Boundary integral formulation for the inverse problem of detecting open cracks

Boundary value problems in mathematical physics are usually well-posed. The detection of open cracks from experimental measurements taken from the surface of a solid is an inverse problem and is ill-posed. Advances in computational techniques for the solution of well-posed problems in solid mechanics have progressed for more than five decades. The same is not true for inverse problems. Computational techniques for the solution of the inverse problem of detecting open cracks may be a helpful tool in the diagnosis, examination and monitoring of structural systems. This paper proposes a boundary integral formulation in conjunction with a first-order optimization technique for the solution of the Inverse ElastoStostatics Problems (IESP) of detecting an open crack inside planar structural bodies. The proposed formulation starts with an initial guess of the open crack geometry defined in terms of design variables or parameters. The sensitivities required in the optimization algorithm are obtained in the boundary integral formulation by implicit differentiation. Three example problems are presented to demonstrate the effectiveness of the proposed formulation in detecting open cracks. Good results are obtained even when the simulated experimental measurements are contaminated with Gaussian errors.