Singularities Occuring in a Bimaterial with Transparent Boundary Conditions

In many engineering problems one has to deal with two materials which can induce local singularities (infinite stresses) as soon as the interface reaches the boundary. This is the case of water pipes with an inner coating for avoiding rusting. Because the wave velocity is smaller in the coating, it appears Love waves which can be used in order to detect defects because they propagate further as in a wave guide. They can be cracks corresponding to a disconnection at the interface between the two materials. In order to detect them, one can use measurements performed at the extremities of the pipes even if the signal in the numerical model, is very much perturbed by the singularities appearing at the interface between the two media at the extremities of the pipe. The phenomenon is amplified when one considers an artificial truncation of the structure and adding transparent boundary conditions in order to avoid reflection for simulating what happens in the full structure. The goal of this paper is to focus on a numerical method which can be used for the analysis of the influence of the singularities on the signal processing analysis. First of all, we give a mathematical description of the singularity met in our problem. Then, we define the extension to our case of the method introduced for cracks by G. Fix. It consists in adding the singular function to finite element functional space used in a classical numerical simulation. The main point of the paper is then to analyze, in a mathematical framework, the error estimates on the coefficients of the singularities with respect to the mesh size. Few numerical tests illustrate the mathematical results obtained for the problem we are dealing with.