Three-dimensional Tetrahedral Meshes Research Articles

High-order compact gas-kinetic schemes for three-dimensional flow simulations on tetrahedral mesh

A general framework for the development of high-order compact schemes has been proposed recently. The core steps of the schemes are composed of the following. 1). Based on a kinetic model equation, from a generalized initial distribution of flow variables construct a time-accurate evolution solution of gas distribution function at a cell interface and obtain the corresponding flux function; 2). Introduce the WENO-type weighting functions into the high-order time-derivative of the cell interface flux function in the multistage multi-derivative (MSMD) time stepping scheme to cope with the possible impingement of a shock wave on a cell interface within a time step, and update the cell-averaged conservative flow variables inside each control volume; 3). Model the time evolution of the gas distribution function on both sides of a cell interface separately, take moments of the inner cell interface gas distribution function to get flow variables, and update the cell-averaged gradients of flow variables inside each control volume; 4). Based on the cell-averaged flow variables and their gradients, develop compact initial data reconstruction to get initial condition of flow distributions at the beginning of next time step. A compact gas-kinetic scheme (GKS) up to sixth-order accuracy in space and fourth-order in time has been constructed on 2D unstructured mesh. In this paper, the compact GKS up to fourth-order accuracy on three-dimensional tetrahedral mesh will be further constructed with the focus on the WENO-type initial compact data reconstruction. Nonlinear weights are designed to achieve high-order accuracy for the smooth Navier-Stokes solution and keep super robustness in 3D computation with strong shock interactions. The fourth-order compact GKS uses a large time step with a CFL number 0.6 in the simulations from subsonic to hypersonic flow. A series of test cases are used to validate the scheme. The high-order compact GKS can be used in 3D applications with complex geometry.

A high order positivity-preserving conservative WENO remapping method on 3D tetrahedral meshes

We propose a high order positivity-preserving conservative remapping method on three-dimensional (3D) tetrahedral meshes, based on the weighted essentially non-oscillatory (WENO) reconstruction method. By precisely computing the overlaps between the meshes before and after the rezoning step in the arbitrary Lagrangian–Eulerian (ALE) framework, our method does not limit the range of mesh movements and has wider applications. This also makes our remapping process simpler to attain high-order accuracy. We use the third order multi-resolution WENO reconstruction procedure in this paper as an example, in which we reconstruct three polynomials of different orders via nested central spatial stencils and distribute nonlinear weights based on the smoothness of the polynomials, ensuring optimal accuracy in the smooth region while avoiding numerical oscillations in the non-smooth region. The multi-resolution WENO procedure involves fewer high-order reconstruction polynomials and can use arbitrary positive linear weights, making it more effective for our 3D remapping problem. We incorporate an efficient local limiting to preserve positivity for the positive physical variables involved in the ALE framework without sacrificing the original high-order accuracy and conservation. A set of numerical tests are provided to verify properties of our remapping algorithm, such as high-order accuracy, conservation, essentially non-oscillatory performance, positivity-preserving and efficiency.

New open computational resource for plasma processes modelling

The paper describes a new open access computing resource nfusion.cs.msu.ru. The resource includes modules for calculating equilibrium, vertical stability, plasma evolution, simulation systems for magnetic diagnostics, as well as a new algorithm for constructing three-dimensional tetrahedral meshes in areas of complex structure. The modules are integrated into a unified software environment designed for numerical support of experiments on tokamak installations. The possibility of accessing the calculation modules on the server via the Internet, data exchange and the issuance of calculation results in the form of files, figures, and tables has been implemented. The resource supports the simultaneous operation of several users and has an information support system in Russian and English.

Расчетный анализ особенностей аэротермодинамики высокоскоростного летательного аппарата X-51

The paper presents a numerical investigation of aero-dynamic coefficients for a model of an X-51-type high-velocity aircraft moving at Mach 6. The simulation made use of the original and modified versions of our custom hydrocodes (UST3D and UST3D-AUSMPW) designed for numerical simulation of aerodynamics and thermodynamics in high-velocity aircraft of arbitrary shapes. Such hydrocodes implement a model of viscous compressible thermally conductive gas described by a non-steady-state spatial system of Navier --- Stokes equations solved over unstructured three-dimensional tetrahedral meshes. The paper considers the theoretical aspects of simulating the aerodynamics and thermodynamics of high-velocity aircraft numerically. We describe the method for computing mass flow through mesh cell boundaries implemented in the modified custom hydrocode version. We performed cross-validation of the results obtained using our custom hydrocodes and compared our hydrocodes in terms of result convergence time. We show that these custom hydrocodes ensure adequately accurate distribution patterns concerning the fields of the values sought, and provide high-precision computation of aerodynamic characteristics as compared to each other

Численное моделирование наземного эксперимента HIFiRE-1

The paper considers the problem of simulating the HIFiRE-1 ground test numerically. The aircraft geometry is represented by either a pointed or a blunted cone combined with a flared cylinder. Our digital simulation investigated the aerodynamics of two aircraft configurations: one featuring a pointed nose, another featuring a blunted nose with a radius of 2.5 mm. We used the UST3D software developed in the Ishlinsky Institute for Problems in Mechanics RAS, to perform our aerodynamic calculations. The software is specifically designed for numerical simulations of aerodynamics and thermodynamics in high-velocity aircraft. It implements a model of viscous compressible thermally conductive gas described by a non-steady-state spatial system of Navier --- Stokes equations solved over unstructured three-dimensional tetrahedral meshes. We compared the numerical simulation results in the form of pressure distribution in the tail segment of the aircraft to the empirical data obtained via ground tests in a wind tunnel. We analysed result convergence as a function of the mesh density used. We used methods of computational aerodynamics to investigate the turbulent flow field over the computation region from the leading shock wave to the far wake for various Mach numbers and attack angles

Automatic Design of 3D Conformal Lightweight Structures Based on a Tetrahedral Mesh

Recent developments of additive manufacturing (AM) have extended its application to the direct fabrication of functional parts. Owing to design flexibility and complexity, design for AM (DFAM) has received increasing attention as a new design method that can overcome traditional manufacturing constraints, and has been applied to multi-components integration, multi-material parts, and lightweight structures. In this study, an automatic design methodology for conformal lightweight structures was developed based on a three-dimensional (3D) tetrahedral mesh. A numerical algorithm was developed to generate lightweight cellular structures via the following steps: (i) definition of a target solid; (ii) discretization of the target volume using a tetrahedral mesh; (iii) construction of a number of struts along the edges of every tetrahedral element; (iv) Boolean operation to unify the generated struts; and (v) preparation of output files for 3D printing and finite element analysis (FEA). This algorithm was then applied to generate conformal cellular structures with various shapes. Effects of lattice design parameters on the relevant density change were discussed. The designed cellular structures were then fabricated by AM, and their mechanical properties were evaluated by compression tests. The fabricated lightweight structures showed high specific stiffness and strength, and could support 10000 times heavier load than their own weight.

Edge-based anisotropic mesh adaptation of unstructured meshes with applications to compressible flows

An adaptation procedure using refinement, coarsening and smoothing with anisotropic considerations for unstructured meshes and its coupling with a compressible flow solver is presented in this work. The developed refinement strategy combines a Riemannian centred split of the edges. The coarsening methodology uses an edge collapse procedure where the collapse point is evaluated using anisotropic quality considerations. An accurate and anisotropic vertex smoothing algorithm that operates along the direction of the edges is presented. Edge and face swapping are also used together with the refinement coarsening and smoothing strategies to enhance anisotropic quality. The developed approach makes use of only simple mesh information and is applicable to two and three-dimensional meshes. The performance of the methodology is analysed for two-dimensional triangular meshes and three-dimensional tetrahedral meshes on analytical test cases and to solve steady-state inviscid compressible flows. The transonic flow past an airfoil, the supersonic flow around a sphere and the flow around a complex real-life application are performed.

Higher-order multi-dimensional limiting process for DG and FR/CPR methods on tetrahedral meshes

Abstract The present paper deals with the robust and accurate multi-dimensional limiting process for higher-order discontinuous Galerkin (DG) and flux resconstruction or correction procedure via reconstruction (FR/CPR) methods on tetrahedral meshes. MLP, which has been originally developed in finite volume method (FVM), provides an accurate, robust and efficient oscillation-control mechanism in multiple dimensions for linear approximation. This limiting philosophy can be hierarchically extended into higher-order Pn approximation. The resulting algorithm has been developed for both DG and FR/CPR methods mostly on two-dimensional triangular meshes. This method can be efficiently extended and implemented into three-dimensional tetrahedral meshes. Through extensive numerical computations and comparisons, it is demonstrated that the proposed limiting approach yields the required order-of-accuracy and outstanding performances in resolving three-dimensional compressible inviscid and viscous flow features.

Разрывный метод Галёркина на трёхмерных тетраэдральных сетках. Использование операторного метода программирования

Разрывный метод Галёркина на трёхмерных тетраэдральных сетках. Использование операторного метода программирования

Разрывный метод Галёркина на трёхмерных тетраэдральных сетках. Применение шаблонного метапрограммирования языка C++

Разрывный метод Галёркина на трёхмерных тетраэдральных сетках. Применение шаблонного метапрограммирования языка C++

Voronoi-based Point-placement for Three-dimensional Delaunay-refinement

An extension of the restricted Delaunay-refinement algorithm for three-dimensional tetrahedral mesh generation is described, in which an off-centre type point-placement scheme is utilised. It is shown that the use of generalised Steiner points, positioned along edges in the associated Voronoi complex, typically leads to improvements in the overall size, quality and grading of the resulting tetrahedral meshes. The new algorithm can be viewed as a Frontal-Delaunay approach – a hybridisation of conventional Delaunay-refinement and advancing-front techniques, in which new vertices are positioned to satisfy both element size- and shape- constraints. The new method is shown to inherit many of the best features of classical Delaunay-refinement and advancing-front type algorithms, combining good practical performance with theoretical robustness. Experimental comparisons show that the new method outperforms classical Delaunay-refinement techniques for a number of three-dimensional benchmark problems.

Revisiting Optimal Delaunay Triangulation for 3D Graded Mesh Generation

This paper proposes a new algorithm to generate a graded three-dimensional tetrahedral mesh. It revisits the class of methods based on optimal Delaunay triangulation (ODT) and proposes a proper way of injecting a background density function into the objective function minimized by ODT. This continuous/analytic point of view leads to an objective function that is continuous and Delaunay consistent, in contrast with the discrete/geometrical point of view developed in previous work. To optimize the objective function, this paper proposes a hybrid algorithm that combines a local search (quasi-Newton) with a global optimization (simulated annealing). The benefits of the method are both improved performances and an improved quality of the result in terms of dihedral angles. This results from the combination of two effects. First, the local search has a faster speed of convergence than previous work due to the better behavior of the objective function, and second, the algorithm avoids getting stuck in a poor local minimum. Experimental results are evaluated and compared using standard metrics.

Parallel multigrid solver of radiative transfer equation for photon transport via graphics processing unit

A graphics processing unit-based parallel multigrid solver for a radiative transfer equation with vacuum boundary condition or reflection boundary condition is presented for heterogeneous media with complex geometry based on two-dimensional triangular meshes or three-dimensional tetrahedral meshes. The computational complexity of this parallel solver is linearly proportional to the degrees of freedom in both angular and spatial variables, while the full multigrid method is utilized to minimize the number of iterations. The overall gain of speed is roughly 30 to 300 fold with respect to our prior multigrid solver, which depends on the underlying regime and the parallelization. The numerical validations are presented with the MATLAB codes at https://sites.google.com/site/rtefastsolver/.

Generation of three-dimensional delaunay meshes from weakly structured and inconsistent data

A method is proposed for the generation of three-dimensional tetrahedral meshes from incomplete, weakly structured, and inconsistent data describing a geometric model. The method is based on the construction of a piecewise smooth scalar function defining the body so that its boundary is the zero isosurface of the function. Such implicit description of three-dimensional domains can be defined analytically or can be constructed from a cloud of points, a set of cross sections, or a “soup” of individual vertices, edges, and faces. By applying Boolean operations over domains, simple primitives can be combined with reconstruction results to produce complex geometric models without resorting to specialized software. Sharp edges and conical vertices on the domain boundary are reproduced automatically without using special algorithms. Refs. 42. Figs. 25.

Parallel multi-frontal solver for p adaptive finite element modeling of multi-physics computational problems

The paper presents a parallel direct solver for multi-physics problems. The solver is dedicated for solving problems resulting from adaptive finite element method computations. The concept of finite element is actually replaced by the concept of the node. The computational mesh consists of several nodes, related to element vertices, edges, faces and interiors. The ordering of unknowns in the solver is performed on the level of nodes. The concept of the node can be efficiently utilized in order to recognize unknowns that can be eliminated at a given node of the elimination tree. The solver is tested on the exemplary three-dimensional multi-physics problem involving the computations of the linear acoustics coupled with linear elasticity. The three-dimensional tetrahedral mesh generation and the solver algorithm are modeled by using graph grammar formalism. The execution time and the memory usage of the solver are compared with the MUMPS solver.

An efficient optimization procedure for tetrahedral meshes by chaos search algorithm

A simple and efficient local optimization-based procedure for node repositioning/smoothing of three-dimensional tetrahedral meshes is presented. The initial tetrahedral mesh is optimized with respect to a specified element shape measure by chaos search algorithm, which is very effective for the optimization problems with only a few design variables. Examples show that the presented smoothing procedure can provide favorable conditions for local transformation approach and the quality of mesh can be significantly improved by the combination of these two procedures with respect to a specified element shape measure. Meanwhile, several commonly used shape measures for tetrahedral element, which are considered to be equivalent in some weak sense over a long period of time, are briefly re-examined in this paper. Preliminary study indicates that using different measures to evaluate the change of element shape will probably lead to inconsistent result for both well shaped and poorly shaped elements. The proposed smoothing approach can be utilized as an appropriate and effective tool for evaluating element shape measures and their influence on mesh optimization process and optimal solution.

3D anisotropic mesh refinement in compliance with a general metric specification

A simple and efficient anisotropic refinement procedure for the three-dimensional tetrahedral element mesh based on successive bisection of edges is proposed. Refinement is done by dividing lines of the mesh based on the lengths calculated on three points as specified by the metric tensor along the direction of the line. To obtain the best mesh quality, the subdivision of line segments is performed in the sequence according to the length of the line segments to be divided. Such an order of priority can be determined by a simple sorting process on all the line segments for which refinement is needed. This list of ordered line segments has to be updated from time to time to take into account the new line segments generated during the subdivision process. From the examples studied, the CPU time for mesh refinement seems to bear a linear relationship with the number of elements generated, with a refinement rate of up to 30 000 elements/s on an IBM Power Station 3BT. Shape optimization procedures can be applied to the refined mesh to further improve the quality of the elements. The refinement scheme is useful as part of a general three-dimensional mesh generation package, or as the mesh refinement module in an adaptive finite element analysis.

The design of improved smoothing operators for finite volume flow solvers on unstructured meshes

Spatial operators used in unstructured finite volume flow solvers are analysed for accuracy using Taylor series expansion and Fourier analysis. While approaching second-order accuracy on very regular grids, operators in common use are shown to have errors resulting in accuracy of only first-, zeroth- or even negative-order on three-dimensional tetrahedral meshes. A technique using least-squares optimization is developed to design improved operators on arbitrary meshes. This is applied to the fourth-order edge sum smoothing operator. The improved numerical dissipation leads to a much more accurate prediction of the Strouhal number for two-dimensional flow around a cylinder and a reduction of a factor of three in the loss coefficient for inviscid flow over a three-dimensional hump. Copyright © 2001 John Wiley & Sons, Ltd.

The design of improved smoothing operators for finite volume flow solvers on unstructured meshes

AbstractSpatial operators used in unstructured finite volume flow solvers are analysed for accuracy using Taylor series expansion and Fourier analysis. While approaching second‐order accuracy on very regular grids, operators in common use are shown to have errors resulting in accuracy of only first‐, zeroth‐ or even negative‐order on three‐dimensional tetrahedral meshes. A technique using least‐squares optimization is developed to design improved operators on arbitrary meshes. This is applied to the fourth‐order edge sum smoothing operator. The improved numerical dissipation leads to a much more accurate prediction of the Strouhal number for two‐dimensional flow around a cylinder and a reduction of a factor of three in the loss coefficient for inviscid flow over a three‐dimensional hump. Copyright © 2001 John Wiley & Sons, Ltd.

Hierarchical tree‐based finite element mesh generation

Hierarchical grid generation and its use as a basis for finite element mesh generation are considered in this paper. The hierarchical grids are generated by recursive subdivision using quadtrees in two dimensions and octrees in three dimensions. A numbering system for efficient storage of the quadtree grid information is examined, tree traversal techniques are devised for neighbour finding, and accurate boundary representation is considered. It is found that hierarchical grids are straightforward to generate from sets of seeding points which lie along domain boundaries. Quadtree grids are triangularized to provide finite element meshes in two dimensions. Three-dimensional tetrahedral meshes are generated from octree grids. The meshes can be generated automatically to model complicated geometries with highly irregular boundaries and can be adapted readily at moving boundaries. Examples are given of two- and three-dimensional hierarchical tree-based finite element meshes and their application to modelling free surface waves. Copyright © 1999 John Wiley & Sons, Ltd.