Wedge Problem Research Articles

Minimal solutions to the local capillary wedge problem

We give sufficient conditions for the existence of minimal capillary graphs over quadrilaterals symmetric with respect to a diagonal. The proof is constructive, making use of the Weierstrass representation theorem for minimal surfaces. In the process, we construct minimal solutions to the local capillary wedge problem for any wedge angle 0 < Φ < n and contact angles γ 1 , γ 2 ∈ (0, π) such that |γ 1 - γ 2 | ≤ π-Φ. When |γ 1 - γ 2 | < π-Φ, the solution presented here has a jump discontinuity at the wedge corner.

On the nonlinear water entry problem of asymmetric wedges

The self-similar solution that characterizes the water impact, with a constant vertical velocity, of a wedge entering the free surface with an arbitrary orientation is derived analytically. The study is carried out by assuming the fluid to be ideal, weightless and with negligible surface tension effects. The solution is based on the complex analysis of nonlinear two-dimensional problems of unsteady free boundary flows and is written in terms of two governing functions, which are the complex velocity and the derivative of the complex potential defined in a parameter domain. The boundary value problem is reduced to the system of an integral and an integro-differential equation in terms of the velocity modulus and of the velocity angle to the free surface, both written as functions of a parameter variable. The system of equations is solved through a numerical procedure which is validated in the case of symmetric wedges. Comparisons with data available in literature are established for this purpose. Results are presented in terms of free surface shape, contact angles at the intersection with the wedge boundary, pressure distribution, force and moment coefficients. For a given wedge angle, the changes induced by the heel angle on the above quantities are discussed. A criterion is proposed to determine the limit conditions beyond which flow separation from the wedge apex occurs. Comparisons with experimental results available in literature are presented.

A curve selection lemma in spaces of arcs and the image of the Nash map

We prove a finiteness result in the space of arcs . As a consequence, we extend to all dimensions the problem of wedges, proposed by Lejeune-Jalabert (Arcs analytiques et résolution minimale des singularités des surfaces quasi-homogénes, Lecture Notes in Mathematics, vol. 777 (Springer, Berlin, 1980), 303–336), and we obtain that an affirmative answer to this problem is equivalent to the surjectivity of the Nash map. This implies, for instance, that the Nash map is bijective for sandwiched surface singularities.

Modified theory of the physical-optics approach to the impedance wedge problem

The problem of a wedge with equal face impedances is examined with a modified theory of physical optics. The surface integral is constructed by use of the impedance boundary condition. The aperture equivalent current is estimated from the behavior of the reflected diffracted field. The integrals obtained are evaluated asymptotically and compared with the exact solution numerically.

Numerical simulation of water entry of twin wedges

The hydrodynamic problem of twin wedges entering water vertically at constant speed is analysed based on the velocity potential theory. The gravity effect on the flow is ignored based on the assumption that the ratio of the entry speed to the acceleration due to gravity is much larger than the time scale of interest. The problem is solved using the complex velocity potential together with the boundary element method through three stages. When the body touches water, the similarity solution is obtained for each wedge in isolation. This is used as the initial solution at the second stage for the time stepping technique for each wedge in a stretched system defined through the ratio of the Cartesian system to the distance the wedge travelled into water. When the disturbed zone of each wedge begins to affect the flow generated by the other wedge, the stretched system is abandoned and the original system is used. At the third stage the full interactions between the two wedges are included. Various results are provided for the wave elevation, pressure distribution and force at different deadrise angles. They are compared with those obtained from a single wedge and the interaction effect is investigated.

The tilted shallow wedge problem

The problem of indentation of a half-plane by a titled, shallow wedge, under frictionless conditions, is studied, using half-plane theory. The contact law itself, together with the pressure distribution and internal state of stress are found. Particular attention is then given to the nature of the pressure and state of stress at the apex, and an asymptotic form appropriate to all angular features within contacts, and therefore of widespread application, is introduced.

Modified theory of physical optics approach to wedge diffraction problems

The problem of diffraction from a perfectly conducting wedge is examined with the modified theory of physical optics (MTPO). The exact wedge diffraction coefficient is compared with the asymptotic edge waves of MTPO integral and related surface currents are evaluated. The scattered electric fields are expressed by using these current components. The total, incident and reflected diffracted fields are compared with the exact series solution of the wedge problem, numerically.

Modified Reynolds equation for coupled stress fluids ? a porous media model

In this paper, the rheological effects of coupled stress fluids on thin film lubrication modeling are developed. Thin porous layers attached to the impermeable substrate are utilized to model the microstructure of bearing surfaces. In the fluid film region, the constitutive equations for coupled stress fluids proposed by Stokes [1] as well as the continuity and momentum equations are applied to model the flow. In the porous region, the Brinkman-extended Darcy equations are applied to model the flow. Under the usual assumption of hydrodynamic lubrication applicable to thin films, the effects of viscous shear and the stress jump boundary condition at the porous media/fluid film interface are included in deriving the modified Reynolds equation. The effects of material properties such as coupled stress parameter Open image in new window viscosity ratio (αi2), thickness of porous layer (Δi), permeability (Ki), and stress jump parameter (βi), on the velocity distributions and load capacities of one-dimensional converging wedge problems are discussed.

The Plane Wedge Problem Loaded at Its Apex ? the Self-similarity Property and the Characteristic Vector

In the present paper the elastostatic problem of a generally anisotropic and angularly inhomogeneous plane wedge loaded at its apex by a concentrated force, is studied in linear elasticity. At first the self-similarity property is formulated and the stress field of the inhomogeneous anisotropic self-similar wedge problem, is deduced. The wedge is radially separated and the plane wedge problem is reformulated by the introduction of a characteristic vector. Furthermore, the angular distribution of the load is determined. The multi-material wedge problem in terms of a formulation based on the isotropic angularly inhomogeneous wedge, is confronted, and necessary conditions that ensure the self-similarity property, are found. Finally, the “similar” elastostatic wedge problems and the involution between stresses, are studied.

Singular Higher Order Complete Vector Bases for Finite Methods

This paper presents new singular curl- and divergence-conforming vector bases that incorporate the edge conditions. Singular bases complete to arbitrarily high order are described in a unified and consistent manner for curved triangular and quadrilateral elements. The higher order basis functions are obtained as the product of lowest order functions and Silvester-Lagrange interpolatory polynomials with specially arranged arrays of interpolation points. The completeness properties are discussed and these bases are proved to be fully compatible with the standard, high-order regular vector bases used in adjacent elements. The curl (divergence) conforming singular bases guarantee tangential (normal) continuity along the edges of the elements allowing for the discontinuity of normal (tangential) components, adequate modeling of the curl (divergence), and removal of spurious modes (solutions). These singular high-order bases should provide more accurate and efficient numerical solutions of both surface integral and differential problems. Sample numerical results confirm the faster convergence of these bases on wedge problems.

A recipe to detect the error in discretization schemes

AbstractThe necessity for a reliable measure of the discretization error arises in adaptive mesh refinement and in moving mesh adaptation. The present work discusses a detector of the discretization error based on the interpolation reconstruction of the operators. The technique presented here is named operator recovery error source detector (ORESD). Its main features are: First, the technique is based on the operators being discretized and does not require any user intervention or any a priori knowledge of the solution or its properties. Second, the ORESD is an a posteriori error indicator, but it is shown to be consistent with the a priori error provided by the modified equation approach. Third, the technique is based on the operators being solved and is tailored to the specific problem at hand. Four, the technique is simple and is based on a small stencil, resulting in a very inexpensive error detection.In the present work, the ORESD is derived and applied to two tutorial examples: divergence and gradient. With the aid of the two examples and using the general derivation, the ORESD is then applied to the gas dynamics equations. Two benchmarks are used to test the performance. First, a shock tube problem is solved (Sod's benchmark) in a Lagrangian and in a Eulerian frame. Second, the Colella's wedge problem is solved using CLAWPACK. Finally, the ORESD is applied to the 2D Poisson equation on a uniform and on a non‐uniform grid to test the application to elliptic problems. In all examples the operator recovery error source detector succeeds in detecting the real sources of error. Copyright © 2004 John Wiley &amp; Sons, Ltd.

Image of the Nash map in terms of wedges

M. Lejeune-Jalabert (Lecture Notes in Math., vol. 777, Springer-Verlag, 1980, pp. 303–336) proposed the following ‘problem of wedges’: let X be a surface over an algebraically closed field k of characteristic zero. Given a wedge φ : Speck[[ξ,t]]→X , whose special arc lifts to the minimal resolution Y of X in an arc transversal to an irreducible component of the exceptional locus in a general point, does φ lift to Y? The main result in this Note is to extend this problem to a problem of wedges in a variety X of any dimension and to prove that, if the wedge problem is true for X, then the Nash problem is true for X. From this, necessary and sufficient conditions are given for an essential divisor to belong to the image of the Nash map, and we conclude that the Nash problem is true for sandwiched surface singularities. To cite this article: A.J. Reguera, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

A Note on Design of Fiber-Nets for Maximum Stiffness

For a specific fiber-net, the design variables are relative fiber density and fiber orientations. With a given amount of fibers, our objective is to maximize the stiffness (minimize the compliance) for a continuum subjected to a given load situation. Analytical optimality criteria are derived, and numerical optimization procedures are presented. Applications to a wedge problem from the literature, Royer-Carfagni (2000), and to a skew plate problem are shown, and the study includes the influence of a basic material to be reinforced. The procedure described can be combined with localized optimal design for density, orientation, and shape, “pointwise” or for design regions.

Singularities in 2D anisotropic potential problems in multi-material corners: Real variable approach

An analysis of singular solutions at corners consisting of several different homogeneous wedges is presented for anisotropic potential theory in plane. The concept of transfer matrix is applied for a singularity analysis first of single wedge problems and then of multi-material corner problems. Explicit forms of eigenequations for evaluation of singularity exponent in the case of multi-material corners are derived both for all combinations of homogeneous Neumann and Dirichlet boundary conditions at faces of open corners and for multi-material planes with singular interior points. Perfect transmission conditions at wedge interfaces are considered in both cases. It is proved that singularity exponents are real for open anisotropic multi-material corners, and a sufficient condition for the singularity exponents to be real for anisotropic multi-material planes is deduced. A case of a complex singularity exponent for an anisotropic multi-material plane is reported, apparently for the first time in potential theory. Simple expressions of eigenequations are presented first for open bi-material corners and bi-material planes and second for a crack terminating at a bi-material interface, as examples of application of the theory developed here. Analytical solutions of these eigenequations are presented for interface cracks with any combination of homogeneous boundary conditions along the interface crack faces, and also for a special case of a crack perpendicular to a bi-material interface. A numerical study of variation of the singularity exponent as a function of inclination of a crack terminating at a bi-material interface is presented.

Torsional load transfer from a rigid shaft to an elastic plane with a slit

A plane elastostatic problem for an elastic wedge loaded by a concentrated moment at its apex provides an example of violation of the Saint-Venant principle for apex angles 2α larger than π. Considering the problem for a truncated wedge, Neuber demonstrated the method of construction of an applicable solution for any apex angles in the range π≤2α≤2π, despite the failure of the Saint-Venant principle. In the present paper the particularly important case of the truncated-wedge problem is examined. The truncated wedge degenerates into a slitted elastic plane, while a rigid circular shaft, acted upon by a torsional moment, is inserted into the plane. The analytical solution of the mixed boundary-value problem is obtained. Numerical results turn out to be in complete agreement with Neuber's results for the slitted elastic plane.

Using numerical experiments to discover theorems in differential equations

This work explores the use of numerical experiments in two specific cases: (1) the discovery of two families of exact solutions to the elastic string equations, and approximately periodic solutions that appear to exist near pseudo-solutions formed from these families; (2) the study of the diffusion-reaction-conduction process in an electrolyte wedge (meniscus corner) of a current-producing porous electrode. This latter work establishes the well-posedness of the electrolyte wedge problem and provides asymptotic expansions for the current density and total current produced by such a wedge. The theme of this paper is the use of computing to discover a result that is difficult or impossible to find without a computer, but which once observed, can then be proven mathematically.

On scattering of an SH-wave by a corner comprised of two different elastic materials

The problem of the scattering of plane SH-waves by a corner comprised of two different elastic materials is considered. This problem is the analogy of the dielectric wedge problem in electromagnetics. A Fourier transform method is used to derive equations which can readily be solved numerically. The unknowns are the Fourier transforms of the displacements and tractions on the interfaces of the wedge. Fredholm integral equations of the second kind with simple continuous kernels are obtained after taking a physically reasonable representation of the unknown quantities. Numerical results for the diffraction coefficients are presented. The method can be generalized for incidence of Stonely, plane P-, or SV-waves.

Decoupled formulation of piezoelectric elasticity under generalized plane deformation and its application to wedge problems

Abstract This paper presents the formulation of piezoelectric elasticity under generalized plane deformation derived from the three-dimensional theory. There are four decoupled classes in the generalized plane deformation formulation, i.e. when l 3 ( μ )= l 2 * ( μ )=0, l 3 ( μ )= l 3 * ( μ )=0, l 3 * ( μ )= l 2 * ( μ )=0 or l 3 ( μ )= l 3 * ( μ )= l 2 * ( μ )=0. Only the inplane fields of the first class and the antiplane field of the second class include the piezoelectric effect. Several examples of wedge problem often encountered in smart structures, such as sensors or actuators are studied to examine the stress singularity near the apex of the structure. The bonded materials to the PZT-4 wedge are PZT-5, graphite/epoxy or aluminum (conductor). The influencing factors on the singular behavior of the electro-elastic fields include the wedge angle, material type, poling direction, and the boundary and interface conditions. The numerical results of the first case are compared with Xu's graphs and some comments are made in detail. In addition, some new results regarding the antiplane stress singularity of the second class are obtained via the case study. The coupled singularity solutions under generalized plane deformation are also investigated to seek the conditions of the weakest or vanishing singular stress fields.

Coupled-mode solutions in generalized ocean environments.

This paper presents the application of the differential equation approach to solving the second-order coupled-mode equations in inhomogeneous ocean environments. The model incorporates sound velocity profile points to construct depth-dependent, piecewise linear, ocean and bottom environments along a range grid. Modal solutions are evaluated in terms of Airy functions. The formalism to evaluate analytically the mode-coupling coefficients is presented. Comparisons to conventional expressions of the coefficients are made. The integro-differential form of the coupled equations is solved using an approach developed in nuclear theory that incorporates the Lanczos method [Knobles, J. Acoust. Soc. Am. 96, 1741-1747 (1994)]. Demonstration of the practicality of this approach is made by applying the results in actual calculations with realistic ocean environments. The formalism to evaluate analytically the mode-coupling coefficients is presented. Several benchmark examples were examined in order to validate the model and are discussed, including propagation over a hill, benchmark wedge problems, and a range-varying sound speed profile benchmark. The importance of this model is also demonstrated by the physical insight gained in having a coupled-mode approach to solving range-dependent problems.

The problem of an inclusion in a three-dimensional elastic wedge

Fundamental solutions for a three-dimensional wedge are used to investigate problems of a thin, rigid, elliptic inclusion in a wedge. A regular asymptotic form is employed which has previously been used in contact problems for a wedge [1] and in problems of a crack in a wedge [2] in the case of an elliptic shape of the contact region or crack. The method is effective in the case of an inclusion which is sufficiently distant from an edge of the wedge when the known exact solution for the space [3] can be taken as the zeroth approximation. A numerical analysis and comparison of different characteristics of wedge problems is carried out.