The LifeV library: engineering mathematics beyond the proof of concept

LifeV is a library for the finite element (FE) solution of partial differential equations in one, two, and three dimensions. It is written in C++ and designed to run on diverse parallel architectures, including cloud and high performance computing facilities. In spite of its academic research nature, meaning a library for the development and testing of new methods, one distinguishing feature of LifeV is its use on real world problems and it is intended to provide a tool for many engineering applications. It has been actually used in computational hemodynamics, including cardiac mechanics and fluid-structure interaction problems, in porous media, ice sheets dynamics for both forward and inverse problems. In this paper we give a short overview of the features of LifeV and its coding paradigms on simple problems. The main focus is on the parallel environment which is mainly driven by domain decomposition methods and based on external libraries such as MPI, the Trilinos project, HDF5 and ParMetis. Dedicated to the memory of Fausto Saleri.Comment: Review of the LifeV Finite Element librar

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

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

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

Sobolev gradients and image interpolation

We present here a new image inpainting algorithm based on the Sobolev gradient method in conjunction with the Navier-Stokes model. The original model of Bertalmio et al is reformulated as a variational principle based on the minimization of a well chosen functional by a steepest descent method. This provides an alternative of the direct solving of a high-order partial differential equation and, consequently, allows to avoid complicated numerical schemes (min-mod limiters or anisotropic diffusion). We theoretically analyze our algorithm in an infinite dimensional setting using an evolution equation and obtain global existence and uniqueness results as well as the existence of an $\omega$-limit. Using a finite difference implementation, we demonstrate using various examples that the Sobolev gradient flow, due to its smoothing and preconditioning properties, is an effective tool for use in the image inpainting problem

Numerical simulation of electrophoresis separation processes

A new Petrov-Galerkin finite element formulation has been proposed for transient convection-diffusion problems. Most Petrov-Galerkin formulations take into account the spatial discretization, and the weighting functions so developed give satisfactory solutions for steady state problems. Though these schemes can be used for transient problems, there is scope for improvement. The schemes proposed here, which consider temporal as well as spatial discretization, provide improved solutions. Electrophoresis, which involves the motion of charged entities under the influence of an applied electric field, is governed by equations similiar to those encountered in fluid flow problems, i.e., transient convection-diffusion equations. Test problems are solved in electrophoresis and fluid flow. The results obtained are satisfactory. It is also expected that these schemes, suitably adapted, will improve the numerical solutions of the compressible Euler and the Navier-Stokes equations

A velocity tracking approach for the Data Assimilation problem in blood flow simulations

preprintSeveral advances have been made in Data Assimilation techniques applied to blood flow modeling. Typically,idealized boundary conditions, only verified in straight parts of the vessel, are assumed. We present ageneral approach, based on a Dirichlet boundary control problem, that may potentially be used in differentparts of the arterial system. The relevance of this method appears when computational reconstructions ofthe 3D domains, prone to be considered sufficiently extended, are either not possible, or desirable, due tocomputational costs. Based on taking a fully unknown velocity profile as the control, the approach uses adiscretize then optimize methodology to solve the control problem numerically. The methodology is appliedto a realistic 3D geometry representing a brain aneurysm. The results show that this DA approach may bepreferable to a pressure control strategy, and that it can significantly improve the accuracy associated totypical solutions obtained using idealized velocity profilesinfo:eu-repo/semantics/submittedVersio