Truly meshless localized type techniques for the steady-state heat conduction problems for isotropic and functionally graded materials

A numerical solution of steady-state heat conduction problems is obtained using the strong form meshless point collocation (MPC) method. The approximation of the field variables is performed using the Moving Least Squares (MLS) and the local form of the multiquadrics Radial Basis Functions (LRBF). The accuracy and the efficiency of the MPC schemes (with MLS and LRBF approximations) are investigated through variation (i) of the nodal distribution type used, i.e. regular or irregular, ensuring the so-called positivity conditions, (ii) of the number of nodes in the total spatial domain (TD), and (iii) of the number of nodes in the support domain (SD). Numerical experiments are performed on representative case studies of increasing complexity, such as, (a) a regular geometry with a constant conductivity and uniformly distributed heat source, (b) a regular geometry with a spatially varying conductivity and non-uniformly distributed heat source, and (c) an irregular geometry in case of insulation of vapor transport tubes, as well. Steady-state boundary conditions of the Dirichlet-, Neumann-, or Robin-type are assumed. The results are compared with those calculated by the Finite Element Method with an in-house code, as well as with analytical solutions and other literature data. Thus, the accuracy and the efficiency of the method are demonstrated in all cases studied.