Accurate,high-order representation of complex three-dimensional surfaces via Fourier continuation analysis

We present a new method for construction of high-order parametrizations of surfaces: starting from point clouds, the method we propose can be used to produce full surface parametrizations (by sets of local charts, each one representing a large surface patch – which, typically, contains thousands of the points in the original point-cloud) for complex surfaces of scientific and engineering relevance. The proposed approach accurately renders both smooth and non-smooth portions of a surface: it yields super-algebraically convergent Fourier series approximations to a given surface up to and including all points of geometric singularity, such as corners, edges, conical points, etc. In view of their C^∞ smoothness (except at true geometric singularities) and their properties of high-order approximation, the surfaces produced by this method are suitable for use in conjunction with high-order numerical methods for boundary value problems in domains with complex boundaries, including PDE solvers, integral equation solvers, etc. Our approach is based on a very simple concept: use of Fourier analysis to continue smooth portions of a piecewise smooth function into new functions which, defined on larger domains, are both smooth and periodic. The “continuation functions” arising from a function f converge super-algebraically to f in its domain of definition as discretizations are refined. We demonstrate the capabilities of the proposed approach for a number of surfaces of engineering relevance.     

High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements

We introduce a new methodology for the numerical solution of Partial Differential Equations in general spatial domains: our algorithms are based on the use of the well-known Alternating Direction Implicit (ADI) approach in conjunction with a certain “Fourier continuation” (FC) method for the resolution of the Gibbs phenomenon. Unlike previous alternating direction methods of order higher than one, which can only deliver unconditional stability for rectangular domains, the present high-order algorithms possess the desirable property of unconditional stability for general domains; the computational time required by our algorithms to advance a solution by one time-step, in turn, grows in an essentially linear manner with the number of spatial discretization points used. In this paper we demonstrate the FC-AD methodology through a variety of examples concerning the Heat and Laplace Equations in two and three-dimensional domains with smooth boundaries. Applications of the FC-AD methodology to Hyperbolic PDEs together with a theoretical discussion of the method will be put forth in a subsequent contribution. The numerical examples presented in this text demonstrate the unconditional stability and high-order convergence of the proposed algorithms, as well the very significant improvements they can provide (in one of our examples we demonstrate a one thousand improvement factor) over the computing times required by some of the most efficient alternative general-domain solvers.     

A spectral FC solver for the compressible Navier–Stokes equations in general domains I: Explicit time-stepping

We present a Fourier continuation (FC) algorithm for the solution of the fully nonlinear compressible Navier–Stokes equations in general spatial domains. The new scheme is based on the recently introduced accelerated FC method, which enables use of highly accurate Fourier expansions as the main building block of general-domain PDE solvers. Previous FC-based PDE solvers are restricted to linear scalar equations with constant coefficients. The FC methodology presented in this text thus constitutes a significant generalization of the previous FC schemes, as it yields general-domain FC solvers for nonlinear systems of PDEs. While not restricted to periodic boundary conditions and therefore applicable to general boundary value problems on arbitrary domains, the proposed algorithm inherits many of the highly desirable properties arising from rapidly convergent Fourier expansions, including high-order convergence, essentially spectrally accurate dispersion relations, and much milder CFL constraints than those imposed by polynomial-based spectral methods—since, for example, the spectral radius of the FC first derivative grows linearly with the number of spatial discretization points. We demonstrate the accuracy and optimal parallel efficiency of the algorithm in a variety of scientific and engineering contexts relevant to fluid-dynamics and nonlinear acoustics.     

A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids

Recently a new high-order formulation for 1D conservation laws was developed by Huynh using the idea of “flux reconstruction”. The formulation was capable of unifying several popular methods including the discontinuous Galerkin, staggered-grid multi-domain method, or the spectral difference/spectral volume methods into a single family. The extension of the method to quadrilateral and hexahedral elements is straightforward. In an attempt to extend the method to other element types such as triangular, tetrahedral or prismatic elements, the idea of “flux reconstruction” is generalized into a “lifting collocation penalty” approach. With a judicious selection of solution points and flux points, the approach can be made simple and efficient to implement for mixed grids. In addition, the formulation includes the discontinuous Galerkin, spectral volume and spectral difference methods as special cases. Several test problems are presented to demonstrate the capability of the method.     

Enhancement of the Debye frequency due to a charged-boson exchange model of high temperature superconductors

A simple model of 1-2-3 superconductors in which electrons (holes) in CuO₂ planes interact via exchange with two kinds of bosons is considered. Namely, via one-phonon exchange (weak coupling-Cooper pairing), and via paired holes on oxygen O⁰ from Cu-O chains. The mechanisms of paired holes exchange (“charged bosons”-“O⁰” exchange) considered here in strong coupling leads to the enhancement of the Fröhlich constant g_f (g²_F→Kg²F), and as a consequence to the enhancement of the Debye frequency ω_D^K=f^Kω_D, f^K 1. In the proposed model the exact expression for the constant K is derived.     

A new method for increasing the laser damage threshold of metal mirrors

In order to increase the damage threshold of metal mirrors we propose to create a special structure on the surface of the mirrors (“photonic surface”). This structure must have the period about λ/2 and will suppress propagation of surface plasmons with the frequency ω₀=2πc/λ along the surface. This structure will also slightly increase the heat removal from the mirror’s surface by the excitation of the thermostimulated plasmon emission from the surface. The heat removal from the surface is estimated and possible implementation of this approach for use with CO₂-lasers (λ=10.6 μm) and Nd-YAG-lasers (λ=1.06 μm) is analyzed.     

The effect of different treatments on fractal forms of gallium arsenide surface

The surface of epitaxial gallium arsenide (as-grown and subjected to standard chemical treatments for producing metal-GaAs electrical contacts) has been studied by atomic force microscopy. It was shown that the surface relief can be characterized by fractal relief in the local approximation. As-grown epitaxial n-GaAs layers have lower values of the mean relief irregularity and roughness and reveal high surface homogeneity. The most marked relief changes are observed due to plasmachemical deposition of silicon dioxide and its removal in a buffered etch and deionized water rinse. The following finishing treatments make the surface more smooth and homogeneous, especially in ammonia-water mixture, which gives almost an ideal (Gaussian) distribution of irregularities and phase contrast in most cases. The fractal dimensionality of the studied surfaces changes in the relatively narrow limits D _f = 2.5−2.8. The values of the local approximation limit L determining the boundary of the applicability of the fractal approach when describing the surfaces change from 0.039 to 10.99 μm depending on the kind of treatment.     

Crack motion in viscoelastic solids: The role of the flash temperature

We present a simple theory of crack propagation in viscoelastic solids. We calculate the energy per unit area, G(v), to propagate a crack, as a function of the crack tip velocity v. Our study includes the non-uniform temperature distribution (flash temperature) in the vicinity of the crack tip, which has a profound influence on G(v). At very low crack tip velocities, the heat produced at the crack tip can diffuse away, resulting in very small temperature increase: in this “low-speed” regime the flash temperature effect is unimportant. However, because of the low heat conductivity of rubber-like materials, already at moderate crack tip velocities a very large temperature increase (of order of 1000 K) can occur close to the crack tip. We show that this will drastically affect the viscoelastic energy dissipation close to the crack tip, resulting in a “hot-crack” propagation regime. The transition between the low-speed regime and the hot-crack regime is very abrupt, which may result in unstable crack motion, e.g. stick-slip motion or catastrophic failure, as observed in some experiments. In addition, the high crack tip temperature may result in significant thermal decomposition within the heated region, resulting in a liquid-like region in the vicinity of the crack tip. This may explain the change in surface morphology (from rough to smooth surfaces) which is observed as the crack tip velocity is increased above the instability threshold.     

Fast elliptic solvers in cylindrical coordinates and the Coulomb collision operator

In this paper, we describe a new class of fast solvers for separable elliptic partial differential equations in cylindrical coordinates (r, θ, z) with free-space radiation conditions. By combining integral equation methods in the radial variable r with Fourier methods in θ and z, we show that high-order accuracy can be achieved in both the governing potential and its derivatives. A weak singularity arises in the Fourier transform with respect to z that is handled with special purpose quadratures. We show how these solvers can be applied to the evaluation of the Coulomb collision operator in kinetic models of ionized gases.     

Modeling and computation of two phase geometric biomembranes using surface finite elements

Biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions. The modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces. This leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints. In this paper we propose a new computational tool for computing equilibria based on an L² relaxation flow for the total energy in which the line energy is approximated by a surface Ginzburg–Landau phase field functional. The relaxation dynamics couple a nonlinear fourth order geometric evolution equation of Willmore flow type for the membrane with a surface Allen–Cahn equation describing the lateral decomposition. A novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface, the mean curvature vector and the surface phase field function. The resulting variational formulation uses H¹ spaces, and we employ triangulated surfaces and H¹ conforming quadratic surface finite elements for approximating solutions. Together with a semi-implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only. Numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells, discocytes and starfishes with lateral phase separation.     

A mixed spectral and finite difference model to study baroclinic annulus waves

A mixed spectral and finite difference model to study finite amplitude baroclinic waves in a differentially heated rotating annulus is presented. The model consists of the full Navier-Stokes equations and the heat equation. The field variables f = f(r, φ z; t) are decomposed into zonally averaged components f_o(r, z; t) and eddy components f(r, φ, z; t), the latter being periodic in f and represented in terms of Fourier series. The unknowns f_o(r, z; t) and f^{c, s}(r, z; t), which are Fourier amplitudes of f′(r, φ, z; t) are governed by two-dimensional primitive equations with the addition of source terms. These equations are solved semi-implicitly by the alternating direction implicit method on variable grids.A simplified model with two Fourier components which permits self-interaction of the chosen wave and the interaction of the wave and the mean fields had been used to repeat a computation done by G. P. Williams, who used a fully three-dimensional finite difference algorithm. We can reproduce almost all of Williams' results in 1/20 of the computing time with the present model. It only requires 1/30 the additional computer storage of Williams' finite difference model over the axisymmetric problem.The potential of the present model for investigation of multiwave interaction as well as the advantages and disadvantages of the two different approaches is discussed.     

Multigrid algorithms for high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Multigrid algorithms are developed for systems arising from high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations on unstructured meshes. The algorithms are based on coupling both p- and h-multigrid (ph-multigrid) methods which are used in nonlinear or linear forms, and either directly as solvers or as preconditioners to a Newton–Krylov method.The performance of the algorithms are examined in solving the laminar flow over an airfoil configuration. It is shown that the choice of the cycling strategy is crucial in achieving efficient and scalable solvers. For the multigrid solvers, while the order-independent convergence rate is obtained with a proper cycle type, the mesh-independent performance is achieved only if the coarsest problem is solved to a sufficient accuracy. On the other hand, the multigrid preconditioned Newton–GMRES solver appears to be insensitive to this condition and mesh-independent convergence is achieved under the desirable condition that the coarsest problem is solved using a fixed number of multigrid cycles regardless of the size of the problem.It is concluded that the Newton–GMRES solver with the multigrid preconditioning yields the most efficient and robust algorithm among those studied.     

On the theory of temporal aberrations for cathode lenses

A new approach to the theory of temporal aberration for cathode lenses is given in the present paper. A definition of temporal aberration is given in which a certain initial energy of electron emission along the axial direction ε_z1 (0ε_z1ε_0max) is considered. A new method to calculate the temporal aberration coefficients of cathode lenses named “direct integral method” is also presented. The “direct integral method” gives new expressions of the temporal aberration coefficients which are expressed in integral forms. The difference between “direct integral method” and “τ-variation method” is that the “τ-variation method” needs to solve the differential equations for the three of temporal geometrical aberration coefficients of second order, while the “direct integral method” only needs to carry out the integral calculation for all of these temporal aberration coefficients of second order.All of the formulae of the temporal aberration coefficients deduced from “direct integral method” and “τ-variation method” have been verified by an electrostatic concentric spherical system model, and contrasted with the analytical solutions. Results show that these two methods have got identical solutions and the solutions of temporal aberration coefficients of the first and second order are the same as with the analytical solutions. Although some forms of the results seem different, but they can be transformed into the same form. Thus, it can be concluded these two methods given by us are equivalent and correct, but the “direct integral method” is related to solve integral equations, which is more convenient for computation and could be suggested for use in practical design.     

Probing Proteins in Solution by Xe NMR Spectroscopy

The interaction of xenon with different proteins in aqueous solution is investigated by ¹²⁹Xe NMR spectroscopy. Chemical shifts are measured in horse metmyoglobin, hen egg white lysozyme, and horse cytochrome c solutions as a function of xenon concentration. In these systems, xenon is in fast exchange between all possible environments. The results suggest that nonspecific interactions exist between xenon and the protein exteriors and the data are analyzed in term of parameters which characterize the protein surfaces. The experimental data for horse metmyoglobin are interpreted using a model in which xenon forms a 1:1 complex with the protein and the chemical shift of the complexed xenon is reported (Locci et al., Keystone Symposia “Frontiers of NMR in Molecular Biology VI”, Jan. 9–15, 1999, Breckenridge, CO, Abstract E216, p. 53; Locci et al., XeMAT 2000 “Optical Polarization and Xenon NMR of Materials”, June 28–30, 2000, Sestri Levante, Italy, p. 46).     

Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits

Let M be a smooth compact manifold of dimension at least 2 and Diff ^r (M) be the space of C ^r smooth diffeomorphisms of M. Associate to each diffeomorphism f;isin; Diff ^r (M) the sequence P _n (f) of the number of isolated periodic points for f of period n. In this paper we exhibit an open set N in the space of diffeomorphisms Diff ^r (M) such for a Baire generic diffeomorphism f∈N the number of periodic points P _n f grows with a period n faster than any following sequence of numbers {a _n } _{n ∈ Z +} along a subsequence, i.e. P _n (f)>a _ni for some n _i →∞ with i→∞. In the cases of surface diffeomorphisms, i.e. dim M≡2, an open set N with a supergrowth of the number of periodic points is a Newhouse domain. A proof of the man result is based on the Gontchenko–Shilnikov–Turaev Theorem [GST]. A complete proof of that theorem is also presented. Received: 27 January 1999 / Accepted: 23 November 1999     

Orientational dynamics of superfluid He: A “two-fluid” model. II. Orbital dynamics

We present a phenomenological theory of the homogeneous orbital dynamics of the class of “separable” anisotropic superfluid phases which includes the ABM state generally identified with ³He-A. The theory is developed by analogy with the spin dynamics described in the first paper of this series; the basic variables are the orientation of the Cooper-pair wavefunction (in the ABM phase, the l-vector) and a quantity K which we visualize as the “pseudo-angular momentum” of the Cooper pairs but which must be distinguished, in general, from the total orbital angular momentum of the system. In the ABM case l is the analog of d in the spin dynamics and K of the “superfluid spin” S_p. Important points of difference from the spin case which are taken into account include the fact that a rotation of l without a simultaneous rotation of the normal-component distribution strongly increases the energy of the system (“normal locking”), and that the equilibrium value of K is zero even for finite total angular momentum. The theory does not claim to handle correctly effects associated with any intrinsic angular momentum arising from particle-hole asymmetry, but it is shown that the magnitude of this quantity can be estimated directly from experimental data and is extremely small; also, the Landau damping does not emerge automatically from the theory, but can be put in in an ad hoc way. With these provisos the theory should be valid for all frequencies irrespective of the value of ωτ. (Δ = gap parameter, τ = quasi-particle relaxation time.) It disagrees with all existing phenomenological theories of comparable generality, although the disagreement with that of Volovik and Mineev is confined to the “gapless” region very close to T_c.The phenomenological equations of motion, which are similar in general form to those of the spin dynamics with damping, involve an “orbital susceptibility of the Cooper pairs” χ_orb(T). We give a possible microscopic definition of the variable K and use it to calculate χ_orb(T) for a general phase of the “separable” type. The theory is checked by inserting the resulting formula in the phenomenological equations for ωτ 1 and comparing with the results of a fully microscopic calculation based on the collisionless kinetic equation; precise agreement is obtained for both the ABM and the (real) polar phase, showing that the complex nature of the ABM phase and the associated “pair angular momentum” is largely irrelevant to its orbital dynamics. We note also that the phenomenological theory gives a good qualitative picture even when ω Δ(T), e.g., for the flapping mode near T_c. Our theory permits a simple and unified calculation of (1) the Cross-Anderson viscous torque in the overdamped regime, (2) the flapping-mode frequency near zero temperature, (3) orbital effects on the NMR, both at low temperatures and near T_c, (4) the orbit wave spectrum at zero temperature (this requires a generalization to inhomogeneous situations which is possible at T = 0 but probably not elsewhere). We also discuss the possibility of experiments of the Einstein-de Haas type. Generally speaking, our results for any one particular application can be also obtained from some alternative theory, but in the case of orbital and spin relaxation very close to T_c (within the “gapless” region) our predictions, while somewhat tentative and qualitative, appear to disagree with those of all existing theories. We discuss briefly how our approach could be extended to apply to more general phases.     

Quantum mechanics with difference operators

A formulation of quantum mechanics with additive and multiplicative (q-) difference operators instead of differential operators is studied from first principles. Borel-quantisation on smooth configuration spaces is used as guiding quantisation method. After a short discussion this method is translated step-by-step to a framework based on difference operators. To restrict the resulting plethora of possible quantisations additional assumptions motivated by simplicity and plausibility are required. Multiplicative difference operators and the corresponding q-Borel kinematics are given on the circle and its N-point discretisation; the connection to q-deformations of the Witt algebra is discussed. For a “natural” choice of the q-kinematics a corresponding q-difference evolution equation is obtained. This study shows general difficulties for a generalisation of a physical theory from a known one to a “new” framework.     

Statistics of ideal randomly branched polymers in a semi-space

We investigate the statistical properties of a randomly branched 3-functional N-link polymer chain without excluded volume, whose one point is fixed at the distance d from the impenetrable surface in a 3-dimensional space. Exactly solving the Dyson-type equation for the partition function Z(N, d )= N^-θe^γN in 3D, we find the “surface” critical exponent θ = , as well as the density profiles of 3-functional units and of dead ends. Our approach enables to compute also the pairwise correlation function of a randomly branched polymer in a 3D semi-space.     

Parametrization of the S0 two nucleon transition matrix

The ¹S₀ two-nucleon transition matrix T is constructed from the symmetric part σ of its half-shell elements. The on-shell component of σ is given by the phase shift, while a wide class of parametrizations is suggested for the off-shell part. Restrictions on the off-shell part of σ arising from the short range and the proper one-pion-exchange tail of the nucleon-nucleon interaction are investigated. Using σ in the ¹S₀ and the Reid soft-core potential in the other partial waves, the binding energy per particle in nuclear matter and ¹⁶O and the ¹⁸O shell-model spectrum are computed. The sensitivity of these nuclear-structure results is tested with respect to (i) smooth off-shell changes in σ, (ii) various assumptions on the high-energy phase shift, (iii) the charge dependence of the phase shift, and (iv) experimental uncertainties in the phase shift.     

Simple type and the boundary of moduli space

We measure, in two distinct ways, the extent to which the boundary region of moduli space contributes to the “simple type” condition of Donaldson theory. Using the natural geometric representative of μ(pt) defined in [L. Sadun, Commun. Math. Phys. 178 (1996) 107–113], the boundary region of moduli space contributes of the homology required for simple type, regardless of the topology or geometry of the underlying 4-manifold. The simple type condition thus reduces to the interior of the (k+1)th ASD moduli space, intersected with two representatives of (4 times) the point class, being homologous to 58 copies of the kth moduli space. This is peculiar, since the only known embeddings of the kth moduli space into the (k+1)th involve Taubes gluing, and the images of such embeddings lie entirely in the boundary region.When using the natural de Rham representatives of μ(pt) considered by Witten [Commun. Math. Phys. 117 (1988) 353], the boundary region contributes of what is needed for simple type, again regardless of the topology or geometry of the underlying 4-manifold. The difference between this and the geometric representative answer is not contradictory, as the contribution of a fixed region to the Donaldson invariants is geometric, not topological.