Unstructured Triangular Meshes Research Articles

A nodal discontinuous Galerkin method for wave propagation in coupled acoustic–elastic media

AbstractThe accurate numerical solution at an acoustic–elastic interface is important for offshore exploration. The solution requires careful implementation for the acoustic–elastic boundary conditions. In this work, we leverage a nodal discontinuous Galerkin method, in which the unstructured uniform triangular meshes are used for the model meshing and an explicit upwind numerical flux derived from the Riemann problem is adopted to handle the boundary conditions at the acoustic–elastic interface. Several numerical results are provided to assess the accuracy and convergence properties of this method. The convergence analysis is carried out in the coupled model with a flat interface, and the accuracy of the proposed method is verified in the curved interface coupled model. Finally, a more complex model with a salt dome, inspired by real geophysical applications, is carried out in this study. The numerical results demonstrate that the proposed nodal discontinuous Galerkin method is effective and accurate for dealing with the coupled acoustic–elastic media with complex geometries.

Two-dimensional magnetotelluric inversion using unstructured triangular mesh implemented in Julia

This study presents a 2-D magnetotelluric inversion code tailored for unstructured triangular meshes, developed using Julia, a high-level, high-performance programming language designed for scientific and numerical computation. The forward modeling engine utilizes a node-based finite element method to solve electromagnetic fields throughout the modeling domain. The inversion process employs the Gauss-Newton optimization algorithm, iteratively updating the model via the minimization of a regularized least-squares objective function. We verified the accuracy of the forward modeling using two reference models and performed a synthetic inversion experiment to validate our inversion method and stress the necessity of accurately handling topography-influenced data. Through its application to constructing an electrical resistivity model beneath the northwestern end of Sumatra Island, Indonesia, we demonstrate the practicality of our code for inverting field datasets.

Great Lakes wave forecast system on high-resolution unstructured meshes

Abstract. Wind-wave forecasts play a crucial role in the North American Great Lakes region towards ensuring the safety of communities, enhancement of the economy, and protection of property. Modeling wind waves in closed and relatively shallow basins with complex bathymetry like the Great Lakes is a challenge that is successfully tackled in part by using variable-resolution triangular unstructured meshes with no limits in terms of computational scalability and maximum resolution in the coastal areas. In this paper, we discuss recent advances in developing unstructured mesh capabilities as part of the spectral wave model WAVEWATCH III, in the context of National Oceanic and Atmospheric Administration (NOAA) operational requirements such as model robustness, efficiency, and accuracy. We revisit the history of developments leading to the transition from rectilinear to curvilinear grids and finally to an unstructured mesh version of NOAA's operational Great Lakes wave modeling system (GLWUv2.0). The article describes the development of the operational GLWUv2.0, from mesh design and scalability analysis to validation and verification for hindcast of storm cases and re-forecast using 4 months of retrospective simulations. In closed Great Lakes basins untouched by swell from distant sources, the atmospheric model's direct impact on wave behavior stands apart, showing reduced forecast accuracy over time, while maintaining consistent precision in accurately wind-hindcasted stormy conditions.

Experimental analysis and numerical simulation of ignition delay time of diesel fuel using a shock tube

The present work consisted in developing a computational routine for prediction and characterization ignition delay time, pressure, temperature generated and energy released in the combustion process in a shock tube. Mathematical modeling considered the high-pressure region of the shock tube known as the conducted section. The Lazarus code compiler was used to generate the unstructured triangular meshes and to implement the computational routines to simulate the propagation of shock waves inside the shock tube. The mathematical modeling used the geometric parameters of the shock tube of the combustion laboratory of the Federal University of Minas Gerais and was based only on the displacement of the shock wave after the diaphragm rupture. The main objective of the work was the development of computational routines to characterize the shock wave parameters of the ignition delay times of conventional diesel to validate the experimental tests carried out in the shock tube conducted by Santana et al. [3] and compare it with the experimental tests and numerical simulations carried out by other authors available in the literature. A linear regression of the experimental tests conducted by Santana et al. [3] with conventional diesel was performed to obtain the Arrhenius equation for numerical simulation of the ignition delay time of diesel under the following conditions: temperatures from 880 to 1300 K, pressures 24 bar and equivalence ratio 1. The results show good prediction between experimental tests and numerical simulations. Were found delay times ranging from 425 to 1890 μs. Considering all temperature range, the difference between the experimental and simulated test was approximately 15%, this difference also can be explained by the measurement in the shock tube, that the ignition delay time was calculated by the time difference between the passage of the shock wave by the pressure sensor and the start of the ignition detected by the luminosity detection sensor. This work is relevant because it was developed computational routine in open software and the results achieved in the simulation were considered satisfactory, since they are in the same order of magnitude as the results found in the experimental tests carried out by Santana et al. [3] and other works.

Computational Fluid Dynamics Analysis of 3D Wind Flow around Rhizophora Mangrove Tree and Drag Force Acting on the Wall of the Tree

Many natural disasters, such as cyclones, typhoons, mudslides, and tsunamis, are currently plaguing the world. These tragedies, which have destroyed the coastal settlements, claim a great number of lives each year. The characteristics and structure of the Mangrove tree have the significant impact on protecting the coast from heavy wind, floods, mudslides, and other kinds of natural disasters. This research work investigates how Mangroves can protect the area near the coast from a fluid dynamics point of view. Complex real world problems require intelligent systems that combine knowledge and techniques. Hence Computational Fluid Dynamics (CFD) technique namely the finite volume method is employed as a tool in this study. In this research, a model Rhizophora Mangrove tree from Pichavaram Mangrove forest, Tamilnadu, India is generated using Ansys Workbench design modeler. A computational domain is created around this tree to reduce the boundary effect. Grid-independent research was carried out using various mesh sizes to determine the best grid resolution for the analysis. An unstructured triangular mesh was generated for the simulation. The focus of this study is to determine the 3D flow around the tree in order to determine the wind velocity, as well as the coefficient of drag and drag force on the wall tree. This research helps to give a clear understanding of wind flow around the 3D Rhizophora Mangrove tree. The velocity profile, pressure distribution, drag force, and drag coefficient around the tree are captured, analysed, and presented in detail. The results show that the Rhizophora mangrove trees can significantly reduce the flow velocity of the wind and will be able to safeguard the coast and communities nearby from natural disasters.

A robust unstructured finite volume scheme for urban flash flood simulation: Duhok city as a case study

ABSTRACT A robust well-balanced finite volume scheme is proposed for simulating the flash floods due to rainfall-runoff process. The solver’s implementation on unstructured triangular meshes is based on a Godunov-type second-order upwind finite volume formulation, and Roe’s average scheme is used to determine the fluxes. The bed slope, friction and rainfall source terms are discretized using an upwind approach. The suggested algorithm uses the slope limiter and Hancock’s predictor-corrector method in order to secure numerical stability. The proposed model overwhelms the drawbacks of many commercial codes that use the finite difference, finite elements and finite volume in their scheme, especially in dealing with rapid change in topography. The prepared scheme solves the two-dimensional shallow water equations over a real topography of Hishkaro basin located in Duhok city in Kurdistan of Iraq where the variation of bed elevation reaches 717 m. The scheme is validated against real observed data in Duhok city provided by the Sewage directory. The data of five observation points over the study area were used to validate the code. The results showed that the max water level reaches 3.67, 2.12, 1.20, 0.67 and 0.3 m at observed points 3, 2, 4, 1 and 5, respectively.

WAVEx: Stabilized finite elements for spectral wind wave models using FEniCSx

The prediction of the wind wave spectrum of the ocean using numerical models are an important tool for researchers, engineers, and communities living in coastal areas. The governing equation of the wind wave models, the Wave Action Balance Equation, presents unique challenges for implementing reliable numerical models because it is highly advective, highly nonlinear and high dimensional. Historically, most operational models have utilized finite difference methods, others have used finite volume methods but relatively few attempts at using finite element methods. In this work, we seek to fill this gap by investigating several different finite element discretizations of the Wave Action Balance Equation. The methods, which include streamline upwind Petrov–Galerkin (SUPG), least squares, and discontinuous Galerkin, are implemented and convergence properties are examined for some simplified 2-D test cases. Then, a new spectral wind wave model, WAVEx, is formulated and implemented for the full problem setting. WAVEx uses continuous finite elements along with SUPG stabilization in geographic/spectral space that allows for fully unstructured triangular meshes in both geographic and spectral space. For propagation in time, a second order fully implicit finite difference method is used. When source terms are active, a second order operator splitting scheme is used to linearize the problem. In the splitting scheme, propagation is solved using the implicit method and the nonlinear source terms are treated explicitly. Several test cases, including analytic tests and laboratory experiments, are demonstrated and results are compared to analytic solutions, observations, as well as output from another model that is used operationally.

Positivity-Preserving Discontinuous Galerkin Methods on Triangular Meshes for Macroscopic Pedestrian Flow Models

The macroscopic models for solving the pedestrian flow problem can be generally classified into two categories as follows: first-order continuum models and high-order continuum models. In first-order continuum models, the density satisfies the mass conservation law, the speed is defined by a flow-density relationship, and the desired directional motion of pedestrians is determined by an Eikonal-type equation. In contrast, in high-order models, the velocity is governed by the momentum conservation law. In this study, we summarize existing first-order and high-order models and rewrite them in the form of unified scalar or system hyperbolic conservation laws. Next, we apply high-order discontinuous Galerkin methods with a positivity-preserving limiter on unstructured triangular meshes to solve the conservation law and a second-order fast-sweeping scheme to solve the Eikonal equations. Our method can efficiently model real-life complex computational regions and avoid nonphysical solutions and simulation blow-ups. Finally, numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed solution algorithm. The numerical results validate the reliability of the proposed numerical method and highlight the advantages of triangular meshes.

Coupling finite elements of class C1 on composite curved meshes for second order elliptic problems

SummaryFinite elements of class are suitable for the computation of magnetohydrodynamics instabilities in tokamak plasmas. In addition, isoparametric approximations allow for a precise alignment of the mesh with the magnetic field line. Mesh alignment is crucial to achieve axisymmetric equilibria accurately. It is also helpful to deal with the anisotropy nature of magnetized plasma flows. In this numerical framework, several practical simulations are now available. They help to understand better the operation of existing devices and predict the optimal strategies for using the international ITER tokamak under construction. However, a mesh‐aligned isoparametric representation suffers from the presence of critical points of the magnetic field (magnetic axis, X‐point). We here explore a strategy that combines aligned mesh out of the critical points with non‐aligned unstructured mesh in a region containing these points. By this strategy, we can avoid highly stretched elements and the numerical difficulties that come with them. The mesh‐aligned interpolation uses bi‐cubic Hemite‐Bézier polynomials on a structured mesh of curved quadrangular elements. On the other hand, we assume reduced cubic Hsieh‐Clough‐Tocher finite elements on an unstructured triangular mesh. Both meshes overlap, and the resulting formulation is a coupled discrete problem solved iteratively by a suitable one‐level Schwarz algorithm. In this paper, we will focus on the Poisson problem on a two‐dimensional bounded regular domain. This elliptic equation is a simplified version of the axisymmetric tokamak equilibrium one at the asymptotic limit of infinite major radius (large aspect ratio).

Accuracy verification of fast adaptive galerkin finite volume solver for thermal and mechanical load crack propagation in triangular meshes

This study presents the details of the Galerkin finite volume method for triangular unstructured meshes and its application for the fracture analysis of 2D elasticity problems under simultaneous thermal and mechanical loading. The heat diffusion and elasticity equations are discretized using the Galerkin finite volume method (which, due to omitting matrix manipulation calculations, computations are considerably faster than FEM and XFEM Solvers). The maximum tangential stress criterion is presented after briefly describing the interaction integral formulation used to calculate the stress intensity factors in thermo-mechanical problems. The accuracy of the computed results of the present strategy under thermo-mechanical loads is demonstrated using some benchmark test cases. First, a thermal stress analysis of a plate with an inclined central crack under thermal boundary conditions is performed. Second, the development of a crack under mechanical stress boundary conditions is modeled. Third, crack propagation under both thermal and mechanical boundary conditions is simulated. The present modeling strategy's results are compared with those reported in previous numerical works to verify the accuracy. The stress intensity factors and predicted crack trajectories are utilized to assess the accuracy of computed results and investigate the quality of crack simulation by the proposed numerical method.

A Robust Adaptive Mesh Generation Algorithm: A Solution for Simulating 2D Crack Growth Problems.

This paper introduces a robust algorithm that efficiently generates high-quality unstructured triangular meshes to model complex two-dimensional crack growth problems within the framework of linear elastic fracture mechanics (LEFM). The proposed Visual Fortran code aims to address key challenges in mesh generation including geometric complexity, required simulation accuracy, and computational resource constraints. The algorithm incorporates adaptive refinement and updates to the mesh structure near the crack tip, resulting in the formation of rosette elements that provide accurate approximations of stress intensity factors (SIFs). By utilizing the maximum circumferential stress theory, the algorithm predicts the new crack path based on these SIFs. Throughout the simulation of crack propagation, a node splitting approach was employed to represent the progression of the crack, while the crack growth path is determined by successive linear extensions for each crack growth increment. To compute stress intensity factors (SIFs) for each increment of crack extension, a displacement extrapolation method was used. The experimental and numerical results demonstrated the algorithm's effectiveness in accurately predicting crack growth and facilitating reliable stress analysis for complex crack growth problems in two dimensions. The obtained results for the SIF were found to be consistent with other analytical solutions for standard geometries.

A new mixed node-based solid-like finite element method (MNS-FEM) for laminated shell structures

A new solid-shell finite element formulation is investigated in this paper. Departing from the S-FEM concept of node-based stress smoothing scheme, a new pattern that introduces new stress variables linked with an extra smoothing region inside the element is proposed. This finding preserves all the advantageous features of the standard NS-FEM but it makes approximate solutions much more accurate. The assumed natural strain (ANS) and the Discrete shear gap (DSG) techniques are employed to alleviate trapezoidal and shear locking. The simple assumptions made allow the analytical computation of the discrete operators based on the node-based backgrounds making the model particularly efficient. Popular benchmark tests are successfully solved for structured and unstructured triangular meshes in linear and buckling analysis. Numerical results show that the proposed method avoids cross-diagonal meshes, which reduces the generality of three-node degenerated shell elements proposed in the literature. This aspect makes the present formulation much more appealing than the published methods available in the literature. Moreover, in the context of the S-FEMs it is the first time in which geometric nonlinear contributions are presented.

A Well-Balanced FVC Scheme for 2D Shallow Water Flows on Unstructured Triangular Meshes

We consider in this work the numerical resolution of a 2D shallow water system with a Coriolis effect and bottom friction stresses on unstructured meshes by a new Finite Volume Characteristics (FVC) scheme, which has been introduced in the preliminary works that will be cited below. Our main goal is to extend this approach to 2D unstructured formalism while preserving the physical and mathematical properties of the system, including the C-property. First, we present our extension by preserving the advantages of the finite volume discretization such as conservation property and the method of characteristics such as elimination of Riemann solvers. Afterward, an approach was applied to the topography source term that leads to a well-balanced scheme satisfying the steady-state condition of still water. A semi-implicit treatment will also be presented in this study to avoid stability problems for the other source terms. Finally, the proposed finite volume method is verified on several benchmark tests and shows good agreement with analytical solutions and experimental results; moreover, it gives a noticeable accuracy and rapidity improvement compared to the original approaches.

Deep learning model for two-fluid flows

Various industries rely on numerical tools to simulate multiphase flows due to the wide occurrence of this phenomenon in nature, manufacturing processes, or the human body. However, the significant computation burden required for such simulations directs the research interest toward incorporating data-based approaches in the solution loop. Although these approaches returned significant results in various domains, incorporating them in the computational fluid dynamics (CFD) field is wrangled by their casting aside of the already known governing constitutional laws along with the natural incompatibility of various models with unstructured irregular discretization spaces. This work suggests a coupling framework, between a traditional finite element CFD solver and a deep learning model, for tackling multiphase fluid flows without migrating the benefits of physics-enriched traditional solvers. The tailored model architecture, along with the coupling framework, allows tackling the required problem with a dynamically adapted unstructured irregular triangular mesh, thus dodging the limitation of traditional convolution neural networks. Moreover, the various ingredients that allowed the model to simulate the complex and computation-demanding Navier–Stokes flow equation, such as relying on a sequential validation dataset while exposing the model training to a noise inherited from the quality of its inferring, along with the proper choice of model inputs, are highlighted and elaborated throughout this paper. To the authors' knowledge, this work is the first of its type to introduce a data-based graph-based approach for solving multiphase flow problems with a level-set interface capturing method.

Floating Point Genetic Algorithm Approach for Optimizing Unstructured Meshes

This paper presents a method for optimizing unstructured triangular meshes using a floating-point genetic algorithm. A mesh generation algorithm based on a modified advancing front method and sets of heuristic rules are used to generate the initial non-smooth triangular meshes for complex shapes. The developed mesh is then smoothed using a floating-point genetic algorithm that is more flexible than the usual binary genetic algorithms, and can handle non-smooth regions containing several local extrema. Three approaches are used in selecting the fitness function to be optimized in the genetic algorithm, namely, the triangle aspect ratio, the maximum angle at each node of the triangular mesh, and a weighted linear combination of both functions. The genetic algorithm has been tested and validated for a number of test cases covering a wide range of complex geometry applications. The results have shown a high degree of improvement in the quality of the smoothed meshes and an ability to handle non-convex regions.

Scale Analysis on Unstructured Grids: Kinetic Energy and Dissipation Power Spectra on Triangular Meshes

AbstractFourier spectra are powerful tools to analyze the scale behavior of turbulent flows. While such spectra are mathematically based on regular periodic data, some state‐of‐the‐art ocean and climate models use unstructured triangular meshes. Observational data is often also available only in an unstructured fashion. In this study, scale analysis specifically for the output of models with triangular meshes is discussed and the representable wavenumbers for Fourier analysis are derived. Aside from using different interpolation methods and oversampling prior to the computation of Fourier spectra, we also consider an alternative scale analysis based on the Walsh–Rademacher basis, that is, indicator functions. It does not require interpolation and can be extended to general unstructured meshes. A third approach based on smoothing filters which focus on grid scales is also discussed. We compare these methods in the context of kinetic energy and dissipation power of a turbulent channel flow simulated with the sea ice‐ocean model FESOM2. One simulation uses a classical viscous closure, another a new backscatter closure. The latter is dissipative on small scales, but anti‐dissipative on large scales leading to more realistic flow representation. All three methods clearly highlight the differences between the simulations as concerns the distribution of dissipation power and kinetic energy over scales. However, the analysis based on Fourier transformation is highly sensitive to the interpolation method in case of dissipation power, potentially leading to inaccurate representations of dissipation at different scales. This highlights the necessity to be cautious when choosing a scale analysis method on unstructured grids.

An efficient hybrid mesh method for seismic traveltime computation with irregular surfaces

Accurate and efficient traveltime computation with irregular surfaces is a prerequisite for traveltime tomography, seismic location, and seismic exploration in mountainous areas. The eikonal equation method is computationally favourable and has gained increasing popularity for obtaining traveltime in recent years. Nowadays most algorithms that can solve eikonal equation with irregular surfaces are based on fully unstructured triangular mesh or surface-fitting mesh. However, these methods have poor performance in terms of computational cost and accuracy. To cope with this problem, we develop a hybrid mesh method based on fast-sweeping-based eikonal solver with rectangular and unstructured triangular mesh for obtaining traveltime with irregular surfaces. In other words, we propose to divide the model into several different regions, which are discretized by rectangular mesh and unstructured triangular mesh respectively. Near the irregular surfaces of model, we use fast sweeping method based on unstructured triangular mesh which releases the full potential of unstructured triangular mesh’s flexibility in accurately describing the irregular geometry of computation domain. For the rest part of the model, we adopt rectangular mesh to make full use of high efficiency of the corresponding high-order fast sweeping algorithm. At the interface of two types of mesh, we need interpolate the known nodes’s traveltimes to get the boundary condition of another mesh, so as to start the fast sweeping method therein. It is found that the accuracy of hybrid mesh method can be increased by 83%, the computation time is reduced by 72%, and the storage requirement is reduced by 72%, compared with the fast sweeping method with full triangular mesh, when the rectangular mesh proportion is 80% in our model. Therefore, this proposed hybrid mesh method can greatly enhance the computational efficacy and serve as an potential forward engine in seismic inversions.

A graph-based algorithm for slicing unstructured mesh files

• This paper describes an efficient algorithm for slicing files with unstructured mesh. • A novel graph-based algorithm was described and implemented for polygon assembly and contour generation. • Up to 10 times speedups were gained by using the new serial algorithm. • A multicore parallel algorithm was implemented to further speedup the slicing of files with a large number of triangles. Process planning is an important step in additive manufacturing processes that involves the creation of a deposition toolpath from input geometry. The most commonly used input geometry file formats are in the form of unstructured triangular meshes. Previous research works have presented different algorithms for slicing an unstructured triangular mesh by a series of uniformly spaced parallel planes. Recent research works show that the use of hashtable data structures results in an optimal algorithm that improves the efficiency of the slicing process. The presented algorithm minimizes the use of hashtable in the slicing process by using graph-based data structures. It was found that the slicing can be sped up by about 10 times by using the graph-based algorithm compared to the previous algorithm. The use of parallel processing on multi-core processors was explored to further speed up the slicing process. The proposed efficient algorithm can be useful in process planning optimization.

A positivity-preserving RD-FV scheme for diffusion problems on triangular meshes

In this paper, we propose a positivity-preserving residual distribution based finite volume scheme (RD-FV) for steady diffusion problems on unstructured triangular meshes. In this scheme, we use the RD method to obtain the auxiliary values of vertex, then adopt the positivity-preserving finite volume scheme to update the cell-centered values for diffusion problems, in which a nonlinear two-point flux approximation is utilized to obtain nonnegative solution. Thus a positivity-preserving conservative property for the RD-FV scheme is guaranteed. Several numerical experiments show the optimal convergence rates for the numerical solution, that is, approximate second-order accuracy for the solution and first-order accuracy for the flux, and demonstrate the positivity-preserving property of our new scheme.

Urban Flood Prediction through GIS-Based Dual-Coupled Hydraulic Models

Propagation of pluvial floods in urban areas, occurring with return time periods of few years, can be well solved using dual models accounting for the mutual relationship between the water level in the streets and the discharges inside the sewer pipes. The extended WEC-flood model (EWEC), based on the use of unstructured triangular meshes and a diffusive formulation of the momentum equations in both the 2D and the 1D lower domains, is presented along with its novelty, limits, and advantages. The model is then applied to a small computational domain in the Palermo area, where only some ‘hard’ data given by one rain gauge has been used for calibration and validation, along with other ‘soft’ data like yes/no surcharge observations and water depths available from photos and interviews. Model input data are mainly geometrical parameters, and calibration parameters are restricted only to average Manning coefficients. In the test case a very good validation has been obtained of three historical events using the EWEC model, with only one average Manning coefficient calibrated using other two historical events.