2D Approximation Research Articles

Three-dimensional unsteady aerodynamics of a H-shaped vertical axis wind turbine over the full operating range

Vertical-axis wind turbines have great potential for harvesting energy with high conversion efficiency both in small-scale distributed installations and in the emerging floating offshore technology; nevertheless, their complex unsteady flow field makes design and optimization very challenging. This paper documents an extensive CFD study on a H-shaped VAWT over the entire range of operation. First, a direct 2D-3D comparison of the prediction capability of CFD models over the entire turbine operational curve is presented, highlighting the range of operation in which 3D effects become crucial and, instead, the range of TSR for which a 2D approximation is acceptable. The simulations are systematically compared to a wide and accurate experimental data set of wind-tunnel experiments in controlled conditions. The combination of CFD results with measured data allowed to assess the turbine performance and time-resolved velocity measurements in the wake. Furthermore, the paper proposes a totally novel approach in analysing the computed non-periodic unsteadiness in the wake. Finally, a tracking of the unsteady tip vortex is also proposed, with remarkable similarity to experimental results documented in the literature. This fact suggests that URANS discretization permits to approximate the dynamic and evolution of the tip vortex, at least in the near wake region.

Atomistic modelling of carbon nanotube networks and analysis of inter filler distance

This paper provides a new approach to model the formation of networks by carbon nanotubes (CNTs) in three dimensions. Network formation by the CNTs plays a vital role in understanding the overall behaviour of the nanocomposites. It is often complicated to analyze the 3D distribution of the fillers inside a nanocomposite experimentally. This 3D atomistic visualization of CNTs through simulations will prove highly beneficial to understand the effect of filler concentration and other experimental parameters. The proposed approach is at the atomistic scale, i.e. each CNT will be built using the appropriate number of carbon atoms depending on its aspect ratio and dimensions. This method aims to be more realistic by eliminating 2D approximation techniques and the cylinder approximation method for the CNTs. Inter filler distances in uniformly distributed short-chopped CNTs are analyzed using this approach. Probable least resistive paths were estimated, and an extended analysis reveals that the minimum filler concentration for initiating electrical conductivity in such short fillers to be around 0.6 vol%.

Numerical Prediction of the Vertical Responses of Planing Hulls in Regular Head Waves

The evaluation of the hydrodynamic performance of planing vessels has always been one of the most attractive study fields in the maritime agenda. Resistance and self-propulsion studies have been performed using experimental and numerical methods by researchers for a long time. As opposed to this, the seakeeping performance of planing hulls is assessed with 2D approximation methods, but limitedly, while the experimental campaign is not cost-effective for several reasons. With this motivation, pitch and heave transfer functions and accelerations were obtained for a monohedral hull and a warped hull using a state of art commercial Reynolds-averaged Navier–Stokes (RANS) solver, in this study. Moreover, 2-DOF (degree of freedom) dynamic fluid–body interaction (DFBI) equations were solved in a coupled manner with an overset mesh algorithm, to find the instantaneous motion of the body. After verification, obtained numerical results at three different Froude numbers and a sufficiently large wave frequency range were compared with the experiments. The results showed that the employed RANS method offers a very accurate prediction of vertical motions and accelerations for planing hulls.

Two‐step method for calculation of eddy current losses in a laminated transformer core

A method for calculating eddy currents and corresponding losses in a laminated transformer core using finite element method is developed. The method is based on the transformation of the 3D model into the corresponding 2D model. The 2D laminated domain coincides with the slice of the 3D laminated domain. Eddy currents are therefore considered as 2D phenomena within lamination, instead of the 1D approximation, thus taking into account the edge effects. By using the 2D mesh, the number of finite elements is drastically reduced. The validity and precision of the method are verified by comparing the results of the simulation with the results obtained using the brute force approach and with the results obtained using the homogenisation method.

Properties of gamma-ray decay lines in 3D core-collapse supernova models, with application to SN 1987A and Cas A

ABSTRACT Comparison of theoretical line profiles to observations provides important tests for supernova explosion models. We study the shapes of radioactive decay lines predicted by current 3D core-collapse explosion simulations, and compare these to observations of SN 1987A and Cas A. Both the widths and shifts of decay lines vary by several thousand kilometres per second depending on viewing angle. The line profiles can be complex with multiple peaks. By combining observational constraints from 56Co decay lines, 44Ti decay lines, and Fe IR lines, we delineate a picture of the morphology of the explosive burning ashes in SN 1987A. For MZAMS = 15−20 M⊙ progenitors exploding with ∼1.5 × 1051 erg, ejecta structures suitable to reproduce the observations involve a bulk asymmetry of the 56Ni of at least ∼400 km s−1 and a bulk velocity of at least 1500 km s−1. By adding constraints to reproduce the UVOIR bolometric light curve of SN 1987A up to 600 d, an ejecta mass around 14 M⊙ is favoured. We also investigate whether observed decay lines can constrain the neutron star (NS) kick velocity. The model grid provides a constraint VNS > Vredshift, and applying this to SN 1987A gives a NS kick of at least 500 km s−1. For Cas A, our single model provides a satisfactory fit to the NuSTAR observations and reinforces the result that current neutrino-driven core-collapse SN models achieve enough bulk asymmetry in the explosive burning material. Finally, we investigate the internal gamma-ray field and energy deposition, and compare the 3D models to 1D approximations.

Formation of combustion wave in lean propane-air mixture with a non-uniform chemical reactivity initiated by nanosecond streamer discharges in the HCCI engine

The propagation of combustion wave, which is formed as a result of impact of a filamentary pulsed periodic non-equilibrium discharge on a mixture that is not ignited by only compression in the HCCI engine, has been simulated. The discharge for a short time locally activated the mixture in the engine cylinder at a certain crankshaft rotation angle at the compression stage. The approach to evaluation of the temperature and composition in the region activated by high-frequency corona discharge has been proposed. Multi-pulse and multi-channel nature of discharge was accounted for in our model. The densities of chemically active species produced during streamer propagation were found using two-dimensional axially symmetric fluid model. The obtained composition and temperature in the activated zone were the initial conditions for modeling the propagation of combustion wave along the cylinder radius in 1D approximation. Special terms were included in equations responsible for the compression along the axis of cylinder to mimic the real change in pressure in cylinder. The calculations were performed for a lean C3H8-air mixture, which is characterized by the presence of low-temperature heat release (LTHR) stage, and intermediate and high temperatures heat release (ITHR and HTHR) stages. It was shown that the ignition timing in activated zone depends on the specific energy input into the streamer channel, the radius of activated region coupled with the volume fraction of activated region treated by discharge, and the moment of discharge initiation relative to the top dead center (TDC). The speed of combustion wave, the occurrence of auto-ignition in the end gas and transition to combustion, accompanied by “knocking” or strong pressure fluctuations are largely determined by the moment of discharge initiation and the specific energy input. The significant effect of discharge is explained by the stimulation of oxidation kinetic mechanisms at LTHR and ITHR stages.

How flat can a horse be? Exploring 2D approximations of 3D crania in equids

Quantitative analyses of morphological variation using geometric morphometrics are often performed on 2D photos of 3D structures. It is generally assumed that the error due to the flattening of the third dimension is negligible. However, despite hundreds of 2D studies, few have actually tested this assumption and none has done it on large animals, such as those typically classified as megafauna. We explore this issue in living equids, focusing on ventral cranial variation at both micro- and macro-evolutionary levels. By comparing 2D and 3D data, we found that size is well approximated, whereas shape is more strongly impacted by 2D inaccuracies, as it is especially evident in intra-specific analyses. The 2D approximation improves when shape differences are larger, as in macroevolution, but even at this level precise inter-individual similarity relationships are altered. Despite this, main patterns of sex, species and allometric variation in 2D were the same as in 3D, thus suggesting that 2D may be a source of ‘noise’ that does not mask the main signal in the data. However, the picture that emerges from this and other recent studies on 2D approximation of 3D structures is complex and any generalization premature. Morphometricians should therefore test the appropriateness of 2D using preliminary investigations in relation to the specific study questions in their own samples. We discuss whether this might be feasible using a reduced landmark configuration and smaller samples, which would save time and money. In an exploratory analysis, we found that in equids results seem robust to sampling, but become less precise and, with fewer landmarks, may slightly overestimate 2D inaccuracies.

Simplified methods for designing thermo-active retaining walls

Thermo-active retaining structures are geotechnical structures employed to provide thermal energy to buildings for space heating and cooling through heat exchanger pipes embedded within the concrete structure. Consequently, the design of these structures needs to consider both the long-term energy efficiency as well as the thermo-mechanical response in terms of stability and serviceability. Transient finite element analyses can be carried out to evaluate the behaviour of thermo-active walls, where the heat exchanger pipes are explicitly modelled, thus requiring three-dimensional (3D) analyses. However, performing long-term 3D finite element analyses is computationally expensive. For this reason, in this study, new approaches are presented that allow the thermal or thermo-mechanical design of thermo-active walls to be carried out by performing two-dimensional (2D) plane strain analyses. Two methods, which are based on different design criteria, are proposed and their performance in replicating the three-dimensional behaviour is assessed. Furthermore, the factors affecting the 2D approximations for the two modelling approaches are evaluated, where particular emphasis is given to the influence of the simulated boundary condition along the exposed face of the retaining wall.

Augmenting 1D conductivity depth sections to include information pertaining to 2D/3D conductors

SummaryConductivity depth data presentations derived using 1D approximations have become a very popular way to present airborne electromagnetic (AEM) data. While data profiles and a single error limit per station can accompany these presentations there are more information contained in data. This work proposes some ideas for augmenting 1D derived data presentations to highlight two and three-dimensional conductors as well as more detailed error level indication. The 1D conductivity depth algorithm used is an extended version of the S-layer differential transform. The depths calculated with the transform is used also present decay constant values corrected for conductive background responses. Fraser filtered X-component data are used to indicate structural complexity and data fit error levels are used to grey out areas on the depth section presentations where the results are less reliable. To demonstrate these ideas a line of Xcite data over the Abra deposit in Western Australia is analysed. The augmentations add valuable information to the conductivity depth section in the sense of drawing attention the where the 1D assumptions are not valid and should encourage interpreters to pursue 2D or 3D modelling techniques for successfully mapping the ore deposit.

Three-dimensional urban flood inundation simulation based on digital aerial photogrammetry

Intense storm surge can lead to severe coastal floods. Flood inundation simulation is useful to identify areas that are most vulnerable to severe storm surges and to address potential flooding hazards. In this paper, both 2D and 3D hydrodynamic models are built separately to quantify the flood danger to a coastal city. A digital city model is generated by integrating building information modeling (BIM) and geographic information systems (GIS) technology based on digital aerial photogrammetry for the 3D hydrodynamic model. The results reveal that 2D hydrodynamic assumptions and approximations weaken if the vertical fluctuations are enormous, especially in urban environments. In comparison, a 3D model coupled with high-resolution topographic data can provide a more realistic and accurate result for the complex flow field. Furthermore, a comparison between using the digital city model and a standard digital elevation model (DEM) in a 3D model has been conducted. The flood inundation is simulated step-wise using the standard DEM, while the wave front propagates in a more realistic manner based on high-resolution topography data. Thus, a 3D hydrodynamic model coupled with the digital city model is recommended for urban flood simulation, which is promising to improve the ability of flood prediction and prevention.

Study of the microwave interaction with supersonic plasma sheets and jet generated by Magneto-Plasma Compressor

In this paper, we present investigation of a method for converting a supersonic plasma axisymmetric jet, created by a Magneto-Plasma Compressor, into flat plasma layers with various thicknesses and electron concentration. This kind of plasma layer has comparatively homogeneous structure and may use for experimental modelling of interaction of X-band radiation with high-speed plasma flow. The results of experimental studies of the microwave radiation transition through these layers are presented. The analysis of the received results within the one-dimensional (1D) approximation is carried out and the plasma concentrations are calculated.

Modelling adsorption using an augmented two-dimensional statistical associating fluid theory: 2D-SAFT-VR Mie

We present an extension of the SAFT-VR Mie approach to model adsorption of molecular fluids based on a two-dimensional (2D) approximation to describe the adsorbed fluid. Analytical results are provided for the first- and second-order perturbation terms of the free energy for the 2D system. The adsorption model is based on the assumption that the particle pair interactions in the adsorbed and bulk phases are described with the same Mie potential exponents and , in contrast with the square-well version of the 2D-SAFT-VR approach in which it is considered necessary to modify the attractive ranges of the SW interactions. This important difference between the two approaches leads to a reduction in the number of molecular parameters to be determined. In order to demonstrate the performance of the 2D-SAFT-VR Mie approach, we present results for the the modelling of carbon dioxide (CO2) and methane (CH4) adsorbed onto dry coal.

A–V formulation for numerical modelling of superconductor magnetization in true 3D geometry

We present the numerical procedure suitable for computing the distribution of electrical currents in a superconductor with geometry extending in all three spatial dimensions when it is exposed to a magnetic field changing in time. Its main advantage is that it solves the problem in terms of vector potential, of the magnetic field. Such an A-formulation is usually the default option for magnetic field calculations in commercial finite element codes. We have used it in the past when a two-dimensional (2D) approximation was found to be sufficient to predict AC losses of single wires, cables, and coils with circular turns. Incorporation of superconductor properties based on the critical state model is straightforward in 2D because the flow of current is restricted to one direction only. The main challenge in extending the method to three dimensions is to cope with the situation when currents could flow in any direction. We have found that assuming that the local current density in a superconductor is always parallel to the local electrical field, and that its modulus is limited by the critical current density, provides sufficient input for accomplishing a new working equation, linking the vector of current density to the time derivative of the magnetic vector potential. Our method successfully passed testing on the benchmark problem of superconducting cube magnetization. It is also used to interpret the magnetization measured on two cylindrical superconducting objects in the transverse field: one was a tube made from bulk BSCCO and the other a REBCO coated conductor tape helically wound on a non-metallic tube. Computed magnetization loops in both cases show good agreement with experiments and confirm the validity of the method.

Solution of Axisymmetric Inhomogeneous Problems with the Markov Chain Monte Carlo

With increasing complexity of EM problems, 1D and 2D axisymmetric approximations in p, z plane are sometimes necessary to quickly solve difficult symmetric problems using limited data storage and within shortest possible time. Inhomogeneous EM problems frequently occur in cases where two or more dielectric media, separated by an interface, exist and could pose challenges in complex EM problems. Simple, fast and efficient numerical techniques are constantly desired. This paper presents the application of simple and efficient Markov Chain Monte Carlo (MCMC) to EM inhomogeneous axisymmetric Laplace’s equations. Two cases are considered based on constant and mixed boundary potentials and MCMC solutions are found to be in close agreement with the finite difference solutions.

Validity of Gross–Pitaevskii solutions of harmonically confined BEC gases in reduced dimensions

By exact numerical solutions of the Gross–Pitaevskii (GP) equation in 3D, we assess the validity of 1D and 2D approximations in the study of Bose–Einstein condensates confined in harmonic trap potentials. Typically, these approximations are performed when one or more of the harmonic frequencies are much greater than the remaining ones, using arguments based on the adiabatic evolution of the initial approximated state. Deviations from the 3D solution are evaluated as a function of both the effective interaction strength and the ratio between the trap frequencies that define the reduced dimension where the condensate is confined. The observables analyzed are both of stationary and dynamical character, namely, the chemical potential, the wave function profiles, and the time evolution of the approximated 1D and 2D stationary states, considered as initial states in the 3D GP equation. Our study, besides setting quantitative limits on approximations previously developed, should be useful in actual experimental studies where quasi-1D and quasi-2D conditions are assumed. From a qualitative perspective, 1D and 2D approximations certainly become valid when the anisotropy is large, but in addition the interaction strength needs to be above a certain threshold.

Nonlinear flexural distribution behavior and ultimate system capacity of skewed steel girder bridges

Bridge designs are routinely developed using simplified 1D approximations of structural behavior and assume linear elastic behavior throughout the structure. More rigorous methods can reveal significant unquantified reserve capacity with 3D system behavior as ductile members experience yielding, but reserve capacity can be overestimated if cracking in a concrete deck is neglected. The primary objective of this study is to characterize the influence of concrete deck nonlinearity on 3D system behavior and capacity of skewed steel girder bridges. Structural behavior is examined throughout the range of potential applied loads, from initial linear elastic behavior to initial yielding and ultimate conditions. Additionally, this study provides a secondary benefit by comparing the applicability of two general load distribution characterization methods: response fractions (displacement, strain, or curvature ratios) versus extracted and integrated results from a rigorous analytical model. Analytical models, accounting for material inelasticity in the steel girders, are constructed for each of two experimentally tested bridges. Skew is varied parametrically from 0° to 60°. Concrete deck is modeled with a tension cracking stress limit consistent with typical design assumptions in United States practice. Results are also presented for the 60° case with linear elastic deck to directly compare the influence of cracking versus non-cracking deck. The analyses indicated that neglecting concrete cracking noticably affected capacity by overestimating slab plate flexure between abutments, resulting in increasingly unconservative reserve capacity evaluations with increasing skew. Bridges with small skews evaluated using only elastic analysis were found to be reasonably well characterized using response fractions feasibly obtained from diagnostic load tests. However, bridges with high skews and bridges evaluated using ultimate capacity from detailed finite element analysis, regardless of skew, will receive unnecessarily conservative flexural capacity ratings if response fractions are relied upon to represent analogous internal load effects.

Improvements in Simulations for Radiotherapy Wedge Filter dose and AAA-Convolution Factor Algorithms

Analytical-convoluted and numerical Gaussian models have been used in recent decades for radiotherapy treatment planning software/calculations, to perform accurately radiation dose delivery –numerical, analytical, or numerical-analytical. The objective of this evoluted-contribution was to obtain an exact dose delivery, 3D analytical-integral-equation solution, for the triple Gaussian model of wedge filters, since previous/initial 2D approximations of other authors, although correct, were not completely exact. Additionally, to set conceptual and mathematical-geometrical differences between the beam modification created by Multi-Leaf Collimator and Wedge Filters, either standard or Conformal. Ever the precision, from mathematical theory algorithms to real laboratory measurements, a series of simulations are presented. The generic triple Gaussian model of Ulmer and Harder sets an Attenuation Exponential Factor, AEF, well approximated in 2 variables, namely, u and z. This evoluted contribution of the research contribution was specially focused on numerical methods and approximation analysis of the integral equation resolution –with extent details about numerical data, Appendix 3. In this paper we set a detailed spatial-spherical geometry discussion/proof towards the determination of a 3D integral form of the delivery dose in water. In other words, with an AEF for magnitude-values of variables u,v, and z. Simulations, based on these new determinations were shown with sharp presentation of the numerical-computational software and functional programming series development. Computing encode techniques are explained with some practical examples for numerical radiotherapy calculus.

Buckling loads for sandwich structures having FRP faces and FGM core

The aim of the present paper is to present a consistent 2D formulation of sandwiches (CS – classical sandwiches and FGMs – functionally gradient materials) that is accurate enough for a prediction of thermo-mechanical buckling load. Various kinematical hypothesis and variational formulations are studied in order to compare them with 3D finite element modelling and to estimate the effectiveness of 2D approximations. The examples deal with the compressed rectangular plates and shallow cylindrical panels. The effects of boundary conditions on both values of thermal and mechanical buckling loads as well on the variations of bifurcation loads with fibre orientations are discussed in details.

Analytic 1D approximation of the divertor broadening S in the divertor region for conductive heat transport

An analytic 1D approximation for the divertor broadening S is introduced, depending only on the electron temperature between X-point and target. It is compared to simulations solving the 2D heat diffusion equation, in order to describe the divertor broadening along a field line solely by the ratio of the perpendicular to the parallel diffusivities. By assuming the temperature dependence of these two diffusivities an integral form of S is derived for the area along the separatrix between X-point and target. Integration along the separatrix results in an approximation for S, being in agreement with the 2D simulations. This approximation is furthermore compared to recent studies, which find a power law with negative exponent to describe S in terms of target temperature. This dependence is not reproduced in a pure conductive description, which instead shows a finite S for zero target temperature. This points to other mechanisms changing the shape of the heat flux profile—by additional widening or radiation losses—not included in the presented reduced approximation.

Efficient and accurate stress recovery procedure and a posteriori error estimator for the stable generalized/extended finite element method

SummaryThis paper presents a new stress recovery technique for the generalized/extended finite element method (G/XFEM) and for the stable generalized FEM (SGFEM). The recovery procedure is based on a locally weighted L2 projection of raw stresses over element patches; the set of elements sharing a node. Such projection leads to a block‐diagonal system of equations for the recovered stresses. The recovery procedure can be used with GFEM and SGFEM approximations based on any choice of elements and enrichment functions. Here, the focus is on low‐order 2D approximations for linear elastic fracture problems. A procedure for computing recovered stresses at re‐entrant corners of any internal angle is also presented. The proposed stress recovery technique is used to define a Zienkiewicz‐Zhu (ZZ) a posteriori error estimator for the G/XFEM and the SGFEM. The accuracy, computational cost, and convergence rate of recovered stresses together with the quality of the ZZ estimator, including its effectivity index, are demonstrated in problems with smooth and singular solutions.