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 analysisNURBSboundary element methodheat conductionradial integration methodIntroduction
Isogeometric analysis (IGA) has drawn extensive attentions in engineering science community since the seminal works of Hughes et al. [1]. 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 [2–6]. To facilitate the implementation of IGA, Bézier extraction [7] 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. [8] and Fu et al. [9] 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. [10]. Wang et al. [11] solved the three-dimensional elasto-plastic problem by boundary element method. On the contrary, boundary element method (BEM) [12–19] 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 [20] and elasticity analysis [21,22]. Since then, IGABEM has been applied successfully to a wide range of fields, such as heat conduction [23], linear elasticity [24,25], fracture mechanics [26–28], electromagnetics [29], acoustics [30–35], structural optimization [34,36–40], 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 [41] 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:
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 [7].
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,i≤k-p;ū-uiui+p-ui,k-p+1≤i≤k0,i≥k+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:
Then, the corresponding control points can be calculated by:
Q=CmT⋯C1TP
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=[ω1⋱ωn]
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=[w1bw2b⋱wn+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:
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*}$$]]>T(x,t)=T¯(x,t)(t>t0,x∈ΓT) t_{0},\mathbf{x}\in \Gamma _{q}\right)
\end{eqnarray*}$$]]>q(x,t)=-k∂T(x,t)∂n(x)=q¯(x,t)(t>t0,x∈Γq)
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)(x∈Ω)
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
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)=-12πr∂r∂n(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,
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:
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
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 [42].
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,i≠j
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 [39].
By substituting Eq. (45) into Eq. (27), the two boundary integral terms can be rewritten as
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. [43] and Gong et al. [44] used exponential transformation to calculate accurately the weak singular integrals. Gao et al. [45] 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 [39].
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
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 ExamplesSquare 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/(kg⋅K), thermal conductivity k=202.4 W/(m⋅K), convective heat transfer coefficient on the right boundary h = 80 W/(m2⋅K) 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/(kg⋅K), thermal conductivity k = 202.4 W/(m⋅K), and boundary convective heat transfer coefficient h = 80 W/(m2⋅K).
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. 4c–4f 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.
Types of radial basis functions
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
e−R
e−R2
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. 7b–7f 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.
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