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One of the important features of the arbitrary Lagrangian-Eulerian (ALE) hydrodynamics methods is the appearance of mesh cells containing fractional amounts of more than one material. To resolve sub-scale material dynamics, interfaces between materials inside a cell have to be calculated from material moments. The moment-of-fluid (MOF) method is one of the most accurate interface reconstruction methods. Unlike other methods, it is local and does not require data from neighboring cells which makes it amenable for emerging computer architectures. We analyze this method for axisymmetric problems. Our analysis closes a few gaps in its theoretical justification and provides useful insight on its properties and practical implementation. The theoretical conclusions are confirmed with numerical experiments.

The results of numerical modeling of the ion-beam dynamics and the dynamics of secondary ions and electrons formed in the ion-optical system of a portable neutron generator are presented. The calculation was performed with the aid of the KARAT code in a two-dimensional flat geometry and axisymmetric geometry. To suppress back electrons, the magnetic field of permanent magnets was used in the flat geometry and a blocking electric field of an additional electrode was used in the axisymmetric geometry. The working configurations of both systems are shown.

Obtaining distributions of current density, magnetic field, and ac loss in coils is essential for practical applications. However, simulating large-scale 2-generation high-temperature-superconducting, YBCO coils is a challenging task. In this paper, we employ a multiscale model to successfully implement electromagnetic modeling of large-scale YBCO solenoid coils in two-dimensional axisymmetric geometry with the approximation of infinite turns (J infinite-turn). Based on the approximation, current penetration, magnetic field, and ac loss of coils can be obtained. The calculation results of multiscale models with J infinite-turn and uniform current density (J uniform) are compared with those from reference models which consider superconducting characteristics on each turn of coils with actual size based on H-formulation. Compared with J uniform, J infinite-turn improves the quality of current density and background magnetic field significantly and demonstrates good accuracy with errors less than 10% in terms of ac loss, whilst speeding up calculation by two orders of magnitudes than reference model. Also, the approximation shows more reliability if the coil is larger with more turns in pancakes.

Control valves, nozzles, and other mechanical flow constricting constructions often introduce highly turbulent flow downstream of vena contracta. When solid particles are present in the fluid stream erosion can occur due to the high fluid velocity and turbulence, both in the converging and diverging section of a constriction. This paper presents an experimental study of such flow structure in a converging–diverging valve geometry. The purpose of the study is to obtain velocity profiles downstream of vena contracta by the use of particle image velocimetry measurements. Profiles and flow structure, with and without introduced swirl component, is investigated and discussed with respect to erosion in such specific geometries.

The behavior of the free surface of a perfectly conducting liquid in an external uniform electric field is considered in the framework of the Hamiltonian formalism for the case of bounded axisymmetric geometry of the system (the fluid is bounded by a cylindrical rigid wall). Taking into account the influence of quadratic nonlinearities, we derive an amplitude equation which describes the evolution of the boundary. Using this equation, we find the condition for the hard excitation of boundary instability that leads to an explosive growth of surface perturbations. The differences in the description of the dynamics of axisymmetric perturbations of the boundary from the cases of plane, square, and hexagonal symmetries of the problem are discussed.

Dynamic compaction (DC) is widely used to improve the mechanical properties of soils and other granular fill material upon which foundations or other structures are to be built. To calculate the inner deformation induced by DC, a computational model based on stochastic medium theory was developed to deduce the amount of deformation in the fill from the geometry of the DC crater. For this model, the tamper-soil system was simplified to an axisymmetric geometry and the probability of any unit of material below the crater being deformed can be calculated. Subsequently, the deformation at any point can be obtained by integration. Input parameters for the model were established by back analysis using the results from a DC field test and published data. Comparing the model results to results from actual and simulated DC programs shows that the computational model is useful for calculating deformation induced by DC without the need to consider the constitutive model of the soil.

International audience; A new method based on topology optimization is proposed for solving a three-dimensional axisymmetric inverse problem regarding the configuration of electric currents used in the electromagnetic casting technique of the metallurgical industry. These electric currents must generate an electromagnetic field such that when certain mass of liquid metal is placed in the field it levitates acquiring a predefined axisymmetric shape. The method proposed is able to obtain a suitable configuration of electric currents, and is very attractive from the computational point of view, since it is based on a sparse convex quadratic programming formulation for which there are very efficient interior-point solution algorithms.

In this report, we are concerned with the solution of Galbrun’s equations in timeharmonicdomain for an arbitrary flow with high order finite element methods. Several equivalentformulations of Galbrun’s equations are proposed and discretized with Discontinuous Galerkinmethod. They are compared with a formulation adapted for continuous Galerkin discretization.Numerically, it has been observed that the tested discretization methods converge correctly foran uniform flow, but no longer for a non-uniform flow. Two kinds of stabilization are proposedin order to restore a nice convergence though original equations are modified. Finally, simplifiedGalbrun’s equations are proposed to coincide with original Galbrun’s equations when the flow isnull. Numerical illustrations are presented in the context of helioseismology.; Dans ce document, nous nous intéressons à la résolution des équations de Galbrunen régime harmonique pour un écoulement quelconque par des méthodes d’éléments finis d’ordreélevé. Nous proposons plusieurs formulations équivalentes des équations de Galbrun discrétiséespar une méthode de Galerkin discontinue, que nous comparons à une formulation adaptée auxéléments finis continus. Nous observons numériquement que les différentes méthodes de discrétisationtestées convergent correctement pour un écoulement uniforme, mais ne convergent paspour un écoulement non-uniforme. Nous proposons deux types de stabilisation qui bien quemodifiant les équations initiales permettent de retrouver une convergence satisfaisante. Nousproposons aussi des équations de Galbrun simplifiées qui coïncident avec les équations de Galbrunoriginales lorsque l’écoulement est nul. Nous montrons des illustrations dans le contexte del’héliosismologie.

This PhD subject comes within actual studies managed by CEA/DAM about ALE (Arbitrary-Lagrangian-Eulerian) schemes (with a splitting of Lagrangian and Remapping steps) in the context of hydrodynamic simulations. These numerical schemes have to respect mass, momentum and total energy conservation, which are the fundamental equations of the studied systems.Space- and Time-Staggered are widely used in industrial codes for their simplicity androbustness despite their known lack of exact energy conservation. This is a major drawbackin presence of strong shocks. Among all existing schemes, none of them meet the expectations of robustness, conservation,thermodynamic consistency (both shocks and relaxations capture), accuracy andadaptibility to complex behaviors. Recently, some novel works have been proposed by A.Llor et. al. in order to make conservative this type of scheme in a Lagrangian context (without remapping step). Current remap methods, necessary in large deformations, donot guarantee simultaneously total energy conservation and thermodynamic consistency.This work aims at extending this conservative Lagrangian space- and time-staggeredscheme to a multi-material ALE methodology, keeping its good properties (conservation,accuracy, thermodynamic consistency, robustness) intact. Classical, but demanding, test cases have been performed (both in plane and axisymmetric 2D geometries) and have been compared to existing numerical methods in order to assess the relevance of our approach.; Ce sujet de thèse s’inscrit dans le cadre des études actuellement menées au CEA/DAM concernant des schémas numériques ALE (Arbitrary-Lagrangian-Eulerian)de type « Lagrange + Projection », dans le contexte des simulations hydrodynamiques mutli-matériaux en grandes déformations. Ces schémas doivent respecter les équations de conservation de la masse, de la quantité de mouvement et de l’énergie totale.Les schémas décalés en temps et en espace sont très utilisés dans les codes industriels. Ils sont robustes et permettent une bonne approximation des comportements complexes, mais sont connus pour ne pas conserver exactement l’énergie totale. Cela pose un problème dans le traitement des chocs, sur maillages raffinés ou dans la simulation des milieux réactifs.En 2016, des travaux originaux on été proposés par A. Llor et. al. pour rendre conservatif ce type de schéma dans un contexte lagrangien (sans projection), notamment en proposant une correction pour retrouver la conservation de l’énergie totale.Le travail de cette thèse a été d’étendre ce schéma lagrangien dans un contexte ALE multi-matériaux (avec interface), en garantissant la conservation de toutes les quantités, le respect du second principe de la thermodynamique et la robustesse. De nombreux cas tests ont été menés (en 2D plan et en 2D axisymétrique) et comparés aux méthodes existantes afin de montrer la pertinence de cette approche.

This PhD subject comes within actual studies managed by CEA/DAM about ALE (Arbitrary-Lagrangian-Eulerian) schemes (with a splitting of Lagrangian and Remapping steps) in the context of hydrodynamic simulations. These numerical schemes have to respect mass, momentum and total energy conservation, which are the fundamental equations of the studied systems.Space- and Time-Staggered are widely used in industrial codes for their simplicity androbustness despite their known lack of exact energy conservation. This is a major drawbackin presence of strong shocks. Among all existing schemes, none of them meet the expectations of robustness, conservation,thermodynamic consistency (both shocks and relaxations capture), accuracy andadaptibility to complex behaviors. Recently, some novel works have been proposed by A.Llor et. al. in order to make conservative this type of scheme in a Lagrangian context (without remapping step). Current remap methods, necessary in large deformations, donot guarantee simultaneously total energy conservation and thermodynamic consistency.This work aims at extending this conservative Lagrangian space- and time-staggeredscheme to a multi-material ALE methodology, keeping its good properties (conservation,accuracy, thermodynamic consistency, robustness) intact. Classical, but demanding, test cases have been performed (both in plane and axisymmetric 2D geometries) and have been compared to existing numerical methods in order to assess the relevance of our approach.; Ce sujet de thèse s’inscrit dans le cadre des études actuellement menées au CEA/DAM concernant des schémas numériques ALE (Arbitrary-Lagrangian-Eulerian)de type « Lagrange + Projection », dans le contexte des simulations hydrodynamiques mutli-matériaux en grandes déformations. Ces schémas doivent respecter les équations de conservation de la masse, de la quantité de mouvement et de l’énergie totale.Les schémas décalés en temps et en espace sont très utilisés dans les codes industriels. Ils sont robustes et permettent une bonne approximation des comportements complexes, mais sont connus pour ne pas conserver exactement l’énergie totale. Cela pose un problème dans le traitement des chocs, sur maillages raffinés ou dans la simulation des milieux réactifs.En 2016, des travaux originaux on été proposés par A. Llor et. al. pour rendre conservatif ce type de schéma dans un contexte lagrangien (sans projection), notamment en proposant une correction pour retrouver la conservation de l’énergie totale.Le travail de cette thèse a été d’étendre ce schéma lagrangien dans un contexte ALE multi-matériaux (avec interface), en garantissant la conservation de toutes les quantités, le respect du second principe de la thermodynamique et la robustesse. De nombreux cas tests ont été menés (en 2D plan et en 2D axisymétrique) et comparés aux méthodes existantes afin de montrer la pertinence de cette approche.

The Method of Dimensionality Reduction (MDR) can be regarded as a formalism for analytical solution of some commonly encountered classes of contact problems using a “mechanical intuition” based on the Winkler foundation model. Such an approach makes it much easier to account for a wide range of physical effects associated with contact interaction (e.g. friction, adhesion, and damping). However, there is still a controversy about the method and its applications (see, e.g., the comment on validity of the MDR-based model of rough contact) – which we believe comes from a misunderstanding of the method itself, and which, in turn, can be reconsidered in view of the recently published book on the MDR. The MDR was originally introduced for Hertz’s problem of axisymmetric frictionless local contact and was generalized subsequently for arbitrary axisymmetric geometry of linearly elastic bodies in unilateral local contact. The latter problem, for which the MDR yields the exact analytical solution, can be viewed as a base case that is used to extend, in a unified manner, the model of local contact by taking into account adhesion, friction, and viscous damping. In what follows, we overview the main concepts of the method starting with the base-case contact problem in which the MDR is rooted, and discuss limitations of the MDR as well. For the sake of their completeness, some criticisms that apply equally to conventional contact mechanics solutions are also considered. It is emphasized that the axisymmetric Hertz-type contact problems with a circular contact area constitute the proven range of validity of the MDR, while the extension of the method to other types of contact (e.g. axisymmetric with a multiply-connected contact area, non-axisymmetric) is a field ripe for research.

An axisymmetric model is suggested to simulate mechanical performance of the left ventricle of the heart. Cardiac muscle is treated as incompressible anisotropic material with active tension directed along muscle fibres. This tension depends on kinetic variables that characterize interaction of contractile proteins and regulation of muscle contraction by calcium ions. For numerical simulation of heartbeats the finite element method was implemented. The model reproduces well changes in ventricle geometry between systole and diastole, ejection fraction, pulse wave of ventricular and arterial pressure typical for normal human heart. The model also reproduces well the dependence of the stroke volume on end-diastolic and arterial pressures (the Frank–Starling law of the heart and Anrep effect). The results demonstrate that our model of cardiac muscle can be successfully applied to multiscale 3D simulation of the heart.

International audience

This document concerns the solution of convected Helmholtz equation for radial or axisymmetric configuration. After setting the considered problem and the associated equations, we propose a numerical method adapted for geometries that only vary radially. Then we propose a numerical method for geometries with an axial symmetry. In these two situations, the solution is computed with finite elements after having been decomposed on a basis of orthogonal modes : spherical harmonics in the radial geometry and Fourier modes in the axisymmetric geometry. The number of required modes depends on the configuration (and especially location) of the source. The two methods are tested on the computation of Green's functions for which analytical solutions are available for quantitative comparison. The methods are then compared with the classical 3D finite elements method, and the performances are assessed for an academic test case, showing the advantages of each method in terms of computation time and memory usage. In the context of helioseismology, the perturbation of the fluid displacement in the sun can be modeled accurately by Galbrun's equations. We show that under some assumptions on the background medium, the Galbrun's equations can be simplified to an equation that has the same variational formulation as the one treated in the first part of this document. Numerical simulations are done in this realistic configuration, and the different methods of resolution are compared.; Ce document traite la résolution de l'équation de Helmholtz convectée pour des géométries radiales ou axisymétriques. Après avoir établi le problème considéré et les équations associées, nous proposons une méthode numérique adaptée pour les géométries qui varient uniquement dans la direction radiale. Ensuite nous proposons une méthode pour les géométries présentant une symétrie axiale. Dans ces deux situations, la solution est calculée avec des éléments finis après l'avoir décomposée sur une base de modes orthogonaux : les harmoniques sphériques pour la géométrie radiale et les modes de Fourier pour la géométrie axi-symétrique. Le nombre de modes nécessaires dépend de la configuration (et en particulier de la localisation) de la source. Les deux méthodes sont testées sur le calcul de la fonction de Green, pour laquelle des solutions analytiques sont disponibles pour une comparaison quantitative. Les méthodes sont ensuite comparées avec les éléments finis classiques en 3D, et les performances sont comparées sur un cas test académique, montrant les avantages de chaque méthode en termes de temps de calcul et d'utilisation mémoire. Dans le contexte de l'héliosismologie, les équations de Galbrun sont une bonne modélisation de la perturbation du déplacement fluide dans le Soleil. Nous montrons que sous certaines hypothèses sur le milieu quasi-stationnaire, les équations de Galbrun peuvent se simplifier en une équation qui admet la même formulation variationnelle que l'équation traitée dans la première partie de ce document. Des simulations numériques sont effectuées dans cette situation réaliste, et les différentes méthodes de résolution sont comparées.

The low-pressure spool in a high-bypass ratio turbofan engine has its unique characteristics. The slightness rotor and the large lumped mass in blade and disk are the structural characteristics while the dynamic coupling between bladed-disk and shaft is the mechanical characteristic. This paper studies on the modeling technology of a flexible rotor system based on the finite element method (FEM). The equivalent-disk method, an equivalent principle of finite element modeling, is put forward. The blades are equivalent to a disk which can not only retain the structural and mechanical characteristics but also control the scale of the model and convert the periodic geometry into axisymmetric geometry. The results show that the equivalent-disk method is helpful to improve the modeling techniques of rotor system and can be used in engineering.

We consider the propagation of a non-Boussinesq gravity current in an axisymmetric configuration (full cylinder or wedge). The current of density ρc is released from rest from a lock of radius r0 and height h0 into an ambient fluid of density ρa in a container of height H. When the Reynolds number is large, the resulting flow is governed by the parameters ρc/ρa and H* = H/h0. We show that the one-layer shallow-water model, carefully combined with a Benjamin-type front condition, provides a versatile formulation for the thickness and speed of the current, without any adjustable constants. The results cover in a continuous manner the range of light ρc/ρa ≪ 1, Boussinesq ρc/ρa ≈ 1, and heavy ρc/ρa ≫ 1 currents in a fairly wide range of depth ratio, H*. We obtain finite-difference solutions for the propagation and show that a self-similar behaviour develops for large times. This reveals the main features, in particular: (a) The heavy current propagates faster and its front is thinner than that for the light counterpart; (b) For large time, t, both the heavy and light currents spread like t1/2, but the thickness profiles display significant differences; (c) The energy-constrained propagation with the thickness of half-ambient-depth (when H* is close to 1) is a very limited occurrence, in contrast to the rectangular geometry counterpart in which this effect plays a major role. The predictions of the simple model are supported by some axisymmetric Navier–Stokes finite-difference simulations.

An approximate solution to Laplace's equation in an axisymmetric geometry is constructed in terms of a pair of two simple functions for the potential in an aperture cut in an infinite conducting tube. The two halves of the tube are held at opposite potentials. Upon differentiation, one aperture potential function provides for a constant axial electric field, and the other recovers the expected edge-condition behavior at the knife-edge of a charged conductor. Continuity of potential and one simple integration, in the spirit of the generalized Ritz-Galerkin method, furnishes integral expressions for the required constants in the expansion. These integrals are evaluated using a combination of analytic regularization and quadrature. The validity of the two-term expansion is verified by comparison with an independent moment method calculation.

Successive combined stamping-forging (CSF) is proposed to produce multi-thickness non-axisymmetric components. This method involves successive compression to create exclusively outward metal flow. Hitherto, the development of CSF has been mostly done for axisymmetric geometry. Using this technique, defect-free rectangular case component with length to thickness ratio of 40 is produced with lower forging pressure. This technology has potential for high throughput production of parts with multiple thicknesses and high width to thickness ratio.

Inviscid gravity currents released from a finite-length lock are known to propagate at a constant speed to a predicted finite distance before decelerating. By extension this should occur in a two-layer fluid with equal upper- and lower-layer depths for an intrusion having the average density of the ambient. The experiments presented here show this is not necessarily the case. The finite-depth thickness of the interface non-negligibly influences the evolution of the intrusion so that it propagates well beyond the predicted constant-speed limit; it propagates without decelerating beyond 22 lock lengths in a rectilinear geometry and beyond 6 lock radii in an axisymmetric geometry. Experiments and numerical simulations demonstrate that the intrusion speed decreases to half the two-layer speed in the circumstance in which the interface spans the domain. The corresponding long mode-2 interfacial wave speed increases rapidly with interfacial thickness, becoming comparable with the intrusion speed when the interfacial thickness is approximately one-quarter the domain height. For somewhat thinner interfacial thicknesses, the intrusion excites solitary waves that move faster than the long-wave speed. The coupling between intrusions and the waves they excite, together with reduced mixing of the current head, result in constant-speed propagation for longer times.

Hydrodynamic instabilities play an important role in the target compression for inertial confinement fusion (ICF). We present the analytical solution of a perturbed isentropic implosion. We compare the analytical solution to the results obtained with perturbation and bi-dimensional Lagrangian hydrodynamic codes. We also compare results from bi-dimensional code and perturbation code on an ICF like test case.

Experimental data for the evaporation coefficients of water have indicated a scatter over four orders of magnitude. One source of uncertainty in the data is probably the temperature at the liquid–vapour interface; recently published results in an axisymmetric geometry show temperature jumps of several °C at the free surface of evaporating water, contradicting the established assumption of a unique temperature in the liquid and vapour phases at the interface. In an apparatus with a two-dimensional geometry, temperature profiles in both phases near the interface during steady state evaporation confirm the earlier measurements and indicate the suitability of the apparatus for measurements close to the interface.

The appropriate augmented Young-Laplace equation for cylindrical and axisymmetric geometry is solved numerically to generate equilibrium drop and bubble shapes for various interaction-potential curves, P ( h ), where h is the distance of the liquid-vapor interface from the solid substrate. Order-of-reduction integration of the augmented Young-Laplace equation for the cylindrical geometry serves as a useful tool in determining how the characteristics of the interaction potential and the associated parameters affect droplet shapes. In particular, a graph of P ( h )+ p c h , where p c is the capillary pressure, versus h codifies the wide variety of drop and bubble shapes that arise from a given interaction-potential isotherm. Examples of shapes calculated are drops or bubbles with and without films, wiggly drops or bubbles, drops or bubbles that exhibit multiple equilibrium shapes, and drops with step structures. Specific conditions are provided for each type of drop shape.

An approach to view factor calculation for 2D axisymmetric geometry is proposed with analytical determination of integration range based on building projection of obscuring element on the plane.

We review mathematical models of tumor growth based on conservation laws in the full system of cells and interstitial liquid. First we deal with tumor cords evolving in axisymmetric geometry, where cells motion is simply passive and compatible with the saturation condition. The model is characterized by the presence of free boundaries with constraints driving the free boundary conditions, which in our opinion are particularly important, especially in the presence of treatments. Then a tumor spheroid is considered in the framework of the so-called two-fluid scheme. In a multicellular spheroid, on the appearance of a fully degraded necrotic core, the analysis of mechanical stresses becomes necessary to determine the motion via momentum balance, requiring the specification of the constitutive law for the “cell fluid.” We have chosen a Bingham-type law that presents considerable difficulties because of the presence of a yield stress, particularly with reference to the determination of an asymptotic configuration. Finally, we report some recent PDE-based models addressing complex processes in multicomponent tumors, more oriented to clinical practice.

We describe a theory for the removal of volatile organic chemicals from an unsaturated soil stratum consisting of highly porous coarse sand layers sandwiching a thin and semipervious lens. Each soil layer is modelled as a periodic array of spherical aggregates formed by solid grains and immobile water trapped by surface tension. Volatile chemicals are vaporized in the mobile air in pores between aggregates, dissolved in the intra-aggregate water, and adsorbed on the surface of soil grains. Using the effective transport equations derived for the aggregated soils, we consider shallow layers with sharp contrast in physical properties. An asymptotic analysis is developed for an axisymmetric geometry, yielding quasi-one-dimensional governing equations for individual layers. At the leading order the flow and the vapour transport are horizontal in the coarse layers but vertical in the semipervious lens. Numerical results are presented for a simple example to demonstrate the significance of the lens permeability, diffusivity and retardation factor, and the aggregate diffusivity in the coarse layers, on the vapour transport during the stages of contamination and air-venting.