ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010

Two Powerful Theorems in Clifford Analysis
View Description Hide DescriptionTwo useful theorems in Euclidean and Hermitean Clifford analysis are discussed: the Fischer decomposition and the Cauchy‐Kovalevskaya extension.

Numerical Energy Preservation of General Hamiltonian Systems
View Description Hide DescriptionA class of partitioned methods, combining collocation with averaged vector fields, is presented. The methods exactly preserve energy for general Hamiltonian systems, they are invariant with respect to linear transformations, and they can be of arbitrarily high order.

Exponential Time Integration of Evolution Equations
View Description Hide DescriptionThe time discretization of semilinear parabolic problems requires integrators that treat the stiff part in an implicit way. Traditionally, linearly implicit methods had been used for this purpose. More recently, exponential integrators proved to be competitive for this kind of problems. In this short note, we will put emphasis on methods that linearize the problem along the numerical trajectory. We will show that the error analysis can be carried out in the framework of logarithmic matrix norms which gives rise to convergence results that are independent of the spatial grid size.

Efficient Solution of Fluid‐Structure Interaction Problems in Computational Hemodynamics
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Ensemble Kalman and Filters
View Description Hide DescriptionThe ensemble Kalman filter has become a popular method for nonlinear data assimilation. Standard ensemble Kalman filter implementations need to be modified to avoid filter divergence due to model and statistical errors. In this communication, we discuss ensemble inflation within the continuous ensemble Kalman filter approach and its link to filtering.

The Acceptance Probability of the Hybrid Monte Carlo Method in High‐Dimensional Problems
View Description Hide DescriptionWe investigate the properties of the Hybrid Monte‐Carlo algorithm in high dimensions. In the simplified scenario of independent, identically distributed components, we prove that, to obtain an G(1) acceptance probability as the dimension d of the state space tends to ∞, the Verlet/leap‐frog step‐size h should be scaled as We also identify analytically the asymptotically optimal acceptance probability, which turns out to be 0.651 (with three decimal places); this is the choice that optimally balances the cost of generating a proposal, which decreases as ℓ increases, against the cost related to the average number of proposals required to obtain acceptance, which increases as ℓ increases.

Fast N‐Body Methods: Why, What, and Which
View Description Hide DescriptionWhy do they matter? Applications abound, even to the extent that fast N‐body methods merit a place in the core of numerical analysis. Why are they fast? The basis for all fast N‐body solvers is (i) a separable approximation for a pairwise interaction kernel and (ii) exploitation of the associativity (and distributivity) of linear transformations. What are the various N‐body methods? Kernel splitting and hierarchical clustering are the two fundamental paradigms. Which is the best N‐body method? For molecular biophysics and structural biology, multilevel summation is suggested.

Solving Differential Equations in R
View Description Hide DescriptionThe open‐source software R has become one of the most widely used systems for statistical data analysis and for making graphs, but it is also well suited for other disciplines in scientific computing. One of the fields where considerable progress has been made is the solution of differential equations. Here we first give an overview of the types of differential equations that R can solve, and then demonstrate how to use R for solving a 2‐Dimensional partial differential equation.

Remarks on the Complexity of the Schrödinger Equation
View Description Hide DescriptionThe solutions of the electronic Schrödinger equation are high‐dimensional objects depending on 3N variables, three for each of the N electrons under consideration. It is therefore rather surprising that simple expansions of the electronic wave functions can be constructed whose convergence rate, measured in terms of the number of determinants involved, is independent of the number of electrons and does not fall below that for a two‐ or even that for a one‐electron system approximated in the same way. In this sense, the complexity of the electronic Schrödinger equation does not exceed that of an equation in three space dimensions. A short report on these developments is given.

SYMPOSIUM: 5th Symposium on Numerical Analysis of Fluid Flow and Heat Transfer
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Turbulence Model Study for Unsteady Cavitating Flows
View Description Hide DescriptionA compressible, multiphase, one‐fluid RANS solver was developed to study cavitating flows. The interaction between turbulence and two‐phase structures is complex and not well known. This constitutes a critical point to accurately simulate unsteady behaviours of cavity sheets. In the present study, different transport‐equation turbulence models are investigated. Numerical results are given for a Venturi geometry and comparisons are made with experimental data.

The Effects of a Constant Bias Force on the Dynamics of a Periodically Forced Spherical Particle in a Newtonian Fluid at Low Reynolds Numbers
View Description Hide DescriptionWe make use of the formulation developed by Lovalenti and Brady [1] for the hydrodynamic force acting upon a spherical particle undergoing arbitrary time dependent motion in an arbitrary time dependent uniform flow field at low Reynolds numbers, to derive an expression for the effects of a constant bias force acting on a periodically forced rigid spherical particle in a Newtonian fluid. We use Newton’s second law to relate the total force acting on the particle to the motion of the particle. The total force is given by: where, is the external force inclusive of both the periodic force and the constant bias force. is the hydrodynamic force derived by Lovalenti and Brady [1] including both unsteady and convective inertia. The equation derived contains a nonlinear history term and is nonlinear. This equation is solved numerically using an adaptive step size Runge—Kutta scheme. We obtain several phase plots (plots between particle displacement and particle velocity), which show the effects of low Reynolds numbers, the periodic force and the effects of the constant bias force on the particle motion. It is observed that at low magnitudes of the periodic forcing the external constant force dominates and the particle moves along the direction of the external constant force. As we increase the magnitude of the periodic forcing, the periodic force is seen to dominate and the particle is seen to oscillate along a mean position with a slight drift along the direction of the periodic force and the external constant force, when they are imposed in the same direction. However the motion of the particle becomes more complicated when the directions of the periodic forcing and external constant force are opposite to each other. We also observe a reflection in phase space when the directions of both the forces are reversed. The phase plots typically are of a half sinusoidal, sinusoidal and a coiled (solenoidal) pattern. These plots include the effects of both periodic force and the constant bias force. As the Reynolds numbers increases the drift of the particle reduces, which indicates the effects of inertia. We present a preliminary analysis of the dynamics in this paper.

Fully Implicit Coupling for Non‐Matching Grids
View Description Hide DescriptionThe efficient solution of flow problems depends on quality meshing the computational domain. In problems with complex geometries or having a large spectrum of time or length scale, the meshing process greatly benefits from the subdivision of the original geometry (domain decomposition) into sub‐domains, that are meshed independently with suitable elements and mesh density. Procedures for solving multiblock meshes can be of two types explicit or implicit. In either case it is essential that the fluxes at the regions interfaces be conserved. In this paper an efficient fully implicit multi‐region coupling discretization procedure is presented. A test problem involving 1, 2, 4 and 8 blocks with a mesh size of about 100,000 elements, is solved to show that the coupling procedure yields the same number of iteration for multiple block as for a single block.

Numerical Simulation of Free Convection in a Porous Annulus of Rhombic Cross Section
View Description Hide DescriptionNumerical solutions are presented for laminar natural convection heat transfer in a fluid saturated porous enclosure between two isothermal concentric cylinders of rhombic cross sections. Simulations are conducted for four values of Raleigh number ( 105, 106, and 107), three values of Darcy number ( 10‐3, and 10‐5), and four values of enclosure gap ( 0.75, 0.5, and 0.25). The porosity and Prandtl number are assigned the values of 0.6 and 0.7, respectively. The results are reported in terms of streamlines, isotherms, and average Nusselt number values. The flow strength and convection heat transfer increase with an increase in Ra, Da, and/or Eg . At low Eg values, the flow in the enclosure is weak and convection heat transfer is low even though the total heat transfer is higher than at higher Eg values due to an increase in conduction heat transfer. Furthermore, predictions indicate the presence of a critical Ra number below which conduction is the dominant heat transfer mode. Convection starts affecting the total heat transfer at Ra values higher than the critical one. The critical Ra decreases with increasing Da, and increases with decreasing Eg.

Numerical Simulation of Channel Flow with Fluid Injection Using MILES Approach
View Description Hide DescriptionA numerical simulation of the unsteady flowfield in a channel with fluid injection through a porous wall is performed using MILES approach. Different states in such flow evolution are identified. Near the head end, the flow is laminar. At the middle of the channel, turbulence appears and then develops until the channel exit. The perturbation necessary to launch the transition process for the MILES simulation is obtained thanks to the time integration scheme used with a CFL value large enough to introduce a numerical destabilization into the flowfield. Computed results are compared with existing experimental data and with a two‐dimensional RANS k‐l simulation, including mean velocity and turbulent profiles. Analysis of the results shows that mean flow properties and transition process are reproduced in good agreement with the experimental data. The three‐dimensional MILES simulation gives more reliable results at the beginning of the transition process compared with the two‐dimensional RANS computation.

Numerical Analysis of the Unsteady Rotor‐Stator Interaction in a Low Pressure Centrifugal Compressor by Using Adamczyk and Proper Orthogonal Decompositions
View Description Hide DescriptionThe aim of this paper is to study the unsteady rotor‐stator interaction in a low pressure centrifugal compressor using the finite volume method to solve the Unsteady Reynolds‐Averaged Navier‐Stokes (URANS) equations. In order to understand better, the rotor‐stator interaction, the unsteady results are processed using both Adamczyk decomposition and Proper Orthogonal Decomposition (POD). Both decompositions show the behavior of unsteady rotor‐stator interaction but the POD modes also show the numerical errors.

Numerical Simulation of Airfoil Vibrations Induced by Compressible Flow
View Description Hide DescriptionThe paper is concerned with the numerical solution of interaction of compressible flow and a vibrating airfoil with two degrees of freedom, which can rotate around an elastic axis and oscillate in the vertical direction. Compressible flow is described by the Euler or Navier‐Stokes equations written in the ALE form. This system is discretized by the semi‐implicit discontinuous Galerkin finite element method (DGFEM) and coupled with the solution of ordinary differential equations describing the airfoil motion. Computational results showing the flow induced airfoil vibrations are presented.

Analysis of the Internal Ventilation for a Motorcycle Helmet
View Description Hide DescriptionThis work deals with a methodology for the numerical simulation of the inner ventilation of a motorcycle helmet, based on a thermo‐fluid‐dynamic model capable of describing evaporation‐related heat transfer phenomena. The final purpose is the enhancment of the comfort of the rider and ultimately his safety. The fluid‐dynamic problem concerns the modelization of the filtration of a flow over a porous medium, while the (decoupled) thermodynamic model is associated with the heat and sweat removal by means of the airflow. The latter is based on a set of evolution equations for the three scalar unknowns temperature, absolute humidity and sweat. Simulations on a sample 2D problem show the applicability of the methodology, highlighting the implicitly‐defined free boundary separating the wet and dry regions as well as the zones where sweat accumulates.

Towards a Formally Path‐Consistent Roe Scheme for the Six‐Equation Two‐Fluid Model
View Description Hide DescriptionWe start from the most common formulation of the six‐equation two‐fluid model, from which we remove the non‐conservative temporal term using an equivalent formulation derived in the literature. We derive a partially analytical, formally path‐consistent Roe scheme, using the flux‐splitting method.
We first expose the model in detail, and split the flux into a convective part, a pressure part, and a non‐conservative part. Then we derive an analytical Jacobian matrix of the fluxes, which allows the model to be written in quasilinear form. Finally, we explain the approach used to express formulas for the Roe‐averaging of the variables. Only a simplified Roe‐condition on the pressure remains. It can be fulfilled numerically, given any equation of state. In the present article, we do not show the full results, but rather explain the approach. The full results will be explained at the conference.