The isogeometric boundary element technique (IGABEM) is presented in this study for steadystate inhomogeneous heat conduction analysis. The physical unknowns in the boundary integral formulations of the governing equations are discretized using nonuniform rational Bspline (NURBS) basis functions, which are utilized to build the geometry of the structures. To speed up the assessment of NURBS basis functions, the Bézier extraction approach is used. To solve the extra domain integrals, we use a radial integration approach. The numerical examples show the potential of IGABEM for dimension reduction and smooth integration of CAD and numerical analysis.
Isogeometric analysis(IGA) has garnered considerable attention in engineering research since Hughes’ pioneering study[
In this study, we use the NURBS method to construct a twodimensional isogeometric model. The Bézier curve, proposed by Pierre Bézier in 1962, paved the path for the application of spline functions for modeling curved surfaces. The application of spline functions in curved surface modeling has been initiated, and NURBS has become the most widely used modeling tool in geometric modeling in the past few decades. NURBS has outstanding advantages for curved surface modeling. It can accurately represent quadratic curved surfaces, thus providing a uniform mathematical representation of regular and free surfaces. It includes a weight factor that affects the shape of curved surfaces, making it easier to control and achieve the required shape.
In this study, we focus on the heat conduction problem. Many researchers have used traditional methods to solve heat conduction problems, such as the singular boundary method for steadystate nonlinear heat conduction problems with temperaturedependent thermal conductivity[
The remainder of this paper is organized as follows.
The principles of NURBS are briefly discussed in this section for completeness. The basis functions of NURBS are defined over a knot vector, which is a monotonic growing real number sequence, represented by
When
In
The derivative of the basis function can be written as
The linear combination of NURBS basis functions and the related control points describes the Bspline curve as
where,
The weight coefficient
where we have the expression for
Similar to Bspline, the NURBS curve is expressed as
The rational form of the NURBS basis function, which consists of the Bspline basis function plus the weight factor
For most materials, the thermal conductivity varies with temperature. For example, the thermal conductivity of a certain composite material at room temperature is 10.85
The steadystate heat transfer equation for an isotropic medium containing an internal heat source with temperaturedependent thermal conductivity is expressed as
In
In
Applying the divisional integration method and Gaussian scattering theorem to the first term, we get
Considering the case where the source point
Two domain integrals and two boundary integrals appear in the unified equation. The boundary integrals can be calculated directly by discretizing the GaussLegendre quadrature. However, the latter two domain integrals are due to the nonlinear thermal conductivity and heat source terms, respectively. For the domain integrals from the nonlinear thermal conductivity, as the integrals contain
We use radial basis functions and primary polynomials to represent
Here,
NTYP  1  2  3  4  5  6 

20 
12 
By applying the radial basis function in
where
The matrix is invertible under the condition that no nodes overlap, and this is obtained by the chain rule shown in
where
The temperature gradient can be expressed as
in which
NTYP  A  B  C  D  E  F 

20 
12 
In
In
and
We continue the simplification of the equation of
The original equation is thus obtained by simplifying and combining similar terms as
in which, we have
The Gaussian product formula is used to solve the radial integral analytically for the inverse complex quadratic radial basis function, which is usually in the form of a tight branch, as shown in
It can be noted that
For
We first consider the domain integral for the heat source, which is the third term on the right hand side of
where
We have considered a threedimensional problem here. Therefore, the power of
The unified integral boundary is discretized into
Substituting
Finally, a unified system of algebraic equations is obtained, as shown in
In
Solving the system of linear equations using the NewtonRaphson iteration, we assume that after the
For the iteration of the (
We need to derive
Substituting
We can find the correction value of the unknown quantity
where
We consider a twodimensional model diagram consisting of a square of 1m side length on the left and a semicircle of radius 0.5m on the right side. Additionally, there are adiabatic left and right boundaries, a constant wall temperature of 100K for the upper boundary, and three boundary conditions of 200K for the lower boundary with convective heat transfer and convective radiative heat transfer. The material density
The original NURBS curve and control point positions are shown in
y/m  200 K  Convection heat transfer  Convective radiant heat transfer  
BEM/K  FLU/K  E/%  BEM/K  FLU/K  E/%  BEM/K  FLU/K  E/%  
0  200.00  200.00  0  138.17  138.88  0.51  138.01  138.96  0.69 
0.1  192.44  193.38  0.49  134.25  134.57  0.24  134.45  135.09  0.48 
0.2  184.61  185.48  0.47  130.54  130.82  0.22  130.72  131.29  0.44 
0.3  176.72  177.53  0.46  126.88  127.13  0.20  127.04  127.55  0.40 
0.4  168.10  168.83  0.43  123.20  123.41  0.18  123.34  123.78  0.36 
0.5  159.40  160.05  0.41  119.51  119.70  0.16  119.64  120.01  0.32 
0.6  149.60  150.16  0.37  115.79  115.94  0.13  115.89  116.20  0.27 
0.7  139.60  139.82  0.34  111.99  112.11  0.10  112.07  112.31  0.21 
0.8  127.69  128.03  0.27  108.10  108.18  0.07  108.16  108.32  0.15 
0.9  114.48  114.67  0.17  104.04  104.08  0.04  104.07  104.15  0.08 
Now, we consider the temperature boundary condition as an example to study the influence of the number of internal nodes on the accuracy of the calculation results. The number of internal points is considered as 14, 42, 66, and 86 for comparison, as shown in
y/m  14 inner points  42 inner points  66 inner points  
BEM/K  86 RESULT  E/%  BEM/K  86 RESULT  E/%  BEM/K  86 RESULT  E/%  
0  197.40  197.64  0.12  197.47  197.64  0.09  197.53931  197.64  0.05 
0.1  192.16  192.95  0.41  192.40  192.95  0.28  192.63783  192.95  0.16 
0.2  184.14  185.86  0.92  184.63  185.86  0.66  185.19521  185.86  0.37 
0.3  175.80  178.58  1.56  176.88  178.58  0.95  177.79231  178.58  0.44 
0.4  167.20  170.63  2.01  168.78  170.63  1.09  169.76849  170.63  0.50 
0.5  157.96  162.05  2.52  160.06  162.05  1.23  161.04216  162.05  0.62 
0.6  148.29  152.48  2.75  150.10  152.48  1.57  151.27515  152.48  0.79 
0.7  137.80  141.79  2.82  139.18  141.79  1.84  140.35611  141.79  1.01 
0.8  126.66  129.41  2.12  127.36  129.41  1.59  128.13763  129.41  0.99 
0.9  114.39  115.72  1.15  114.77  115.72  0.83  115.12064  115.72  0.52 
Let us consider a twodimensional model diagram of rectangular and semielliptical models. The upper region is a rectangle of length 2m and width 1m, the lower region is a semiellipse and the analytical equation is
In this study, the isogeometric boundary element technique (IGABEM) is applied to solve the twodimensional nonhomogeneous steady state heat conduction problem, and NURBS is used to construct a smooth geometric model. The radial integration method is used to transform the domain integral caused by the variable coefficient into the boundary integral. The numerical results show that the developed algorithm can effectively solve nonhomogeneous steadystate heat conduction problems with variable coefficients. In the future, we will extend the algorithm to transient analysis.
The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.