Model reduction for driven PDEs: Application to polymer constitutive equations - Mellor, Daniel (2024)
In polymer dynamics, the direct derivation of equations for stress response is often very difficult due to complex dynamics arising from interactions between long-chain molecules. One useful approach is to map from an expensive molecular constitutive equation to a cheaper model using model reduction. There is an ever-growing need for computationally cheap polymer models as many important applications such as modelling polydispersity or computational fluid dynamics require a vast number of evaluations of the model. Currently, many of these applications use the Rolie-Poly model which has several known weaknesses. We develop a new data-driven way of carrying out model reduction for constitutive equations. This reduction is achieved by choosing a number of slow-moving variables as coarse-grained variables. A mapping between these coarse-grained variables and the full configuration is then created using a data-driven approach. With this mapping, we then evolve these coarse-grained variables, but not with directly derived differential equations. Instead, we utilise the mapping to map back to the full model and calculate derivatives using the original model. The result of this is a model that can take large timesteps but retains greater accuracy. Using this framework on the GLaMM model for polymer dynamics, we derive a new model reduction with the accuracy of the GLaMM model and the speed of the Rolie-Poly model, with the only minor limitation being the span of training data. This model is sufficiently fast to be adapted for use in many of the aforementioned applications.