Inverse Problems in Elastostatics

An e last os tat ic problem is well-posed when the geometry of the general 3-D multiply-connected object is known and either displacement vectors, u. or surface traction vectors, p. are specified every where on the surface of the object. The elastostaiic problem becomes ill-posed when either:

a) a part of the object’s geometry is not known, or b) when both u and p arc unknown on certain parts of the surface. Both types of inverse problems can be solved only if additional information is provided. This information should be in the form of over-specified boundary conditions where both u and p are simultaneously provided at least on certain surfaces of the 3-D body.

The inverse determination of locations, sizes and shapes of unknown interior voids subject to overspecified stress-strain outer surface field is a common inverse design problem in elasticity [276)-(279|. The general approach is to formulate a cost function that measures a sum of least squares differences in the surface values of given and computed stresses or deforma­tions for a guessed configuration of voids. This cost function is then minimized using any of the standard optimization algorithms by perturbing the number, sizes, shapes and locations of the guessed voids. Thus, the process is identical to the already described inverse design of coolant How passages subject to over-specified surface thermal conditions 1280]. It should be pointed out that this approach to inverse design of interior cavities and voids can generate interior con­figurations that are potentially non-unique.