1,229 research outputs found
Reduced order methods for modeling and computational reduction
No full textThis monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics. Several topics are covered, including: design, optimization, and control theory in real-time with applications in engineering; data assimilation, geometry registration, and parameter estimation with special attention to real-time computing in biomedical engineering and computational physics; real-time visualization of physics-based simulations in computer science; the treatment of high-dimensional problems in state space, physical space, or parameter space; the interactions between different model reduction and dimensionality reduction approaches; the development of general error estimation frameworks which take into account both model and discretization effects. This book is primarily addressed to computational scientists interested in computational reduction techniques for large scale differential problems
Analysis of the Yosida method for the incompressible Navier–Stokes equations
No full textThe Yosida method was introduced in (Quarteroni et al., to appear) for the numerical approximation of the incompressible unsteady Navier–Stokes equations. From the algebraic viewpoint, it can be regarded as an inexact factorization of the matrix arising from the space and time discretization of the problem. However, its differential interpretation resides on an elliptic stabilization of the continuity equation through the Yosida regularization of the Laplacian (see (Brezis, 1983, Ciarlet and Lions, 1991)). The motivation of this method as well as an extensive numerical validation were given in (Quarteroni et al., to appear).
In this paper we carry out the analysis of this scheme. In particular, we consider a first-order time advancing unsplit method. In the case of the Stokes problem, we prove unconditional stability and moreover that the splitting error introduced by the Yosida scheme does not affect the overall accuracy of the solution, which remains linear with respect to the time step. Some numerical experiments, for both the Stokes and Navier–Stokes equations, are presented in order to substantiate our theoretical results
M for Models
No full textIn this short notice I illustrate the role of mathematical models and their impact on our daily life. In particular I provide a couple of instances of applications: one to the enhancement of sports performances, the others to the improvement of our knowledge of the human cardio-circulatory system
Mathematics and food: a tasty binomium
No full textThe pleasure of eating, the art of cuisine, the science of nutrition,
and the technology for food preparation, represent various facets of the most
basic of human needs, that of finding every day the energy to supply to our
body. Food processing has for a long time evolved from an artisanal activity
to large industry, with a progressive involvement of multinational factories
operating at a planetary level. Surprisingly as it may be, over the past few
years, a tight bond has been consolidated between the food industry and
mathematics, i.e., the science that has always been (erroneously!) considered
as the farthest from the primary human needs
Analysis of a geometrical Multiscale Model based on the coupling of ODE's and PDE's for blood flow simulations
No full textIn hemodynamics, local phenomena, such as the perturbation of flow pattern in a specific vascular region, are strictly related to the global features of the whole circulation (see, e.g., [L. Formaggia et al., Comput. Vis. Sci., 2 (1999), pp. 75--83]). In [A. Quarteroni, S. Ragni, and A. Veneziani, Comput. Vis. Sci., 4 (2001), pp. 111--124] we have proposed a heterogeneous model, where a local, accurate, three-dimensional description of blood flow by means of the Navier--Stokes equations in a specific artery is coupled with a systemic, zero-dimensional, lumped model of the remainder of circulation. This is a geometrical multiscale strategy, which couples an initial-boundary value problem to be used in a specific vascular region with an initial-value problem in the rest of the circulatory system. It has been successfully adopted to predict the outcome of a surgical operation (see [K. Laganà et al., Biorheology, 39 (2002), pp. 359--364, G. Dubini et al., Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, Spain, 2000]). However, its interest goes beyond the context of blood flow simulations. In this paper we provide a well-posedness analysis of this multiscale model by proving a local-in-time existence result based on a fixed-point technique. Moreover, we investigate the role of matching conditions between the two submodels for the numerical simulation
Modeling the heart and the circulatory system
No full textThe book comprises contributions by some of the most respected scientists in the field of mathematical modeling and numerical simulation of the human cardiocirculatory system. The contributions cover a wide range of topics, from the preprocessing of clinical data to the development of mathematical equations, their numerical solution, and both in-vivo and in-vitro validation. They discuss the flow in the systemic arterial tree and the complex electro-fluid-mechanical coupling in the human heart. Many examples of patient-specific simulations are presented. This book is addressed to all scientists interested in the mathematical modeling and numerical simulation of the human cardiocirculatory system
Numerical models for differential problems
No full textIn this text, we introduce the basic concepts for the numerical modelling of partial differential equations. We consider the classical elliptic, parabolic and hyperbolic linear equations, but also the diffusion, transport, and Navier-Stokes equations, as well as equations representing conservation laws, saddle-point problems and optimal control problems. Furthermore, we provide numerous physical examples which underline such equations. We then analyze numerical solution methods based on finite elements, finite differences, finite volumes, spectral methods and domain decomposition methods, and reduced basis methods.
In particular, we discuss the algorithmic and computer implementation aspects and provide a number of easy-to-use programs.
The text does not require any previous advanced mathematical knowledge of partial differential equations: the absolutely essential concepts are reported in a preliminary chapter. It is therefore suitable for students of bachelor and master courses in scientific disciplines, and recommendable to those researchers in the academic and extra-academic domain who want to approach this interesting branch of applied mathematics
A mathematical model of the human heart
No full textIn this paper, we present a mathematical model able to simulate the cardiac function. We first describe the basic physical principles behind the mathematical equations, then we illustrate a few examples of application to problems of clinical relevance.CMCSCI
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