New Method for Solving the Inverse Thermal Conduction Problem (θ-Scheme Combined with CG Method under Strong Wolfe Line Search)

Abstract

Most thermal researchers have solved thermal conduction problems (inverse or direct) using several different methods. These include the usual discretization methods, conventional and special estimation methods, in addition to simple synchronous gradient methods such as finite elements, including finite and special quantitative methods. Quantities found through the finite difference methods, i.e., explicit, implicit or Crank–Nicolson scheme method, have also been adopted. These methods offer many disadvantages, depending on the different cases; when the solutions converge, limited range stability conditions. Accordingly, in this paper, a new general outline of the thermal conduction phenomenon, called (θ-scheme), as well as a gradient conjugate method that includes strong Wolfe conditions has been used. This approach is the most useful, both because of its accuracy (16 decimal points of importance) and the speed of its solutions and convergence; by addressing unfavorable adverse problems and stability conditions, it can also have wide applications. In this paper, we applied two approaches for the control of the boundary conditions: constant and variable. The θ-scheme method has rarely been used in the thermal field, though it is unconditionally more stable for θ∈ [0.5, 1]. The simulation was carried out using Matlab software.

Keywords

• thermal conduction
• inverse problem
• θ-scheme
• CG method
• Wolfe conditions
• unconditionally stable