Corner Conditions Research Articles

Statistical Corner Conditions of Interconnect Delay (Corner LPE Specifications)

Timing closure in LSI design is becoming more and more difficult. But the conventional interconnect RC extraction method has over-margins caused by its corner conditions settings. In this paper, statistical corner conditions using the independence of variations between process parameters and between interconnect layers are proposed, with examinations using the measurement data. As a result of the method, the fast-to-slow guardband decreases by half in average, compared to the conventional method. The proposed method is ready for implementation to LPE tools.

Classical Solutions for Nonelliptic Euler–Lagrange Equations via Continuation

We consider a higher-gradient model in one-dimensional nonlinear elasticity. Therefore we consider a physically reasonable stored-energy density W such that $W(\nu)$ goes to infinity for $\nu\searrow 0$ and $\nu\nearrow\infty$. We prescribe a parameter-dependent body-forcing. Our main goal is to show under a certain sign condition for the body-force that global solution branches of the singular perturbed problem converge to a global branch of weak solutions of vanishing capillarity. Moreover, the solution satisfies the first and the second Weierstrass–Erdmann corner conditions, and the strain field is uniformly pointwise positive.

ERRATUM: "POLYHOMOGENEOUS SOLUTIONS OF NONLINEAR WAVE EQUATIONS WITHOUT CORNER CONDITIONS"

ERRATUM: "POLYHOMOGENEOUS SOLUTIONS OF NONLINEAR WAVE EQUATIONS WITHOUT CORNER CONDITIONS"

POLYHOMOGENEOUS SOLUTIONS OF NONLINEAR WAVE EQUATIONS WITHOUT CORNER CONDITIONS

The study of Einstein equations leads naturally to Cauchy problems with initial data on hypersurfaces which closely resemble hyperboloids in Minkowski space-time, and with initial data with polyhomogeneous asymptotics, that is, with asymptotic expansions in terms of powers of ln r and inverse powers of r. Such expansions also arise in the conformal method for analysing wave equations in odd space-time dimension. In recent work it has been shown that for non-linear wave equations, or for wave maps, polyhomogeneous initial data lead to solutions which are also polyhomogeneous provided that an infinite hierarchy of corner conditions holds. In this paper we show that the result is true regardless of corner conditions.

Finite element formulation with high-order absorbing boundary conditions for time-dependent waves

The Hagstrom–Warburton high-order absorbing boundary conditions (ABCs) are considered. They are based on a high-order form of the Higdon ABCs using auxiliary variables and constitute a modification of the previously proposed Givoli–Neta ABCs. Here the Hagstrom–Warburton ABCs, which were originally used in a finite difference scheme, are incorporated into a finite element formulation. Exterior time-dependent problems are considered with rectangular computational domains. Special corner conditions are used in conjunction with the ABCs to make the truncated problem well-posed. The properties of the Hagstrom–Warburton and Givoli–Neta formulations are compared, and the relations between the two formulations are established. Numerical examples demonstrate the performance of the Hagstrom–Warburton finite element scheme.

Automatic triangulation over three-dimensional parametric surfaces based on advancing front method

A new mesh generation method is proposed for creating adaptive triangle meshes over three-dimensional parametric surfaces based on the advancing front method. The method is advanced from two opposite boundaries instead of the completed edges. This method solves the problem that will lead to triangulation procedure failure or quality decrease of the surface triangulation results because of poor surface corner conditions. A robust and refined triangular element formation procedure is established, which proceeds directly in three dimensions and then the element is mapped into the parametric space in order to carry out the succeeding functions in two-dimensional space, such as validity checking and convergence checking. A simple but effective convergence checking scheme is also presented in this paper and misjudgement of the convergence points can be avoided. Numerical experiments demonstrate that the triangulation results with high quality can be obtained easily by using the method.

Mathematical analysis of absorbing boundary conditions for the wave equation: the corner problem

Our goal in this work is to establish the existence and the uniqueness of a smooth solution to what we call in this paper the corner problem, that is to say, the wave equation together with absorbing conditions at two orthogonal boundaries. First we set the existence of a very smooth solution to this initial boundary value problem. Then we show the decay in time of energies of high order—higher than the order of the boundary conditions. This result shows that the corner problem is strongly well-posed in spaces smaller than in the half-plane case. Finally, specific corner conditions are derived to select the smooth solution among less regular solutions. These conditions are required to derive complete numerical schemes.

On enforcement of corner conditions in triangular differential quadrature

This paper is concerned with the enforcement of corner conditions in the recently proposed triangular differential quadrature method. The sensitivity of solution to some corner conditions is exemplified and a reduced quadrature technique is introduced to overcome the sensitivity. Numerical examples in the context of bending and vibration of Mindlin plates are studied to validate the proposed reduced quadrature technique. It is shown that the technique is effective to cope with sensitivity of solution to corner conditions. The effect of the order of the reduced quadrature on accuracy of solution is also studied. It is found that satisfactory convergence is usually achieved when the order of reduced quadrature employed at a corner is larger than one half of the order of the triangular differential quadrature approximation in the entire triangle but two orders less than the order of the triangular differential quadrature approximation.

A New Theory and the Efficient Methods of Solution of Strong, Pathwise, Stochastic Variational Problems

This paper has two major objectives. The first objective is to give a new, natural extension of the classical calculus of variations to a stochastic setting. Most significantly, it appears that this is the first time in the literature that a random objective functional is used rather than its mean. This is accomplished by using an appropriate class of variations. Our second major objective is to give an efficient method of solution for these problems. To do this we derive the Euler equation in this setting which is the primary necessary condition for a critical point or extremal solution. We also derive transversality and corner conditions. These results, which are the three basic necessary conditions for an extremal, are sufficient to construct closed form solutions, when they exist. Our results hold for $n \ge 1$ dependent variables. Of particular interest are several example problems. They illustrate random objective functionals that are functions of a Brownian path, pathwise extremals of these functionals, and the use of necessary conditions to find solutions in closed form. We also illustrate that our theory can generalize the usual stochastic control theory by considering individual paths and not the mean value of the objective functional. Finally, these results illustrate steps towards a complete stochastic variational theory for random objective functionals by extending deterministic ideas of the first author. The three necessary conditions of this paper allow a general stochastic control theory with general equality and inequality constraints.

Mantle wedge control on back-arc crustal accretion.

At mid-ocean ridges, plate separation leads to upward advection and pressure-release partial melting of fertile mantle material; the melt is then extracted to the spreading centre and the residual depleted mantle flows horizontally away. In back-arc basins, the subducting slab is an important control on the pattern of mantle advection and melt extraction, as well as on compositional and fluid gradients. Modelling studies predict significant mantle wedge effects on back-arc spreading processes. Here we show that various spreading centres in the Lau back-arc basin exhibit enhanced, diminished or normal magma supply, which correlates with distance from the arc volcanic front but not with spreading rate. To explain this correlation we propose that depleted upper-mantle material, generated by melt extraction in the mantle wedge, is overturned and re-introduced beneath the back-arc basin by subduction-induced corner flow. The spreading centres experience enhanced melt delivery near the volcanic front, diminished melting within the overturned depleted mantle farther from the corner and normal melting conditions in undepleted mantle farther away. Our model explains fundamental differences in crustal accretion variables between back-arc and mid-ocean settings.

Unified formulation of radiation conditions for the wave equation

AbstractA family of radiation boundary conditions for the wave equation is derived by truncating a rational function approximation of the corresponding plane wave representation, and it is demonstrated how these boundary conditions can be formulated in terms of fictitious surface densities, governed by second‐order wave equations on the radiating surface. Several well‐established radiation boundary conditions appear as special cases, corresponding to different choices of the coefficients in the rational approximation. The relation between these choices is established, and an explicit formulation in terms of selected directions with ideal transmission is presented. A mechanical interpretation of the fictitious surface densities enables identification of suitable conditions at corners and boundaries of the radiating surface. Numerical examples illustrate excellent results with one or two fictitious layers with suitable corner and boundary conditions. Copyright © 2001 John Wiley & Sons, Ltd.

A numerical method for the heat equation with non‐smooth corner conditions

AbstractIn this paper, we show that a direct application of a high‐order finite difference method to the solution of the heat equation involving non‐smooth corner conditions does not yield high‐order convergence. We also show that, it is sometimes possible to improve the convergence rate by combining the finite difference method with the numerical calculation of a simple integral. Numerical results are presented to compare the convergence of this new approach with that of the Crank–Nicolson method. Copyright © 2001 John Wiley & Sons, Ltd.

FREE VIBRATION ANALYSIS OF ISOSCELES TRIANGULAR MINDLIN PLATES BY THE TRIANGULAR DIFFERENTIAL QUADRATURE METHOD

The triangular differential quadrature method is applied to the free flexural vibration of isosceles triangular Mindlin plates. The first six frequencies are sought and the convergence of triangular differential quadrature method in vibrational analysis of triangular Mindlin plates is examined. In comparison with available results, good to excellent agreement is achieved for triangular plates with combinations of clamped and simply supported boundary conditions. The implementation of various other boundary conditions in triangular differential quadrature analysis is also discussed and the sensitivity of solution to the implementation of corner conditions in vibrational analysis is highlighted.

Un résultat d'existence et d'unicité pour l'équation de Helmholtz avec conditions aux limites absorbantes d'ordre

In this Note, we prove the existence and uniqueness of the solution to the Helmholtz equation with second order absorbing boundary conditions in a rectangle. The problem is completed by appropriate corner conditions. The formal approach makes use of a regularity result for the solution of the homogeneous problem. To this aim, one has to consider explicitly the singularities of the Laplacian

The Equal Marginal Value Principle: A Graphical Analysis with Environmental Applications

The typical undergraduate intermediate microeconomics textbook uses geometric arguments accompanied by two-dimensional diagrams to present the principles of cost minimization and profit maximization (see Pindyck and Rubinfeld 1995). These tools are sufficient to explain the behavior of a firm with one choice variable-for example, the level of production of a single product from a single factory. More realistic models, where firms have multiple products, factories, and technologies, are usually not presented because they require the use of more sophisticated mathematical techniques, such as Lagrange multipliers. In this article, I develop a simple diagrammatic exposition of optimization principles for problems with two choice variables. In the basic model, the choice variables correspond to production levels in two different factories.' In addition to illustrating the familiar equal marginal value principle, the basic model makes more advanced topics, such as corner conditions and the role of convexity in optimization, accessible to a wide variety of students. An extension of the model illustrates the fundamental relationship between profit maximization and cost minimization. Other extensions are used to analyze important environmental issues, such as the cost effectiveness of policy proposals to reduce auto emissions, the economics of garbage disposal, and the economics of pollution abatement. THE MODEL Consider a firm that produces a single product in two different factories. Let q, be the quantity produced in the first factory and q2 be the quantity produced in the second. The factories have cost functions CI(qi) and C2(q2) and associated marginal cost functions MCI(ql) and MC2(q2), respectively.2 The firm wants to produce q, units of the product so that the total cost of production is minimized. The solution may be found graphically by placing the marginal costs curves back-to-back so that they share the same y axis, but quantity produced increases in opposite directions along the x axis. The total cost associated with any particular division Y, q q of production between the two factories is found by placing a ruler of length q, along the x axis so that the right edge is directly under qf and then calculating the area bounded by the marginal cost curves and the ruler.

Implicit integration and consistent linearization for yield criteria of the Mohr-Coulomb type

In this paper, we discuss the efficient treatment of yield criteria that are of the Mohr–Coulomb type for elastic and plastic isotropy. On the basis of the fully implicit method, we derive the explicit expression for the integrated stress along with a flow rule that represents volumetric non-associativity. The integration algorithm covers all the possible cases of regular, corner and apex solutions including the suitable indicator for each case. We also establish the consequent consistently linearized tangent stiffness modulus tensor, which is shown to appear in the form of an additive modification of the continuum tangent stiffness tensor. The convergence properties of the consistent tangent stiffness tensor are compared with its feasible approximations. The results indicate the strongly sensitivity to the proper treatment of the corner conditions at the establishment of the ATS-tensor.

Vector absorbing boundary conditions for nodal or mixed finite elements

We present 2D and 3D vector absorbing boundary conditions for unbounded microwave problems. Coupling with vector formulation leads to a non-symmetric system matrix. We show bow it is possible to symmetrise these conditions with the same degree of precision. Edge and corner conditions are also presented for rectangular boundaries.

Three‐dimensional finite element for analysing thin plate/shell structures

AbstractA three‐dimensional (3‐D) hexahedron finite element is presented for the analysis of thin plate/shell structures. The element employs an explicit algebraic definition of six uniform (continuum) strains, six rigid body modes and classical Lagrange‐Germain‐Kirchhoff thin plate bending modes. Nine additional stiffness factors are used to control higher‐order hourglass modes. The element may be used for plate/shell analyses where the flat plate assumptions are appropriate. Also it can easily be adapted to form transition elements to lower order 2‐D elements, or to higher‐order 3‐D continuum elements.The stiffness matrix satisfies the geometric isotropy requirement, passes the patch test, and gives essentially identical response to either applied transverse corner forces or to twisting moments applied on the corner, a requirement of Kirchhoff's corner conditions for a classical thin plate. Several examples are presented to demonstrate the performance of this finite element.

A unified nonlinear formulation for plate and shell theories

A general geometrically exact nonlinear theory for the dynamics of laminated plates and shells under-going large-rotation and small-strain vibrations in three-dimensional space is presented. The theory fully accounts for geometric nonlinearities by using the new concepts of local displacements and local engineering stress and strain measures, a new interpretation and manipulation of the virtual local rotations, an exact coordinate transformation, and the extended Hamilton principle. Moreover, the model accounts for shear coupling effects, continuity of interlaminar shear stresses, free shear-stress conditions on the bonding surfaces, and extensionality. Because the only differences among different plates and shells are the initial curvatures of the coordinates used in the modeling and all possible initial curvatures are included in the formulation, the theory is valid for any plate or shell geometry and contains most of the existing nonlinear and shear-deformable plate and shell theories as special cases. Five fully nonlinear partial-differential equations and corresponding boundary and corner conditions are obtained, which describe the extension-extension-bending-shear-shear vibrations of general laminated two-dimensional structures and display linear elastic and nonlinear geometric coupling among all motions. Moreover, the energy and Newtonian formulations are completely correlated in the theory.

Analysis of sidewall fracturing and spalling during coal-pillar extraction : Vervoort, A; Jack, B J S Afr Inst Min MetallV92, N1, Jan 1992, P3-11

During pillar extraction, sidewall fracturing and spalling at some collieries constitute a severe strata-control problem. In order to obtain a better understanding of this phenomenon, a survey of pillar corner conditions before and during pillar extraction was conducted at one particular site. For the various successive mining steps, the vertical stresses on the pillar corners were calculated by use of a three-dimensional boundary element program. A significant correlation was found between the calculated vertical stress prior to the extraction of the panel and the sidewall conditions. During the extraction of the panel, a significant deterioration of the sidewall conditions at a pillar occurred only when the adjacent pillar was mined, causing an increase in vertical stress on the remaining pillar.