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The book is mainly addressed to young graduate students in engineering and natural sciences who start to face numerical simulation, either at a research level or in the field of industrial applications. The main subjects covered are: Biomechanics, Stochastic Calculus, Geophysical flow simulation and Shock-Capturing numerical methods for Hyperbolic Systems of Partial Differential Equations. The book can also be useful to researchers or even technicians working at an industrial environment, who are interested in the state-of-the-art numerical techniques in these fields. Moreover, it gives an overview of the research developed at the French and Spanish universities and in some European scientific institutions. This book can be also useful as a textbook at master courses in Mathematics, Physics or Engineering.





Fundamental Aspects in Modelling the Constitutive Behaviour of Fibered Soft Tissues

Fibered soft tissues like ligament, tendons, cartilage or those composing the cardiovascular system among others are characterized by a complex behaviour derived from their specific internal composition and architecture that has to be considered when trying to simulate their response under physiological or pathological external loads, their interaction with external implants or during and after surgery. The evaluation of the acting stresses and strains on these tissues is essential in predicting possible failure (i.e., aneurisms, atherosclerotic plaques, ligaments rupture) or the evolution of their microstructure under changing mechanical environment (i.e. cardiac aging, atherosclerosis, ligament remodeling). As structural materials, fibered soft tissues undergo large deformations even under physiological loads and are almost incompressible and highly anisotropic, mainly due to the directional distribution of the different composing families of collagen fibers. In addition, they are non-linearly elastic under slowly-acting loads, viscoelastic, due both to the moving internal fluid in some tissues (i.e. cartilage) or to the inherent viscoelasticity of the solid matrix. They are also subjected to non-negligible initial stresses due to the growth and remodeling processes that act along their whole live. Finally, they are susceptible to suffer damage induced by the rupture of the fibers or tearing of the surrounding matrix. All these aspects should be considered for a full description of the constitutive behaviour of these materials, requiring an appropriate mathematical formulation and finite element implementation to get efficient simulations useful for a better understanding of their phsyiological function, the effect of pathologies or surgery as well as for surgery planning and design of implants among many other usual applications. In this work, formulations of all the different phenomena commented above in fibered soft tissues are presented. The effect of each of these aspects is analyzed in simplified examples to demonstrate the applicability of the models. Finally, different applications of clinical interest are discussed.
Begoña Calvo, Estefanía Peña

Some Remarks on Avalanches Modelling: An Introduction to Shallow Flows Models

The main goal of these notes is to present several depth-averaged models with application in granular avalanches. We begin by recalling the classical Saint-Venant or Shallow Water equations and present some extensions like the Saint-Venant–Exner model for bedload sediment transport. The first part is devoted to the derivation of several avalanche models of Savage–Hutter type, using a depth-averaging procedure of the 3D momentum and mass equations. First, the Savage–Hutter model for aerial avalanches is presented. Two other models for partially fluidized avalanches are then described: one in which the velocities of both the fluid and the solid phases are assumed to be equal, and another one in which both velocities are unknowns of the system. Finally, a Savage–Hutter model for submarine avalanches is derived. The second part is devoted to non-newtonian models, namely viscoplastic fluids. Indeed, a one-phase viscoplastic model can also be used to simulate fluidized avalanches. A brief introduction to Rheology and plasticity is presented in order to explain the Herschel–Bulkley constitutive law. We finally present the derivation of a shallow Herschel–Bulkley model.
Enrique D. Fernández-Nieto, Paul Vigneaux

Introduction to Stochastic Calculus and to the Resolution of PDEs Using Monte Carlo Simulations

I give a pedagogical introduction to Brownian motion, stochastic calculus introduced by Itô in the fifties, following the elementary (at least not too technical) approach by Föllmer [Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French), pp. 143–150. Springer, Berlin, 1981]. Based on this, I develop the connection with linear and semi-linear parabolic PDEs. Then, I provide and analyze some Monte Carlo methods to approximate the solution to these PDEs. This course is aimed at master students, Ph.D. students and researchers interesting in the connection of stochastic processes with PDEs and their numerical counterpart. The reader is supposed to be familiar with basic concepts of probability (say first chapters of the book Probability essentials by Jacod and Protter [Probability Essentials, 2nd edn. Springer, Berlin, 2003]), but no a priori knowledge on martingales and stochastic processes is required.
Emmanuel Gobet

Structure-Preserving Shock-Capturing Methods: Late-Time Asymptotics, Curved Geometry, Small-Scale Dissipation, and Nonconservative Products

We consider weak solutions to nonlinear hyperbolic systems of conservation laws arising in compressible fluid dynamics and we describe recent work on the design of structure-preserving numerical methods. We focus on preserving, on one hand, the late-time asymptotics of solutions and, on the other hand, the geometrical effects that arise in certain applications involving curved space. First, we study here nonlinear hyperbolic systems with stiff relaxation in the late time regime. By performing a singular analysis based on a Chapman–Enskog expansion, we derive an effective system of parabolic type and we introduce a broad class of finite volume schemes which are consistent and accurate even for asymptotically late times. Second, for nonlinear hyperbolic conservation laws posed on a curved manifold, we formulate geometrically consistent finite volume schemes and, by generalizing the Cockburn–Coquel–LeFloch theorem, we establish the strong convergence of the approximate solutions toward entropy solutions.
Philippe G. LeFloch



Gradient Calculus for a Class of Optimal Design Problems in Engineering

This chapter reviews some recent works in which the analysis and control of partial differential equations are applied to optimal design in some problems appearing in aerodynamics and elasticity. From a mathematical point of view, the idea is to apply a descent algorithm to a cost functional defined on a part of the boundary. More specifically, we focus here on problems where the cost functional is defined on the part of the boundary to be optimized. This is the case, for instance, when the goal is to improve the lift or the drag in aerodynamic problems or to uniformize the tangential stresses along the boundary of a elastic material.
Carlos Castro

Medical Image Processing: Mathematical Modelling and Numerical Resolution

Medical image processing is an interdisciplinary research field attracting expertise from applied mathematics, computer sciences, engineering, statistics, physics, biology and medicine. In this context we shall present an introduction to basic techniques and concepts as well as more advanced methods to promote interests for further study and research in the field.
Emanuele Schiavi, Juan Francisco Garamendi, Adrián Martín

On Probabilistic Analytical and Numerical Approaches for Divergence Form Operators with Discontinuous Coefficients

In this paper we review some recent results on stochastic analytical and numerical approaches to parabolic and elliptic partial differential equations involving a divergence form operator with a discontinuous coefficient and a compatibility transmission condition.
In the one-dimensional case existence and uniqueness results for such PDEs can be obtained by stochastic methods. The probabilistic interpretation of the solutions allows one to develop and analyze a low complexity Monte Carlo numerical resolution method. In addition, it allows to get accurate pointwise estimates for the derivatives of the solutions from which sharp convergence rate estimates are deduced for the stochastic numerical method.
A stochastic approach is also developed for the linearized Poisson–Boltzmann equation in Molecular Dynamics. As in the one-dimensional case, the probabilistic interpretation of the solution involves the solution of a SDE including a non standard local time term related to the discontinuity interface. We present an extended Feynman–Kac formula for the Poisson–Boltzmann equation. This formula justifies various probabilistic numerical methods to approximate the free energy of a molecule and bases error analyzes.
We finally present probabilistic interpretations of the non-linearized Poisson–Boltzmann equation in terms of backward stochastic differential equations.
Denis Talay
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