Unstructured Polyhedral Meshes Research Articles

Coherent mesh representation for parallel I/O of unstructured polyhedral meshes

AbstractThis paper presents a new mesh data layout for parallel I/O of linear unstructured polyhedral meshes. The new mesh representation infers coherence across entities of different topological dimensions, i.e., grid cells, faces, and points. The coherence due to cell-to-face and face-to-point connectivities of the mesh is formulated as a tree data structure distributed across processors. The mesh distribution across processors creates consecutive and contiguous slices that render an optimized data access pattern for parallel I/O. A file format using the coherent mesh representation, developed and tested with OpenFOAM, enables the usability of the software at unprecedented scales. Further implications of the coherent and sliceable mesh representation arise due to simplifications in partitioning and diminished pre- and post-processing overheads.

Second-order accurate normal reconstruction from volume fractions on unstructured meshes with arbitrary polyhedral cells

This paper introduces a novel method for the computation of accurate normal fields from volume fractions on unstructured polyhedral meshes. For surfaces of considerable curvature variation, the method is second-order accurate with respect to symmetric volume differences and first-order accurate with respect to normal field angular deviation. For this purpose, an averaged normal is locally computed, i.e. in each mesh cell, by fitting a plane in a least squares sense with appropriate weights to the volume fraction data of neighbouring cells while explicitly accounting for volume conservation in the cell at hand. The resulting minimization problem is solved approximately by means of a Newton-type method. Moreover, employing the Reynolds transport theorem allows for both assessing the regularity of the derivatives of the least squares error functional with respect to the free parameters and to exactly compute these derivatives at small additional costs. Since the divergence theorem implies that the volume fraction can be written as a sum of face-based quantities, our method considerably simplifies the numerical procedure for applications in three spatial dimensions while demonstrating an inherent ability to robustly deal with unstructured meshes.We discuss the theoretical foundations, regularity and appropriate error measures, along with the details of the numerical algorithm. Finally, numerical results for convex and non-convex hypersurfaces embedded in cuboidal and tetrahedral meshes are presented together with a convergence study considering two different error measures. Moreover, our findings are complemented by profound insights into the minimization procedure.

Third-order accurate initialization of volume fractions on unstructured meshes with arbitrary polyhedral cells

This paper introduces a novel method for the efficient and accurate computation of volume fractions on unstructured polyhedral meshes, where the phase boundary is an orientable hypersurface, implicitly given as the iso-contour of a sufficiently smooth level-set function. Locally, i.e. in each mesh cell, we compute a principal coordinate system in which the hypersurface can be approximated as the graph of an osculating paraboloid. A recursive application of the Gaussian divergence theorem then allows to analytically transform the volume integrals into curve integrals associated to the polyhedron faces, which can be easily approximated numerically by means of standard Gauss-Legendre quadrature. This face-based formulation enables the applicability to unstructured meshes and considerably simplifies the numerical procedure for applications in three spatial dimensions. We discuss the theoretical foundations and provide details of the numerical algorithm. Finally, we present numerical results for convex and non-convex hypersurfaces embedded in cuboidal and tetrahedral meshes, showing both high accuracy and third- to fourth-order convergence with spatial resolution. The proposed algorithm outperforms existing methods in terms of both accuracy and execution time.

A coupled level set and volume of fluid method for three dimensional unstructured polyhedral meshes for boiling flows

For the numerical modelling of boiling flows with accurate interface dynamics, a new coupled level set and volume of fluid (CLSVOF) method for three-dimensional unstructured polyhedral meshes has been presented in this paper. An unsplit multidirectional method is used for advecting volume fraction. In this method, the cell vertex velocities interpolated from the cell centres are used to create the flux polyhedron. A total variational diminishing (TVD) scheme is used for the advection of level set function. The level set field is geometrically reinitialized after every time step. The present CLSVOF method is tested against several two as well as three-dimensional advection test cases for both structured and unstructured polyhedral meshes to determine its accuracy and performance. In terms of advection test cases, the suggested method shows good agreement with results of previous studies that used interface reconstruction techniques of higher order. The proposed method is then used to simulate two as well as three-dimensional saturated film boiling over a flat planar surface for both unstructured polyhedral and structured meshes. The obtained results are in accordance with the results available in literature. The present CLSVOF method is also demonstrated for saturated film boiling on a horizontal cylinder in both two and three dimensions. In comparison to the earlier numerical studies, the present results are in good accordance with the available semi-empirical correlations. • A CLSVOF method is developed for boiling flows on 3D unstructured polyhedral meshes. • The proposed boiling flow solver can be used for any complex geometries. • The solver is validated for two-phase flows with and without phase change. • For boiling flows, the present results agree well with semi-empirical correlations.

Sequential implicit vertex approximate gradient discretization of incompressible two-phase Darcy flows with discontinuous capillary pressure

This work deals with sequential implicit schemes for incompressible and immiscible two-phase Darcy flows which are commonly used and well understood in the case of spatially homogeneous capillary pressure functions. To our knowledge, the stability of this type of splitting schemes solving sequentially a pressure equation followed by the saturation equation has not been investigated so far in the case of discontinuous capillary pressure curves at different rock type interfaces. It will be shown here to raise severe stability issues for which stabilization strategies are investigated in this work. To fix ideas, the spatial discretization is based on the Vertex Approximate Gradient (VAG) scheme accounting for unstructured polyhedral meshes combined with an Hybrid Upwinding (HU) of the transport term and an upwind positive approximation of the capillary and gravity fluxes. The sequential implicit schemes are built from the total velocity formulation of the two-phase flow model and only differ in the way the conservative VAG total velocity fluxes are approximated. The stability, accuracy and computational cost of the sequential implicit schemes studied in this work are tested on oil migration test cases in 1D, 2D and 3D basins with a large range of capillary pressure parameters for the drain and barrier rock types. It will be shown that usual splitting strategies fail to capture the right solutions for highly contrasted rock types and that it can be fixed by maintaining locally the pressure saturation coupling at different rock type interfaces in the definition of the conservative total velocity fluxes. The numerical investigation of the sequential schemes is also extended to the widely used finite volume Two-Point Flux Approximation spatial discretization.

Drag induced aerodynamic braking for racing motorcycles

Winglets are small aerodynamic structures attached to the front fairing of a racing motorcycle to increase traction. It increases the control of the rider and maneuverability while accelerating as it reduces the event of front wheel lift. This paper discusses the speed of the motorcycle and braking force given by the rider. Based on the inputs it would change the orientation of the winglet which would produce a drag force on the bike and increases the rate of deceleration. CFD simulation is performed by STARCCM + software with unstructured polyhedral mesh and AKN k-ε turbulence model is used on the winglet surface with different orientations (0 and 90 degree) to find the type of airflow that creates optimum drag and maximum pressure, in which winglet can withstand without being damaged. The procedural way of numerical approach and it results may provide design assistance for the improvement of the aerodynamic winglets for racing motor vehicles.

A smoothed finite element method for octree-based polyhedral meshes with large number of hanging nodes and irregular elements

Techniques are now available to generate high-quality polyhedral elements offering a good alternative way for accurate stress analysis for solids. However, distorted polyhedral elements cannot be effectively used in standard finite element methods (FEM) without extra complicated treatments, due to the hanging nodes on the element edges and complex shape of polyhedral elements. On the contrary, the hanging nodes and complex shape of the elements can be naturally dealt with in the smoothed finite element method (S-FEM). This paper, utilizes the octree method to automatically and efficiently generate unstructured polyhedral meshes, using the standard tessellation language (STL). We develop three S-FEM models for octree meshes, using all existing node-based, edge-based and face-based smoothing domains. A subdomain smoothing technique is developed to re-cut the smoothing domain (SD), which simplifies the process of smoothing domain formation. The shapes of the subdomains have some common geometries that are much easier to handle, making use of the good features offered by the S-FEM formulation. The introduction of subdomains does not affect the results of the calculations, and the properties of the three S-FEMs are still preserved. Several demonstration examples are presented to verify the efficiency of the present S-FEM models.

Numerical analysis of the flow behavior in a helically coiled once through steam generator

The once-through steam generator (OTSG) is an essential equipment with a key position in the primary side and the secondary side usually used in small modular reactors. It is valuable for designers of the OTSG to investigate the flow behavior in the primary side. A three-dimensional (3D) computational fluid dynamics (CFD) model is conducted in this paper to study the flow behaviors in the primary side of OTSG. The unstructured polyhedral meshes were used in this paper, and ANSYS FLUENT was the solving tool to solve the governing equations by using Realizable k − ɛ turbulence model with enhanced wall treatment. Through a series of cases, the velocity profiles and pressure drop through the primary side of the OTSG were calculated, and the influences of different inlet parameters and different structure designs on the coolant behaviors and flow distributions were studied. Ultimately some suggestions for improvements were proposed to achieve a more rational design with more uniform flow distributions and smaller pressure drops. It was found that the flow distribution in the primary side of an OTSG is a property of its structure, on which inlet conditions have little effect. The position of inlet nozzle has a significant impact on the flow distribution.

Simulation of the Francis-99 Hydro Turbine During Steady and Transient Operation

Numerical simulation of the Francis-99 hydroturbine with correlation to experimental measurements are presented. Steady operation of the hydroturbine is analyzed at three operating conditions: the best efficiency point (BEP), high load (HL), and part load (PL). It is shown that global quantities such as net head, discharge and efficiency are well predicted. Additionally, time-averaged velocity predictions compare well with PIV measurements obtained in the draft tube immediately downstream of the runner. Differences in vortex rope structure between operating points are discussed.Unsteady operation of the hydroturbine from BEP to HL and from BEP to PL are modeled. It is shown that simulation methods used to model the steady operation produce predictions that correlate well with experiment for transient operation.Time-domain unsteady simulation is used for both steady and unsteady operation. The full-fidelity geometry including all components is meshed using an unstructured polyhedral mesh with body-fitted prism layers. Guide vane rotation for transient operation is imposed using fully-conservative, computationally efficient mesh morphing. The commercial solver STAR-CCM+ is used for all portions of the analysis including meshing, solving and post-processing.

A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation

We present a new family of mimetic finite difference schemes for solving elliptic partial differential equations in the primal form on unstructured polyhedral meshes. These mimetic discretizations are built to satisfy local consistency and stability conditions. The consistency condition is an exactness property, i.e., the mimetic schemes are exact when the solution is a polynomial of an assigned degree. The stability condition ensures the well-posedness of the method. The degrees of freedom are the solution moments on mesh faces and inside mesh cells. Higher order schemes are built using higher order moments. The developed schemes are verified numerically on diffusion problems with constant and spatially variable (possibly, discontinuous) tensorial coefficients.

Large eddy simulation of a randomly stacked nuclear pebble bed

This paper reports large eddy simulation results for a randomly stacked bed of spherical pebbles, using a second-order accurate, cell-centered finite volume method on an unstructured polyhedral mesh. The selected flow configuration represents the core of a high temperature reactor, in which nuclear fuel is embedded in the pebbles. The geometrical arrangement consists of approximately 30 pebbles, which are randomly stacked and in contact with each other. Extensive analyses of flow and thermal fields are performed to derive valuable insights on the flow characteristics. The predicted flow-field is fairly complex and exhibits highly unsteady and three-dimensional turbulent behavior with strong rotational and cross flow regions. The flow, while moving over the pebbles shows attenuation and enhancement in the turbulence levels, and eventually yields to flow separation and its subsequent reattachment. Consequently, a wide range of temperatures over the pebbles is observed; this includes the appearance of hot-spots especially around contact areas. The occurrence of high- and low-velocity streak regions, and the corresponding temperature fields are examined and quantified. The reported analyses can be used for validation of low-order turbulence modeling applications to complex pebble flow configurations.

ComPASS : a tool for distributed parallel finite volume discretizations on general unstructured polyhedral meshes

The objective of the ComPASS project is to develop a parallel multiphase Darcy flow simulator adapted to general unstructured polyhedral meshes (in a general sense with possibly non planar faces) and to the parallelization of advanced finite volume discretizations with various choices of the degrees of freedom such as cell centres, vertices, or face centres. The main targeted applications are the simulation of CO2 geological storage, nuclear waste repository and reservoir simulations. The CEMRACS 2012 summer school devoted to high performance computing has been an ideal framework to start this collaborative project. This paper describes what has been achieved during the four weeks of the CEMRACS project which has been focusing on the implementation of basic features of the code such as the distributed unstructured polyhedral mesh, the synchronization of the degrees of freedom, and the connection to scientific libraries including the partitioner METIS, the visualization tool PARAVIEW, and the parallel linear solver library PETSc. The parallel efficiency of this first version of the ComPASS code has been validated on a toy parabolic problem using the Vertex Approximate Gradient finite volume spacial discretization with both cell and vertex degrees of freedom, combined with an Euler implicit time integration.

A fast mesh deformation method using explicit interpolation

A novel mesh deformation algorithm for unstructured polyhedral meshes is developed utilizing a tree-code optimization of a simple direct interpolation method. The algorithm is shown to provide mesh quality that is competitive with radial basis function based methods, with markedly better performance in preserving boundary layer orthogonality in viscous meshes. The parallelization of the algorithm is described, and the algorithm cost is demonstrated to be O( n log n). The parallel implementation was used to deform meshes of 100 million nodes on nearly 200 processors demonstrating that the method scales to large mesh sizes. Results are provided for a simulation of a high Reynolds number fluid–structure interaction case using this technique.

Conservative interpolation on unstructured polyhedral meshes: An extension of the supermesh approach to cell-centered finite-volume variables

A method is presented for conservatively transferring, or remapping, cell-centered variable fields from one mesh to another with second-order accuracy. The method is generally applicable to any polyhedral source or target mesh. Like the work of Farrell et al. [1], which was designed for finite-element computations, the proposed methodology uses a logical supermesh consisting of the intersections of polyhedra from both meshes. The resulting transfer process is well-suited for finite-volume methods that rely on cell-centered variables. The accuracy and efficacy of the new remapping process is demonstrated with numerical experiments and a computational fluid dynamics test.

The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes

We extend the mimetic finite difference (MFD) method to the numerical treatment of magnetostatic fields problems in mixed div– curl form for the divergence-free magnetic vector potential. To accomplish this task, we introduce three sets of degrees of freedom that are attached to the vertices, the edges, and the faces of the mesh, and two discrete operators mimicking the curl and the gradient operator of the differential setting. Then, we present the construction of two suitable quadrature rules for the numerical discretization of the domain integrals of the div– curl variational formulation of the magnetostatic equations. This construction is based on an algebraic consistency condition that generalizes the usual construction of the inner products of the MFD method. We also discuss the linear algebraic form of the resulting MFD scheme, its practical implementation, and discuss existence and uniqueness of the numerical solution by generalizing the concept of logically rectangular or cubic meshes by Hyman and Shashkov to the case of unstructured polyhedral meshes. The accuracy of the method is illustrated by solving numerically a set of academic problems and a realistic engineering problem.

A monotone nonlinear finite volume method for advection–diffusion equations on unstructured polyhedral meshes in 3D

We present a new monotone finite volume method for the advection–diffusion equation with a full anisotropic discontinuous diffusion tensor and a discontinuous advection field on 3D conformal polyhedral meshes. The proposed method is based on a nonlinear flux approximation both for diffusive and advective fluxes and guarantees solution non-negativity. The approximation of the diffusive flux uses the nonlinear two-point stencil described in [Danilov and Vassilevski, Russ. Numer. Anal. Math. Modelling 24: 207–227, 2009]. Approximation of the advective flux is based on the second-order upwind method with a specially designed minimal nonlinear correction [Lipnikov, Svyatskiy, and Vassilevski, J. Comp. Phys. 229: 4017–4032, 2010]. The second-order convergence rate and monotonicity are verified with numerical experiments.

Convergence analysis of the high-order mimetic finite difference method

We prove second-order convergence of the conservative variable and its flux in the high-order MFD method. The convergence results are proved for unstructured polyhedral meshes and full tensor diffusion coefficients. For the case of non-constant coefficients, we also develop a new family of high-order MFD methods. Theoretical result are confirmed through numerical experiments.

3D conforming power diagrams for radial LGR in CPG reservoir grids

Mesh generation becomes a crucial step in reservoir flow simulation of new generation. The mesh must faithfully represent the architecture of the reservoir and its heterogeneity. In (Flandrin et al. in IJNME 65(10):1639–1672, 2006) a three-dimensional hybrid mesh model was proposed to capture the radial characteristics of the flow around the wells. In this hybrid mesh, the reservoir is described by a non-uniform Cartesian structured mesh and the drainage areas around the wells are represented by structured radial circular meshes. Unstructured polyhedral meshes are used to connect these two kinds of structured grids. The construction of these transition meshes is based on 3D power diagrams (Aurenhammer in SIAM J Comput 16(1):78–96, 1987) to ensure finite volume properties such as mesh conformity, dual orthogonality and cell convexity. In this paper, we propose an extension of this hybrid model to the case where the reservoir is described by a corner point geometry (CPG) grid. At first, the CPG grid is mapped, in a reference space, into a non-uniform Cartesian grid by minimizing the mapping deformation. Then, a hybrid mesh is generated in this reference space using the previous method. Finally, this mesh is mapped back into the real space. Some quality criterions are introduced to measure and improve the quality of the polyhedral transition mesh.

CONVERGENCE OF MIMETIC FINITE DIFFERENCE METHOD FOR DIFFUSION PROBLEMS ON POLYHEDRAL MESHES WITH CURVED FACES

New mimetic finite difference discretizations of diffusion problems on unstructured polyhedral meshes with strongly curved (non-planar) faces are developed. The material properties are described by a full tensor. The optimal convergence estimates, the second order for a scalar variable (pressure) and the first order for a vector variable (velocity), are proved.

The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes

We study the mimetic finite difference discretization of diffusion-type problems on unstructured polyhedral meshes. We demonstrate high accuracy of the approximate solutions for general diffusion tensors, the second-order convergence rate for the scalar unknown and the first order convergence rate for the vector unknown on smooth or slightly distorted meshes, on non-matching meshes, and even on meshes with irregular-shaped polyhedra with flat faces. We show that in general the meshes with non-flat faces require more than one flux unknown per mesh face to get optimal convergence rates.