A lattice Boltzmann method for simulation of multi-species shock accelerated flows

A numerical scheme for simulating multi-species shock accelerated flows using lattice Boltzmann approach has been proposed. It uses the moment conservation approach of Yang, Shu, and Wu and extends it to multi-species fluid problems. The multi-species method of Wang et al. has been modified by use of a predictor–corrector approach. This has helped in preventing the pressure oscillations while handling multi-species. Simulation of 2D shock cylinder interaction with this solver has shown good agreement with the experimental data and could capture material discontinuity and unsteady shocks. The simulation of a single mode Richtmyer–Meshkov instability showed that the solver is able to capture the development of spike and bubble as per the experimental findings of Aure and Jacobs. The dissipation in the proposed scheme was further reduced by the use of fifth-order weighted essentially non-oscillatory (WENO). Validated with multiple problems, this method has been found to capture shock instability with good accuracy with a check on pressure oscillations.     

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).     

One‐dimensional shock‐capturing for high‐order discontinuous Galerkin methods

Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent structure (a numerical flux based on a suitable approximate Riemann solver introduces some stabilization) suggests that they are specially adapted to capture shocks. However, numerical fluxes are not sufficient to stabilize the solution in the presence of shocks. Thus, slope limiter methods, which are extensions of finite volume methods, have been proposed. These techniques require, in practice, mesh adaption to localize the shock structure. This is is more obvious for large elements typical of high‐order approximations. Here, a new approach based on the introduction of artificial diffusion into the original equations is presented. The order is not systematically decreased to one in the presence of the shock, large high‐order elements can be used, and several linear and nonlinear tests demonstrate the efficiency of the proposed methodology. Copyright © 2012 John Wiley & Sons, Ltd.     

Hybrid mesh generation with embedded surfaces using a multiple marching direction approach

This paper proposes a method for the creation of hybrid meshes with embedded surfaces for viscous flow simulations as an extension of the multiple marching direction approach (AIAA J. 2007; 45 (1):162–167). The multiple marching direction approach enables to place semi‐structured elements around singular points, where valid semi‐structured elements cannot be placed using conventional hybrid mesh generation methods. This feature is discussed first with a couple of examples. Elements sometimes need to be clustered inside a computational domain to obtain more accurate results. For example, solution features, such as shocks, vortex cores and wake regions, can be extracted during the process of adaptive mesh generation. These features can be represented as surface meshes embedded in a computational domain. Semi‐structured elements can be placed around the embedded surface meshes using the multiple marching direction approach with a pretreatment method. Tetrahedral elements can be placed easily instead. A couple of results are presented to demonstrate the capability of the mesh generation method. Copyright © 2008 John Wiley & Sons, Ltd.     

A high‐order flux reconstruction adaptive mesh refinement method for magnetohydrodynamics on unstructured grids

We report our recent development of the high‐order flux reconstruction adaptive mesh refinement (AMR) method for magnetohydrodynamics (MHD). The resulted framework features a shock‐capturing duo of AMR and artificial resistivity (AR), which can robustly capture shocks and rotational and contact discontinuities with a fraction of the cell counts that are usually required. In our previous paper, 36 we have presented a shock‐capturing framework on hydrodynamic problems with artificial diffusivity and AMR. Our AMR approach features a tree‐free, direct‐addressing approach in retrieving data across multiple levels of refinement. In this article, we report an extension to MHD systems that retains the flexibility of using unstructured grids. The challenges due to complex shock structures and divergence‐free constraint of magnetic field are more difficult to deal with than those of hydrodynamic systems. The accuracy of our solver hinges on 2 properties to achieve high‐order accuracy on MHD systems: removing the divergence error thoroughly and resolving discontinuities accurately. A hyperbolic divergence cleaning method with multiple subiterations is used for the first task. This method drives away the divergence error and preserves conservative forms of the governing equations. The subiteration can be accelerated by absorbing a pseudo time step into the wave speed coefficient, therefore enjoys a relaxed CFL condition. The AMR method rallies multiple levels of refined cells around various shock discontinuities, and it coordinates with the AR method to obtain sharp shock profiles. The physically consistent AR method localizes discontinuities and damps the spurious oscillation arising in the curl of the magnetic field. The effectiveness of the AMR and AR combination is demonstrated to be much more powerful than simply adding AR on finer and finer mesh, since the AMR steeply reduces the required amount of AR and confines the added artificial diffusivity and resistivity to a narrower and narrower region. We are able to verify the designed high‐order accuracy in space by using smooth flow test problems on unstructured grids. The efficiency and robustness of this framework are fully demonstrated through a number of two‐dimensional nonsmooth ideal MHD tests. We also successfully demonstrate that the AMR method can help significantly save computational cost for the Orszag‐Tang vortex problem.     

Adaptive remeshing for ductile fracture prediction in metal formingRemaillage adaptatif pour la prévision de la rupture ductile en mise en forme

The analysis of mechanical structures using the Finite Element Method in the framework of large elastoplastic strain, needs frequent remeshing of the deformed domain during computation. Indeed, the remeshing is due to the large geometrical distortion of finite elements and the adaptation to the physical behavior of the solution. This paper gives the necessary steps to remesh a mechanical structure during large elastoplastic deformations with damage. An important part of this process is constituted by geometrical and physical error estimates. The proposed method is integrated in a computational environment using the ABAQUS/Explicit solver and the BL2D-V2 adaptive mesher. To cite this article: H. Borouchaki et al., C. R. Mecanique 330 (2002) 709–716.     

Collocated discrete least‐squares (CDLS) meshless method: Error estimate and adaptive refinement

Meshless methods are new approaches for solving partial differential equations. The main characteristic of all these methods is that they do not require the traditional mesh to construct a numerical formulation. They require node generation instead of mesh generation. In other words, there is no pre‐specified connectivity or relationships among the nodes. This characteristic make these methods powerful. For example, an adaptive process which requires high computational effort in mesh‐dependent methods can be very economically solved with meshless methods. In this paper, a posteriori error estimate and adaptive refinement strategy is developed in conjunction with the collocated discrete least‐squares (CDLS) meshless method. For this, an error estimate is first developed for a CDLS meshless method. The proposed error estimator is shown to be naturally related to the least‐squares functional, providing a suitable posterior measure of the error in the solution. A mesh moving strategy is then used to displace the nodal points such that the errors are evenly distributed in the solution domain. Efficiency and effectiveness of the proposed error estimator and adaptive refinement process are tested against two hyperbolic benchmark problems, one with shocked and the other with low gradient smooth solutions. These experiments show that the proposed adaptive process is capable of producing stable and accurate results for the difficult problems considered. Copyright © 2007 John Wiley & Sons, Ltd.     

The discontinuous profile method for simulating two‐phase flow in pipes using the single component approximation

Godunov‐type algorithms are very attractive for the numerical solution of discontinuous flows. The reconstruction of the profile inside the cells is crucial to scheme performance. The non‐linear generalization of the discontinuous profile method (DPM) presented here for the modelling of two‐phase flow in pipes uses a discontinuous reconstruction in order to capture shocks more efficiently than schemes using continuous functions. The reconstructed profile is used to define the Riemann problem at cell interfaces by averaging of the components of the variable in the base of eigenvectors over their domain of dependence. Intercell fluxes are computed by solving the Riemann problem with an approximate‐state solver. The adapted treatment of boundary conditions is essential to ensure the quality of the computational results and a specific procedure using virtual cells at both extremities of the computational domain is required. Internal boundary conditions can be treated in the same way as external ones. Application of the DPM to test cases is shown to improve the quality of computational results significantly. Copyright © 2001 John Wiley & Sons, Ltd.     

Mesh adaptivity for the transport equation led by variational multiscale error estimators

Recently, we developed an explicit a posteriori error estimator especially suited for fluid dynamics problems solved with a stabilized method. The technology is based upon the theory that inspired stabilized methods, namely, the variational multiscale theory. The salient features of the formulation are that it can be readily implemented in existing codes, it is a very economical procedure, and it yields very accurate local error estimates uniformly from the diffusive to the advective regime. In this work, the variational multiscale error estimator is applied to develop adaptive strategies for the advection–diffusion‐reaction equation. The performance of L₁ and L₂ local error norms combined with three strategies to adapt the mesh is investigated. Emphasis is placed on flows with sharp boundary and interior layers but also attention is given to diffusion‐dominated flows. Computational results show that the method generates meshes with a smooth transition of the element size, which capture all the flow features. Copyright © 2011 John Wiley & Sons, Ltd.     

Effects of mesh resolution on hypersonic heating prediction

Aeroheating prediction is a challenging and critical problem for the design and optimization of hypersonic vehicles. One challenge is that the solution of the Navier-Stokes equations strongly depends on the computational mesh. In this letter, the effect of mesh resolution on heat flux prediction is studied. It is found that mesh-independent solutions can be obtained using fine mesh, whose accuracy is confirmed by results from kinetic particle simulation. It is analyzed that mesh-induced numerical error comes mainly from the flux calculation in the boundary layer whereas the temperature gradient on the surface can be evaluated using a wall function. Numerical schemes having strong capability of boundary layer capture are therefore recommended for hypersonic heating prediction.     

Numerical capture of wing tip vortex improved by mesh adaptation

This paper presents a mesh adaptation procedure linked to a finite volume solver, the goal of which is to increase the precision of the numerical simulation of a wing tip vortex flow. The adaptation scheme is applied to hexahedron meshes and hybrid meshes made up of tetrahedrons and prisms. To evaluate the ability of each type of element to capture the physics of a tip vortex, a specific test case is studied and results obtained numerically from this test case are compared with experimental results. The error estimator of the adaptation scheme is derived from a solution scalar variable. It is shown that the element anisotropy as well as the adaptation algorithms used have an impact on the precision of the solution. Adaptation of hexahedrons allows a better capture of the tip vortex far from the vortex root, even though the adaptation of those hexahedrons barely changes the number of nodes used to achieve a specified precision, contrary to the adaptation of hybrid meshes. Copyright © 2009 John Wiley & Sons, Ltd.     

Fluid–structure interaction analysis of flexible turbomachinery

A method for the performance computation of an expandable-impeller pump is developed and validated. Large deformations of the highly flexible pump impellers result in a strong coupling between the impeller and fluid flow. The computational method therefore requires simultaneous solution of fluid flow and structural response. OpenFOAM provides the flow and mesh motion solvers and is coupled to an author-developed structural solver in a tightly coupled approach using a fixed-point iteration. The structural deformations are time-dependent because the material exhibits stress relaxation. The time-constant of the relaxation, however, is very large, thereby allowing quasi-steady simulations. A water-tunnel test of a viscoelastic hydrofoil is employed to validate the solver. Simulations of the test problem show good agreement with the experimental results and demonstrate the need for several sub-iterations of the solver even for the quasi-steady simulations.     

Mesh adaptation using C1 interpolation of the solution in an unstructured finite volume solver

The goal of this paper is to show the effectiveness of a newly developed estimate of the truncation error calculated based on C¹ interpolation of the solution weighted by the adjoint solution as the adaptation indicator for an unstructured finite volume solver. We will show that adjoint‐based mesh adaptation based on the corrected functional using the new developed truncation error estimate is capable of adapting the mesh to improve the accuracy of the functional and the convergence rate. Both discrete and continuous adjoint solutions are used for adaptation. Results are significantly better with new truncation error estimate than with previously used estimates.     

Error sensitivity to refinement: a criterion for optimal grid adaptation

Most indicators used for automatic grid refinement are suboptimal, in the sense that they do not really minimize the global solution error. This paper concerns with a new indicator, related to the sensitivity map of global stability problems, suitable for an optimal grid refinement that minimizes the global solution error. The new criterion is derived from the properties of the adjoint operator and provides a map of the sensitivity of the global error (or its estimate) to a local mesh refinement. Examples are presented for both a scalar partial differential equation and for the system of Navier–Stokes equations. In the last case, we also present a grid-adaptation algorithm based on the new estimator and on the \(FreeFem++\) software that improves the accuracy of the solution of almost two order of magnitude by redistributing the nodes of the initial computational mesh.     

Robustness versus accuracy in shock‐wave computations

Despite constant progress in the development of upwind schemes, some failings still remain. Quirk recently reported (Quirk JJ. A contribution to the great Riemann solver debate. International Journal for Numerical Methods in Fluids 1994; 18 : 555–574) that approximate Riemann solvers, which share the exact capture of contact discontinuities, generally suffer from such failings. One of these is the odd–even decoupling that occurs along planar shocks aligned with the mesh. First, a few results on some failings are given, namely the carbuncle phenomenon and the kinked Mach stem. Then, following Quirk's analysis of Roe's scheme, general criteria are derived to predict the odd–even decoupling. This analysis is applied to Roe's scheme (Roe PL, Approximate Riemann solvers, parameters vectors, and difference schemes, Journal of Computational Physics 1981; 43 : 357–372), the Equilibrium Flux Method (Pullin DI, Direct simulation methods for compressible inviscid ideal gas flow, Journal of Computational Physics 1980; 34 : 231–244), the Equilibrium Interface Method (Macrossan MN, Oliver. RI, A kinetic theory solution method for the Navier–Stokes equations, International Journal for Numerical Methods in Fluids 1993; 17 : 177–193) and the AUSM scheme (Liou MS, Steffen CJ, A new flux splitting scheme, Journal of Computational Physics 1993; 107 : 23–39). Strict stability is shown to be desirable to avoid most of these flaws. Finally, the link between marginal stability and accuracy on shear waves is established. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical simulation of compressible two-phase flow using a diffuse interface method

In this article, a high-resolution diffuse interface method is investigated for simulation of compressible two-phase gas–gas and gas–liquid flows, both in the presence of shock wave and in flows with strong rarefaction waves similar to cavitations. A Godunov method and HLLC Riemann solver is used for discretization of the Kapila five-equation model and a modified Schmidt equation of state (EOS) is used to simulate the cavitation regions. This method is applied successfully to some one- and two-dimensional compressible two-phase flows with interface conditions that contain shock wave and cavitations. The numerical results obtained in this attempt exhibit very good agreement with experimental results, as well as previous numerical results presented by other researchers based on other numerical methods. In particular, the algorithm can capture the complex flow features of transient shocks, such as the material discontinuities and interfacial instabilities, without any oscillation and additional diffusion. Numerical examples show that the results of the method presented here compare well with other sophisticated modeling methods like adaptive mesh refinement (AMR) and local mesh refinement (LMR) for one- and two-dimensional problems.     

Incorporating spatially variable bottom stress and Coriolis force into 2D,a posteriori,unstructured mesh generation for shallow water models

An enhanced version of our localized truncation error analysis with complex derivatives (LTEA?CD ) a posteriori approach to computing target element sizes for tidal, shallow water flow, LTEA+CD , is applied to the Western North Atlantic Tidal model domain. The LTEA + CD method utilizes localized truncation error estimates of the shallow water momentum equations and builds upon LTEA and LTEA?CD‐based techniques by including: (1) velocity fields from a nonlinear simulation with complete constituent forcing; (2) spatially variable bottom stress; and (3) Coriolis force. Use of complex derivatives in this case results in a simple truncation error expression, and the ability to compute localized truncation errors using difference equations that employ only seven to eight computational points. The compact difference molecules allow the computation of truncation error estimates and target element sizes throughout the domain, including along the boundary; this fact, along with inclusion of locally variable bottom stress and Coriolis force, constitute significant advancements beyond the capabilities of LTEA. The goal of LTEA + CD is to drive the truncation error to a more uniform, domain‐wide value by adjusting element sizes (we apply LTEA + CD by re‐meshing the entire domain, not by moving nodes). We find that LTEA + CD can produce a mesh that is comprised of fewer nodes and elements than an initial high‐resolution mesh while performing as well as the initial mesh when considering the resynthesized tidal signals (elevations). Copyright © 2008 John Wiley & Sons, Ltd.     

Two‐dimensional anisotropic Cartesian mesh adaptation for the compressible Euler equations

Simulating transient compressible flows involving shock waves presents challenges to the CFD practitioner in terms of the mesh quality required to resolve discontinuities and prevent smearing. This paper discusses a novel two‐dimensional Cartesian anisotropic mesh adaptation technique implemented for transient compressible flow. This technique, originally developed for laminar incompressible flow, is efficient because it refines and coarsens cells using criteria that consider the solution in each of the cardinal directions separately. In this paper, the method will be applied to compressible flow. The procedure shows promise in its ability to deliver good quality solutions while achieving computational savings. Transient shock wave diffraction over a backward step and shock reflection over a forward step are considered as test cases because they demonstrate that the quality of the solution can be maintained as the mesh is refined and coarsened in time. The data structure is explained in relation to the computational mesh, and the object‐oriented design and implementation of the code is presented. Refinement and coarsening algorithms are outlined. Computational savings over uniform and isotropic mesh approaches are shown to be significant. Copyright © 2004 John Wiley & Sons, Ltd.     

Versatile anisotropic mesh adaptation methodology applied to pure quantity of interest error estimator. Steady,laminar incompressible flow

We introduce a new flexible mesh adaptation approach to efficiently compute a quantity of interest by the finite element method. Efficiently, we mean that the method provides an evaluation of that quantity up to a predetermined accuracy at a lower computational cost than other classical methods. The central pillar of the method is our scalar error estimator based on sensitivities of the quantity of interest to the residuals. These sensitivities result from the computation of a continuous adjoint problem. The mesh adaptation strategy can drive anisotropic mesh adaptation from a general scalar error contribution of each element. The full potential of our error estimator is then reached. The proposed method is validated by evaluating the lift, the drag, and the hydraulic losses on a 2D benchmark case: the flow around a cylinder at a Reynolds number of 20.     

A simple shock‐capturing technique for high‐order discontinuous Galerkin methods

This article presents a novel shock‐capturing technique for the discontinuous Galerkin (DG) method. The technique is designed for compressible flow problems, which are usually characterized by the presence of strong shocks and discontinuities. The inherent structure of standard DG methods seems to suggest that they are especially adapted to capture shocks because of the numerical fluxes based on suitable approximate Riemann solvers, which, in practice, introduces some stabilization. However, the usual numerical fluxes are not sufficient to stabilize the solution in the presence of shocks for large high‐order elements. Here, a new basis of shape functions is introduced. It has the ability to change locally between a continuous or discontinuous interpolation depending on the smoothness of the approximated function. In the presence of shocks, the new discontinuities inside an element introduce the required stabilization because of numerical fluxes. Large high‐order elements can therefore be used and shocks captured within a single element, avoiding adaptive mesh refinement and preserving the locality and compactness of the DG scheme. Several numerical examples for transonic and supersonic flows are studied to demonstrate the applicability of the proposed approach. Copyright © 2011 John Wiley & Sons, Ltd.