Calculation of radiation fluxes to the surface of a space vehicle with the aid of a discrete ordinates method

A method and results of numerical modeling of radiative heating of the back surface of the MSRO (Mars Sample Return Orbiter) space vehicle of the European Space Agency are presented. To determine radiation heat fluxes, the method of discrete ordinates on unstructured tetrahedral grids is used. The radiative model is based on the radiation-transfer equation in a multigroup approximation. Numerical calculation has been performed for the most thermally stressed point of the assumed trajectory of the entry of an MSRO-type space vehicle into the Mars atmosphere. A comparison with the discrete ordinates method on structured grids is made. Good agreement between the results of calculations on structured and unstructured grids is demonstrated. The level of radiation heat fluxes to the back surface of the MSRO space vehicle is predicted. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 80, No. 2, pp. 71–78, March–April, 2007.     

Finite element multigrid solution of Euler flows past installed aero-engines

A finite element based procedure for the solution of the compressible Euler equations on unstructured tetrahedral grids is described. The spatial discretisation is accomplished by means of an approximate variational formulatin, with the explicit addition of a matrix form of artificial viscosity. The solution is advanced in time by means of an explicit multi-stage time stepping procedure. The method is implemented in terms of an edge based representation for the tetrahedral grid. The solution procedure is accelerated by use of a fully unstructured multigrid algorithm. The approach is applied to the simulation of the flow past an installed aero-engine nacelle, at three different free stream conditions. Comparison is made between the numerical predictions and experimental pressure observations.     

Performance comparison of data‐reordering algorithms for sparse matrix–vector multiplication in edge‐based unstructured grid computations

Several performance improvements for finite‐element edge‐based sparse matrix–vector multiplication algorithms on unstructured grids are presented and tested. Edge data structures for tetrahedral meshes and triangular interface elements are treated, focusing on nodal and edges renumbering strategies for improving processor and memory hierarchy use. Benchmark computations on Intel Itanium 2 and Pentium IV processors are performed. The results show performance improvements in CPU time ranging from 2 to 3. Copyright © 2005 John Wiley & Sons, Ltd.     

An adaptive solution of the 3-D Euler equations on an unstructured grid

Summary A new algorithm to generate the unstructured grid on a curved surface is developed. The advancing front method is used to generate the tetrahedral meshes in the space. An adaptive grid technique is used to enhance the calculation efficiency. The AUSM⁺ (Advection Upstream Splitting Method) scheme which was developed on a structured grid has been extended to be used to the spatial discretization of a cell-centered finite volume formulation on the unstructured grid. A second order spatial accuracy is achieved by applying a novel cell reconstruction procedure which can prevent the solution from exhibiting spurious oscillations without adding a limiter. A 3-D Euler solver for an adaptive tetrahedral grid and numerical results for several cases are presented.     

Variational Delaunay approach to the generation of tetrahedral finite element meshes

We describe an algorithm which generates tetrahedral decomposition of a general solid body, whose surface is given as a collection of triangular facets. The principal idea is to modify the constraints in such a way as to make them appear in an unconstrained triangulation of the vertex set à priori. The vertex set positions are randomized to guarantee existence of a unique triangulation which satisfies the Delaunay empty‐sphere property. (Algorithms for robust, parallelized construction of such triangulations are available.) In order to make the boundary of the solid appear as a collection of tetrahedral faces, we iterate two operations, edge flip and edge split with the insertion of additional vertex, until all of the boundary facets are present in the tetrahedral mesh. The outcome of the vertex insertion is another triangulation of the input surfaces, but one which is represented as a subset of the tetrahedral faces. To determine if a constraining facet is present in the unconstrained Delaunay triangulation of the current vertex set, we use the results of Rajan which re‐formulate Delaunay triangulation as a linear programming problem. Copyright © 2001 John Wiley & Sons, Ltd.     

Node placement for triangular mesh generation by Monte Carlo simulation

A new approach of node placement for unstructured mesh generation is proposed. It is based on the Monte Carlo method to position nodes for triangular or tetrahedral meshes. Surface or volume geometries to be meshed are treated as atomic systems, and mesh nodes are considered as interacting particles. By minimizing system potential energy with Monte Carlo simulation, particles are placed into a near‐optimal configuration. Well‐shaped triangles or tetrahedra can then be created after connecting the nodes by constrained Delaunay triangulation or tetrahedrization. The algorithm is simple, easy to implement, and works in an almost identical way for 2D and 3D meshing. Copyright © 2005 John Wiley & Sons, Ltd.     

Simple finite element‐based computation of distance functions in unstructured grids

A distance field is a representation of the closest distance from a point to a given surface. Distance fields are widely used in applications ranging from computer vision, physics and computer graphics and have been the subject of research of many authors in the last decade. Most of the methods for computing distance fields are devoted to Cartesian grids while little attention has been paid to unstructured grids. Finite element methods are well known for their ability to deal with partial differential equations in unstructured grids. Therefore, we propose an extension of the fast marching method for computing a distance field in a finite element context employing the element interpolation to hold the Eikonal property (∥?φ∥ = 1). A simple algorithm to develop the computations is also presented and its efficiency demonstrated through various unstructured grid examples. We observed that the presented algorithm has processing times proportional to the number of mesh nodes. Copyright © 2007 John Wiley & Sons, Ltd.     

An upwind kinetic flux vector splitting method on general mesh topologies

An upwind flux vector splitting algorithm which utilizes the moments of the Boltzmann equation to derive the Euler equations for inviscid compressible flow has been used with a variety of grid types. Although the upwind approach offers the potential for accurate flow simulations, it is necessary to ensure that such procedures can be utilized on realistic grids. In this paper, an upwind algorithm is used with structured multiblock grids, unstructured grids of triangles and hybrid structured/unstructured grids to solve realistic compressible flow problems in two dimensions.     

Graded tetrahedral finite element meshes

Vertices in the body centred cubic (bcc) lattice are used to create a tetrahedral spatial decomposition. With this spatial decomposition an octree approach is combined with Delaunay triangulations to decompose solids into tetrahedral finite element meshes. Solids must have their surfaces triangulated and the vertices in the triangulation are finite element nodes. Local densities of interior tetrahedra are controlled by the densities of surface triangles. Accuracy of the decomposition into finite elements depends on the accuracy of the surface triangulation which can be constructed with state of the art computer aided design systems.     

Triangulation of cubic panorama for view synthesis

An unstructured triangulation approach, new to our knowledge, is proposed to apply triangular meshes for representing and rendering a scene on a cubic panorama (CP). It sophisticatedly converts a complicated three-dimensional triangulation into a simple three-step triangulation. First, a two-dimensional Delaunay triangulation is individually carried out on each face. Second, an improved polygonal triangulation is implemented in the intermediate regions of each of two faces. Third, a cobweblike triangulation is designed for the remaining intermediate regions after unfolding four faces to the top/bottom face. Since the last two steps well solve the boundary problem arising from cube edges, the triangulation with irregular-distribution feature points is implemented in a CP as a whole. The triangular meshes can be warped from multiple reference CPs onto an arbitrary viewpoint by face-to-face homography transformations. The experiments indicate that the proposed triangulation approach provides a good modeling for the scene with photorealistic rendered CPs.     

An analysis of monotonicity conditions in the mixed hybrid finite element method on unstructured triangulations

In this paper we consider the mixed hybrid finite element method on unstructured triangular grids and evaluate its monotonicity properties by using a non standard set of basis functions for the velocity approximation space. The mixed hybrid discretization of the steady‐state diffusion equation produces a system matrix that depends only on the inner product of the outward normals to the edges of the triangulation and not on the choice of the velocity space basis. This property is used to study the characteristics of the system matrix. It is well known that this matrix is of type M if the angles of the triangulation are not bigger than π/2. An M‐matrix has a nonnegative inverse, i.e. all the elements are nonnegative. This implies the existence of a discrete maximum principle and thus monotonicity of the discretization. We show that, when the triangulation is of Delaunay type and satisfies the property that no circumcenters of boundary elements with Dirichlet conditions lie outside the domain, the inverse of the final matrix is always positive, even in the presence of obtuse angles. Copyright © 2008 John Wiley & Sons, Ltd.     

Higher‐resolution convection schemes for flow in porous media on highly distorted unstructured grids

Higher‐resolution schemes are presented for convective flow approximation on highly distorted unstructured grids. The schemes are coupled with continuous full‐tensor Darcy‐flux approximations. A sequence of non‐uniform and distorted grid formulations are developed and compared for a range of unstructured meshes with variable grid spacing. The higher‐order schemes are constructed using non‐uniform grid slope limiters such that they are stable with a local maximum principle, ensuring that solutions are free of spurious oscillations. Benefits of the resulting schemes are demonstrated for classical test problems in reservoir simulation including cases with full‐tensor permeability fields. The test cases involve a range of unstructured grids with variations in grid spacing, orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher‐order formulations are shown to effectively reduce numerical diffusion, leading to improved resolution of concentration and saturation fronts. Copyright © 2008 John Wiley & Sons, Ltd.     

Generation of unstructured tetrahedral meshes by advancing front technique

An algorithm for generating unstructured tetrahedral meshes by the advancing front technique is presented. Emphasis is placed on the construction of tetrahedral elements. Several measures are employed to prevent difficult situations. A control line/surface scheme is used to specify element size. Numerical examples are provided to show the performance of the algorithm.     

THREE-DIMENSIONAL FINITE ELEMENT MESHING BY INCREMENTAL NODE INSERTION

This paper describes an efficient algorithm for fully automated three-dimensional finite element meshing which is applicable to non-convex geometry and non-manifold topology. This algorithm starts with sparsely placed nodes on the boundaries of a geometric model and a corresponding 3-D Delaunay triangulation. Nodes are then inserted incrementally by checking the tetrahedral mesh geometry and topological compatibility between Delaunay triangulation and the geometric model. Topological compatibility is checked in a robust manner by a method which relies more on a mesh's topology than its geometry. The node placement strategy is tightly coupled to an incremental Delaunay triangulation algorithm, and results in a low growth rate of computational time.     

High‐order finite volume schemes on unstructured grids using moving least‐squares reconstruction. Application to shallow water dynamics

This paper introduces the use of moving least‐squares (MLS) approximations for the development of high‐order finite volume discretizations on unstructured grids. The field variables and their successive derivatives can be accurately reconstructed using this mesh‐free technique in a general nodal arrangement. The methodology proposed is used in the construction of two numerical schemes for the shallow water equations on unstructured grids: a centred Lax–Wendroff method with added shock‐capturing dissipation, and a Godunov‐type upwind scheme, with linear and quadratic reconstructions. This class of mesh‐free techniques provides a robust and general approximation framework which represents an interesting alternative to the existing procedures, allowing, in addition, an accurate computation of the viscous fluxes. Copyright © 2005 John Wiley & Sons, Ltd.     

Delaunay versus max-min solid angle triangulations for three-dimensional mesh generation

The Delaunay triangulation has been used in several methods for generating finite element tetrahedral meshes in three-dimensional polyhedral regions. Other types of three-dimensional triangulations are possible, such as a triangulation satisfying a local max-min solid angle criterion. In this paper, we present experimental results to show that max-min solid angle triangulations are better than Delaunay triangulations for finite element tetrahedral meshes, since the former type of triangulations contains tetrahedra of better shape than the latter type. We also describe how mesh points are generated and triangulated in our tetrahedral mesh generation method.     

Multigrid solution of the 3-D compressible euler equations on unstructured tetrahedral grids

A low storage, computationally efficient algorithm for the solution of the compressible Euler equations on unstructured tetrahedral meshes is developed. The algorithm takes the form of a centred scheme with the explicit addition of a high accuracy artificial viscosity and the solution is advanced to steady state by means of a multi-stage time stepping method. The side based data structure which is employed enables a clear connection to be established between the proposed algorithm and upwind cell vertex schemes for unstructured meshes. The computational efficiency of the procedure is improved by incorporating an unstructured multigrid acceleration procedure. A number of flows of practical interest are analysed to demonstrate the numerical performance of the proposed approach.     

Calculation of a hypersonic flow over bodies of complex configuration on unstructured tetrahedral meshes using the AUSM scheme

An approach to solving the three-dimensional Navier-Stokes equations on tetrahedral computational meshes on the basis of splitting by physical processes is considered. An algorithm of numerical implementation of the suggested method for solving three-dimensional problems of aerothermodynamics of freeconfiguration hypersonic flying vehicles (HFVs) is elaborated. The finite-volume method is applied to approximate the gasdynamics equations. The fluxes on the boundaries of the computational elements are calculated using the AUSM scheme. A computer code aimed at numerical simulation of the three-dimensional aerothermodynamics of the structural elements and the integral configurations of the HFV on the basis of the Euler and Navier-Stokes equations is developed. The algorithms developed are tested using the benchmark problem of a viscous hypersonic perfect gas flow over a sphere. The results of comparison of the computational data found using the suggested approach on the unstructured different-size meshes with the numerical solutions found on structured grids with application of the computational code NERAT are presented. The computational model of the flow of viscous and inviscid perfect gas developed is applied to investigate the aerothermodynamics of a model of an unmanned experimental aircraft X-43 of complex configuration.     

Hamiltonian Paths Through Two- and Three-Dimensional Grids

This paper addresses the existence of Hamiltonian paths and cycles in two-dimensional grids consisting of triangles or quadrilaterals, and three-dimensional grids consisting of tetrahedra or hexahedra. The paths and cycles may be constrained to pass from one element to the next through an edge, through a vertex, or be unconstrained and pass through either. It was previously known that an unconstrained Hamiltonian path exists in a triangular grid under very mild conditions, and that there are triangular grids for which there is no through-edge Hamiltonian path. In this paper we prove that a through-vertex Hamiltonian cycle exists in any triangular or tetrahedral grid under very mild conditions, and that there exist quadrilateral and hexahedral grids for which no unconstrained Hamiltonian path exists. The existence proofs are constructive, and lead to an efficient algorithm for finding a through-vertex Hamiltonian cycle.