An extension of implicit Monte Carlo diffusion: Multigroup and the difference formulation

Implicit Monte Carlo (IMC) and Implicit Monte Carlo Diffusion (IMD) are approaches to the numerical solution of the equations of radiative transfer. IMD was previously derived and numerically tested on grey, or frequency-integrated problems [1]. In this research, we extend Implicit Monte Carlo Diffusion (IMD) to account for frequency dependence, and we implement the difference formulation [2] as a source manipulation variance reduction technique. We derive the relevant probability distributions and present the frequency dependent IMD algorithm, with and without the difference formulation. The IMD code with and without the difference formulation was tested using both grey and frequency dependent benchmark problems. The Su and Olson semi-analytic Marshak wave benchmark was used to demonstrate the validity of the code for grey problems [3]. The Su and Olson semi-analytic picket fence benchmark was used for the frequency dependent problems [4]. The frequency dependent IMD algorithm reproduces the results of both Su and Olson benchmark problems. Frequency group refinement studies indicate that the computational cost of refining the group structure is likely less than that of group refinement in deterministic solutions of the radiation diffusion methods. Our results show that applying the difference formulation to the IMD algorithm can result in an overall increase in the figure of merit for frequency dependent problems. However, the creation of negatively weighted particles from the difference formulation can cause significant numerical instabilities in regions of the problem with sharp spatial gradients in the solution. An adaptive implementation of the difference formulation may be necessary to focus its use in regions that are at or near thermal equilibrium.     

A 3D mesh deformation technique for irregular in‐flight ice accretion

Aircraft holding around busy airports may be requested to sustain as much as 45 min of icing before landing or being diverted to another airport. In this paper, a three‐dimensional mesh deformation scheme, based on a structural frame analogy, is proposed for the numerical simulation of ice accretion during extended exposure to adverse weather conditions. The goal is to provide an approach that is robust and efficient enough to delay or altogether avoid re‐meshing while preserving (enforcing) nearly orthogonal elements at the highly distorted ice surface. Robustness is achieved by suitably modifying the axial and torsional stiffness components of the frame elements in order to handle large and irregular grid displacements typical of in‐flight icing. Computational efficiency is obtained by applying the mesh displacement to an automatically selected small subset of the entire computational domain. The methodology is validated first in the case of deformations typical of fluid‐structure interaction problems, including wing bending, a helicopter rotor in forward flight, and the twisting of a high‐lift wing configuration. The approach is then assessed for aero‐icing on two swept wings and compared against experimental measurements where available. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal control for reinitialization in finite element level set methods

A new optimal control problem that incorporates the residual of the Eikonal equation into its objective is presented. The formulation of the state equation is based on the level set transport equation but extended by an additional source term, correcting the solution so as to minimize the objective functional. The method enforces the constraint so that the interface cannot be displaced at least in the continuous setting. The system of first‐order optimality conditions is derived, linearized, and solved numerically. The control also prevents numerical instabilities, so that no additional stabilization techniques are required. This approach offers the flexibility to include other desired design criteria into the objective functional. The methodology is evaluated numerically in three different examples and compared with other PDE‐based reinitialization techniques. Copyright © 2016 John Wiley & Sons, Ltd.     

Higher-order streamfunction-vorticity formulation of 2D steady-state Navier-Stokes equations

Based on regular boundary element method and a kind of linearity invariance under homotopy, a kind of numerical scheme of 2D steady-state Navier-Stokes equation in streamfunction-vorticity formulation is described. The flow inside a square cavity is used to illustrate this numerical scheme.     

Substructuring preconditioners for the three fields domain decomposition method

We study a class of preconditioners based on substructuring, for the discrete Steklov-Poincaré operator arising in the three fields formulation of domain decomposition in two dimensions. Under extremely general assumptions on the discretization spaces involved, an upper bound is provided on the condition number of the preconditioned system, which is shown to grow at most as ( and denoting, respectively, the diameter and the discretization mesh-size of the subdomains). Extensive numerical tests--performed on both a plain and a stabilized version of the method--confirm the optimality of such bound.

     

Equilibrium schemes for scalar conservation laws with stiff sources

We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a so-called equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients.

     

Network formed by traces of random walks

We propose a model of time evolving networks in which a kind of transport between vertices generates new edges in the graph. We call the model “Network formed by traces of random walks”, because the transports are represented abstractly by random walks. Our numerical calculations yield several important properties observed commonly in complex networks, although the graph at initial time is only a one-dimensional lattice. For example, the distribution of vertex degree exhibits various behaviors such as exponential, power law like, and bi-modal distribution according to change of probability of extinction of edges. Another property such as strong clustering structure and small mean vertex–vertex distance can also be found. The transports represented by random walks in a framework of strong links between regular lattice is a new mechanisms which yields biased acquisition of links for vertices.     

A kinetic formulation for a model coupling free surface and pressurised flows in closed pipes

The aim of this paper is to present a kinetic formulation of a model for the coupling of transient free surface and pressurised flows. Firstly, we revisit the system of Saint-Venant equations for free surface flow: we state some properties of Saint-Venant equations, we propose a kinetic formulation and we verify that this kinetic formulation leads to a Gibbs equilibrium that minimises (in some general case) an energy and preserves the still water steady state. Secondly, we propose a model for pressurised flows in a Saint-Venant-like conservative formulation. We then propose a kinetic formulation and we verify that this kinetic formulation leads to a Gibbs equilibrium that minimises in any case an energy and preserves the still water steady state. Finally, we propose a dual model that couples these two types of flow.     

BECOOL: Ballooning eigensolver with COOL finite elements

An incompressible variational ideal ballooning mode equation is discretized with the COOL finite element discretization scheme using basis functions composed of variable order Legendre polynomials. This reduces the second order ordinary differential equation to a special block pentadiagonal matrix equation that is solved using an inverse vector iteration method. A benchmark test of BECOOL (Ballooning Eigensolver using COOL finite elements) with second order Legendre polynomials recovers precisely the eigenvalues computed by the VVBAL shooting code. Timing runs reveal the need to determine an optimal lower order case. Eigenvalue convergence runs show that cubic Legendre polynomials construct the optimal ballooning mode equation for intensive computations.     

Elastic line deformed on a pseudo-hypersurface by an external field in pseudo-Euclidean spaces

We derive intrinsic formulation for elastic line deformed on a pseudo-hypersurface by an external field in the pseudo-Euclidean spaces E_v~n.This formulation determines elastic line deformed on a pseudo-hypersurface.