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Publication# Implicit coupling of one-dimensional and three-dimensional blood flow models with compliant vessels

Alfio Quarteroni, Simone Deparis, Adelmo Cristiano Innocenza Malossi, Paolo Crosetto

2013

Journal paper

2013

Journal paper

Abstract

Simulating arterial trees in the cardiovascular system can be made by the help of different models, depending on the outputs of interest and the desired degree of accuracy. In particular, one-dimensional fluid-structure interaction models for arteries are very effective in reproducing the physiological pressure wave propagation and in providing quantities like pressure and velocity, averaged on the cross section of the arterial lumen. In locations where one-dimensional models cannot capture the complete flow dynamics, e.g., in presence of stenoses and aneurysms, three-dimensional coupled fluid-structure interaction models are necessary to evaluate, for instance, critical factors responsible for pathologies which are associated to hemodynamics. In this work we formalize and investigate the geometrical multiscale problem, where heterogeneous fluid-structure interaction models for arteries are implicitly coupled. We introduce new coupling algorithms, describe their implementation and investigate on simple geometries the numerical reflections that occur at the interface between the heterogeneous models. We also simulate on a supercomputer a three-dimensional abdominal aorta under physiological conditions, coupled with up to six one-dimensional models representing the surrounding arterial branches. Finally, we compare CPU times and number of coupling iterations for different algorithms and time discretizations.

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Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it - for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it - for example, both a latitude and longitude are required to locate a point on the surface of a sphere.

Three-dimensional space

In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, the Euclidean n-space of dimension n=3 that models physical space. More general three-dimensional spaces are called 3-manifolds. Technically, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n-dimensional Euclidean space.

One-dimensional space

In physics and mathematics, a sequence of n numbers can specify a location in n-dimensional space. When n = 1, the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the number line, where the position of each point on it can be described by a single number. In algebraic geometry there are several structures that are technically one-dimensional spaces but referred to in other terms. A field k is a one-dimensional vector space over itself.

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