Regularized collocation Trefftz method for void detection in two-dimensional steady-state heat conduction problems

Abstract

Abstract. We propose the use of the Trefftz collocation method for the solution of inverse geometric problems and, in particular, the determination of the boundary of a void. As was the case in the solution of such problems using the method of fundamental solutions, the algorithm for imaging the interior of the medium also makes use of radial polar parametrization of the unknown void shape in two dimensions. The centre of this radial polar parametrization is considered to be unknown. The feasibility of this new method is illustrated by several numerical examples highlighting its advantages and shortcomings.

Trefftz method for an inverse geometry problem in steady-state heat conduction

“…The computations were performed on numerically simulated (synthetic) temperature, as in the reference papers [6][7][8][9][10][11]. Such an approach allows convenient and accurate evaluation of the results because the exact solution is known.…”

“…The modified collocation Trefftz method for a boundary detection problem was used in [6,7] and the same technique was applied for an inverse geometric problem governed by the biharmonic equation [8] and the modified Helmholtz equation [9]. The regularized collocation Trefftz approach was successfully tested on different shapes in a two-dimensional void detection problem [10]. The method of fundamental solutions, which may be viewed as one of the Trefftz methods, also belongs to computational techniques effective in IGPs, see e.g.…”

Numerical Solutions of Direct and Inverse Stokes Problems by the Method of Fundamental Solutions and the Laplacian Decomposition

“…In order to avoid time-consuming mesh generation and numerical quadrature in conventional numerical schemes, many so-called meshless methods have been developed recently, such as the MFS [16][17][18][19], the Trefftz method [20][21][22][23], the method of approximate particular solutions [24][25][26], the local RBFCM [27][28][29], the generalized finite-difference method [30,31], smoothed-particle hydrodynamics [32,33], etc. Among them, the MFS and the Trefftz method are two of the most promising boundary-type meshless methods, since only boundary nodes are needed for numerical simulation.…”

Trefftz method for a polynomial-based boundary identification in two-dimensional Laplacian problems

“…Although purely computational aspects are crucial to many publications in this area, researchers usually indicate possible industrial applications of their theoretical results. Illustrative examples could be non-destructive detection of defects like voids, cracks or inclusions (Cheng and Chang, 2003a,b; Karageorghis et al, 2014), locating the 1150 • C isotherm in a blast furnace hearth (Gonzalez and Goldschmit, 2006), or non-destructive evaluation of corrosion (Lesnic et al, 2002;Mera and Lesnic, 2005;Liu, 2008) -to name but a few.…”

“…Karageorghis et al (2014) proposed collocation TM for reconstruction of the void(s) whose centre(s) is (are) unknown, solving the proceeding non-linear least squares problem with a use of a suitable predefined routine in MATLAB. The collocation Trefftz method, modified by Liu (2007) and further referred as to modified collocation Trefftz method (MCTM), was applied by Fan et al (2012) to a Laplacian problem to infer the spatial position of the unknown part of the boundary from the Dirichlet condition.…”