Finite Prism Research Articles

Mean wake evolution behind low aspect-ratio wall-mounted finite prisms

The evolution in the mean wake of wall-mounted prisms with varying depth-ratio (length-to-width) between 0.016 and 4 is examined numerically over a range of Reynolds numbers (Re = 50 - 500). The aspect-ratio (height-to-width) of the prism is limited to 1. This study aims to ascertain the mechanism of the evolution of mean wake topology due to changing depth-ratio. The mean wake topology is classified into Dipole, Multipole, and Quadrupole types as functions of Reynolds number and depth-ratio. The threshold depth-ratio for the wake evolution changes with Reynolds number. The Multipole-type wake appears as an evolutionary or intermediate wake pattern between Dipole and Quadrupole-type wakes. Increasing the depth-ratio leads to the enhancement of the downwash flow, resulting in the mean wake evolving from Quadrupole to Dipole wake. This suggests that the downwash flow dominates in dictating the wake topology downstream of the prism. Enhancing the downwash flow (with increase in depth-ratio) results in the reattachment of the flow on the prism surfaces, which further results in the suppression of velocity gradients and streamwise momentum transfer downstream of the prism trailing edge. This suppression of gradients and momentum leads to the deterioration of the base vortex, which evolves the wake from Quadrupole (at a small depth-ratio) to Dipole-type (at a large depth-ratio).

FFT-based multiscale scheme for homogenisation of heterogeneous plates including damage and failure

This paper introduces a new approach that combines FFT-based multiscale homogenisation with the Lippman–Schwinger equation to efficiently and accurately analyse plate structures with periodic micro-structures. For the first time, the plate periodic Lippman–Schwinger equation is derived to address the auxiliary cell problem. The proposed method comprises two key components: (i) a 3D plate formulation using a novel finite prism approach, and (ii) solving the Lippman–Schwinger equations by means of Green’s functions. Solutions for both the classical and first-order plate theories are derived and implemented for linear and nonlinear problems, completed with an open-source code provided.The efficiency and accuracy of the proposed method are assessed through several case studies, including complex woven composites. It is demonstrated that proposed method achieves a comparable level of accuracy to 3D solid problems while significantly reducing the required computing effort (i.e., milliseconds in computing time as opposed to hours required for 3D solid models). Moreover, nonlinear progressive damage problem involving an actual plate woven composite model is investigated. The results obtained show good agreement with experimental measurements for the progressive damage in plain woven composite, further highlighting the effectiveness of the proposed method.

XFEM based method for buckling analysis of thin-walled beams

Buckling of thin-walled beams involves complex deformation modes, making their analyses challenging. Most of the methods available for the computation of buckling loads of these elements rely on beam theories, which provide high computational efficiency, yet are limited in accuracy and versatility. On the other hand, the finite element method provides reliable solutions for a variety of structures, albeit with higher computational costs.This paper presents a new approach for the buckling analysis of thin-walled beams. The proposed model is developed by utilizing the eXtended Finite Element Method (XFEM). Global enrichment functions, based on the solutions of beam theory, are added to a standard 3D Finite Element (FE) coarse mesh in the longitudinal direction to enhance its performance and reduce the number of degrees of freedom. We consider the enrichment functions to be a set of modes derived from dynamic free vibrations of a beam or those derived from buckling analysis. The enriched nodes can be viewed as connected lines along the sections of the beam, enabling the computation of lateral–torsional buckling.While our XFEM method can be viewed as a semi-analytical approach, it does overcome well-known limitations of similar methods, e.g., the Finite Prism Method (FPM). For example, our method can in principle be extended to beams with non-prismatic cross-sections (e.g., tapered beams), open cross-sections, or those with perforations. However, these extensions are not discussed in the current paper. Furthermore, the XFEM method can also be applied to beams with different boundary conditions along the beam, such as lateral restraints. Additionally, one can improve the solution by sectional and/or longitudinal mesh refinements.The performance of the proposed method is investigated, and the results are compared with solutions obtained from FE models with a very fine mesh. It is shown that the proposed approach enables modeling thin-walled beams for buckling analysis using coarse FE mesh. In some cases, the number of degrees of freedom in XFEM is less than 5% of those required in the 3D FE model to achieve the same error in computing the first buckling load.

Efficient XFEM approach for the analysis of thin-walled beams

This paper presents a new approach to the analysis of thin-walled beams. The proposed model is developed by utilizing the eXtended Finite Element Method (XFEM). Global enrichment functions are added to a standard 3D Finite Element (FE) coarse mesh to enhance its performance and reduce the number of degrees of freedom. These analytical functions provide an approximate solution in the longitudinal direction of the beam. The resulting model bridges the gap between the Finite Prism Method (FPM), which provides an efficient approximation to beam bending with analytical functions but has also well-known limitations, and a detailed 3D FE model, which can accurately resolve thin-walled structures but is computationally demanding.Our method can be easily extended for beams with non-prismatic cross-sections (e.g., tapered beams) or with perforations, which makes it more general than other semi-analytical methods, such as the FPM. However, these extensions are not in the scope of this paper. Additionally, one can improve the solution by sectional and/or longitudinal mesh refinements.The performance of the model is investigated on two benchmark problems. The results show that the proposed method captures different global and sectional deformation modes of thin-walled beams when subjected to torsion, including shear-lag and distortion cross-section. Furthermore, convergence studies demonstrate the superior performance of the proposed method as compared to the FPM, and a 3D FE model with a coarse mesh.

Mechanisms of wake asymmetry and secondary structures behind low aspect-ratio wall-mounted prisms

The wake of a wall-mounted finite prism is numerically studied and characterized with an aspect ratio (height-to-width) of$1$and varying depth ratios (length to width) of between$0.016$and$4$at Reynolds numbers of 50–500. The prism is immersed in a laminar boundary layer. The minimum depth ratio considered here accounts for the special case of a wall-mounted very thin prism (similar to a flat plate), which is used to establish the mechanism and evolution of the wake associated with free-end effects and the shear-layer dynamics in small aspect-ratio prisms. The onset of an unsteady wake behind a very thin prism at a Reynolds number of$200$is characterized by symmetric shedding of hairpin-like vortices. A unique asymmetric wake pattern appears at lower depth ratios starting at a Reynolds number of 250, which transitions to an symmetric wake with increasing depth ratio. The threshold depth ratio for this symmetric transition increases with Reynolds number. The asymmetric wake results from alternate shear-layer peel-off from either side of the prism, which itself is attributed to the out-of-phase shedding of tip vortices at a lower Strouhal number ($St_{sh}/2$) that interact with the detaching side shear layers. Alternate shedding of tip vortices form secondary vortex structures that are fed by the excess vorticity resulting from shear-layer detachment from either side of the prism. Increasing the depth ratio leads to simultaneous shedding of the tip vortices, which restores the commonly observed wake symmetric patterns. Thus, we identify and characterize the formation and interaction mechanisms of symmetric and asymmetric wakes during the transition process with increasing Reynolds number for different depth-ratio prisms.

A semi-analytical FE method for the 3D bending analysis of nonhomogeneous orthotropic toroidal shells

Based on Reissner's mixed variational theorem (RMVT), the authors develop a semi-analytical finite element (FE) method for a three-dimensional (3D) bending analysis of nonhomogeneous orthotropic, complete and incomplete toroidal shells subjected to uniformly-distributed loads. In this formulation, the toroidal shell is divided into several finite annular prisms (FAPs) with quadrilateral cross-sections, where trigonometric functions and serendipity polynomials are used to interpolate the circumferential direction and meridian-radial surface variations in the primary field variables of each individual prism, respectively. The material properties of the toroidal shell are considered to be nonhomogeneous orthotropic over the meridian-radial surface, such that homogeneous isotropic toroidal shells, laminated cross-ply toroidal shells, and single- and bi-directional functionally graded toroidal shells can be included as special cases in this work. Implementation of the current FAP methods shows that their solutions converge rapidly, and the convergent FAP solutions closely agree with the 3D elasticity solutions available in the literature.

Wind loading of a finite prism: aspect ratio, incidence and boundary layer thickness effects

A systematic set of low-speed wind tunnel experiments was performed at Re = 6.5x104 and 1.1x105 to study the mean wind loading experienced by surface-mounted finite-height square prisms for different aspect ratios, incidence angles, and boundary layer thicknesses. The aspect ratio of the prism was varied from AR = 1 to 11 in small increments and the incidence angle was changed from α = 0° to 45° in increments of 1°. Two different boundary layer thicknesses were used: a thin boundary layer with δ/D = 0.8 and a thick boundary layer with δ/D = 2.0–2.2. The mean drag and lift coefficients were strong functions of AR, α, and δ/D, while the Strouhal number was mostly influenced by α. The critical incidence angle, at which the prism experiences minimum drag, maximum lift, and highest vortex shedding frequency, increased with AR, converged to a value of αc = 18° ± 2° once AR was sufficiently high, and was relatively insensitive to changes in δ/D. A local maximum value of mean drag coefficient was identified for higher-AR prisms at low α. The overall behaviour of the force coefficients and Strouhal number with AR suggests the possibility of three flow regimes.

Derivative using complex variable conjugate approach for analytic signal of magnetic field anomaly due to 2D finite prism

The complex variable conjugate approach has been derived analyticaly for derivative computation. Computational results are then used in calculating the amplitude of analytic signal. It is the square root of the square of the total magnetic field anomaly derivative. The total magnetic fields are generated by the upper and lower parts of a 2D finite prism, and subtraction of both parts yields the magnetic field anomaly. While the approach is obtained by truncating the Taylor series expansion of the total magnetic field function in argument of the complex conjugate in the h2 order-term. Truncating the series does not significantly affect the computational results. This is because when step-size h get smaller, and at h≤10−2, then the errors due to the series truncation became 0 (Kp→0). For derivative computation, the approach has precision on the order of 10−17 to 10−12 towards analytical settlement. On the order of h≤10−2, the approach is insensitive to the selection of step-size h for small numbers, so it can be done arbitrarily without any particular treatment or requires a complicated combination of numbers. The computational results of the analytic signal amplitude show that the positive and negative polarity on the magnetic profile is transformed into a positive profile only. This can facilitate the interpretation of actual magnetic data, especially in determining the causative source position of anomalies.

A semianalytical finite element method for stress and deformation analyses of bi-directional functionally graded truncated conical shells

On the basis of Reissner’s mixed variational theorem, the authors develop a semianalytical finite annular prism method (FAPM) for three-dimensional (3D) stress and deformation analyses of bi-directional functionally graded (FG) truncated conical shells with various boundary conditions subjected to either uniformly or sinusoidally distributed loads. The material properties of the FG-truncated conical shell are assumed to obey a bi-directional power-law distribution of the volume fractions of the constituents through the meridian–thickness surface, the effective material properties of which are estimated using the rule of mixtures. Implementation of the current FAPMs shows their solutions converge rapidly and that the convergent solutions are in excellent agreement with the 3D solutions available in the literature.

Features of the flow over a finite length square prism on a wall at various incidence angles

Wake characteristics of the flow over a finite square prism at different incidence angles were experimentally investigated using an open-loop wind tunnel. A finite square prism with a width D = 15 mm and a height H = 7D was vertically mounted on a horizontal flat plate. The Reynolds number was varied from 6.5X103 to 28.5X103 and the incidence angle a was changed from 0 to 45. The ratio of boundary layer thickness to the prism height was about s/H = 7%. The time-averaged velocity, turbulence intensity and the vortex shedding frequency were obtained through a single-component hotwire probe. Power spectrum of the streamwise velocity fluctuations revealed that the tip and base vortices shed at the same frequency as that of spanwise vortices. Furthermore, the results showed that the critical incidence angle corresponding to the maximum Strouhal number and minimum wake width occurs at acr = 15o which is equal to that reported for an infinite prism. There is a reduction in the size of the wake region along the height of the prism when moving away from the ground plane towards the free end.

Transient flows past arrays of yawed finite prisms

Transient flows past arrays of yawed finite-length prisms in tandem and staggered arrangements are simulated. Large eddy simulation method is employed to characterize spatial and temporal characteristics of flows in the vicinity of prisms. Drag coefficients of prisms are determined for different spacing and yaw angle of prisms. Three-dimensional effects are investigated for different spacings of prisms in a tandem array. Drag coefficient of downstream prism is significantly lower compared to that of upstream prism when prisms are tightly spaced. Drag coefficient of downstream prism predicted by simulations in three-dimensional geometry is recovered fully as spacing approaches 10 times the height of the prism while drag coefficient predicted by simulations in two-dimensional geometry is only about 30% of the upstream prism even when two prisms are separated by 20 times the height of the prism. The wake flow patterns do not influence the drag coefficient strongly for the staggered arrays of yawed prisms. Prisms can be placed in a tighter arrangement without a significant penalty of drag reduction. Mathematical models and numerical methods employed are validated by comparing simulation results against experimental results. Drag exerted on these prisms can directly be related to power generation by devices containing translating blades of marine-current applications. This study demonstrates that arrangement, spacing, and yaw angle can be used to design and optimize energy harvesting devices and other offshore structures.

Finite prism method based topology optimization of beam cross section for buckling load maximization

The use of the finite element method (FEM) for buckling topology optimization of a beam cross section requires large numerical cost due to the discretization in the length direction of the beam. This investigation employs the finite prism method (FPM) as a tool for linear buckling analysis, reducing degrees of freedom of three-dimensional nodes of FEM to those of two-dimensional nodes with the help of harmonic basis functions in the length direction. The optimization problem is defined as the maximization problem of the lowest eigenvalue, for which a bound variable is introduced and set as the design objective to treat mode switching phenomena of multiple eigenvalues. The use of the bound formulation also helps the proposed optimization to treat beams having local plate buckling modes as the fundamental modes as well as beams having global buckling modes. The axial stress is calculated according to the distribution of material modulus which is interpolated using the SIMP approach. Optimization problems finding cross-section layouts from rectangular, L-shaped and generally-shaped design domains are solved for various beam lengths to ascertain the effectiveness of the proposed method.

The effect of a splitter plate on the flow around a finite prism

The effect of a wake-mounted splitter plate on the flow around a surface-mounted finite-height square prism was investigated experimentally in a low-speed wind tunnel. Measurements of the mean drag force and vortex shedding frequency were made at Re=7.4×104 for square prisms of aspect ratios AR=9, 7, 5 and 3. Measurements of the mean wake velocity field were made with a seven-hole pressure probe at Re=3.7×104 for square prisms of AR=9 and 5. The relative thickness of the boundary layer on the ground plane was δ/D=1.5–1.6 (where D is the side length of the prism). The splitter plates were mounted vertically from the ground plane on the wake centreline, with a negligible gap between the leading edge of the plate and rear of the prism. The splitter plate heights were always the same as the heights of prisms, while the splitter plate lengths ranged from L/D=1 to 7. Compared to previously published results for an “infinite” square prism, a splitter plate is less effective at drag reduction, but more effective at vortex shedding suppression, when used with a finite-height square prism. Significant reduction in drag was realized only for short prisms (of AR≤5) when long splitter plates (of L/D≥5) were used. In contrast, a splitter plate of length L/D=3 was sufficient to suppress vortex shedding for all aspect ratios tested. Compared to previous results for finite-height circular cylinders, finite-height square prisms typically need longer splitter plates for vortex shedding suppression. The effect of the splitter plate on the mean wake was to narrow the wake width close to the ground plane, stretch and weaken the streamwise vortex structures, and increase the lateral entrainment of ambient fluid towards the wake centreline. The splitter plate has little effect on the mean downwash. Long splitter plates resulted in the formation of additional streamwise vortex structures in the upper part of the wake.

An Experimental Investigation of Aspect Ratio and Incidence Angle Effects for the Flow Around Surface-Mounted Finite-Height Square Prisms

The flow around a surface-mounted finite-height square prism was investigated using a low-speed wind tunnel. The experiments were conducted at a Reynolds number of Re = 7.3 × 104 for prism aspect ratios of AR = 3, 5, 7, 9, and 11 and incidence angles from α = 0 deg to 45 deg. The thickness of the boundary layer on the ground plane relative to the side length was δ/D = 1.5. Measurements of the vortex shedding frequency were made with a single-component hot-wire probe, and measurements of the mean drag and lift forces were obtained with a force balance. For all aspect ratios and incidence angles, the mean drag coefficient and Strouhal number were lower than those of an infinite prism, while the mean lift coefficient was of nearly similar magnitude. As the aspect ratio was increased from AR = 3 to 11, the force coefficients and Strouhal number slowly approached the infinite-square-prism data. The mean drag coefficient and Strouhal number for the finite prism were less sensitive to changes in incidence angle compared to the infinite square prism. The critical incidence angle, corresponding to minimum mean drag coefficient, minimum (most negative) mean lift coefficient, and maximum Strouhal number, shifted to a higher incidence angle compared to the infinite square prism, with values ranging from αcritical = 15 deg to 18 deg; this shift was greatest for the prisms of higher aspect ratio. The behavior of the force coefficients and Strouhal number for the prism of AR = 3 was distinct from the other prisms (with lower values of mean drag coefficient and mean lift coefficient magnitude, and a different Strouhal number trend), suggesting the critical aspect ratio was between AR = 5 and AR = 3 in these experiments. In the wall-normal direction, the power spectra for AR = 11 and 9 tended to have weaker and/or more broad-banded vortex shedding peaks near the ground plane and near the free end at α = 0 deg and 15 deg. For AR = 7 to 3, well-defined vortex shedding peaks were detected along the entire height of the prisms. For AR = 11 and 9, at α = 30 deg and 45 deg, vortex shedding peaks were absent in the power spectra in the upper part of the wake.

Three-dimensional free vibration analysis of functionally graded carbon nanotube-reinforced composite plates with various boundary conditions

A unified formulation of finite prism methods (FPMs) based on Reissner's mixed variational theorem is developed for the three-dimensional (3D) free vibration analysis of functionally graded (FG) carbon nanotube-reinforced composite (CNTRC) plates and laminated fiber-reinforced composite (FRC) plates, the edge conditions of which are considered such that one pair of opposite edges is simply supported and the other pair may be combinations of free, clamped or simply supported edges. The single-walled carbon nanotubes (CNTs) and polymer are regarded as the reinforcements and matrices, respectively, to produce the CNTRC plate. Four different distributions of CNTs varying in the thickness direction are considered (i.e. the uniformly distributed, and FG V-, rhombus-, and X-type variations), and the through-thickness distributions of effective material properties of the CNTRC plate are determined using the rule of mixtures. In the formulation, the CNTRC/FRC plate is divided into a number of finite prisms in the domain, in which the trigonometric functions and Lagrange polynomials are used to interpolate the y-direction and plane variations for the primary variables of each individual prism, respectively, and the related orders used for expansion of assorted primary variables in the thickness coordinate can be freely chosen. It is shown that the eight- and nine-node quadratic FPM solutions of frequency parameters of simply supported, CNTRC plates and laminated FRC plates are in excellent agreement with the exact 3D solutions available in the literature, and with the ones obtained using the ANSYS software for those plates with various boundary conditions.

An rmvt-based finite rectangular prism method for the 3d analysis of sandwich fgm plates with various boundary conditions

An rmvt-based finite rectangular prism method for the 3d analysis of sandwich fgm plates with various boundary conditions

Wave scattering by a prism made of a lossy metamaterial

The results of rigorous calculation of the scattering pattern of a Gaussian beam incident onto a finite prism made of a metamaterial are presented for different values of the refractive index nr < 0 and loss factor ν of the medium. It is shown that geometric optics can correctly predict the tilt of the scattering-pattern maximum, while consideration for the medium’s loss factor 0.001 ≤ |ν| ≤ 0.1 affects the amplitudes of the transmitted and reflected fields.

Two‐dimensional inverse filtering for the rectification of the magnetic gradiometry signal

ABSTRACTThe magnetic difference, the quantity measured by magnetic gradiometers, is considered to be the convolution between a function that controls the anomaly pattern and another that controls the strength signal. These are called shape and amplitude functions, respectively. They are distinct and analytically determined; thus, after the assessment of a suitable model, its shape function can be inverted to serve as a filter. If this filter is next convolved with the measured field, a series of amplitude functions is recovered, provided that the subsurface structures can be simulated by a combination of a number of models similar to the one whose anomaly was inverted. The recovered series is essentially the subsurface distribution of the amplitude function.Alternatively, the scheme can be viewed as a transformation of the original field of magnetic differences. The signals, transformed in this context, comprise rectified monopolar versions of the originals, positioned directly above the centre of the targets. Furthermore, their amplitude is a measure of the magnetization of the targets. Thus, the new anomalies can be viewed as a kind of magnetization or susceptibility mapping. Inversion filters are computed in the Wiener mode by inverting the shape function of the anomalies caused by vertical sided finite prisms. This particular model is appropriate for a wide variety of targets that are commonly met in archaeological prospecting. Converting the magnetic signal from the dual lobe pattern to single monopolar anomalies cause them to resemble the usual outcome of resistivity exploration. In this context, this work aims to form a scheme that makes the magnetic gradiometry outcome directly comparable to the outcome of resistivity mapping. Furthermore, it can be applied without contradicting or excluding any other processing operation.The method is implemented in a FORTRAN program that is reasonably user‐ friendly. The efficiency of the scheme is tested both on synthetic and real data sets.

Real time discrete techniques for distortion and microstructure of long product sections

A combined suite of in house finite element (FE) and finite prism (FP) models has been developed to address real time predictions of temperature, distortion and structure during cooling of long product sections. The present paper also describes various applications of cooling, and compares the developed approaches with conventional three-dimensional and generalised plane strain formulation elements and experimental results. These models have also been used to simulate the effectiveness of alternative cooling strategies (e.g. spray cooling) in reducing levels of thermal distortion.

Fast Three-Dimensional Quasi-Static Analysis of Helical Gears Using the Finite Prism Method

This paper describes a model for analyzing external and internal helical gears. The finite prism method is used to study the elastic behavior of the structure. Contact deformations are also included in the model. Load sharing, pressure distribution, meshing stiffness and three-dimensional tooth fillet stresses are calculated at each instant. An experimental validation of the numerical model is also presented.