Unstructured Tetrahedral Elements Research Articles

Angular-spatial discontinuous Galerkin method for radiative heat transfer with a participating medium in complex three-dimensional geometries

The angular-spatial discontinuous Galerkin method (ASDGM) is a numerical technique developed to solve the radiative transfer equation (RTE) in complex three-dimensional geometries with absorbing, emitting, and scattering media. The ASDGM discretizes both the angular and spatial domains of the RTE using structured quadrilateral elements for the angular domain and unstructured tetrahedral elements for the spatial domain. In the solution process, the Riemann upwind solver separates the input and output information at the tetrahedral element boundary. Moreover, the discontinuous Galerkin (DG) sweep algorithm optimizes the order of the unstructured tetrahedral elements and reduces memory usage caused by the coupling of the discretization in the angular and spatial domains. The performance of the ASDGM has been evaluated using several radiative transfer cases in complex three-dimensional geometries, which demonstrate its good accuracy compared to available data from the literature. Additionally, the computational time in the angular domain for high-order DG is lower than that of the finite element method for the same discrete angles. Furthermore, the DG sweep algorithm provides a faster convergence rate in the spatial domain compared to the original sweep algorithm. Therefore, the ASDGM with the DG sweep algorithm is a preferable method for solving radiative transfer problems in complex three-dimensional geometries.

Deep Electrical Structure of the Qilian Orogenic Belt with Dynamic Implications for the Northeastern Margin of the Tibetan Plateau: Revealed by 3D Magnetotelluric Inversion Using Unstructured Tetrahedral Elements

Abstract We present the results of a magnetotelluric (MT) array across the Qilian orogenic belt to elucidate the uplift mechanism of the northeastern Tibetan Plateau (Qinghai-Tibet Plateau). The array extends from the Qaidam basin in the south to the southern edge of the Alxa block in the north. Using the three-dimensional (3D) inversion of MT data based on unstructured tetrahedral elements, the electrical structure 100 km below the orogenic belt is obtained. The results show that there are high-resistivity bodies in the lithospheric mantle of the North Qilian and Hexi Corridor, which may represent the trace of southward subduction of the Asian lithosphere. Besides, there are partially molten bodies with low resistivity in the middle and lower crust below the Qilian orogenic belt, which may be caused by tectonic heat. The melt fraction of low-resistivity bodies is 2-5%, which indicates that the crustal flow from the Qiangtang and Songpan-Ganzi blocks is unable to penetrate beneath the Qilian orogenic belt. The low-resistivity bodies beneath the Qilian orogenic belt decouples the upper crust from the middle-lower crust. Owing to the continuous compression, the decoupled middle-lower crust has subsequently driven the northward movement of the upper crust, resulting in the uplift of the Qilian orogenic belt.

Stress-strain distribution in intact L4-L5 vertebrae under the influence of physiological movements: A finite element (FE) investigation

This study is aimed at finding the stress and strain distribution in functional spinal unit of L4-L5 occurring due to physiological body movements under five loading conditions, namely compression, flexion, extension, lateral bending and torsion. To this purpose, 3D finite element (FE) model has been generated using 4-noded unstructured tetrahedral elements considered both for bones and intervertebral disc, and 1D tension-only spring elements for ligaments. The analyses were performed for a compression load of 500 N and for other load cases, a moment of 10 N-m along with a preload of 500 N was applied. The model was validated against in-vitro experimental data obtained from literature and FE analysis data for a range of motion (RoM) corresponding to various loading conditions. The highest stress was predicted in the case of torsion though the angular deformation was highest in case of flexion.

Three-dimensional modeling of non-planar fatigue crack growth in spur gear tooth using tetrahedral finite elements

In this paper, a numerical methodology for prediction of crack growth path and life of structures containing cracks and its application to three different case studies on spur gear tooth are presented. Three-dimensional non-planar fatigue crack propagation analyses are carried out using FRAC3D, which is part of Fracture and Crack Propagation Analysis System (FCPAS), employing fully unstructured tetrahedral elements along the crack front and in the whole model. Fatigue crack growth path predictions are performed using recently developed crack growth criteria that combine different equations used to predict crack deflection angle and to determine equivalent stress intensity factor. Comparisons of obtained results from numerical simulations with available data from the literature in terms of crack growth trajectories show very close agreement. Therefore, it is concluded that FCPAS is capable of performing three-dimensional non-planar crack growth modeling using fully tetrahedral elements in the whole finite element model, which requires less user intervention, time and effort for modeling. Moreover, the presented methodology and the recently developed crack growth criteria can be used for three-dimensional crack growth analysis.

MATLAB PDE toolbox for 2-D and 3-D neutron diffusion simulations

This research extends MATLAB PDE Toolbox to model nuclear reactors by solving the 2-D and 3-D multigroup neutron diffusion equations using the finite element method. The developed scripts can simulate reactors with rectangular and hexagonal lattices and have the potential to handle complex core configurations, benefitting from the constructive solid geometry. The meshing subroutine decomposes the domain into unstructured triangular or tetrahedral elements that are either linear or quadratic. With many versatile MATLAB functions, data analysis and visualization can be accomplished within the same software environment. The proposed toolbox provides a graphical user interface for novices and simple models and a programmatic workflow for experienced users and complex problems. Multiple benchmark tests show that the developed framework produces accurate results with acceptable computation speed. The developed MATLAB reactor simulation engine is license friendly and readily accessible by researchers and students, making it an ideal research and educational tool.

Radiused Edge Blade Tip for a Wider Operating Range in Wells Turbine

The narrow operating range of a Wells turbine restricts its energy extraction capability from the ocean waves. In this work, the concept of a radiused edge tip blade (RETB) was introduced to overcome such an issue. The RET modifies the tip and changes the tip leakage flow behaviour. The flow through the turbine annulus was simulated numerically and compared with the existing experimental results of the reference turbine. Three-dimensional Reynolds-averaged Navier–Stokes equations with a two-equation turbulence closure model available in ANSYS CFX 14.5 was used for the simulations. The computational domain was discretized with unstructured tetrahedral elements, and the grid independence study gave an optimal grid. The RETB altered the tip leakage flow characteristics and delayed the stall inception. The RETB enhanced the relative operating range by 25% and peak torque by 37%.

Matrix-Free Method for Transient Maxwell-Thermal Cosimulation in Arbitrary Unstructured Meshes

Existing electrical–thermal cosimulation methods require solving a system matrix when handling inhomogeneous materials and irregular geometries discretized into unstructured meshes. In this paper, a matrix-free method is developed for cosimulating full-wave Maxwell’s equations and the thermal diffusion equation in the time domain. The method is free of matrix solutions regardless of the shape of the element used for space discretization. A theoretical stability analysis is also developed for the coupled electrical and thermal analysis, which is nonlinear in nature. Numerical experiments on both unstructured tetrahedral and triangular prism element meshes have validated the accuracy and efficiency of the proposed method.

An edge-based finite element method for 3D marine controlled-source electromagnetic forward modeling with a new type of second-order tetrahedral edge element

This paper presents a second order edge-based finite element method based on a new type of second order edge element for numerical modeling of 3D controlled source electromagnetic data in an anisotropic conductive medium. We adopt the anomalous potential field formulation to avoid the effect of sou rce’s singularity. The computational domain was subdivided into unstructured tetrahedral elements. The sparse linear system was solved using the Induced dimensional reduction (IDR(s)) method with an incomplete LU preconditioner. We have applied the developed algorithm to compute a typical MCSEM response over several 3D models, we compare the numerical results from our method and FEM solution based on conventional first and second order element. The results of comparison shown that our algorithm can significantly reduce the computation cost without loss to much accuracy.

Shutdown dose rate analysis with unstructured tetrahedral element based R2S method using deterministic transport solver AETIUS

We have been developing an in-house deterministic neutron and gamma transport solver, AETIUS, which uses an unstructured tetrahedral mesh for spatial discretization such as Attila. We make the most of the spatial discretization of AETIUS for a shutdown dose rate analysis.This is similar to the MCR2S unstructured mesh method with the exception that the deterministic solver AETIUS is used for the neutron and gamma transport calculations instead of MCNP(X). AETIUS provides neutron spectra for every unstructured tetrahedral element belonging to the activation zones. Activation calculations are then carried out using FISPACT-2007, and the resulting gamma spectra of each element are used as decay gamma sources for the AETIUS gamma transport calculation to obtain a shutdown dose map.We calculated the shutdown dose rate on the ITER port plug computational benchmark, and our results are the most similar to those of the FDS R2S method.

3D electrical conductivity tomography of volcanoes

Electrical conductivity tomography is a well-established galvanometric method for imaging the subsurface electrical conductivity distribution. We characterize the conductivity distribution of a set of volcanic structures that are different in terms of activity and morphology. For that purpose, we developed a large-scale inversion code named ECT-3D aimed at handling complex topographical effects like those encountered in volcanic areas. In addition, ECT-3D offers the possibility of using as input data the two components of the electrical field recorded at independent stations. Without prior information, a Gauss-Newton method with roughness constraints is used to solve the inverse problem. The roughening operator used to impose constraints is computed on unstructured tetrahedral elements to map complex geometries. We first benchmark ECT-3D on two synthetic tests. A first test using the topography of Mt. St Helens volcano (Washington, USA) demonstrates that we can successfully reconstruct the electrical conductivity field of an edifice marked by a strong topography and strong variations in the resistivity distribution. A second case study is used to demonstrate the versatility of the code in using the two components of the electrical field recorded on independent stations along the ground surface. Then, we apply our code to real data sets recorded at (i) a thermally active area of Yellowstone caldera (Wyoming, USA), (ii) a monogenetic dome on Furnas volcano (the Azores, Portugal), and (iii) the upper portion of the caldera of Kīlauea (Hawai'i, USA). The tomographies reveal some of the major structures of these volcanoes as well as identifying alteration associated with high surface conductivities. We also review the petrophysics underlying the interpretation of the electrical conductivity of fresh and altered volcanic rocks and molten rocks to show that electrical conductivity tomography cannot be used as a stand-alone technique due to the non-uniqueness in interpreting electrical conductivity tomograms. That said, new experimental data provide evidence regarding the strong role of alteration in the vicinity of preferential fluid flow paths including magmatic conduits and hydrothermal vents.

Adaptive finite element modelling of three-dimensional magnetotelluric fields in general anisotropic media

We present a goal-oriented adaptive finite element (FE) modelling algorithm for 3-D magnetotelluric fields in generally anisotropic conductivity media. The model consists of a background layered structure, containing anisotropic blocks. Each block and layer might be anisotropic by assigning to them 3×3 conductivity tensors. The second-order partial differential equations are solved using the adaptive finite element method (FEM). The computational domain is subdivided into unstructured tetrahedral elements, which allow for complex geometries including bathymetry and dipping interfaces. The grid refinement process is guided by a global posteriori error estimator and is performed iteratively.The system of linear FE equations for electric field E is solved with a direct solver MUMPS. Then the magnetic field H can be found, in which the required derivatives are computed numerically using cubic spline interpolation. The 3-D FE algorithm has been validated by comparisons with both the 3-D finite-difference solution and 2-D FE results. Two model types are used to demonstrate the effects of anisotropy upon 3-D magnetotelluric responses: horizontal and dipping anisotropy. Finally, a 3D sea hill model is modelled to study the effect of oblique interfaces and the dipping anisotropy.

The computational fluid dynamic modeling of downwash flow field for a six-rotor UAV

The downwash flow field of the multi-rotor unmanned aerial vehicle (UAV), formed by propellers during operation, has a significant influence on the deposition, drift and distribution of droplets as well as the spray width of the UAV for plant protection. To study the general characteristics of the distribution of the downwash airflow and simulate the static wind field of multi-rotor UAVs in hovering state, a 3D full-size physical model of JF01-10 six-rotor plant protection UAV was constructed using SolidWorks. The entire flow field surrounding the UAV and the rotation flow fields around the six rotors were established in UG software. The physical model and flow fields were meshed using unstructured tetrahedral elements in ANSYS software. Finally, the downwash flow field of UAV was simulated. With an increased hovering height, the ground effect was reduced and the minimum current velocity increased initially and then decreased. In addition, the spatial proportion of the turbulence occupied decreased. Furthermore, the appropriate operational hovering height for the JF01-10 is considered to be 3 m. These results can be applied to six-rotor plant protection UAVs employed in pesticide spraying and spray width detection.

A Dynamic $p$ -Adaptive DGTD Algorithm for Electromagnetic and Multiphysics Simulations

In the time-domain simulation of electromagnetic and multiphysics problems, the distributions of physical quantities of interest vary in both space and time. To achieve a good spatial resolution, high-order basis functions can be used to expand the unknown quantities, which is known as the $p$ -refinement. However, a global and static $p$ -refinement will increase the computational cost significantly. In this paper, a dynamic $p$ -adaptation algorithm is proposed based on the discontinuous Galerkin time-domain method, which is able to determine and adjust the basis order in a given discretization element in real time of the simulation. Based on the relation between the nodal and modal approximations defined on unstructured tetrahedral elements, an error estimator, which is very cheap to compute, is developed to determine the proper basis order to achieve a desired numerical accuracy. The dynamic $p$ -adaptation algorithm proposed in this paper is able to capture the fast varying physics by changing the order of basis functions wherever and whenever needed. Several numerical examples adopted from multiple physical disciplines are presented to demonstrate the accuracy, efficiency, and flexibility of the proposed algorithm in the simulation of electromagnetic and multiphysics problems.

Three-dimensional resistivity structure of Asama Volcano revealed by data-space magnetotelluric inversion using unstructured tetrahedral elements

Three-dimensional resistivity structure of Asama Volcano revealed by data-space magnetotelluric inversion using unstructured tetrahedral elements

Two-Dimensional Evaluation of ATHAM-Fluidity, a Nonhydrostatic Atmospheric Model Using Mixed Continuous/Discontinuous Finite Elements and Anisotropic Grid Optimization

AbstractThis paper presents the first attempt to apply the compressible nonhydrostatic Active Tracer High-Resolution Atmospheric Model–Fluidity (ATHAM-Fluidity) solver to a series of idealized atmospheric test cases. ATHAM-Fluidity uses a hybrid finite-element discretization where pressure is solved on a continuous second-order grid while momentum and scalars are computed on a first-order discontinuous grid (also known as ). ATHAM-Fluidity operates on two- and three-dimensional unstructured meshes, using triangular or tetrahedral elements, respectively, with the possibility to employ an anisotropic mesh optimization algorithm for automatic grid refinement and coarsening during run time. The solver is evaluated using two-dimensional-only dry idealized test cases covering a wide range of atmospheric applications. The first three cases, representative of atmospheric convection, reveal the ability of ATHAM-Fluidity to accurately simulate the evolution of large-scale flow features in neutral atmospheres at rest. Grid convergence without adaptivity as well as the performances of the Hermite–Weighted Essentially Nonoscillatory (Hermite-WENO) slope limiter are discussed. These cases are also used to test the grid optimization algorithm implemented in ATHAM-Fluidity. Adaptivity can result in up to a sixfold decrease in computational time and a fivefold decrease in total element number for the same finest resolution. However, substantial discrepancies are found between the uniform and adapted grid results, thus suggesting the necessity to improve the reliability of the approach. In the last three cases, corresponding to atmospheric gravity waves with and without orography, the model ability to capture the amplitude and propagation of weak stationary waves is demonstrated. This work constitutes the first step toward the development of a new comprehensive limited area atmospheric model.

A goal-oriented adaptive finite element method for 3D resistivity modeling using dual-error weighting approach

A goal-oriented adaptive finite element (FE) method for solving 3D direct current (DC) resistivity modeling problem is presented. The model domain is subdivided into unstructured tetrahedral elements that allow for efficient local mesh refinement and flexible description of complex models. The elements that affect the solution at each receiver location are adaptively refined according to a goal-oriented posteriori error estimator using dual-error weighting approach. The FE method with adapting mesh can easily handle such structures at almost any level of complexity. The method is demonstrated on two synthetic resistivity models with analytical solutions and available results from integral equation method, so the errors can be quantified. The applicability of the numerical method is illustrated on a resistivity model with a topographic ridge. Numerical examples show that this method is flexible and accurate for geometrically complex situations.

Method of adaptive artificial viscosity for solving the Navier–Stokes equations

A numerical technique based on the method of adaptive artificial viscosity is proposed for solving the viscous compressible Navier–Stokes equations in two dimensions. The Navier–Stokes equations is approximated on unstructured meshes, namely, on triangular or tetrahedral elements. The monotonicity of the difference scheme according to the Friedrichs criterion is achieved by adding terms with adaptive artificial viscosity to the scheme. The adaptive artificial viscosity is determined by satisfying the maximum principle conditions. An external flow around a cylinder at various Reynolds numbers is computed as a numerical experiment.

Influence of Mesh Density on Airflow and Particle Deposition in Sinonasal Airway Modeling

There are methodological ambiguities in the literature on mesh refinement analysis for computational fluid dynamics (CFD) modeling of physiologically realistic airflow dynamics and particle transport in the human sinonasal cavity. To investigate grid independence in discretization of the (sino)nasal geometry, researchers have considered CFD variables such as pressure drop, velocity profile, wall shear, airflow, and particle deposition fractions. Standardization in nasal geometry is also lacking: unilateral or bilateral nasal cavities with and without paranasal sinuses have been used. These methodological variants have led to inconsistencies in establishing grid-independent mesh densities. The aim of this study is to provide important insight in the role of mesh refinement analysis on airflow and particle deposition in sinonasal airway modeling. A three-dimensional reconstruction of the complete sinonasal cavity was created from computed tomography images of a subject who had functional endoscopic sinus surgery. To investigate airflow grid independence, nine different tetrahedral mesh densities were generated. For particle transport mesh refinement analysis, hybrid tetrahedral-prism elements with near-wall prisms ranging from 1 to 6 layers were implemented. Steady-state, laminar inspiratory airflow simulations under physiologic pressure-driven conditions and nebulized particle transport simulations were performed with particle sizes ranging from 1-20 μm. Mesh independence for sinonasal airflow was achieved with approximately 4 million unstructured tetrahedral elements. The hybrid mesh containing 4 million tetrahedral cells with three prism layers demonstrated asymptotic behavior for sinonasal particle deposition. Inclusion of boundary prism layers reduced deposition fractions relative to tetrahedral-only meshes. To ensure numerically accurate simulation results, mesh refinement analyses should be performed for both airflow and particle transport simulations. Tetrahedral-only meshes overpredict particle deposition and are less accurate than hybrid tetrahedral-prism meshes.

3-D inversion of magnetotelluric data using unstructured tetrahedral elements: applicability to data affected by topography

3-D inversion of magnetotelluric data using unstructured tetrahedral elements: applicability to data affected by topography

Recent developments in high‐performance computing and simulation: distributed systems, architectures, algorithms, and applications

Recent developments in high‐performance computing and simulation: distributed systems, architectures, algorithms, and applications