Isogeometric methods for thermal analysis with spatially varying thermal conductivity under general boundary and other constraints

This paper presents some results on steady-state thermal analysis with variable thermal conductivity under general boundary conditions and other internal constraints using isogeometric methods. Non-Uniform Rational B-splines (NURBS) serve as basis functions for representing both the geometry of the physical domains and the solution. While both isogeometric collocation method and Galarkin formulation are discussed for facilitating comparisons, the main emphasis of the presented work is on isogeometric collocation method (IGA-C) for thermal analysis. To obtain the final solution, the respective partial differential equation (PDE) is discretized in its strong form at a number of collocation sites in IGA-C, as opposed to Galerkin formulations that involve a costly process of numerical integration in building up the system equations. The proposed method on IGA-C for thermal analysis can be easily implemented due to the simplicity of IGA-C in setting up the system equations. In addition to general boundary conditions of the respective PDE, other arbitrary constraints can also be easily incorporated into the final system of equations for producing desired solutions. Numerical examples with different kinds of spatially varying thermal conductivity along with other additional constraints and heat sources are provided to demonstrate the effectiveness of the proposed methods. The results show that the proposed methods are capable of conveniently handling arbitrary boundary and other additional constraints when solving thermal PDEs and can produce stable and accurate solutions with expected convergence.