Non-structured Meshes Research Articles

A semi-Lagrangian scheme for mean curvature motion with nonlinear Neumann conditions

A numerical method for mean curvature motion in bounded domains with nonlinear Neumann boundary conditions is proposed and analyzed. It consists of a semi-Lagrangian scheme in the main part of the domain as proposed by Carlini, Falcone and Ferretti, combined with a finite difference scheme in small layers near the boundary to cope with the boundary condition. The consistency of the new scheme is proved for nonstructured triangular meshes in dimension two. The monotonicity of a regularized version of the scheme with some additional vanishing artificial viscosity is studied. Details on the implementation are given. Numerical tests are presented.

Numerical Analysis of Air Flow Past the 2415-3S Airfoil for an Unmanned Aerial Vehicle with Internal Propulsion System

This paper deals with the prediction of pressure and velocity fields on the 2415-3S airfoil which will be used for and unmanned aerial vehicle with internal propulsion system and in this way analyze the air flow through an internal duct of the airfoil using computational fluid dynamics. The main objective is to evaluate the effect of the internal air flow past the airfoil and how this affects the aerodynamic performance by means of lift and drag forces. For this purpose, three different designs of the internal duct were studied; starting from the base 2415-3S airfoil developed in previous investigation, basing on the hypothesis of decreasing the flow separation produced when the propulsive airflow merges the external flow, and in this way obtaining the best configuration. For that purpose, an exhaustive study of the mesh sensitivity was performed. It was used a non-structured mesh since the computational domain is tridimensional and complex. The selected mesh contains approximately 12.5 million elements. Both the computational domain and the numerical solution were made with commercial CAD and CFD software respectively. Air, incompressible and steady was analyzed. The boundary conditions are in concordance with experimental setup in the AF 6109 wind tunnel. The k-e model is utilized to describe the turbulent flow process as followed in references. Results allowed obtaining pressure and velocity contours as well as lift and drag coefficients and also the location of separation and reattachment regions in some cases for zero degrees of angle of attack on the internal and external surfaces of the airfoil. Finally, the selection of the configuration with the best aerodynamic performance was made, selecting the option without curved baffles.

A Micromagnetic Solver for Large-Scale Patterned Media Based on Non-Structured Meshing

The paper presents a micromagnetic solver, where the long-range contributions to the magnetostatic field are computed by a multipole expansion technique and the spatial distribution of the exchange field is reconstructed via a finite difference scheme able to handle non-structured meshes. In this way, large samples having complex geometries can be simulated. The numerical accuracy, the computational efficiency and the flexibility of the proposed solver are investigated by comparison to standard approaches.

Efficient 3D numerical prediction of the pressure wave generated by high-speed trains entering tunnels

The topic of this paper is to present a new methodology for the three-dimensional numerical simulation of the entrance of high-speed trains in a tunnel. The movement of the train is made thanks to a technique of sliding meshes and a conservative treatment of the faces between two domains. All parts of the development are thought with the aim to reduce the computational time. In particular, non-reflecting boundary conditions for non-structured three-dimensional meshes are developed in order to limit the calculation domain. Validations of the methodology are presented on different test cases.

DEVELOPMENT OF THE SPACE CODE FOR NUCLEAR POWER PLANTS

The Korean nuclear industry is developing a thermal-hydraulic analysis code for safety analysis of pressurized water reactors (PWRs). The new code is called the Safety and Performance Analysis Code for Nuclear Power Plants (SPACE). The SPACE code adopts advanced physical modeling of two-phase flows, mainly two-fluid three-field models which comprise gas, continuous liquid, and droplet fields and has the capability to simulate 3D effects by the use of structured and/or nonstructured meshes. The programming language for the SPACE code is C++ for object-oriented code architecture. The SPACE code will replace outdated vendor supplied codes and will be used for the safety analysis of operating PWRs and the design of advanced reactors. This paper describes the overall features of the SPACE code and shows the code assessment results for several conceptual and separate effect test problems.

Численное решение квазигидродинамических уравнений на неструктурированных треугольных сетках

Предложен метод численного решения квазигидродинамических уравнений на неструктурированных треугольных сетках. В качестве сетки была использована триангуляция Делоне. Система уравнений аппроксимировалась с помощью метода конечных объемов. Граница области аппроксимировалась прямоугольными треугольниками. На основе данного алгоритма была разработана программа и проведена серия тестов, результаты которых показали, что данный алгоритм дает результаты, которые хорошо совпадают с результатами расчетов, выполненных на регулярных сетках.

MULTI2D – a computer code for two-dimensional radiation hydrodynamics

Simulation of radiation hydrodynamics in two spatial dimensions is developed, having in mind, in particular, target design for indirectly driven inertial confinement energy (IFE) and the interpretation of related experiments. Intense radiation pulses by laser or particle beams heat high-Z target configurations of different geometries and lead to a regime which is optically thick in some regions and optically thin in others. A diffusion description is inadequate in this situation. A new numerical code has been developed which describes hydrodynamics in two spatial dimensions (cylindrical R – Z geometry) and radiation transport along rays in three dimensions with the 4 π solid angle discretized in direction. Matter moves on a non-structured mesh composed of trilateral and quadrilateral elements. Radiation flux of a given direction enters on two (one) sides of a triangle and leaves on the opposite side(s) in proportion to the viewing angles depending on the geometry. This scheme allows to propagate sharply edged beams without ray tracing, though at the price of some lateral diffusion. The algorithm treats correctly both the optically thin and optically thick regimes. A symmetric semi-implicit (SSI) method is used to guarantee numerical stability. Program summary Program title: MULTI2D Catalogue identifier: AECV_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AECV_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 151 098 No. of bytes in distributed program, including test data, etc.: 889 622 Distribution format: tar.gz Programming language: C Computer: PC (32 bits architecture) Operating system: Linux/Unix RAM: 2 Mbytes Word size: 32 bits Classification: 19.7 External routines: X-window standard library (libX11.so) and corresponding heading files (X11/*.h) are required. Nature of problem: In inertial confinement fusion and related experiments with lasers and particle beams, energy transport by thermal radiation becomes important. Under these conditions, the radiation field strongly interacts with the hydrodynamic motion through emission and absorption processes. Solution method: The equations of radiation transfer coupled with Lagrangian hydrodynamics, heat diffusion and beam tracing (laser or ions) are solved, in two-dimensional axial-symmetric geometry ( R – Z coordinates) using a fractional step scheme. Radiation transfer is solved with angular resolution. Matter properties are either interpolated from tables (equations-of-state and opacities) or computed by user routines (conductivities and beam attenuation). Restrictions: The code has been designed for typical conditions prevailing in inertial confinement fusion (ns time scale, matter states close to local thermodynamical equilibrium, negligible radiation pressure, …). Although a wider range of situations can be treated, extrapolations to regions beyond this design range need special care. Unusual features: A special computer language, called r94, is used at top levels of the code. These parts have to be converted to standard C by a translation program (supplied as part of the package). Due to the complexity of code (hydro-code, grid generation, user interface, graphic post-processor, translator program, installation scripts) extensive manuals are supplied as part of the package. Running time: 567 seconds for the example supplied.

A Probabilistic Extended Finite Element Approach: Application to the Prediction of Bone Crack Propagation

The Extended Finite Element Method (XFEM), has become a well-known tool to simulate crack propagation problems using non-structured meshes avoiding the remeshing process usually needed in this type of problems and allowing the inclusion of appropriate shape functions that reflect the asymptotic displacement field, near the crack tip, via a partition of unity fracture approach. However, in this kind of numerical applications, all the variables involved have been considered as deterministic (defined by a single given value), despite the well-known uncertainty associated to many of them (external loads, geometry and material properties, among others). The combination of the XFEM and probabilistic techniques is here proposed and formulated allowing treating fracture mechanics problems from a probabilistic point of view. We present the implementation of this probabilistic extended finite element method and apply it to the prediction of the appearance and propagation of a femur’s neck fracture under probabilistic loads.

Arbitrarily smooth generalized finite element approximations

This paper presents a procedure to build C k , k arbitrarily large, generalized finite element (FE) shape functions defined on non-structured finite element meshes. The functions have the same support as corresponding global FE Lagrangian shape functions. Meshes with both convex and non-convex clouds (set of elements sharing a vertex node), can be used. The so-called R-functions are used to build C k FE-based partition of unity functions with non-convex support. A technique to combine C 0 Lagrangian FE shape functions with the proposed C k partition of unity is presented. The technique allows the use of C k generalized FE shape functions in parts of the computational domain where their high smoothness is required, as in the case of problems with distributional boundary conditions, and the less computationally demanding C 0 generalized FE shape functions elsewhere in the domain. A linear elasticity problem with a concentrated moment is solved using the proposed C k generalized FE method. Higher order distributional boundary conditions can also be handled by the method. A detailed convergence analysis is presented for this class of problems as well as for problems in energy space. The integrability of the functions using standard Gauss–Legendre rules is also investigated.

The role of wave propagation in hydrocyclone operations I: An axisymmetric streamfunction formulation for a conical hydrocyclone

The objective of this work is to study the role of some wavelike motion observed in the interior of a conical hydrocyclone used in mineral processing operations. In this paper, we obtained a stable numerical solution of the equations of motion, using a streamfunction-vorticity formulation on non-structured mesh. The numerical solution is obtained in a finite element context, where a stabilization is imposed through an unusual finite element method (USFEM) scheme. Finally, an adaptive mesh refinement is used for to enhance the solution.

Dry Granular Flow Modelling Including Erosion and Deposition

In this paper, we considered dense avalanches made of dry and cohesion-less snow as gravitational granular flows. We used an effective friction law derived from the recent progress on granular behaviour research to represent momentum loss. We formulated a simple erosion and deposition model where the eroded and deposited mass fluxes are estimated from the knowledge of the starting and stopping conditions on a given slope angle. We adopted the shallow water hypothesis and introduced the new formulation of erosion and deposition processes. We solved the equations thanks to an accurate finite volumes solver scheme, using a non-structured mesh. This allows an easy integration of defence structures and singularities. Each component of this model was tested using analytical solutions and experimental laboratory data. Finally, we used the February 1999 Taconnaz avalanche event to test this model on a real snow avalanche case. These tests gave promising results.

Complexity control in the topology optimization of continuum structures

A general mesh independent filter as a mean to control the complexity of topology optimization designed structures is discussed. A new mesh-independent filter, applied over the move-limits of the sequential linear programming is proposed, and it is shown that its use alleviates common problems in the continuum topology optimization, like checkerboarding, mesh dependency, as well as effects associated to non-structured meshes, like numerical anisotropy. The structural optimization formulation adopted in this work is the minimization of a penalized function of the volume, with constraints on the compliance of each load case. Aspects of this penalized objective function are discussed, and several numerical examples are shown.

A cell-centered finite volume scheme on general meshes for the Stokes equations in two space dimensions

This Note presents a new finite volume scheme for the Stokes equations on general non-structured meshes. A convergence result is presented, and an error estimate is given when the solution is regular enough. To cite this article: R. Eymard, R. Herbin, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Improvements to compressible Euler methods for low‐Mach number flows

In the present study improvements to numerical algorithms for the solution of the compressible Euler equations at low Mach numbers are investigated. To solve flow problems for a wide range of Mach numbers, from the incompressible limit to supersonic speeds, preconditioning techniques are frequently employed. On the other hand, one can achieve the same aim by using a suitably modified acoustic damping method. The solution algorithm presently under consideration is based on Roe's approximate Riemann solver [Roe PL. Approximate Riemann solvers, parameter vectors and difference schemes. Journal of Computational Physics 1981; 43: 357–372] for non-structured meshes. The numerical flux functions are modified by using Turkel's preconditioning technique proposed by Viozat [Implicit upwind schemes for low Mach number compressible flows. INRIA, Rapport de Recherche No. 3084, January 1997] for compressible Euler equations and by using a modified acoustic damping of the stabilization term proposed in the present study. These methods allow the compressible Euler equations at low-Mach number flows to be solved, and they are consistent in time. The efficiency and accuracy of the proposed modifications have been assessed by comparison with experimental data and other numerical results in the literature. Copyright © 2000 John Wiley & Sons, Ltd.

Smoothed surface gradients for panel methods

A weak form to compute the dipolar and monopolar surface gradients, related to a low-order panel method, is shown. The flow problem is formulated by means of a three-dimensional potential model and the discretization is based on Morino's formulation for the perturbation velocity potential. On the body surface, this representation reduces to a boundary integral equation with the source (or monopolar) and the doublet (or dipolar) densities. The first of the two is found by application of the boundary flow condition, and the second one is the unknown over the body surface. A lower panel method is used for the analytic integrations of both the monopolar and dipolar influence coefficients. The surface velocity field is computed after solving the linear system, with a strong and a weak form of the Stokes theorem, which is oriented to fairly non-structured panel meshes. The proposed method is validated by comparing the numerical results with analytical ones for an isolated sphere and includes a prediction over a car-like configuration.

Improvements to compressible Euler methods for low-Mach number flows

Improvements to numerical algorithms for the solution of the compressible Euler equations at low Mach numbers are investigated. To solve flow problems for a wide range of Mach numbers, from the incompressible limit to supersonic speeds, preconditioning techniques are frequently employed. On the other hand, one can achieve the same aim by using a suitably modified acoustic damping method. The solution algorithm presently under consideration is based on Roe's approximate Riemann solver for non-structured meshes. The numerical flux functions are modified by using Turkel's preconditioning technique proposed by Viozat for compressible Euler equations and by using a modified acoustic damping of the stabilization term proposed in the present study. These methods allow the compressible Euler equations at low-Mach number flows to be solved, and they are consistent in time. The efficiency and accuracy of the proposed modifications have been assessed by comparison with experimental data and other numerical results in the literature

Advance in overset grid schemes - From Chimera to DRAGON grids

A new approach that uses nonstructured mesh to replace the arbitrarily overlapped structured regions of embedded grids is presented. The present methodology uses the Chimera overlapped structured grids that are independently generated and body fitted, yielding a high-quality grid readily accessible for efficient solution schemes. A hybrid grid scheme is then proposed that maximizes the advantages of the Chimera scheme and adapts the strengths of the unstructured grid while at the same time keeps its weaknesses minimal. This new adaptation to the Chimera thinking is the direct replacement of arbitrary grid overlapping by nonstructured (DRAGON) grid. The nonstructured grid region sandwiched between the structured grids is limited in size, resulting in only a small increase in memory and computational effort. The DRAGON grid method has three important advantages: 1) preserving strengths of the Chimera grid, 2) eliminating difficulties sometimes encountered in the Chimera scheme, and 3) making grid communication in a fully conservative and consistent manner insofar as the governing equations are concerned. Numerical results on representative two-dimensional steady and unsteady inviscid flows are demonstrated.

Godunov type method on non-structured meshes for three-dimensional moving boundary problems

This paper presents a numerical method for the computation of compressible flows in domains whose boundaries move in a well defined predictable manner. The method uses the space-time formulation by Godunov while the discretization is conducted on non-structured tetrahedral meshes, using Roe's approximate Riemann solver, an implicit time stepping and a MUSCL-type interpolation. The computation of the geometrical parameters required to take into account the movement of the boundaries is described. Examples including the calculation of the flow in the cylinder of an internal combustion engine illustrates the possibilities of the method.

Unstructured multigridding by volume agglomeration: Current status

We describe a multigrid (MG) method for solving the Euler equations as applied to non-structured meshes in two (triangles) and three dimensions (tetrahedra). The main idea is to coarsen the given mesh by using topological neighboring relations. It is applied to upwind solvers relying on the MUSCL methodology. Two MG schemes are presented: an explicit Runge-Kutta FAS, and an implicit correction scheme. Transonic external flow computations are described for illustration.