Extension of the spectral distribution method to many-quasiparticle subspaces

The full BCS Hamiltonian is given as a quasiparticle-conserving part H_c and a quasiparticle non-conserving one, H_nc. The moments of H_c can be calculated and propagated to different K-subspaces (subspaces of states with a well-defined number, K, of quasiparticles) by French's method. Here we give the propagation formulae for the second moment of H_ac in terms of a small number of basic quantities. The spectral distribution method, applied so far to particle systems, can be therefore transcribed to quasiparticle ones. We derive also expressions for the partial widths, and study the extent to which the number of quasiparticles is conserved. Then we discuss a possible truncation of the quasiparticle basis and end up with a discussion on the behaviour of centroid energies and widths in different cases.     

Efficient quadrature-free high-order spectral volume method on unstructured grids: Theory and 2D implementation

An efficient implementation of the high-order spectral volume (SV) method is presented for multi-dimensional conservation laws on unstructured grids. In the SV method, each simplex cell is called a spectral volume (SV), and the SV is further subdivided into polygonal (2D), or polyhedral (3D) control volumes (CVs) to support high-order data reconstructions. In the traditional implementation, Gauss quadrature formulas are used to approximate the flux integrals on all faces. In the new approach, a nodal set is selected and used to reconstruct a high-order polynomial approximation for the flux vector, and then the flux integrals on the internal faces are computed analytically, without the need for Gauss quadrature formulas. This gives a significant advantage over the traditional SV method in efficiency and ease of implementation. For SV interfaces, a quadrature-free approach is compared with the Gauss quadrature approach to further evaluate the accuracy and efficiency. A simplified treatment of curved boundaries is also presented that avoids the need to store a separate reconstruction for each boundary cell. Fundamental properties of the new SV implementation are studied and high-order accuracy is demonstrated for linear and non-linear advection equations, and the Euler equations. Several well known inviscid flow test cases are utilized to show the effectiveness of the simplified curved boundary representation.     

Extension of the finite volume particle method to viscous flow

The finite volume particle method (FVPM) is a mesh-free method for fluid dynamics which allows simple and accurate implementation of boundary conditions and retains the conservation and consistency properties of classical finite volume methods. In this article, the FVPM is extended to viscous flows using a consistency-corrected smoothed particle hydrodynamics (SPH) approximation to evaluate velocity gradients. The accuracy of the viscous FVPM is improved by a higher-order discretisation of the inviscid flux combined with a second-order temporal discretisation. The higher-order inviscid FVPM is validated for a 1-D shock tube problem, in which it demonstrates an enhanced shock capturing ability. For two-dimensional simulations, a small arbitrary Lagrange–Euler correction to fully Lagrangian particle motion is beneficial in maintaining a favourable particle distribution over long simulation times. The viscous FVPM is validated for two-dimensional Poiseuille, Taylor–Green and lid-driven cavity flows, and good agreement is achieved with analytic or reference numerical solutions. These results establish the viability of FVPM as a tool for mesh-free simulation of viscous flows in engineering.     

A spectral invariant representation of spectral reflectance

Spectral image acquisition as well as color image is affected by several illumination factors such as shading, gloss, and specular highlight. Spectral invariant representations for these factors were proposed for the standard dichromatic reflection model of inhomogeneous dielectric materials. However, these representations are inadequate for other characteristic materials like metal. This paper proposes a more general spectral invariant representation for obtaining reliable spectral reflectance images. Our invariant representation is derived from the standard dichromatic reflection model for dielectric materials and the extended dichromatic reflection model for metals. We proof that the invariant formulas for spectral images of natural objects preserve spectral information and are invariant to highlights, shading, surface geometry, and illumination intensity. It is proved that the conventional spectral invariant technique can be applied to metals in addition to dielectric objects. Experimental results show that the proposed spectral invariant representation is effective for image segmentation.     

基于边界元方法的超导核磁共振成像设备高阶轴向匀场线圈优化算法

在磁共振成像设备中,为了消除目标区域内的高阶谐波磁场分量,传统方法采用无源匀场,但该方法匀场精度较低,针对性较差,适用于全局匀场,而有源匀场则可以通过优化线圈分布来产生所需要的特定的磁场分布.但是,由于匀场线圈线型的复杂度会随着线圈阶数的增加而增加,难以满足设计需要,因此本文提出了一种用于磁共振成像超导匀场线圈系统的多变量非线性优化设计方法.该方法基于边界元方法,将匀场线圈所产生的磁场与目标磁场之间的偏差作为目标函数,线匝间距、线圈半径等作为约束条件,通过非线性优化算法,得到满足设计要求的线圈分布.通过一个中心磁场为0.5 T的开放式双平面磁共振成像超导轴向匀场线圈的设计案例,说明本方法具有计算效率高、灵活性好的特点.     

A spectral boundary integral method for flowing blood cells

A spectral boundary integral method for simulating large numbers of blood cells flowing in complex geometries is developed and demonstrated. The blood cells are modeled as finite-deformation elastic membranes containing a higher viscosity fluid than the surrounding plasma, but the solver itself is independent of the particular constitutive model employed for the cell membranes. The surface integrals developed for solving the viscous flow, and thereby the motion of the massless membrane, are evaluated using an O(NlogN)$O(N\log N)$ particle-mesh Ewald (PME) approach. The cell shapes, which can become highly distorted under physiologic conditions, are discretized with spherical harmonics. The resolution of these global basis functions is, of course, excellent, but more importantly they facilitate an approximate de-aliasing procedure that stabilizes the simulations without adding any numerical dissipation or further restricting the permissible numerical time step. Complex geometry no-slip boundaries are included using a constraint method that is coupled into an implicit system that is solved as part of the time advancement routine. The implementation is verified against solutions for axisymmetric flows reported in the literature, and its accuracy is demonstrated by comparison against exact solutions for relaxing surface deformations. It is also used to simulate flow of blood cells at 30% volume fraction in tubes between 4.9 and 16.9 μm in diameter. For these, it is shown to reproduce the well-known non-monotonic dependence of the effective viscosity on the tube diameter.     

A high-order boundary integral method for surface diffusions on elastically stressed axisymmetric rods

Many applications in materials involve surface diffusion of elastically stressed solids. Study of singularity formation and long-time behavior of such solid surfaces requires accurate simulations in both space and time. Here we present a high-order boundary integral method for an elastically stressed solid with axi-symmetry due to surface diffusions. In this method, the boundary integrals for isotropic elasticity in axi-symmetric geometry are approximated through modified alternating quadratures along with an extrapolation technique, leading to an arbitrarily high-order quadrature; in addition, a high-order (temporal) integration factor method, based on explicit representation of the mean curvature, is used to reduce the stability constraint on time-step. To apply this method to a periodic (in axial direction) and axi-symmetric elastically stressed cylinder, we also present a fast and accurate summation method for the periodic Green’s functions of isotropic elasticity. Using the high-order boundary integral method, we demonstrate that in absence of elasticity the cylinder surface pinches in finite time at the axis of the symmetry and the universal cone angle of the pinching is found to be consistent with the previous studies based on a self-similar assumption. In the presence of elastic stress, we show that a finite time, geometrical singularity occurs well before the cylindrical solid collapses onto the axis of symmetry, and the angle of the corner singularity on the cylinder surface is also estimated.     

Hierarchical reconstruction for spectral volume method on unstructured grids

The hierarchical reconstruction (HR) [Y.-J. Liu, C.-W. Shu, E. Tadmor, M.-P. Zhang, Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction, SIAM J. Numer. Anal. 45 (2007) 2442–2467; Z.-L. Xu, Y.-J. Liu, C.-W. Shu, Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO type linear reconstruction and partial neighboring cells, J. Comput. Phys. 228 (2009) 2194–2212] is applied to a piecewise quadratic spectral volume method on two-dimensional unstructured grids as a limiting procedure to prevent spurious oscillations in numerical solutions. The key features of this HR are that the reconstruction on each control volume only uses adjacent control volumes, which forms a compact stencil set, and there is no truncation of higher degree terms of the polynomial. We explore a WENO-type linear reconstruction on each hierarchical level for the reconstruction of high degree polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed. We demonstrate that the hierarchical reconstruction can generate essentially non-oscillatory solutions while keeping the resolution and desired order of accuracy for smooth solutions.     

Constrained high-order statistical blind deconvolution of spectral data

A constrained high-order statistical algorithm is proposed to blindly deconvolute the measured spectral data and estimate the response function of the instruments simultaneously. In this algorithm, no priorknowledge is necessary except a proper length of the unit-impulse response. This length can be easily set to be the width of the narrowest spectral line by observing the measured data. The feasibility of this method has been demonstrated experimentally by the measured Raman and absorption spectral data.     

A new high-order discontinuous Galerkin spectral finite element method for Lagrangian gas dynamics in two-dimensions

This paper presents a new high-order cell-centered Lagrangian scheme for two-dimensional compressible flow. The scheme uses a fully Lagrangian form of the gas dynamics equations, which is a weakly hyperbolic system of conservation laws. The system of equations is discretized in the Lagrangian space by discontinuous Galerkin method using a spectral basis. The vertex velocities and the numerical fluxes through the cell interfaces are computed consistently in the Eulerian space by virtue of an improved nodal solver. The nodal solver uses the HLLC approximate Riemann solver to compute the velocities of the vertex. The time marching is implemented by a class of TVD Runge–Kutta type methods. A new HWENO (Hermite WENO) reconstruction algorithm is developed and used as limiters for RKDG methods to maintain compactness of RKDG methods. The scheme is conservative for the mass, momentum and total energy. It can maintain high-order accuracy both in space and time, obey the geometrical conservation law, and achieve at least second order accuracy on quadrilateral meshes. Results of some numerical tests are presented to demonstrate the accuracy and the robustness of the scheme.     

Extension of the doubling method to inhomogeneous sources

This paper gives a general theory for applying the doubling method to spatially inhomogeneous radiation sources whose angular and spatial variations separate. In particular, inhomogeneous sources of thermal radiation may be efficiently treated by the methods herein, as well as direct and specularly-reflected beams of radiation which do not lie along a quadrature direction for the intensity. Doubling rules for a linear-in-optical-depth source (which may approximate a source of thermal radiation) and for an exponential-in-optical-depth source are derived as special cases.     

Full tunability of laser femtosecond high-order harmonics in the ultraviolet spectral range

We demonstrate experimentally the full tunability of a coherent femtosecond source in the whole ultraviolet spectral region. The experiment relies on the technique of high-order harmonic generation driven by a near-infrared parametric laser source in krypton gas. By tuning the drive wavelength in the range between 1100 to 1900?nm, we generated intense harmonics from near to extreme ultraviolet. A number of photons per shot of the order of 10⁷ has been measured for the first harmonic orders. Many novel scientific prospects are expected to benefit from the use of such a table-top tunable source.     

Extension of Kleiser and Schumann’s influence-matrix method for generalized velocity boundary conditions

In 1980, Kleiser and Schumann introduced a novel influence-matrix method to treat the incompressibility and no-slip boundary conditions when solving the Navier–Stokes equations. They also outlined the related “tau” error correction technique which is essential for the high accuracy direct numerical simulation (DNS) of turbulent flows. However, their method is not valid for Robin type velocity boundary conditions (i.e., B(u) = αu + βu′ − γ = 0). In this note, a new influence-matrix method is introduced where the boundary condition and “tau” correction are enforced in one step using an extended influence matrix. The new method is simple and easy to be implemented. It broadens the applicability of the Kleiser and Schumann method. Examples with the new method show excellent agreement with data in the literature and the velocity field is divergence free up to machine precision.     

Simplifying the spectral analysis of the volume operator

The volume operator plays a central role in both the kinematics and dynamics of canonical approaches to quantum gravity which are based on algebras of generalized Wilson loops. We introduce a method for simplifying its spectral analysis, for quantum states that can be realized on a cubic three-dimensional lattice. This involves a decomposition of Hilbert space into sectors transforming according to the irreducible representations of a subgroup of the cubic group. As an application, we determine the complete spectrum for a class of states with six-valent intersections.