Quadrilateral Grids Research Articles

Multiscale Finite-Volume CVD-MPFA Formulations on Structured and Unstructured Grids

This paper presents the development of finite-volume multiscale methods for quadrilateral and triangular unstructured grids. Families of Darcy-flux approximations have been developed for consistent approximation of the general tensor pressure equation arising from Darcy's law together with mass conservation. The schemes are control-volume distributed (CVD) with flow variables and rock properties sharing the same control-volume location and are comprised of a multipoint flux family formulation (CVD-MPFA). The schemes are used to develop a CVD-MPFA based multiscale finite-volume (MSFV) formulation applicable to both structured and unstructured grids in two dimensions. The basis functions are a key component of the MSFV method, and are a set of local solutions, usually defined subject to Dirichlet boundary conditions. A generalization of the Cartesian grid Dirichlet basis functions described in [P. Jenny, S. H. Lee, and H. A. Tchelepi, J. Comput. Phys., 187 (2003), pp. 47--67] is presented here for unstructured grids. Whilst the transition from a Cartesian grid to an unstructured grid is largely successful, use of Dirichlet basis functions can still lead to pressure fields that exhibit spurious oscillations in areas of strong heterogeneity. New basis functions are proposed in an attempt to improve the pressure field solutions where Neumann boundary conditions are imposed almost everywhere, except corners which remain specified by Dirichlet values.

Syntheses, structures, and properties of transition metal coordination polymers based on a long semirigid tetracarboxylic acid and multidentate N-donor ligands

Six transition metal coordination polymers based on a semirigid tetracarboxylic acid and the multidentate N-donor ligands have been synthesized by the hydrothermal method, namely, {[Co(H2obda) (μ2-H2O) (H2O)2]·2H2O}n (1), {[Co(obda)0.5(bpe) (H2O)2]·3H2O}n (2), {[Zn(H2obda) (H2O)4]·H4obda·6H2O}n (3), {[Zn(bpy) (H2O)4]·H2obda}n (4), {[Ni(bpy) (H2O)4]·H2obda}n (5) and {[Cu(H2obda) (bpy)2]}n (6) (H4obda = 1,4-bis(4-oxy-1,2-benzene dicarboxylic acid)benzene, bpe = 1,2-Bis(4-pyridyl)ethylene), bpy = 4,4′-bipyridine). Compounds 1–6 were structurally characterized by the elemental analyses, infrared spectra, and single crystal X-ray diffractions. Compounds 1–2 exhibit the 2D quadrilateral and polygonal layered grid structures, respectively; a 3D supramolecular structure of 2 has been build via π···π and hydrogen bonds interactions. Compounds 3–6 reveal the 1D zigzag and linear chains structures, respectively; furthermore, 3–5 display the diverse 3D supramolecular structures via hydrogen bonds, respectively. The 1-D infinite water chain in 3 has been found between the lattice water molecules. In addition, the thermogravimetric analyses of 1–6, magnetic property of 1, and photoluminescence of 3–4 have been investigated, respectively.

A cell-centered multipoint flux approximation method with a diamond stencil coupled with a higher order finite volume method for the simulation of oil–water displacements in heterogeneous and anisotropic petroleum reservoirs

The simulation of fluid flow and transport in heterogeneous and anisotropic oil and gas reservoirs poses a great challenge from the numerical point of view, due to the modeling of complex depositional environments, including inclined laminated layers, channels, fractures and faults and the modeling of deviated wells, making it difficult to build and handle the Reservoir Characterization Process (RCM), particularly by using structured meshes (cartesian or corner point), which is the current pattern in petroleum reservoir simulators. Under certain hypotheses, the mathematical model that describes the fluid flow in petroleum reservoirs includes an elliptic equation with heterogeneous, possibly discontinuous, coefficients for the pressure field and a non-linear hyperbolic equation for the saturation field. In the present paper, these equations are solved via an Implicit Pressure Explicit Saturation (IMPES) procedure. To solve these equations, we use a full cell-centered finite volume formulation. The pressure equation is discretized by a non-orthodox Multipoint Flux Approximation Method with a Diamond type stencil (MPFA-D) and used for the first time for the solution of two-phase flow problems in heterogeneous porous media. It is very robust and capable of reproducing piecewise linear solutions exactly by means of a linear preserving interpolation with explicit weights that avoids the solution of locally defined systems of equations. For the solution of the saturation equation, we use a Monotone Upstream Centered Scheme for Conservation Laws (MUSCL) method based on a gradient reconstruction obtained by a least square technique in which monotonicity is reinforced by an appropriate slope limiter. The method can be used with general polygonal meshes, even though we restrict ourselves to conforming triangular and quadrilateral grids. In order to validate and show the robustness of our formulation, we solve some problems including heterogeneous and anisotropic reservoirs and displacements with high mobility ratios. Our results compare well with others found in literature.

A well-balanced positivity-preserving central-upwind scheme for shallow water equations on unstructured quadrilateral grids

We introduce a new second-order central-upwind scheme for shallow water equations on the unstructured quadrilateral grids. We propose a new technique for bottom topography approximation over quadrilateral cells as well as an efficient water surface correction procedure which guarantee the positivity of the computed fluid depth. We also design a new quadrature for the discretization of the source term, using which the new scheme exactly preserves “lake at rest” steady states. We demonstrate these features of the new scheme as well as its high resolution and robustness and its potential advantages over the triangular central-upwind scheme in a number of numerical examples.

Numerical study of underwater explosion near a free surface and a structural object on unstructured grid

Underwater explosions are very complicated problem possessing many challenges for numerical modeling. In the present paper, a two-dimensional underwater explosion near a free surface and a structural object is numerically investigated using a 6-equation diffuseinterface model that is solved by a HLLC-type scheme with MUSCL-Hancock method based on the least-squares reconstruction and Barth and Jespersen’s limiter on quadrilateral unstructured grid. A long-time computation is performed to capture the important physics of the underwater explosion: initial underwater shock propagation and its interaction with material interfaces, expansion and contraction of the explosive gas bubble, and water jetting near a structure on the collapse of the gas bubble. These phenomena are clearly observed by the appropriate flow visualization of the computed results.

Stable finite element pair for Stokes problem and discrete Stokes complex on quadrilateral grids

In this paper, we first construct a nonconforming finite element pair for the incompressible Stokes problem on quadrilateral grids, and then construct a discrete Stokes complex associated with that finite element pair. The finite element spaces involved consist of piecewise polynomials only, and the divergence-free condition is imposed in a primal formulation. Combined with some existing results, these constructions can be generated onto grids that consist of both triangular and quadrilateral cells.

A mixed nonoverlapping covolume method on quadrilateral grids for elliptic problems

A covolume method is proposed for the mixed formulation of second-order elliptic problems. The solution domain is divided by a quadrilateral grid, corresponding to which a nonoverlapping dual grid is constructed. The velocity and pressure are approximated by the lowest-order Raviart–Thomas space on quadrilaterals. We prove its first order optimal rate of convergence for the approximate velocities in the H(div)-norm as well as for the approximate pressures in the L2-norm. A second order error estimate between a suitable projection of the exact velocity (or pressure) and the approximate velocity (or approximate pressure) is also presented. Numerical experiments are provided to illustrate the error behavior of the scheme.

Assessment of Cell-centered and Cell-vertex Finite VolumeApproaches for Computation of 2D Structural Dynamicson Arbitrary Quadrilateral Grids

Assessment of Cell-centered and Cell-vertex Finite VolumeApproaches for Computation of 2D Structural Dynamicson Arbitrary Quadrilateral Grids

A simple, efficient, and high-order accurate curved sliding-mesh interface approach to spectral difference method on coupled rotating and stationary domains

This paper presents a simple, efficient, and high-order accurate sliding-mesh interface approach to the spectral difference (SD) method. We demonstrate the approach by solving the two-dimensional compressible Navier–Stokes equations on quadrilateral grids. This approach is an extension of the straight mortar method originally designed for stationary domains [7,8]. Our sliding method creates curved dynamic mortars on sliding-mesh interfaces to couple rotating and stationary domains. On the nonconforming sliding-mesh interfaces, the related variables are first projected from cell faces to mortars to compute common fluxes, and then the common fluxes are projected back from the mortars to the cell faces to ensure conservation. To verify the spatial order of accuracy of the sliding-mesh spectral difference (SSD) method, both inviscid and viscous flow cases are tested. It is shown that the SSD method preserves the high-order accuracy of the SD method. Meanwhile, the SSD method is found to be very efficient in terms of computational cost. This novel sliding-mesh interface method is very suitable for parallel processing with domain decomposition. It can be applied to a wide range of problems, such as the hydrodynamics of marine propellers, the aerodynamics of rotorcraft, wind turbines, and oscillating wing power generators, etc.

基于壳单元的包装箱强度有限元分析

包装箱承受产品重力、气压和过载等载荷。使用hypermesh软件对箱体进行了四边形壳单元建模，并对箱体结构进行了强度计算分析，得出了箱体的应力和变形云图，对设计中的薄弱环节进行了改进，并对改进后的包装箱强度进行了计算和验证。 The packing box bears product gravity, air pressure and overload. The box is divided by quadrila-teral sheel element grid with hypermesh software, and intensity finite-element-analysis of the box is carried on. Stress and deformation cloud of the box is calculated. Then the weakness of the box is analyzed and improved. And the intensity of the improved box is also calculated and validated.

Numerical Study on Homogeneous Condensation in a Vortex

The homogeneous water condensation in a vortex is numerically studied in the present work by using an extended method of moments (EMOM). The EMOM is based on the liquid phase so that the gas phase and the liquid phase can be treated separately, having their own governing equations respectively. The EMOM is integrated into a well validated numerical method-VAS2D (two-dimensional & axisymmetric vectorized adaptive solver), which is a MUSCL-type finite-volume method developed for the unstructured quadrilateral grids. The interactions between the two phases are realized through the source terms including the exchange of momentum and heat. The new numerical method is then applied to a vortex flow with a mixture of nitrogen and water vapor. Several cases with different initial situations are investigated. The evolution of the flow field is discussed. The case with the undersaturated gas apart from the nucleation region has a smooth diffusion of the droplets. But for the case with the supersaturated gas, the droplets continuously grow when they move to the new supersaturated region under the effect of centrifugal force. It leads to a more complex interaction between the vortex and the condensation.

The spectral element method (SEM) on variable-resolution grids: evaluating grid sensitivity and resolution-aware numerical viscosity

Abstract. We evaluate the performance of the Community Atmosphere Model's (CAM) spectral element method on variable-resolution grids using the shallow-water equations in spherical geometry. We configure the method as it is used in CAM, with dissipation of grid scale variance, implemented using hyperviscosity. Hyperviscosity is highly scale selective and grid independent, but does require a resolution-dependent coefficient. For the spectral element method with variable-resolution grids and highly distorted elements, we obtain the best results if we introduce a tensor-based hyperviscosity with tensor coefficients tied to the eigenvalues of the local element metric tensor. The tensor hyperviscosity is constructed so that, for regions of uniform resolution, it matches the traditional constant-coefficient hyperviscosity. With the tensor hyperviscosity, the large-scale solution is almost completely unaffected by the presence of grid refinement. This later point is important for climate applications in which long term climatological averages can be imprinted by stationary inhomogeneities in the truncation error. We also evaluate the robustness of the approach with respect to grid quality by considering unstructured conforming quadrilateral grids generated with a well-known grid-generating toolkit and grids generated by SQuadGen, a new open source alternative which produces lower valence nodes.

Accuracy-preserving boundary flux quadrature for finite-volume discretization on unstructured grids

In this paper, we derive a general accuracy-preserving boundary flux quadrature formula for the second- and third-order node-centered edge-based finite-volume discretizations on triangular and tetrahedral grids. It is demonstrated that the general formula reduces to some well-known formulas for the second-order scheme in the case of linear fluxes. Some special boundary grids are also examined. In particular, a simple one-point quadrature formula, which is typically used for quadrilateral grids, is shown to be exact for quadratic fluxes on triangular grids with uniformly spaced straight boundaries. Numerical results are presented to demonstrate the accuracy of the general formula, and accuracy deterioration caused by incompatible boundary flux quadrature formulas. In general, the third-order accuracy is lost everywhere in the domain unless the third-order accuracy is maintained at boundary nodes. It is also demonstrated that the third-order scheme does not require high-order curved elements for curved boundaries but requires accurate surface normal vectors defined, which can be estimated by a quadratic interpolation from a given grid, to deliver the designed third-order accuracy.

Discontinuous finite element solution of the radiation diffusion equation on arbitrary polygonal meshes and locally adapted quadrilateral grids

In this paper, we propose a piece-wise linear discontinuous (PWLD) finite element discretization of the diffusion equation for arbitrary polygonal meshes. It is based on the standard diffusion form and uses the symmetric interior penalty technique, which yields a symmetric positive definite linear system matrix. A preconditioned conjugate gradient algorithm is employed to solve the linear system. Piece-wise linear approximations also allow a straightforward implementation of local mesh adaptation by allowing unrefined cells to be interpreted as polygons with an increased number of vertices. Several test cases, taken from the literature on the discretization of the radiation diffusion equation, are presented: random, sinusoidal, Shestakov, and Z meshes are used. The last numerical example demonstrates the application of the PWLD discretization to adaptive mesh refinement.

Buckling of spherical cap gridshells: A numerical and analytical study revisiting the concept of the equivalent continuum

The use of gridshell structures in architecture and structural engineering has risen in the past decade, yet fundamental research on the mechanics of such structures is lacking. The majority of gridshells built use either quadrilateral or triangular grids. In this paper, we assess the effect of the grid shape (triangular or quadrilateral), the grid spacing, and the span-to-height ratio on the failure load of spherical cap gridshells. The failure load is calculated using finite element methods and analytical methods derived from the equivalent continuum. Both a linear eigenvalue and a nonlinear analysis using geometric imperfections are performed and compared. While a triangular grid is expected to be greater in structural capacity, it is found that for shallower shells there is less of a structural advantage to using a triangular grid. Further, the triangular grid is highly sensitive to imperfections whereas a quadrilateral grid is not. The equivalent continuum defined by area equivalence provides a conservative estimate for the buckling load and multiple definitions of the equivalent continuum can be used to establish a practical design tool in the early phases of design. By understanding the fundamental mechanical behavior of gridshells, design guidelines aimed to maximize their capacity and efficiency and intended to facilitate the discussion between architect and engineer are proposed.

Analysis of adaptive mesh refinement for IMEX discontinuous Galerkin solutions of the compressible Euler equations with application to atmospheric simulations

The resolutions of interests in atmospheric simulations require prohibitively large computational resources. Adaptive mesh refinement (AMR) tries to mitigate this problem by putting high resolution in crucial areas of the domain. We investigate the performance of a tree-based AMR algorithm for the high order discontinuous Galerkin method on quadrilateral grids with non-conforming elements. We perform a detailed analysis of the cost of AMR by comparing this to uniform reference simulations of two standard atmospheric test cases: density current and rising thermal bubble. The analysis shows up to 15 times speed-up of the AMR simulations with the cost of mesh adaptation below 1% of the total runtime. We pay particular attention to the implicit–explicit (IMEX) time integration methods and show that the ARK2 method is more robust with respect to dynamically adapting meshes than BDF2. Preliminary analysis of preconditioning reveals that it can be an important factor in the AMR overhead. The compiler optimizations provide significant runtime reduction and positively affect the effectiveness of AMR allowing for speed-ups greater than it would follow from the simple performance model.

Discontinuous diffusion synthetic acceleration for [formula omitted] transport on 2D arbitrary polygonal meshes

In this paper, a Diffusion Synthetic Acceleration (DSA) technique applied to the Sn radiation transport equation is developed using Piece-Wise Linear Discontinuous (PWLD) finite elements on arbitrary polygonal grids. The discretization of the DSA equations employs an Interior Penalty technique, as is classically done for the stabilization of the diffusion equation using discontinuous finite element approximations. The penalty method yields a system of linear equations that is Symmetric Positive Definite (SPD). Thus, solution techniques such as Preconditioned Conjugate Gradient (PCG) can be effectively employed. Algebraic MultiGrid (AMG) and Symmetric Gauss–Seidel (SGS) are employed as conjugate gradient preconditioners for the DSA system. AMG is shown to be significantly more efficient than SGS. Fourier analyses are carried out and we show that this discontinuous finite element DSA scheme is always stable and effective at reducing the spectral radius for iterative transport solves, even for grids with high-aspect ratio cells. Numerical results are presented for different grid types: quadrilateral, hexagonal, and polygonal grids as well as grids with local mesh adaptivity.

Reprint of: A conservative multi-tracer transport scheme for spectral-element spherical grids

Atmospheric models used for practical climate simulation must be capable handling the transport of hundreds of tracers. For computational efficiency conservative multi-tracer semi-Lagrangian type transport schemes are appropriate. Global models based on high-order Galerkin approach employ highly non-uniform spectral-element grids, and semi-Lagrangian transport is a challenge on those grids. A conservative semi-Lagrangian scheme (SPELT — SPectral-Element Lagrangian Transport) employing a multi-moment compact reconstruction procedure is developed for non-uniform quadrilateral grids. The scheme is based on a characteristic semi-Lagrangian method that avoids complex and expensive upstream area computations. The SPELT scheme has been implemented in the High-Order Method Modeling Environment (HOMME), which is based on a cubed-sphere grid with spectral-element spatial discretization. Additionally, we show the (strong) scalability and multi-tracer efficiency using several benchmark tests. The SPELT solution can be made monotonic (positivity preserving) by combining the flux-corrected transport algorithm, which is demonstrated on a uniform resolution grid. In particular, SPELT can be efficiently used for non-uniform grids and provides accurate and stable results for high-resolution meshes.

Numerical comparison of robustness of some reduction methods in rough grids

In this article, we present three nonsymmetric mixed hybrid methods and compare with some recently developed reduction methods which are suitable for the single‐phase Darcy flow problem with full anisotropic and highly heterogeneous permeability on general quadrilateral grids. The methods reviewed are multipoint flux approximation (MPFA), multipoint flux mixed finite element method, mixed‐finite element with broken method, MPFA‐type mimetic finite difference method, and symmetric mixed‐hybrid finite element method. The numerical experiments of these methods on different distorted meshes are compared, as well as their differences in performance of fluxes are discussed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1484–1506, 2014

A fine-scale spatial population distribution on the High-resolution Gridded Population Surface and application in Alachua County, Florida

Geospatial techniques, using Geographic Information Systems and remote sensing data, have become more commonly used with dasymetric modeling of fine-scale demographic data. In this study, we apply a dasymetric approach using the Heuristic Sampling Method for 2010 parcel data to disaggregate population counts from the 2010 U.S. Census into a quadrilateral grid composed of 30 × 30 m cells covering the Alachua County, Florida. The final output, termed the High-resolution Gridded Population Surface (HGPS), is compared to a land cover-based population product (LCPP) and the detail of each product is assessed. Results suggest that the HGPS provides increased spatial heterogeneity and more detail in the boundaries of populated areas over the use of census blocks or land cover lots. For an example of the final output, we use a case study at the Cabot–Koppers Superfund Site to demonstrate the advantages of the HGPS over the LCPP. The HGPS is expected to serve as a more accurate input in various research fields, such as public health, crime analysis, and climate change. The approach outlined provides an improved means of producing spatially-explicit population grids where fine-scale ancillary data, such as parcel data, is available.