Publications by Luca Formaggia

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Mathematical modeling of human physiopathology is a tremendously ambitious task. It encompasses the modeling of most diverse compartments such as the cardiovascular, respiratory, skeletal and nervous systems, as well as the mechanical and biochemical interaction between blood flow and arterial walls, or electrocardiac processes and the electric conduction into biological tissues. Mathematical models can be set up to simulate both vasculogenesis (the aggregation and organisation of endothelial cells disperse

Cardiovascular diseases have a major impact in Western countries. Mathematical models and numerical simulations can help the understanding of physiological and pathological processes, complementing the information provided to medical doctors by medical imaging and other non-invasive means, and opening the possibility of a better diagnosis and more in-depth surgical planning.This book offers a mathematically sound and up-to-date foundation to the training of researchers, and serves as a useful reference for

In this work, we address the numerical solution of fluid-structure interaction problems. This issue is particularly difficulty to tackle when the fluid and the solid densities are of the same order, for instance as it happens in hemodynamic applications, since fully implicit coupling schemes are required to ensure stability of the resulting method. Thus, at each time step, we have to solve a highly nonlinear coupled system, since the fluid domain depends on the unknown displacement of the structure. Standar

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-k

We prove a priori anisotropic estimates for the L2 and H1 interpolation error on linear finite elements. The full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions. This new H1 error estimate does not require the "maximal angle condition". The analysis has been carried out for the 2D case, but may be extended to three dimensions. Numerical experiments have been carried out t

Parallel computation of blood flow in a 1D model of human arterial network has been carried out employing a Taylor Galerkin finite element method. Message passing interface libraries have been used on Origin 2000 SGI machine. A Greedy strategy for load-distribution has been devised and data-flow graphs necessary for parallelization have been generated. The performance of parallel implementation measured in terms of speedup and efficiency factors is found to be good. Further, the parallel code is used in sim

In this work we discuss the reliability of the coupling among three-dimensional (3D) and one-dimensional (1D) models that describe blood flowing into the circulatory tree. In particular, we study the physical consistency of the 1D model with respect to the ...

Moving from the anisotropic interpolation error estimates derived in Formaggia and Perotto, (2001), we provide here both a-priori and a-posteriori estimates for a generic elliptic problem. The a-priori result is deduced by following the standard finite element theory. For the a-posteriori estimate, the analysis extends to anisotropic meshes the theory presented in Becker and Rannacher, (1995, 1996, 1999). Numerical test-cases validate the derived results

We investigate a one-dimensional model of blood flow in human arteries. In particular, we consider the case when an abrupt variation of mechanical characteristic of artery is caused by the presence of a vascular prosthesis. The derivations of the model and of a numerical scheme adopted for its solution are detailed. Numerical experiments show the effectiveness of the model.

We investigate some domain decomposition techniques to solve large scale aerodynamics problems on unstructured grids. Where implicit time advancing scheme are used, a large sparse linear system have to be solved at each step. To obtain good scalability and CPU times, a good preconditioner is needed for the parallel iterative solution of these systems. For the widely-used Schwarz technique this can be achieved by a coarse level operator. Since many of the current coarse operators are difficult to implement o

We develop an a-posteriori error estimation for the finite element simulation of an advection-diffusion problem with anisotropic meshes. The error analysis is carried out by exploiting some recent interpolation error estimates for anisotropic meshes within the a-posteriori analysis framework proposed by Becker and Rannacher (1995, 1996, 1997) and Rannacher (1998). The target application is the transport of a solute (like oxygen or lipids) by the blood stream in a large artery

Applications of mesh adaptation techniques could be found in the numerical solution of PDE's or in the optimal triangulation of surfaces for shape representation or graphic display. The scope of this work is to verify through numerical experiments the effectiveness of some algorithms for the control of the L^2 error norm for piecewise linear approximation on 2D unstructured triangular meshes. The analysis could be extended to parametric surfaces and to the 3D case.

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 faciliti ...

Luca Formaggia

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Complex Systems in Biomedicine

Cardiovascular Mathematics

Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions

One-dimensional models for blood flow in arteries

New anisotropic a priori error estimates

On parallel computation of blood flow in human arterial network based on 1-D modelling

On the physical consistency between three-dimensional and one-dimensional models in haemodynamics

Anisotropic error estimates for elliptic problems

A one-dimensional model for blood flow: Application to vascular prosthesis

Algebraic coarse grid operators for domain decomposition based preconditioners

An anisotropic a-posteriori error estimate for a convection-diffusion problem

Function approximation on triangular grids: some numerical results using adaptive techniques

The LifeV library: engineering mathematics beyond the proof of concept

Advances on numerical modelling of blood flow problems

The circulatory system: from case studies to mathematical modeling

Anisotropic estimation for finite element methods

Numerical simulation of the arterial pulse propagation using one-dimensional models

Error estimation for finite element methods

Mathematical Modelling and Numerical Simulation of the Cardiovascular System