Mesh Of Triangular Elements Research Articles

A Calculation Method for the Hyperspectral Imaging of Targets Utilizing a Ray-Tracing Algorithm

This paper proposes a hyperspectral imaging simulation method based on a ray-tracing algorithm. The algorithm combines calculations based on solar and atmospheric visible light radiation as well as the spectral bidirectional reflection distribution function (BRDF) of the target surface material and can create its own scenarios for simulation calculations on demand. Considering the presence of multiple scattering between the target and background, using the ray-tracing algorithm enables the precise computation of results involving multiple scattering. To validate the accuracy of the algorithm, we compared the simulated results with the theoretical values of the visible light scattering intensity from a Lambertian sphere. The relative error obtained was 0.8%. Subsequently, a complex scene of engineering vehicles and grass was established. The results of different observation angles and different coating materials were calculated and analyzed. In summary, the algorithm presented in this paper has the following advantages. Firstly, it is applicable to geometric models composed of any triangular mesh elements and accurately computes the effects of multiple scattering. Secondly, the algorithm combines the spectral BRDF information of materials and improves the efficiency of multiple scattering calculations using nonuniform sampling. The computed hyperspectral scattering data can be applied to simulate airborne or space-borne remote sensing data.

A Comparative Study of Finite Element Method and Hybrid Finite Element Method–Spectral Element Method Approaches Applied to Medium-Frequency Transformers with Foil Windings

This study aims to improve the computational efficiency of the frequency domain analysis of medium-frequency transformers (MFTs) with the presence of large clearance distances and fine foil windings. The winding loss and magnetic energy in MFTs in the medium-frequency range are calculated utilizing a finite element method (FEM) using common triangular and alternative rectilinear mesh elements. Additionally, in order to improve the computational efficiency of the calculations, a spectral element method (SEM) is coupled with a FEM, thus creating a hybrid FEM–SEM formulation. In such a hybrid approach, the FEM is used to calculate the current density distribution in the two-dimensional (2D) cross-section of the foil conductors to achieve reliable accuracy, and the SEM is adopted in the nonconducting clearance distances of the winding window to reduce the system of equations. The comparative analysis of the calculated resistance and reactance of the under-study models showed that the FEM with rectilinear mesh elements and the FEM–SEM model outperformed the FEM with triangular mesh elements in terms of accuracy and computational cost. The hybrid FEM–SEM model enables a reduced system of equations for modeling the electromagnetic behavior of MFTs. This research provides valuable insights into both the computational approaches and meshing challenges in the analysis of MFTs and offers a foundation for future research on the design and optimization of MFTs.

Entropy Production Analysis in an Octagonal Cavity with an Inner Cold Cylinder: A Thermodynamic Aspect

Understanding fluid dynamics and heat transfer is crucial for designing and improving various engineering systems. This study examines the heat transfer characteristics of a buoyancy-driven natural convection flow that is laminar and incompressible. The investigation also considers entropy generation (Egen) within an octagonal cavity subject to a cold cylinder inside the cavity. The dimensionless version of the governing equations and their corresponding boundary conditions have been solved numerically using the finite element method, employing triangular mesh elements for discretization. The findings indicated that incorporating a cold cylinder inside the octagonal cavity resulted in a higher heat transfer (HT) rate than in the absence of a cold cylinder. Furthermore, using the heat flux condition led to a higher average Nusselt number (Nuavg) and a lower Bejan number (Be) than the isothermal boundary condition. The results also showed that HT and Egen were more significant in the Al2O3-H2O nanofluid than the basic fluids such as air and water, and HT increased as χ increased. The current research demonstrates that employing the heat flux condition and incorporating nanoparticles can enhance the rate of HT and Egen. Furthermore, the thermo-fluid system should be operated at low Ra to achieve greater HT effectiveness for nanofluid concerns.

Application of immersed boundary method to the simulation of three-dimensional flow in solid rocket motors

The immersed boundary method (IBM) is currently utilized in the simulation of two-dimensional axisymmetric flow in solid rocket motors. In this paper, the IBM is applied to three-dimensional flow fields, keeping the grain surface fixed. Based on the Cartesian grid, a three-dimensional Euler flow solver is developed using the finite difference method. All boundaries of the flow field are processed using the IBM, including the slip walls, mass flow inlet, pressure outlet, and rotational periodic boundary. Specific implementation of these boundary conditions and the mesh generation process are described. Using the ray-casting approach and the alternating digital tree data structure, an efficient method is proposed to determine the intersection relationship between a rectangular volume grid cell and a triangular surface mesh element. The numerical results of Taylor–Culick flow verify that the developed solver has more than one-order accuracy in space. To conduct the validation of the established method, three typical grains are selected for flow simulations, namely, the perforated cylindrical grain with burning on both the ends and the inner surface, the end-slotted end-burning grain, and the finocyl grain, respectively. The simulated results are compared with those of the zero-dimensional interior ballistics, the two-dimensional axisymmetric IBM, and the body-fitted grid method, verifying the fidelity of the developed three-dimensional flow solver.

Energy dissipation by grain boundary replacement during grain growth

The changes in both the grain boundary area and grain boundary energy that occur during grain growth have been measured in polycrystalline Ni using high energy diffraction microscopy. In addition to the reduction of grain boundary area, the average grain boundary energy decreases as higher energy grain boundaries are replaced by lower energy grain boundaries. This energy dissipation mechanism influences grain boundary migration and might explain the absence of a correlation between grain boundary curvature and migration velocity.Classical studies of isotropic grain growth in polycrystals have been based on the idea that grain boundary (GB) migration is driven by the product of the GB energy and curvature. [1,2] While support for this foundational concept is certainly found in studies of bicrystals [3] and simulations [4], recent experimental evidence contradicts this idea. When measured curvatures and velocities were compared in Ni [5] and SrTiO3, [6] they were found to be uncorrelated. Similar observations were reported for α-Fe, [7] where the measured constant of proportionality between curvature and velocity did not behave in expected ways. The purpose of this letter is to present evidence that, during grain growth, the replacement of higher energy GBs by lower energy boundaries dissipates energy by a mechanism that is not related to curvature. This reduction of the average GB energy represents an additional driving force for grain growth.For a GB network comprised of N triangular mesh elements of area ai, the total free energy (F) is:

Validation of a robust method for quantification of three-dimensional growth of the thoracic aorta using deformable image registration.

PurposeAccurate assessment of thoracic aortic aneurysm (TAA) growth is important for appropriate clinical management. Maximal aortic diameter is the primary metric that is used to assess growth, but it suffers from substantial measurement variability. A recently proposed technique, termed vascular deformation mapping (VDM), is able to quantify three‐dimensional aortic growth using clinical computed tomography angiography (CTA) data using an approach based on deformable image registration (DIR). However, the accuracy and robustness of VDM remains undefined given the lack of ground truth from clinical CTA data, and, furthermore, the performance of VDM relative to standard manual diameter measurements is unknown.MethodsTo evaluate the performance of the VDM pipeline for quantifying aortic growth, we developed a novel and systematic evaluation process to generate 76 unique synthetic CTA growth phantoms (based on 10 unique cases) with variable degrees and locations of aortic wall deformation. Aortic deformation was quantified using two metrics: area ratio (AR), defined as the ratio of surface area in triangular mesh elements and the magnitude of deformation in the normal direction (DiN) relative to the aortic surface. Using these phantoms, we further investigated the effects on VDM's measurement accuracy resulting from factors that influence the quality of clinical CTA data such as respiratory translations, slice thickness, and image noise. Lastly, we compare the measurement error of VDM TAA growth assessments against two expert raters performing standard diameter measurements of synthetic phantom images.ResultsAcross our population of 76 synthetic growth phantoms, the median absolute error was 0.063 (IQR: 0.073–0.054) for AR and 0.181 mm (interquartile range [IQR]: 0.214–0.143 mm) for DiN. Median relative error was 1.4% for AR and 3.3% for DiN at the highest tested noise level (contrast‐to‐noise ratio [CNR] = 2.66). Error in VDM output increased with slice thickness, with the highest median relative error of 1.5% for AR and 4.1% for DiN at a slice thickness of 2.0 mm. Respiratory motion of the aorta resulted in maximal absolute error of 3% AR and 0.6 mm in DiN, but bulk translations in aortic position had a very small effect on measured AR and DiN values (relative errors <1%). VDM‐derived measurements of magnitude and location of maximal diameter change demonstrated significantly high accuracy and lower variability compared to two expert manual raters (p<0.03 across all comparisons).ConclusionsVDM yields an accurate, three‐dimensional assessment of aortic growth in TAA patients and is robust to factors such as image noise, respiration‐induced translations, and differences in patient position. Further, VDM significantly outperformed two expert manual raters in assessing the magnitude and location of aortic growth despite optimized experimental measurement conditions. These results support validation of the VDM technique for accurate quantification of aortic growth in patients and highlight several important advantages over diameter measurements.

An unstructured mesh algorithm for simulation of hydraulic fracture

The paper describes a flexible computational method that can be used to represent the complex geometric evolution of a fluid-driven fracture front using an unstructured triangular element mesh. A specific motivation for the method is to allow subsequent treatment of non-planar fracture growth problems. Elastic interactions between the crack opening displacements and the surrounding medium are calculated using displacement discontinuity boundary element influence functions. A novel feature of the approach is the use of an adjustable region of tip elements to accommodate both time-dependent crack edge movements and to ensure correct velocity-dependent asymptotic behavior for viscous and viscous-toughness flow conditions. The adjustable tip element region is represented using a special-purpose data structure that allows the iteration of the moving edge element vertices to be linked directly to the flow rate and the crack opening solution values that are determined in each time step. The fringe region is periodically replaced with a set of fixed vertex elements as the fracture surface evolves. The proposed scheme has been evaluated using available analytic and experimental results for planar fracture propagation. The method is found to be both accurate and robust for a range of choices of geometric fracture shapes and time step sizes.

A Novel Subdomain 2D/Q-2D Finite Element Method for Power/Ground Plate-Pair Analysis

Upon excitation by a surface magnetic current, a power/ground plate-pair supports only $\mathrm{TM}^{z}$ modes. This means that the magnetic field has only azimuthal components permitting a simple but effective domain decomposition method (DDM) to be used. In the proximity of an antipad, field interactions are rigorously modeled by a quasi-two-dimensional (Q-2D) finite element method (FEM) making use of three-dimensional (3D) triangular prism mesh elements. Since high-order $\mathrm{TM}^{z}$ modes are confined in the close proximity of the antipad, field interactions in the region away from the antipad only involve the fundamental mode and are rigorously modeled by a 2D FEM. This approach reduces 3D computation domain into a hybrid 2D/Q-2D domain. The discretization of this hybrid domain results in a global matrix system consisting of two globally coupled matrix equations pertinent to 2D and Q-2D domains. In this article, these two matrix equations are “decoupled” using a Riemann solver and the information exchange between the two domains is facilitated using numerical flux. The resulting decoupled two matrix equations are iteratively solved using the Gauss–Seidel algorithm. The accuracy, efficiency, and robustness of the proposed DDM are verified by four representative examples.

New approaches for Delaunay triangulation and optimisation

New techniques are presented for Delaunay triangular mesh generation and element optimisation. Sample points for triangulation are generated through mapping (a new approach). These sample points are later triangulated by the conventional Delaunay method. Resulting triangular elements are optimised by addition, removal and relocation of mapped sample points (element nodes). The proposed techniques (generation of sample points through mapping for Delaunay triangulation and mesh optimisation) are demonstrated by using Mathematica software. Simulation results show that the proposed techniques are able to form meshes that consist of triangular elements with aspect ratio of less than 2 and minimum skewness of more than 45°.

A new high-order spectral difference method for simulating viscous flows on unstructured grids with mixed-element meshes

In the present study, a spectral difference (SD) method is developed for viscous flows on meshes with a mixture of triangular and quadrilateral elements. The standard SD method for triangular elements, which employs Lagrangian interpolating functions for fluxes, is not stable when the designed accuracy of spatial discretization is third order or higher. Unlike the standard SD method, the method examined here uses vector interpolating functions in the Raviart-Thomas (RT) spaces to construct continuous flux functions on reference elements. The spectral-difference Raviart-Thomas (SDRT) method was originally proposed by Balan et al. [1] and implemented on triangular-element meshes for invisid flow only. Our present results demonstrated that the SDRT method is stable and high-order accurate in two dimensions (2D) for a number of test problems by using triangular-, quadrilateral-, and mixed-element meshes for both inviscid and viscous flows. A stability analysis is also performed for mixed elements. We found our current SDRT scheme is stable for fourth-order accurate spatial discretizations. However, a stable fifth-order SDRT scheme remains to be identified.

Biomechanical evaluation of pin placement of external fixator in treating tranverse tibia fracture: Analysis on first and second cortex of cortical bone

Biomechanical perspective of external fixator is one of the greatest factor to consider in successfully treating bone fracture. This is due to the fact that mechanical behavior of the structure can be analyzed and optimized in order to avoid mechanical failure, increase bone fracture healing rate and prevent pre-term screw loosening. There are three significant factors that affect the stability of external fixator which are the placement of pin at the bone, configuration and components of external fixator. These factors lead to one question: what is the optimum pin placement in which exerts optimum stability? To date, literature on above mentioned factors is limited. Therefore, we conducted a study to evaluate the uniplanar-unilateral external fixator for two different pin placement techniques in treating transverse tibia fracture via finite element method. The study was started off with the development of transverse tibia fracture using Mimics software. Computed tomography (CT) data image was utilized to develop three dimensional tibia bone followed by crafting fracture on the bone. Meanwhile, the external fixator was developed using SolidWork software. Both tibia bone and external fixator were meshed in 3-matic software with triangular mesh element. Simulation of this configuration was took place in a finite element software, Marc.Mentat software. A load of 400 N was applied to the proximal tibia bone in order to simulate stance phase of a gait cycle. From the findings, the pin placement at the second cortex of bone provided optimum stability in terms of stress distribution and displacement, which should be considered for better treatment for transverse tibia fracture. On the other hand, the pin placement at first cortex should be avoided to prevent many complications.

An Efficient Mode-Based Domain Decomposition Hybrid 2-D/Q-2D Finite-Element Time-Domain Method for Power/Ground Plate-Pair Analysis

Generally, with a surface magnetic current excitation, only the $\mathrm {TM}_{\mathrm {zmn}}$ modes are activated between the power/ground plate pair. As a result, the magnetic field only has azimuthal component while the longitude component is zero. To make full use of this quasi-2-D (Q-2D) property, a Q-2D finite-element time-domain (FETD) method combined with the basic 3-D triangular prism mesh elements is firstly proposed to solve the magnetic field wave equation. It is further noted that the higher order modes are confined in the proximity of the antipads (labeled as via domain) due to the exponential attenuation property along the propagating direction. Therefore, in the region sufficiently far from the antipad (marked as plate-pair domain), only the fundamental mode needs to be considered. In this way, a Q-2D FETD analysis is required in the via domain while in the plate-pair domain, only a 2-D FETD solver is needed. As a result, the proposed algorithm is actually a hybrid 2-D and Q-2D FETD method. Besides, to handle arbitrarily shaped antipads, the wave-port excitation is implemented, and the corresponding mode-based S-parameter extraction methodology is developed according to the orthogonal property of different eigenmodes. The efficiency, accuracy, and generality of the proposed approach are verified by several representative examples.

Time-Domain Method Having a Naturally Diagonal Mass Matrix Independent of Element Shape for General Electromagnetic Analysis—2-D Formulations

In this paper, we present a new time-domain method that has a naturally diagonal mass matrix and thereby a strict linear computational complexity per time step, regardless of whether the discretization is a structured grid or an unstructured mesh. This property is obtained independent of the element shape used for discretization. No interpolations, projections, and mass lumping are required. The accuracy and stability of the proposed method are both theoretically guaranteed. In addition, no dual mesh is needed and the tangential continuity of the fields is satisfied across the element interface. The flexible framework of the proposed method also allows for a straightforward extension to higher order accuracy in both electric and magnetic fields. Numerical experiments have been conducted on a variety of unstructured triangular-element meshes. Correlations with analytical solutions and the time-domain finite-element method have validated the accuracy and generality of the proposed new time-domain method.

Debris flow damage incurred to buildings: an in situ back analysis

In an attempt to conduct a back analysis of the damage caused to a 19th century masonry structure due to the October 2009 flash flood/debris flow event in Scaletta Zanclea, a flood discharge hydrograph is reconstructed in the ungauged conditions. The hydrograph for the solid discharge is then estimated by scaling up the liquid volume to the estimated debris volume. The debris flow diffusion is simulated by solving the differential equations for a single‐phase two‐dimensional (2D) flow that employs triangular mesh elements, also taking into account the channelling of the flow through the buildings. The damage to the building is modelled using 2D finite shell elements and is based on the maximum hydraulic action caused by the debris flow. The boundary conditions provided by the openings, floor slabs, orthogonal wall panels and the foundations are also modelled. The reconstruction of the event and the damage to the case‐study building confirm the location of the damage induced by the event.

Adjoint-based hp -adaptivity on anisotropic meshes for high-order compressible flow simulations

Abstract We present an efficient adjoint-based hp -adaptation methodology on anisotropic meshes for high order Discontinuous Galerkin schemes applied to (nonlinear) convection-diffusion problems, including the compressible Euler and Navier-Stokes equations. The refinement strategy is based on a recently proposed interpolation error estimate [Dolejsi V. Anisotropic hp-adaptive method based on interpolation error estimates in the lq-norm. Applied Numerical Mathematics 2014;82:80-114.], as well as an adjoint-based error estimate. Using the two error estimates, we determine the size and the shape of the triangular mesh elements on the desired mesh to be used for the subsequent adaptation steps. This is done using the concept of mesh-metric duality, where metric tensors encode information about mesh elements. The metric tensors that correspond to the desired mesh elements can be passed to a metric-conforming mesh generator to create the required anisotropic mesh. The effectiveness of the adaptation methodology is demonstrated using numerical experiments, involving a scalar model problem with a strong boundary layer, as well as various compressible flow test cases.

Analytic Back Calculation of Debris Flow Damage Incurred to a Masonry Building: The Case of Scaletta Zanclea 2009 Event

In an attempt to do a back analysis of the damages caused to a nineteenth century masonry structure due to the October 2009 flash flood/debris flow event in Scaletta Zanclea, the flood discharge hydrograph is reconstructed in the ungauged conditions. The hydrograph for the solid discharge is then estimated by scaling up the liquid volume to the estimated debris volume. The debris flow diffusion is simulated by solving the differential equations for a single-phase 2D flow employing triangular mesh elements, taking into account also the channelling of the flow through the buildings. The damage to the building is modelled, based on the maximum hydraulic actions caused by the debris flow, using 2D finite shell elements to model the building, boundary conditions provided by the openings, floor slab, orthogonal wall panels, and the foundation. The reconstruction of the event and the damages to the case-study building confirms the location of the damages induced by the event.

A new method for automated discontinuity trace mapping on rock mass 3D surface model

This paper presents an automated discontinuity trace mapping method on a 3D surface model of rock mass. Feature points of discontinuity traces are first detected using the Normal Tensor Voting Theory, which is robust to noisy point cloud data. Discontinuity traces are then extracted from feature points in four steps: (1) trace feature point grouping, (2) trace segment growth, (3) trace segment connection, and (4) redundant trace segment removal. A sensitivity analysis is conducted to identify optimal values for the parameters used in the proposed method. The optimal triangular mesh element size is between 5cm and 6cm; the angle threshold in the trace segment growth step is between 70° and 90°; the angle threshold in the trace segment connection step is between 50° and 70°, and the distance threshold should be at least 15 times the mean triangular mesh element size. The method is applied to the excavation face trace mapping of a drill-and-blast tunnel. The results show that the proposed discontinuity trace mapping method is fast and effective and could be used as a supplement to traditional direct measurement of discontinuity traces.

The comparative analysis of the two dimensional Laplace equation using the Galerkin finite element method with the exact solution for various domains (circular and rectangular) with triangular elemental meshing

Laplace Equation is used in many areas of studies such as potential theory and the fundamental forces of nature, Newtonian theory of gravity and electrostatics. It is used in Probability theory and Markov Chains as well as potential flows in fluid mechanics. Laplace Equation is used in various research areas and for this reason, to determine an accurate solution to this equation is of importance. In this study, the Finite Element Method is used to approximate the solution of the 2D Laplace Equation for two regions, circular and rectangular domains. These are compared to the exact solutions of the systems subjected to various physical restrictions; boundary conditions. This was done by using a mesh generator in Matlab, Distmesh, to obtain a mesh of triangular elements and then using Matlab to plot the exact and the approximated solutions as well as to determine the errors; norm. For these domains, the number of elements in the mesh was incremented and it was noted there was a convergence of the approximated to the actual solution. The boundary conditions were altered to observe the changes in the regions' field variable distribution (intensity values) of the Matlab plots.

An edge-based finite element method (ES-FEM) with adaptive scaled-bubble functions for plane strain limit analysis

This paper deals with critical roles of cubic bubble functions for the edge-based finite element method (ES-FEM) formulation within the framework of the kinematic theorem for predicting the plastic collapse loads of structures. We show that the bubble function can be scaled by a non-zero coefficient alpha, while the volumetric locking is entirely eliminated. Based on this finding, the present method is designed, in a very small value, with a bubble function that is maximum at the center of the element. This can lead to the loss of the partition of unity. For easy reference, the present method is termed as αbES-FEM. The significant advantage of the present method lies in the rigorous treatment of the volumetric locking-free that can occur in the fully plastic range. The αbES-FEM works well with high efficiency by using triangular meshes. It achieves both simplicity and computational efficiency for implementation into packages of plastic limit analyses. In case of αbES-FEM, the locking issue can be solved conveniently by two schemes: (1) the usual piecewise linear displacements enriched with a cubic bubble function on a primal mesh of triangular elements and (2) a projection operator of strain rates through a dual mesh associated with the edges in the mesh. The optimization formulation of finite element limit analysis is written in the form of a second-order cone programming (SOCP). The well-established interior-point solvers can be exploited efficiently. The αbES-FEM using a small number of integration points enables to solve the large-scale optimization problems efficiently. In addition, an adaptive meshing procedure based on an alternative indicator of dissipation is also derived to further enhance the quality of the solution without increasing significantly the number of degrees of freedom of the model. Numerical results show the robustness of the proposed method.

A multi-moment finite volume method for incompressible Navier–Stokes equations on unstructured grids: Volume-average/point-value formulation

A robust and accurate finite volume method (FVM) is proposed for incompressible viscous fluid dynamics on triangular and tetrahedral unstructured grids. Differently from conventional FVM where the volume integrated average (VIA) value is the only computational variable, the present formulation treats both VIA and the point value (PV) as the computational variables which are updated separately at each time step. The VIA is computed from a finite volume scheme of flux form, and is thus numerically conservative. The PV is updated from the differential form of the governing equation that does not have to be conservative but can be solved in a very efficient way. Including PV as the additional variable enables us to make higher-order reconstructions over compact mesh stencil to improve the accuracy, and moreover, the resulting numerical model is more robust for unstructured grids.We present the numerical formulations in both two and three dimensions on triangular and tetrahedral mesh elements. Numerical results of several benchmark tests are also presented to verify the proposed numerical method as an accurate and robust solver for incompressible flows on unstructured grids.