Alfio Quarteroni

On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels

For the analysis of flows in compliant vessels, we propose an approach to couple the original 3D equations with a convenient 1D model. This multiscal- e strategy allows for a dramatic reduction of the computational complexity and is suitable for «absorbing» outgoing pressure waves. In particular, it is of utmost interest for the description of blood motion in the arterial system.

Mathematical and numerical models for coupling surface and groundwater flows

In this paper we present some results about the coupling of Navier-Stokes and Shallow Water equations for surface flows and Darcys equation for groundwater flows. We discuss suitable interface conditions and show the well-posedness of the coupled problem in the case of a linear Stokes problem. An iterative method to compute the solution is proposed. At each step this method requires the solution of one problem in the fluid part and one in the porous medium. Finally we introduce the Steklov-Poincare equation associated to the coupled problem.

One-dimensional models for blood flow in arteries

In this paper a family of one-dimensional nonlinear systems which model the blood pulse propagation in compliant arteries is presented and investigated. They are obtained by averaging the Navier-Stokes equation on each section of an arterial vessel and using simplified models for the vessel compliance. Different differential operators arise depending on the simplifications made on the structural model. Starting from the most basic assumption of pure elastic instantaneous equilibrium, which provides a well-known algebraic relation between intramural pressure and vessel section area, we analyse in turn the effects of terms accounting for inertia, longitudinal pre-stress and viscoelasticity. The problem of how to account for branching and possible discontinuous wall properties is addressed, the latter aspect being relevant in the presence of prosthesis and stents. To this purpose a domain decomposition approach is adopted and the conditions which ensure the stability of the coupling are provided. The numerical method here used in order to carry out several test cases for the assessment of the proposed models is based on a finite element Taylor-Galerkin scheme combined with operator splitting techniques.

2d and 3D elastic wave propagation by a pseudo-spectral domain decomposition method

A new numerical method is presented for propagating elastic waves in heterogeneous earth media, based on spectral approximations of the wavefield combined with domain decomposition techniques. The flexibility of finite element techniques in dealing with irregular geologic structures is preserved, together with the high accuracy of spectral methods. High computational efficiency can be achieved especially in 3D calculations, where the commonly used finite-difference approaches are limited both in the frequency range and in handling strongly irregular geometries. The treatment of the seismic source, introduced via a moment tensor distribution, is thoroughly discussed together with the aspects associated with its numerical implementation. The numerical results of the present method are successfully compared with analytical and numerical solutions, both in 2D and 3D.

Numerical Treatment of Defective Boundary Conditions for the Navier--Stokes Equations

We present a formulation for accommodating defective boundary conditions for the incompressible Navier--Stokes equations where only averaged values are prescribed on measurable portions of the boundary. In particular we consider the case where the flow rate is imposed on several domain sections. This methodology has an interesting application in the numerical simulation of flow in blood vessels, when only a reduced set of boundary data are generally available for the upstream and downstream sections.

A relaxation procedure for domain decomposition methods using finite elements

SummaryWe present the convergence analysis of a new domain decomposition technique for finite element approximations. This technique was introduced in [11] and is based on an iterative procedure among subdomains in which transmission conditions at interfaces are taken into account partly in one subdomain and partly in its adjacent. No global preconditioner is needed in the practice, but simply single-domain finite element solvers are required. An optimal strategy for an automatic selection of a relaxation parameter to be used at interface subdomains is indicated. Applications are given to both elliptic equations and incompressible Stokes equations.

An Iterative Procedure with Interface Relaxation for Domain Decomposition Methods

A domain decomposition method for second-order elliptic problems is considered. An iterative procedure that reduces the problem to a sequence of mixed boundary value problems on each subdomain is proposed. At each iteration, a relaxation is accomplished at the subdomain interfaces. In several circumstances, a value of the relaxation parameter that yields exact convergence in a finite number of iterations is explicitly found. Moreover, when such a value is not available, an appropriate strategy for the automatic selection of the relaxation parameter at each iteration is indicated and analyzed.This iterative method is then applied to the spectral collocation approximation of the differential problem. The same kind of convergence results are proven. Many numerical experiments show the effectiveness of the method proposed here.

Mathematical Modelling and Numerical Simulation of the Cardiovascular System

Publisher Summary The development of mathematical models, algorithms and numerical simulation tools for the investigation of the human cardiovascular system has received a great impulse in the past years. This chapter addresses the problem of developing models for the numerical simulation of the human circulatory system. It particularly focuses on the problem of hemodynamics in large human arteries. There are several important aspects, which require the use of sophisticated mathematical and numerical tools, such as the reconstruction of geometries from medical data; the transport of biochemicals in blood and vessel wall tissue; the heart dynamics; and blood rheology. Besides, the need of validating the models calls for development of accurate in-vivo measurement techniques. The number and complexity of the mathematical, numerical and technological problems involved makes the development of tools for accurate, reliable and efficient simulations of the human cardiovascular system one of the challenges of the next decades.

Factorization methods for the numerical approximation of Navier-Stokes equations

We investigate a general approach for the numerical approximation of incompressible Navier-Stokes equations based on splitting the original problem into successive subproblems which are cheaper to solve. The splitting is obtained through an algebraic approximate factorization of the matrix arising from space and time discretization of the original equations. Several schemes based on approximate factorization are investigated. For some of these methods a formal analogy with well known time advancing schemes, such as the projection Chorin-Temams, can be pointed out. Features and limits of this analogy (that was earlier introduced in B. Perot, J. Comp. Phys. 108 (1993) 51-8) are addressed. Other, new methods can also be formulated starting from this approach: in particular, we introduce the so called Yosida method, which can be investigated in the framework of quasi-compressibility schemes. Numerical results illustrating the different performances of the different methods addressed are presented for a couple of test cases

A vision and strategy for the virtual physiological human in 2010 and beyond

European funding under framework 7 (FP7) for the virtual physiological human (VPH) project has been in place now for nearly 2 years. The VPH network of excellence (NoE) is helping in the development of common standards, open-source software, freely accessible data and model repositories, and various training and dissemination activities for the project. It is also helping to coordinate the many clinically targeted projects that have been funded under the FP7 calls. An initial vision for the VPH was defined by framework 6 strategy for a European physiome (STEP) project in 2006. It is now time to assess the accomplishments of the last 2 years and update the STEP vision for the VPH. We consider the biomedical science, healthcare and information and communications technology challenges facing the project and we propose the VPH Institute as a means of sustaining the vision of VPH beyond the time frame of the NoE.