Identification of fields of elastic properties using reduced bases

The development of full-field measurement techniques in experimental mechanics has provided a large amount of data that can be used to analyze the mechanical behavior and calibrate model parameters. Heterogeneous identification, which aims to determine fields of properties, benefits greatly from these spatially dense measurements. The methods used to identify homogeneous properties can be applied to identify heterogeneous properties. However, representing a field requires a large number of parameters, which creates additional challenges. First, the computational cost of the methods increases with the increase in the dimension of the parameter space. Second, the ratio of the number of measurements to the number of unknowns is much lower than for homogeneous identification. This lower ratio increases the impact of measurement uncertainties on the identification result and can even create underdetermined problems.To address these challenges, an approach based on the equilibrium gap method is proposed. This method utilizes measured displacement fields and discretized equilibrium equations to construct the cost function to be minimized. For a linear model, the cost function is quadratic, resulting in a low computational cost for its minimization. Minimizing the equilibrium residual with a conventional norm 2 results in a problem that is highly sensitive to measurement uncertainties. However, by taking measurement uncertainty into account via a covariance weighting, it is possible to mitigate this sensitivity. The covariance matrix is calculated by propagating the measurement uncertainties to the residuals. Furthermore, it is shown that with covariance weighting, the Equilibrium Gap Method residual approximates the Finite Element Method Updating residual, which compares simulated displacements to measured displacements. Applications on synthetic and real experiments show that the Equilibrium Gap Method gives results close to FEMU in a fraction of the computation time.The use of a reduced basis to represent heterogeneous material parameters is proposed in order to control the effects of uncertainties on the solution. A reduced basis using a correlation assumption between parameter values for nearby points is constructed using the random field theory. The size of the basis is related to that of the heterogeneities it can represent. Therefore, the basis size acts as a regularization parameter, controlling the number of unknowns. The optimal basis size is the one that offers the best compromise between interpolation errors (predominant with a small reduced basis) and errors due to uncertainties (predominant with a large reduced basis). Three heuristics are proposed to find the optimal size of the reduced basis.The proposed approach is ultimately applied to a complete experiment containing a sequence of measured displacement fields. A reduced spatio-temporal basis is used to represent the heterogeneous parameters that vary over time. The set of residuals for all measured fields is concatenated to construct the minimized cost function. Temporal and spatial variations are identified separately, which transforms the nonlinear least squares minimization problem into a sequence of linear least squares problems.