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. Copyright © 2011 John Wiley & Sons, Ltd.     

On the usage of NURBS as interface representation in free‐surface flows

When simulating free‐surface flows using the finite element method, there are many cases where the governing equations require information which must be derived from the available discretized geometry. Examples are curvature or normal vectors. The accurate computation of this information directly from the finite element mesh often requires a high degree of refinement—which is not necessarily required to obtain an accurate flow solution. As a remedy and an option to be able to use coarser meshes, the representation of the free surface using non‐uniform rational B‐splines (NURBS) curves or surfaces is investigated in this work. The advantages of a NURBS parameterization in comparison with the standard approach are discussed. In addition, it is explored how the pressure jump resulting from surface tension effects can be handled using doubled interface nodes. Numerical examples include the computation of surface tension in a two‐phase flow as well as the computation of normal vectors as a basis for mesh deformation methods. For these examples, the improvement of the numerical solution compared with the standard approaches on identical meshes is shown. Copyright © 2011 John Wiley & Sons, Ltd.     

A fixed‐grid b‐spline finite element technique for fluid–structure interaction

We present a fixed‐grid finite element technique for fluid–structure interaction problems involving incompressible viscous flows and thin structures. The flow equations are discretised with isoparametric b‐spline basis functions defined on a logically Cartesian grid. In addition, the previously proposed subdivision‐stabilisation technique is used to ensure inf–sup stability. The beam equations are discretised with b‐splines and the shell equations with subdivision basis functions, both leading to a rotation‐free formulation. The interface conditions between the fluid and the structure are enforced with the Nitsche technique. The resulting coupled system of equations is solved with a Dirichlet–Robin partitioning scheme, and the fluid equations are solved with a pressure–correction method. Auxiliary techniques employed for improving numerical robustness include the level‐set based implicit representation of the structure interface on the fluid grid, a cut‐cell integration algorithm based on marching tetrahedra and the conservative data transfer between the fluid and structure discretisations. A number of verification and validation examples, primarily motivated by animal locomotion in air or water, demonstrate the robustness and efficiency of our approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust multilevel methods for quadratic finite element anisotropic elliptic problems

This paper discusses a class of multilevel preconditioners based on approximate block factorization for conforming finite element methods employing quadratic trial and test functions. The main focus is on diffusion problems governed by a scalar elliptic partial differential equation with a strongly anisotropic coefficient tensor. The proposed method provides a high robustness with respect to non‐grid‐aligned anisotropy, which is achieved by the interaction of the following components: (i) an additive Schur complement approximation to construct the coarse‐grid operator; (ii) a global block (Jacobi or Gauss–Seidel) smoother complementing the coarse‐grid correction based on (i); and (iii) utilization of an augmented coarse grid, which enhances the efficiency of the interplay between (i) and (ii). The performed analysis indicates the high robustness of the resulting two‐level method. Moreover, numerical tests with a nonlinear algebraic multilevel iteration method demonstrate that the presented two‐level method can be applied successfully in the recursive construction of uniform multilevel preconditioners of optimal or nearly optimal order of computational complexity. Copyright © 2013 John Wiley & Sons, Ltd.     

Nonexistence of global solutions to new ordinary differential inequality and applications to nonlineardispersive equations

A new ordinary differential inequality without global solutions is proposed. Comparison with similar differential inequalities in the well‐known concavity method is performed. As an application, finite time blow up of the solutions to nonlinear Klein–Gordon equation and generalized Boussinesq equation is proven. The initial energy is arbitrary high positive. The structural conditions on the initial data generalize the assumptions used in the literature for the time being. Copyright © 2015 John Wiley & Sons, Ltd.     

Split least‐squares finite element methods for non‐Fickian flow in porous media

In this article, we introduce two least‐squares finite element procedures for parabolic integro‐differential equations arising in the modeling of non‐Fickian flow in porous media. By selecting the least‐squares functional properly the presented procedure can be split into two independent subprocedures, one subprocedure is for the primitive unknown and the other is for the flux. The optimal order convergence analysis is established. Numerical examples are given to show the efficiency of the introduced schemes. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Resolution estimates for simple one-dimensional flames

The accurate resolution of flame structure is critically important to the direct numerical simulation of reacting flows. While grid-resolution estimates are readily available for cold flows, premixed flames appear to have received relatively little attention. In this paper, a premixed flame characterised by single-step chemistry at moderate-to-high Zel'dovitch numbers (β) is analysed, and its structure is used to provide estimates for sufficient grid resolution. It is found that the critical region of the flame exhibits a weak inverse Zel'dovitch number dependence, and that heuristic methods of resolution estimation based on flame thickness grow relatively less meaningful as β → ∞. Resolution estimates for second- and fourth-order finite-difference schemes are provided.     

Dynamic analysis on generalized linear elastic body subjected to large scale rigid rotations

The dynamic analysis of a generalized linear elastic body undergoing large rigid rotations is investigated. The generalized linear elastic body is described in kine-matics through translational and rot...     

Generalized matching theorems for closed coverings of convex sets

In this paper we use fixed point and coincidence theorems due to Park [8] to give matching theorems concerning closed coverings of nonempty convex sets in a real topological vector space. Our new results extend previously given ones due to Ky Fan [2], [3], Shih [10], Shih and Tan [11], and Park [7].