Mathematical analysis of parallel convective exchangers with general lateral boundary conditions using generalized graetz modes

We propose a mathematical analysis of parallel convective exchangers for any general but longitudinally invariant domains. We analyze general Dirichlet or Neumann prescribed boundary conditions at the outer solid domain. Our study provides general mathematical expressions for the solution of convection/diffusion problems. Explicit form of generalized solutions along longitudinal coordinate are found from convoluting elementary base Graetz mode with the applied sources at the boundary. In the case of adiabatic zero flux counter-current configuration, we recover the longitudinally linearly varying solution associated with the zeroth eigenmode which can be considered as the fully developed behavior for heat-exchangers. We also provide general expression for the infinite asymptotic behavior of the solutions which depends on simple parameters such as total convective flux, outer domain perimeter and the applied boundary conditions. Practical considerations associated with the numerical precision of truncated mode decomposition is also analyzed in various configurations for illustrating the versatility of the formalism. Numerical quantities of interest are investigated, such as fluid/solid internal and external fluxes

Transports couplés en géométries complexes

Ces travaux s'intéressent aux questions de transports non stationnaires et de transferts stationnaires de chaleur et de masse par convection-diffusion au sein de géométries complexes. Par complexe, nous entendons d'une part pour le transport que le fluide est convecté au sein d'une cavité de section quelconque lentement variable dans la direction longitudinale, c'est à dire ayant des variations longitudinales grandes devant hauteur et largeur moyennes. Nous considérons d'autre part le transfert au sein de domaines non-axisymétriques dans lesquels sont plongés un ou plusieurs tubes où le fluide porteur s'écoule. Pour ce qui concerne le transfert, ce travail a consisté à montrer comment étendre le principe, valider l'utilisation, et illustrer l'efficacité d'une décomposition en mode de Graetz pour la prédiction des échanges dans des configurations réalistes d'échangeurs. Cette décomposition permet de formuler le problème initial 3D comme un problème aux valeurs propres généralisées en 2D dont la résolution numérique est drastiquement moins coûteuse. Nous généralisons la notion de mode de Graetz à des conditions aux limites latérales quelconques et, en particulier pour le cas d'échangeurs équilibrés où nous avons mis en évidence un nouveau mode linéairement variables dans la direction longitudinale. Nous mettons en oeuvre le calcul de ces modes de Graetz dans le cas de configurations semi-infinies pour traiter, par exemple, des configurations transversalement périodiques (types plancher chauffant) et montrons qu'un faible nombre de modes suffit pour donner une très bonne approximation des transferts. Dans le cas d'échangeurs finis couplé avec des tubes en entrée/sortie, nous montrons comment déterminer les amplitudes des modes de Graetz dans les différents domaines par la minimisation d'une fonctionnelle associée aux conditions d'entrée sorties retenues. Ces modes permettent l'étude paramétrique systématique des champs de température, des flux de chaleurs entre les domaines fluides et solides ainsi que des rendements thermiques d'un échangeur à deux tubes. Nos résultats indiquent que la longueur d'échange caractéristique est gouvernée par le premier mode de Graetz généralisé à grand nombre de Péclet. Nous montrons aussi, en particulier, qu'un échangeur symétrique possède un spectre symétrique, et une évolution amont/aval symétrique. Dans le cas de la dispersion de Taylor, nous avons établi une forme conservative 3D des équations de dispersion de Taylor en géométrie variable généralisant le cas 2D déjà connu. Nous avons ensuite implémenté en éléments finis puis validé numériquement ces équations de dispersion en 2D et 3D. Nous montrons que les variations longitudinales 3D de la cavité peuvent considérablement augmenter la dispersion longitudinale.This work interest is about stationary transfer and non-stationary transport by convection-diffusion onto complex geometries. For transport issues, complex refers to convection into flattened cavity of arbitrary transverse shape, slowly varying along the longitudinal direction. In the context of transfer, complex refers to non-axisymmetric domains of arbitrary transverse shape along which one or several parallel tubes convect heat or mass. For the transfer problem, this work extends the principle, validates the use, and illustrates the efficiency of Graetz modes decompositions for exchanges prediction in realistic exchangers configurations. This decomposition permits to formulate the initial 3D problem as a generalysed 2D eigenvalue problem, the numerical evaluation of which is drastically reduced. We generalyze Graetz modes solutions for arbitrary applied lateral boundary conditions. In the particular case of balanced exchangers, we bring to the fore a new neutral mode whose longitudinal variations are linear as opposed to classical Graetz modes displaying exponential decay. The numerical computation of those modes for semi-infinite configurations with lateral periodic boundary conditions shows that a few number of those provides a very good approximation for exchanges. In the case of finite exchangers coupled with inlet/oulet tubes, we show how to evaluate the amplitudes of Graetz modes in the various domains (inlet, exchanger, outlet) from functional minimization associated with input/output boundary conditions. The evaluation of these amplitudes permit a systematic parametric study of temperature fields, heat fluxes between fluid and solid, and hot/cold performance of a couple-tube exchanger. Our results indicate that the typical exchange length is governed by the first Graetz mode at large P\'eclet number. We also show that a symmetric exchanger has a symmetric spectrum and a upward/backward symmetric evolution. In the case transport we elaborate theoretically the conservative form of 3D Taylor dispersion equations into variable cavities which generalyzes the framework already known in 2D. We numerically implement these averaged dispersion equations with finite element, and validate in 2D the obtained results. We show that 3D longitudinal variations of a cavity has a strong impact on the longitudinal dispersion

Numerical computation of 3D heat transfer in complex parallel heat exchangers using generalized Graetz modes

We propose and develop a variational formulation dedicated to the simulation of parallel convective heat exchanger that handles possibly complex input/output conditions as well as connection between pipes. It is based on a spectral method that allows to re-cast three-dimensional heat exchangers into a two-dimensional eigenvalue problem, named the generalized Graetz problem. Our formulation handles either convective, adiabatic, or prescribed temperature at the entrance or at the exit of the exchanger. This formulation is robust to mode truncation, offering a huge reduction in computational cost, and providing insights into the most contributing structure to exchanges and transfer. Several examples of heat exchangers are analyzed, their numerical convergence is tested and the numerical efficiency of the approach is illustrated in the case of Poiseuille flow in tubes

Analyse de la convection-diffusion entre deux tubes parallèles plongés dans un domaine cylindrique

Nous étudions la convection-diffusion tri-dimensionelle entre tubes parallèles par une formulation théorique bi-dimensionnelle précédement proposée. L’implémentation de cette formulation bi-dimensionnelle par éléments finis permet de calculer une vaste classe de configurations physique, hydrodynamiques et géométriques. Nous nous attachons à l’étude du champ de température et de l’évolution des flux en fonction du nombre de Péclet Pe, l’écart entre les deux tubes d, le rayon des tubes r et les vitesses des écoulements au sein des tubes

Inverse ECG problem using the factorization method

Inverse Problem in Electrocardiography viaFactorization Method of Boundary ValueProblemsJulien Bouyssier, Nejib Zemzemi, Jacques Henry,CARMEN team, Inria Bordeaux Sud-Ouest200 avenue de la vieille tour, 33405 Talence CedexElectrocardiographic Imaging (ECGI) is a new imaging technique thatnoninvasively images cardiac electrical activity on the heart surface. InECGI, a multi-electrode vest records body-surface potential maps (BSPMs);then, using geometrical information from CT-scans and a mathematical al-gorithm, electrical potentials, electrograms and isochrones are reconstructedon the heart surface. The reconstruction of cardiac activity from BSPMs isan ill-posed inverse problem. In this work, we present an approach basedon an invariant embedding method: the factorization method of boundaryvalues problems [1, 2]. The idea is to embed the initial problem into a familyof similar problems on subdomains bounded by a moving boundary from thetorso skin to the epicardium surface. For the direct problem this method pro-Inverse Problem in Electrocardiography via FactorizationMethod of Boundary Value Problems :How reconstruct epicardial potential maps from measurements of the torso ?Julien Bouyssier, Nejib Zemzemi, Jacques [email protected], [email protected], [email protected] and goalSolve the inverse problem in electrocardiography frommeasurements of the torso.factorizationmethod to compute epicardial potential maps.simplified presentation of the method by considering acylindrical geometry of our problem.Conclusions and perspectivesConclusions :Direct optimal estimation of t and ! before using any discretisation :=# Analyse ill-posedness and propose a better regularization and discretizationEquations for P and Q depend only of the geometry :=# Not necessary to repeat resolution at every time step of cardiac cyclePerspectives :Apply the method to 3D case where the moving boundary S will be a deformed surface :=# First : model of spheres=# Then : realistic geometries : how compute 3D surfaces ? + numerical cost ?. This method calculates Neumman-Dirichlet and Dirichlet-Neumannoperators on the moving boundary using Riccati equations. Mathematicalanalysis allows to write an optimal estimation of the epicardial potentialbased on a quadratic criterion. The analysis of the of the inverse problemill-posedness allows to compare different regularisation terms and choose abetter one. For numerical simulations we first construct a synthetical databased on the ECG solver [3]. The electrical potential on the torso boundaryis then extracts from the forward solution to be used as an input of the in-verse problem. The first obtained results using this method in 3D show thatwe can capture the wave front. Whereas the amplitude of theinverse problem solution is too low compared to the forward solution.References[1] Jacques Henry and Angel Manuel Ramos , La methode de factorisationdes problemes aux limites, book (in preparation), (2013)[2] Fadhel Jday, Completion de donnees frontieres : la methode de plonge-ment invariant, PHD thesis, (2012)[3] Nejib Zemzemi, Etude theorique et numerique de l'activite electrique ducoeur: Applications aux electrocardiogrammes, PHD thesis, (2009)

Inverse ECG problem using the factorization method

International audienceInverse Problem in Electrocardiography viaFactorization Method of Boundary ValueProblemsJulien Bouyssier, Nejib Zemzemi, Jacques Henry,CARMEN team, Inria Bordeaux Sud-Ouest200 avenue de la vieille tour, 33405 Talence CedexElectrocardiographic Imaging (ECGI) is a new imaging technique thatnoninvasively images cardiac electrical activity on the heart surface. InECGI, a multi-electrode vest records body-surface potential maps (BSPMs);then, using geometrical information from CT-scans and a mathematical al-gorithm, electrical potentials, electrograms and isochrones are reconstructedon the heart surface. The reconstruction of cardiac activity from BSPMs isan ill-posed inverse problem. In this work, we present an approach basedon an invariant embedding method: the factorization method of boundaryvalues problems [1, 2]. The idea is to embed the initial problem into a familyof similar problems on subdomains bounded by a moving boundary from thetorso skin to the epicardium surface. For the direct problem this method pro-Inverse Problem in Electrocardiography via FactorizationMethod of Boundary Value Problems :How reconstruct epicardial potential maps from measurements of the torso ?Julien Bouyssier, Nejib Zemzemi, Jacques [email protected], [email protected], [email protected] and goalSolve the inverse problem in electrocardiography frommeasurements of the torso.factorizationmethod to compute epicardial potential maps.simplified presentation of the method by considering acylindrical geometry of our problem.Conclusions and perspectivesConclusions :Direct optimal estimation of t and ! before using any discretisation :=# Analyse ill-posedness and propose a better regularization and discretizationEquations for P and Q depend only of the geometry :=# Not necessary to repeat resolution at every time step of cardiac cyclePerspectives :Apply the method to 3D case where the moving boundary S will be a deformed surface :=# First : model of spheres=# Then : realistic geometries : how compute 3D surfaces ? + numerical cost ?. This method calculates Neumman-Dirichlet and Dirichlet-Neumannoperators on the moving boundary using Riccati equations. Mathematicalanalysis allows to write an optimal estimation of the epicardial potentialbased on a quadratic criterion. The analysis of the of the inverse problemill-posedness allows to compare different regularisation terms and choose abetter one. For numerical simulations we first construct a synthetical databased on the ECG solver [3]. The electrical potential on the torso boundaryis then extracts from the forward solution to be used as an input of the in-verse problem. The first obtained results using this method in 3D show thatwe can capture the wave front. Whereas the amplitude of theinverse problem solution is too low compared to the forward solution.References[1] Jacques Henry and Angel Manuel Ramos , La methode de factorisationdes problemes aux limites, book (in preparation), (2013)[2] Fadhel Jday, Completion de donnees frontieres : la methode de plonge-ment invariant, PHD thesis, (2012)[3] Nejib Zemzemi, Etude theorique et numerique de l'activite electrique ducoeur: Applications aux electrocardiogrammes, PHD thesis, (2009)

Coupled transport onto complex geometries

TOULOUSE3-BU Sciences (315552104) / SudocSudocFranceF

Inverse problem in electrocardography via the factorization method of boundary value problems

We present a new mathematical approach for solving the inverse problem in electrocardiography. This approach is based on the factorization of boundary value problems method. In this paper we derive the mathematical equations and test this method on synthetical data generated on realistic heart and torso geometries using the state-of-the-art bidomain model in the heart coupled to the Laplace equation in the torso. We measure the accuracy of the inverse solution using spatial Relative Error (RE) and Correlation Coefficient (CC)

MATHEMATICAL ANALYSIS OF PARALLEL CONVECTIVE EXCHANGERS WITH GENERAL LATERAL BOUNDARY CONDITIONS USING GENERALISED GRAETZ MODES

Abstract. We propose a mathematical analysis of parallel convective exchangers for any general but longitudinally invariant domains. We analyse general Dirichlet or Neumann prescribed boundary conditions at the outer solid domain. Our study provides general mathematical expressions for the solution of convection/diffusion problems. Explicit form of generalised solutions along longitudinal coordinate are found from convoluting elementary base Graetz mode with the applied sources at the boundary. In the case of adiabatic zero flux counter-current configuration we recover the longitudinally linearly varying solution associated with the zeroth eigenmode which can be considered as the fully developed behaviour for heat-exchangers. We also provide general expression for the infinite asymptotic behaviour of the solutions which depends on simple parameters such as total convective flux, outer domain perimeter and the applied boundary conditions. Practical considerations associated with the numerical precision of truncated mode decomposition is also analysed in various configurations for illustrating the versatility of the formalism. Numerical quantities of interest are investigated, such as fluid/solid internal and external fluxes. 1

Inverse problem in electrocardiography via factorization method of boundary problems : how reconstruct epicardial potential maps from measurements of the torso ?

Poster Workshop Liryc, 24 et 25 Octobre 2013, BordeauxDans ces travaux on s'intéresse au problème inverse en électrocardiographie en utilisant une méthode de plongement invariant : la méthode de factorisation de problèmes aux limites. Ceci revient à plonger le problème initial en une famille de problèmes similaires sur des sous domaines limités par une frontière mobile (suivant un axe d'évolution que nous définirons) balayant le thorax depuis la peau jusqu'à l'épicarde. Pour le problème direct cette méthode permet d'obtenir une formulation équivalente avec deux problèmes de Cauchy évoluant sur cette surface mobile et qui sont à résoudre successivement dans des directions opposées. La méthode permet de calculer les opérateurs Neumann-Dirichlet et Dirichlet-Neumann sur cette surface mobile qui vérifient des équations de Riccati. Dans le cadre du problème inverse cette analyse permet d'écrire directement avant discrétisation l'équation vérifiée sur l'épicarde par l'estimation optimale du potentiel épicardiaque au sens d'un critère quadratique. Elle permet d'analyser le caractère mal posé du problème inverse et donc de discrétiser et de régulariser au mieux ce problème. Un des avantages de cette méthode est que si l'on souhaite calculer le potentiel à différents temps du cycle cardiaque, il n'est pas nécessaire de refaire la résolution de toutes les équations à chaque instant.In this work, we present a new approach solving the inverse problem of electrocardiography. This approach is based on an invariant embedding method: the factorization method of boundary values problems. The idea is to embed the initial problem into a family of similar problems on subdomains bounded by a moving boundary (along a axis of evolution that we define) from the torso skin to the epicardium surface. For the direct problem this method provides an equivalent formulation with two Cauchy problems evolving on this moving boundary and which have to be solved successively in opposite directions. This method calculates Neumman-Dirichlet and Dirichlet-Neumann operators on this moving boundary that satisfy Riccati equations. Regarding the inverse problem, mathematical analysis allows to write an optimal estimation of the epicardial potential based on a quadratic criterion. Then, we can analyse the ill-posed behaviour of the inverse problem and propose a better regularization and discretization of the problem. One of the advantages of this method is the computation of the potential at different times during cardial cycle : it is not necessary to repeat the resolution of all the equations at every time