Multigrid preconditioning of steam generator two-phase mixture balance equations in the Genepi software

International audienceWithin the framework of averaged two-phase mixture flow simulations of PWR Steam Generators (SG), this paper provides a geometric version of a pseudo-FMG FAS preconditioning of the balance equations used in the CEA Genepi code. The 3D steady-state flow is reached by a transient computation using a fractional step algorithm and a projection method. Our application is based on the PVM package. The difficulties of applying geometric FAS multigrid methods to the balance equations solver are addressed. The effects on the convergence behaviour of the numerical parameters are investigated. An original parallel red-black pseudo-FMG FAS multigrid algorithm is also presented. The use of dynamic multigrid cycles leads to perceptible improvements in the computation convergences. Numerical tests (academic and industrial simulations) underline a noticeable computation speed-up, essentially for a large number of freedom degrees: the speed-up reached for 2 or 3 grids ranges between 2 and 3

FICTITIOUS DOMAIN SIMULATIONS FOR THE TWO-PHASE FLOW ENERGY BALANCE OF THE CLOTAIRE STEAM GENERATOR MOCK-UP

International audienceThe context of this paper is the numerical simulation of a nuclear component, in the framework of the NEPTUNE platform, by a fictitious domain method. We consider industrial simulations of twophase flows with the homogeneous equilibrium (or relaxed) model. The introduction of no remeshing fictitious domain method is mainly motivated by free surface studies, fluid-structure interactions or fast Cartesian mesh solvers. As a first step toward a full fictitious domain simulation, this paper focuses on the fictitious domain computation of the energy balance equation of a nuclear component. Considering the steam generator, this equation is solved by a finite-element volume-penalization method. We recall the model used for the energy balance equation and review the modelizations and the computations. An industrial simulation of the steam generator mock-up CLOTAIRE is presented in order to appreciate the accuracy and the limits of the fictitious domain approach. Exploring L2-norm error along some vertical enthalpy profiles, we claim that the relative error introduced by the fictitious domain method is globally decreasing with the space step and can be lower than 10-3 for space steps around the U-tube diameter size. We conclude that we can reach an enough precision for the industrial applications and benefit from numerical advantages due to the use of Cartesian meshes

Fictitious domain methods for two-phase flow energy balance computations in nuclear components

This paper is dedicated to the numerical simulation of nuclear components (cores and steam generators) by fictitious domain methods. The fictitious domain approach consists in immersing the physical domain under study in a Cartesian domain, called the fictitious domain, and in performing the numerical resolution on this fictitious domain. The calculation times are then efficiently reduced by the use of fast solvers. In counterpart, one has to handle with an immersed boundary, generally non-aligned with the Cartesian mesh, which can be non-trivial. The two fictitious domain methods compared here on industrial simulations and developed by Ramière et al. deal with an approximate immersed interface directly derived from the uniform Cartesian mesh. All the usual immersed boundary conditions (Dirichlet, Robin, Neumann), possibly mixed, are handled through a unique formulation of the fictitious problem. This kind of approximation leads to first-order methods in space that exhibit a good ratio of the precision of the approximate solution over the CPU time, which is very important for industrial simulations. After a brief recall of the fictitious domain method with spread interface (Ramière et al., CMAME 2007) and the fictitious domain method with immersed jumps (Ramière et al., JCP 2008), we will focus on the numerical results provided by these methods applied to the energy balance equation in a steam generator. The advantages and drawbacks of each method will be pointed out. Generally speaking, the two methods confirm their very good efficiency in terms of precision, convergence, and calculation time in an industrial context

MULTIGRID PRECONDITIONING OF THE STEAM GENERATOR TWO-PHASE MIXTURE BALANCE EQUATIONS IN THE GENEPI SOFTWARE

International audienceIn the framework of the two-phase fluid simulations of the steam generators of pressurized water nuclear reactors, we present in this paper a geometric version of a pseudo-Full MultiGrid (pseudo-FMG) Full Approximation Storage (FAS) preconditioning of balance equations in the GENEPI code. In our application, the 3D steady state flow is reached by a transient computation using a semi-implicit fractional step algorithm for the averaged two-phase mixture balance equations (mass, momentum and energy for the secondary flow). Our application, running on workstation clusters, is based on a CEA code-linker and the PVM package. The difficulties to apply the geometric FAS multigrid method to the momentum and mass balance equations are addressed. The use of a sequential pseudo-FMG FAS two-grid method for both energy and mass/momentum balance equations, using dynamic multigrid cycles, leads to perceptibly improvements in the computation convergences. An original parallel red-black pseudo-FMG FAS three-grid algorithm is presented too. The numerical tests (steam generator mock-up simulations) underline the sizable increase in speed of convergence of the computations, essentially for the ones involving a large number of freedom degrees (about 100 thousand cells). The two-phase mixture balance equation residuals are quickly reduced: the reached speed-up stands between 2 and 3 following the number of grids. The effects on the convergence behavior of the numerical parameters are investigated

Computation of two-phase flow in steam generator using Domain Decomposition and local zoom methods

International audienceWe present flow simulations in the Steam Generator of a pressurised water nuclear reactor using Domain Decomposition (DDM) and local zoom methods on workstation cluster. Concerning the DDM, we use a Dirichlet-Neumann approach jointly with FEM for averaged mixture balance equations. The algorithm, based on parallel or sequential iteration-by-subdomain method, works with overlapping or nonoverlapping subdomains and with conforming or nonconforming meshing. With DDM, the computational problem size is easily enhanced to about 100,000 mesh cells and the CPU time is strongly reduced. Concerning the Local Zoom computations, the used Local Defect Correction Method (LDC), in 3D local hierarchical multigrid context, is shown. The LDC computation results are compared with the classical full domain computation results (with high or low spatial resolution). We conclude to an improvement of the accuracy on the full domain with a high coherence between the zoom and the full domain

Local Zoom Computation of Two-Phase Flows in Steam Generators using a Local Defect Correction Method

International audienceA local defect method used to perform local zoom computations within the framework of the steady-state two-phase flow simulations of pressurized water reactor (PWR) steam generators is described. The particular local defect correction (LDC) formulation used, jointly with the finite-element method (FEM), a projection algorithm, and the Crank-Nicholson scheme for nonlinear averaged mixture balance equations, is discussed, as well as the particular geometry involved (3-D local hierarchical multigrid). In the case of vortices located at an inner interface, the boundary conditions are dynamically managed in an approach "à la adaptive Dirichlet Neumann." Cluster workstations, using a master-slaves context (through a code linker) and the PVM package, are used. Results concerning the simulation of a mock-up are provided. The parallel and sequential LDC computation results are compared with the results of classical full-domain computations. The conclusion describes the improvements in the accuracy of the corrected region and on the coherence between the zoom and the full domain

Penalized Direct Forcing and Projection Schemes for Navier-Stokes

This note presents a new method of direct forcing to deal with obstacles in incompressible flows. It mixes projection schemes and velocity L2 penalty schemes. The penalized direct forcing term is distributed in the velocity prediction and the correction equations. It leads to a natural treatment in the correction equation of the boundary conditions in pressure around obstacles. A numerical experiment provided an illustration of the method

Fictitious domain methods to solve convection-diffusion problems with general boundary conditions

International audienceSince a few years, fictitious domain methods have been arising for Computational Fluid Dynamics. The main idea of these methods consists in immersing the original physical domain in a geometrically bigger and simply-shaped other one called fictitious domain. As the spatial discretization is then performed in the fictitious domain, simple structured meshes can be used. The aim of this paper is to solve convection-diffusion problems with fictitious domain methods which can easily simulate free-boundary with possibly deformations of the boundary without increasing the computational cost. Two fictitious domain approaches performing either a spread interface or a thin interface are introduced. These two approaches require neither the modification of the numerical scheme near the immersed interface nor the use of Lagrange multipliers. Several ways to impose general embedded boundary conditions (Dirichlet, Robin or Neumann) are presented. The spread interface approach is computed using a finite element method as a finite volume method is used for the thin interface approach. The numerical schemes conserve the first- order accuracy with respect to the discretization step as observed in the numerical results reported here. The spread interface approach is then combined with a local adaptive mesh refinement algorithm in order to increase the precision in the vicinity of the immersed boundary. The results obtained are full of promise, more especially as convection-diffusion equations are the core of the resolution of Navier-Stokes equations

A Second Order Penalized Direct Forcing for Hybrid Cartesian/Immersed Boundary Flow Simulations

International audienceFlows around complex stationary/moving solids take an important place in life-science context or in many engineering applications. Usually, these problems are solved by body-fitted approaches on unstructured meshes with boundary conditions directly imposed on the domain boundary. Another way is using immersed boundary (IB) techniques: the physical domain is immersed in a fixed fictitious one of simpler geometry on Cartesian grids. It allows to use efficient, fast and accurate numerical methods avoiding the tedious task of re-meshing in case of time varying geometry. In contrast, one needs specific methods to take into account the IB conditions (IBC). Here, we propose a second order penalized direct forcing method for unsteady incompressible flows with Dirichlet's IBC. It consists in adding a penalized forcing term to the initial problem, applied only on Cartesian nodes near the IB, in order to bring back the variable to the imposed one. Regarding Navier-Stokes solvers using a projection scheme, the forcing term is distributed both in the velocity prediction and in the correction equations. It leads to a natural way to prescribe the pressure boundary conditions around obstacles. Numerical experiments, performed for laminar flows around static/moving solids, assess the validity and illustrate the ability of our method, showing in particular a quadratic convergence rate

Numerical Platon: a unified linear equation solver interface for industrial softwares

This paper describes a tool called Numerical Platon developed by the French Atomic Energy Commission (CEA). It provides an interface to a set of parallel linear equation solvers for high-performance computers that may be used in industrial software written in various programming languages. This tool was developed as part of considerable efforts by the CEA Nuclear Energy Division in the past years to promote massively parallel software and on-shelf parallel tools to help develop new generation simulation codes. After the presentation of the package architecture and the available algorithms, we show examples of how Numerical Platon is used in CEA codes