A boundary element formulation for the inverse elastostatics problem (iesp) of flaw detection

Abstract

A boundary integral formulation is presented for the detection of flaws in planar structural members from the displacement measurements given at some boundary locations and the applied loading. Such inverse problems usually start with an initial guess for the flaw location and size and proceed towards the final configuration in a sequence of iterative steps. A finite element formulation will require a remeshing of the object corresponding to the revised flaw configuration in each iteration making the procedure computationally expensive and cumbersome. No such remeshing is required for the boundary element approach. The inverse problem is written as an optimization problem with the objective function being the sum of the squares of the differences between the measured displacements and the computed displacements for the assumed flaw configuration. The geometric condition that the flaw lies within the domain of the object is imposed using the internal penalty function approach in which the objective function is augmented by the constraint using a penalty parameter. A first-order regularization procedure is also implemented to modify the objective function in order to minimize the numerical fluctuations that may be caused in the numerical procedure due to errors in the experimental measurements for displacements. The flaw configuration is defined in terms of geometric parameters and the sensitivities with respect to these parameters are obtained in the boundary element framework using the implicit differentiation approach. A series of numerical examples involving the detection of circular and elliptical flaws of various sizes and orientations are solved using the present approach. Good predictions of the flaw shape and location are obtained.

Numerical boundary identification for Helmholtz-type equations

“…An important class of inverse problems in engineering consists of the inverse geometrical problems which can be divided into the following subclasses: shape and design optimisation [3][4][5][6][7], identification of defects, e.g. cracks, cavities or inclusions, [8][9][10][11][12][13][14] and identification of an unknown boundary [15][16][17][18][19][20][21][22][23][24].…”

Boundary integral equation for inclusion and cavity shape sensitivity in harmonic elastodynamics

“…The most usual ones are based on a definition of the complete geometry by splines of all kinds and orders. In identification problems, the geometry is usually defined by simple geometrical entities, in turn defined by a few parameters (like ellipses defined by the coordinates of the center, the axes length and an angle of orientation [2,13,19]). …”

Optimization algorithms for identification inverse problems with the boundary element method

“…Problem F consists of a 100 £ 50 rectangle with different boundary conditions and an elliptical interior¯aw. This problem was solved by Bezerra et al [3]. Here only two parameters are allowed from a starting circle, allowing horizontal and vertical displacement.…”