A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids

Recently a new high-order formulation for 1D conservation laws was developed by Huynh using the idea of “flux reconstruction”. The formulation was capable of unifying several popular methods including the discontinuous Galerkin, staggered-grid multi-domain method, or the spectral difference/spectral volume methods into a single family. The extension of the method to quadrilateral and hexahedral elements is straightforward. In an attempt to extend the method to other element types such as triangular, tetrahedral or prismatic elements, the idea of “flux reconstruction” is generalized into a “lifting collocation penalty” approach. With a judicious selection of solution points and flux points, the approach can be made simple and efficient to implement for mixed grids. In addition, the formulation includes the discontinuous Galerkin, spectral volume and spectral difference methods as special cases. Several test problems are presented to demonstrate the capability of the method.     

Charge-conserving FEM–PIC schemes on general grids

In this article, we aim at proposing a general mathematical formulation for charge-conserving finite-element Maxwell solvers coupled with particle schemes. In particular, we identify the finite-element continuity equations that must be satisfied by the discrete current sources for several classes of time-domain Vlasov–Maxwell simulations to preserve the Gauss law at each time step, and propose a generic algorithm for computing such consistent sources. Since our results cover a wide range of schemes (namely curl-conforming finite element methods of arbitrary degree, general meshes in two or three dimensions, several classes of time discretization schemes, particles with arbitrary shape factors and piecewise polynomial trajectories of arbitrary degree), we believe that they provide a useful roadmap in the design of high-order charge-conserving FEM–PIC numerical schemes.     

Numerical analyses of radiative heat transfer in any arbitrarily-shaped axisymmetric enclosures

A numerical approach for the treatment of radiative heat transfer in any irregularly-shaped axisymmetric enclosure filled with absorbing, emitting and scattering gray media is developed. Radiative transfer equation (RTE) is formulated for a general axisymmetric geometrical configurations, and the discretized equation is conducted using an unstructured meshes, generated by an appropriate computer algorithm, and the control volume finite element method which frequently adopted in CFD problems. A computer procedure has been done to solve the discretized RTE and to examine the accuracy and the computational efficiency of the proposed numerical approach. By using this computer algorithm, five test cases, a cylindrical enclosure with absorbing and emitting medium, a diffuser shaped axisymmetric enclosure, a finite axisymmetric cylindrical enclosure with a curved wall, a furnace with axially varying medium temperature and a rocket nozzle, are treated and the obtained results agree very well with other published works. Furthermore, the developed computer procedure has an accurate CPU time and it can be coupled easily with CFD codes.     

Convergence of two-dimensional staggered central schemes on unstructured triangular grids

In this paper, we present a convergence analysis of a two-dimensional central finite volume scheme on unstructured triangular grids for hyperbolic systems of conservation laws. More precisely, we show that the solution obtained by the numerical base scheme presents, under an appropriate CFL condition, an optimal convergence to the unique entropy solution of the Cauchy problem.     

Numerical investigation of cavitating flows for marine propulsors using an unstructured mesh technique

In the present study, the cavitating flows around marine propulsors have been numerically investigated by using a multi-phase RANS flow solver based on pseudo-compressibility and a homogeneous mixture model using unstructured meshes. To handle the relative motion between the rotating rotor and the stator, an overset mesh technique was adopted. The mass transfer rate between the liquid and vapor phases was determined by Merkle’s cavitation model based on the difference between the local and vapor pressure. The calculations were made for the P4381 marine propeller with different cavitation numbers at several advancing ratios. It was shown that the vapor structure, such as cavitation size and shape, was well captured at cavitating flow conditions. It was observed that the cavitation breakdown behavior was also well captured by the present method. Good agreement was obtained between the present results and the experiment for the integrated blade loadings, such as thrust and torque. The calculations were also made for a water-jet pump configuration at several flow conditions, and the cavitation breakdown behaviors for total headrise, power and thrust were validated by comparing the results with the experiment. The blade area covered by the cavitation and the shape of tip leakage cavitation were also compared with the experiment. Reasonable agreement between the predicted results and the experiment was obtained.     

Finite Element Modeling of Liquid Deuterium Flow and Heat Transfer in a Cold-neutron Source

A finite element simulation of flow and heat transfer in the moderator cell of a cold-neutron source (CNS), in which liquid deuterium subject to internal heat generation is flowing, is reported. The numerical scheme consists of a stabilized equal-order method. A time-accurate approach is adopted to resolve the large-scale eddies of the flow, with a Smagorinsky's model for the subgrid-scale effects. The thermal coupling follows a staggered strategy, with SUPG-type upwinding. A specific wall-law is developed that accounts for the correct partition of the heat deposited at the wall by radiation between the liquid deuterium and the helium gas flowing at the outer side of the wall. The average flow and thermal structure are presented. The turbulent fluctuations are both illustrated in physical space and decomposed into spectral components. The wavenumber spectrum suggests that adequate resolution of the large-scale eddies has been attained with just 200,000 nodes, while a DNS analysis would have required at least 1010 nodes. Usefulness of the approach in the design process of the CNS is highlighted.     

Development and comparison of different spatial numerical schemes for the radiative transfer equation resolution using three-dimensional unstructured meshes

In the present work four different spatial numerical schemes have been developed with the aim of reducing the false-scattering of the numerical solutions obtained with the discrete ordinates (DOM) and the finite volume (FVM) methods. These schemes have been designed specifically for unstructured meshes by means of the extrapolation of nodal values of intensity on the studied radiative direction. The schemes have been tested and compared in several 3D benchmark test cases using both structured orthogonal and unstructured grids.     

A piecewise bearing capacity method of unstructured terrain considering characteristics of soil mechanic and wheel geometry

Bearing capacity of the unstructured terrain considering the effects of the wheel geometry and soil mechanic properties is analyzed in this paper. Two-dimensional pressure-sinkage simulations are conducted to evaluate the degrees of similarity between the flat plate and wheel in terms of their ultimate bearing characteristics of Terzaghi theory. The results show that these degrees of similarity are mainly reflected in the soil in-depth direction and the corresponding failure behaviors. Based on the approximation of the ultimate bearing capacity between the wheel and flat plate, a piecewise bearing capacity evaluation method with the effects of the soil mechanic properties and three-dimensional wheel geometry is proposed. The pressure-sinkage values of the proposed model show a satisfactory agreement with the experimental ones. The proposed model performs better than the semi-empirical models, as it considers more soil bearing features and needs less fitting parameters to assess the unstructured terrain.     

A three-dimensional electrostatic particle-in-cell methodology on unstructured Delaunay–Voronoi grids

The mathematical formulation and computational implementation of a three-dimensional particle-in-cell methodology on unstructured Delaunay–Voronoi tetrahedral grids is presented. The method allows simulation of plasmas in complex domains and incorporates the duality of the Delaunay–Voronoi in all aspects of the particle-in-cell cycle. Charge assignment and field interpolation weighting schemes of zero- and first-order are formulated based on the theory of long-range constraints. Electric potential and fields are derived from a finite-volume formulation of Gauss’ law using the Voronoi–Delaunay dual. Boundary conditions and the algorithms for injection, particle loading, particle motion, and particle tracking are implemented for unstructured Delaunay grids. Error and sensitivity analysis examines the effects of particles/cell, grid scaling, and timestep on the numerical heating, the slowing-down time, and the deflection times. The problem of current collection by cylindrical Langmuir probes in collisionless plasmas is used for validation. Numerical results compare favorably with previous numerical and analytical solutions for a wide range of probe radius to Debye length ratios, probe potentials, and electron to ion temperature ratios. The versatility of the methodology is demonstrated with the simulation of a complex plasma microsensor, a directional micro-retarding potential analyzer that includes a low transparency micro-grid.     

Adaptive variational curve smoothing based on level set method

This paper presents an adaptive method for variational curve smoothing based on level set implementation. A suitable cost functional is minimized via solving the derived Euler–Lagrangian equation, of which the discretization is conducted on unstructured triangular meshes by employing a simple and effective finite volume scheme. Through adaptive refinement of the mesh, the geometry features of the given curve can be well resolved in a cost-effective way. Various numerical experiments demonstrate the effectiveness and efficiency of the proposed approach.