177 research outputs found
Reduced-Order Modeling based on Approximated Lax Pairs
A reduced-order model algorithm, based on approximations of Lax pairs, is
proposed to solve nonlinear evolution partial differential equations. Contrary
to other reduced-order methods, like Proper Orthogonal Decomposition, the space
where the solution is searched for evolves according to a dynamics specific to
the problem. It is therefore well-suited to solving problems with progressive
waves or front propagation. Numerical examples are shown for the KdV and FKPP
(nonlinear reaction diffusion) equations, in one and two dimensions
Approximated Lax Pairs for the Reduced Order Integration of Nonlinear Evolution Equations
A reduced-order model algorithm, called ALP, is proposed to solve nonlinear
evolution partial differential equations. It is based on approximations of
generalized Lax pairs. Contrary to other reduced-order methods, like Proper
Orthogonal Decomposition, the basis on which the solution is searched for
evolves in time according to a dynamics specific to the problem. It is
therefore well-suited to solving problems with progressive front or wave
propagation. Another difference with other reduced-order methods is that it is
not based on an off-line / on-line strategy. Numerical examples are shown for
the linear advection, KdV and FKPP equations, in one and two dimensions
Identification of weakly coupled multiphysics problems. Application to the inverse problem of electrocardiography
This work addresses the inverse problem of electrocardiography from a new
perspective, by combining electrical and mechanical measurements. Our strategy
relies on the defini-tion of a model of the electromechanical contraction which
is registered on ECG data but also on measured mechanical displacements of the
heart tissue typically extracted from medical images. In this respect, we
establish in this work the convergence of a sequential estimator which combines
for such coupled problems various state of the art sequential data assimilation
methods in a unified consistent and efficient framework. Indeed we ag-gregate a
Luenberger observer for the mechanical state and a Reduced Order Unscented
Kalman Filter applied on the parameters to be identified and a POD projection
of the electrical state. Then using synthetic data we show the benefits of our
approach for the estimation of the electrical state of the ventricles along the
heart beat compared with more classical strategies which only consider an
electrophysiological model with ECG measurements. Our numerical results
actually show that the mechanical measurements improve the identifiability of
the electrical problem allowing to reconstruct the electrical state of the
coupled system more precisely. Therefore, this work is intended to be a first
proof of concept, with theoretical justifications and numerical investigations,
of the ad-vantage of using available multi-modal observations for the
estimation and identification of an electromechanical model of the heart
Fast reconstruction of 3D blood flows from Doppler ultrasound images and reduced models
This paper deals with the problem of building fast and reliable 3D
reconstruction methods for blood flows for which partial information is given
by Doppler ultrasound measurements. This task is of interest in medicine since
it could enrich the available information used in the diagnosis of certain
diseases which is currently based essentially on the measurements coming from
ultrasound devices. The fast reconstruction of the full flow can be performed
with state estimation methods that have been introduced in recent years and
that involve reduced order models. One simple and efficient strategy is the
so-called Parametrized Background Data-Weak approach (PBDW). It is a linear
mapping that consists in a least squares fit between the measurement data and a
linear reduced model to which a certain correction term is added. However, in
the original approach, the reduced model is built a priori and independently of
the reconstruction task (typically with a proper orthogonal decomposition or a
greedy algorithm). In this paper, we investigate the construction of other
reduced spaces which are built to be better adapted to the reconstruction task
and which result in mappings that are sometimes nonlinear. We compare the
performance of the different algorithms on numerical experiments involving
synthetic Doppler measurements. The results illustrate the superiority of the
proposed alternatives to the classical linear PBDW approach
A Quasi-Newton Algorithm Based on a Reduced Model for Fluid-Structure Interaction Problems in Blood Flows
International audienceWe propose a quasi-Newton algorithm for solving fluid-structure interaction problems. The basic idea of the method is to build an approximate tangent operator which is cost effective and which takes into account the so-called added mass effect. Various test cases show that the method allows a significant reduction of the computational effort compared to relaxed fixed point algorithms. We present 2D and 3D fluid-structure simulations performed either with a simple 1D structure model or with shells in large displacements
Simulation numérique du système cardiovasculaire
Les progrès réalisés en mathématiques appliquées permettent aujourd’hui d’envisager la simulation sur ordinateur de certains compartiments du système cardiovasculaire. Nous proposons de faire un point sur quelques modèles, en nous focalisant sur la simulation de l’écoulement du sang dans des artères déformables et sur la simulation de la contraction du myocarde sous l’effet de la propagation d’un signal électrique. Nous tentons également de présenter des applications possibles de ce type de travaux.In this article, we aim at giving a non-technical overview of some mathematical models currently used in the numerical simulation of the cardiovascular system. A hierarchy of models for blood flows in large arteries is briefly described as well as an electromechanical model for the heart. We discuss some possible applications of the numerical simulations of such models, for example the optimization of prostheses. We also address the issue of the data assimilation for the calibration of the models
Variational formulation of the Generalized Navier Boundary Condition.
In this paper, we propose an Arbitrary Lagrangian Eulerian (ALE) formulation of the Generalized Navier Boundary Condition introduced in~\cite{qian-wang-sheng-03,qian-wang-sheng-06} to model the displacement of the contact line of an interface in two-fluid flows. Owing to these boundary conditions, it is possible to circumvent the incompatibility between the classical no-slip boundary condition and the fact that the contact line of the interface on the wall is actually moving. We present some results on the stability of the numerical scheme in energy norm. We show the validity of the approach by numerical experiments on two-fluid flows in narrow channels
Semi-Implicit Roe-Type Fluxes for Low-Mach Number Flows
Two semi-implicit methods based on the splitting of the Euler equations flux into fluid and acoustic parts applied to low Mach number flows are presented. The first method is based on the splitting of slow and fast eigenvalues of the jacobian matrix of the fluxes and a semi-implicit scheme is constructed by introducing only the fast eigenvalues in the implicit matrices. The second method is based on the splitting of the Euler flux by separating the terms in velocity and the terms in pressure ; this system is solved by a fractional step method. A semi-implicit scheme is obtained by using a linearised implicit scheme for the acoustic step only. These two methods are applied to the convection of a density pulse for Mach numbers equal to 0.1 and 0.01. Accuracy and efficiency of the different schemes are compared
Explicit coupling schemes for a fluid-fluid interaction problem arising in hemodynamics
International audienceIn this work we propose a new approach to the loosely coupled time-marching of a fluid-fluid interaction problems involving the incompressible Navier-Stokes equations. The methods combine a specific explicit Robin-Robin treatment of the interface coupling with a weakly consistent interface pressure stabilization in time. A priori energy estimates guaranteeing stability of the splitting are obtained for a total pressure formulation of the coupled problem. The performance of the proposed schemes is illustrated on several numerical experiments related to simulation of aortic blood flow
Sigma transformation and ALE formulation for three-dimensional free surface flows
International audienceIn this paper we establish a link between the sigma transformation approach and the arbitrary Lagrangian–Eulerian (ALE) approach. For that purpose we introduce the ALE-sigma (ALES) approach, which consists in an ALE interpretation of the sigma transformation. Taking advantage of this new approach, we propose a general ALES transformation, allowing for a great adaptability of the vertical discretization and therefore overcoming some drawbacks of the classical sigma transformation. Numerical results are presented, showing the advantages of this general coordinate system, as, for example, a better representation of horizontal stratifications
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