Prismatic Meshes Research Articles

A discontinuous finite element baroclinic marine model on unstructured prismatic meshes

We describe the time discretization of a three-dimensional baroclinic finite element model for the hydrostatic Boussinesq equations based upon a discontinuous Galerkin finite element method. On one hand, the time marching algorithm is based on an efficient mode splitting. To ensure compatibility between the barotropic and baroclinic modes in the splitting algorithm, we introduce Lagrange multipliers in the discrete formulation. On the other hand, the use of implicit–explicit Runge–Kutta methods enables us to treat stiff linear operators implicitly, while the rest of the nonlinear dynamics is treated explicitly. By way of illustration, the time evolution of the flow over a tall isolated seamount on the sphere is simulated. The seamount height is 90% of the mean sea depth. Vortex shedding and Taylor caps are observed. The simulation compares well with results published by other authors.

A discontinuous finite element baroclinic marine model on unstructured prismatic meshes

We describe the space discretization of a three-dimensional baroclinic finite element model, based upon a discontinuous Galerkin method, while the companion paper (Comblen et al. 2010a) describes the discretization in time. We solve the hydrostatic Boussinesq equations governing marine flows on a mesh made up of triangles extruded from the surface toward the seabed to obtain prismatic three-dimensional elements. Diffusion is implemented using the symmetric interior penalty method. The tracer equation is consistent with the continuity equation. A Lax–Friedrichs flux is used to take into account internal wave propagation. By way of illustration, a flow exhibiting internal waves in the lee of an isolated seamount on the sphere is simulated. This enables us to show the advantages of using an unstructured mesh, where the resolution is higher in areas where the flow varies rapidly in space, the mesh being coarser far from the region of interest. The solution exhibits the expected wave structure. Linear and quadratic shape functions are used, and the extension to higher-order discretization is straightforward.

Superconvergence and Extrapolation Analysis of a Nonconforming Mixed Finite Element Approximation for Time-Harmonic Maxwell’s Equations

In this paper, a nonconforming mixed finite element approximating to the three-dimensional time-harmonic Maxwell’s equations is presented. On a uniform rectangular prism mesh, superclose property is achieved for electric field E and magnetic filed H with the boundary condition E×n=0 by means of the asymptotic expansion. Applying postprocessing operators, a superconvergence result is stated for the discretization error of the postprocessed discrete solution to the solution itself. To our best knowledge, this is the first global superconvergence analysis of nonconforming mixed finite elements for the Maxwell’s equations. Furthermore, the approximation accuracy will be improved by extrapolation method.

A modified advancing layers mesh generation for thin three-dimensional objects with variable thickness

A method of generating modified advancing layers mesh is proposed. In this paper the mesh generation process of semi-unstructured prismatic/tetrahedral mesh is presented for relatively thin three-dimensional geometries with variable thickness, as in the case of injection molding analysis. Prismatic meshes are generated by offsetting initial surface triangular meshes. During the mesh generation process, mesh quality is efficiently improved with the use of a new node relocation method. Finally, tetrahedral meshes are automatically generated in the rest of the domain. The mesh generating capability of the proposed algorithm is demonstrated with the several practical test cases.

振動する翼周りの流れに対するハイブリッド型非構造移動格子有限体積法(流体工学,流体機械)

Computing a flow around a pitching airfoil, the unstructured moving-grid finite-volume method is developed in this paper. The method has been proposed for various moving boundary problems. However, it is limited for compressible inviscid flows. In the case of calculating a viscous flow at high Reynolds number as a flow around a pitching airfoil, it is necessary to solve a boundary layer. So, a hybrid gird system of a prismatic mesh for boundary layer region and tetrahedral mesh for other regions is suitable. In this paper, the unstructured hybrid moving-grid finite-volume method is formulated in four dimensional space-time unified domain to satisfy a geometric conservation law on such a moving grid in addition to satisfy a physical conservation law. The method is apply to a flow around a pitching NACA0012 airfoil, then the results show that the method is effective for a high Reynolds number viscous flow around a moving body.

Some issues and insights for gravity and magnetic modelling at the region to continent scale

The form of the Earth is much closer to a sphere than a flat slab, so why is gravity and magnetic modelling almost exclusively carried out within a flat Earth or Cartesian framework? The use of a Cartesian coordinate system can be attributed in part to the absence of alternatives in the commercial modelling and visualisation software packages and the need for additional computational resources if modelling is carried out in a spherical or ellipsoidal reference system. These are not sufficiently good reasons, however, to blindly continue with this practice. Gravity modelling is a particularly widespread activity, being used for basic gravity data processing (e.g., Bouguer and terrain corrections - e.g., LaFehr, 1991a, 1991b, 1998; Talwani, 1998; Hinze et al, 2005), geodesy applications (e.g., geoid calculations), and geological interpretation. To what degree are the outcomes of these activities affected by the choice of Earth representation? In the exploration geophysics community, there is a standard answer to the question ?when do curvature effects become significant? which is ?when the models are larger than 166.7 km in horizontal extent?. This criterion can be traced back to Hayford and Bowie (1912). What was the basis for choosing this distance? Is this a criterion that can be applied to all modelling applications or just the original application of Hayford and Bowie (1912)? Although there is general acknowledgment that the curvature of the Earth is important when performing gravity or magnetic modelling of long traverses or large regions, there are few studies of the errors involved if a Cartesian (or rectangular) coordinate reference system is used. Examples that demonstrate the magnitude of curvature effects on gravity response can be found in Hayford and Bowie (1912), Takin and Talwani (1966), Johnson and Litehiser (1972), Qureshi (1976), Miku?ka et al. (2006), and Cavsak (2008). In the following, results of simple calculations for a regular mesh of prismatic elements are presented to quantify the differences in vertical gravity and vertical gravity gradient response for equivalent representations of source elements in Cartesian or spherical coordinate reference systems. It is noted that outcomes for vertical gravity gradient response are the same as those that would be obtained for total magnetic intensity response (TMI). Although this work is based on models comprised of a regular array of prismatic mesh elements, the outcomes can be related to those that would be obtained for models based on arbitrary 3D polyhedra. There are clear extensions of methodology that would allow modellers to estimate the significance of curvature effects for any specific application. This will help to answer the question that is prompted by Figure 1, i.e., ?How significant are the differences if I carry out gravity or magnetic modelling for a portion of the Earth using a Cartesian framework rather than a spherical framework??

Error estimation of a quadratic finite volume method on right quadrangular prism grids

In this paper, we develop a finite volume element method with affine quadratic bases on right quadrangular prism meshes for three-dimensional elliptic boundary value problems. The optimal H 1 -norm error estimate of second order accuracy is proved under certain assumptions about the meshes. Numerical results are presented to illustrate the theoretical analysis.

Hybrid three-dimensional mesh generation from quad-dominant surface meshes

A hybrid mesh generation algorithm using a modified plastering method for three-dimensional objects with variable thickness is presented. The method starts with a quad-dominant surface mesh and generates layered elements with variable thickness by using adaptive offsetting, resulting in hex-tet dominant mesh generation. Hexahedral and prismatic meshes are generated by the inward offsetting of the initial boundary mesh. In order to generate a conforming mesh, pyramid elements are constructed on top of hexahedral elements, and tetrahedral elements are generated for the remaining domain by using an advancing front method. This method is applied to several different geometries, and the effectiveness of the proposed algorithm is demonstrated.

Discrete maximum principle for FE solutions of the diffusion-reaction problem on prismatic meshes

In this paper we analyse the discrete maximum principle (DMP) for a stationary diffusion-reaction problem solved by means of prismatic finite elements. We derive geometric conditions on the shape parameters of the prismatic partitions which guarantee validity of the DMP. The presented numerical tests show the sharpness of the obtained conditions.

Tracer Conservation for Three-Dimensional, Finite-Element, Free-Surface, Ocean Modeling on Moving Prismatic Meshes

Abstract Large-scale free-surface ocean models designed to run over climatic time scales are required to globally conserve the volume and any tracer up to machine precision. In addition, local consistency is critical and requires that the discrete tracer equation preserve constants in a closed domain and if there is no tracer source or sink. Local consistency, together with monotonicity, will ensure that no spurious tracer extrema occur. A three-dimensional, finite-element, shallow-water model is presented. The mesh is unstructured in the horizontal, extruded in the third dimension, and made up of multiple layers of prisms. In addition, the mesh is allowed to move in the vertical and adapts itself to the free-surface motions. It is shown that achieving consistency requires a discrete compatibility between the tracer and continuity equations. In addition, to ensure global tracer conservation in a consistent way, a discrete compatibility between the tracer, continuity, and free-surface elevation equations must be fulfilled. It is suggested that this compatibility constraint, together with the use of a numerically stable scheme, severely restricts the choice of usable finite-element spatial discretizations. A consistent and conservative time-stepping algorithm is described for which a unique time step is used. It is suggested that future research is needed in order to design a consistent and conservative split-explicit algorithm. Some illustrative test cases are presented in which the method is shown to satisfy all conservation properties. A few experiments in which consistency breaks down are carried out, and the consequences of this breakdown process are investigated.

Homogenized mixed finite element method for diffusion equations on prismatic meshes

In this paper, we propose a new mixed finite element (FE) method for diffusion equations on a sequence of two hierarchical polyhedral meshes: a coarse and a fine one. The method is motivated by applications of diffusion equations in basin modelling when the computational domain is a union of ‘horizontal’ layered subdomains and the layers may degenerate. We consider in detail only the case of meshes with distorted and degenerated ‘vertical’ prismatic mesh cells. The discretization on the coarse mesh is derived from the macro-hybrid mixed FE method on the fine mesh with a new definition of the underlying FE spaces. The new mixed FE spaces are subspaces of classical spaces and they are designed by using special homogenization/coarsening algorithms. The accuracy and efficiency of the method is illustrated by numerical results for 2D and 3D diffusion equations.

Evaluation of hexahedral, prismatic and hybrid mesh styles for simulating respiratory aerosol dynamics

In simulating biofluid flow domains, structured hexahedral meshes are often associated with high quality solutions. However, extensive time and effort are required to generate these meshes for complex branching geometries. This study evaluates potential mesh configurations that may maintain the advantages of the structured hexahedral style while providing significant savings in grid construction time and complexity. Specifically, the objective of this study is to evaluate the performance of unstructured hexahedral, prismatic and hybrid meshes based on grid convergence and local particle deposition fractions in a bifurcating model of the respiratory tract. A grid convergence index (GCI) has been implemented to assess the mesh-independence of solutions in cases where true grid halving is not feasible. Localized and total deposition values have been evaluated for particles ranging from 1 through 10 μm in planar and out-of-plane geometries. Structured hexahedral, unstructured hexahedral and prismatic meshes were found to provide GCI values of approximately 5% and nearly identical velocity fields. In contrast, the hexahedral–tetrahedral hybrid model resulted in GCI values that were significantly higher in comparison to the other meshes. The resulting velocity field for the hybrid configuration differed from the hexahedral and prismatic solutions by up to an order of magnitude at some locations. Considering the deposition of 10 μm particles in the planar configuration, all meshes considered provided relatively close agreement (2–20% difference) with an available experimental study. For all particle sizes considered, local and total deposition results for the structured and unstructured hexahedral meshes were similar. In contrast, the prismatic and hybrid geometries resulted in significantly higher deposition rates when compared to the hexahedral meshes for particles less than 10 μm. As a result, only the unstructured hexahedral mesh was found to provide overall performance similar to the structured hexahedral configuration with the advantage of a significant savings in construction time. These results emphasize the importance of aligning control volume gridlines with the predominant flow direction in biofluid applications that involve long and thin internal flow domains. Future studies are needed to assess other forms of the hybrid configuration and the effects of other element styles.

Separation of Horizontal and Vertical Dependencies in a Surface/Volume Integral Equation Approach to Model Quasi 3-D Structures in Multilayered Media

A new and efficient integral equation approach is presented to model heterogeneous dielectric volumes in a multilayered environment. The dielectric volumes can be vertically large, they can cross dielectric interfaces and they should be "quasi 3-D," which means that the contour faces should be placed either horizontally or vertically in the multilayered environment. In this way, a perpendicular prismatic mesh can be fitted on the volumes and the 3-D displacement currents can be expanded in generalized rooftop functions with separated horizontal x, y- and vertical z-dependencies. This makes it possible to evaluate all z, z' reaction integrals fully analytically in the spectral domain and ensures an efficient implementation. The formulation of the source-field relations is adapted to the quasi 3-D geometries as a hybrid dyadic-mixed potential form. Additionally, the electromagnetic coupling between dielectric volumes and metal sheets is included using a coupled volume/surface formulation. This turns the implementation into a complete full wave solver for planar antennas containing both finite and infinite dielectric regions. In order to illustrate and validate the presented approach, two patch antennas with a local volumetric inhomogeneity under the patch are numerically analyzed in a multilayered environment. Eventually, a matrix fill time comparison is used to demonstrate the improvement in computation efficiency

Numerical analysis of a two-level preconditioner for the diffusion equation with an anisotropic diffusion tensor

We develop and analyze a new two-level preconditioner for algebraic systems arising from finite volume discretization of 3D anisotropic diffusion problems on prismatic meshes. The preconditioner is based on partitioning of the mesh in the (x,y)-plane into non-overlapping subdomains and on a special coarsening algorithm. The condition number of the preconditioned matrix does not depend on the coefficients in the diffusion operator. Numerical experiments confirm the theoretical results and reveal the competitiveness of the new preconditioner with respect to the algebraic multigrid preconditioner.

Performance of a pressure-based unstructured code and a multi-block structured code for heat transfer and fluid flow

This paper reports a multi-block unstructured method for incompressible flows and the performance comparison with a multi-block structured method. The unstructured collocated grid is composed of arbitrary cell shapes. Pressure and velocity are stored at cell centres; Rhie and Chow interpolation is used to prevent checker boarding. The SIMPLE algorithm is used for pressure-velocity coupling. The code is calibrated with benchmark cases using hexahedral and prism meshes, and is shown to perform very well for all cases considered. In addition, the performances of the structured and unstructured codes are compared.

242 カルマン渦放出周波数の格子依存性(S41-1 移動境界・連成(1),S41 移動境界・連成・振動と騒音問題)

Karman vortex sheets from a two dimensional circular cylinder at low Reynolds number are calculated using node-centered Finite Volume Method and unstructured mesh. In order to prepare for a coupling analysis between flow and structure oscillation, dependence of mesh density for frequency of vortex shedding are investigated in this paper. The results show that it is possible to capture a periodic vortex shedding using a sufficient quantity of unstructured hybrid mesh (mixture of prism and hexagon mesh).

Three-Dimensional Modeling of Mold Filling in Microelectronics Encapsulation Process

In this paper, a fully three-dimensional (3-D) numerical model is developed to simulate the mold-filling behavior in the plastic encapsulation of microelectronics. The conventional Hele-Shaw approximation is inadequate to analyze such a complex process owing to the 3-D nature inherent in the molding compound flow between leadframe and mold cavity. The developed methodology combines the efficiency of SIMPLE-based finite volume method (FVM) and the robustness of VOF volume-tracking method to solve the two-phase flow field in complex mold geometry. An efficient method for automatic generation of prismatic mesh for plastic packages is also presented. The molding process of a TSOP II 54L LOC package is studied. Short-shot experiments are conducted to investigate the filling patterns at several different flow times. The close agreements between experimental data and simulated results demonstrate the applicability of the present computational model for practical plastic encapsulation simulations.

Streamlining Fluid Dynamics

More-intuitive pre-processors and advanced solvers are making computational fluid dynamics (CFD) software easier to use, more accurate, and faster. CFD techniques involve the solution of the Navier-Stokes equations that describe fluid-flow processes. Using MSC/ PATRAN as a starting point, AEA Technology plc, Harwell, Oxfordshire, England, has developed a pre-processor for its software that is fully computer-aided design (CAD)-compatible and works with native CAD databases such as CADDS 5, CATIA, Euclid3, Pro /ENG INEER, and Unigraphics. The simplicity of modeling complex geometries in CFX allows more details to be included in models, such as gangways between coaches, bogies, and even some parts of the pantograph. CFX 5's coupled solver offers a radically different approach that solves all the hydrodynamic equations as a single system. CFX 5 has demonstrated its ability to deliver much faster pre-processing and shorter run times, thus increasing productivity for its users. CFX 5.2 should be a further step forward in commercial CFD, with its mixed element types combining the accuracy of prismatic meshes adjacent to surfaces with the speed and geometric flexibility of tetrahedral elements in the remainder of the grid.

Hybrid prismatic/tetrahedral grid generation for viscous flows around complex geometries

A method for the generation of hybrid prismatic/tetrahedral grids is described for complex three-dimensional geometries, including multibody domains, and employed for viscous flows around an aircraft. The prisms cover the region close to each body's surface, while tetrahedra are created elsewhere. A special method is presented that allows the generation of single-block, nonoverlapping prismatic meshes even in complex geometries that contain narrow gaps and cavities. Examples of cases treated by this method include wing-engine configurations as well as multi-element wings. The second development is a combined octree/advancing front method for the generation of the tetrahedra of the hybrid mesh. The main feature of this octree-based tetrahedra generator is that it does not require the creation of a background mesh by the user for the determination of the grid-spacing and stretching parameters. These are determined via an automatically generated octree. The hybrid grid generator is robust, geometry independent, and requires no user intervention. The suitability of hybrid meshes for capturing viscous flow phenomena is demonstrated by simulation of viscous flows around a high-speed civil transport type of aircraft configuration. Reduction in computer resources has been substantial, allowing flow simulations to be performed on workstations rather than supercomputers.