Inverse boundary traction reconstruction with the bem

A boundary integral formulation is presented for the solution of the inverse elastostatics problem (IESP) of reconstructing missing boundary tractions in two-dimensional structural members. Traction reconstruction may involve the determination of the location of the traction distribution in addition to its extent and amplitude. The missing boundary tractions are rebuilt from measured quantities such as displacements, strains or stresses. These quantities may be obtained from sensors located at some internal or boundary points of the object. The proposed formulation starts with an initial guess for the magnitude, extent and location of the missing boundary tractions and proceeds towards the final traction distribution in a sequence of iterative steps. 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 quantities at each sensor location and the corresponding computed quantities for the assumed boundary traction distribution. The constraints that the missing traction distribution lies within a certain portion of the boundary of the object are imposed. This is done using the step retraction and inverse penalty function approach in which the objective function is augmented by the constraint equations using a penalty parameter. The unknown traction distribution and its location are defined in terms of load and 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 reconstruction of linear, parabolic and trigonometric boundary tractions, respectively, are solved using the present approach. The effect of Gaussian errors in the sensors is also studied. Good reconstruction of the missing boundary tractions is obtained for the examples studied. The advantages of the present boundary element formulation over the corresponding finite element formulations are also outlined.