An inverse model and mathematical solution for inferring viscoelastic properties and dynamic deformations of heterogeneous structures

We proposed an adjoint-based inverse method and mathematical solutions using Lagrange multiplier theorem for inferring dynamic viscoelastic properties of heterogeneous structures with displacement observations to improve computation reliability and efficiency. Existing methods such as the one using finite-difference approximated gradient may not be efficient and accurate enough for considering complex situations of the coupled effects of dynamic loading, material deformation memory, and structural heterogeneity. The proposed method derives both gradient and Hessian mathematically to satisfy the first-order necessary and second-order sufficient optimal conditions, which has great advantages compared to other approaches. We also proposed robust numerical algorithms for accurate and fast computations, including a regularization to control reasonable parameter ranges, a Newton's method with Hessian function to determine search direction for fast convenience, and a modified Armijo rule to find a stable step length. We developed a Galerkin time-domain finite-element method for numerical solutions of resulting partial differential equations, and validated the model for a layered structure under dynamic loading tests. Results indicate that the proposed method has greatly improved computation accuracy and speed as compared to existing approaches. It may be applied to a broad range of heterogeneous materials and structures at different length and time scales.