Tangential Derivatives Research Articles

Comment on “Continuum modes in rotating plasmas: General equations and continuous spectra for large aspect ratio tokamaks” [Phys. Plasmas 18, 092103 (2011)

It is shown that some of the main results of the recent paper by Lakhin and Ilgisonis [Phys. Plasmas 18, 092103 (2011)], viz. the derivation of the equations for the continuous spectra of poloidally and toroidally rotating plasmas and their special solution for large aspect ratio tokamaks with large parallel flows were obtained before by Goedbloed, Beliën, van der Holst, and Keppens [Phys. Plasmas 11, 28 (2004)]. A further rearrangement of the system of equations for the coupled Alfvén and slow continuous spectra clearly exhibits: (a) coupling through a single tangential derivative, which is a generalization of the geodesic curvature; (b) the “transonic” transitions of the equilibrium, which need to be carefully examined in order to avoid entering hyperbolic flow regimes where the stability formalism breaks down. A critical discussion is devoted to the implications of this failure, which is generally missed in the tokamak literature, possibly as a result of the wide-spread use of the sonic Mach number of gas dynamics, which is an irrelevant and misleading parameter in “transonic” magnetohydrodynamics. Once this obstacle in understanding is removed, further application of the theory of trans-slow Alfvén continuum instabilities to both tokamaks, with possible implications for the L-H transition, and astrophysical objects like “fat” accretion disks, with a possible new route to magnetohydrodynamic turbulence, becomes feasible.

High order integral equation method for diffraction gratings

Conventional integral equation methods for diffraction gratings require lattice sum techniques to evaluate quasi-periodic Green's functions. The boundary integral equation Neumann-to-Dirichlet map (BIE-NtD) method in Wu and Lu [J. Opt. Soc. Am. A 26, 2444 (2009)], [J. Opt. Soc. Am. A 28, 1191 (2011)] is a recently developed integral equation method that avoids the quasi-periodic Green's functions and is relatively easy to implement. In this paper, we present a number of improvements for this method, including a revised formulation that is more stable numerically, and more accurate methods for computing tangential derivatives along material interfaces and for matching boundary conditions with the homogeneous top and bottom regions. Numerical examples indicate that the improved BIE-NtD map method achieves a high order of accuracy for in-plane and conical diffractions of dielectric gratings.

The integral equation method for electromagnetic scattering problem at oblique incidence

We consider the scattering of electromagnetic waves scattered by an infinitely long impedance cylinder at oblique incidence, which is modeled as a system of a pair of the two-dimensional Helmholtz equations with coupled oblique boundary conditions. The solvability of such a scattering problem is proven by using the boundary integral equation method. By expressing the scattered fields in the form of single-layer potentials, our oblique scattering problem is transformed to a system of two integral equations. It is not a usual Fredholm system of the second kind as that in the case of normal incidence, since the system involves the tangential derivatives of the single-layer potential. By relating it to the Cauchy integral operator, we show that this system of operators is of Fredholm type with index 0. Therefore, the solvability of the integral system follows from the uniqueness of its solutions due to the Fredholm theory. A numerical scheme for solving the integral equations is also presented with some numerics. The numerical results illustrate the validity and efficiency of the proposed method.

Efficient implementation of high order inverse Lax–Wendroff boundary treatment for conservation laws

In [20], two of the authors developed a high order accurate numerical boundary condition procedure for hyperbolic conservation laws, which allows the computation using high order finite difference schemes on Cartesian meshes to solve problems in arbitrary physical domains whose boundaries do not coincide with grid lines. This procedure is based on the so-called inverse Lax–Wendroff (ILW) procedure for inflow boundary conditions and high order extrapolation for outflow boundary conditions. However, the algebra of the ILW procedure is quite heavy for two dimensional (2D) hyperbolic systems, which makes it difficult to implement the procedure for order of accuracy higher than three. In this paper, we first discuss a simplified and improved implementation for this procedure, which uses the relatively complicated ILW procedure only for the evaluation of the first order normal derivatives. Fifth order WENO type extrapolation is used for all other derivatives, regardless of the direction of the local characteristics and the smoothness of the solution. This makes the implementation of a fifth order boundary treatment practical for 2D systems with source terms. For no-penetration boundary condition of compressible inviscid flows, a further simplification is discussed, in which the evaluation of the tangential derivatives involved in the ILW procedure is avoided. We test our simplified and improved boundary treatment for Euler equations with or without source terms representing chemical reactions in detonations. The results demonstrate the designed fifth order accuracy, stability, and good performance for problems involving complicated interactions between detonation/shock waves and solid boundaries.

Nonconstant Continuous Functions whose Tangential Derivative Vanishes along a Smooth Curve

AbstractWe provide a simple example showing that the tangential derivative of a continuous function ϕ can vanish everywhere along a curve while the variation of ϕ along this curve is nonzero. We give additional regularity conditions on the curve and/or the function that prevent this from happening.

Ocular Q-factor: an approach to eye aberrations analysis

An approach is proposed to distinguish between the types of aberrations originating from the beam declination in the radial and orthogonal directions. These orthogonal components are derived from the wavefront error map as partial radial and tangential derivatives. We propose defining the ratio of the standard deviation of the radial component to the standard deviation of the tangent component as the eye shape quality factor Q, or ocular Q-factor. Studies showed that this varies with changing the accommodation state, each eye having its own characteristics even for the same patient. The Q-factor becomes lower after surgeries for vision correction. It is higher in the leading eye than in the lagging eye. In four groups of our preliminary study, the Q-factor showed statistical values practically unchanged within the age range of 20–60 years, but becoming lower after the age of 80. The software package can be used as a plug-in to the clinical aberrometers.

Calculus of tangent sets and derivatives of set-valued maps under metric subregularity conditions

In this paper we give some calculus rules for tangent sets in the sense of Bouligand and Ursescu, as well as for corresponding derivatives of set-valued maps. Both first and second-order objects are envisaged and the assumptions we impose in order to get the calculus are in terms of metric subregularity of the assembly of the initial data. This approach is different from those used in alternative recent papers in literature and allows us to avoid compactness conditions. An application to a special type of vector optimization problems, where the objective is given as the sum of two multifunctions, is presented. Furthermore, also as application, a special attention is paid for the case of perturbation set-valued maps which naturally appear in optimization problems.

Dynamic element discretization method for solving 2D traction boundary integral equations

A sufficient condition for the existence of element singular integral of the traction boundary integral equation for elastic problems requires that the tangential derivatives of the boundary displacements are Hölder continuous at collocation points. This condition is violated if a collocation point is at the junction between two standard conforming boundary elements even if the field variables themselves are Hölder continuous there. Various methods are proposed to overcome this difficulty. Some of them are rather complicated and others are too different from the conventional boundary element method. A dynamic element discretization method to overcome this difficulty is proposed in this work. This method is novel and very simple: the form of the standard traction boundary integral equation remains the same; the standard conforming isoparametric elements are still used and all collocation points are located in the interior of elements where the continuity requirements are satisfied. For boundary elements with boundary points where the field variables themselves are singular, such as crack tips, corners and other boundary points where the stress tensors are not unique, a general procedure to construct special elements has been developed in this paper. Highly accurate numerical results for various typical examples have been obtained.

Decay estimates of planar stationary waves for damed wave equations with nonlinear convection in multi-dimensional half space

This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space R + n : (I) { u t t − Δ u + u t + div f ( u ) = 0 , t > 0 , x = ( x 1 , x ′ ) ∈ R + n ( : = R + × R n − 1 ) , u ( 0 , x ) = u 0 ( x ) → u + , as x 1 → + ∞ , u t ( 0 , x ) = u 1 ( x ) , u ( t , 0 , x ′ ) = u b , x ′ = ( x 2 , x 3 , ⋯ , x n ) ∈ R n − 1 . For the non-degenerate case f 1 ′ ( u + ) < 0 , it was shown in [10] that the above initial-boundary value problem (I) admits a unique global solution u( t,x) which converges to the corresponding planar stationary wave ϕ( x 1) uniformly in x 1 ∈ R + as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. And in [10] Ueda, Nakamura, and Kawashima proved the algebraic decay estimates of the tangential derivatives of the solution u( t,x) for t → + ∞ by using the space-time weighted energy method initiated by Kawashima and Matsumura [5] and improved by Nishihkawa [7]. Moreover, by using the same weighted energy method, an additional algebraic convergence rate in the normal direction was obtained by assuming that the initial perturbation decays algebraically. We note, however, that the analysis in [10] relies heavily on the assumption that f 1 ′ ( u b ) < 0 . The main purpose of this paper is devoted to discussing the case of f 1 ′ ( u b ) ≥ 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.

Evaluation of Green's boundary formula and its first‐ and second‐order derivatives in Legendre series

AbstractThis work presents a numerical method for evaluating Green's boundary formula and its first‐ and second‐order derivatives in two dimensions. Analysis includes both the normal and tangential derivatives, which contain logarithmic and strong singularities, as well as hypersingularities of order up to r−3. The singular integrands are expanded by the Laurent series. Regularized equations are obtained by further expressing the unknown functions as a truncated Fourier–Legendre series, and applying the subtraction–addition technique. The resulting singular integrals are then evaluated analytically. The proposed regularization is conducted before any discretization. The regularized equations can be solved by directly applying standard quadrature rules over the entire integration domain. Test examples consist of the Neumann and Dirichlet problems related to an inverse elliptic cylinder, and a mixed boundary value problem related to a square cylinder. Comparing numerical results with analytical solutions demonstrates a very good accuracy, confirming the effectiveness of the proposed method. Copyright © 2009 John Wiley &amp; Sons, Ltd.

Boundary integral equation Neumann-to-Dirichlet map method for gratings in conical diffraction

Boundary integral equation methods for diffraction gratings are particularly suitable for gratings with complicated material interfaces but are difficult to implement due to the quasi-periodic Green's function and the singular integrals at the corners. In this paper, the boundary integral equation Neumann-to-Dirichlet map method for in-plane diffraction problems of gratings [Y. Wu and Y. Y. Lu, J. Opt. Soc. Am. A26, 2444 (2009)] is extended to conical diffraction problems. The method uses boundary integral equations to calculate the so-called Neumann-to-Dirichlet maps for homogeneous subdomains of the grating, so that the quasi-periodic Green's functions can be avoided. Since wave field components are coupled on material interfaces with the involvement of tangential derivatives, a least squares polynomial approximation technique is developed to evaluate tangential derivatives along these interfaces for conical diffraction problems. Numerical examples indicate that the method performs equally well for dielectric or metallic gratings.

A high order moving boundary treatment for compressible inviscid flows

We develop a high order numerical boundary condition for compressible inviscid flows involving complex moving geometries. It is based on finite difference methods on fixed Cartesian meshes which pose a challenge that the moving boundaries intersect the grid lines in an arbitrary fashion. Our method is an extension of the so-called inverse Lax–Wendroff procedure proposed in [17] for conservation laws in static geometries. This procedure helps us obtain normal spatial derivatives at inflow boundaries from Lagrangian time derivatives and tangential derivatives by repeated use of the Euler equations. Together with high order extrapolation at outflow boundaries, we can impose accurate values of ghost points near the boundaries by a Taylor expansion. To maintain high order accuracy in time, we need some special time matching technique at the two intermediate Runge–Kutta stages. Numerical examples in one and two dimensions show that our boundary treatment is high order accurate for problems with smooth solutions. Our method also performs well for problems involving interactions between shocks and moving rigid bodies.

ℎ𝑝-Discontinuous Galerkin methods for the Helmholtz equation with large wave number

In this paper we develop and analyze some interior penalty h p hp -discontinuous Galerkin ( h p hp -DG) methods for the Helmholtz equation with first order absorbing boundary condition in two and three dimensions. The proposed h p hp -DG methods are defined using a sesquilinear form which is not only mesh-dependent (or h h -dependent) but also degree-dependent (or p p -dependent). In addition, the sesquilinear form contains penalty terms which not only penalize the jumps of the function values across the element edges but also the jumps of the first order tangential derivatives as well as jumps of all normal derivatives up to order p p . Furthermore, to ensure the stability, the penalty parameters are taken as complex numbers with positive imaginary parts, so essentially and practically no constraint is imposed on the penalty parameters. It is proved that the proposed h p hp -discontinuous Galerkin methods are stable (hence, well-posed) without any mesh constraint. For each fixed wave number k k , sub-optimal order (with respect to h h and p p ) error estimates in the broken H 1 H^1 -norm and the L 2 L^2 -norm are derived without any mesh constraint. The error estimates as well as the stability estimates are improved to optimal order under the mesh condition k 3 h 2 p − 2 ≤ C 0 k^3h^2p^{-2}\le C_0 by utilizing these stability and error estimates and using a stability-error iterative procedure, where C 0 C_0 is some constant independent of k k , h h , p p , and the penalty parameters. To overcome the difficulty caused by strong indefiniteness (and non-Hermitian nature) of the Helmholtz problems in the stability analysis for numerical solutions, our main ideas for stability analysis are to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in [19, 20, 33], which enable us to derive stability estimates and error bounds with explicit dependence on the mesh size h h , the polynomial degree p p , the wave number k k , as well as all the penalty parameters for the numerical solutions.

Formula omitted] B-spline interpolation to a closed mesh

This paper focuses on interpolating vertices and normal vectors of a closed quad-dominant mesh 1 1 A mesh is quad-dominant if it contains quad facets as its majority. The quantity of quad facets is much more than the quantity of triangular and other multi-sided facets. G 2 -continuously using regular Coons B-spline surfaces, which are popular in industrial CAD/CAM systems. We first decompose all non-quadrangular facets into quadrilaterals. The tangential and second-order derivative vectors are then estimated on each vertex of the quads. A least-square adjustment algorithm based on the homogeneous form of G 2 continuity condition is applied to achieve curvature continuity. Afterwards, the boundary curves, the first- and the second-order cross-boundary derivative curves are constructed fulfilling G 2 continuity and compatibility conditions. Coons B-spline patches are finally generated using these curves as boundary conditions. In this paper, the upper bound of the rank of G 2 continuity condition matrices is also strictly proved to be 2 n − 3 , and the method of tangent-vector estimation is improved to avoid petal-shaped patches in interpolating solids of revolution. Several examples demonstrate its feasibility.

Theoretical study of a Bénard–Marangoni problem

In this paper we prove the existence of strong solutions for the stationary Bénard–Marangoni problem in a finite domain flat on the top, bifurcating from the basic heat conductive state. The Bénard–Marangoni problem is a physical phenomenon of thermal convection in which the effects of buoyancy and surface tension are taken into account. This problem is modelled with a system of partial differential equations of the type Navier–Stokes and heat equation. The boundary conditions include crossed boundary conditions involving tangential derivatives of the temperature and normal derivatives of the velocity field. To define tangential derivatives at the boundary, intended in the trace sense, it is necessary order two derivatives in the interior of the domain and thus the boundary term contains as high derivatives as the interior term. We overcome this difficulty by considering the weak formulation, and transforming the boundary integral into an equivalent integral defined in the whole domain. This allows us to reformulate the weak problem with a temperature having only order one weak derivatives. Concerning regularity results, we obtain strong solutions for the stationary Bénard–Marangoni problem.

Sharp pointwise estimates for directional derivatives of harmonic functions in a multidimensional ball

A representation of the sharp constant in a pointwise estimate for the absolute value of the directional derivative of a harmonic function in a multidimensional ball is obtained under the assumption that the boundary values of the function belong to L p . This representation is specified in the cases of radial and tangential derivatives. It is proved for p = 1 and p = 2 that the maximum of the absolute value of the directional derivative of a harmonic function with a fixed L p -norm of its boundary values is attained at the radial direction. This confirms D. Khavinson’s conjecture for p = 1 and p = 2. Bibliography: 11 titles.

Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws

We develop a high order finite difference numerical boundary condition for solving hyperbolic conservation laws on a Cartesian mesh. The challenge results from the wide stencil of the interior high order scheme and the fact that the boundary intersects the grids in an arbitrary fashion. Our method is based on an inverse Lax-Wendroff procedure for the inflow boundary conditions. We repeatedly use the partial differential equation to write the normal derivatives to the inflow boundary in terms of the time derivatives and the tangential derivatives. With these normal derivatives, we can then impose accurate values of ghost points near the boundary by a Taylor expansion. At outflow boundaries, we use Lagrange extrapolation or least squares extrapolation if the solution is smooth, or a weighted essentially non-oscillatory (WENO) type extrapolation if a shock is close to the boundary. Extensive numerical examples are provided to illustrate that our method is high order accurate and has good performance when applied to one and two-dimensional scalar or system cases with the physical boundary not aligned with the grids and with various boundary conditions including the solid wall boundary condition. Additional numerical cost due to our boundary treatment is discussed in some of the examples.

Inverse resistivity problem: Geoelectric uncertainty principle and numerical reconstruction method

Mathematical model of vertical electrical sounding by using resistivity method is studied. The model leads to an inverse problem of determination of the unknown leading coefficient (conductivity) of the elliptic equation in R 2 in a slab. The direct problem is obtained in the form of mixed BVP in axisymmetric cylindrical coordinates. The additional (available measured) data is given on the upper boundary of the slab, in the form of tangential derivative. Due to ill-conditionedness of the considered inverse problem the logarithmic transformation is applied to the unknown coefficient and the inverse problem is studied as a minimization problem for the cost functional, with respect to the reflection coefficient. The Conjugate Gradient method (CGM) is applied for the numerical solution of this problem. Computational experiments were performed with noise free and random noisy data.

Solving conical diffraction grating problems with integral equations

Off-plane scattering of time-harmonic plane waves by a plane diffraction grating with arbitrary conductivity and general surface profile is considered in a rigorous electromagnetic formulation. Integral equations for conical diffraction are obtained involving, besides the boundary integrals of the single and double layer potentials, singular integrals, the tangential derivative of single-layer potentials. We derive an explicit formula for the calculation of the absorption in conical diffraction. Some rules that are expedient for the numerical implementation of the theory are presented. The efficiencies and polarization angles compared with those obtained by Lifeng Li for transmission and reflection gratings are in a good agreement. The code developed and tested is found to be accurate and efficient for solving off-plane diffraction problems including high-conductive gratings, surfaces with edges, real profiles, and gratings working at short wavelengths.

On boundary regularity for the stress in problems of linearized elasto-plasticity

We investigate regularity properties of the stress tensor near the boundary for models of elasto-plasticity in arbitrary dimension. Focusing on special geometries, namely on balls and infinite strips, we obtain L 2-estimates for the tangential derivatives of the stress tensor near the boundary. We indicate why these estimates may fail for more general domains. In addition, we establish L 2-estimates for (the trace of) the stress tensor on the boundary.