Parameter identification in constitutive models via optimization with a posteriori error control

In this paper we outline a computational technique for the calibration of macroscopic constitutive laws with automatic error control. In the most general situation the state variables of the constitutive law, as well as the material parameters, are spatially non-homogeneous. The experimental observations are given in space–time. Based on an appropriate dual problem, we compute a posteriori the discretization error contributions from approximations of the parameter, state and costate fields in space–time for an arbitrarily chosen goal-oriented error measure of engineering significance. Such a measure can be used in an adaptive strategy (not discussed in this paper) to meet a predefined error tolerance. An important observation is that the Jacobian matrix associated with the resulting Newton method is used (in principle) in solving the dual problem. Rather than treating the Jacobian in a monolithic fashion, we utilize a sequential solution strategy, whereby the FE-topology of the discretized state problem is used repeatedly. Moreover, the proposed solution strategy lends itself naturally to the computation of first and second order sensitivities, which are obtained with little extra computational effort. Numerical results are given for the prototype model of confined aquifer flow with spatially non-homogeneous permeability. The efficiency of the optimization strategy and the effectivity of the error computation are assessed