Solid Wedges Research Articles

Multidimensional transonic shock waves and free boundary problems

We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for potential flow, and related nonlinear PDEs of mixed elliptic–hyperbolic type. The transonic shock problems include the problem of steady transonic flow past solid wedges, the von Neumann problem for shock reflection–diffraction, and the Prandtl–Meyer problem for unsteady supersonic flow onto solid wedges. We first show how these longstanding multidimensional transonic shock problems can be formulated as free boundary problems for the compressible Euler equations and related nonlinear PDEs of mixed type. Then we present an effective nonlinear method and related ideas and techniques to solve these free boundary problems. The method, ideas, and techniques should be useful to analyze other longstanding and newly emerging free boundary problems for nonlinear PDEs.

Flight stability of wedges

Recent experiments have shown that cones of intermediate apex angles display orientational stability with apex leading in flight. Here we show in experiments and simulations that analogous results hold in the two-dimensional context of solid wedges or triangular prisms in planar flows at Reynolds numbers Re∼102 to 103. Slender wedges are statically unstable with apex leading and tend to flip over or tumble, and broad wedges oscillate or flutter due to dynamical instabilities, but those of apex half angles between about 40° and 55° maintain stable posture during flight. The existence of “Goldilocks” shapes that possess the “just right” angularity for flight stability is thus robust to dimensionality. We also show that the stability is robust to moderate changes in shape and Reynolds number.

Anterior-Posterior (AP) Calcaneal Profile View: A Novel Radiographic Image to Assess Varus Malalignment.

Assessing and correcting malalignment is important when treating calcaneus fractures. The Harris axial view is commonly utilized to assess varus deformity but may be inherently inaccurate due to its tangential nature. The anterior-posterior (AP) calcaneal profile view is a novel radiographic view that is easily obtained with demonstrated increased accuracy for assessing calcaneal axial alignment. Five nonpaired ankle cadaveric specimens were used in this investigation. Oblique osteotomies were created in relation to the long axis, and varus deformities were produced by inserting solid radiolucent wedges into the osteotomies to create models of 10, 20, and 30 degrees of angulation of the calcaneal tuberosity. Specimens were imaged using both the Harris axial view and the AP calcaneal profile view. For cadavers with 10 degrees of actual varus angulation, the mean Harris axial view angle and the AP calcaneal profile view angle were 10.9 ± 4.8 (range, 5.5-16.0) degrees and 13.0 ± 5.5 (range, 7.3-20.9) degrees, respectively. For cadavers with 20 degrees of actual varus angulation, the mean Harris view angle and the AP calcaneal profile view angle were 11.5 ± 2 (range, 8.2-13.6) degrees and 18.1 ± 4.8 (range, 11.7-23.5) degrees, respectively (P = .005). On pairwise comparison with Bonferroni correction, there was a significant difference between the Harris axial view angle and both the AP calcaneal profile view angle (P = .012) and actual angulation (P = .011). For cadavers with 30 degrees of actual varus angulation, the mean Harris axial view angle and the AP calcaneal profile view angle were 18.3 ± 4.3 (range, 13.3-23.6) degrees and 28.3 ± 2.9 (range, 24.4-31.1) degrees, respectively (P < .001). On pairwise comparison with Bonferroni correction, there was a significant difference between the Harris axial view angle and both the AP calcaneal profile view angle (P = .001) and actual angulation (P < .001). There was no significant difference between the AP calcaneal profile view angle and actual angulation (P > .999). The AP calcaneal profile view is a novel radiographic view that is easily obtained with demonstrated increased accuracy for assessing calcaneal axial alignment. While both views demonstrated similar measurement error for lesser degrees of varus malalignment, the AP calcaneal profile view demonstrated more accurate measurement of increasing heel varus compared with the Harris view. The AP calcaneal profile view could be used in addition to other radiographic views when treating displaced, intra-articular calcaneus fractures to help optimize correction of hindfoot alignment.

Parametric Investigation of the Effects of Localization and Slenderness on the Stress–Strain Response and Confinement Efficiency in FRP-Wrapped Concrete Cylinders

In order to improve the efficiency of fiber reinforced plastics (FRP) confinement as a method to repair and strengthen concrete structures, a parametric analysis was carried out to investigate the effects of cylinder slenderness and the stiffness of the confinement on the localization pattern, the stress–strain response and the effectiveness of the confinement. FRP-wrapped concrete cylinders under axial compression were modeled in a high-resolution finite element model. Concrete was modeled as a Mohr–Coulomb material. The bi-linear stress–strain structural responses concur with published experimental data. Localization along discrete shear planes results in a failure mechanism that causes non-uniform hoop stresses in the FRP wrap due to the movement of solid wedges in the mechanism. A characteristic length for localization was identified and found in agreement with published experimental observations. The confinement efficiency shows a clear dependence on the confinement level and a weak dependence on slenderness above the characteristic length. A simple mechanistic model is proposed for the second branch of the bi-linear stress–strain response curve. The results of this study can be used to estimate the confinement efficiency factor and refine the design recommendations of Equation 12.1 of ACI 440.2R17.

The Captain’s View: A Novel Radiographic View to Assess Axial Alignment of the Calcaneus

Category:Hindfoot, TraumaIntroduction/Purpose:The axial alignment of the calcaneus has paramount importance in the management of these fractures. The Harris view has long stood as the recommended radiograph to assess axial alignment. However, given the obliquity at which the radiograph is obtained, it doesn´t represent a true axial view and is subject to inaccuracies secondary to rotational malpositioning of the foot and mismeasurement of angulation. Multiple reports have described the axial alignment as a surgical outcome, but usually this assessment of the residual deformity have no described method.The objectives of this study are to evaluate the capacity of Harris view to assess axial alignment in a cadaveric model and to describe the use of a true AP view of the calcaneus that we have named Captain´s view.Methods:Five below knee amputated fresh-frozen cadaveric specimens were used in the study. For each specimen, the soft tissues over the lateral side were removed to access to the lateral wall. A small wedge of the cuboid was removed to visualize the center of the calcaneocuboid articular surface. LCA-guide and a cannulated drill were used to create a tunnel in the axis of the calcaneus. An oblique osteotomy was performed in order to simulate a non-comminuted fracture. Varus deformity was created by inserting solid radiolucent wedges into the osteotomy to create models of 10, 20, and 30 degrees of varus angulation.Harris and Captain views were obtained for each specimen with 0 (control), 10, 20, and 30 degrees of varus malalignment. Measurements of the deformity were made digitally on each fluoroscopic image.Results:The average degrees of varus in Harris views were 10,9 (5,5-16); 11,5 (8,2-13,6); and 18,3 (13,3-23,6) for 10,20 and 30 degrees of deformity respectively.The average degrees of varus in Captain´s view were 13,0 (7,3-20,9); 18,4 (11,7-23,5); and 28,2 (24,4-31,1) for 10,20 and 30 degrees of deformity respectively.The average degrees of error for varus deformity in Harris views were 4,1 (41%); 8,4 (42%) and 11,6 (39%) for 10,20 and 30 degrees of deformity respectively.The average degrees of error for varus deformity in Captain´s views were 4,8 (48%); 3,6 (18%) and 2,8 (8%) for 10,20 and 30 degrees of deformity respectively.Conclusion:The results of this study show a high rate of mismeasurement for both radiographic views. Despite the average angles have a clear correlation with the severity of varus, the wide range of error observed between specimens make this assessment unreliable and inaccurate. We observed an improvement of accuracy of captain´s view for more severe deformities, but not with Harris views which maintain a 40% mismeasurement in all the settings. Therefore, intraoperative Harris views should not be used in isolation to evaluate axial alignment and Captain´s view provides an additional perspective that can be useful to rule out severe deformities.

Numerical and experimental investigation of oblique shock wave reflection off a water wedge

Shock wave interaction with solid wedges has been an area of much research in past decades, but so far very few results have been obtained for shock wave reflection off liquid wedges. In this study, numerical simulations are performed using the inviscid Euler equations and the stiffened gas equation of state to study the transition angles, reflection patterns and triple point trajectory angles of shock reflection off solid and water wedges. Experiments using an inclined shock tube are also performed and schlieren photography results are compared to simulations. Results show that the transition angles for the water wedge cases are within 5.3 % and 9.2 %, for simulations and experiments respectively, compared to results obtained with the theoretical detachment criterion for solid surfaces. Triple point trajectory angles are measured and compared with analytic solutions, agreement within $1.3^{\circ }$ is shown for the water wedge cases. The transmitted wave in the water observed in the simulation is quantitatively studied, and two different scenarios are found. For low incident shock Mach numbers, $M_{s}=1.2$ and 2, no shock wave is formed in the water but a precursor wave is induced ahead of the incident shock wave and passes the information from the water back into the air. For high incident shock Mach numbers, $M_{s}=3$ and 4, precursor waves no longer appear but instead a shock wave is formed in the water and attached to the Mach stem at every instant. The temperature field in the water is measured in the simulation. For strong incident shock waves, e.g. $M_{s}=4$, the temperature increment in the water is up to 7.3 K.

Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems

When an upstream steady uniform supersonic flow impinges onto a symmetric straight-sided wedge, governed by the Euler equations, there are two possible steady oblique shock configurations if the wedge angle is less than the detachment angle—the steady weak shock with supersonic or subsonic downstream flow (determined by the wedge angle that is less than or greater than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which satisfy the entropy condition. The fundamental issue—whether one or both of the steady weak and strong shocks are physically admissible solutions—has been vigorously debated over the past eight decades. In this paper, we survey some recent developments on the stability analysis of the steady shock solutions in both the steady and dynamic regimes. For the static stability, we first show how the stability problem can be formulated as an initial-boundary value type problem and then reformulate it into a free boundary problem when the perturbation of both the upstream steady supersonic flow and the wedge boundary are suitably regular and small, and we finally present some recent results on the static stability of the steady supersonic and transonic shocks. For the dynamic stability for potential flow, we first show how the stability problem can be formulated as an initial-boundary value problem and then use the self-similarity of the problem to reduce it into a boundary value problem and further reformulate it into a free boundary problem, and we finally survey some recent developments in solving this free boundary problem for the existence of the Prandtl-Meyer configurations that tend to the steady weak supersonic or transonic oblique shock solutions as time goes to infinity. Some further developments and mathematical challenges in this direction are also discussed.

Laser-generated ultrasonic pulse shapes at solid wedges

Laser pulses focused near the tip of an elastic wedge generate acoustic waves guided at its apex. The shapes of the acoustic wedge wave pulses depend on the energy and the profile of the exciting laser pulse and on the anisotropy of the elastic medium the wedge is made of. Expressions for the acoustic pulse shapes have been derived in terms of the modal displacement fields of wedge waves for laser excitation in the thermo-elastic regime and for excitation via a pressure pulse exerted on the surface. The physical quantity considered is the local inclination of a surface of the wedge, which is measured optically by laser-probe-beam deflection. Experimental results on pulse shapes in the thermo-elastic regime are presented and confirmed by numerical calculations. They pertain to an isotropic sharp-angle wedge with two wedge-wave branches and to a non-reciprocity phenomenon at rectangular silicon edges.

An electromechanical impedance-based method for tensile force estimation and damage diagnosis of post-tensioning systems

We propose an effective methodology using electromechanical impedance characteristics for estimating the remaining tensile force of tendons and simultaneously detecting damages of the anchorage blocks. Once one piezoelectric patch is attached on the anchor head and the other is bonded on the bearing plate, impedance responses are measured through these two patches under varying tensile force conditions. Then statistical indices are calculated from the impedances, and two types of relationship curves between the tensile force and the statistical index (TE Curve) and between statistical indices of two patches (SR Curve) are established. Those are considered as database for monitoring both the tendon and the anchorage system. If damage exists on the bearing plate, the statistical index of patch on the bearing plate would be out of bounds of the SR curve and damage can be detected. A change in the statistical index by damage is calibrated with the SR curve, and the tensile force can be estimated with the corrected index and the TE Curve. For validation of the developed methodology, experimental studies are performed on the scaled model of an anchorage system that is simplified only with 3 solid wedges, a 3-hole anchor head, and a bearing plate. Then, the methodology is applied to a real scale anchorage system that has 19 strands, wedges, an anchor head, a bearing plate, and a steel duct. It is observed that the proposed scheme gives quite accurate estimation of the remaining tensile forces. Therefore, this methodology has great potential for practical use to evaluate the remaining tensile forces and damage status in the post-tensioned structural members.

Free boundary problems in shock reflection/diffraction and related transonic flow problems.

Shock waves are steep wavefronts that are fundamental in nature, especially in high-speed fluid flows. When a shock hits an obstacle, or a flying body meets a shock, shock reflection/diffraction phenomena occur. In this paper, we show how several long-standing shock reflection/diffraction problems can be formulated as free boundary problems, discuss some recent progress in developing mathematical ideas, approaches and techniques for solving these problems, and present some further open problems in this direction. In particular, these shock problems include von Neumann's problem for shock reflection-diffraction by two-dimensional wedges with concave corner, Lighthill's problem for shock diffraction by two-dimensional wedges with convex corner, and Prandtl-Meyer's problem for supersonic flow impinging onto solid wedges, which are also fundamental in the mathematical theory of multidimensional conservation laws.

A Smoothed Particle Hydrodynamics model for 3D solid body transport in free surface flows

This study has developed a 3D Smoothed Particle Hydrodynamics (SPH) numerical scheme to reproduce the transport of rigid bodies in free surface flows (e.g. floods, surface waves). It is based on the Euler–Newton equations for body dynamics, implemented through the SPH formalism. This scheme has been coupled to a Weakly Compressible (WC)-SPH model for the main flow, based on the semi-analytic approach (Di Monaco et al. 2011). The SPH boundary treatment of Adami et al. (2012) has been implemented and adapted to free-slip conditions to model the “fluid–solid body” coupling terms. On the other hand, the “solid–solid” (“body–body” and “body–frontier”) interactions are represented by the “boundary force particles” of Monaghan (2005). This technique has been implemented by introducing some modifications to represent the impingements of entire bodies (not only isolated particles), even at low velocities.The new model has been validated on a sequence of 2D and 3D test cases. They involve preliminary tests both with single and multiple bodies and frontiers, four water entries of falling solid wedges, and the propagation of a 3D dam break front. This is driven by the regular lift of a mobile gate, transports a floating body (6 degrees of freedom) and impacts two fixed obstacles and several solid frontiers. This last phenomenon has also been experimentally realized during this study. Validation refers to comparisons vs. measurements, Unsteady Reynolds-Averaged Navier–Stokes (URANS) results, other SPH models, analytical and theoretical solutions.

Asymmetric impact between liquid and solid wedges

The hydrodynamic problem of impact between a solid wedge and a liquid wedge is analysed. The liquid is assumed to be ideal and incompressible; gravity and surface tension effects are ignored. The flow generated by the impact is assumed to be irrotational and therefore can be described by the velocity potential theory. The solution procedure is based on the analytical derivation of the complex-velocity potential in a parameter plane and the function mapping conformally the parameter plane onto the similarity plane. The mapping function is found as a combination of the derivatives of the complex potential in the similarity and parameter planes, through the integral equations for mixed and homogeneous boundary-value problems in terms of the velocity modulus and the velocity angle with the fluid boundary, together with the dynamic and kinematic boundary conditions. These equations are solved through a numerical method. The procedure is first verified through comparisons with some known results. Simulations are then made for a variety of cases, and detailed results are presented in terms of the free surface shape, streamlines, pressure distribution on the wetted solid surface, and contact angles between the free surface and the body surface.

SU-FF-T-201: Effect of Mini-Phantom Material On In-Air Dosimetry Quantities

Purpose: Recent studies suggest that miniphantom material and thickness affect the measured values of in‐air output ratio. These results prompt us to investigate the effect of mini‐phantom depth and material on several in‐air quantities. Method and Materials: Beam data for two photon beams (6x and 15x) were compared for several in‐air quantities: off‐axis ratio (OAR); central axis wedge factor (CAX WF); distance factor (DF) dependence and off‐axis wedge factor (OAX WF). Several miniphantom materials (Lucite, copper,lead and graphite) were used with a radiological depth ranging from 2 to 25g/cm2. Results: OAR decreases with increasing miniphantom thickness and off‐ axis distance x, for the same mini‐phantom material. For open field, the maximum difference is 5% and 8% for 6x and 15x, respectively. CAX for WF were measured for 15°, 45° and 60° wedge angles, for both virtual and solid wedges. WF increases up to 11% with mini‐phantom thickness, when solid wedges are used. Virtual wedges have an effect of less than 1% on WF. Maximum errors were found for 450 solid wedge. DF dependence was measured for several SAD's for 6 and 15x, open and 60° virtual and solid wedges. DF agreed within 3% regardless of beam energy, material and SAD. OAX WF were measured at a few off‐axis points along both wedge and non‐wedge gradient directions for 6x. WF varies by up to 2.6% in the toe direction, by up to 1.1% in the heel direction and by up to 3.3% in the non‐wedge direction. Conclusion: Mini‐phantom material has an effect on measured in‐air dosimetric quantities. Ideally, a water‐equivalent miniphantom should be used. When the miniphantom thickness increases (even for water equivalent material), the value of WF, OAR, DF increase by 11%, 8%, 3%, respectively. The error also increases with Z value of mini‐phantom and photon energy.

Graphical representation of stereochemical configuration (IUPAC Recommendations 2006)

Abstract Stereochemical configuration is determined by the relationship of atoms in three-dimensional space, yet remains most commonly represented in two-dimensional media such as printed publications or computer screens. Recommendations are provided for the display of three-dimensional stereochemical information in two-dimensional diagrams in ways that avoid ambiguity and are likely to be understood correctly by all viewers. Examples are provided for all types of stereochemical configuration, with explanation of which styles are preferred and which should be avoided. Principal recommendations include: Know your audience: Diagrams that have a wide audience should be drawn as simply as possible. Avoid ambiguous drawing styles. Avoid the use of perspective diagrams and class-specific drawing styles (Fischer projections, Haworth projections, etc.) when structures are to be interpreted by computers. Use solid wedges to indicate bonds that project above the plane of the paper and hashed wedges to indicate bonds that project below the plane of the paper; in both cases, the bonds must be oriented with the narrow end at the stereogenic center. Avoid connecting stereogenic centers with a stereobond.

Taxonomic significance of spine morphology in the echinoid genera Diadema and Echinothrix

Abstract. The spine morphology of all established species of Diadema and Echinothrix, including 2 color morphs of E. calamaris, were examined externally and internally via transverse sectioning to identify diagnostic species features and to assess the morphological relationship between species. Forty‐nine different morphological characters were measured and analysed using ordination by multi‐dimensional scaling (MDS) and cluster analysis. Specimens of Diadema paucispinum and D. setosum had very distinct spine structures. In D. paucispinum, the spines were more robust than those of other species of Diadema. This was evident in the spine's internal structure, with large, closely packed solid wedges, a small axial cavity, and rings of trabeculae throughout the spine's length. The spines in D. setosum were distinctive because of their length in relation to test size and the reduced flaring of their verticillations. The spines of other members of this genus were very similar to each other. Without careful sectioning, the spines from specimens of D. antillarum, D. ascensionis, D. mexicanum and D. savignyi were difficult to differentiate. The internal structures of spines for each species did, however, possess a combination of features that differentiated the species. Such features included the shape, orientation, and number of solid wedges, the presence or absence of spokes and rings of trabeculae between the solid wedges, and the presence or absence of tissue within the axial cavity. Individuals of Diadema palmeri also had spines morphologically similar to other species, however, the red pigmentation of these spines (in life and when preserved) made them easily distinguishable. The spine structures of the 2 species of Echinothrix were starkly different, while the white and brown color morphs of E. calamaris had morphologically distinctive ambulacral and interambulacral spines. The blunt, open‐tipped interambulacral spines, with reticular tissue present in the axial cavity of the white color morph, were easily distinguished from the pointed, closed‐tipped spines, with a hollow axial cavity found in the brown color morph. Such differences indicate that the brown color morph is either a subspecies or a separate species. Taken together the data show that each species has significant morphological differences in the structure of the spines. It is evident from our data that spine morphology is a useful tool to differentiate these commonly confused species.

Simulation of 3-D radiation beam patterns propagated through a planar interface from ultrasonic phased array transducers

Phased array transducers are quite often mounted on solid wedges with specific angles in many practical ultrasonic inspections of thin plates <10 mm in their thickness or welded joints with convex crowns. For the reliable application of phased array techniques with testing set-up, it is essential to have thorough understanding on the characteristics of radiation beam pattern produced in the interrogated medium. To address such a need, this paper proposes a systematic way to calculate full 3-D radiation beam patterns produced in the interrogated solid medium by phased array transducers mounted on a solid wedge. In order to investigate the characteristics of radiation beam patterns in steel, simulation is carried out for 7.5 MHz array transducers mounted on an acrylic wedge with the angle of 15.45° with various of steering angles and/or focal planes.

A three-dimensional UTD heuristic diffraction coefficient for complex penetrable wedges

A three-dimensional heuristic diffraction coefficient, useful to predict the electromagnetic field scattered from penetrable wedges of complex geometry, is presented. The model, based on a modification of the exact uniform theory of diffraction (UTD) metallic wedge diffraction coefficient, allows the analysis of a large class of penetrable objects with an adequate degree of accuracy. Numerical results concerning the scattering from solid and hollow penetrable wedges show the suitability of the proposed diffraction coefficient to analyze complex structures of large practical interest. A good agreement with exact numerical results already available in the literature has been found.

Scattering of Rayleigh and Stoneley waves by two adhering elastic wedges

The method of Sommerfeld integrals is used to study propagation of Rayleigh and Stoneley waves in a system of two bonded solid wedges with a common vertex. Numerical results are obtained for configurations with a wide range of angles of steel and aluminum wedges.

Rigorous solutions of acoustic wave diffraction by penetrable wedges

A well-known problem of acoustic wave propagation in two fluid wedges with different densities and wave speeds is addressed by a recently developed approach to more general problems of wave propagation in bi-material solid wedges. The obtained solutions are valid for arbitrary frequencies and the approach is also extendable to wedge structures in which one or both wedges support propagation of shear waves.

On the velocities of localized vibration modes in immersed solid wedges

The approximate theory of localized elastic waves in immersed solid wedges earlier developed for wedges with small values of the apex angle [V. V. Krylov, Proc. IEEE Ultrason. Symp., Cat. #94, CHO 793–796 (1994)] has predicted that the effect of water loading results in velocity decrease for wedge modes travelling in the subsonic regime of wave propagation. The results of this theory, in particular the absolute values of wedge wave velocity calculated for slender Plexiglas wedges, agree well with the corresponding experiments. The present study demonstrates that for relative values of wedge wave velocity, as compared with those for wedges in vacuum, this theory provides good quantitative agreement with the experiments on Plexiglas samples also for large values of the apex angle. In addition to this, a generalization of the theory is undertaken to describe the effect of heavier wedge material and a supersonic regime of wave propagation. The corresponding results show good agreement with the existing velocity measurements in immersed brass wedges.