Remaillage adaptatif pour la mise en forme des tôles minces

Problems associated with finite element simulation of the forming processes are characterized by large elastoplastic deformations, evolutive contact with friction, geometrical nonlinearities inducing a severe distortion of the computational mesh of the domain. In this case, frequent remeshing of the deformed domain during computation are necessary to obtain an accurate solution and complete the computation until the termination of the numerical simulation process. This Note presents a new adaptive remeshing method of thin sheets for numerical simulation of metal forming processes. The proposed method is based on geometrical criteria and does not use the geometry of the forming tools. It is integrated in a computational environment using the ABAQUS solver. Numerical examples are given to show the efficiency of our approach. To cite this article: L. Moreau et al., C. R. Mecanique 333 (2005).     

近似黎曼解对高超声速气动热计算的影响研究

高超声速流场计算一般采用TVD型格式,这些格式中,大多采用了不同形式的近似黎曼解. 通过分析和数值验证,论述了激波捕捉格式中近似黎曼解的耗散性质,说明其对高超声速热流计算的影响. 数值实验证明,采用低耗散格式可大大提高热流计算精度,降低热流计算对网格的依赖程度,从而获得精确的热流数值解.      

基于Nitsche方法与拟牛顿求解的二维接触问题等几何分析

在等几何框架内,基于Nitsche方法推导了二维无摩擦弹性接触列式,采用基于BFGS逆更新的拟牛顿迭代格式求解.提出了Nitsche接触列式中罚系数的经验公式和拟牛顿求解时迭代的初始化方法,研究了基于割线刚度阵的修正方法以克服因接触面变化而导致的迭代发散.所提出的接触分析方法在粗糙网格下也能精确描述接触边界,列式推导简单,计算量小.算例表明了接触列式和求解方法的有效性.     

A machine learning based solver for pressure Poisson equations

When using the projection method (or fractional step method) to solve the incompressible Navier-Stokes equations, the projection step involves solving a large-scale pressure Poisson equation (PPE), which is computationally expensive and time-consuming. In this study, a machine learning based method is proposed to solve the large-scale PPE. An machine learning (ML)-block is used to completely or partially (if not sufficiently accurate) replace the traditional PPE iterative solver thus accelerating the solution of the incompressible Navier-Stokes equations. The ML-block is designed as a multi-scale graph neural network (GNN) framework, in which the original high-resolution graph corresponds to the discrete grids of the solution domain, graphs with the same resolution are connected by graph convolution operation, and graphs with different resolutions are connected by down/up prolongation operation. The well trained ML-block will act as a general-purpose PPE solver for a certain kind of flow problems. The proposed method is verified via solving two-dimensional Kolmogorov flows (Re = 1000 and Re = 5000) with different source terms. On the premise of achieving a specified high precision (ML-block partially replaces the traditional iterative solver), the ML-block provides a better initial iteration value for the traditional iterative solver, which greatly reduces the number of iterations of the traditional iterative solver and speeds up the solution of the PPE. Numerical experiments show that the ML-block has great advantages in accelerating the solving of the Navier-Stokes equations while ensuring high accuracy.     

Implicit schemes for unsteady Euler equations on triangular meshes

An implicit finite element method is presented for the solution of steady and unsteady inviscid compressible flows on triangular meshes under transonic conditions. The method involves a first-order time-stepping scheme with a finite element discretization that reduces to central differencing on a rectangular mesh. On a solid wall the slip condition is prescribed and the pressure is obtained from an approximation of the normal momentum equation. With this solver no artificial viscosity is added to ensure the success of the calculation. Numerical examples are given for steady and unsteady cases.     

Development of a hybrid finite volume/element solver for incompressible flows

In this paper, we report our development of an implicit hybrid flow solver for the incompressible Navier–Stokes equations. The methodology is based on the pressure correction or projection method. A fractional step approach is used to obtain an intermediate velocity field by solving the original momentum equations with the matrix‐free implicit cell‐centred finite volume method. The Poisson equation derived from the fractional step approach is solved by the node‐based Galerkin finite element method for an auxiliary variable. The auxiliary variable is closely related to the real pressure and is used to update the velocity field and the pressure field. We store the velocity components at cell centres and the auxiliary variable at cell vertices, making the current solver a staggered‐mesh scheme. Numerical examples demonstrate the performance of the resulting hybrid scheme, such as the correct temporal convergence rates for both velocity and pressure, absence of unphysical pressure boundary layer, good convergence in steady‐state simulations and capability in predicting accurate drag, lift and Strouhal number in the flow around a circular cylinder. Copyright © 2007 John Wiley & Sons, Ltd.     

An approximate‐state Riemann solver for the two‐dimensional shallow water equations with porosity

PorAS, a new approximate‐state Riemann solver, is proposed for hyperbolic systems of conservation laws with source terms and porosity. The use of porosity enables a simple representation of urban floodplains by taking into account the global reduction in the exchange sections and storage. The introduction of the porosity coefficient induces modified expressions for the fluxes and source terms in the continuity and momentum equations. The solution is considered to be made of rarefaction waves and is determined using the Riemann invariants. To allow a direct computation of the flux through the computational cells interfaces, the Riemann invariants are expressed as functions of the flux vector. The application of the PorAS solver to the shallow water equations is presented and several computational examples are given for a comparison with the HLLC solver. Copyright © 2009 John Wiley & Sons, Ltd.     

An efficient numerical method for reactive flow with general equation of states

This paper presents an efficient method to simulate the reactive flow for general equation of states with the compressible fluid model coupled with reactive rate equation. The important aspect is to deal with mixture of different phases in one cell, which will inevitably happen in the Eulerian method for reactive flow. Physical variables such as the pressure,velocity, and speed of sound in each cell need to be reconstructed for the Harten‐Lax‐Leer‐Contact (HLLC) Riemann solver, which will result in nonlinear algebra equations, and these reconstructed variables are used to obtain the flux. Numerical examples of stable and unstable detonations with different equation of states demonstrate the accuracy and efficiency of this method. Copyright © 2016 John Wiley & Sons, Ltd.     

A second-order positivity-preserving well-balanced finite volume scheme for Euler equations with gravity for arbitrary hydrostatic equilibria

We present a well-balanced finite volume scheme for the compressible Euler equations with gravity, where the approximate Riemann solver is derived using a Suliciu relaxation approach. Besides the well-balanced property, the scheme is robust with respect to the physical admissible states. General hydrostatic solutions are captured up to machine precision by deriving, for a given initial value problem, suitable time-independent functions and using them in the discretization of the source term. The first-order scheme is extended to a second-order scheme by reconstructing in equilibrium variables while preserving the well-balanced and robustness properties. Numerical examples are performed to demonstrate the accuracy, well-balanced, and robustness properties of the presented scheme for up to three space dimensions.     

The vibration and stability analysis of moderate thick plates by the method of lines

The method of lines based on Hu Hai-chang's theory for the vibration and stability of moderate thick plates is developed. The standard nonlinear ordinary differential equation (ODE) system for natural frequencies and critical load is given by use of ODE techniques, and then any indicated eigenvalue could be obtained directly from ODE solver by employing the so-called initial eigenfunction technique instead of the mode orthogonality condition.Numerical examples show that the present method is very effective and reliable.     

HLLC scheme for the preconditioned pseudo-compressibility Navier–Stokes equations for incompressible viscous flows

A new HLLC (Harten-Lax-van leer contact) approximate Riemann solver with the preconditioning technique based on the pseudo-compressibility formulation for numerical simulation of the incompressible viscous flows has been proposed, which follows the HLLC Riemann solver (Harten, Lax and van Leer solver with contact resolution modified by Toro) for the compressible flow system. In the authors' previous work, the preconditioned Roe's Riemann solver is applied to the finite difference discretisation of the inviscid flux for incompressible flows. Although the Roe's Riemann solver is found to be an accurate and robust scheme in various numerical computations, the HLLC Riemann solver is more suitable for the pseudo-compressible Navier--Stokes equations, in which the inviscid flux vector is a non-homogeneous function of degree one of the flow field vector, and however the Roe's solver is restricted to the homogeneous systems. Numerical investigations have been performed in order to demonstrate the efficiency and accuracy of the present procedure in both two- and three-dimensional cases. The present results are found to be in good agreement with the exact solutions, existing numerical results and experimental data.     

An extension of Osher's Riemann solver for chemical and vibrational non-equilibrium gas flows

In this paper we study an extension of Osher's Riemann solver to mixtures of perfect gases whose equation of state is of the form encountered in hypersonic applications. As classically, one needs to compute the Riemann invariants of the system to evaluate Osher's numerical flux. For the case of interest here it is impossible in general to derive simple enough expressions which can lead to an efficient calculation of fluxes. The key point here is the definition of approximate Riemann invariants to alleviate this difficulty. Some of the properties of this new numerical flux are discussed. We give 1D and 2D applications to illustrate the robustness and capability of this new solver. We show by numerical examples that the main properties of Osher's solver are preserved; in particular, no entropy fix is needed even for hypersonic applications.     

A parallel fast multipole BEM and its applications to large-scale analysis of 3-D fiber-reinforced composites

In this paper, an adaptive boundary element method (BEM) is presented for solving 3-D elasticity problems. The numerical scheme is accelerated by the new version of fast multipole method (FMM) and parallelized on distributed memory architectures. The resulting solver is applied to the study of representative volume element (RVE) for short fiber-reinforced composites with complex inclusion geometry. Numerical examples performed on a 32-processor cluster show that the proposed method is both accurate and efficient, and can solve problems of large size that are challenging to existing state-of-the-art domain methods.The project supported by the National Natural Science Foundation of China (10472051).The English text was polished by Yunming Chen.     

Dynamic Evaluation of Mesh Resolution and Its Application in Hybrid LES/RANS Methods

In this work, we investigate a resolution evaluation criterion based on the ratio between turbulent length-scales and grid spacing within the context of dynamic resolution evaluation in hybrid LES/RANS simulations. A modified version of the commonly used length-scale criterion is adopted. The modified length-scale criterion is evaluated for a plane channel flow and compared to the criterion based on two-point correlations. Simulation results show qualitative agreement between the two criteria and physical predictions from both resolution indicators. These observations are confirmed by simulations of flows over periodic hills. It is further demonstrated that the length-scale based criterion is relatively less sensitive on variation of model parameters compared to criteria based on resolved percentage of turbulent quantities. The improved resolution criterion is applied in a dual-mesh hybrid LES/RANS solver. Numerical simulations with the hybrid solver suggest that the interactions between the length-scale resolution indicator and the solution are moderate, and that favorable comparisons with benchmark results are obtained. In summary, we demonstrate that the modified length-scale based resolution indicator performs satisfactorily in both pure LES and hybrid simulations. Therefore, it is selected as a promising candidate to provide reliable predictions of resolution adequacy for individual cells in hybrid LES/RANS simulations.     

A general approximate‐state Riemann solver for hyperbolic systems of conservation laws with source terms

An approximate‐state Riemann solver for the solution of hyperbolic systems of conservation laws with source terms is proposed. The formulation is developed under the assumption that the solution is made of rarefaction waves. The solution is determined using the Riemann invariants expressed as functions of the components of the flux vector. This allows the flux vector to be computed directly at the interfaces between the computational cells. The contribution of the source term is taken into account in the governing equations for the Riemann invariants. An application to the water hammer equations and the shallow water equations shows that an appropriate expression of the pressure force at the interface allows the balance with the source terms to be preserved, thus ensuring consistency with the equations to be solved as well as a correct computation of steady‐state flow configurations. Owing to the particular structure of the variable and flux vectors, the expressions of the fluxes are shown to coincide partly with those given by the HLL/HLLC solver. Computational examples show that the approximate‐state solver yields more accurate solutions than the HLL solver in the presence of discontinuous solutions and arbitrary geometries. Copyright © 2006 John Wiley & Sons, Ltd.     

A linearized Riemann solver for the steady supersonic Euler equations

A very simple linearization of the solution to the Riemann problem for the steady supersonic Euler equations is presented. When used locally in conjunction with the Godunov method, computing savings by a factor of about four relative to the use of exact Riemann solvers can be achieved. For severe flow regimes, however, the linearization loses accuracy and robustness. We then propose the use of a Riemann solver adaptation procedure. This retains the accuracy and robustness of the exact Riemann solver and the computational efficiency of the cheap linearized Riemann solver. Numerical results for two- and three-dimensional test problems are presented.     

Calculation of Ship Sinkage and Trim Using a Finite Element Method and Unstructured Grids

An unstructured grid-based, parallel free-surface flow solver has been extended to account for sinkage and trim effects in the calculation of steady ship waves. The overall scheme of the solver combines a finite-element, equal-order, projection-type three-dimensional incompressible flow solver with a finite element, two-dimensional advection equation solver for the free surface equation. The sinkage and trim, wave profiles, and wave drag computed using the present approach are in good agreement with experimental measurements for two hull forms at a wide range of Froude numbers. Numerical predictions indicate significant differences between the wave drag for a ship fixed in at-rest position and free to sink and trim, in agreement with experimental observations.     

Parallel SOR methods with a parabolic-diffusion acceleration technique for solving an unstructured-grid Poisson equation on 3D arbitrary geometries

This paper presents a parallel algorithm for the finite-volume discretisation of the Poisson equation on three-dimensional arbitrary geometries. The proposed method is formulated by using a 2D horizontal block domain decomposition and interprocessor data communication techniques with message passing interface. The horizontal unstructured-grid cells are reordered according to the neighbouring relations and decomposed into blocks using a load-balanced distribution to give all processors an equal amount of elements. In this algorithm, two parallel successive over-relaxation methods are presented: a multi-colour ordering technique for unstructured grids based on distributed memory and a block method using reordering index following similar ideas of the partitioning for structured grids. In all cases, the parallel algorithms are implemented with a combination of an acceleration iterative solver. This solver is based on a parabolic-diffusion equation introduced to obtain faster solutions of the linear systems arising from the discretisation. Numerical results are given to evaluate the performances of the methods showing speedups better than linear.     

Reordering and incomplete preconditioning in serial and parallel adaptive mesh refinement and coarsening flow solutions

The effects of reordering the unknowns on the convergence of incomplete factorization preconditioned Krylov subspace methods are investigated. Of particular interest is the resulting preconditioned iterative solver behavior when adaptive mesh refinement and coarsening (AMR/C) are utilized for serial or distributed parallel simulations. As representative schemes, we consider the familiar reverse Cuthill–McKee and quotient minimum degree algorithms applied with incomplete factorization preconditioners to CG and GMRES solvers. In the parallel distributed case, reordering is applied to local subdomains for block ILU preconditioning, and subdomains are repartitioned dynamically as mesh adaptation proceeds. Numerical studies for representative applications are conducted using the object‐oriented AMR/C software system libMesh linked to the PETSc solver library. Serial tests demonstrate that global unknown reordering and incomplete factorization preconditioning can reduce the number of iterations and improve serial CPU time in AMR/C computations. Parallel experiments indicate that local reordering for subdomain block preconditioning associated with dynamic repartitioning because of AMR/C leads to an overall reduction in processing time. Copyright © 2011 John Wiley & Sons, Ltd.     

Analysis of a meshless solver for high Reynolds number flow

In this paper, the extension of an upwind least‐square based meshless solver to high Reynolds number flow is explored, and the properties of the meshless solver are analyzed both theoretically and numerically. Existing works have verified the meshless solver mostly with inviscid flows and low Reynolds number flows, and in this work, we are interested in the behavior of the meshless solver for high Reynolds number flow, especially in the near‐wall region. With both theoretical and numerical analysis, the effects of two parameters on the meshless solver are identified. The first one is the misalignment effect caused by the significantly skewed supporting points, and it is found that the meshless solver still yields accurate prediction. It is a very interesting property and is opposite to the median‐dual control volume based vertex‐centered finite volume method, which is known to give degraded result with stretched triangular/tetrahedral cells in the near‐wall region. The second parameter is the curvature, and according to theoretical analysis, it is found in the region with both large aspect ratio and curvature, and the streamwise residual is less affected; however, the wall‐normal counterpart suffers from accuracy degradation. In this paper, an improved method that uses a meshless solver for the streamwise residual and finite difference for wall‐normal residual is developed. This method is proved to be less sensitive to the curvature and provides improved accuracy. This work presents an understanding of the meshless solver for high Reynolds number flow computation, and the analysis in this paper is verified with a series of numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.