Numerical simulations of 1D inverse heat conduction problems using overdetermined RBF-MLPG method

Abstract

This paper proposes a numerical method to deal with the one-dimensional inverse heat conduction problem (IHCP). The initial temperature, a condition on an accessible part of the boundary and an additional temperature measurements in time at an arbitrary location in the domain are known, and it is required to determine the temperature and the heat flux on the remaining part of the boundary. Due to the missing boundary condition, the solution of this problem does not depend continuously on the data and therefore its numerical solution requires special care especially when noise is present in the measured data. In the proposed method, the time variable is eliminated by using finite differences approximation. The method uses a weak formulation of the problem to enjoy the stability condition. To avoid the numerical integration on the whole domain, the weak form equations are constructed on local subdomains. The approximate solution is assumed to be a linear combination of Multi Quadric (MQ) radial basis function (RBF) constructed on nodal points in the domain and on the boundary. Since the problem is known to be ill-posed, Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system.

A local meshless collocation method for solving Landau–Lifschitz–Gilbert equation

“…They derived six MLPG formulations depending on various test functions applied and marked them MLPG1-MLPG6. The success of the MLPG method has been reported in solving several engineering problems; see [14,7,[15][16][17][18][19][20] and the references therein. A local meshless method based on the use of Radial Basis Functions (RBF) and collocation approach was proposed in [21].…”

“…(1.5) via a local meshless method. Meshless methods [7][8][9][10][11][15][16][17][18][19][20] are very attractive and effective for solving boundary value problems, because they involve simple preprocessing, arbitrary node distribution and flexibility of placing nodes at arbitrary locations, straightforward adaptive refinement, versatility in solving large deformation and also have the high order continuity and the ability to treat the evolution of non-smooth solutions, which is very useful to solve PDE problems. Many of them are derived from a weak-form formulation on a global domain or a set of local subdomains.…”