Semi-infinite Wedge Research Articles

A generalized antiplane singularity within a semi-infinite wedge of arbitrary angle

The state of stress arising within an elastic wedge generated by an antiplane singularity present within it is studied. An analytical solution is derived with a unified generic approach. The singularity may represent either an anti-plane concentrated force or a screw dislocation. To validate the general solution, two degenerate cases are presented. Further, the image force present on the screw dislocation due to the wedge free boundary is obtained. It is found that when the screw dislocations are placed in the vicinity of the wedge surface, the image force will drive dislocations to the free boundary where they will be annihilated. This implies that a dislocation-free zone may exist along the free surface of the wedge. To demonstrate the application of the fundamental solutions, a formulation for a slip band under anti-plane loading with Green’s function method is provided. The solutions developed in this study may be used as building blocks to model the damage of material near a V-notch under versatile anti-plane load conditions or torsional loading.

Numerical geometric acoustics: An eikonal-based approach for modeling sound propagation in 3D environments

We present algorithms for solving high-frequency acoustic scattering problems in complex domains. The eikonal and transport partial differential equations from the WKB/geometric optic approximation of the Helmholtz equation are solved recursively to generate boundary conditions for a tree of eikonal/transport equation pairs, describing the phase and amplitude of a geometric optic wave propagating in a complicated domain, including reflection and diffraction. Edge diffraction is modeled using the uniform theory of diffraction. For simplicity, we limit our attention to domains with piecewise linear boundaries and a constant speed of sound. The domain is discretized into a conforming tetrahedron mesh. For the eikonal equation, we extend the jet marching method to tetrahedron meshes. Hermite interpolation enables second order accuracy for the eikonal and its gradient and first order accuracy for its Hessian, computed using cell averaging. To march the eikonal on an unstructured mesh, we introduce a new method of rejecting unphysical updates by considering Lagrange multipliers and local visibility. To handle accuracy degradation near caustics, we introduce several fast Lagrangian initialization algorithms. We store the dynamic programming plan uncovered by the marcher in order to propagate auxiliary quantities along characteristics. We introduce an approximate origin function which is computed using the dynamic programming plan, and whose 1/2-level set approximates the geometric optic shadow and reflection boundaries. We also use it to propagate geometric spreading factors and unit tangent vector fields needed to compute the amplitude and evaluate the high-frequency edge diffraction coefficient. We conduct numerical tests on a semi-infinite planar wedge to evaluate the accuracy of our method. We also show an example with a more realistic building model with challenging architectural features. Finally, we demonstrate a simple approach to extending the method to handle nonconstant speeds of sound by modifying the semi-Lagrangian updates to account for a varying speed.

A wedge of arbitrary angle interacting with a generalized singularity

In this paper, the state of stress induced within a semi-infinite wedge of arbitrary angle by a generalized singularity is studied, and its analytical solution is derived. The wedge may, usefully, be a V-shaped notch if the internal wedge angle is larger than π, and the generalized singularity may represent either a concentrated force or an edge dislocation. Several scenarios are studied to validate the general solution. As an interaction example, the wedge interacting with an edge dislocation is considered, and the driving force on the dislocation is obtained in terms of the vector J-integral. The solutions developed in this study give basic information about the state of stress induced in the wedge, which, as well as representing crystalline defects, may be used as building blocks to model the damage at the V-notch root under complex load conditions, as well as to model elastic edge contacts. In addition, it is proved that the stress distribution behavior at the vertex of wedge is categorically determined by its open angle.

Double Hurwitz numbers: polynomiality, topological recursion and intersection theory

Double Hurwitz numbers enumerate branched covers of {{{mathbb {C}}}}{{{mathbb {P}}}}^1 with prescribed ramification over two points and simple ramification elsewhere. In contrast to the single case, their underlying geometry is not well understood. In previous work by the second- and third-named authors, the double Hurwitz numbers were conjectured to satisfy a polynomiality structure and to be governed by the topological recursion, analogous to existing results concerning single Hurwitz numbers. In this paper, we resolve these conjectures by a careful analysis of the semi-infinite wedge representation for double Hurwitz numbers. We prove an ELSV-like formula for double Hurwitz numbers, by deforming the Johnson–Pandharipande–Tseng formula for orbifold Hurwitz numbers and using properties of the topological recursion under variation of spectral curves. In the course of this analysis, we unveil certain vanishing properties of Omega -classes.

On the evolutions of triple point structure in wedge-stabilized oblique detonations

The oblique detonation induced by a two-dimensional semi-infinite wedge is simulated numerically with the Navier–Stokes equations and a detailed H2/air reaction model based on the open-source program-Adaptive Mesh Refinement in Object-oriented C++. A spatially seventh-order-accurate weighted essentially non-oscillatory scheme is adopted for the convective flux discretization. The formation and evolution of the oblique detonation induced by wedges at different angles and inflow conditions are investigated, and a prediction model for the oblique detonation flow field is proposed. The results show that the formation of the oblique detonation flow field can be divided into two processes. The first process is similar to the oblique shock flow field with unreactive inflow. When the inflow passes through the wedge, the oblique shock wave starts to form at the tip, followed by the unstable curved shock surface and triple point. In this process, a thin reaction layer is formed on the wedge front, but the thickness of the reaction layer is almost constant. The second process is similar to the process of deflagration to detonation. As the reaction rate increases, the deflagration front is fixed on the wedge, the reaction layer thickens, and the deflagration front gradually approaches the oblique shock wave. When the deflagration front is coupled with the oblique shock wave, the oblique detonation is formed. Moreover, a theoretical prediction model for the triple point location is proposed. Compared with the numerical simulation results, the theoretical model prediction for the position of the transition point of the oblique shock wave–oblique detonation wave is relatively acceptable.

Cracking Pattern of Indented Ice Sheet Bending Failures

AbstractIce sheet bending failures have been investigated extensively for ice loads on conical offshore structures and icebreakers in arctic regions. Most previous theoretical studies focus on bending failures of semi-infinite level ice, ice wedges, or finite-sized rectangular ice floes. For indented ice sheet bending failures, analytical ice load models have been developed by assuming a radial-before-circumferential cracking pattern. Recently, real-time simulations of ice-structure interactions are gaining increasing traction due to their great application potential. The analytical or semi-analytical models implemented into the real-time simulator significantly influence the accuracy of real-time simulations. Against this backdrop, the cracking pattern assumption needs to be more critically examined, and the criterion for cracking pattern determinations is in demand for utilizing different models for different cracking patterns in real-time simulations. Motivated by this need, the current paper establishes the cracking pattern determination criterion for indented ice sheet bending failures, based on the theory of plates on elastic foundations and normalized formulae. It is found that large indentation lengths and radii of structure waterline curvature induce a circumferential-before-radial cracking pattern. Conversely, small indentation lengths and radii of structure waterline curvature result in a radial-before-circumferential cracking pattern.

Wave structure of oblique detonation disturbed by an expansion wave from a bended tunnel

Oblique detonation waves (ODWs) have attracted increasing attention in recent years due to their potential application to air-breathing hypersonic propulsion. Most of the previous studies on them were based on a semi-infinite wedge, considering the ODW itself but ignoring its interaction with geometric confinement. In this study, the ODWs based on bended tunnel geometry had been studied with a detailed hydrogen–air chemical reaction model. An expansion wave was introduced by the bended upper-wall, and then interacted with the original ODW surface, resulting in a series of complicated wave configurations along different positions of the upper expansion wave. When the expansion wave was induced far upstream before the original ODW surface, the ODW surface would be partially decoupled. And if the expansion wave has located downstream enough, the reflection of the ODW surface on the wall would induce an unsteady ODW. However, a modest distance between the ODW and expansion wave created a structure that had never been reported before. This novel structure was found to be combined by two high-temperature regions, one was from the original ODW reaction zone while the other append near the upper wall after the turning point. The new high-temperature region was found to be overlapped by a recirculation region, suggesting a balance between flow and heat release.

Linear stability of the flow of a second order fluid past a wedge

The linear stability analysis of Rivlin–Ericksen fluids of second order is investigated for boundary layer flows, where a semi-infinite wedge is placed symmetrically with respect to the flow direction. Second order fluids belong to a larger family of fluids called order fluids, which is one of the first classes proposed to model departures from Newtonian behavior. Second order fluids can model non-zero normal stress differences, which is an essential feature of viscoelastic fluids. The linear stability properties are studied for both signs of the elasticity number K, which characterizes the non-Newtonian response of the fluid. Stabilization is observed for the temporal and spatial evolution of two-dimensional disturbances when K > 0 in terms of increase of critical Reynolds numbers and reduction of growth rates, whereas the flow is less stable when K < 0. By extending the analysis to three-dimensional disturbances, we show that a positive elasticity number K destabilizes streamwise independent waves, while the opposite happens for K < 0. We show that, as for Newtonian fluids, the non-modal amplification of streamwise independent disturbances is the most dangerous mechanism for transient energy growth, which is enhanced when K > 0 and diminished when K < 0.

Numerical study on transition structures of oblique detonations with expansion wave from finite-length cowl

The oblique detonation wave (ODW) triggered by a semi-infinite cowl or wedge has been widely studied, but the effects of finite-length cowls require further clarification to enhance their applicability. Based on two-dimensional inviscid Euler equations and a two-step induction–reaction model, two structures with smooth and abrupt transitions are simulated, and their interactions with the cowl-induced expansion wave are investigated. The expansion waves located downstream and far away from the initiation zone do not affect the macroscopic structures of the ODW. However, when the expansion waves move upstream, the structures evolve into a decoupled shock and a reactive front. Alongside findings from previous studies, these results indicate that the initiation mechanism, rather than the transition type, determines the near-quenching evolution of the structure. Moreover, given the same parameter settings for the chemical and inflow gas dynamics, these numerical results show that the abrupt transition evolves into a smooth transition under an expansion wave disturbance. This demonstrates that the local wave interaction plays a key role in the ODW structures.

X-Ray Transmission through Infinite Dielectric Wedge-Shaped Objects

The problem of transmission and reflection of X-rays (0.1 A ≤ λ ≤ 10 A) from an infinite amorphous homogeneous wedge-shaped dielectric plate has been analytically solved. The obtained solution is a “zero” step in solving the problem of X-ray diffraction on a semi-infinite amorphous homogeneous dielectric wedge within the heuristic geometric diffraction theory.

A completely coated wedge crack interacting with a screw dislocation

We employ conformal mapping and analytic continuation to study the elastic field induced by a screw dislocation interacting with a completely coated semi-infinite wedge crack under mode III loading conditions. The screw dislocation can be located either inside the matrix or in the coating itself. Explicit expressions for the local wedge stress intensity factor and the image force acting on the screw dislocation are obtained. The dislocation emission criterion from the completely coated wedge crack is established.

The Effect of Chemical Reactivity on the Formation of Gaseous Oblique Detonation Waves

High-fidelity numerical simulations using a Graphics Processing Unit (GPU)-based solver are performed to investigate oblique detonations induced by a two-dimensional, semi-infinite wedge using an idealized model with the reactive Euler equations coupled with one-step Arrhenius or two-step induction-reaction kinetics. The novelty of this work lies in the analysis of chemical reaction sensitivity (characterized by the activation energy Ea and heat release rate constant kR) on the two types of oblique detonation formation, namely, the abrupt onset with a multi-wave point and a smooth transition with a curved shock. Scenarios with various inflow Mach number regimes M0 and wedge angles θ are considered. The conditions for these two formation types are described quantitatively by the obtained boundary curves in M0–Ea and M0–kR spaces. At a low M0, the critical Ea,cr and kR,cr for the transition are essentially independent of the wedge angle. At a high flow Mach number regime with M0 above approximately 9.0, the boundary curves for the three wedge angles deviate substantially from each other. The overdrive effect induced by the wedge becomes the dominant factor on the transition type. In the limit of large Ea, the flow in the vicinity of the initiation region exhibits more complex features. The effects of the features on the unstable oblique detonation surface are discussed.

Diffraction simulation from wedges to finite-sized plates based on the physical theory of diffraction

Efficient predictions of sound diffraction around objects are of critical significance in room-acoustic simulations. An advanced diffraction theory has recently been investigated for potential applications in room acoustics for some semi-infinite, canonical wedges and for finite-sized rectangular plates [Rozynova and Xiang, J. Acoust. Soc. Am. 144 (to be published)]. The physical theory of diffraction (PTD) still relies on both geometrical and physical principles, yet emphasizes the physical one. Important features of the PTD approach are its computational efficiency and the high degree of accuracy for the diffracted sound field. This paper reviews the fringe field predictions of canonical semi-infinite wedges and further discusses solutions of diffraction problems on finite, rigid rectangular plates. The PTD is applied to approximate the solutions of a finite-sized, rigid rectangular plate that achieves high numerical efficiency. The PTD simulation allows sound diffraction contributions to be determined independently from two pairs of edges of the rigid plate, while ignoring the edge waves around the corner in far-field. This paper uses numerical implementations of the PTD predictions to demonstrate the simulation efficiency of the PTD in finite-sized objects. The numerical simulations are also validated by some preliminary experimental results carried out using an acoustic goniometer.

Harer–Zagier formula via Fock space

The goal of this note is to provide a very short proof of Harer-Zagier formula for the number of ways of obtaining a genus g Riemann surface by identifying in pairs the sides of a (2d)-gon, using semi-infinite wedge formalism operators.

Wall-crossing formulae and strong piecewise polynomiality for mixed Grothendieck dessins d'enfant, monotone, and double simple Hurwitz numbers

We derive explicit formulae for the generating series of mixed Grothendieck dessins d'enfant/monotone/simple Hurwitz numbers, via the semi-infinite wedge formalism. This reveals the strong piecewise polynomiality in the sense of Goulden–Jackson–Vakil, generalising a result of Johnson, and provides a new explicit proof of the piecewise polynomiality of the mixed case. Moreover, we derive wall-crossing formulae for the mixed case. These statements specialise to any of the three types of Hurwitz numbers, and to the mixed case of any pair.

A numerical study on the instability of oblique detonation waves with a two-step induction–reaction kinetic model

In this study, the surface instability of oblique detonation waves (ODW) formed by two-dimensional, semi-infinite wedges is investigated numerically by solving the unsteady Euler equations with a two-step induction–reaction kinetic model. The chemical kinetic model introduces two length scales, namely, induction and reaction lengths, which can be varied independently to change the sensitivity of the chemical reaction and also the shape of the reaction zone structure. The present numerical results elucidate that both smooth and cellular ODW surfaces may appear after the initiation, and the surface becomes unstable when the reaction zone length decreases while keeping the induction zone the same as observed in normal detonation wave propagation. To investigate the degree of instability quantitatively, the oscillations of post-shock pressure inside the reaction zone are examined, and analyzed using Fast Fourier Transformation (FFT) to get the power spectral density (PSD). Results suggest that there are two types of unstable surfaces, one is dominated by random disturbances, without distinct large amplitude unstable modes, on the ODW surface due to the upstream perturbations interacting with the incoming flow and continuous generation within the structure, while the other formed from the inherent disturbances convected from upstream in the initiation region and later developed into dominant unstable modes via an apparent bifurcation pattern. Equivalent to normal detonations, the stability parameter χ as defined by the ratio of induction length over the reaction length multiplied by the global reduced activation energy can also be used to describe qualitatively the trends of the ODW surface instability observed in this study.

A high-order doubly asymptotic continued-fraction solution for frequency-domain analysis of vector wave propagation in unbounded domains

The scaled boundary finite element (SBFE) equation in dynamic stiffness of unbounded domains is efficiently solved by the continued fraction expansion, which has a large convergence range and a high convergence rate. In this paper, a doubly asymptotic continued fraction solution algorithm for frequency-domain analysis of vector wave propagation in unbounded domains of arbitrary geometry is developed based on the high-frequency (singly) asymptotic approach. The exact solution over the whole frequency range can be calculated rapidly with increasing order of expansion. The coefficient matrices of the solution are determined recursively by satisfying the dynamic stiffness at both high (ω→∞) and low (ω→0) frequency limits. The solutions of out-plane motion of a semi-infinite wedge, in-plane motion of a semi-infinite wedge and a 3-D homogeneous half space in the frequency domain are presented. The results of three numerical examples demonstrate that the doubly asymptotic algorithm is more stable and superior than the high-frequency asymptotic approach.

Effects of inflow Mach number on oblique detonation initiation with a two-step induction-reaction kinetic model

Oblique detonations induced by two-dimensional, semi-infinite wedges are simulated by solving numerically the reactive Euler equations with a two-step induction-reaction kinetic model. Previous results obtained with other models have demonstrated that for the low inflow Mach number M0 regime past a critical value, the wave in the shocked gas changes from an oblique reactive wave front into a secondary oblique detonation wave (ODW). The present numerical results not only confirm the existence of such critical phenomenon, but also indicate that the structural shift is induced by the variation of the main ODW front which becomes sensitive to M0 near a critical value. Below the critical M0,cr, oscillations of the initiation structure are observed and become severe with further decrease of M0. For low M0 cases, the non-decaying oscillation of the initiation structure exists after a sufficiently long-time computation, suggesting the quasi-steady balance of initiation wave systems. By varying the heat release rate controlled by kR, the pre-exponential factor of the second reaction step, the morphology of initiation structures does not vary for M0 = 10 cases but varies for M0 = 9 cases, demonstrating that the effects of heat release rate become more prominent when M0 decreases. The instability parameter χ is introduced to quantify the numerical results. Although χ cannot reveal the detailed mechanism of the structural shift, a linear relation between χ and kR exists at the critical condition, providing an empirical criterion to predict the structural variation of the initiation structure.

Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces

Efficient implementation of the method of images is addressed for 3D multiple scattering models (MSM) for spheres with perfectly-reflecting flat surfaces. The Helmholtz equation is solved by the spherical-coordinate separation-of-variables approach and addition theorems for spherical wavefunctions. The unknown coefficients of outgoing waves from image objects are related with those of real counterparts through ‘image conditions’ which are derived in this article. The method of images is applied to concave part of wedge-shaped acoustic domains with apex angles of π/n rad for a positive integer n, which includes half-space and corners. Image conditions make the MSM numerically more efficient: memory space is reduced by 4n2 times; matrix is populated 2n- times faster for infinite or 2D wedges. Savings are 16n2 times in memory and 4n times in speed for semi-infinite or 3D wedges. Image conditions are valid regardless of the type of scatterers as long as they are spheres and submerged in acoustic domains; they are also suitable for the modified Helmholtz equation and radiation problems. However, specific formulae of image conditions depend on definitions of the spherical harmonics. Image conditions for rigid flat surfaces are verified by measurements of 13 balls in an anechoic chamber for configurations of half-space, 2D & 3D corners and 3D wedge with n=3. Image conditions for pressure-release flat interfaces are validated by the boundary element method (BEM) for the pulsation mode of an underwater air sphere in half-space and for scattering by a sphere in an ocean environment with the wedge angle of 1.2° by n=150. Agreement is very good between the MSM and measurements and is impeccable between the MSM and BEM.

Efficient solution for the diffraction of elastic SH waves by a wedge: Performance of various exact, asymptotic and simplified solutions

The diffraction of horizontally polarized shear waves by a semi-infinite wedge in frequency and time domains is studied. In particular, this work focus on the performance of different solutions, including the classical contributions from Macdonald, Sommerfeld and Kouyoumjian & Pathak. In addition, two fully analytical, simplified solutions are proposed using arguments from the so-called geometrical theory of diffraction. The main advantage of the two proposed solutions is the fact that the resulting solutions can be scaled to problems with arbitrary and complex geometries. Moreover, it is found that one of the proposed new solutions is highly efficient in terms of accuracy and computational speed as compared to alternative formulations (approximately 1000 times faster than the Macdonald and Kouyoumjian & Pathak solutions), thus, this important characteristic renders this solution ideal for implementation in GPUs (Graphics Processor Units) for multiscale modeling applications.