Existence of a solution to the quasi-variational inequality problem arising in a modelfor sand surface evolution has been an open problem for a long time. Another long-standingopen problem concerns determining the dual variable, the flux of sand pouring down theevolving sand surface, which is also of practical interest in a variety of applications ofthis model. Previously, these problems were solved for the special case in which theinequality is simply variational. Here, we introduce a regularized mixed formulationinvolving both the primal (sand surface) and dual (sand flux) variables. We derive,analyse and compare two methods for the approximation, and numerical solution, of thismixed problem. We prove subsequence convergence of both approximations, as the meshdiscretization parameters tend to zero; and hence prove existence of a solution to thismixed model and the associated regularized quasi-variational inequality problem. One ofthese numerical approximations, in which the flux is approximated by thedivergence-conforming lowest order Raviart–Thomas element, leads to an efficient algorithmto compute not only the evolving pile surface, but also the flux of pouring sand. Resultsof our numerical experiments confirm the validity of the regularization employed.

We review the optimal design of an arterial bypass graft following either a (i) boundaryoptimal control approach, or a (ii) shape optimization formulation. The main focus isquantifying and treating the uncertainty in the residual flow when the hosting artery isnot completely occluded, for which the worst-case in terms of recirculation effects isinferred to correspond to a strong orifice flow through near-complete occlusion.Aworst-case optimal control approach is applied to the steady Navier-Stokes equations in 2Dto identify an anastomosis angle and a cuffed shape that are robust with respect to apossible range of residual flows. We also consider a reduced order modelling frameworkbased on reduced basis methods in order to make the robust design problem computationallyfeasible. The results obtained in 2D are compared with simulations in a 3D geometry butwithout model reduction or the robust framework.

We consider the solution of second order elliptic PDEs in Rdwith inhomogeneous Dirichlet data by means of an h–adaptive FEM withfixed polynomial order p ∈ N. As model example serves the Poissonequation with mixed Dirichlet–Neumann boundary conditions, where the inhomogeneousDirichlet data are discretized by use of an H1 / 2–stableprojection, for instance, the L2–projection forp = 1 or the Scott–Zhang projection for general p ≥ 1.For error estimation, we use a residual error estimator which includes the Dirichlet dataoscillations. We prove that each H1 / 2–stable projectionyields convergence of the adaptive algorithm even with quasi–optimal convergence rate.Numerical experiments with the Scott–Zhang projection conclude the work.

We analyze a numerical model for the Signorini unilateral contact, based on the mortarmethod, in the quadratic finite element context. The mortar frame enables one to usenon-matching grids and brings facilities in the mesh generation of different components ofa complex system. The convergence rates we state here are similar to those alreadyobtained for the Signorini problem when discretized on conforming meshes. The matching forthe unilateral contact driven by mortars preserves then the proper accuracy of thequadratic finite elements. This approach has already been used and proved to be reliablefor the unilateral contact problems even for large deformations. We provide however somenumerical examples to support the theoretical predictions.

Consider time-harmonic electromagnetic wave scattering from a biperiodic dielectricstructure mounted on a perfectly conducting plate in three dimensions. Given thatuniqueness of solution holds, existence of solution follows from a well-known Fredholmframework for the variational formulation of the problem in a suitable Sobolev space. Inthis paper, we derive a Rellich identity for a solution to this variational problem undersuitable smoothness conditions on the material parameter. Under additional non-trappingassumptions on the material parameter, this identity allows us to establish uniqueness ofsolution for all positive wave numbers.

In this article, we propose an integrated model for oxygen transfer into the blood,coupled with a lumped mechanical model for the ventilation process. Objectives.We aim at investigating oxygen transfer into the blood at rest or exercise. Thefirst task consists in describing nonlinear effects of the oxygen transfer under normalconditions. We also include the possible diffusion limitation in oxygen transfer observedin extreme regimes involving parameters such as alveolar and venous blood oxygen partialpressures, capillary volume, diffusing capacity of the membrane, oxygen binding byhemoglobin and transit time of the red blood cells in the capillaries. The second taskconsists in discussing the oxygen concentration heterogeneity along the path length in theacinus. Method. A lumped mechanical model is considered: a double-balloonmodel is built upon physiological properties such as resistance of the branches connectingalveoli to the outside air, and elastic properties of the surrounding medium. Then, wefocus on oxygen transfer: while the classical [F.J. Roughton and R.E. Forster, J.Appl. Physiol. 11 (1957) 290–302]. approach accounts for thereaction rate with hemoglobin by means of an extra resistance between alveolar air andblood, we propose an alternate description. Under normal conditions, the Hill’s saturationcurve simply quantifies the net oxygen transfer during the time that venous blood stays inthe close neighborhood of alveoli (transit time). Under degraded and/or exerciseconditions (impaired alveolar-capillary membrane, reduced transit time, high altitude)diffusion limitation of oxygen transfer is accounted for by means of the nonlinearequation representing the evolution of oxygen partial pressure in the plasma during thetransit time. Finally, a one-dimensional model is proposed to investigate the effects oflongitudinal heterogeneity of oxygen concentration in the respiratory tract during theventilation cycle, including previous considerations on oxygen transfer. Results.This integrated approach allows us to recover the right orders of magnitudes interms of oxygen transfer, at rest or exercise, by using well-documented data, without anyparameter tuning or curve fitting procedure. The diffusing capacity of thealveolar-capillary membrane does not affect the oxygen transfer rate in the normal regimebut, as it decreases (e.g. because of emphysema) below a critical value,it becomes a significant parameter. The one-dimensional model allows to investigate thescreening phenomenon, i.e. the possibility that oxygen transfer might besignificantly affected by the fact that the exchange area in the peripheral acinus poorlyparticipates to oxygen transfer at rest, thereby providing a natural reserve of transfercapacity for exercise condition. We do not recover this effect: in particular we showthat, at rest, although the oxygen concentration is slightly smaller in terminal alveoli,transfer mainly occurs in the acinar periphery.

A distributed-parameter (one-dimensional) anatomically detailed model for the arterialnetwork of the arm is developed in order to carry out hemodynamics simulations. This workfocuses on the specific aspects related to the model set-up. In this regard, stringentanatomical and physiological considerations have been pursued in order to construct thearterial topology and to provide a systematic estimation of the involved parameters. Themodel comprises 108 arterial segments, with 64 main arteries and 44 perforator arteries,with lumen radii ranging from 0.24 cm – axillary artery- to 0.018 cm – perforatorarteries. The modeling of blood flow in deformable vessels is governed by a well-known setof hyperbolic partial differential equations that accounts for mass and momentumconservation and a constitutive equation for the arterial wall. The variationalformulation used to solve the problem and the related numerical approach are described.The model rendered consistent pressure and flow rate outputs when compared with patientrecords already published in the literature. In addition, an application todimensionally-heterogeneous modeling is presented in which the developed arterial networkis employed as an underlying model for a three-dimensional geometry of a branching pointto be embedded in order to perform local analyses.

In vivo visualization of cardiovascular structures ispossible using medical images. However, one has to realize that the resulting 3Dgeometries correspond to in vivo conditions. This entails an internalstress state to be present in the in vivo measured geometry ofe.g. a blood vessel due to the presence of the blood pressure. In orderto correct for this in vivo stress, this paper presents an inverse methodto restore the original zero-pressure geometry of a structure, and to recover thein vivo stress field of the final, loaded structure. The proposedbackward displacement method is able to solve the inverse problem iteratively using fixedpoint iterations, but can be significantly accelerated by a quasi-Newton technique inwhich a least-squares model is used to approximate the inverse of the Jacobian. The hereproposed backward displacement method allows for a straightforward implementation of thealgorithm in combination with existing structural solvers, even if the structural solveris a black box, as only an update of the coordinates of the mesh needs to beperformed.

We consider optimal control problems for the bidomain equations of cardiacelectrophysiology together with two-variable ionic models, e.g. theRogers–McCulloch model. After ensuring the existence of global minimizers, we provide arigorous proof for the system of first-order necessary optimality conditions. The proof isbased on a stability estimate for the primal equations and an existence theorem for weaksolutions of the adjoint system.

The reliable and effective assimilation of measurements and numerical simulations inengineering applications involving computational fluid dynamics is an emerging problem assoon as new devices provide more data. In this paper we are mainly driven by hemodynamicsapplications, a field where the progressive increment of measures and numerical toolsmakes this problem particularly up-to-date. We adopt a Bayesian approach to the inclusionof noisy data in the incompressible steady Navier-Stokes equations (NSE). The purpose isthe quantification of uncertainty affecting velocity and flow related variables ofinterest, all treated as random variables. The method consists in the solution of anoptimization problem where the misfit between data and velocity - in a convenient norm -is minimized under the constraint of the NSE. We derive classical point estimators, namelythe maximum a posteriori – MAP – and the maximum likelihood – ML – ones.In addition, we obtain confidence regions for velocity and wall shear stress, a flowrelated variable of medical relevance. Numerical simulations in 2-dimensional andaxisymmetric 3-dimensional domains show the gain yielded by the introduction of a completestatistical knowledge in the assimilation process.

A second-order in time finite-difference scheme using a modified predictor–corrector method is proposed for the numerical solution of the generalized Burgers–Fisher equation. The method introduced, which, in contrast to the classical predictor–corrector method is direct and uses updated values for the evaluation of the components of the unknown vector, is also analysed for stability. Its efficiency is tested for a single-kink wave by comparing experimental results with others selected from the available literature. Moreover, comparisons with the classical method and relevant analogous modified methods are given. Finally, the behaviour and physical meaning of the two-kink wave arising from the collision of two single-kink waves are examined.

This paper is concerned with the initial boundary value problem of a class of nonlinear wave equations and reaction–diffusion equations with several nonlinear source terms of different signs. For the initial boundary value problem of the nonlinear wave equations, we derive a blow up result for certain initial data with arbitrary positive initial energy. For the initial boundary value problem of the nonlinear reaction–diffusion equations, we discuss some probabilities of the existence and nonexistence of global solutions and give some sufficient conditions for the global and nonglobal existence of solutions at high initial energy level by employing the comparison principle and variational methods.

The aim of this work is to compare a new uncoupled solver for the cardiac Bidomain model with a usual coupled solver. The Bidomain model describes the bioelectric activity of the cardiac tissue and consists of a system of a non-linear parabolic reaction-diffusion partial differential equation (PDE) and an elliptic linear PDE. This system models at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue. The evolution equation is coupled through the non-linear reaction term with a stiff system of ordinary differential equations (ODEs), the so-called membrane model, describing the ionic currents through the cellular membrane. A novel uncoupled solver for the Bidomain system is here introduced, based on solving twice the parabolic PDE and once the elliptic PDE at each time step, and it is compared with a usual coupled solver. Three-dimensional numerical tests have been performed in order to show that the proposed uncoupled method has the same accuracy of the coupled strategy. Parallel numerical tests on structured meshes have also shown that the uncoupled technique is as scalable as the coupled one. Moreover, the conjugate gradient method preconditioned by Multilevel Hybrid Schwarz preconditioners converges faster for the linear systems deriving from the uncoupled method than from the coupled one. Finally, in all parallel numerical tests considered, the uncoupled technique proposed is always about two or three times faster than the coupled approach.

One of the current debate about simulating the electrical activity in the heart is thefollowing: Using a realistic anatomical setting, i.e. realisticgeometries, fibres orientations, etc., is it enough to use a simplified 2-variablephenomenological model to reproduce the main characteristics of the cardiac actionpotential propagation, and in what sense is it sufficient? Using a combination ofdimensional and asymptotic analysis, together with the well-known Mitchell − Schaeffermodel, it is shown that it is possible to accurately control (at least locally) thesolution while spatial propagation is involved. In particular, we reduce the set ofparameters by writing the bidomain model in a new nondimensional form. The parameters ofthe bidomain model with Mitchell − Schaeffer ion kinetics are then set and shown to be inone-to-one relation with the main characteristics of the four phases of a propagatedaction potential. Explicit relations are derived using a combination of asymptotic methodsand ansatz. These relations are tested against numerical results. We illustrate how theserelations can be used to recover the time/space scales and speed of the action potentialin various regions of the heart.

This paper is devoted to the general problem of reconstructing the cost from theobservation of trajectories, in a problem of optimal control. It is motivated by thefollowing applied problem, concerning HALE drones: one would like them to decide bythemselves for their trajectories, and to behave at least as a good human pilot. Thisapplied question is very similar to the problem of determining what is minimized in humanlocomotion. These starting points are the reasons for the particular classes of controlsystems and of costs under consideration. To summarize, our conclusion is that in general,inside these classes, three experiments visiting the same values of the control are neededto reconstruct the cost, and two experiments are in general not enough. The method isconstructive.

The proof of these results is mostly based upon the Thom’s transversality theory.

This study is partly supported by FUI AAP9 project SHARE, and by ANR Project GCM, program“blanche”, project number NT09-504490.

In this paper, we consider the approximation of second order evolution equations. It is well known that the approximated system by finite element or finite difference is not uniformly exponentially or polynomially stable with respect to the discretization parameter, even if the continuous system has this property. Our goal is to damp the spurious high frequency modes by introducing numerical viscosity terms in the approximation scheme. With these viscosity terms, we show the exponential or polynomial decay of the discrete scheme when the continuous problem has such a decay and when the spectrum of the spatial operator associated with the undamped problem satisfies the generalized gap condition. By using the Trotter–Kato Theorem, we further show the convergence of the discrete solution to the continuous one. Some illustrative examples are also presented.