Hexagons and spiral defect chaos in non-Boussinesq convection at low Prandtl numbers

We study the stability and dynamics of non-Boussinesq convection in pure gases ?CO2 and SF6? with Prandtl numbers near Pr? 1 and in a H2-Xe mixture with Pr= 0.17. Focusing on the strongly nonlinear regime we employ Galerkin stability analyses and direct numerical simulations of the Navier-Stokes equations. For Pr ? 1 and intermediate non-Boussinesq effects we find reentrance of stable hexagons as the Rayleigh number is increased. For stronger non-Boussinesq effects the usual, transverse side-band instability is superseded by a longitudinal side-band instability. Moreover, the hexagons do not exhibit any amplitude instability to rolls. Seemingly, this result contradicts the experimentally observed transition from hexagons to rolls. We resolve this discrepancy by including the effect of the lateral walls. Non-Boussinesq effects modify the spiral defect chaos observed for larger Rayleigh numbers. For convection in SF6 we find that non-Boussinesq effects strongly increase the number of small, compact convection cells and with it enhance the cellular character of the patterns. In H2-Xe, closer to threshold, we find instead an enhanced tendency toward roll-like structures. In both cases the number of spirals and of targetlike components is reduced. We quantify these effects using recently developed diagnostics of the geometric properties of the patterns.

Reentrant hexagons in non-Boussinesq convection.

While non-Boussinesq hexagonal convection patterns are known to be stable close to threshold (i.e. for Rayleigh numbers R ? Rc ), it has often been assumed that they are always unstable to rolls for slightly higher Rayleigh numbers. Using the incompressible Navier?Stokes equations for parameters corresponding to water as the working fluid, we perform full numerical stability analyses of hexagons in the strongly nonlinear regime ( ? (R ? Rc )/Rc = O(1)). We find ?re-entrant? behaviour of the hexagons, i.e. as is increased they can lose and regain stability. This can occur for values of as low as = 0.2. We identify two factors contributing to the re-entrance: (i) far above threshold there exists a hexagon attractor even in Boussinesq convection as has been shown recently and (ii) the non-Boussinesq effects increase with . Using direct simulations for circular containers we show that the re-entrant hexagons can prevail even for sidewall conditions that favour convection in the form of competing stable rolls. For sufficiently strong non-Boussinesq effects hexagons even become stable over the whole -range considered, 0 6 6 1.5.

Differential invariants of second-order ordinary differential equations

The notion of a differential invariant for systems of second-order differential equations on a manifold M with respect to the group of vertical automorphisms of the projection is de?ned and the Chern connection attached to a SODE allows one to determine a basis for second-order differential invariants of a SODE.

A second order in time modified Lagrange-Galerkin finite element method for the incompressible Navier-Stokes equations

We introduce a second order in time modified Lagrange--Galerkin (MLG) method for the time dependent incompressible Navier--Stokes equations. The main ingredient of the new method is the scheme proposed to calculate in a more efficient manner the Galerkin projection of the functions transported along the characteristic curves of the transport operator. We present error estimates for velocity and pressure in the framework of mixed finite elements when either the mini-element or the $P2/P1$ Taylor--Hood element are used.

Involutivity of the Hamilton-Cartan equations of a second-order Lagrangian admitting a first-order Hamiltonian formalism

Involutivity of the Hamilton-Cartan equations of a second-order Lagrangian admitting a first-order Hamiltonian formalism

Reentrant and whirling hexagons in non-Boussinesq convection

We review recent computational results for hexagon patterns in non- Boussinesq convection. For sufficiently strong dependence of the fluid parameters on the temperature we find reentrance of steady hexagons, i.e. while near onset the hexagon patterns become unstable to rolls as usually, they become again stable in the strongly nonlinear regime. If the convection apparatus is rotated about a vertical axis the transition from hexagons to rolls is replaced by a Hopf bifurcation to whirling hexagons. For weak non-Boussinesq effects they display defect chaos of the type described by the two-dimensional (2D) complex Ginzburg-andau equation. For stronger non-Boussinesq effects the Hopf bifurcation becomes subcritical and localized bursting of the whirling amplitude is found. In this regime the cou- pling of the whirling amplitude to (small) deformations of the hexagon lattice becomes important. For yet stronger non-Boussinesq effects this coupling breaks up the hexagon lattice and strongly disordered states characterized by whirling and lattice defects are obtained.

Hexagonal patterns in a model for rotating convection

We study a model equation that mimics convection under rotation in a fluid with temperature- dependent properties (non-Boussinesq (NB)), high Prandtl number and idealized boundary conditions. It is based on a model equation proposed by Segel [1965] by adding rotation terms that lead to a Kuppers-Lortz instability [Kuppers & Lortz, 1969] and can develop into oscillating hexagons. We perform a weakly nonlinear analysis to find out explicitly the coefficients in the amplitude equation as functions of the rotation rate. These equations describe hexagons and os- cillating hexagons quite well, and include the Busse?Heikes (BH) model [Busse & Heikes, 1980] as a particular case. The sideband instabilities as well as short wavelength instabilities of such hexagonal patterns are discussed and the threshold for oscillating hexagons is determined.

Defect chaos and bursts: Hexagonal rotating convection and the complex Ginzburg-Landau equation

We employ numerical computations of the full Navier-Stokes equations to investigate non-Boussinesq convection in a rotating system using water as the working fluid. We identify two regimes. For weak non- Boussinesq effects the Hopf bifurcation from steady to oscillating (whirling) hexagons is supercritical and typical states exhibit defect chaos that is systematically described by the cubic complex Ginzburg-Landau equation. For stronger non-Boussinesq effects the Hopf bifurcation becomes subcritical and the oscil- lations exhibit localized chaotic bursting, which is modeled by a quintic complex Ginzburg-Landau equation.

Design equations for reinforced concrete members strengthened in shear with external FRP reinforcement formulated in an evolutionary multi-objective framework

Methods for predicting the shear capacity of FRP shear strengthened RC beams assume the traditional approach of superimposing the contribution of the FRP reinforcing to the contributions from the reinforcing steel and the concrete. These methods become the basis for most guides for the design of externally bonded FRP systems for strengthening concrete structures. The variations among them come from the way they account for the effect of basic shear design parameters on shear capacity. This paper presents a simple method for defining improved equations to calculate the shear capacity of reinforced concrete beams externally shear strengthened with FRP. For the first time, the equations are obtained in a multiobjective optimization framework solved by using genetic algorithms, resulting from considering simultaneously the experimental results of beams with and without FRP external reinforcement. The performance of the new proposed equations is compared to the predictions with some of the current shear design guidelines for strengthening concrete structures using FRPs. The proposed procedure is also reformulated as a constrained optimization problem to provide more conservative shear predictions.

Efficient adaptive methods based on adjoint equations and truncation error estimation for unstructured grids

El propósito de esta tesis es la implementación de métodos eficientes de adaptación de mallas basados en ecuaciones adjuntas en el marco de discretizaciones de volúmenes finitos para mallas no estructuradas. La metodología basada en ecuaciones adjuntas optimiza la malla refinándola adecuadamente con el objetivo de mejorar la precisión de cálculo de un funcional de salida dado. El funcional suele ser una magnitud escalar de interés ingenieril obtenida por post-proceso de la solución, como por ejemplo, la resistencia o la sustentación aerodinámica. Usualmente, el método de adaptación adjunta está basado en una estimación a posteriori del error del funcional de salida mediante un promediado del residuo numérico con las variables adjuntas, “Dual Weighted Residual method” (DWR). Estas variables se obtienen de la solución del problema adjunto para el funcional seleccionado. El procedimiento habitual para introducir este método en códigos basados en discretizaciones de volúmenes finitos involucra la utilización de una malla auxiliar embebida obtenida por refinamiento uniforme de la malla inicial. El uso de esta malla implica un aumento significativo de los recursos computacionales (por ejemplo, en casos 3D el aumento de memoria requerida respecto a la que necesita el problema fluido inicial puede llegar a ser de un orden de magnitud). En esta tesis se propone un método alternativo basado en reformular la estimación del error del funcional en una malla auxiliar más basta y utilizar una técnica de estimación del error de truncación, denominada _ -estimation, para estimar los residuos que intervienen en el método DWR. Utilizando esta estimación del error se diseña un algoritmo de adaptación de mallas que conserva los ingredientes básicos de la adaptación adjunta estándar pero con un coste computacional asociado sensiblemente menor. La metodología de adaptación adjunta estándar y la propuesta en la tesis han sido introducidas en un código de volúmenes finitos utilizado habitualmente en la industria aeronáutica Europea. Se ha investigado la influencia de distintos parámetros numéricos que intervienen en el algoritmo. Finalmente, el método propuesto se compara con otras metodologías de adaptación de mallas y su eficiencia computacional se demuestra en una serie de casos representativos de interés aeronáutico. ABSTRACT The purpose of this thesis is the implementation of efficient grid adaptation methods based on the adjoint equations within the framework of finite volume methods (FVM) for unstructured grid solvers. The adjoint-based methodology aims at adapting grids to improve the accuracy of a functional output of interest, as for example, the aerodynamic drag or lift. The adjoint methodology is based on the a posteriori functional error estimation using the adjoint/dual-weighted residual method (DWR). In this method the error in a functional output can be directly related to local residual errors of the primal solution through the adjoint variables. These variables are obtained by solving the corresponding adjoint problem for the chosen functional. The common approach to introduce the DWR method within the FVM framework involves the use of an auxiliary embedded grid. The storage of this mesh demands high computational resources, i.e. over one order of magnitude increase in memory relative to the initial problem for 3D cases. In this thesis, an alternative methodology for adapting the grid is proposed. Specifically, the DWR approach for error estimation is re-formulated on a coarser mesh level using the _ -estimation method to approximate the truncation error. Then, an output-based adaptive algorithm is designed in such way that the basic ingredients of the standard adjoint method are retained but the computational cost is significantly reduced. The standard and the new proposed adjoint-based adaptive methodologies have been incorporated into a flow solver commonly used in the EU aeronautical industry. The influence of different numerical settings has been investigated. The proposed method has been compared against different grid adaptation approaches and the computational efficiency of the new method has been demonstrated on some representative aeronautical test cases.

Interpretation of the pressuremeter test using numerical models based on deformation tensor equations

The pressuremeter test in boreholes has proven itself as a useful tool in geotechnical explorations, especially comparing its results with those obtained from a mathematical model ruled by a soil representative constitutive equation. The numerical model shown in this paper is aimed to be the reference framework for the interpretation of this test. The model analyses variables such as: the type of response, the initial state, the drainage regime and the constitutive equations. It is a model of finite elements able to work with a mesh without deformation or one adapted to it.

On the generative equations of fractal self-similarity in granular media and the related PSD models

The physical appearance of granular media suggests the existence of geometrical scale invariance. The paper discuss how this physico-empirical property can be mathematically encoded leading to different generative models: a smooth one encoded by a differential equation and another encoded by an equation coming from a measure theoretical property.

Chaos in non lineal Alfven waves using the DNLS equations

The electro-dynamical tethers emit waves in structured denominated Alfven wings. The Derivative Nonlineal Schrödinger Equation (DNLS) possesses the capacity to describe the propagation of circularly polarized Alfven waves of finite amplitude in cold plasmas. The DNLS equation is truncated to explore the coherent, weakly nonlinear, cubic coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. In this article is presented a theoretical and numerical analysis when the growth rate of the unstable wave is next to zero considering two damping models: Landau and resistive. The DNLS equation presents a chaotic dynamics when is consider only three wave truncation. The evolution to chaos possesses three routes: hard transition, period-doubling and intermittence of type I.

A dynamic programming approach to the formulation and solution of finite element equations

A method for formulating and algorithmically solving the equations of finite element problems is presented. The method starts with a parametric partition of the domain in juxtaposed strips that permits sweeping the whole region by a sequential addition (or removal) of adjacent strips. The solution of the difference equations constructed over that grid proceeds along with the addition removal of strips in a manner resembling the transfer matrix approach, except that different rules of composition that lead to numerically stable algorithms are used for the stiffness matrices of the strips. Dynamic programming and invariant imbedding ideas underlie the construction of such rules of composition. Among other features of interest, the present methodology provides to some extent the analyst's control over the type and quantity of data to be computed. In particular, the one-sweep method presented in Section 9, with no apparent counterpart in standard methods, appears to be very efficient insofar as time and storage is concerned. The paper ends with the presentation of a numerical example