Variational Average-Atom in Quantum Plasmas (VAAQP) – Application to radiative properties

We present an application of the Variational Average-Atom in Quantum Plasmas (VAAQP) model and numerical code to dense plasmas radiative properties. We propose an approximate Detailed Configuration Accounting (DCA) approach starting from the variational model of configurations in jellium. This leads to a simplified model which is qualitatively different from those which usually stem from the atom-in-cell approach. It is also shown that, with some additional approximations, the DCA calculation can be handled by use of the Gaussian approximation to perform a statistical approach to the spectrum. Our purpose here is to show that this model provides a simple way to calculate the radiative properties and is likely to give realistic results.     

On the generalized virial theorem and Eshelby tensors

The formal relationships between the scalar and tensorial virials and Eshelby tensors have been presently investigated. The key idea is to evaluate the Eshelby stress from discrete or atomistic simulations for a structured body, conceived as a numerical homogenization method to reconstitute the macroscopic continuum behavior in multiscale modelling approaches. Extending first the writing of the scalar virial to a material format, it is shown that the average of the elaborated scalar material virial is the trace of the (material) Eshelby stress. The spatial and material virials are further related to eachother in the framework of hyperelasticity, and a tensorial extension of the material virial is provided. Interpretation of those results from the microscopic point of view shows that Eshelby stress may be identified and calculated at the discrete level from the average of the virial tensor. Consideration of the material version of the virial theorem further leads to express Eshelby stress versus the average of the internal tensorial material virial and of the kinetic energy. The average scalar virial is further identified to the grand potential in a thermodynamic context. A definition of the material scalar virial for a second order continuum is lastly proposed, based on the identification of a second order Eshelby stress and in line with the second order Cauchy–Born rule.     

On the generalized virial theorem for systems with variable mass

We presently extend the virial theorem for both discrete and continuous systems of material points with variable mass, relying on developments presented in Ganghoffer (Int J Solids Struct 47:1209–1220, 2010). The developed framework is applicable to describe physical systems at very different scales, from the evolution of a population of biological cells accounting for growth to mass ejection phenomena occurring within a collection of gravitating objects at the very large astrophysical scales. As a starting basis, the field equations in continuum mechanics are written to account for a mass source and a mass flux, leading to a formulation of the virial theorem accounting for non-constant mass within the considered system. The scalar and tensorial forms of the virial theorem are then written successively in both Lagrangian and Eulerian formats, incorporating the mass flux. As an illustration, the averaged stress tensor in accreting gravitating solid bodies is evaluated based on the generalized virial theorem.     

3-D numerical simulation of fracture processes in heterogeneous brittle materials

By using the lattice model combined with finite element methods and statistical techniques, a numerical approach is developed to establish mechanical models of three-dimensional heterogeneous brittle materials. A special numerical code is introduced, in which a lattice model and statistical approaches are used to simulate the initial heterogeneity of material properties. The size of displacement-load step is adaptively determined so that only few elements would fail in each load step. When the tensile principal strain in an element exceeds the ultimate strain of this element, the element is considered broken and its Young's modulus is set to be very low. Some important behaviors of heterogeneous brittle materials are indicated using this code. Load-displacement curves and figures of three-dimensional fracture patterns are also numerically obtained, which are similar to those observed in laboratory tests.     

Shear-horizontal (SH) surface waves propagating around stratified piezoelectric cylinders

Theoretical results describing the properties of SH (shear-horizontal) surface waves propagating around piezoelectric cylinders with superimposed layers are presented. Mode orthogonality, virial theorem and equality between group and energy velocities are proved. Integral formulas for the energy velocity are much more compact than the differential expressions for the group velocity, and this provides a convenient way for computing the latter. The results obtained are illustrated by numerical calculations and graphs for a set of typical cylindrical waveguides.     

The relativistic virial theorem in plasma EOS calculations

A method is presented for accurate calculation of equation of state (EOS) for warm dense matter. The method extends an approach presented recently, based on the adjustment of the correlation energy to impose consistency between two pressure representations: the volume derivative of the free energy and the relativistic virial theorem. In this work we show that the free energy of any neutral system obeys a fundamental differential equation, which bypasses the correlation specifics and serves as a basis to enhance EOS approximations. Specifically, we start with LDA calculations and improve the results significantly using this equation with a boundary condition at the zero pressure point. The method retains the emphasis on thermal excitations, but connects to the appropriate results at low temperatures. It effectively compensates for simplifications, including the use of a spherical model to account for global solid structure effects. EOS and opacities are calculated on the same footing for low to high Z elements and in large domains of density and temperature without recourse to parametric fitting procedures. Excellent agreement is obtained with experiments. Finally the method is applied successfully to calculate EOS and opacities for mixtures. Results for C–H mixture are compared with other calculations.     

Velocity-based reciprocal theorems in elastodynamics and BIEM implementation issues

Reciprocal theorems in elastodynamics are introduced as extensions of respective theorems from elastostatics. Inasmuch as the latter is a subset of the former, the aim here is to present an elastodynamic reciprocal theorem that also includes elastostatics as a special case when the time variable becomes irrelevant. This is accomplished by introducing a velocity-based reciprocal theorem, whose basic properties are subsequently explored. The next step is to use this theorem and formulate a numerical approach based on boundary integral equation statements and compare them with existing formulations based on conventional reciprocity relations. The applications presented here involve the standard mechanical oscillator and the unidimensional axial element as two simple, yet important problems of structural dynamics. Along with the numerical results, a thorough stability analysis of the corresponding time-stepping algorithms is formulated. In both cases, the superior performance of the methodologies built on velocity-based reciprocal theorems is clearly demonstrated.     

Approche statistique du tenseur des contraintes par le viriel généralisé

The classical virial theorem, applied to particles without internal structure, is used to make a direct statistical estimation of a fluid pressure avoiding a thermodynamical potential derivation. Generalising this theorem to structured media leads to a direct statistical estimation of the stress tensor.     

Partial stresses in heterogeneous media by a direct statistical approachContraintes partielles en milieux hétérogènes par une approche statistique

The total stress tensor in a structured or non-structured medium can be obtained by a direct statistical approach using the generalized virial theorem, without any reference to a potential function, as soon as positions, velocities and interactions of the particles are given by Molecular Dynamics. However, as shown here, it would be wrong to apply these results to a given class of particles in an heterogeneous medium without adding a cross internal virial tensor to the self internal virial tensor and the partial kinetic energy tensor relative to this class of particles. To cite this article: P. Jouanna, L. Pèdesseau, C. R. Mecanique 332 (2004).     

三维显式有限元程序及炸药冲击起爆应用

冲击和爆炸数值模拟在军事工程和民用领域都应用广泛。基于显式有限元理论,本文介绍了自主研发的三维冲击和起爆流体动力学程序。该程序的功能包含了材料本构、状态方程、炸药反应速率方程和接触碰撞算法,能够模拟复杂条件下结构冲击和炸药起爆问题。由于采用面向对象的编程方法,程序结构相对简单并且容易修改。首先,进行了Taylor杆碰撞数值模拟,并将计算结果与实验结果以及DYNA2D结果进行了比较;接着,基于Lee-Tarver点火增长模型,模拟了冲击加载下裸炸药和带壳炸药的冲击起爆问题。计算结果表明,三维程序计算结果与实验值和商业软件结果都非常吻合,该程序能够较准确地模拟三维结构的冲击和炸药起爆问题。同时,基于面向对象开发的三维计算程序实现了模块化功能,易于后续开发,为冲击和爆炸计算程序的开发提供了参考。     

Freezing of Water in a Differentially Heated Cubic Cavity

An experimental and numerical study has been made of transient natural convection of water freezing in a cube-shaped cavity. The effect of the heat transfer through the side walls is studied in two configurations: with the cavity surrounded by air and with the cavity immersed in an external water bath of constant temperature. The experimental data for the velocity and temperature fields are obtained using liquid crystal tracers. The transient development of the ice/water interface is measured. The collected data are used as an experimental benchmark and compared with numerical results obtained from a Finite-difference code with boundary fitted grid generation. The computational model has been adopted to simulate as closely as possible the physical experiment. Hence, fully variable fluid properties are implemented in the code, and, to improve modelling of the thermal boundary conditions, the energy equation is also solved inside the bounding walls. Although the general behaviour of the calculated ice front and its volume matches observations, several details of the flow structure do not. Observed discrepancies between experimental and numerical results indicate the necessity of verifying and improving the usual assumptions for modelling ice formation.     

Exploring the origin, the nature, and the dynamical behavior of distant stars in galaxy models

We explore the regular or chaotic nature of orbits moving in the meridional plane of an axially symmetric galactic gravitational model with a disk, a dense spherical nucleus, and some additional perturbing terms corresponding to influence from nearby galaxies. In order to obtain this, we use the Smaller ALingment Index (SALI) technique integrating extensive samples of orbits. Of particular interest is the study of distant, remote stars moving in large galactocentric orbits. Our extensive numerical experiments indicate that the majority of the distant stars perform chaotic orbits. However, there are also distant stars displaying regular motion as well. Most distant stars are ejected into the galactic halo on approaching the dense and massive nucleus. We study the influence of some important parameters of the dynamical system, such as the mass of the nucleus and the angular momentum, by computing in each case the percentage of regular and chaotic orbits. A second-order polynomial relationship connects the mass of the nucleus and the critical angular momentum of the distant star. Some heuristic semi-theoretical arguments to explain and justify the numerically derived outcomes are also given. Our numerical calculations suggest that the majority of distant stars spend their orbital time in the halo where it is easy to be observed. We present evidence that the main cause for driving stars to distant orbits is the presence of the dense nucleus combined with the perturbation caused by nearby galaxies. The origin of young O and B stars observed in the halo is also discussed.     

Multicomponent fluid flow analysis using a new set of conservation equations

In this work hydrodynamics of multicomponent ideal gas mixtures have been studied. Starting from the kinetic equations, the Eulerian approach is used to derive a new set of conservation equations for the multicomponent system where each component may have different velocity and kinetic temperature. The equations are based on the Grad's method of moment derived from the kinetic model in a relaxation time approximation (RTA). Based on this model which contains separate equation sets for each component of the system, a computer code has been developed for numerical computation of compressible flows of binary gas mixture in generalized curvilinear boundary conforming coordinates. Since these equations are similar to the Navier-Stokes equations for the single fluid systems, the same numerical methods are applied to these new equations. The Roe's numerical scheme is used to discretize the convective terms of governing fluid flow equations. The prepared algorithm and the computer code are capable of computing and presenting flow fields of each component of the system separately as well as the average flow field of the multicomponent gas system as a whole. Comparison of the present code results with those of a more common algorithm based on the mixture theory in a supersonic converging-diverging nozzle provides the validation of the present formulation. Afterwards, a more involved nozzle cooling problem with a binary ideal gas (helium-xenon) is chosen to compare the present results with those of the ordinary mixture theory. The present model provides the details of the flow fields of each component separately which is not available otherwise. It is also shown that the separate fluids treatment, such as the present study, is crucial when considering time scales on the order of (or shorter than) the intercollisions relaxation times.     

Density effects in plasmas: Detailed atomic calculations and analytical expressions

In warm dense plasmas, the free-electron and ion spatial distribution may strongly affect atomic structure. To account for such effects we have implemented a potential correction based on the uniform electron gas model in the Flexible Atomic Code (FAC). This code has been applied to obtain energies, wave-functions and radiative rates modified by the plasma environment. In hydrogen-like ions, these numerical results have been successfully compared to an analytical calculation based on first-order perturbation theory. In the case of multi-electron ions, we observe level crossings in agreement with another recent model calculation.     

Ho's theorem in global-local mode interaction of pin-jointed bar structures

This paper is focused on the interaction phenomena among a global critical mode and some local Eulerian critical modes in pin-jointed structures. These phenomena are framed within Koiter's theory of elastic instability, by an asymptotic reduction into cubic systems. The aim is to present an algorithm for the appraisal of the lowest critical load characterizing the structure under the effect of small imperfections. First of all, the Ho's theorem, concerning the definition of the most dangerous imperfection, is presented and discussed. Then, a FEM code aimed at the determination of the most dangerous shape for the imperfection, and at performing the related sensitivity analysis, is implemented, by superimposing a proper FE beam model (able to model Eulerian instability) to a non-linear FE model for spatial pin-jointed structures. Some numerical results having a practical interest are presented and discussed.     

Experimental and numerical investigation of a viscoplastic Carbopol gel injected into a prototype 3D mold cavity

A numerical model for free-surface flow of a viscoplastic liquid into a cavity is presented. This flow is regarded as a basic model of injection molding, which is a widely used processing technology. Model experiments of the injection process are performed with a water-based gel with shear-thinning behavior. The filling process is visualized by tracing the free surface of the gel within the cavity. Filling times of the cavity are deduced from the experimental observations. The filling process is also analyzed by means of numerical simulation.The flow equations are integrated according to the finite-volume method. The volume-of-fluid method is employed in order to describe the flow of two incompressible, immiscible phases, the phase interface is resolved by the method of geometric reconstruction or alternatively by the method of surface compression. The Herschel–Bulkley model is used in order to describe the shear-thinning behavior of the gel and the effects of a yielding point. The governing equations of the flow are solved by means of the commercial code Fluent as well as the Open Source code OpenFOAM.The results of the numerical simulations are analyzed in detail. They are compared with the experimental findings. Cavity filling times in the experiments and the simulations are in good agreement. Different patterns of the filling flow depending on the injection parameters are evident in the experiments and the simulations. They are characterized and arranged with respect to the similarity parameters of the flow configuration. Again, the results of the simulation are found to agree well with the experimental observations.     

Limit analysis and homogenization of porous materials with Mohr–Coulomb matrix. Part II: Numerical bounds and assessment of the theoretical model

This second part of the two-part study is devoted to the numerical Limit Analysis of a hollow sphere model with a Mohr–Coulomb matrix and its use for the assessment of theoretical results. Brief background and fundamental of the static and kinematic approaches in the context of numerical limit analysis are first recalled. We then present the hollow sphere model, together with its axisymmetric FEM discretization and its mechanical position. A conic programming adaptation of a previous iterative static approach, based on a piecewise linearization (PWL) of the plasticity criterion, was first realized. Unfortunately, the resulting code, no more than the PWL one, did not allow sufficiently refined meshes for loss of convergence of the conic optimizer. This problem was solved by using the projection algorithm of Ben Tal and Nemriovski (BTN) and the (interior point) linear programming code XA. For the kinematic approach, a first conic adaptation appeared also inefficient. Then, an original mixed (but fully kinematic) approach dedicated to the general Mohr–Coulomb axisymmetric problem was elaborated. The final conic mixed code appears much more robust than the classic one when using the conic code MOSEK, allowing us to take into account refined numerical meshes. After a fine validation in the case of spherical cavities and isotropic loadings (for which the exact solution is known) and comparison to previous (partial) results, numerical lower and upper bounds (a posteriori verified) of the macroscopic strength are provided. These bounds are used to assess and validate the theoretical results of the companion (part I) paper. Effects of the friction angle as well as that of the porosity are illustrated.