Using a DPG method to validate DMA experimental calibration of viscoelastic materials

Abstract: A discontinuous Petrov-Galerkin (DPG) method is used to solve the time-harmonic equations of linear viscoelasticity. It is based on a "broken" primal variational formulation, which is very similar to the classical primal variational formulation used in Galerkin methods, but has additional "interface" variables at the boundaries of the mesh elements. Both the classical and broken formulations are proved to be well-posed in the infinite-dimensional setting, and the resulting discretization is proved to be stable. A full $hp$-convergence analysis is also included, and the analysis is verified using computational simulations. The method is particularly useful as it carries its own natural arbitrary-$p$ a posteriori error estimator, which is fundamental for solving problems with localized solution features. This proves to be useful when validating calibration models of dynamic mechanical analysis (DMA) experiments. Indeed, different DMA experiments of epoxy and silicone resins were successfully validated to within $5\%$ of the quantity of interest using the numerical method.