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

## TL;DR

This paper introduces a discontinuous Petrov-Galerkin (DPG) method for solving viscoelastic equations, demonstrating its stability, convergence, and effectiveness in validating DMA experimental calibration of viscoelastic materials.

## Contribution

The paper develops a stable, convergent DPG method with an a posteriori error estimator for viscoelastic problems, applied to validate DMA calibration models.

## Key findings

- Successfully validated DMA experiments within 5% accuracy
- Proved well-posedness and stability of the DPG formulation
- Verified $hp$-convergence through computational simulations

## 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.