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This thesis presents a work at the interface between applied mathematics and biomedical engineering. The work's main subject is the estimation of blood flows and quantities of medical interest in diagnosing certain diseases concerning the cardiovascular system. We propose a complete pipeline, providing the theoretical foundations for state estimation from medical data using reduced-order models, and addressing inter-patient variability. Extensive numerical tests are shown in realistic 3D scenarios that verify the potential impact of the work in the medical comunnity.; Cette thèse présente un travail à l’interface entre les mathématiques appliquées et l’ingénierie biomédicale. Le sujet principal en est l’estimation des écoulements sanguins et de quantités d’intérêt pour le diagnostic de certaines maladies cardiovasculaires. Nous proposons une procédure complète, dont nous détaillons les fondements théoriques, permettant l’estimation d’état à partir de données médicales en utilisant des techniques de réduction de modèle, et en prenant en compte la problématique de la variabilité inter-patients. De nombreux test numériques en 3D sont exposés afin de vérifier le potentiel de cette étude dans le domaine médical.

This thesis presents a work at the interface between applied mathematics and biomedical engineering. The work’s main subject is the estimation of blood flows and quantities of medical interest in diagnosing certain diseases concerning the cardiovascular system. We propose a complete pipeline, providing the theoretical foundations for state estimation from medical data using reduced-order models, and addressing inter-patient variability. Extensive numerical tests are shown in realistic 3D scenarios that verify the potential impact of the work in the medical comunnity.; Cette thèse présente un travail à l’interface entre les mathématiques appliquées et l’ingénierie biomedicale. Le sujet principal en est l’estimation des écoulements sanguins et de quantités d’intérêt pour le diagnostic de certaines maladies cardiovasculaires. Nous proposons une procédure complète, dont nous détaillons les fondements théoriques, permettant l’estimation d’état à partir de données médicales en utilisant des techniques de réduction de modèle, et en prenant en compte la problématique de la variabilité inter-patients. De nombreux test numériques en 3D sont exposés afin de vérifier le potentiel de cette étude dans le domaine médical.

The objective of traditional goal-oriented strategies is to construct an optimal mesh that mini- mizes the problem size needed to achieve a user prescribed tolerance error for a given quantity of interest (QoI). Typical geophysical resistivity measurement acquisition systems can easily record electromagnetic (EM) fields. However, depending upon the application, EM fields are sometimes loosely related to the quantity that is to be inverted (conductivity or resistivity), and therefore they become inadequate for inversion. In the present work, we study the impact of the selection of the QoI in our inverse problem. We focus on two different acquisition systems: marine controlled source electromagnetic (CSEM), and magnetotellurics (MT). For both ap- plications, numerical results illustrate the benefits of employing adequate QoI. Specifically, the use as QoI of the impedance matrix on MT measurements provides significant computational savings, since one can replace the existing absorbing boundary conditions (BCs) by a homoge- neous Dirichlet BC to truncate the computational domain, something that is not possible when considering EM fields as QoI.

In this paper, we study a new approach in a posteriori error estimation, in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm. These so-called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs. We present here the theory with respect to a class of elliptic boundary-value problems, and in particular, show how to obtain accurate estimates as well as upper and lower bounds on the error. We also study the new concept of goal-oriented adaptivity, which embodies mesh adaptation procedures designed to control error in specific quantities. Numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates.

[EN] Goal oriented error estimation and adaptive procedures are essential for the accurate and efficient evaluation of finite element numerical simulations that involve complex domains. By locally improving the approximation quality, for example, by using the extended finite element method (XFEM), we can solve expensive problems which could result intractable otherwise. Here, we present an error estimation technique for enriched finite element approximations that is based on an equilibrated recovery technique, which considers the stress intensity factor as the quantity of interest. The locally equilibrated superconvergent patch recovery is used to obtain enhanced stress fields for the primal and dual problems defined to evaluate the error estimate., This work was supported by the EPSRC grant EP/G042705/1 "Increased Reliability for Industrially Relevant Automatic Crack Growth Simulation with the eXtended Finite Element Method". Stephane Bordas also thanks partial funding for his time provided by the European Research Council Starting Independent Research Grant (ERC Stg Grant Agreement No. 279578) "RealTCut Towards real time multiscale simulation of cutting in non-linear materials with applications to surgical simulation and computer guided surgery". This work has been carried out within the framework of the research project DPI2010-20542 of the Ministerio de Ciencia e Innovacion (Spain). The financial support of the FPU program (AP2008-01086), the funding from Universitat Politecnica de Valencia and Generalitat Valenciana (PROMETEO/2012/023) are also acknowledged.

Excavation processes can frequently manifest critical issues regarding permanent damages in surrounding buildings. The employment of bulkheads or pile-sheets in dense urban areas cannot overlook a proper monitoring of the excavation process. It can be necessary to combine different kinds of information and computations since excavation processes gather together dissimilar issues and require a wide assortment of solution strategies. Aim of this work is to propose a strategy for monitoring purposes, based on the employment of Bayesian networks. The proposed strategy lays on the statistics of soil parameters and external loads which are available in the literature. Computing the results of land survey measurements of the site, a Bayesian network is able to update the model's parameters and to estimate many quantities of interest. A proper choice of them can be employed in order to set a limit state condition aiming to avoid permanent damages in surrounding buildings. The proposed procedure has been applied to a real excavation site. The inference process shows how, for the considered site, an efficient reliability analysis cannot leave out of consideration the detection of ongoing soil displacements. This is because geotechnical tests are not sufficient to properly define the model features. Furthermore, it takes advantage of soft evidences which are provided by employed survey methodology. Eventually, the proposed strategy aims to be fast and efficient in order to promptly give advice of critical issues while excavation is carried out.

During the last years the use of error estimators which measure the error in a quantity of interest defined by the analyst, instead of the energy norm, have become increasingly popular as they provide an error indicator for goal oriented adaptivity procedures. In this paper we propose an a posteriori recovery-based error estimation procedure which considers the stress intensity factor K typical of singular problems as the quantity of interest in finite element (FE) approximations. In general, error estimators in quantities of interest have been based on residual techniques and, although recovery techniques have been often preferred when considering the error in energy norm due to their robustness and simplicity, so far, there is no available procedure which considers an equilibrated recovery technique that can be used in standard FE frameworks. In [1] a standard SPR recovery technique is used to obtain an error measure of the J-integral, which is closely related to the value of the SIF. However, it does not consider any equilibrium constraints or the singularity near the crack tip, thus the obtained recovered stress field is not well suited for this kind of problems. The technique proposed herein relies on the enhanced superconvergent patch recovery technique presented in [2] to evaluate highly accurate recovered stress fields of the primal and dual problems, which are then used to obtain a sharp error estimate. The primal problem is simply the problem under analysis. To formulate the dual problem we consider the linear interaction integral representing K to obtain the applied loads of the dual FE approximation to solve. The high accuracy of the recovered stress fields for both the primal and dual solutions is obtained by decomposing the raw stress field obtained from the finite element approximations into singular and smooth parts, and enforcing the fulfilment of boundary and internal equilibrium equations. The results indicate an accurate estimation of the error in K for benchmark problems with exact solution.

International audience; Local error estimators known as in quantities of interest enable to obtain directly the error of discretization committed on a quantity with a physical meaning, in a definite zone. Such an estimator is build starting from of the error on the solution of the primal problem, the one of the dual problem and an error estimator based on the explicit residual. Some numerical examples performed by Code_Aster allow to validate this estimator but also to illustrate its use in industrial cases.; Des estimateurs d'erreurs locaux dits en quantités d'intérêt permettent d'obtenir directement l'erreur de discrétisation commise sur une quantité ayant un sens physique, dans une zone définie. On construit un tel estimateur à partir de l'erreur sur la solution du problème primal, celle du problème dual et un estimateur d'erreur basé sur les résidus explicites. Des exemples numériques traités par Code_Aster permettent de valider cet estimateur mais aussi d'illustrer son utilisation sur des structures industrielles.

Over the last ten years there has been an increase on the use of goal-oriented error estimates aimed to quantify the local error on a (non)linear quantity of interest (QoI) that might result relevant for design purposes. Residual-based error estimators have been used recursively to obtain upper and lower bounds of the error in quantities of interest for finite element approximations. In this paper, we present a recovery technique for 2D linear elasticity problems, based on the superconvergent patch recovery (SPR), which provides recovered displacement and stress fields that are then utilised to evaluate practical upper and lower error bounds in QoI.

We present an "a posteriori" error analysis in quantities of interest for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. The multiscale method is based on a macro-to-micro formulation, where the macroscopic physical problem is discretized in a macroscopic finite element space, and the missing macroscopic data are recovered on-the-fly using the solutions of corresponding microscopic problems. We propose a new framework that allows to follow the concept of the (single-scale) dual-weighted residual method at the macroscopic level in order to derive a posteriori error estimates in quantities of interests for multiscale problems. Local error indicators, derived in the macroscopic domain, can be used for adaptive goal-oriented mesh refinement. These error indicators rely only on available macroscopic and microscopic solutions. We further provide a detailed analysis of the data approximation error, including the quadrature errors. Numerical experiments confirm the efficiency of the adaptive method and the effectivity of our error estimates in the quantities of interest. (C) 2013 Wiley Periodicals, Inc.

Traditionally, recovery based error estimators in linear elasticity have considered the evaluation of an enhanced stress field obtained from the raw Finite Element (FE) stress solution. Instead of that, one can also obtain a recovered displacement field from the FE displacements. Herein, we describe a superconvergent patch recovery of the displacement field which considers the local fulfilment of boundary and internal equilibrium equations, Dirichlet constraints and, for singular problems, the splitting of the displacement and stress fields into singular and smooth parts. Numerical tests using problems with known analytical solution have been carried out to validate the proposed technique for error estimation and error bounding in energy norm and quantities of interest.

There is an increasing interest on the use of goal-oriented error estimates which help to measure and control the local error on a linear or non-linear quantity of interest (QoI) that might result relevant for design purposes (e.g. the mean stress value in a particular area, displacements, the stress intensity factor for fracture problems,⋯). In general, residual-based error estimators have been used to obtain upper and lower bounds of the error in quantities of interest for finite element approximations. In this work, we propose a novel a posteriori recovery technique to obtain an upper error bound of the QoI. We use a recovery procedure based on the superconvergent patch recovery (SPR) technique to obtain nearly statically admissible recovered stress fields for the primal and dual problems. This recovery technique was previously used to obtain upper bounds of the error in energy norm and has been used in this paper to obtain a computable version of the upper bound for the quantity of interest.