Fluid-Structure Interaction Problems in Hemodynamics:Parallel Solvers, Preconditioners, and Applications

In this work we aim at the description, study and numerical investigation of the fluid-structure interaction (FSI) problem applied to hemodynamics. The FSI model considered consists of the Navier-Stokes equations on moving domains modeling blood as a viscous incompressible fluid and the elasticity equation modeling the arterial wall. The fluid equations are derived in an arbitrary Lagrangian-Eulerian (ALE) frame of reference. Several existing formulations and discretizations are discussed, providing a state of the art on the subject. The main new contributions and advancements consist of: A description of the Newton method for FSI-ALE, with details on the implementation of the shape derivatives block assembling, considerations about parallel performance, the analytic derivation of the derivative terms for different formulations (conservative or not) and for different types of boundary conditions. The implementation and analysis of a new category of preconditioners for FSI (applicable also to more general coupled problems). The framework set up is general and extensible. The proposed preconditioners allow, in particular, a separate treatment of each field, using a different preconditioning strategy in each case. An estimate for the condition number of the preconditioned system is proposed, showing how preconditioners of this type depend on the coupling, and explaining the good performance they exhibit when increasing the number of processors. The improvement of the free (distributed under LGPL licence) parallel finite elements library LifeV. Most of the methods described have been implemented within this library during the period of this PhD and all the numerical tests reported were run using this framework. The simulation of clinical cases with patient-specific data and geometry, the comparison on simulations of physiological interest between different models (rigid, FSI, 1D), discretizations and methods to solve the nonlinear system. A methodology to obtain patient-specific FSI simulations starting from the raw medical data and using a set of free software tools is described. This pipeline from imaging to simulation can help medical doctors in diagnosis and decision making, and in understanding the implication of indicators such as the wall shear stress in the pathogenesis

Development of a parallel CFD solver with application to arterial flows.

In this research, the finite element method (FEM) was used to solve the nonlinear, incompressible, transient, three dimensional Navier-Stokes equations in their non-conservative form. Linear tetrahedron elements were employed with the elegant, equal order interpolation for both pressure and velocity. The characteristic based split scheme was formulated in a fully implicit manner to circumvent the time step restrictions of the classical explicit formulations. The monolithic (single step, fully coupled solution procedure for pressures and velocity) form of the CBS scheme was also derived and its suitability was positively demonstrated. Casting the CBS scheme in a monolithic framework, results in the generation of a pressure stabilization term in the mass conservation equation, thereby circumventing the LBB restriction by the elimination of the zero pressure block. An account of all the steps involved in discretizing the Navier-Stokes equations (both in split and monolithic frameworks) was presented in meticulous detail, which included the derivation of the convective and pressure stabilization terms, linearization of the non-linear terms and the consequent derivation of the highly efficient analytical jacobian matrix, along with the temporal and spatial discretizations of the corresponding terms. The monolithic and the split version of the CBS scheme were integrated into a parallel, scalable and extensible Fortran90 software called IFENs. The development of IFENs started during the course of this research and all of its components have been designed and implemented by the author of this thesis. Multi processor parallelism was achieved using the Intel implementation of the most widely used/preferred, Message Passing Interface (MPI) standard. The parallel support needed for the use of a variety of parallel, linear, iterative solvers belonging to the Krylov subspace family (e.g. GMRES and its variants, CG, BiCG, BiCG- stab, etc.), parallel non linear solvers belonging to the Newton-Krylov family (line search newton, trust region newton, nonlinear GMRES, etc.) and parallel preconditioners (incomplete LU, Additive Shwarz Method - ASM, algebraic multigrid, etc.), was provided by the incorporation of PETSc into IFENs. PETSc is a state of the art, non-trivial toolkit, which represents a collection of several parallel libraries useful in high performance scientific computing. Keeping in mind the specific requirements of IFENs, a custom mesh partitioner was implemented. It operated on meshes that were renumbered using bandwidth reducing algorithms like Revere Cuthill Mckee. The possibility of using established domain decomposition libraries like ParMETIS was explored and demonstrated to be counter productive for the demands of this research. After the preliminary testing and validation of the procedures adopted before and during the execution of IFENs, large, high definition domains representative of human arteries (specifically, carotid bifurcations, found in the neck) were considered and the complete incompressible set of Navier-Stokes equations were solved for pressure and velocity fields. During the tenure of this research more than 1000 recorded parallel test cases were executed to test various components of IFENs, as well as various simulations representative of a wide variety of problems. IFENs can easily handle meshes with tens of millions of elements. The largest mesh used for the purpose of this research contained 14.58 million tetrahedrons and 2.489 million nodes, which on average required just 7 minutes per timestep, while executing the classical split framework of the CBS scheme. Results from the simulation of 9 carotid meshes, representative of 4 carotid geometries were presented and found to be in good agreement with the available ultrasound data. The flow fields were analysed and post processed using different techniques for each case. The haemodynamic wall parameters like time averaged wall shear stress and oscillatory shear index were calculated and mapped onto the corresponding boundary nodes. The region in the carotid bifurcation susceptible to the deposition of plaques and consequent stenosis were pointed out and other anomalies were highlighted

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Microgravity Science and Applications: Program Tasks and Bibliography for Fiscal Year 1996

NASA's Microgravity Science and Applications Division (MSAD) sponsors a program that expands the use of space as a laboratory for the study of important physical, chemical, and biochemical processes. The primary objective of the program is to broaden the value and capabilities of human presence in space by exploiting the unique characteristics of the space environment for research. However, since flight opportunities are rare and flight research development is expensive, a vigorous ground-based research program, from which only the best experiments evolve, is critical to the continuing strength of the program. The microgravity environment affords unique characteristics that allow the investigation of phenomena and processes that are difficult or impossible to study an Earth. The ability to control gravitational effects such as buoyancy driven convection, sedimentation, and hydrostatic pressures make it possible to isolate phenomena and make measurements that have significantly greater accuracy than can be achieved in normal gravity. Space flight gives scientists the opportunity to study the fundamental states of physical matter-solids, liquids and gasses-and the forces that affect those states. Because the orbital environment allows the treatment of gravity as a variable, research in microgravity leads to a greater fundamental understanding of the influence of gravity on the world around us. With appropriate emphasis, the results of space experiments lead to both knowledge and technological advances that have direct applications on Earth. Microgravity research also provides the practical knowledge essential to the development of future space systems. The Office of Life and Microgravity Sciences and Applications (OLMSA) is responsible for planning and executing research stimulated by the Agency's broad scientific goals. OLMSA's Microgravity Science and Applications Division (MSAD) is responsible for guiding and focusing a comprehensive program, and currently manages its research and development tasks through five major scientific areas: biotechnology, combustion science, fluid physics, fundamental physics, and materials science. FY 1996 was an important year for MSAD. NASA continued to build a solid research community for the coming space station era. During FY 1996, the NASA Microgravity Research Program continued investigations selected from the 1994 combustion science, fluid physics, and materials science NRAS. MSAD also released a NASA Research Announcement in microgravity biotechnology, with more than 130 proposals received in response. Selection of research for funding is expected in early 1997. The principal investigators chosen from these NRAs will form the core of the MSAD research program at the beginning of the space station era. The third United States Microgravity Payload (USMP-3) and the Life and Microgravity Spacelab (LMS) missions yielded a wealth of microgravity data in FY 1996. The USMP-3 mission included a fluids facility and three solidification furnaces, each designed to examine a different type of crystal growth