CMES CMES CMES Computer Modeling in Engineering & Sciences 1526-1506 1526-1492 Tech Science Press USA 12821 10.32604/cmes.2021.012821 Article Isogeometric Boundary Element Analysis for 2D Transient Heat Conduction Problem with Radial Integration Method Isogeometric Boundary Element Analysis for 2D Transient Heat Conduction Problem with Radial Integration Method Isogeometric Boundary Element Analysis for 2D Transient Heat Conduction Problem with Radial Integration Method Chen Leilei 1 Li Kunpeng 1 Peng Xuan 2 Lian Haojie 34 haojie.lian@outlook.com Lin Xiao 5 Fu Zhuojia 6 College of Architecture and Civil Engineering, Xinyang Normal University, Xinyang, 464000, China Department of Mechanical Engineering, Suzhou University of Science and Technology, Suzhou, 215009, China Key Laboratory of In-Situ Property-Improving Mining of Ministry of Education, Taiyuan University of Technology, Taiyuan, 030024, China Institute for Computational Engineering, Faculty of Science, Technology and Communication, University of Luxembourg, Luxembourg The York Management School, University of York, York, YO10 5GD, UK College of Mechanics and Materials, Hohai University, Nanjing, 211100, China *Corresponding Author: Haojie Lian. Email: haojie.lian@outlook.com 03 10 2020 126 1 125 146 16 07 2020 04 09 2020 © 2021 Chen et al. 2021 Chen et al. This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents an isogeometric boundary element method (IGABEM) for transient heat conduction analysis. The Non-Uniform Rational B-spline (NURBS) basis functions, which are used to construct the geometry of the structures, are employed to discretize the physical unknowns in the boundary integral formulations of the governing equations. Bézier extraction technique is employed to accelerate the evaluation of NURBS basis functions. We adopt a radial integration method to address the additional domain integrals. The numerical examples demonstrate the advantage of IGABEM in dimension reduction and the seamless connection between CAD and numerical analysis.

Isogeometric analysis NURBS boundary element method heat conduction radial integration method
Introduction

Isogeometric analysis (IGA) has drawn extensive attentions in engineering science community since the seminal works of Hughes et al. . As an alternative to the traditional analysis based on Lagrange polynomial, IGA uses the same spline functions that are used in the geometric expression in CAD, for example Non-Uniform Rational B-splines (NURBS), as the basis functions to approximate the unknown field. Hence, IGA is able to perform numerical analysis directly from CAD without cumbersome preprocessing procedure and geometric errors. Additionally, IGA offers high-order continuity and flexible refinement schemes which are particularly attractive in numerical simulation. Due to the aforementioned salient features, IGA has been successfully applied in many fields . To facilitate the implementation of IGA, Bézier extraction  technique is proposed that enables one to develop IGA codes in existing finite element codes, without changing the program framework. By using the extraction operator, the complicated iterative process for evaluating B-spline basis functions is eliminated, which improves the efficiency of simulation significantly.

Although the concept of IGA was originally proposed in the context of finite element methods, its application in engineering practice is restricted because FEM relies on volume parameterization which conflicts with the boundary representation in CAD field. Mierzwiczak et al.  and Fu et al.  applied the isogeometric boundary element method to the heat conduction problem, and proposed the origin intensity factor to eliminate the singularity of the integral. The order reduction of POD model for transient heat transfer was studied by Li et al. . Wang et al.  solved the three-dimensional elasto-plastic problem by boundary element method. On the contrary, boundary element method (BEM)  only needs boundary parameterization and thus is compatible with CAD models naturally. The isogeometric analysis with boundary element method (IGABEM) was first applied to potential problems  and elasticity analysis [21,22]. Since then, IGABEM has been applied successfully to a wide range of fields, such as heat conduction , linear elasticity [24,25], fracture mechanics , electromagnetics , acoustics , structural optimization [34,3640], etc.

In this paper, we are aimed to extend the application of IGABEM to heat conduction problems which are an increasingly important topic in many engineering fields. The radial integration method proposed by Gao  is incorporated to solve the additional domain integrals. The problems of constant and variable coefficients, transient and steady state of temperature field are studied. The rest of the paper is arranged as follows: Section 2 gives an overview of the B-spline and NURBS basis. Bézier extraction is detailed in Section 3. In Section 4, the boundary integral equation for heat conduction is introduced and the discrete formula derivation of the boundary integral equation, as well as the time marching method are introduced in detail. Section 5 provides two sets of numerical examples to verify the algorithm, followed by the conclusion in Section 6.

NURBS

For the sake of completeness, the fundamentals of NURBS are briefed in this section. NURBS are generalized from B-splines, whose basis functions are defined over a knot vector which is a monotonic increasing real number sequence, denoted by U={u1,u2,,um} where ui is the i-th knot in the vector U. B-spline basis function is evaluated iteratively as:

Ni,0(u)={1,ifuiu<ui+10,otherwise

Ni,p=u-uiui+p-uiNi,p-1(u)+ui+p+1-uui+p+1-ui+1Ni+1,p-1(u),

where p is the order of the polynomial in B-spline basis functions, and n is the number of the basis functions.

The derivative expression of the basis function can be derived as follows:

Ni,p=pui+p-uiNi,p-1-pui+p+1-ui+1Ni+1,p-1(u).

The B-spline curve is described by the linear combination of NURBS basis functions and the corresponding control points,

x(u)= i=1nNi,p(u)Pi,

where Pi is the Cartesian coordinates of control points.

Non-uniform rational B-spline (NURBS) basis functions are obtained by introducing the weight coefficient ω to rationalize the B-spline basis,

Ri,p(u)=Ni,p(u)ωiW(u),

where

W(u)= j=1nNj,pωj.

Similar to B-splines, the NURBS curve is expressed by:

x(u)= i=1nRj,p(u)Pi

Since the NURBS basis function adopts the rational form composed of the B-spline basis function and the weight factor ω, it is able to exactly express conic curves, such as ellipses and circles.

Bézier Extraction Operation

The computation of IGABEM can be accelerated using Bézier extraction. The idea is to extract linear operators and map the Bernstein polynomial basis functions of Bézier elements to global B-spline basis functions. The linear transformation is defined by a matrix called an extraction operator. The transpose of the extraction operator maps the control points of the global B-spline to that of Bézier curves. The process of Bézier extraction operation is proposed in this work, also see .

To perform Bézier decomposition, all internal knots of a knot vector are repeated until their multiplicity equals p. For example, we insert ū after the k-th knot of the original knot vector Ξ={u1,u2,,un+p+1}. The new knot ξ={u1,u2,,uk,ū,uk+1,un+p+1} with k > p. The introduction of new knots will bring about changes in the basis functions and the generation of new control points. The relationship between the original control point P and the new control point Q can be deduced:

Qi={P1,i=1αiPi+(1-αi)Pi-1,0<i<mPn,i=m

where m = n + 1. The expression for calculating factor αi in Eq. (8) is as follows:

αi={1,ik-p;ū-uiui+p-ui,k-p+1ik0,ik+1

After the knot is inserted, it brings about a change in the control point. Let {ū1,ū2,,ūm} be the set of insertion knots required for the Bézier decomposition of the B-spline. Then for each new knot, ūj, j=1,2,,m, using Eq. (9), we define αij,i=1,2,,n+j, to be the ith α. So the expression for the conversion matrix C is as follows:

Cj=[α11-α2000α21-α30000α31-α40000αn+j-11-αn+j ]

Then, the corresponding control points can be calculated by:

Q=CmTC1TP

We rewrite Eq. (11) with matrix form:

Q=CTP

To decompose a B-spline curve defined in the knot vector {0,0,0,0.25,0.5,0.75,1,1,1} with Bézier extraction, we will insert the knots {0.25,0.25,0.5,0.5,0.75,0.75} into the existing knot vector. Without changing the geometry, the B-spline curve is expressed using the Bézier basis function as follows:

X(u)=i=1nNi,pPi= i=1mBi,nQi

Using Eqs. (12), (13) can be expressed as follows:

X(u)=QTB(u)=(CTP)TB(u)=PTCB(u)=PTN(u)

Since P is arbitrary, we can get the following equation

N(u)=CB(u)

It should be noted that only the knot vector determines the C matrix. In other words, the extraction operator is a product of parameterization and does not depend on control points or basis functions. Therefore, we can apply the extraction operator directly to NURBS. Define weights as diagonal matrices

W=[ω1ωn ]

Converting Eq. (5) to matrix form, we have

R(u)=1W(u)WN(u)

Using Eq. (15), the NURBS basis function can be expressed in terms of the Bernstein basis as

R=1ωTNWN=1ωTCBWCB

The NURBS curve is expressed as:

T(u)=1ωTCB(u)PTWCB(u)

In Eq. (19), B(u) can be used to express the NURBS curve. Let ωb=CTω, and define Wb to be the diagonal matrix

Wb=[w1bw2bwn+mb ]

Eq. (19) can be written as follows:

T(u)=1(ωb)TB(u)PTWCB(u)

The expression of the Bézier control points can refer to the method of homogeneous coordinates:

Pb=(Wb)-1CTWP

Both sides are multiplied by Wb at the same time:

WbPb=CTWP

After solving the weight factor ωb and the control point Pb after the Bézier operation, we arrive at the final Bézier representation of the NURBS:

T(u)=1Wb(u)(WbPb)TB(u)=i=1n+mPibωibBi(u)Wb(u)

Thus, we can express NURBS equivalently with a series of C0 Bézier elements.

Governing Equation

The governing partial differential equation of transient heat conduction in isotropic medium with internal heat source is expressed by:

t_{0},~\mathbf{x}\in \Omega \right) \label{eqn-25} \end{equation}$$]]> k2T(x,t)+b(x,t)=ρcxT(x,t)t(t>t0,xΩ) where T(x, t) represents the temperature at point x at time t, k is the thermal conductivity, b(x, t) shows the heat source value at point x at time t, ρ is the density of the material, and cx is specific heat capacity. The following three kinds of boundary conditions are considered: t_{0},\mathbf{x}\in \Gamma _{T}\right) \end{eqnarray*}$$]]> T(x,t)=T¯(x,t)(t>t0,xΓT) t_{0},\mathbf{x}\in \Gamma _{q}\right) \end{eqnarray*}$$]]> q(x,t)=-kT(x,t)n(x)=q¯(x,t)(t>t0,xΓq) t_{0},\mathbf{x}\in \Gamma _{f}\right) \label{eqn-26} \end{equation}$$]]> Tf(x,t)=T¯f(x,t)(t>t0,xΓf)

where γT represents the dirichlet boundary, γq the neumann boundary and γf convective boundary condition. n(x) is the unit external normal direction vector at the boundary point.

The transient problem is time-dependent, so the initial conditions need to be prescribed: T(x,t0)=T0(x)(xΩ) where T0 indicates initial temperature on the domain.

Boundary Element Method of Transient Temperature Field

Through weighted residual operation and partial integration operation, Eq. (25) can be expressed as the following boundary-domain integral equation:

c(x)T^(x,t)+(H*T^)(x)=(G*q)(x)+(G̃b)(x)-(G̃Ṫ)(x)

where the coefficient c(x)=1 when the source point x is located in the domain, c(x)=0.5 when the source point x is on the boundary of structure, and the integrals are

{(H*T^)(x):=ΓH(x,y)T^(y,t)dΓ(y)(G*q)(x):=ΓG(x,y)q(y,t)dΓ(y)(G̃b)(x):=ΩG(x,ỹ)b(ỹ,t)dΩ(ỹ)(G̃Ṫ)(x):=ΩG(x,ỹ)ρ̃T^.(ỹ,t)dΩ(ỹ)

where T^=kT indicates normalized temperature, T^.=T^t, ρ̃=ρcx/k is reciprocal of thermal diffusivity, G(x, y) and H(x, y) are Green’s function and it’s derivative, as follows:

{G(x,y)=12πln(1r)H(x,y)=G(x,y)n(y)=-12πrrn(y)

In this work, we do not consider the source. Thus, Eq. (27) can be rewritten as

c(x)T^(x,t)+(H*T^)(x)=(G*q)(x)-(G̃Ṫ)(x)

By observing Eq. (28), we can find that the last term on the right hand of Eq. (30) is domain integral term rather than boundary integral term. In this work, the radial integral method is used to transform domain integral into boundary integral to retain the advantage of dimension reduction of IGABEM.

We firstly approximate unknown temperature gradient T^. by interpolation calculation. The commonly used interpolation function is radial basis function (RBF). We select a number of collocation points and interpolation to get the approximate formula of variables T^.,

T^.(ỹ,t)= i=1Nβi(t)fi(r(ỹ))

where N is the number of collocation points, βi is the undetermined coefficient at the i-th collocation point, fi(r) is the radial basis function at the i-th collocation point, r=|ỹ-xi|, and xi is the coordinates of the i-th collocation point.

It is worth noting that collocation points can be boundary points or internal points. Generally, combining boundary points and internal points to form collocation points can effectively improve the approximation accuracy. In addition, the accuracy of approximation can be further improved by combining radial basis functions and polynomials, which can be expressed in combination with quadratic polynomials,

T^.(ỹ,t)= i=1Nβi(t)fi(r(ỹ))+γ0(t)+ i=12γi(t)ỹi+ i=12 j=i2γij(t)ỹiỹj

where ỹi and ỹj are Cartesian coordinate components of the point ỹ. The following conditions are met:

{ i=1Nβi=0 i=1Nβix1i= i=1Nβix2i=0 i=1Nβix1ix1i= i=1Nβix2ix2i= i=1Nβix1ix2i=0

It is particularly important to determine these coefficients β and γ in Eq. (32). Although it is not easy to directly express these coefficients, we can get the implicit expression of these coefficients through a series of transformation operations. By setting collocation point as ỹ in Eq. (32) and assembling them into matrix-vector formulation, we can get the following expression:

fv=T^.

where

f=[f11f12f1N1x11x21x11x21x11x11x21x21f21f22f2N1x12x22x12x22x12x12x22x22fN1fN2fNN1x1Nx2Nx1Nx2Nx1Nx1Nx2Nx2N111000000x11x12x1N000000x21x22x2N000000x11x21x12x22x1Nx2N000000x11x11x12x12x1Nx1N000000x21x21x22x22x2Nx2N000000 ]

and

v=[β1β2βNγ0γ1γ2γ11γ12 ]T

T^.=[T^.1T^.2T^.N0 ]T

where fij = fj(r) in Eq. (35), r=|xi-xj|, and T^.i denotes the temperature derivative at the i-th collocation point.

When the collocation point is not repeated, the matrix f is invertible. Thus, the coefficient vector v can be expressed as:

v=f-1T^.

By substituting Eq. (33) into the last term on the right of Eq. (27) and using the radial integration method, the following boundary integral formula can be obtained

(G̃Ṫ)(x):=I0+I1+I2

where

{I0= i=1NβiΓBi(x,y)r(x,y)rn(y)dΓ(y)I1= i=12γiΓr,kB̃1(x,y)r(x,y)rn(y)dΓ(y)I2=(γ0+ i=12γixi)ΓB0(x,y)r(x,y)rn(y)dΓ(y)

and

{Bi(x,y)=0r(x,y)G(x,y^)ρ̃fi(R)r(x,y^)dr(y^)B̃1(x,y)=0r(x,y)G(x,y^)ρ̃r2(x,y^)dr(y^)B0(x,y)=0r(x,y)G(x,y^)ρ̃r(x,y^)dr(y^)

It can be found that the domain integral in Eq. (39) is successfully transformed into the boundary integral. For the detailed derivation process of the above results, also see .

It is worth noting that these coefficients in Eq. (40) are unknown, so it is impossible to obtain the results of I0, I1, and I2 directly through the boundary integral calculation. In fact, these coefficients do not need to be calculated directly.

When the source point x is set as the i-th collocation point, the terms I0, I1, and I2 can be rewritten as a formulas of vector and vector multiplication, respectively. By combining the three terms, we can get the following formulas

(G̃Ṫ)(x):=Aiv

where the vector Ai contains the result of the boundary integrals in Eq. (40). By substituting Eq. (38) into Eq. (42), we can obtain the following formulation

(G̃Ṫ)(x):=Aif-1T^.=CiT^.

where Ci=Aif-1. When the point x in Eq. (43) is chosen as collocation point, by assembling the equations at all points, we could get a formulation of matrix vector multiplication.

Discretization of Boundary Integral Equation

First, we discretize the boundary into Ne non-overlapping NURBS elements,

Γ=e= 1 Ne,Γi Γj=0,ij

NURBS basis functions are used for approximation. In any NURBS element e, variables can be approximately expressed as follows:

{T^(ξ)= l=1mRl,p(ξ)T^lq(ξ)= l=1mRl,p(ξ)ql

where m is the number of control points, ξ[-1,1] represents local parametric coordinates, T^l and ql are normalized temperature and its flux at the l-th collocation points. In this work, the Greville abscissa is used to define the positions of collocation points in parameter space, also see .

By substituting Eq. (45) into Eq. (27), the two boundary integral terms can be rewritten as

{(H*T^)(x):= e=1Ne l=1m[-11H(x,y(ξ))Rl,p(ξ)Je(ξ)dξ]T^l(G*q)(x):= e=1Ne l=1m[-11G(x,y(ξ))Rl,p(ξ)Je(ξ)dξ]ql

where Je=dΓ/dξ denotes Jacobian. Since the last term in Eq. (30) is calculated by radial basis function approximation, only geometric interpolation is needed in the final boundary integrals. Therefore, we only need to replace dΓ in Eq. (40) with Jedξ. On the other hand, the integrals in the above equations are solved with Gauss integral method. In this work, the number of Gauss quadrature points is set as eight.

After collecting the equations for all collocation points and expressing them in matrix forms, one can obtain the following system of linear algebraic equations:

HT^-Gq=-CT^.

In this work, the finite difference method is used for solution of the above time domain equations.

It is worth noting that weakly singular integrals exist in Eq. (30). For BEM, it is very important to deal with this kind of singular integral effectively, because it directly determines the accuracy of BEM. Qu et al.  and Gong et al.  used exponential transformation to calculate accurately the weak singular integrals. Gao et al.  used the subtraction of singularity technique to overcome the singularity problem. In this work, the subtraction of singularity technique is also used for calculation of weak singular integrals, see .

Time Marching Method for Solving Transient Heat Conduction Problems

Eq. (27) contains a set of integral equations about time. First, we assume that the total time period is from t0 to tm. And then, divide it into m intervals. Thus, the ith moment can be expressed as:

ti=t0+tm-t0mi

Herein, T^i denotes the temperature at ti, and T^i+1 denotes the temperature at ti+1. When the time marching method is used to solve the equations, we can obtain the derivative of temperature with respect to time, as follows:

T^.=T^i+1-T^iδt

where the time step δt is equal to tm-t0m. After interpolation approximation, we can get the temperature T^ and its derivative q at t[ti,ti+1], as follows:

{T^=βT^i+1+(1-β)T^iq=βqi+1+(1-β)qi

where β changes from 0 to 1. When β=0, it means forward difference scheme is used, when β=1, it means backward difference scheme is used, otherwise it means central difference scheme is used. By substituting Eq. (50) into Eq. (30), we can obtain the following formulations

{(H*T^)(x):=β(H*T^i+1)(x)+(1-β)(H*T^i)(x)(G*q)(x):=β(G*qi+1)(x)+(1-β)(G*qi)(x)

Thus, Eq. (47) can be rewritten as

βHT^i+1+(1-β)HT^i-βGqi+1-(1-β)Gqi=-CT^i+1-T^iδt

After merging the similar terms, we can get the following expression:

PT^i+1-βGqi+1=-QT^i+(1-β)Gqi

where

{P=βH+C/δtQ=(1-β)H-C/δt

When solving the unknown temperature T^i+1 or its derivative qi+1 at ti+1, the temperature T^i and its derivative qi at ti are all known. Moving all the unknown terms of Eq. (53) to the left-hand side and all the known terms to the right-hand side by considering the boundary conditions, one finally obtain the following system of linear equations:

Bxi+1=yi+1

where B is the coefficient matrix, xi+1 is the vector of unknown variable values at the collocation points, yi+1 is the known vector. All the unknown state values can be obtained after Eq. (55) is solved.

Numerical Examples Square Example

Consider a two-dimensional flat plate, the initial temperature is 100 K, the upper and lower boundaries are adiabatic, the left boundary temperature is maintained at 100 K. Convective boundary condition is applied to the right boundary, and the ambient temperature is 400 K, as shown in the Fig. 1. Material density ρ=271 kg/m3, specific heat capacity cx = 871 J/(kgK), thermal conductivity k=202.4 W/(mK), convective heat transfer coefficient on the right boundary h = 80 W/(m2K) and thermal emissivity ε=1. The boundary element model is as shown in the Fig. 1b. Each boundary is divided into 16 equally spaced linear elements, with a total of 64 boundary elements, 64 boundary nodes and 15 internal points.

Plate model diagram and boundary conditions. (a) Plate boundary conditions. (b) Plate model diagram

Fig. 2 shows the distribution of temperature along the lower boundary of plate when t = 100, 200, 400 and 600 s. In this figure, the calculation results obtained by using IGABEM are compared with those of the finite volume method (Fluent), and it demonstrates the correctness and validity of this algorithm proposed in this work.

Temperature distribution along the lower boundary of the plate at different times
An Example of Elliptic Plate

Consider an elliptical plate with a long radius of 2 m and a short radius of 1 m. The initial temperature is 100 K, and the ambient temperature of the plate boundary is 400 K. Material density ρ=271 kg/m3, specific heat capacity cx = 871 J/(kgK), thermal conductivity k = 202.4 W/(mK), and boundary convective heat transfer coefficient h = 80 W/(m2K).

The shape of ellipse and the distribution of internal points are shown in Fig. 3. Fig. 4a shows the initial NURBS curve and the position of control points. After normalizing the knot vector, the parameter space vector of boundary element can be obtained, which is expressed as Ξ̃=[0,0.25,0.5,0.75,1]. Therefore, four boundary elements are formed, and the parameter space intervals are [0, 0.25);[0.25, 0.5);[0.5, 0.75);[0.75, 1) respectively. By splitting each NURBS elements equally and inserting new knots, the new control point sequence, knot vector and weight factor vector after refinement can be obtained. For example, insert a new control point on the middle point of each initial NURBS cell parameter space to get the refined control point sequence shown in Fig. 4b, as shown by the black circle point “inserted knots.” After Bézier extraction operation, a new set of control point sequence is obtained, as shown in “BE operation knots.” For “initial knots” in Fig. 4a and “inserted knots” in Fig. 4b, we can find that inserting a new point will also change the position of the original point, but still describe the same curve. However, Bézier extraction operation does not change the position of the original control points, but inserts a new control point at the middle point of some elements. Then, the new control point sequence and Bézier extraction sequence when the initial NURBS cells are divided into 3, 4, 5 and 6 subunits are given in turn, as shown in Figs. 4c4f respectively.

The NURBS curve and internal points for elliptic Geometric control points and NURBS curves of elliptical model with different refinement levels. (a) Initial. (b) Refine2. (c) Refine3. (d) Refine4. (e) Refine5. (f) Refine6

In this work, the finite difference method is used to solve the time-domain equation. Therefore, we first investigate the influence of time step on the calculation results, as shown in Tab. 1. The temperature values at nine points on the x axis are calculated, where different time step is used, for example δt=5, 6, 7, 8, 9, and 10. From this table, we can find that the results with different time steps have small deviation. In fact, too large step value may lead to large error of temperature gradient value, even the calculation is not correct at all. If the time step is too small, it will also lead to inaccurate calculation of temperature gradient value. Therefore, it is very important to select an appropriate time step value. It can be seen from Tabs. 1 and 2 that it is appropriate to set the time step from 5 to 10.

Temperature values at several internal points with different time step values in 200 s
NSTEP NODE 1 NODE 2 NODE 3 NODE 4 NODE 5 NODE 6 NODE 7 NODE 8 NODE 9
( −1.6, 0) ( −1.2, 0) ( −0.8, 0) ( −0.4, 0) (0, 0) (0.4, 0) (0.8, 0) (1.2, 0) (1.6, 0)
5 136.156 117.783 109.520 106.530 105.821 106.535 109.522 117.784 136.159
6 135.715 117.454 109.308 106.386 105.691 106.386 109.309 117.456 135.717
7 135.273 117.127 109.096 106.238 105.562 106.238 109.097 117.128 135.275
8 136.043 117.776 109.588 106.621 105.908 106.621 109.588 117.778 136.045
9 135.603 117.450 109.376 106.472 105.778 106.472 109.377 117.452 135.605
10 135.969 117.773 109.632 106.678 105.966 106.678 109.633 117.775 135.971
Temperature values at several internal points with different time step values in 400 s
NSTEP NODE 1 NODE 2 NODE 3 NODE 4 NODE 5 NODE 6 NODE 7 NODE 8 NODE 9
( −1.6, 0) ( −1.2, 0) ( −0.8, 0) ( −0.4, 0) (0, 0) (0.4, 0) (0.8, 0) (1.2, 0) (1.6, 0)
5 170.298 148.477 134.829 127.678 125.473 127.678 134.830 148.479 170.300
6 169.674 147.887 134.314 127.229 125.049 127.229 134.316 147.889 169.678
7 170.076 148.287 134.688 127.575 125.384 127.575 134.690 148.289 170.080
8 170.185 148.404 134.808 127.694 125.502 127.694 134.809 148.407 170.187
9 169.559 147.815 134.294 127.246 125.079 127.247 134.295 147.817 169.563
10 170.108 148.356 134.794 127.705 125.521 127.705 134.795 148.358 170.112

In this work, we use the radial basis method to approximate the temperature gradient. Therefore, it is significant to investigate the influence of different types of radial basis functions on the numerical results. Herein, eight different types of radial basis functions are given, as shown in Tab. 3. Tab. 4 shows the comparison of temperature values at several calculation points when using different radial basis functions, where the time step is 5. By observing Tab. 4, it can be found that the deviation of calculation results in 200 s is very small when using different types of radial basis functions, more results in 400 s are also found in Tab. 5. It verifies the stability of the algorithm proposed.

NTYP 1 2 3 4 5 6 7 8
R (1+R2)12 (1+R2)-12 (1+R2)-52 (1+R2)-72 (1+R)−1 eR eR2
Comparison of calculation results with different types of radial basis functions in 200 s
NTYP NODE 1 NODE 2 NODE 3 NODE 4 NODE 5 NODE 6 NODE 7 NODE 8 NODE 9
( −1.6, 0) ( −1.2, 0) ( −0.8, 0) ( −0.4, 0) (0, 0) (0.4, 0) (0.8, 0) (1.2, 0) (1.6, 0)
1 136.417 117.892 109.492 106.436 105.70 106.436 109.491 117.892 136.416
2 136.541 118.114 109.663 106.374 105.545 106.370 109.648 118.109 136.559
3 136.156 117.783 109.521 106.535 105.82 106.535 109.522 117.784 136.159
4 135.574 117.337 109.233 106.515 105.737 106.515 109.232 117.338 135.575
5 135.595 117.413 109.232 106.419 105.539 106.419 109.232 117.414 135.595
6 135.991 117.575 109.300 106.550 105.903 106.549 109.300 117.575 135.992
7 135.878 117.538 109.341 106.606 105.948 106.607 109.341 117.537 135.877
8 135.530 117.347 109.292 106.615 105.954 106.615 109.291 117.347 135.529
Comparison of calculation results with different types of radial basis functions in 400 s
NTYP NODE 1 NODE 2 NODE 3 NODE 4 NODE 5 NODE 6 NODE 7 NODE 8 NODE 9
( −1.6, 0) ( −1.2, 0) ( −0.8, 0) ( −0.4, 0) (0, 0) (0.4, 0) (0.8, 0) (1.2, 0) (1.6, 0)
1 170.288 148.415 134.783 127.719 125.568 127.719 134.783 148.415 170.288
2 170.101 148.329 134.821 127.816 125.700 127.813 134.812 148.319 170.107
3 170.298 148.477 134.829 127.678 125.473 127.678 134.830 148.479 170.301
4 170.851 148.993 134.882 127.143 124.591 127.143 134.882 148.993 170.851
5 171.047 149.173 134.912 126.998 124.356 126.998 134.913 149.173 171.050
6 170.625 148.741 134.815 127.396 125.048 127.395 134.816 148.741 170.625
7 170.633 148.781 134.852 127.383 125.004 127.384 134.852 148.781 170.632
8 170.524 148.743 134.853 127.370 124.965 127.370 134.853 148.744 170.524

Figs. 5a and 5b respectively show the comparison of temperature values at points on the x axis and y axis based on traditional BEM and IGABEM algorithm developed in this paper. From these two graphs, it can be found that the results of the IGABEM algorithm are consistent with the results of conventional BEM, which verifies the correctness of the algorithm developed in this work.

Temperature distribution on X-axis and Y-axis, respectively. (a) Temperature of X-axis symmetry point. (b) Temperature of positive half axis of Y-axis
An Example of Octagonal Leaf

In this section, we consider an example of octagonal leaf, as shown in Fig. 6. The initial temperature is 100 K, and the ambient temperature of the structural boundary is 600 K. Other physical parameters are consistent with the previous example. The new control point sequence and Bézier extraction sequence when the initial NURBS cells are divided into 2, 3, 4, 5 and 6 subunits are given in turn, as shown in Figs. 7b7f respectively.

The NURBS curve and internal points for octagonal model Generation of NURBS curve control points for octagon model. (a) Initial. (b) Refine2. (c) Refine3. (d) Refine4. (e) Refine5. (f) Refine6

Similar to the previous example, we investigate the influence of different radial basis functions on the calculation results, as shown in Tab. 6 for 200 s and in Tab. 7 for 400 s. It shows that there is a small deviation in the calculation results when different radial basis functions are used, which verifies the stability of the algorithm in this work.

Comparison of calculation results with different types of radial basis functions in 200 s for octagonal leaf
NTYP NODE 1 NODE 2 NODE 3 NODE 4 NODE 5 NODE 6 NODE 7
( −0.6, 0) ( −0.2, 0) (0, 0) (0.4, 0) (0, 0.5) (0, −0.3) (0, −0.7)
1 135.010 111.721 109.027 119.731 126.151 115.106 145.490
2 135.254 112.784 110.301 120.490 126.821 115.988 145.524
3 135.570 112.847 110.281 120.632 126.975 116.130 145.897
4 135.335 111.355 108.550 119.775 126.252 114.895 146.002
5 135.087 109.657 106.608 118.714 125.596 113.462 146.163
6 134.604 108.323 105.146 117.715 124.901 112.269 145.893
7 134.215 107.510 104.270 117.068 124.414 111.519 145.591
8 134.725 109.683 106.691 118.533 125.347 113.404 145.643
Comparison of calculation results with different types of radial basis functions in 400 s for octagonal leaf
NTYP NODE 1 NODE 2 NODE 3 NODE 4 NODE 5 NODE 6 NODE 7
( −0.6, 0) ( −0.2, 0) (0, 0) (0.4, 0) (0, 0.5) (0, −0.3) (0, −0.7)
1 197.535 168.822 165.221 179.613 187.604 173.319 209.127
2 197.461 169.004 165.495 179.734 187.701 173.441 209.002
3 197.869 169.333 165.777 180.052 188.015 173.806 209.420
4 198.077 169.287 165.685 180.174 188.180 173.813 209.715
5 198.025 169.040 165.398 179.983 188.084 173.606 209.729
6 197.647 168.557 164.890 179.494 187.650 173.136 209.365
7 197.342 168.207 164.522 179.133 187.315 172.786 209.064
8 197.590 168.590 164.942 179.512 187.604 173.151 209.280

The influence of time step parameters on the calculation accuracy can be found in Tab. 8 for 200 s and in Tab. 9 for 400 s. Similar to the previous results, the change of time step parameters from 5 to 10 has little effect on the calculation results, which verifies the stability of the algorithm in this work.

Temperature at internal points with different time step values in 200 s for octagonal leaf model
NSTEP NODE 1 NODE 2 NODE 3 NODE 4 NODE 5 NODE 6 NODE 7
( −0.6, 0) ( −0.2, 0) (0, 0) (0.4, 0) (0, 0.5) (0, −0.3) (0, −0.7)
5 135.335 111.355 108.550 119.775 126.252 114.895 146.002
6 134.721 111.012 108.241 119.303 125.704 114.502 145.315
7 134.112 110.675 107.943 118.834 125.159 114.114 144.629
8 135.381 111.702 108.934 119.978 126.376 115.187 145.960
9 134.771 111.361 108.630 119.507 125.831 114.795 145.276
10 135.413 111.930 109.187 120.112 126.459 115.379 145.935
Temperature at internal points with different time step values in 400 s for octagonal leaf model
NSTEP NODE 1 NODE 2 NODE 3 NODE 4 NODE 5 NODE 6 NODE 7
( −0.6, 0) ( −0.2, 0) (0, 0) (0.4, 0) (0, 0.5) (0, −0.3) (0, −0.7)
5 198.077 169.287 165.685 180.174 188.180 173.813 209.715
6 196.833 168.046 164.445 178.926 186.929 172.568 208.479
7 197.713 168.968 165.370 179.833 187.825 173.484 209.341
8 197.985 169.272 165.677 180.124 188.108 173.783 209.602
9 196.744 168.037 164.444 178.881 186.859 172.544 208.367
10 197.925 169.263 165.676 180.093 188.060 173.764 209.527
Conclusion

This paper applied IGABEM to the heat conduction problem of temperature field. IGABEM can accurately represent the geometric model. In addition to the advantages of dimensionality reduction, IGABEM can also realize the seamless integration of CAD modeling and numerical analysis. The use of Bézier extraction technique further simplifies the implementation of IGA. This method transformed the basis functions of NURBS into Bernstein polynomials and maintained the consistency of geometric models. The main feature of this method was that the iterative calculation of NURBS basis function was avoided in the process of element physical interpolation calculation of BEM, which can effectively improve the computational efficiency. The domain integral term was successfully converted to the boundary integral in IGABEM with radial integration method. The present method provides a powerful tool for fast and high fidelity simulation of transient heat conduction problems commonly encountered in numerous industrial sectors.

Funding Statement: This research was funded by National Natural Science Foundation of China (NSFC) under Grant Nos. 11702238, 51904202, and 11902212, and Nanhu Scholars Program for Young Scholars of XYNU.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.