Computational Methods for Coupled Multiphysics Transport Problems
Advisor: Professor Arif Masud
Abstract
Coupled multiphysics problems involve multiple interacting physical phenomena that span a spectrum of spatial, material, and temporal scales. Examples include targeted drug delivery, convective heat transfer, and curing process of polymer materials, to name a few. These systems are modeled by coupling Navier-Stokes equations with advection-diffusion equations for scalar transport. Numerical methods for solving nonlinear equations face challenges such as nonlinearity of convective velocity, mathematical instabilities related to inf-sup condition, steep convective gradients causing numerical instabilities, cross-coupling effects leading to anisotropy, and high computational costs. To this end, this dissertation develops numerical methods that address the above technical bottlenecks by employing Variational Multiscale (VMS) framework and Physics Informed Neural Networks (PINNs).
The VMS approach embeds computational intelligence to the modeling framework by facilitating development of closure-models for the subgrid physics. The launching point is additive decomposition of unknown solution fields into coarse- and fine-scales, resulting in a hierarchically coupled system of equations. The coarse-scale equations capture the physics resolved by the numerical discretization, while the fine-scale equations operate on the residuals of the coarse system, facilitating the development of residual-based closure models. A systematic approach for variationally deriving analytical expressions for these closure models is developed. These models represent the missing physics, which are variationally injected into the coarse-scale equations, resulting in stabilized formulations with enhanced accuracy and stability properties. The final formulations feature a stabilization tensor that preserves solution field cross-coupling without relying on ad-hoc parameters. Implemented via the finite element method, these stabilized formulations achieve optimal mathematical convergence rates in space and time. Moreover, the consistently linearized stiffness matrix yields quadratic convergence rates in solving nonlinear coupled equations, thus achieving accuracy comparable to or better than other methods with reduced computational costs.
The fundamental nature of the aforementioned methods makes them suitable for a wide range of
applications, from engineering and physical sciences to biological science. In this dissertation, numerical methods were applied to simulate spatiotemporal dynamics of drug-coated nanoparticles in microvasculature and buoyancy-induced thermal convection in density-stratified flows. In each case, the momentum balance equations were coupled with a convection-diffusion equation for a scalar field—such as drug concentration or temperature—addressing unique challenges relating to stability, accuracy, and multiscale physics. The computational method provides an advanced strategy to optimize nanoparticle design for targeted drug delivery under various hemodynamic conditions in patient-specific arterial systems. The method also provides insights into the underlying physical mechanisms observed in experiments and enhancing the capabilities of the thermal management devices such as heat exchangers and air conditioners at reduced computational cost. Numerical results were validated against reference numerical and experimental data, and excellent agreement is achieved in all the cases.
In the end, this dissertation also presents a machine learning based surrogate modeling approach that can be employed in engineering design or clinical translation applications. A robust physics informed neural network is developed to solve thermally coupled incompressible Navier–Stokes equations. Accuracy in predicting the pressure field within coupled nonlinear equations presents a challenge for PINNs. A pressure stabilization term, formulated as a pressure Poisson equation, is proposed to augment the residuals for PINNs. This novel physics-informed augmentation improves the accuracy of the pressure field by an order of magnitude.
