Constrained least-squares solution and regularization in inverse boundary element analysis of photoelastic models involving contact

Existing work by the present team has involved the development of an inverse boundary element approach for fitting unknown boundary conditions to full-field stress data obtained using photoelasticity, treating the displacements on the unknown region as the solution variables. However, the recovered boundary conditions sometimes show unrealistic features such as negative contact pressures or shear tractions which exceed realistic frictional values; noisy or incomplete input data make these problems worse and can make the results oscillatory. The work described here exploits available constrained least-squares techniques in order to express the contact conditions as linear inequality constraints upon the displacement solution. A variant of Tikhonov regularization is also used to penalize against oscillatory traction distributions on the unknown boundary. The chosen implementation, solved using the LSSOL algorithm, can handle the issue of rigid-body motion without the need for manual imposition of restraints. The technique is illustrated using simulated and real examples, demonstrating that the technique correctly reconstructs the individual principal stresses.