat generation. They presented an exact implementation of a source of heat generation concentrated on a point or a line in the BEM formulation. Shiah et al. (2005) analyzed two-dimensional thermo-mechanical problems containing point sources using the BEM. They could solve the problem with boundary-only discretization. In another research, Shiah et al. (2006) used the direct domain mapping (DDM) technique to analyze 2D and 3D heat conduction problems in composites consisting of multi- ple anisotropic media with embedded point heat sources. Hematiyan et al. (2011) presented a formulation based on the BEM for analysis of two and three dimensional thermo-elastic pro- blems involving point, line and area heat sources. They only employed boundary discretiza- tion in their formulation. However, their proposed formulation considered only straight line and flat surface heat sources with a linear variation of the heat source intensity. Moham- madi et al. (2016) used the BEM for analysis of two- and three-dimensional thermo-elastic problems involving arbitrary curved line heat sources. They effectively solved the problem without considering any internal points/cells; but they did not consider curved surface heat sources. The present work uses the MFS to analyze problems of 2D/3D heat conduction involving internal concentrated heat sources. In this paper, theMFS, a widely applied meshless method, is shown to be very efficient for the analysis on account of which the benefit is that no inter- nal points/nodes are required for the modeling. Similar to the BEM, the MFS is applicable when a fundamental solution of the problem is known. However, the important advantage of the MFS over the BEM is that the MFS is an integration-free method and it can be easi- ly implemented for problems especially in three-dimensional and irregular domains. The basic idea of the MFS is to approximate the solution as a linear combination of fundamental so- lutions. The singularities (sources) of the fundamental solutions are located outside the phy- sical domain of the problem. The MFS solutions exactly satisfy governing equations of the problem and approximately satisfy boundary conditions. In the study carried out by Fair- weather and Karageorghis (1998), the development of the MFS in the past three decades was explained. The equation governing steady-state heat conduction in a medium with a heat source is the standard Poisson equation. To solve Poisson’s equation using the MFS, a particular so- lution corresponding to the heat source term in addition to a homogeneous solution of the Laplace equation should be found. Two important methods proposed to calculate this par- ticular solution are the Atkinson method (1985) and the dual reciprocity method (DRM) (Partridge et al., 1992). In Atkinson’s method, the particular solution is taken to be a New- ton potential and is obtained by evaluating a domain integral. Poullikkas et al. (1998) used this method for solving inhomogeneous harmonic and biharmonic problems. In the DRM, the particular solution is approximated by a series of basis solutions. As an example, Golberg (1995) used this method to solve Poisson’s equation without a boundary or domain discre- tization. In this work, the MFS is formulated for 2D/3D problems of heat conduction involving in- ternal heat sources concentrated on curved lines/surfaces. Although the MFS has been widely used for analysis of heat conduction problems in different conditions (e.g. Ahmadabadi et al., 2009; Kołodziej et al., 2010; Mierzwiczak and Kołodziej, 2012); however, to the authors’ know- ledge, theMFS formulation for analysis of internal curve line/surface heat sources has not been presented yet. The method presented here can be simply employed without considering any internal points/nodes and, therefore, it preserves the attractiveness of theMFS as a boundary- -type mesh-free method. Several two- and three-dimensional numerical examples are presented at the end to show that the proposed formulation is very efficient to yield accurate results in comparison with the BEM and FEM. An efficient analysis of steady-state heat conduction... 1125 2. MFS formulation for steady-state heat conduction in a domain including heat sources Consider an isotropic mediumΩ with its boundary Γ (Fig. 1). In the presence of heat sources, the governing equation of steady-state heat conduction can be expressed as follows ∇ 2τ(x)=− s(x) k x∈Ω (2.1) where ∇2 represents the Laplace operator, τ is temperature, k is thermal conductivity, and s(x) is a known function describing the heat source distribution. Fig. 1. DomainΩ, boundary Γ and pseudo boundary Γ ′ Boundary condition in a generalized form can be written as follows f1τ +f2 ∂τ ∂n = f3 on Γ (2.2) where f1, f2 and f3 are given functions on the boundary and n is the normal direction. In theMFS, the solution to Poisson equation (2.1) is approximated by a linear combination of fundamental solutions of the Laplace equation and a particular solution τ(x) = N ∑ j=1 ajτ ∗(x,ξj)+ τp(x) (2.3) where ξj and aj are the known location and unknown intensity of the j-th source located on the pseudo-boundaryΓ ′ (Fig. 1), respectively. x is a point in the domain or on the boundary of the solution domain, andN is the number of sources. τ∗ represents the fundamental solution to the Laplace operator that is given as follows τ∗(x,ξj)=        −1 2π ln(r(x,ξj)) for 2D 1 4πr(x,ξj) for 3D (2.4) where r(x,ξj) is the distance between the field point x and the source point ξj. τp(x) is the particular solution to equation (2.1) associated to the heat source function s(x) that can be concentrated on a part of the domain or distributed over the entire domain. The particular solution canbeobtainedbyconstructing theassociatedNewtonpotential in the followingdomain integral form τp(x)= 1 k ∫ Ω s(ξ)τ∗(x,ξ) dV (ξ) (2.5) 1126 M.Mohammadi et al. Efficient evaluation of this domain integral is very important in theMFS tomaintain the attrac- tiveness of themethod. If one can evaluate the domain integral in Eq. (2.5) without considering any internal cells/points, the attractiveness of theMFS is preserved. The constants aj (with the unit m ◦C in the SI system) are unknown intensities of the sources and they have to be found. To find these unknowns, we consider N boundary points y1,y2, . . . ,yN that are a priori located onΓ and collocate the correspondingboundary condition at these points. From Eqs. (2.2) and (2.3), the following equation is obtained N ∑ j=1 aj [ f1(yi)τ ∗(yi,ξj)+f2(yi) ∂τ∗(yi,ξj) ∂n ] = f3(yi) − [ f1(yi)τp(yi)+f2(yi) ∂τp(yi) ∂n ] i=1,2, . . . ,N (2.6) which represents a system of N linear equations with N unknowns. In general, one can write system (2.6) as follows AX=F (2.7) where the components of the matrix A ∈ RN×N and the vectors F ∈ RN and X ∈ RN are expressed as follows Aij = f1(yi)τ ∗(yi,ξj)+f2(yi) ∂τ∗(yi,ξj) ∂n Fi = f3(yi)− [ f1(yi)τp(yi)+f2(yi) ∂τp(yi) ∂n ] Xi = ai (2.8) By selecting a suitable configuration for boundary and source points, Eq. (2.7) can be solved by standard methods such as the Gaussian elimination method. In the next Section, the method for computation of the particular solution τp(x) using Eq. (2.5) for the special case of heat sources concentrated on a curved line/surface is described. 3. Formulations for heat sources concentrated on a curved line/surface In this Section, particular solutions associated to curved line/surface heat sources in the MFS are presented. For the 2D case, curved line sources, while for the 3D case, both curved line and curved surface sources are considered. 3.1. Curved line heat source in 2D problems At first, the formulation for a curved line heat source with a quadratic shape is presented. It is also assumed that the intensity of the source has a quadratic variation along the heat source. An arbitrary curved line heat source can bemodeled by several quadratic heat sources. A domain including a general curved line heat source and a part of the source modeled as a quadratic line heat source is illustrated in Fig. 2. Each quadratic line heat source is discretized by three points. The intensity per unit length of the source at the starting point (x1,y1), middle point (x2,y2), and end point (x3,y3) are represented by g1, g2, g3, respectively. Assuming a quadratic variation for the intensity of the heat source, s(x) can be given as follows s(η)=N1g1+N2g2+N3g3 (3.1) An efficient analysis of steady-state heat conduction... 1127 Fig. 2. An arbitrary curved line heat source modeled as several quadratic line heat sources where the quadratic shape functionsNi are N1 = 1 2 η(η−1) N2 =−(η+1)(η−1) N3 = 1 2 η(η+1) (3.2) where η is a dimensionless local coordinate aligned with the quadratic segment that varies from −1 to 1. Domain integral (2.5) for the quadratic line heat source can be expressed as follows τp(x)= ∫ L s(ξ) k τ∗(x,ξ) dl (3.3) where dl is an infinitesimal element along the quadratic line heat source. Substituting τ∗ in the 2D case from Eq. (2.4) into Eq. (3.3) results in τp(x)= −1 2πk ∫ l s(ξ)ln[r(x,ξ)] dl (3.4) where r(x,ξ) = √ (x−xs)2+(y−ys)2 is the distance between the field point x = (x,y) and the source points ξ= (xs,ys) on the quadratic line heat source. xs and ys can be expressed in terms of the three points of the quadratic line heat source as follows xs =(N1x1+N2x2+N3x3) ys =(N1y1+N2y2+N3y3) (3.5) Using Eqs. (3.5), the infinitesimal element dl in Eq. (3.4) can be expressed as dl= √ dx2s +dy 2 s = Jdη (3.6) where J is Jacobian which can be expressed as J = √ ( x1 dN1 dη +x2 dN2 dη +x3 dN3 dη )2 + ( y1 dN1 dη +y2 dN2 dη +y3 dN3 dη )2 (3.7) Substituting Eqs. (3.1) and (3.6) into Eq. (3.4) results in τp(x)= −1 2πk 1 ∫ −1 (N1g1+N2g2+N3g3)ln[r(η)]J dη (3.8) The integral in Eq. (3.8) can be calculated using conventional numerical integration methods such as the Gaussian quadrature method (GQM). It should be noted that if the field point 1128 M.Mohammadi et al. x = (x,y) is exactly on the line source, the integral in Eq. (3.8) will be weakly singular with a finite value. In other words, in the two-dimensional case, the temperature has a finite value at points exactly on the curved line heat source. In this case, the integral in Eq. (3.8) can be calculated by various methods such as the weighted Gaussian integration (Stroud and Secrest, 1996), transformation of variable (Telles, 1987) and subtraction of singularity (Aliabadi 2002) method. In this research, the weighted Gaussian integration method is used. 3.2. Curved line heat source in three-dimensional problems Similar to 2D, the intensity per unit length of the quadratic line heat source at the starting point (x1,y1,z1), middle point (x2,y2,z2) and end point (x3,y3,z3) are assumed g1, g2 and g3, respectively. Substituting τ∗ in the 3D case from Eq. (2.4) into Eq. (3.3) results in τp(x)= 1 4πk ∫ L s(ξ) r(x,ξ) dl (3.9) where r(x,ξ) = √ (x−xs)2+(y−ys)2+(z−zs)2 is the distance between the field point x= (x,y,z) and the source point ξ= (xs,ys,zs) on the line heat source. xs, ys, and zs can be expressed as follows xs =(N1x1+N2x2+N3x3) ys =(N1y1+N2y2+N3y3) zs =(N1z1+N2z2+N3z3) (3.10) The infinitesimal element dl in Eq. (3.9) can be expressed as dl= √ dx2s +dy 2 s +dz 2 s = Jdη (3.11) where J = √ √ √ √ √ ( 3 ∑ j=1 xj dNj dη )2 + ( 3 ∑ j=1 yj dNj dη )2 + ( 3 ∑ j=1 zj dNj dη )2 (3.12) Therefore, Eq. (3.9) can be written as follows τp(x)= 1 4πk 1 ∫ −1 N1g1+N2g2+N3g3 r(η) J dη (3.13) Similar to the 2D case, the integral in equation (3.13) can be evaluated using standardnumerical integration methods such as the GQM. According to the integral in equation (3.13), it is clear that if the field pointx=(x,y,z) is exactly on the curved line source, the integral in Eq. (3.13) will be a strongly singular integral without any finite value. In other words, in the 3D case, the temperature at points on the curved line source does not have a finite value. 3.3. Curved surface heat source in 3D problems We consider a heat source distributed over a curved surface in a 3D domain. The shape of the surface source and its intensity function are assumedarbitrarily and sufficiently complicated. The surface of the heat source is discretized by several quadrilateral surfaces. Each quadrilateral surface heat source has a quadratic shape with a quadratic variation of the intensity over it. An efficient analysis of steady-state heat conduction... 1129 Fig. 3. A quadratic surface heat source In this part, the formulation for treatment of a quadratic surface heat source is presented. A quadratic surface heat source, which is described by 8 points, is shown in Fig. 3. The intensity per unit area of the quadratic surface heat source is written as follows s(ξ1,ξ2)= 8 ∑ i=1 Ni(ξ1,ξ2)gi (3.14) where g1,g2, . . . ,g8 are the intensities per unit area at 8 points of the source, ξ1 and ξ2 are local coordinates which have a variation between −1 and 1 in the source. The shape functions Ni in terms of ξ1 and ξ2 are expressed as follows (Becker, 1992) N1 = −1 4 (1− ξ1)(1− ξ2)(1+ ξ1+ξ2) N2 = 1 2 (1− ξ21)(1− ξ2) N3 = −1 4 (1+ ξ1)(1− ξ2)(1− ξ1+ξ2) N4 = 1 2 (1+ ξ1)(1− ξ 2 2) N5 = −1 4 (1+ ξ1)(1+ ξ2)(1− ξ1−ξ2) N6 = 1 2 (1− ξ21)(1+ ξ2) N7 = −1 4 (1− ξ1)(1+ ξ2)(1+ ξ1−ξ2) N8 = 1 2 (1− ξ1)(1− ξ 2 2) (3.15) The domain integral in Eq. (2.5) associated to the quadratic surface heat source is given as follows τp(x)= ∫ A s(ξ) k τ∗(x,ξ) dA (3.16) where dA is an infinitesimal area element on the quadratic surface heat source. Substituting τ∗ in the 3D case from Eq. (2.4) into Eq. (3.16) results in τp(x)= 1 4πk ∫ A s(ξ) r(x,ξ) dA (3.17) where r(x,ξ) is the distance between the field point x = (x,y,z) and the source points ξ=(xs,ys,zs) on the quadratic surface heat source. xs, ys and zs can be expressed in terms of the 8 shape functions as follows 1130 M.Mohammadi et al. xs(ξ1,ξ2)= 8 ∑ i=1 Ni(ξ1,ξ2)xi ys(ξ1,ξ2)= 8 ∑ i=1 Ni(ξ1,ξ2)yi zs(ξ1,ξ2)= 8 ∑ i=1 Ni(ξ1,ξ2)zi (3.18) The infinitesimal area element dA can be written as follows (Becker, 1992) dA= J(ξ1,ξ2)dξ1dξ2 (3.19) where J = √ (Jx) 2+(Jy) 2+(Jz) 2 (3.20) and Jx = ∂ys ∂ξ1 ∂zs ∂ξ2 − ∂zs ∂ξ1 ∂ys ∂ξ2 Jy = ∂zs ∂ξ1 ∂xs ∂ξ2 − ∂xs ∂ξ1 ∂zs ∂ξ2 Jz = ∂xs ∂ξ1 ∂ys ∂ξ2 − ∂ys ∂ξ1 ∂xs ∂ξ2 (3.21) Substituting Eqs. (3.19) and (3.14) into Eq. (3.17) results in τp(x)= 1 4πk 1 ∫ −1 1 ∫ −1 ∑8 i=1Ni(ξ1,ξ2)gi r(ξ1,ξ2) J(ξ1,ξ2) dξ1dξ2 (3.22) The integral in Eq. (3.22) can be calculated using standard 2D numerical integration methods such as the GQM. In the case that thefieldpointx is exactly on the surfaceof theheat source, the integral inEq. (3.22) will be weakly singular with a finite value. In other words, in the three-dimensional case, temperatures at points on a surfaceheat sourcehave finite values. In this case, the integral inEq. (3.22), which isweakly singular, canbe calculated byvariousmethods suchas the transformation of variable and subtraction of singularitymethod (Aliabadi, 2002). In this research, themethod of transformation of the variable is used for these cases. 4. Numerical examples In this Section, two 2D and two 3D examples containing different kinds of curved heat sources are presented. In each example, the results computed by the presentedMFS in comparisonwith theBEMandFEMare presented. Source codes are developed inMATLAB software for analysis of the examples using theMFS and BEM. ANSYS package is used for analysis of the examples using the FEM. The computations are implemented on a laptop with an Intel(R) (Intel, Inc., Santa Clara, CA, USA) Core(TM) i7-2670QM CPU of 2.20GHz, on 64-bit Windows operating system with 8.00 GBRAM. In all examples, the thermal conductivity is k=60W/(m◦C). 4.1. A circular domain including a circular heat source In this example, according to Fig. 4, a circular domain with R = 0.5 is considered. This problem is analyzed under the Dirichlet boundary condition with τB = 10 ◦C. A curved heat source which is distributed over a circle with the radius r=0.25m is considered. The strength of the heat source is considered to be constant over the circle and equal to s=4000W/m. The pseudo boundary Γ ′ is considered to be a circle with radius R′ = 2.5m (5 times of R). Only An efficient analysis of steady-state heat conduction... 1131 4 sources are considered on this pseudo-boundary. The circular heat source is modeled by only four quadratic line heat sources. The obtained results by the proposedMFS are compared with those of the BEM (32 linear boundary elements) and FEM (9461 quadratic elements) presented in (Mohammadi et al., 2016). The temperature results along the vertical diameter of the circle are shown in Fig. 5. As can be seen, the presentedMFS formulation yields very accurate results. Fig. 4. A circular domain including a circular heat source Fig. 5. Temperature on the vertical diameter (y-axis) of the circle obtained by the FEM, BEMandMFS 4.2. Arectangular domain includingaheat sourcewithanelliptical shapeandnon-uniform intensity In this example, a heat conduction problem over a 0.15×0.3m rectangle containing a curved heat source is considered. Figure 6a shows the geometry and thermal boundary conditions of the problem. An elliptical curved line heat source centered at (0.085,0.065) is considered in the domain. The lengths of the horizontal and vertical radii of the ellipse are r1 = 0.04m and r2 = 0.02m, respectively. The heat source intensity is considered to be a function of β ∈ [0,2π] as follows s=40000(1+cosβ)W/m (4.1) where β is the angular coordinate on the heat source measured from a horizontal axis passing through the center of the ellipse (Fig. 6a). 1132 M.Mohammadi et al. Fig. 6. A rectangle with an elliptical heat source: (a) geometry and boundary conditions, (b) configuration of collocation and source points In the proposed MFS, the elliptical heat source is modeled by only eight quadratic heat sources. 48 source points and 48 collocation points are considered for the MFS analysis. The configuration of collocation and source points is depicted in Fig. 6b. The locations of source points are determined according to themethod suggested byHematiyan et al. (2018). The ratio of the distance from the source point to its corresponding collocation point to the distance from the same source point to the neighboring collocation point is 0.85. By this configuration, a solution without undesired oscillation is obtained (Hematiyan et al., 2018). Fig. 7. The FEM, BEM andMFS results for temperature along the line AB of the rectangle with an elliptical heat source The temperature results along the line AB (Fig. 6a) are depicted in Fig. 7. In this figure, the results based on the presented formulation are compared with those of the BEM (48 linear boundaryelements) andFEM(4356 quadratic elements) presented in (Mohammadi et al., 2016). As it can be seen, the presented MFS and the BEM yield accurate results. The computational times (in seconds) for solving this problem using the proposed MFS and the BEM based on An efficient analysis of steady-state heat conduction... 1133 (Mohammadi et al., 2016) have been 2.11 and 6.24, respectively. The reported results indicate that the proposedMFS is more efficient than the BEM for analysis of this example. 4.3. A cubic domain including two circular heat sources with non-uniform intensity In this example, as shown in Fig. 8, a cube with edges of L= 10m, including two circular heat sources, is considered. All faces are kept at τ = 0◦C. The radius of both circular heat sources is r=2.5m. Fig. 8. A cube with two circular heat sources Thefirst circular heat source is centered at (5,6,5) and the second one is centered at (5,4,5). The strengths of the sources are considered to be functions ofβ∈ [0,2π] with the following forms s1 =10000(1+cosβ)W/m s2 =20000(1+cosβ)W/m (4.2) where β is the angular coordinate on the heat sources as shown in Fig. 8. In the proposedMFS, each circular heat source is modeled by only four quadratic line heat sources. 100 collocation points are considered on each face of the cube. Therefore, 600 colloca- tion points with 600 corresponding source points which are located on the cube with edges of L′ = 14m are considered. The configuration of the collocation points and their corresponding source points on a face of the cube is shown in Fig. 9. Fig. 9. The configuration of collocation and source points on a face of the cube The temperature results along the line x = z = 5 and the line x = y = 5 are shown in Fig. 10. In this figure, the obtained results by the proposed MFS are compared with those of the BEM (600 constant elements) and FEM (56669 3D quadratic elements) presented in 1134 M.Mohammadi et al. (Mohammadi et al., 2016). As it can be seen, the presented MFS and the BEM formulation yield very accurate results. However, the runningCPU time for theBEM is almost 1.8 times the MFS. The computations take 48.1s for the presentedMFS while they take 87.0s for the BEM. Fig. 10. Temperature on the line (a) x= z=5 and (b) x= y=5 of the cube with circular heat sources obtained by the FEM, BEM andMFS 4.4. A spherical domain including a cylindrical heat source with a non-uniform intensity In the last example, a spherical domain with the radius R = 1m, centered at (0,0,0) is considered.The surface of the sphere is kept at τ =0◦C.A surface heat sourcewith a cylindrical shape is included in the domain. The radius, center of the base, and the height of the cylindrical heat source are 0.2m, (0,0,0), and 0.8m, respectively. A cut-out part of the spherical domain and the cylindrical heat source is shown in Fig. 11a. The intensity per unit area of the source is considered to be a function of β ∈ [0,2π] (angular coordinate on the heat source, measured from the x-axis) and the y-coordinate with the following form s=20000y(1+cosβ)W/m 2 (4.3) In theproposedMFS, thecylindrical heat source ismodeledbyonly eightquadrilateral quadratic heat sources. The pseudo boundary Γ ′ is considered to be a sphere with the radiusR′ =1.4m. 98 collocation points and 98 sources are considered for the MFS analysis of the problem. The configuration of collocation and source points are shown in Fig. 11b. Fig. 11. The spherical domain including a cylindrical heat source: (a) a cut-out part of the domain, (b) the configuration of collocation and source points So far, this kind of problem has not been solved by the BEM. The obtained results by the presentedMFSare comparedwith those of theFEM.Thecommercial software package,ANSYS, is employed for the FE analysis. In the FE analysis, the heat source which is concentrated over An efficient analysis of steady-state heat conduction... 1135 a cylindrical surface should be modeled as a cylindrical volume with a small thickness. The inner and outer radius of this cylindrical volume are considered as ri =0.18m and r0 =0.22m, respectively. Thefinite element discretization of thedomainwith 3Dquadratic elements is shown in Fig. 12. The whole domain is discretized with 48178 elements and 64869 nodes. In order to visualize the position of the curved surface heat source inside the domain, only a cut-out part of the FEmesh is shown in Fig. 12a. The nodal arrangement of the mesh is depicted in Fig. 12b. Fig. 12. The finite element discretization of the sphere with a cylindrical heat source: (a) 48178 elements, (b) 64869 nodes The temperature results on the y- and z-axes are depicted in Fig. 13. As it can be observed, the MFS results are in an excellent agreement with the FEM solutions. noteworthy is the fact that the modeling of the problem in the proposedMFS is much simpler than in the FEM. Fig. 13. The FEM, andMFS results for temperature along (a) the y-axis and (b) the z-axis in the sphere with a cylindrical heat source 5. Conclusions A formulation based on the MFS for analysis of 2D and 3D heat conduction problems in iso- tropic media containing heat sources concentrated on arbitrary curved lines/surfaces has been presented. The shape and the variation of the intensity of the sources can be arbitrarily and sufficiently complicated. For 2D problems, curved line heat sources while for 3D problems, curved surface heat sour- ces have been considered. The equations derived for 2D curved line heat sources showed that temperature at points exactly on the source had a finite value. The formulation for 3D curved line heat sources showed that the temperature at points exactly on those sources had an infinite value but the temperature had a finite value at points on the curved surface heat source. For reliablemodelingof a concentratedheat source in theFEM, the source shouldbemodeled as a separated region with a small thickness. Moreover, a large number of internal elements and 1136 M.Mohammadi et al. nodes should also be considered for the modeling of the source. However, these sources can be effectively modeled in the BEM as well as the proposed MFS without considering any internal cells or points. To show the performance of the presented MFS formulation, four numerical examples have been given. It was observed that the proposed method gave accurate results even with a small number of source points and it was found that the computational cost of the presented method was much smaller than the BEM. Somemodified/improved versions of theMFS such as the singular boundarymethod (Gu et al., 2012) have been presented too. The proposed formulation for the implementation of curved heat sources can be employed for these methods too. References 1. 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