Finite element method with nonlocal boundary condition for solving the nondestructive testing problem of wood moisture content

摘要

In this paper, we use the finite element method (FEM) with nonlocal boundary condition for solving the nondestructive testing problem of wood moisture content based on a planar capacitance sensor model (i.e. the DtN-FEM for solving the mathematical model which is described by the exterior problems of a class of 3D Laplace equation with complicated boundary conditions). The original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a Dirichlet-to-Neumann (DtN) mapping represented in terms of the Fourier expansion series. For numerical computation, a series of approximate problems with higher accuracy can be derived if one truncates the series term in the variational formulation, which is equivalent to the reduced problem. The error estimate is presented to show how the error depends on the finite element discretization and the accuracy of the approximate problem. Based on the numerical results, the relationship between the dielectric constant (DC) of tested wood and the capacitance value of the sensor is discussed. Finally, we applied a least squares fitting method (LSFM) to reconstruct the wood moisture content (WMC) from the data measured with a planar capacitance sensor. Compared with popular statistical methods, the hybrid experimental–computational method is more convenient and faster, and a large number of experiments are avoided, the costs of testing are reduced.