A Parallel Computing Procedure for the Lower and Upper Bounds on the Functionals of Solutions to PDE: Application to the J-Integral in Functionally Graded Materials

A parallel computing procedure for computing the bounds on the {\it J}-integral in functionally graded materials is presented based on a Neumann element a-posteriori error bound. The finite element solution of {\it J}-integral is first obtained on a coarser finite element mesh, then a-posteriori bounding procedure based on the finite element error estimate is used to compute the lower and upper bounds on the {\it J}-integral. The computation of the error estimate is performed by solving independent elemental Neumann sub problems decomposed from the finite element model, thus the computing procedure is parallel and potential to solve large scale structural problems. An example is given in the end of paper to compute the lower and upper bounds on the {\it J}-integral of functionally graded materials.