Advancements In Finite Element Methods For Newtonian And Non-Newtonian Flows

This dissertation studies two important problems in the mathematics of computational fluid dynamics. The first problem concerns the accurate and efficient simulation of incompressible, viscous Newtonian flows, described by the Navier-Stokes equations. A direct numerical simulation of these types of flows is, in most cases, not computationally feasible. Hence, the first half of this work studies two separate types of models designed to more accurately and efficient simulate these flows. The second half focuses on the defective boundary problem for non-Newtonian flows. Non-Newtonian flows are generally governed by more complex modeling equations, and the lack of standard Dirichlet or Neumann boundary conditions further complicates these problems. We present two different numerical methods to solve these defective boundary problems for non-Newtonian flows, with application to both generalized-Newtonian and viscoelastic flow models. Chapter 3 studies a finite element method for the 3D Navier-Stokes equations in velocity- vorticity-helicity formulation, which solves directly for velocity, vorticity, Bernoulli pressure and helical density. The algorithm presented strongly enforces solenoidal constraints on both the velocity (to enforce the physical law for conservation of mass) and vorticity (to enforce the mathematical law that div(curl)= 0). We prove unconditional stability of the velocity, and with the use of a (consistent) penalty term on the difference between the computed vorticity and curl of the computed velocity, we are also able to prove unconditional stability of the vorticity in a weaker norm. Numerical experiments are given that confirm expected convergence rates, and test the method on a benchmark problem. Chapter 4 focuses on one main issue from the method presented in Chapter 3, which is the question of appropriate (and practical) vorticity boundary conditions. A new, natural vorticity boundary condition is derived directly from the Navier-Stokes equations. We propose a numerical scheme implementing this new boundary condition to evaluate its effectiveness in a numerical experiment. Chapter 5 derives a new, reduced order, multiscale deconvolution model. Multiscale deconvolution models are a type of large eddy simulation models, which filter out small energy scales and model their effect on the large scales (which significantly reduces the amount of degrees of freedom necessary for simulations). We present both an efficient and stable numerical method to approximate our new reduced order model, and evaluate its effectiveness on two 3d benchmark flow problems. In Chapter 6 a numerical method for a generalized-Newtonian fluid with flow rate boundary conditions is considered. The defective boundary condition problem is formulated as a constrained optimal control problem, where a flow balance is forced on the inflow and outflow boundaries using a Neumann control. The control problem is analyzed for an existence result and the Lagrange multiplier rule. A decoupling solution algorithm is presented and numerical experiments are provided to validate robustness of the algorithm. Finally, this work concludes with Chapter 7, which studies two numerical algorithms for viscoelastic fluid flows with defective boundary conditions, where only flow rates or mean pressures are prescribed on parts of the boundary. As in Chapter 6, the defective boundary condition problem is formulated as a minimization problem, where we seek boundary conditions of the flow equations which yield an optimal functional value. Two different approaches are considered in developing computational algorithms for the constrained optimization problem, and results of numerical experiments are presented to compare performance of the algorithms

Multiscale Fluid-Structure Interaction Models Development and Applications to the 3D Elements of a Human Cardiovascular System

Cardiovascular diseases (CVD) are the number one cause of death of humans in the United States and worldwide. Accurate, non-invasive, and cheaper diagnosis methods have always been on demand as cardiovascular monitoring increase in prevalence. The primary causes of the various forms of these CVDs are atherosclerosis and aneurysms in the blood vessels. Current noninvasive methods (i.e., statistical/medical) permit fairly accurate detection of the disease once clinical symptoms are suggestive of the existence of hemodynamic disorders. Therefore, the recent surge of hemodynamics models facilitated the prediction of cardiovascular conditions. The hemodynamic modeling of a human circulatory system involves varying levels of complexity which must be accounted for and resolved. Pulse-wave propagation effects and high aspect-ratio segments of the vasculature are represented using a quasi-one-dimensional (1D), non-steady, averaged over the cross-section models. However, these reduced 1D models do not account for the blood flow patterns (recirculation zones), vessel wall shear stresses and quantification of repetitive mechanical stresses which helps to predict a vessel life. Even a whole three-dimensional (3D) modeling of the vasculature is computationally intensive and do not fit the timeline of practical use. Thus the intertwining of a quasi 1D global vasculature model with a specific/risk-prone 3D local vessel ones is imperative. This research forms part of a multiphysics project that aims to improve the detailed understanding of the hemodynamics by investigating a computational model of fluid-structure interaction (FSI) of in vivo blood flow. First idealized computational a 3D FSI artery model is configured and executed in ANSYS Workbench, forming an implicit coupling of the blood flow and vessel walls. Then the thesis focuses on an approach developed to employ commercial tools rather than in-house mathematical models in achieving multiscale simulations. A robust algorithm is constructed to combine stabilization techniques to simultaneously overcome the added-mass effect in 3D FSI simulation and mathematical difficulties such as the assignment of boundary conditions at the interface between the 3D-1D coupling. Applications can be of numerical examples evaluating the change of hemodynamic parameters and diagnosis of an abdominal aneurysm, deep vein thrombosis, and bifurcation areas

Partitioned Algorithms for Fluid-Structure Interaction Problems in Haemodynamics

We consider the fluid-structure interaction problem arising in haemodynamic applications. The finite elasticity equations for the vessel are written in Lagrangian form, while the Navier-Stokes equations for the blood in Arbitrary Lagrangian Eulerian form. The resulting three fields problem (fluid/ structure/ fluid domain) is formalized via the introduction of three Lagrange multipliers and consistently discretized by p-th order backward differentiation formulae (BDFp). We focus on partitioned algorithms for its numerical solution, which consist in the successive solution of the three subproblems. We review several strategies that all rely on the exchange of Robin interface conditions and review their performances reported recently in the literature. We also analyze the stability of explicit partitioned procedures and convergence of iterative implicit partitioned procedures on a simple linear FSI problem for a general BDFp temporal discretization

Computational methods in cardiovascular mechanics

The introduction of computational models in cardiovascular sciences has been progressively bringing new and unique tools for the investigation of the physiopathology. Together with the dramatic improvement of imaging and measuring devices on one side, and of computational architectures on the other one, mathematical and numerical models have provided a new, clearly noninvasive, approach for understanding not only basic mechanisms but also patient-specific conditions, and for supporting the design and the development of new therapeutic options. The terminology in silico is, nowadays, commonly accepted for indicating this new source of knowledge added to traditional in vitro and in vivo investigations. The advantages of in silico methodologies are basically the low cost in terms of infrastructures and facilities, the reduced invasiveness and, in general, the intrinsic predictive capabilities based on the use of mathematical models. The disadvantages are generally identified in the distance between the real cases and their virtual counterpart required by the conceptual modeling that can be detrimental for the reliability of numerical simulations.Comment: 54 pages, Book Chapte