Quadrilateral Grids Research Articles

Thermal effect on the flow induced by a single-dielectric-barrier-discharge plasma actuator under steady actuation

The thermal effect of a single-dielectric-barrier-discharge plasma actuator under steady actuation is numerically investigated. A new actuator model is proposed and validated using experimental data. A discrete Galerkin method based on high-order flux reconstruction schemes is employed to solve the flow governing equations and the actuator model equations on unstructured quadrilateral grids. By comparing the induced heated and cold flow fields of the actuator with and without a plasma thermal source, its thermal effect is revealed. The actuator generates a thermal wall jet with rich vorticity, forming a monopolar starting vortex with a high-temperature and low-density core. Over time, the starting vortex becomes unstable and transforms into a dipole. Actuator heating enhances jet velocity and width, as well as vortex stability, while slowing down vorticity generation. The relative change in density and temperature fields due to actuator heating is four orders of magnitude greater than that without actuator heating. Additionally, the actuator heating causes the background thermodynamic fields to increase approximately linearly with time. Two stages in the actuator's thermal effect are distinguished due to time accumulation. Initially, the actuator heating minimally affects the monopolar starting vortex motion, and the temperature and density fields are treated as passive variables driven by the velocity field. During this stage, the momentum and thermal effects of the actuator can be studied separately. However, after the starting vortex becomes unstable, the actuator heating significantly impacts its motion and morphology, and these two effects are coupled with each other.

Multi-objective optimization of assembly free-form cable-stiffened reticulated shell considering the influence of initial states

In this study, we propose a geometric model of free-form grid structures based on the sliding translation method. Furthermore, an end-flattened inserted-plate joint could be applied to a quadrilateral free-form grid structure. The height of the control point was considered as the optimization variable, while the strain energy, geometric integrated index, and ultimate bearing capacity were considered as the optimization objectives. The results demonstrated the influence of the initial prestress and joint stiffness using multi-objective optimization. Based on the effects of the initial prestress, joint stiffness, and initial geometric imperfections, we propose a multi-objective optimization method for the assembly of a free-form cable-stiffened reticulated shell. The feasibility of the optimization method was verified using examples. Compared with the results of the shape optimization, structures having superior mechanical properties and mesh quality were obtained via the combined method for optimizing the shape and initial prestress, which verified the necessity and effectiveness of the combined optimization method. The mechanical properties of the optimized structures were improved by increasing the joint stiffness; in particular, the ultimate bearing capacity was improved. On using the multi-objective optimization method while considering the initial geometric imperfections, the properties of the structure were improved effectively, and the mechanical properties can be guaranteed even after geometric imperfections are applied.

Multipoint stress mixed finite element methods for linear viscoelasticity with weak symmetry

In this paper, we propose two Multipoint Stress Mixed Finite Element (MSMFE) methods for linear viscoelasticity with weak symmetry on quadrilateral grids. The methods are constructed based on the lowest order Brezzi–Douglas–Marini mixed finite element spaces for elastic and viscous stress, piecewise constant velocity and piecewise constant (linear) vorticity. A special quadrature rule is applied for local stress and vorticity elimination. This results in a positive definite cell centered velocity–vorticity or only velocity system at each time step. Unconditional energy-dissipation of the MSMFE methods is proved rigorously. The accuracy of all the numerical solutions in their nature norms are established to be first order space convergence, both for the semi-discrete and fully-discrete formulations. Numerical results validate the theory results of the proposed methods.

QuadGridSIM: A quadrilateral grid‐based method for high‐performance and robust trajectory similarity analysis

AbstractMeasuring trajectory similarity is a fundamental algorithm in trajectory data mining, playing a key role in trajectory clustering, pattern mining, and classification, for instance. However, existing trajectory similarity measures based on vector representation have challenges in achieving both fast and accurate similarity measurements. On one hand, most existing methods have a high computational complexity of O(n × m), resulting in low efficiency. On the other hand, many of them are sensitive to trajectory sampling rates and lack of accuracy. This article proposes QuadGridSIM, a quadrilateral grid‐based method for trajectory similarity analysis, which enables high‐performance trajectory similarity measure without the cost of low effectiveness. Specifically, we first realize the multiscale coding representation of trajectory data based on quadrilateral discrete grids. Then, a novel trajectory similarity measure is defined to reduce the computational complexity of O(n). Several effectiveness properties of QuadGridSIM are further optimized, including the spatial overlap, directionality, symmetry, and robustness to sampling rate variations. Experimental results based on real‐world and simulated taxi trajectory data indicate that QuadGridSIM outperforms most of the other tested algorithms developed previously in terms of effectiveness, particularly in its robustness regarding trajectory sampling rates. Furthermore, QuadGridSIM exhibits superior performance and is approximately one order of magnitude faster than previous methods in the literature. QuadGridSIM provides a solution to the low‐efficiency problem of massive trajectory similarity analysis and can be applied in many application scenarios, such as route recommendation and suspect detection.

A monotone finite volume element scheme for diffusion equations on arbitrary polygonal grids

We develop a monotone finite volume element scheme for the diffusion problem on arbitrary polygonal grids inspired by the scheme in the literature ([35]). The main contributions of this paper include three aspects. Firstly, some auxiliary lines or points are introduced for any polygonal element so that it can be partitioned into two or more triangles, and one novel type of its control volume is concomitantly designed. In this paper, we mainly discuss quadrilateral grids. Secondly, some piecewise linear finite element spaces are imported for the trial spaces. Fictitious elements, continuous extensions and the nonlinear two-point flux idea can all be inherited from the triangular case. Therefore, the monotonicity of the scheme is easily proved only by the assembling of some triangles' element stiff matrices. Thirdly, the agreeable adaptability for distorted grids is completely borned. The strict convexity restriction on the grids can also be removed, the concave or severely distorted polygonal (such as quadrilateral) grids are transformed as the convex and general regular triangles. Finally, numerical results confirm its monotonicity, accuracy and stability.

New Quality Measures for Quadrilaterals and New Discrete Functionals for Grid Generation

In this paper, we review some grid quality metrics and define some new quality measures for quadrilateral elements. The curved elements are not discussed. Usually, the maximum value of a quality measure corresponds to the minimum value of the energy density over the grid. We also define new discrete functionals, which are implemented as objective functions in an optimization-based method for quadrilateral grid generation and improvement. These functionals are linearly combined with a discrete functional whose domain has an infinite barrier at the boundary of the set of unfolded grids to preserve convex grid cells in each step of the optimization process.

Isoparametric Finite Element Method to Generate Structured Grid for Numerical Flow Simulation

Eight-nodded quadrilateral finite element is used to generate structured grid for a numerical simulation. The computational domain is sub-divided into number of quadrilateral regions that display the geometry of each sub-domain. The inner and the outer boundaries of the computational domain are described in terms of the surface coordinates. An algebraic homotopy procedure is used to generate grid clustering in order to resolve boundary layer. The structured grid is linked with the flow solver based on finite volume of space discretiztion scheme with multi-stage Runge-Kutta time stepping technique. Examples are illustrated to demonstrate the grid generation procedure and data processing for a forward facing aero-disc spike attached to a hemispherical blunt body at Mach 6 and over a heat shield of a satellite launch vehicle at Mach 0.8 at an angle of attack 5 deg. The present grid generation method is convenient for checking the grid independence test by varying the stretching factor. The quadrilateral grid generated by finite element, vector plot of velocity and contour plots of the computed flow field data are easily drawn with the help of MATLAB.

Progressive Collapse Analysis of Single-Layer Latticed Domes With Fabricated Joints

Progressive collapse accidents of single-layer latticed domes seriously threaten public safety. The progressive collapse-resisting capacity (PCRC) is gradually becoming an essential requirement in the design of spatial structures. Currently, the joint system used in spatial structures can be divided into two categories: welded joints and fabricated joints. The semi-rigidity of fabricated joints may have a significant influence on the PCRC of single- layer latticed domes. In this study, the PCRCs of single-layer fabricated latticed domes with identical span and rise-to-span ratio are evaluated based on the critical progressive collapse load (CPCL). The collapse mechanism of single-layer latticed domes with different grid forms is revealed by progressive analysis. In addition, the effects of joint rigidity, grid form and type of initial failure are further evaluated. The results indicate that the collapse mechanism of single-layer latticed domes includes the radial propagation type and the circumferential propagation type. The latticed domes with quadrilateral grids have much lower PCRC than those with triangular grids. The CPCLs of weld-jointed latticed domes are 5% to 22% higher than those of the fabricated latticed domes with the same grid form. The above influence should be fully considered in the progressive collapse-resisting design of latticed domes with fabricated joints.

Towards a robust and portable pipeline for quad meshing: Topological initialization of injective integer grid maps

Integer Grid Maps (IGMs) are a class of mappings characterized by integer isolines that align up to unit translations and rotations of multiples of 90 degrees. They are widely used in the context of remeshing, to lay a quadrilateral grid onto the mapped surface. The presence of both discrete and continuous degrees of freedom makes the computation of IGMs extremely challenging. In particular, solving for all degrees of freedom altogether leads to a mixed-integer problem that is known to be NP-Hard. Such a problem can only be solved heuristically, occasionally failing to produce a valid quadrilateral mesh. In this paper we propose a simple topological construction that allows to reduce the problem of computing a valid IGM to the one of mapping a topological disk to a convex domain. This is a much easier problem to deal with, because it completely removes the integer constraints, permitting to obtain a provably injective parameterization that is guaranteed to incorporate all the correct integer transitions with a simple linear solve. Not only the proposed algorithm is easy to implement, but it is also independent from costly numerical solvers that are unavoidable in existing quadmeshing pipelines, preventing their exploitation in open source or low-budget projects. Despite provably correct, the so generated maps contain a considerable amount of geometric distortion and a poor quad connectivity, making this technique more suitable for a robust initialization rather than for the computation of an application-ready IGM. In the article we present the details of our construction, also analyzing its geometric and topological properties.

Research and Construction of a Global Hexagonal Marine Gravity Gradient Reference Map for Navigation

A high-precision marine gravity gradient reference map is key to enabling underwater gravity gradient matching navigation. At present, the construction of the reference maps is based on quadrilateral geographic grids. However, quadrilateral grids lead to detriangulation at high latitudes, which limits the global applicability of such maps to underwater gravity navigation. To circumvent the limitations of quadrilateral grids, a hexagonal grid is introduced for constructing the reference map. This paper analyzes the characteristics of the icosahedral Snyder equal area aperture 4 hexagon (ISEA4H) and H3 grid systems and selects an appropriate grid system. In addition, we calculate and analyze the grid model errors and matching positioning errors of hexagonal and quadrilateral grids at the same resolution. The experimental results show that the grid model and matching positioning errors of a hexagonal grid system are more than 14% and 15% less than those of a quadrilateral grid system, respectively, indicating the feasibility and effectiveness of applying hexagonal grids to gravity gradient matching navigation. Given the low construction efficiency of a marine hexagonal grid gravity gradient reference map, we propose an efficient CPU+GPU hybrid parallel scheme. A global total tensor marine hexagonal grid gravity gradient reference map model is then constructed.

A unification of least-squares and Green–Gauss gradients under a common projection-based gradient reconstruction framework

We propose a family of gradient reconstruction schemes based on the solution of over-determined systems by orthogonal or oblique projections. In the case of orthogonal projections, we retrieve familiar weighted least-squares gradients, but we also propose new direction-weighted variants. On the other hand, using oblique projections that employ cell face normal vectors we derive variations of consistent Green–Gauss gradients, which we call Taylor–Gauss gradients. The gradients are tested and compared on a variety of grids such as structured, locally refined, randomly perturbed, unstructured, and with high aspect ratio. The tests include quadrilateral and triangular grids, and employ both compact and extended stencils, and observations are made about the best choice of gradient and weighting scheme for each case. On high aspect ratio grids, it is found that most gradients can exhibit a kind of numerical instability that may be so severe as to make the gradient unusable. A theoretical analysis of the instability reveals that it is triggered by roundoff errors in the calculation of the cell centroids, but ultimately is due to truncation errors of the gradient reconstruction scheme, rather than roundoff errors. Based on this analysis, we provide guidelines on the range of weights that can be used safely with least squares methods to avoid this instability.

Grid refinement in ICON v2.6.4

Abstract. This article describes the implementation of grid refinement in the ICOsahedral Nonhydrostatic (ICON) modeling system. It basically follows the classical two-way nesting approach known from widely used mesoscale models like MM5 or WRF, but it differs in the way feedback from fine grids to coarser grids is applied. Moreover, the ICON implementation supports vertical nesting in the sense that the upper boundary of a nested domain may be lower than that of its parent domain. Compared to the well-established implementations on quadrilateral grids, new methods had to be developed for interpolating the lateral boundary conditions from the parent domain to the child domain(s). These are based on radial basis functions (RBFs) and partly apply direct reconstruction of the prognostic variables at the required grid points, whereas gradient-based extrapolation from parent to child grid points is used in other cases. The runtime flow control is written such that limited-area domains can be processed identically to nested domains except for the lateral boundary data supply. To demonstrate the functionality and quality of the grid nesting in ICON, idealized tests based on the Jablonowski–Williamson test case (Jablonowski and Williamson, 2006) and the Schär mountain wave test case (Schär et al., 2002) are presented. The results show that the numerical disturbances induced at the nest boundaries are small enough to be negligible for real applications. This is confirmed by experiments closely following the configuration used for operational numerical weather prediction at DWD, which demonstrate that a regional refinement over Europe has a significant positive impact on the forecast quality in the Northern Hemisphere.

A Novel Reliability Analysis Approach under Multiple Failure Modes Using an Adaptive MGRP Model

In this paper, a novel MRGP-SS method is proposed to deal with the reliability analysis problems under multiple failure modes. First, a random moving quadrilateral grid sampling (RMQGS) method is proposed to improve the randomness and uniformity of initial samples. Second, an adaptive procedure, which combines the multiple response Gaussian process (MRGP) model and the novel active learning functions, is proposed to efficiently and accurately produce surrogate models for failure surfaces. In this regard, two novel learning functions are introduced to adapt to different iterative cycles, one is employed to correct the quality of samples, and the other is used to search for the samples closest to the limit state surface. Third, the subset simulation (SS) is integrated into the adaptive MRGP model to estimate the failure probability under multiple failure modes with fewer function calls and time consumption. Numerical and engineering case studies are finally provided to demonstrate the effectiveness of the proposed method.

B-spline surface based 3D reconstruction method for deflectometry.

In the field of optical three-dimension (3-D) measurement, reconstruction usually is completed by the integration of a two-dimensional (2-D) gradient data set. Position and posture of camera and shape of the surface under test determine the location of gradient data which usually is on quadrilateral grids. This paper proposes a B-spline surface-based 3D reconstruction method for deflectometry, which reconstructs the surface under test with its 2-D gradient data set. The 2-D gradient data set consists of gradient data and the 2-D location of the gradient data in the camera coordinate system. The 2-D gradient data set is first transferred to the cameras' virtual image plane, so it locates on rectangular grids. Then, based on the properties of the B-spline basis function and characteristics of the camera, linear equations are derived to solve control points along the virtual image plane. The solved control points reconstruct the surface under test in the camera coordinate system. The property of the B-spline basis function determines the relationship between the depth of the surface and its derivative. The characteristic of the camera determines the relationship between the depth of the surface and the 2-D location of the gradient data. Meanwhile, the accuracy of the 2-D location can also be improved by the linear equations. Finally, simulated and actual experiments show that the proposed method is accurate and efficient at reconstructing surfaces in deflectometry.

On the tensor viscosity based on Gauss quadrature: A comparison of robustness, efficiency, and connection with hourglass control

From the numerical quadrature perspective, we study the tensor viscosity for the staggered Lagrangian hydrodynamics. Although this approach is analogous to the tensor viscosities in [5, Campbell&Shashkov2001] and [14, Kolev&Rieben2009], we gain some novel insights into such tensor viscosity. The instability originates from the singularity of the Jacobian determinant, and there's a naturally trade-off between efficiency and robustness of the tensor viscosity. Based on the four-point Gauss quadrature, a novel tensor viscosity is proposed which overcomes the instability in some degenerate cases of the quadrilateral grid. On the other hand, the proposed viscosity improves the numerical precision due to the exactness of the tensor product Gauss quadrature. A more compact tensor viscosity form is presented by the reduced integration technique, which is identical to the compact tensor viscosity in [29, Wendroff2010]. By analyzing the discrepancy between the two viscosities, we clarify why the tensor viscosity based on four-point quadrature can suppress hourglass motion. Furthermore, we demonstrate that the tensor viscosity proposed in [18, Lipnikov&Shashkov2010] is equivalent to the combination of the compact tensor viscosity and Flanagan-Belytschko type anti-hourglass force in a quadrilateral grid. The numerical experiments illustrate the efficiency of the proposed tensor viscosities. By varying the hourglass control strategies, we verify the propositions of the existing tensor viscosities.

MPAS-Seaice (v1.0.0): sea-ice dynamics on unstructured Voronoi meshes

Abstract. We present MPAS-Seaice, a sea-ice model which uses the Model for Prediction Across Scales (MPAS) framework and spherical centroidal Voronoi tessellation (SCVT) unstructured meshes. As well as SCVT meshes, MPAS-Seaice can run on the traditional quadrilateral grids used by sea-ice models such as CICE. The MPAS-Seaice velocity solver uses the elastic–viscous–plastic (EVP) rheology and the variational discretization of the internal stress divergence operator used by CICE, but adapted for the polygonal cells of MPAS meshes, or alternatively an integral (“finite-volume”) formulation of the stress divergence operator. An incremental remapping advection scheme is used for mass and tracer transport. We validate these formulations with idealized test cases, both planar and on the sphere. The variational scheme displays lower errors than the finite-volume formulation for the strain rate operator but higher errors for the stress divergence operator. The variational stress divergence operator displays increased errors around the pentagonal cells of a quasi-uniform mesh, which is ameliorated with an alternate formulation for the operator. MPAS-Seaice shares the sophisticated column physics and biogeochemistry of CICE and when used with quadrilateral meshes can reproduce the results of CICE. We have used global simulations with realistic forcing to validate MPAS-Seaice against similar simulations with CICE and against observations. We find very similar results compared to CICE, with differences explained by minor differences in implementation such as with interpolation between the primary and dual meshes at coastlines. We have assessed the computational performance of the model, which, because it is unstructured, runs with 70 % of the throughput of CICE for a comparison quadrilateral simulation. The SCVT meshes used by MPAS-Seaice allow removal of equatorial model cells and flexibility in domain decomposition, improving model performance. MPAS-Seaice is the current sea-ice component of the Energy Exascale Earth System Model (E3SM).

A Method for Digital Terrain Reconstruction Using Longitudinal Control Lines and Sparse Measured Cross Sections

Using longitudinal control lines and sparse measured cross sections with large spaces, a new method for quickly reconstructing digital terrains in natural riverways is presented. The longitudinal control lines in a natural riverway, mainly including the river boundaries, the thalweg, the dividing lines of floodplains and main channel, and the water edges, can be obtained by interpreting satellite images, remote sensing images or site surveys. Then, the longitudinal control lines are introduced into quadrilateral grid generation as auxiliary lines that can control longitudinal riverway trends and reflect transverse terrain changes. Then, by the equal cross-sectional area principle at the same water level, all measured cross sections are reasonably fitted. On the above basis, by virtue of the fitted cross-sectional data and the weighted distance method, the terrain interpolations along the longitudinal grid lines are conducted to obtain the elevation data of all grid nodes. Finally, according to the readable text formats of MIKE21 and SMS, the gridded digital terrain and connection information are output by computer programming to achieve good construction of the data exchange channels and fully exploit the special advantages of various software programs for digital terrain visualization and further utilization.

An Occupancy Information Grid Model for Path Planning of Intelligent Robots

Commonly used robot map models include occupancy grid maps, topological maps, and semantic maps. Among these, an occupancy grid map is mainly represented as a quadrilateral grid. This paper proposes an occupancy information grid for intelligent robots by exploiting the advantages of the occupancy grid map and spatial information grid. In terms of geometric structure, a regular hexagonal grid is used instead of a regular quadrilateral grid. In terms of attribute structure, the single obstacle attribute is replaced by the grid terrain characteristics, grid element attributes, and grid edge attributes. Thus, the occupancy information grid model is transformed into a new data structure describing the spatial environment, and it can be effectively applied to map construction and path planning of intelligent robots. For the map construction application of intelligent robots, this paper describes the basic process of laser sensor-based grid model construction. For the path planning application of intelligent robots, this paper extends the A* algorithm based on a regular hexagonal grid. Additionally, map construction and path planning applications for intelligent robots are experimentally verified. Several experimental results were obtained. First, the experimental results confirmed the theoretical conclusion that the minimum sampling density of the hexagonal structure was 13.4% lower than that of the quadrilateral structure. Second, the regular hexagonal grid is clearly more advantageous in describing environmental scenes, which can ameliorate the "undercompleteness" phenomenon. Third, there were large differences in the planning paths based on two types of grids, as shown by the fact that the distance of the planning paths obtained by the regular hexagonal grid was reduced by at least 10.8% and at most 15.6% compared with the regular quadrilateral grid.

Water and Salt Transports in the Hengsha Channel of Changjiang Estuary

In a multilevel bifurcated estuary, the channels between the bifurcated branches play important roles in the exchanges of water and salt. In the Changjiang Estuary, the Hengsha Channel (HC) connects the North Channel (NC) and the North Passage (NP). In this paper, based on a two-way nesting unstructured quadrilateral grid, finite-differencing, three-dimensional estuarine and coastal ocean model, the tidal and seasonal variations in the water and salt transports in the HC were simulated, and their dynamic mechanism was analyzed. The residual water and salt transports in the HC both flow southward from the NC to the NP. In wet season, the residual water transport in the HC is 677 m3/s during neap tide and 245 m3/s during spring tide, and the residual salt transport is 0. In dry season, the residual water and salt transports in the HC are 1278 m3/s and 0.38 t/s during neap tide, respectively, and 1328 m3/s and 12.61 t/s during spring tide. Affected by the northerly wind and the southeastward baroclinic gradient force, the water and salt fluxes in dry season are much larger than those in wet season. The dynamic mechanism responsible for the water transport in the HC was numerically simulated and analyzed.

Dynamically taming focal fields of femtosecond lasers for fabricating microstructures

Micromachining based on femtosecond lasers usually requires accurate control of the sample movement, which may be very complex and costly. Therefore, the exploration of micromachining without sample movement is valuable. Herein, we have illustrated the manipulation of optical fields by controlling the polarization or phase to vary periodically and then realized certain focal traces by real-time loading of the computer-generated holograms (CGHs) on the spatial light modulator. The focal trace is composed of many discrete focal spots, which are generated experimentally by using the real-time dynamically controlled CGHs. With the designed focal traces, various microstructures such as an ellipse, a Chinese character “Nan”, and an irregular quadrilateral grid structure are fabricated in the z-cut LiNbO3 wafers, showing good qualities in terms of continuity and homogeneity. Our method proposes a movement free solution for micromachining samples and completely abandons the high precision stage and complex movement control, making microstructure fabrication more flexible, stable, and cheaper.