This paper proposes an isogeometric boundary element (IGABEM) with the aim to solve the 2D steady-state heat conduction problems under spatially varying conductivity and internal heat source. The IGABEM boundary-domain integral equation is derived underlying the divergence theorem of Gauss and the Laplace equation. The non-uniform rational B-spline (NURBS) basis used in the CAD/CAE industries is applied to approximate both the boundary of problem domain and the unknown physical quantities, and the radial integration method (RIM) is employed to deal with the domain integrals induced by the non-homogeneous thermal conductivity and the internal heat source. The present boundary-domain integral equation is firstly divided into several NURBS patches, upon which the IGABEM can be further defined. The main advantage of the NURBS basis in comparison with the standard piecewise polynomial basis is that there is almost no error in the boundary approximation with IGABEM, which plays a significant role in controlling the accuracy of numerical calculation. Four numerical examples consisting of square regions with cubic and exponential thermal conductivities (with or without internal heat source), annulus region with constant and quadratic conductivities as well as complex region with varying thermal conductivity and complex heat source are provided. The accuracy and convergence of the proposed method are assessed by comparing the present solution with the existing results obtained by several other researchers or ANSYS results, and excellent performance is achieved.

本文提出了一种等几何边界元（IGABEM），旨在解决空间变化的电导率和内部热源下的二维稳态导热问题。IGABEM边界域积分方程是在高斯和拉普拉斯方程的发散定理的基础上得出的。CAD / CAE行业中使用的非均匀有理B样条（NURBS）基础用于近似问题域和未知物理量的边界，并采用径向积分方法（RIM）来处理该域由非均匀热导率和内部热源引起的积分。首先将本边界域积分方程分为几个NURBS面片，然后可以进一步定义IGABEM。与标准分段多项式基础相比，NURBS基础的主要优势在于，使用IGABEM进行边界逼近几乎没有误差，这在控制数值计算的准确性方面起着重要作用。提供了四个数值示例，这些示例由具有立方和指数导热率的正方形区域（带有或不带有内部热源），具有恒定和二次导热率的环形区域以及具有不同导热率和复杂热源的复杂区域组成。通过将本解决方案与其他研究人员获得的现有结果或ANSYS结果进行比较，评估了所提出方法的准确性和收敛性，并获得了出色的性能。