A finite element analysis of optimal variable thickness sheets

A quasi-mixed finite element (FE) method for maximum stiffness of variable thickness sheets is analyzed. The displacement is approximated with nine node Lagrange quadrilateral elements, and the thickness is approximated as elementwise constant. One is guaranteed that the FE displacement solutions will converge in H-1 (Omega), but in an example it is shown that, in general, one cannot expect any subsequence of the FE thickness solutions to converge in any L-p (Omega)-norm. However, under a regularity and biaxiality assumption on the optimal stress field, uniqueness of the optimal thickness function as well as convergence in L-p (Omega) (1 less than or equal to p < infinity) of FE thickness solutions are proven.