Computational methods for inverse deformations in quasi-incompressible finite elasticity

This paper presents a formulation for incorporating quasi-incompressibility in inverse design problems for finite elastostatics where deformed configurations and Cauchy tractions are known. In the recent paper of Govindjee and Mihalic [1996, Comput. Methods Appl. Mech. Engng., 136, 47–57.] a method for solving this class of inverse problems was presented for compressible materials; here we extend this work to the important case of nearly incompressible materials. A displacement-pressure mixed formulation is combined with a penalty method to enforce the quasi-incompressible constraint without locking. Numerical examples are presented and compared to known solutions; further examples present practical applications of this research to active problems in elastomeric component design.