Quadrilateral Grids Research Articles

One-Dimensional/Two-Dimensional Coupling Approach with Quadrilateral Confluence Region for Modeling River Systems

We study shallow water flows in river systems. An accurate description of such flows can be obtained using the two-dimensional (2-D) shallow water equations, which can be numerically solved by a shock-capturing finite-volume method. This approach can, however, be inefficient and computationally unaffordable when a large river system with many tributaries and complex geometry is to be modeled. A popular simplified approach is to model flow in each uninterrupted section of the river (called a reach) as one-dimensional (1-D) and connect the reaches at the river junctions. The flow in every reach can then be modeled using the 1-D shallow water equations, whose numerical solution is dramatically less computationally expensive compared with solving its 2-D counterpart. Even though several point-junction models are available, most of them prove to be sufficiently accurate only in the case of a smooth flow though the junction. We propose a new 1-D/2-D river junction model, in which each reach of the river is modeled by the 1-D shallow water equations, while the confluence region, where the mixing of flows from the different directions occurs, is modeled by the 2-D ones. We define the confluence region to be a trapezoid with parallel vertical sides. This allows us to take into account both the average width of each reach and the angle between the directions of flow of the tributary and the principal river at the confluence. We choose a trapezoidal confluence region as it is consistent with the 1-D model of the river. We implement well-balanced positivity preserving second-order semi-discrete central-upwind schemes developed in Kurganov and Petrova (Commun Math Sci 5:133–160, 2007) for the 1-D shallow water equations and in Shirkhani et al. (Comput Fluids 126: 25–40, 2016) for the 2-D shallow water equations using quadrilateral grids. For the 2-D junction simulations in the confluence region we choose a very coarse 2-D mesh as the goal of our model is not to resolve the fine details of complex 2-D vortices that form around the junction, but to efficiently compute average water depth and velocity in the connected 1-D reaches. A special ghost cell technique is developed for coupling the reaches to the confluence region, which is one of the most important parts of a good 1-D/2-D coupling method. The proposed approach leads to very significant computational savings compared to numerically solving the full 2-D problem. We perform several numerical experiments to demonstrate plausibility of the proposed 1-D/2-D coupling model.

Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra

We develop higher order multipoint flux mixed finite element (MFMFE) methods for solving elliptic problems on quadrilateral and hexahedral grids that reduce to cell-based pressure systems. The methods are based on a new family of mixed finite elements, which are enhanced Raviart–Thomas spaces with bubbles that are curls of specially chosen polynomials. The velocity degrees of freedom of the new spaces can be associated with the points of tensor-product Gauss–Lobatto quadrature rules, which allows for local velocity elimination and leads to a symmetric and positive definite cell-based system for the pressures. We prove optimal [Formula: see text]th order convergence for the velocity and pressure in their natural norms, as well as [Formula: see text]st order superconvergence for the pressure at the Gauss points. Moreover, local postprocessing gives a pressure that is superconvergent of order [Formula: see text] in the full [Formula: see text]-norm. Numerical results illustrating the validity of our theoretical results are included.

A High Order Control Volume Finite Element Method for Transient Heat Conduction Analysis of Multilayer Functionally Graded Materials with Mixed Grids

This paper describes a new two-dimensional (2-D) control volume finite element method (CV-FEM) for transient heat conduction in multilayer functionally graded materials (FGMs). To deal with the mixed-grid problem, 9-node quadrilateral grids and 6-node triangular grids are used. The unknown temperature and material properties are stored at the node. By using quadratic triangular grids and quadratic quadrilateral grids, the present method offers greater geometric flexibility and the potential for higher accuracy than the linear CV-FEM. The properties of the FGMs are described by exponential, quadratic and trigonometric grading functions. Some numerical tests are studied to demonstrate the performance of the developed method. First, the present CV-FEM with mixed high-order girds provides a higher accuracy than the linear CV-FEM based on the same grid size. Second, the material properties defined location is proved to have a significant effect on the accuracy of the numerical results. Third, the present method provides better numerical solutions than the conventional FEM for the FGMs in conjunction with course high-order grids. Finally, the present method is also capable of analysis of transient heat conduction in multilayer FGM.

Simulation of multi-component multi-phase fluid flow in two-dimensional anisotropic heterogeneous porous media using high-order control volume distributed methods

In this paper, flow of multi-component two-phase fluids in highly heterogeneous anisotropic two-dimensional porous media is studied using computational methods suitable for unstructured triangular and/or quadrilateral grids. The physical model accounts for miscibility and compressibility of fluids while gravity and capillary effects are neglected. The governing equations consist of a pressure equation together with a system of mass conservation equations. For solving pressure equation, a new method called Control Volume Distributed Finite Element Method (CVDFEM) is introduced which uses Control Volume Distributed (CVD) vertex-centered grids. It is shown that the proposed method is able to approximate the pressure field in highly anisotropic and heterogeneous porous media fairly accurately. Moreover the system of mass conservation equations is solved using various upwind and central schemes. These schemes are extended from one-dimensional to two-dimensional unstructured grids. Using a series of numerical test cases, comparison is made between different approaches for approximation of the hyperbolic flux function. Semi one-dimensional high-order data reconstruction procedures are employed to decrease stream-wise numerical diffusion. The results suggest that the Modified Dominant Wave (MDW) scheme outperforms other hyperbolic schemes studied in this paper from both accuracy and computational cost points of view.

A goal-oriented and self-adaptive mesh refinement approach for the even parity neutron transport equation

This survey focuses on a goal-oriented approach for the adaptive mesh refinement (AMR) of even parity neutron transport equation on the unstructured triangular or quadrilateral grids. Two a posteriori error estimates are compared for conducting the refinement process. User specified quantity of interest is set as the source in the adjoint equation to obtain a map of importance for steering the AMR procedure toward the desired computational target. Both forward and adjoint even parity equations are solved and the discontinuity of the flux gradient across the border of an element is used for the evaluation of element's error. The self-adaptivity is provided through a fully automated cycle in which the h-refinement is performed until the error in the quantity of interest drops below a user prescribed threshold. Several numerical test cases are investigated to assess the performance of the proposed approach in both fixed source and eigenvalue search problems against uniform mesh refinement.

Hexahedral constraint-free finite element method on meshes with hanging nodes

Nonconforming grids with hanging nodes are frequently used in adaptive finite element computations. In all earlier works on such methods, proper constraints should be enforced on degrees of freedom on edges/faces with hanging nodes to keep continuity, which yield numerical computations much complicated. In 2014, Zhao et al. (2014) presented quadrilateral constraint-free finite element methods on quadrilateral grids with hanging nodes. This paper further develops a hexahedral constraint-free finite element method on hexahedral grids with hanging nodes, which is of greater challenge than the two-dimensional case. Residual-based a posteriori error reliability and efficiency are also established in this paper.

Efficient gradient stencils for robust implicit finite-volume solver convergence on distorted grids

Two gradient-stencil augmentation techniques are discussed: symmetric and F-decreasing augmentations, for efficient and robust implicit finite-volume-solver convergence on distorted unstructured grids. The former augments a face-neighbor stencil with extra cells to increase the symmetry of the stencil as much as possible, and the latter adds further extra cells to decrease the reciprocal of the Frobenius norm of a scaled least-squares matrix to minimize the lower bound of the magnitude of the gradient. These techniques are proposed as efficient ways of overcoming a known stability issue with a face-neighbor stencil. It is demonstrated that the F-decreasing augmentation in combination with the symmetric augmentation yields robust and efficient gradient stencils on highly distorted quadrilateral and triangular grids.

A high‐order solver for simulating vortex‐induced vibrations using the sliding‐mesh spectral difference method and hybrid grids

SummaryWe present a high‐order solver for simulating vortex‐induced vibrations (VIVs) at very challenging situations, for example, VIVs of a row of very closely placed objects with large relative displacements. This solver works on unstructured hybrid grids by employing the high‐order tensor‐product spectral difference method for quadrilateral grids and the Raviart‐Thomas spectral difference method for triangular grids. To deal with the challenging situations where a traditional conforming moving mesh is incapable, we split a computational domain into nonoverlapping subdomains, where each interior subdomain encloses an object and moves freely with respect to its neighbors. A nonuniform sliding‐mesh method that ensures high‐order accuracy is developed to deal with sliding interfaces between subdomains. A monolithic approach is adopted to seamlessly couple the fluid and solid vibration equations. Moreover, the solver is parallelized to further improve its efficiency on distributed‐memory computers. Through a series of numerical tests, we demonstrate that this solver is high‐order accurate for both inviscid and viscous flows and has good parallel efficiency, making it ideal for VIV studies.

Minimal consistent finite element space for the biharmonic equation on quadrilateral grids

Abstract This study presents a finite element space comprising piecewise quadratic polynomials on quadrilateral grids. This space provides a minimal-degree consistent discretisation for the biharmonic equation.

Validation of a two-dimensional gas-kinetic scheme for compressible natural convection on structured and unstructured meshes

Gas-kinetic schemes (GKS) have been developed as a kinetic Finite-Volume approach to computational fluid dynamics. The GKS a priori allows to obtain approximate solutions of the fully compressible Navier-Stokes equations. In our contribution we show simulation results of compressible natural convection at large temperature differences and low Mach numbers beyond the applicable range of the Boussinesq approximation. The simulations were performed on non-uniform quadrilateral and unstructured triangular grids. The dependence of the critical Rayleigh number on the temperature difference for compressible Rayleigh-Bénard convection, predicted by theory, is accurately reproduced. Moreover, heat transfer in a buoyancy driven square cavity with differentially heated sides at large Rayleigh numbers and large temperature differences is investigated. Temperature and velocity profiles as well as Nusselt numbers show good agreement with benchmark results in literature. After validating the scheme for thermal compressible convection, we investigate unsteady natural convection at a Rayleigh number of Ra=5⋅109 and at a large temperature difference of Th/Tc=4. We find that compressibility has a leading influence on the stability of the boundary layers, such that the flow at the heated wall becomes unstable, whereas the flow at the cooled wall remains stable. This phenomenon has not yet received much attention.

On a robust and accurate localized artificial diffusivity scheme for the high-order flux-reconstruction method

The high-order flux reconstruction (FR) method on unstructured hexahedral grids is coupled with the localized artificial diffusivity (LAD) scheme for aiming at accurate simulation of shock-turbulence interaction. In order to overcome known robustness issues particularly with the high-order (r>0) derivative formulation, important properties of the artificial bulk viscosity (ABV) profile affecting the solution are investigated first. For the purpose of comparison, an approximated Gaussian filter and modified restriction-prolongation (RP) filters are developed and tested for the present FR-LAD approach. Then, we propose a multidimensional RP filter on unstructured quadrilateral and hexahedral grids to address the issues on non-Cartesian grids. It is shown that a simple extension of a one-dimensional RP filter for two-dimensional grids may result in insufficient smoothing of ABV with r=2, and that the proposed multidimensional RP filter can provide smooth ABV with both r=0 and r=2. The proposed FR-LAD scheme is tested for typical shock-related problems, including the 1D shock tube, 1D shock-entropy wave interaction, 2D steady shock flows, and 2D shock-vortex interaction. The FR-LAD scheme has favorable properties of subcell shock capturing with the length scale of O(h/p) and superior preservation of high-order accuracy for smooth flows. Finally, LES of an overexpanded supersonic jet is performed to demonstrate its capability for practical applications. The proposed FR-LAD scheme can be an attractive candidate for LES or DNS of compressible flows involving shocks, contact discontinuities, turbulence, and their interactions on unstructured meshes.

Time integration of unsteady nonhydrostatic equations with dual time stepping and multigrid methods

A time integration method based on dual time stepping coupled with a multigrid and implicit iterative solver is discussed for overcoming the temporal stiffness associated with modeling unsteady atmospheric flows. Adopting the dual time stepping allows solving an unsteady flow problem as a steady-state one by introducing a pseudo-time derivative into the governing equations. Consequently, the model uses a large time step and leads to a reduction in computational time. In the framework of the dual time stepping, a full approximation storage multigrid and lower–upper symmetric Gauss–Seidel implicit solver are employed to accelerate convergence in a structured, quadrilateral grid. The discretization of a physical-time derivative is based on a second-order backward differentiation formula or a family of singly diagonally implicit Runge–Kutta methods. The spatial terms of nonhydrostatic equations are discretized with a second-order Roe scheme in the finite volume method. To assess the performance of the model in terms of accuracy and efficiency, which are indicated by the normalized error norms and wall-clock time, respectively, three idealized tests are performed with the unsteady flow problems of isentropic vortex, rising thermal bubble and inertia-gravity waves. The results of numerical experiments consistently demonstrate that the dual time stepping coupled with explicit-first, singly diagonally implicit Runge–Kutta and multigrid methods is more efficient than the three-stage, third-order explicit Runge–Kutta method with comparable accuracy. The numerical experiment also indicates that the dual time stepping is more efficient when a stretched grid is employed. The efficiency in the framework of the dual time stepping is dependent on various numerical factors, including a multigrid level and cycle, Butcher coefficients, an implicit solver and a physical-time step.

A coupled volume of fluid and level set method based on analytic PLIC for unstructured quadrilateral grids

We develop a coupled volume of fluid and level set (VOSET) method for unstructured quadrilateral grids to simulate incompressible two-phase interfacial flows in irregular domains. In the method, an analytic piecewise linear interface calculation (PLIC) is first proposed and its solution speed is much faster (about five times) than that of Brent's iterative method; then an iterative geometric operation by which the level set function near interfaces can be calculated, is extended to unstructured quadrilateral grids; moreover, the volume fraction advection is solved by a Lagrangian-Eulerian advection scheme. Finally, the present method is validated by the single vortex flow problem, bubble rising problem and falling droplet problem. Our simulation results show good agreement with those in previous studies.

The discontinuous Galerkin spectral element methods for compressible flows on two-dimensional mixed grids

This paper presents a practical algorithm for constructing high order discontinuous Galerkin spectral element methods (DGSEM) on mixed triangular and quadrilateral grids. The traditional DGSEM belongs to the collocation-type nodal discontinuous Galerkin method which is computationally efficient on one-dimensional and tensor-product grids. This work generalizes DGSEM to triangular grids using symmetric quadrature points in sphere close packed arrangement. The dispersion and dissipation analysis shows that the scheme is stable for linear problems on triangles. A problem independent limiting procedure is proposed for capturing the shock waves. The main components are a modified KXRCF shock detector, a smoothness indicator and a high-resolution WENO limiter based on candidate polynomials constructed by newly developed projection approach. The implementation of the DGSEM on the mixed grids is also presented. Numerical examples on triangular and mixed grids verify the scheme's high order accuracy and robustness in computing inviscid and viscous compressible flows.

A higher-efficient non-hydrostatic finite volume model for strong three-dimensional free surface flows and sediment transport

In order to accurately simulate strong three-dimensional (3-D) free surface flows and sediment transport, the fully 3-D non-hydrostatic pressure models are developed based on the incompressible Navier-Stokes equations and convection–diffusion equation of sediment concentration with the mixing triangle and quadrilateral grids. The governing equations are discretized with the unstructured finite volume method in order to provide conservation properties of mass and momentum, and flexibility with practical application. It is shown that it is first-order accurate on nonuniform plane two-dimensional (2-D) grids and second-order accurate on uniform plane grids. A third-order approximation of the vertical velocity at the top-layer is applied. In such a way, free surface zero stress boundary condition is satisfied maturely, and very few vertical layers are needed to give an accurate solution even for complex discontinuous flow and short wave simulation. The model is applied to four examples to simulate strong 3-D free surface flows and sediment transport where non-hydrostatic pressures have a considerable effect on the velocity field. The newly developed model is verified against analytical solutions with an excellent agreement.

Mechanical properties and failure behavior of the sandwich structures with carbon fiber-reinforced X-type lattice truss core

As we known, composite sandwich structure with advantages of high specific strength/modulus, high temperature resistance, damping, etc., which is widely used in various fields. And there is more space in its core to supply for multi-functional filler, which provides the possibility for special engineering applications (such as heat insulation, wave absorption, etc.). In this paper, a new type of lightweight composite lattice structure (called X-type lattice structure) is proposed based on the stretching dominated idea of quadrilateral grid and lattice strut. The mechanical properties and failure behavior of X-type lattice structures are investigated based on theoretical and experimental methods. The experiment results show that the mechanical properties of the X-type lattice sandwich structure is more superior than those of other pyramidal lattice sandwich structures. In theoretical analysis, we found the fact that the shear equivalent stiffness of X-type lattice structure is independent of the loading direction, while the shear strength is related to the loading direction. Meanwhile, its minimum shear strength will be reached at the 0 degrees loading direction, and the maximum shear strength will be obtained when the loading direction is 45 degrees.

Experimental investigation on bending and shear performance of two-way aluminum alloy gusset joints

Two-way aluminum alloy gusset joints (AAGJs) are considered to be employed in cable-stiffened aluminum alloy single-layer reticulated shell structures meshed with quadrilateral grids. In this study, a series of experiments on six AAGJs with and without shear connectors were conducted to investigate their semi-rigid behavior. Two joints were subjected to pure bending moments and others to shear loads, then their corresponding failure modes were investigated which were mainly beam rupture, beam web buckling, and bolt failure. The moment-rotation curves, shear load-rotation curves, and strain-load curves were also obtained, indicating that AAGJs are typical semi-rigid joints. Shear connectors can optimize the load transferring path and improve ductility under bending moment while enhancing shear performance significantly. A simplified theoretical AAGJ model was also established and parameterized according to the experimental results.

Higher resolution total velocity Vt and Va finite-volume formulations on cell-centred structured and unstructured grids

Novel cell-centred finite-volume formulations are presented for incompressible and immiscible two-phase flow with both gravity and capillary pressure effects on structured and unstructured grids. The Darcy-flux is approximated by a control-volume distributed multipoint flux approximation (CVD-MPFA) coupled with a higher resolution approximation for convective transport. The CVD-MPFA method is used for Darcy-flux approximation involving pressure, gravity, and capillary pressure flux operators. Two IMPES formulations for coupling the pressure equation with fluid transport are presented. The first is based on the classical total velocity Vt fractional flow (Buckley Leverett) formulation, and the second is based on a more recent Va formulation. The CVD-MPFA method is employed for both Vt and Va formulations. The advantages of both coupled formulations are contrasted. The methods are tested on a range of structured and unstructured quadrilateral and triangular grids. The tests show that the resulting methods are found to be comparable for a number of classical cases, including channel flow problems. However, when gravity is present, flow regimes are identified where the Va formulation becomes locally unstable, in contrast to the total velocity formulation. The test cases also show the advantages of the higher resolution method compared to standard first-order single-point upstream weighting.

A two-step finite volume method to discretize heterogeneous and anisotropic pressure equation on general grids

A novel Two-Step cell-centered Finite Volume Method (TSFVM) is developed in this work to discretize the heterogeneous and anisotropic pressure equation on triangular and quadrilateral grids in 2D and hexahedral and tetrahedral grids in 3D. Physical properties such as permeability and porosity are piece-wise constant on each grid cell. In the first step, the Galerkin Finite Element Method (FEM) is utilized to compute pressure solutions at all cell vertices. In the second step, pressure values at cell vertices are used to derive continuous two-point flux stencils for cell faces. Mass conservation equations are then written for each cell to obtain a system of linear equations that can be solved for pressure at cell centers. Extensive numerical experiments are carried out to test the performance of our TSFVM. In particular, we compare TSFVM with the classical Multipoint Flux Approximation (MPFA-O) method as well as a more recently developed MPFA method with full pressure support called enhanced MPFA (eMPFA). The results show that the TSFVM compares well with eMPFA for challenging test cases for which MPFA-O breaks down. Specifically, and as a significant step forward, our TSFVM is quite robust for challenging problems involving heterogeneous and highly anisotropic permeability tensors when both MPFA-O and eMPFA suffer from unphysical oscillations. Finally, the numerical convergence study demonstrates that TSFVM has comparable convergence behavior to MPFA-O method for both homogeneous and discontinuous permeability fields.

A residual-based h-adaptive reconstructed discontinuous Galerkin method for the compressible Euler equations on unstructured grids

In this paper, a residual-based h-adaptive high-order reconstructed DG (rDG) method based on the hybrid reconstruction strategy is developed for solving the compressible Euler equations on unstructured grids. The proposed method combines the advantages of high-order rDG discretization with appropriate residual-based error estimation techniques and h-adaptive refinement strategies, which exhibits its superior potential compared to the underlying DG methods. To be specific, a third-order hybrid rDG(p1p2) method has been carefully designed and evaluated on incompatible quadrilateral grids with hanging nodes in order to preserve 2-exactness property during the implementation of mesh refinement and coarsening. A residual-based error estimator is used as local error indicator during the h-adaptive procedures. A number of test cases are presented to assess the performance of the high-order h-adaptive rDG method. The hybrid reconstruction strategy combined with h-adaptive techniques presented in this work demonstrates promise for improving the level of accuracy and reducing the computational cost for numerical simulations of compressible inviscid flows.