Arbitrary Polyhedral Cells Research Articles

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 finite volume coupled level set and volume of fluid method with a mass conservation step for simulating two‐phase flows

AbstractThis article presents a finite volume method for simulating two‐phase flows using a level set approach coupled with volume of fluid method capable of simulating sharp fluid interfaces. The efficiency of the method is a result of the fact that the interface is calculated in order to satisfy mass conservation with no explicit interface reconstruction step and the mass fluxes across cell‐faces are corrected to respect the recovered volume fraction. The mass‐conservation correction step proposed here, is utilized using an iterative algorithm which solves a reaction‐diffusion equation for the mass correction of the level set. The re‐sharpened volume fraction is used for the new volumetric fluxes at each cell which are calculated through the proposed algorithm that guarantees that they satisfy mass conservation. The algorithm is not limited in representing the interface with the 0.5‐contour and is applicable for arbitrary polyhedral cells. Good accuracy and mass conservation are achieved when compared to other conservative approaches.

Uncollided flux techniques for arbitrary finite element meshes

The uncollided angular flux can be difficult to compute accurately in discrete-ordinate radiation transport codes, especially in weakly-scattering configurations with localized sources. It has long been recognized that an analytical or semi-analytical treatment of the uncollided flux, coupled with a discrete-ordinate solution for the collided flux, can yield dramatic improvements in solution accuracy and computational efficiency. In this paper, we present such an algorithm for the semi-analytical calculation of the uncollided flux. This algorithm is unique in several aspects: (1) it applies to arbitrary polyhedral cells (and can be thus coupled with collided flux solvers that support arbitrary polyhedral meshes without the need for explicit tetrahedral re-meshing), (2) it provides accurate uncollided solutions near sources, (3) it is devised with parallel implementation in mind, and (4) it minimizes the total number of traced rays and maintains a reasonable ray density on each local subdomain. This paper provides a complete derivation of the algorithm and demonstrates its important features on a set of simple examples and a standard transport benchmark. Assessment of its parallel performance will be the subject of a subsequent paper.

Mixed finite volume method for linear thermoelasticity at all Poisson’s ratios

ABSTRACTA mixed (displacement/pressure) method for modeling both compressible and incompressible linear thermoelastic body problems is described. The method uses a form of Hooke’s law that introduces the pressure as an additional dependent variable and removes the numerical problems associated with incompressibility. A cell-centerd finite volume discretization by arbitrary polyhedral cells, an implicit temporal discretization scheme, both second-order accurate and segregated solution procedure make the method computationally efficient.The results of calculations are compared with analytical or finite element solutions showing a very good agreement. The present method is also compared with the displacement finite volume method and shown that it is accurate and stable for all Poisson’s ratios, while the performance of the displacement method rapidly deteriorates as the Poisson’s ratio approaches the incompressibility limit.

A computational method for sharp interface advection

We devise a numerical method for passive advection of a surface, such as the interface between two incompressible fluids, across a computational mesh. The method is called isoAdvector, and is developed for general meshes consisting of arbitrary polyhedral cells. The algorithm is based on the volume of fluid (VOF) idea of calculating the volume of one of the fluids transported across the mesh faces during a time step. The novelty of the isoAdvector concept consists of two parts. First, we exploit an isosurface concept for modelling the interface inside cells in a geometric surface reconstruction step. Second, from the reconstructed surface, we model the motion of the face–interface intersection line for a general polygonal face to obtain the time evolution within a time step of the submerged face area. Integrating this submerged area over the time step leads to an accurate estimate for the total volume of fluid transported across the face. The method was tested on simple two-dimensional and three-dimensional interface advection problems on both structured and unstructured meshes. The results are very satisfactory in terms of volume conservation, boundedness, surface sharpness and efficiency. The isoAdvector method was implemented as an OpenFOAM® extension and is published as open source.

Implementation of a second-order upwind method in a semi-implicit two-phase flow code on unstructured meshes

A two-phase flow analysis code, CUPID, has been developed for a realistic simulation of thermal–hydraulic phenomena in nuclear reactor components. In the CUPID code, a two-fluid three-field model is adopted and the governing equations are solved on unstructured meshes. To obtain the numerical solution, the semi-implicit method of the RELAP5 code was used with some modifications for a cell-centered finite volume method. In this work, a second-order upwind method was implemented for the convective terms of the CUPID code. To get the slopes of the convective quantities, we adopted the Frink’s reconstruction method (1994, AIAA Paper 94-0061) and modified it for an application to arbitrary polyhedral cells. To stabilize the numerical solutions, the Barth and Jesperson’s slope limiter was used. In order to evaluate the enhanced accuracy and ensure the robustness of the implemented scheme, numerical tests were performed using conceptual single- and two-phase flow problems, which include a strong phase change and very heterogeneous phase distributions.

An efficient and robust particle-localization algorithm for unstructured grids

An efficient and robust particle-localization algorithm for unstructured grids is presented. Given a particle position and the cell containing that position, the algorithm determines the cell which contains a nearby position. The algorithm is based on tracking a particle along its trajectory by computing the intersections of the trajectory and the cell faces. Compared to previously published particle-localization algorithms, the new algorithm has several advantages. First, it can be applied to grids consisting of arbitrary polyhedral cells. Second, the algorithm is not limited to small particle displacements. Third, the interaction of particles with boundaries is dealt with correctly and naturally. Fourth, the algorithm is more efficient than other published algorithms. A modified version of the present algorithm is also presented which can detect inconsistencies and take appropriate corrective action. With these modifications, the computational cost becomes comparable to other published algorithms which cannot detect inconsistencies without resorting to fall-back algorithms such as exhaustive or Octree search.

Foreword to the Special Issue on Computational Heat Transfer

Foreword to the Special Issue on Computational Heat Transfer

Computation of flow-induced motion of floating bodies

A computational procedure for the prediction of motion of rigid bodies floating in viscous fluids and subjected to currents and waves is presented. The procedure is based on a coupled iterative solution of equations of motion of a rigid body with up to six degrees of freedom and the Reynolds-averaged Navier–Stokes equations describing the two- or three-dimensional fluid flow. The fluid flow is predicted using a commercial CFD package which can use moving grids made of arbitrary polyhedral cells and allows sliding interfaces between fixed and moving grid blocks. The computation of body motion is coupled to the CFD code via user-coding interfaces. The method is used to compute the 2D motion of floating bodies subjected to large waves and the results are compared to available experimental data, showing favorable agreement.

Simulation of Flow-Induced Ship Motions in Waves using a RANSE Method

A coupled method for the computation of viscous free-surface flows and flowinduced motion predicts the fluid flow using a finite-volume method which can accommodate moving grids made of arbitrary polyhedral cells. The computation of ship motion is coupled to the ow solver at each iteration. Two dfferent methods, namely inlet or apping wall boundary, are employed to generate waves in the numerical tank. Small amplitude and large amplitude waves are produced and compared to corresponding theories or experiments, showing favorable agreement. The method is applied to compute the motion of ships subjected to waves agreeing well with experimental data.

Improved Formulation for Geometric Properties of Arbitrary Polyhedra

T is necessary to compute various geometric properties in a e nite volume discretization scheme used in computational e uid dynamics. These properties often include face areas and normal vectors, face centers, cell volumes and cell centroids, etc. In a computational grid composed of relatively simple cell types, such as tetrahedra, hexahedra, and prisms, it is relatively easy to compute these properties. For a computational grid with arbitrary polyhedral cells, such as cut cells generated through cell cutting in an adaptive Cartesian grid method, 1 it is not trivial to calculate the geometric properties,especiallythecellcentroids.InarecentpaperbyBruner, 2 aformulawasgiventocomputethecellcentroidofan N-facedpolyhedron. In Bruner’ s formula, the cell centroid is computed through a surface integral over the N faces bounding the cell. The surface integral, however, requires the evaluation of a Gaussian quadrature. The computation of the quadrature needs three quadrature points for a triangle and four points for a quadrilateral. The computation of the Gaussian quadrature for an arbitrary polygonal face is not straightforward. In this Note, a new formulation is derived for the cell centroid of an N-faced polyhedron for which each face is an arbitrary planar polygon. The new centroid formulation does not need the evaluation of a Gaussian quadrature. Only surface areas, normals, and face centers are required.