A method for solving ill-posed integral equations of the first kind

Improperly-posed (or Hadamard-incorrect) problems may arise when numerical solutions are extremely sensitive to a discretization process. The nonconformal contact problem in three-dimensional elastostatics falls into this category. It is shown how such contact stress problems may be formulated and successfully solved using the “Functional Regularization Method” of Tychonov. The Functional Regularization Method requires the use of a parameter, called the Regularization Parameter. Although no general rules for the choice of such a parameter appear to exist, we have determined appropriate bounds on the parameter for a wide class of contact problems (including that of Hertz). The method developed should be capable of extension to more general ill-posed problems. It is also shown that refinements in the discretization process such as reduced mesh lengths or higher order quadrature formulas may postpone, but do not necessarily remove the numerical difficulties associated with the physics of the problem.