Triangular Grid Research Articles

Local-inertial shallow water model on unstructured triangular grids

Two-dimensional shallow water models are widely used for flood risk assessment. Application of these models to large-scale urban areas significantly increases the computational cost as it requires high-resolution simulations to capture the complex hydrodynamic processes. Hence, simplified models that are based on local-inertial formulations have been proposed by different researchers. Although these models are successful in simulating floods, their applications are limited by the use of structured grids and instability issues. Structured grids do not have the flexibility of providing different mesh resolutions to model the computational domain. Importantly, in the case of structured grids, the use of a single grid or the combination of finer and coarser grids inevitably increases the overall computational time. To overcome all these problems, an unstructured grid-based local-inertial model is developed. The performance of the model is rigorously evaluated by solving analytical test cases and simulating an urban flood in Glasgow, UK. Finally, the model is applied to simulate a catastrophic flood event that occurred in Chennai, India. The simulated water depths and inundation extents are compared with observed data or full-2D model results. The investigations show that the developed model has the potential for simulating floods on a large-scale with high-resolution grids at a comparatively lower computational cost than its predecessors for a similar range of accuracy.

Mesh generation and optimization from digital rock fractures based on neural style transfer

The complex geometric features of subsurface fractures at different scales makes mesh generation challenging and/or expensive. In this paper, we make use of neural style transfer (NST), a machine learning technique, to generate mesh from rock fracture images. In this new approach, we use digital rock fractures at multiple scales that represent ‘content’ and define uniformly shaped and sized triangles to represent ‘style’. The 19-layer convolutional neural network (CNN) learns the content from the rock image, including lower-level features (such as edges and corners) and higher-level features (such as rock, fractures, or other mineral fillings), and learns the style from the triangular grids. By optimizing the cost function to achieve approximation to represent both the content and the style, numerical meshes can be generated and optimized. We utilize the NST to generate meshes for rough fractures with asperities formed in rock, a network of fractures embedded in rock, and a sand aggregate with multiple grains. Based on the examples, we show that this new NST technique can make mesh generation and optimization much more efficient by achieving a good balance between the density of the mesh and the presentation of the geometric features. Finally, we discuss future applications of this approach and perspectives of applying machine learning to bridge the gaps between numerical modeling and experiments.

Estimation of layer coefficients of cubipod homogeneous low-crested structures using physical and numerical model placement tests

Homogeneous low-crested structures (HLCSs) on hard seabed are designed to protect beaches and regenerate coral reefs. The height of a HLCS depends on the placement grid which determines the crest freeboard, wave transmission and concrete consumption. In real seafloor conditions, it is not easy to define feasible placement grids for HLCSs on uneven sea bottoms. In this study, the parameters of the numerical model Bullet Physics Engine (BPE) are calibrated and validated using the results of small-scale physical model placement tests of five-layer Cubipod HLCSs on horizontal rigid bottom. The BPE model showed a low sensitivity to variations in the calibrated parameters; the numerical model estimated the layer coefficients with global mean relative errors of 1.04% and 1.39% in the triangular and rectangular placements grids, respectively. Once the numerical model was calibrated, new numerical and physical model tests on a 4% rigid bottom slope were compared for validation. A five-layer Cubipod HLCS on a 4% bottom slope was simulated using the BPE numerical model showing a global mean relative error of 2.75% compared to the small-scale physical model tests. A good agreement was found between numerical and physical model tests of five-layer Cubipod HLCSs on both horizontal as well as 4% rigid bottom slope. The BPE numerical model was found a suitable tool to estimate the structure height of HLCSs and to optimize placement grids of HLCSs on real cases with hard sea bottom.

A blended transient/quasistatic Lagrangian framework for salt tectonics simulations with stabilized tetrahedral finite elements

AbstractWe propose a Lagrangian solid mechanics framework for the simulation of salt tectonics and other large‐deformation geomechanics problems at the basin scale. Our approach relies on general elastic‐viscoplastic constitutive models to characterize the deformation of geologic strata, in contrast with the majority of published works on the subject, which utilize nonlinear Stokes flow models. By means of multiscale asymptotics, we also show that the inertia term in the momentum balance equation can be safely neglected, if the goal is to track the Earth's crust deformation over long periods of time. Our time integration strategy is a blended transient/quasistatic approach, in that it consists of a constitutive stress update, subject to the constraint that the stresses must satisfy static equilibrium. In addition, we use stabilized finite element methods specifically built for triangular and tetrahedral grids, which can also perform well under incompressibility constraints. Our approach offers computational geologists the following advantages: (1) improved flexibility in the choice of subsurface constitutive models with respect to the nonlinear Stokes flow; (2) improved efficiency over transient dynamics algorithms used in this context in the past, which are forced to resolve seismic events over geologic time scales; and (3) improved robustness in large strain computations over quadrilateral/hexahedral finite elements. We demonstrate the performance of the proposed approach with simulations of passive diapirism.

Mechanical Properties and Failure Analysis of Lattice Grid Sandwich Structure

In this study, quare, kagome and triangular grid sandwich structures about carbon fibre reinforced epoxy (CF/EP) composite, were designed and manufactured. Mechanical properties of these designed sandwich structures were performed by three-point bending and finite element analysis (FEA). The bending stiffness and core shear stress were calculated and then compared. Results obtained indicate that stress concentration was found at loading regions that cracks occurred on the interface. The face yielding and core shear failure also play a key role in the sandwich grid structure. The results show that the carbon fibre fabricated triangular grid core has a much stiffer and stronger character than other structures.

Numerical investigation of performance analysis of Triangular Solar air heater using Computational Fluid Dynamics (CFD)

Computational fluid dynamics based research is performed to investigate the thermal efficiency of triangular solar air heater (TSAH) passage. A systematic analysis of the device output parameters was carried out controlled under all test conditions including solar intensity, mass flow rate and air temperature at the inlet section. The glazing side of the triangular section is subjected to the solar intensity 746 W/m2. The three-dimensional TSAH model is developed and simulated using the FVM method with CFD code. The flow control equations have been solved with commercial ANSYS 14.5 software.A uniform triangular grid is generated according to the configuration using tetrahedron. The k-ε, RNG turbulence model with enhanced wall treatment function is used to solve the energy transport equations for turbulent flow. The optimal values of the geometric parameters are obtained on the basis of the performance index in the Reynolds number range from 2263 to 18103.It is observed the value of thermal efficiency increases with respect to velocity of air. It also increases the absorber and glass plate temperature.

Application of the Discontinuous Galerkin Method to the Study of the Dynamics of Temperature and Pressure Changes in a Formation with an Injection Well and a Hydraulic Fracture

Introduction. In this article, the problem of temperature distribution in an oil-bearing formation with a hydraulic fracture and a vertical injection well is numerically modeled. Materials and Methods. To describe the process of temperature distribution in the formation under the action of the fluid injected into the formation, the Fourier-Kirchhoff equation of convective heat transfer is used. To solve this equation, the discontinuous Galerkin method on staggered unstructured grids is used. To describe the process of pressure change in the formation under the action of the injection well, an equation is used that is obtained based on the continuity equation and Darcy’s law. To solve it, the discontinuous Galerkin method on an unstructured triangular grid is used. To parallelize the numerical algorithm, the MPI library is used. Results. The article presents a numerical algorithm and the results of modeling the dynamics of the temperature fields in an oil reservoir with a hydraulic fracture and a vertical injection well. Discussion and Conclusion. A numerical algorithm based on the discontinuous Galerkin method for math modeling of the temperature and pressure fields in a oil-bearing formation with a hydraulic fracture and injection well was developed and implemented. The results obtained for the distribution of temperature and pressure in the fracture are adequate and in good agreement with the specified initial-boundary conditions. Further work in this direction involves modeling on tetrahedral unstructured meshes for a more accurate study of the ongoing processes.

Vector Arithmetic in the Triangular Grid.

Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: it is not closed under vector-addition, which gives a challenge. The points of the triangular grid are represented by zero-sum and one-sum coordinate-triplets keeping the symmetry of the grid and reflecting the orientations of the triangles. This system is expanded to the plane using restrictions like, at least one of the coordinates is an integer and the sum of the three coordinates is in the interval [−1,1]. However, the vector arithmetic is still not straightforward; by purely adding two such vectors the result may not fulfill the above conditions. On the other hand, for various applications of digital grids, e.g., in image processing, cartography and physical simulations, one needs to do vector arithmetic. In this paper, we provide formulae that give the sum, difference and scalar product of vectors of the continuous coordinate system. Our work is essential for applications, e.g., to compute discrete rotations or interpolations of images on the triangular grid.

Triangular and Octagonal Array Grammars

Hexagonal Array Grammars (HAG) were introduced by K.G. Subramanian [4]. HAG increased the capacity of generating the hexagonal arrays on triangular grid. In this paper, using the Triangular Array Grammars (TAG) and Octagonal Array Grammars (OAG), we increasing the generative capacity for triangular and octagonal arrays. A system of the family of Triangular Array Language (TAL) and Octagonal Array Language (OAL) are presented.

Study of the Saffman–Taylor Instability in an Oil Reservoir Formation in Two Dimensions

The paper is devoted to the simulation of oil displacement by water and the formation of the Saffman–Taylor instability. The problem is solved in two-dimensional formulation. A circular domain with one injection well and eight production wells located along the contour around the injection well is considered as geometry. To study the patterns of oil displacement by water, hydrostatic pressure, oil and water seepage rates, and oil saturation are calculated. The graphical analysis considers mainly the oil saturation field. The hydrostatic pressure field is calculated by solving the steady-state piezoconductivity equation; the field of the oil-water seepage rate is calculated using the linear Darcy filtration law; and the oil saturation field is calculated from the solution of the advection transport equation. The two-phase nature of the flow considered in the problem lies in the fact that the oil and water phases have their own relative phase permeabilities calculated using the Brooks–Corey model. The equations are solved numerically using the finite volume method. An irregular triangular grid is used to discretize the computational domain. As a result of the simulation, it is established that the type of the Saffman–Taylor instability, due to its randomness, strongly depends on the computational grid. After the production wells are flooded, the displacement front stabilizes. The instability increases as the ratio of dynamic viscosities of oil and water increases.

A comprehensive analytical model for global buckling analysis of general grid cylindrical structures with various cell geometries

In this paper, we present an analytical method based on equivalent continuum homogenization for the global buckling analysis of the general grid lattice cylindrical structures. In the proposed scheme, grid structures with arbitrary cell geometries can be analyzed by obtaining their effective cell stiffness based on force and moment analysis of the struts. The grid structures are assumed to be composed of curvilinear axial, circumferential, and oblique unidirectional composite ribs. To evaluate the results of the presented analytical method, a parametric finite element code is derived to generate the desired geometry of the grid structures and their buckling loads are obtained and compared with the analytical method. The effects of various parameters, including the number of ribs, their thickness and elastic properties, and helical angle of the oblique ribs are studied for hexagonal, triangular, and mixed grid shells. The results are compared with the available data published in the literature with a good agreement.

An efficient reconstruction algorithm for diffusion on triangular grids using the nodal discontinuous Galerkin method

High-energy-density (HED) hydrodynamics studies such as those relevant to inertial confinement fusion and astrophysics require highly disparate densities, temperatures, viscosities, and other diffusion parameters over relatively short spatial scales. This presents a challenge for high-order accurate methods to effectively resolve the hydrodynamics at these scales, particularly in the presence of highly disparate diffusion. A significant volume of engineering and physics applications use an unstructured discontinuous Galerkin (DG) method developed based on the finite element mesh generation and algorithmic framework. This work discusses the application of an affine reconstructed nodal DG method for unstructured grids of triangles. Solving the diffusion terms in the DG method is non-trivial due to the solution representations being piecewise continuous. Hence, the diffusive flux is not defined on the interface of elements. The proposed numerical approach reconstructs a smooth solution in a parallelogram that is enclosed by the quadrilateral formed by two adjacent triangle elements. The interface between these two triangles is the diagonal of the enclosed parallelogram. Similar to triangles, the mapping of parallelograms from a physical domain to a reference domain is an affine mapping, which is necessary for an accurate and efficient implementation of the numerical algorithm. Thus, all computations can still be performed on the reference domain, which promotes efficiency in computation and storage. This reconstruction does not make assumptions on choice of polynomial basis. Reconstructed DG algorithms have previously been developed for modal implementations of the convection–diffusion equations. However, to the best of the authors’ knowledge, this is the first practical guideline that has been proposed for applying the reconstructed algorithm on a nodal discontinuous Galerkin method with a focus on accuracy and efficiency. The algorithm is demonstrated on a number of benchmark cases as well as a challenging substantive problem in HED hydrodynamics with highly disparate diffusion parameters.

Resonant oscillations in the western part of the Peter the Great Gulf in the Sea of Japan

The research aims to study the resonant properties of the water area of the Peter the Great Gulf, in the Sea of Japan. The coast of the gulf is exposed to the threat of tsunamis due to relatively high seismic activity in the Sea of Japan. On May 26, 1983, a tsunami hit several bays of the gulf with wave heights reaching 5 m. As a response, the coasts were equipped with three sophisticated sea level measuring stations. Field measurements with installation of autonomous pressure gauges were carried out August 18–26, 2016, to study the peculiarities of free surface oscillations in the western part of the Peter the Great Gulf. Spectral analysis of the recorded data and the data obtained from the Posyet automated post of the Russian Tsunami Warning Service showed, in particular, that there were several clearly pronounced peaks at periods of 40, 48, 79 and 89 min. These oscillations are coincident for the Sea of Japan and the studied water area with a wide entrance. A finite-volume numerical model was used to calculate free oscillations in the computational domain covering the Sea of Japan region. We performed model calculations with an irregular triangular grid. This allowed us to describe both the outer water area of the Sea of Japan and separate small-scale relief elements in the water area under study, which are important for the formation of local seiches. Isophases and spatial patterns of several resonant oscillations spectra at 15–105 min periods were obtained for the Peter the Great Gulf area. Model calculations carried out in this work showed an evident spatial structure of resonant oscillations. Such research, including field measurements in adjacent coastal waters, is required to identify and monitor potentially hazardous coastal areas. The amplification of resonant oscillations depends on how effectively the specific atmospheric disturbance generates corresponding eigenmodes of the Sea of Japan. In some bays, the amplitudes of the Sea of Japan eigen oscillations can increase due to resonant properties of coastal water areas, even with a low quality factor. If the amplitudes are large, such a double resonance can lead to a meteorological tsunami.

A Large-Eddy Simulation Study on the Diurnally Evolving Nonlinear Trapped Lee Waves over a Two-Dimensional Steep Mountain

AbstractThe diurnally evolving trapped lee wave over a small-scale two-dimensional steep mountain is investigated in large-eddy simulations based on a fully compressible and nonhydrostatic model [Icosahedral Nonhydrostatic (ICON)] with triangular grids of 50-m-edge length. An idealized atmospheric profile derived from a realistic case is designed to account for influences from the stagnant layer near the surface, the stability of the atmospheric boundary layer (ABL) and the upper-level jet. First, simulations were done to bridge from the linear regime to the nonlinear regime by increasing the mountain height, which showed that larger-amplitude lee waves with longer wavelength can be produced in the nonlinear regime than in the linear regime. Second, the effects of the stagnant layer near the surface and the ABL stability were explored, which showed that the stagnant layer or the stable ABL can play a similar wave-absorbing role in the nonlinear regime as in linear theories or simulations. Third, the role of the upper-level jet was explored, indicating that a stronger (weaker) upper-level jet can help to produce longer (shorter) lee waves. The stable ABL with a stagnant layer can more (less) efficiently absorb the longer (shorter) lee waves due to the stronger (weaker) jet, so that the wave response is more sensitive to the wave-absorption layer when an upper-level jet is present. Finally, the momentum budget was analyzed to explore the interaction between the upper and lower levels of the troposphere, which showed that the momentum flux due to the upward-propagating waves and trapped waves varies with the upper-level jet strength and low-level stagnancy and ABL stability.

Effects of coastal reclamation on the storm surge in the Bohai Bay

In recent years, the Bohai Bay has experienced frequent reclamation activities and significant changes in the coastline with the rapid development of economy. Coastline changes will influence the propagation of tide and ocean waves, and the storm surges may get worse as well. Based on the ADCIRC storm surge model, an unstructured triangular grid model driven by tide and wind was established to simulate the storm surge in Bohai Sea. The numerical simulation of hydrodynamic environment during typhoon NO.9711 under four coastline scenarios which are 1980, 2004, 2008 and 2015 had been carried out to investigate how coastal reclamation affects storm surges. The results reveal that the storm surge level and hazard risk in the reclamation areas has increased along with the changes of the coastline.

Observation and simulation of wind waves near a typical reef lagoon in South China Sea

Due to the inference of the uneven shallow water seabed and the surrounding islands, the wind-generated waves around or in a reef lagoon are rather complicated, and critical to the safety of floating structures deployed near islands or inside a lagoon. This paper aims to find a feasible analysis tool for the wave simulations near islands and reefs. The proposed three methods of grid techniques of WAVEWATCH III (WW3) are assessed by using on-site measured data which was collected and accumulated for about 5 years since August 2014 by a wave observation system deployed inside and outside a reef lagoon in South China Sea. In the assessments, the wave statistics including the correlation coefficients, root mean square errors, and their variances are used to quantify the precisions of the simulation results of the significant wave heights, mean wave periods, and peak wave directions at two sites. Among the three methods, the Multi-scale Zone and Multi-scale grid Technique (MZMGT) established on unstructured triangular grids exhibits better results in terms of the accuracy and CPU cost. In addition, the bimodal feature of wave spectra was observed at both sites of the reef lagoon in different typhoon events. The wave characteristics inside the reef lagoon and open sea are also analyzed.

Impact of downstream lakes on dam break wave attenuation

In this study, the effect of downstream lakes or ponds on dam break flow was investigated. To do this, a robust finite volume method with Harten-Lax-van Leer contact surface Riemann’s solver technique for flux evaluation has been used. For space discretization, a triangular mesh grid was used and a complete software with mesh generator was developed. A reservoir with a long downstream channel was assumed, and many scenarios such as a single lake with various lengths, widths and positions along the channel and multiple consecutive lakes as symmetric and non-symmetric with respect to the channel centerline were considered. The numerical modeling of dam break flow along the channel showed that the downstream lake can decrease the speed and depth of the dam break wave front and the amount of this wave attenuation was very dependent on the storage volume of the lake and vortexes which were established in the lake.

M-Bonacci graceful labeling

We introduce new labeling called m-bonacci graceful labeling. A graph G on n edges is m-bonacci graceful if the vertices can be labeled with distinct integers from the set such that the derived edge labels are the first n m-bonacci numbers. We show that complete graphs, complete bipartite graphs, gear graphs, triangular grid graphs, and wheel graphs are not m-bonacci graceful. Almost all trees are m-bonacci graceful. We give m-bonacci graceful labeling to cycles, friendship graphs, polygonal snake graphs, and double polygonal snake graphs.

Theta Functions and Optimal Lattices for a Grid Cells Model

Certain types of neurons called “grid cells” have been shown to fire on a triangular grid when an animal is navigating on a two-dimensional environment, whereas recent studies suggest that the face...

The triangle grid, the evolution of layered shells since the beginning of the 19th century

AbstractThis paper presents the evolution of and influence that the use of a triangular grid has had on the transformation of layered grid structures during the last two centuries. Historical facts, technological advances, economic and social crises among other factors influence the process of change and evolution of the grid systems in architecture to make construction processes more efficient by reducing the consumption of materials and qualified labor. Geometry and structure innovate styles and processes that are perfected over the years, achieving lighter and more stable structures. For the purpose of this paper historical data will be collected that allows the development of a theory that shows the influence of various periods based on technological advances. This research shows four historical periods in the evolution of layered grid structures. Being the triangular grid a widely used methodology in the construction of a diversity of geometric proposals and innovative construction systems for the solution of a diversity of buildings from the first domes built in the early 19th century to the free-form structures built from the late 20th century to the present day.