Interpolation Formula Research Articles

Application of compact schemes in the CUSP framework for strong shock–vortex interaction

We present a methodology for application of compact schemes for unsteady compressible flow problems involving strong shocks and vortices. A compact interpolation formula is combined with a solver based on the ‘convective upwind and split pressure’ (CUSP) scheme [32] in conjunction with a limiter to restrict the numerical fluxes at the cell faces. The interpolation scheme is derived in a manner so that its use in a first derivative calculation results in high accuracy in the wave number space. This is established by the modified wavenumber and the numerical group velocity of the scheme. Use of compact schemes in compressible flow with shocks has mostly been limited to shocks at lower Mach number range. We test our solver by computing a Mach 7 shock interacting with a strong vortex. Results from the lower resolution version of the solver match well with existing data from an ENO scheme based solver. The high resolution scheme shows the break-up of the vortex into smaller vortices in great details, including roll-up of Kelvin–Helmholtz instability generated small scale vortices from slip lines. The presented method can be seamlessly integrated into existing finite volume or finite difference compressible solvers when such high resolution capability is required.

Memory‐Efficient Simulation of Frequency‐DependentQ

Memory‐variable methods have been widely applied to approximate frequency‐independent quality factor Q in numerical simulation of wave propagation. The frequency‐independent model is often appropriate for frequencies up to about 1 Hz but at higher frequencies is inconsistent with some regional studies of seismic attenuation. We apply the memory‐variable approach to frequency‐dependent Q models that are constant below, and follow a power‐law above, a chosen transition frequency. We present numerical results for the corresponding memory‐variable relaxation times and weights, obtained by nonnegative least‐squares fitting of the Q ( f ) function, for a range of exponent values; these times and weights can be scaled to an arbitrary transition frequency and a power‐law prefactor, respectively. The resulting memory‐variable formulation can be used with numerical wave‐propagation solvers based on methods such as finite differences (FDs) or spectral elements and may be implemented in either conventional or coarse‐grained form. In the coarse‐grained approach, we fit effective Q for low‐ Q values (<200) using a nonlinear inversion technique and use an interpolation formula to find the corresponding weighting coefficients for arbitrary Q . A 3D staggered‐grid FD implementation closely approximates the frequency–wavenumber solution to both a half‐space and a layered model with a shallow dislocation source for Q as low as 20 over a bandwidth of two decades. We compare the effects of different power‐law exponents using a finite‐fault source model of the 2008 M w 5.4 Chino Hills, California, earthquake and find that Q ( f ) models generally better fit the strong‐motion data than the constant Q models for frequencies above 1 Hz. Online Material: Figures comparing finite difference and frequency–wavenumber seismograms for an elastic layered‐model point source simulation. Median spectral acceleration centered at 1 s and Fourier amplitude centered at 0.25 and 2.25 Hz for strong ground motion recordings and synthetics from the Chino Hills earthquake.

The investigation of polarized reflectivity of explosively driven dense plasma

The analysis of the response of dense plasma to electromagnetic waves of moderate intensity can be used as a tool to investigate the validity of the physical models describing the behavior of matter under extreme conditions. Within this work, the new experimental data on oblique incidence of polarized electromagnetic wave are presented. The plasma composition was calculated within a chemical picture and the integration of Maxwell equations to construct the spatial profile of the density of charge carriers of plasma was based on an interpolation formula for dc conductivity.

Avoiding negative elastic moduli when using Lagrange interpolation for material grading in finite element analysis

Polynomial interpolations, one of the most common interpolations used in finite element methods (FEMs), are a workhorse of many FEM codes. These interpolations are readily available for all kinds of elements, and using them for modeling the variation of elastic moduli in graded elements is thus both convenient and natural. Yet, like all polynomial interpolations, they can be prone to oscillations that can result in regions of negative elastic modulus in the element, even with only positive nodal values of elastic moduli. The result of these negative modulus regions, even if the region is small, can be unexpected singularities in the solution. This defeats the purpose of using polynomial interpolations for capturing material grading in the element. We demonstrate the issue using three-node quadratic Lagrange interpolations of material grading in otherwise isoparametric p-type elements and show how to avoid this problem.

Decimation in Time and Space of Finite-Difference Time-Domain Schemes: Standard Isotropic Lossless Model

Finite-difference time-domain (FDTD) schemes permit changes in the grid density on selected regions of the wave propagation domain, which can reduce the computational load of the simulations. One possible alternative to varying the spatial density is to change the simulation temporal rate. This idea looks attractive when the wave signals exhibit pronounced bandwidth fluctuations across time. This is particularly true in sound synthesis, where a physically based acoustic resonator can be conveniently modeled using such schemes. To overcome the computational constraints that must be met by real-time distributed resonator models, this paper deals with the decimation in time and space of isotropic lossless finite-difference time-domain schemes holding conventional Nyquist-Shannon limits on the bandwidth of the wave signals. Formulas for the reconstruction of these signals at runtime over the interpolated grid are provided for both the 1D and 2D orthogonal case, depending on the ideal boundary conditions (either Neumann or Dirichlet) holding at each side of the grid in connection with the domain side lengths (either even or odd). Together, the boundaries and size determine the type of Discrete Cosine Transform used in the corresponding interpolation formula. Numerical artifacts arising as a consequence of decimating in space in 2D are discussed in terms of dispersion error and aliasing. Considerations concerning the temporal reconstruction of components lying at the decimated Nyquist frequency are addressed in the conclusion.

New consequences of the static scaling hypothesis at low temperatures

The general interpolation formulas, which generalize the classical consequences of the static scaling hypothesis for phase transitions with variation in the temperature and field, have been proposed. The classical results have been derived from these formulas as a limiting partial case for the asymptotic proximity to a positive critical temperature. It has also been shown that some consequences of the scaling hypothesis also remain correct at a zero critical temperature; however, for example, the Essam–Fisher equality changes its form in the latter case, namely, the numeral two on the right-hand side is replaced by unity.

Rhie–Chow interpolation in strong centrifugal fields

Rhie–Chow interpolation formulas are derived from the Navier–Stokes and continuity equations. These formulas are generalized to gas dynamics in strong centrifugal fields (as high as 106 g) occurring in gas centrifuges.

On the interpolation formula for the bound state energies of atomic systems

By using results of highly accurate computations of the total energies of a large number of few-electron atoms we construct a few interpolation formulas which can be used to approximate the total energies of bound atomic states. In our procedure the total energies of atomic states $E$ are represented as a function of the electric charge of atomic nucleus $Q$ and the total number of bound electrons $N_e$. Some general properties of the $E(Q, N_e)$ function are investigated. The knowledge of the $E(Q, N_e)$ function allows one to determine the total (and binding) energies of these states in arbitrary atoms and ions with different $Q$ and $N_e$.

On the Q-dependence of the lowest-order QED corrections and other properties of the ground 11S-states in the two-electron ions

Formulas and expectation values which are need to determine the lowest-order QED corrections (∼α3) and corresponding recoil (or finite mass) corrections in the two-electron helium-like ions are presented. Other important properties of the two-electron ions are also determined to high accuracy, including the expectation values of the quasi-singular Vinti operator and 〈reN−2〉 and 〈ree−2〉 expectation values. Elastic scattering of fast electrons by the two-electron ions in the Born approximation is considered. Interpolation formulas are derived for the bound state properties of the two-electron ions as the function of the nuclear electric charge Q.

On the numerical stability of Floater-Hormann’s rational interpolant

In this note we extend the definition of the first barycentric formula for Lagrange interpolation to Floater-Hormann interpolants and present an algorithm to evaluate it which is backward stable on the entire real line. We also discuss in detail the numerical stability of the second barycentric formula for Floater-Hormann interpolants.

Algorithms that eliminate the effects of calibration artefact and trial-imposed offsets of Masimo oximeter in BOOST-NZ trial.

The displayed readings of Masimo pulse oximeters used in the Benefits Of Oxygen Saturation Targeting (BOOST) II and related trials in very preterm babies were influenced by trial-imposed offsets and an artefact in the calibration software. A study was undertaken to implement new algorithms that eliminate the effects of offsets and artefact. In the BOOST-New Zealand trial, oxygen saturations were averaged and stored every 10s up to 36weeks' post-menstrual age. Two-hundred and fifty-seven of 340 babies enrolled in the trial had at least twoweeks of stored data. Oxygen saturation distribution patterns corresponding with a +3% or -3% offset in the 85-95% range were identified together with that due to the calibration artefact. Algorithms involving linear and quadratic interpolations were developed, implemented on each baby of the dataset and validated using the data of a UK preterm baby, as recorded from Masimo oximeters with the original software and a non-offset Siemens oximeter. Saturation distributions obtained were compared for both groups. There were a flat region at saturations 85-87% and a peak at 96% from the lower saturation target oximeters, and at 93-95 and 84% respectively from the higher saturation target oximeters. The algorithms lowered the peaks and redistributed the accumulated frequencies to the flat regions and artefact at 87-90%. The resulting distributions were very close to those obtained from the Siemens oximeter. The artefact and offsets of the Masimo oximeter's software had been addressed to determine the true saturation readings through the use of novel algorithms. The implementation would enable New Zealand data be included in the meta-analysis of BOOST II trials, and be used in neonatal oxygen studies.

Uncertain eigenvalue analysis by the sparse grid stochastic collocation method

In this paper, the eigenvalue problem with multiple uncertain parameters is analyzed by the sparse grid stochastic collocation method. This method provides an interpolation approach to approximate eigenvalues and eigenvectors’ functional dependencies on uncertain parameters. This method repetitively evaluates the deterministic solutions at the pre-selected nodal set to construct a high-dimensional interpolation formula of the result. Taking advantage of the smoothness of the solution in the uncertain space, the sparse grid collocation method can achieve a high order accuracy with a small nodal set. Compared with other sampling based methods, this method converges fast with the increase of the number of points. Some numerical examples with different dimensions are presented to demonstrate the accuracy and efficiency of the sparse grid stochastic collocation method.

A Variable Order Fractional Differential-Based Texture Enhancement Algorithm with Application in Medical Imaging.

Texture enhancement is one of the most important techniques in digital image processing and plays an essential role in medical imaging since textures discriminate information. Most image texture enhancement techniques use classical integral order differential mask operators or fractional differential mask operators using fixed fractional order. These masks can produce excessive enhancement of low spatial frequency content, insufficient enhancement of large spatial frequency content, and retention of high spatial frequency noise. To improve upon existing approaches of texture enhancement, we derive an improved Variable Order Fractional Centered Difference (VOFCD) scheme which dynamically adjusts the fractional differential order instead of fixing it. The new VOFCD technique is based on the second order Riesz fractional differential operator using a Lagrange 3-point interpolation formula, for both grey scale and colour image enhancement. We then use this method to enhance photographs and a set of medical images related to patients with stroke and Parkinson’s disease. The experiments show that our improved fractional differential mask has a higher signal to noise ratio value than the other fractional differential mask operators. Based on the corresponding quantitative analysis we conclude that the new method offers a superior texture enhancement over existing methods.

High-Order Finite-Volume Transport on the Cubed Sphere: Comparison between 1D and 2D Reconstruction Schemes

Abstract This paper presents two finite-volume (FV) schemes for solving linear transport problems on the cubed-sphere grid system. The schemes are based on the central-upwind finite-volume (CUFV) method, which is a class of Godunov-type method for solving hyperbolic conservation laws, and combines the attractive features of the classical upwind and central FV methods. One of the CUFV schemes is based on a dimension-by-dimension approach and employs a fifth-order one-dimensional (1D) Weighted Essentially Nonoscillatory (WENO5) reconstruction method. The other scheme employs a fully two-dimensional (2D) fourth-order accurate reconstruction method. The cubed-sphere grid system imposes several computational challenges due to its patched-domain topology and nonorthogonal curvilinear grid structure. A high-order 1D interpolation procedure combining cubic and quadratic interpolations is developed for the FV schemes to handle the discontinuous edges of the cubed-sphere grid. The WENO5 scheme is compared against the fourth-order Kurganov–Levy (KL) scheme formulated in the CUFV framework. The performance of the schemes is compared using several benchmark problems such as the solid-body rotation and deformational-flow tests, and empirical convergence rates are reported. In addition, a bound-preserving filter combined with an optional positivity-preserving filter is tested for nonsmooth problems. The filtering techniques considered are local, inexpensive, and effective. A fourth-order strong stability preserving explicit Runge–Kutta time-stepping scheme is used for integration. The results show that schemes are competitive to other published FV schemes in the same category.

Lagrange interpolation and Newton-Cotes formulas for functions with boundary layer components on piecewise-uniform grids

The problem of interpolation of a one-variable function considered as a solution to a boundary value problem for an equation with a small parameter ɛ in the highest derivative is investigated. The application of a Lagrange polynomial for such a function on a uniform grid may result in serious errors. For a Shishkin grid, ɛ-uniform error estimates of Lagrange polynomial interpolation are obtained. The Shishkin grid is modified to increase the interpolation accuracy. For Newton-Cotes formulas on such grids, ɛ-uniform error estimates are obtained. The results of numerical experiments confirm the theoretical estimates.

A note on the quasi-periodic solutions of the modified Boussinesq hierarchy

Based on the theory of trigonal curve and the properties of three kinds of the Abel differentials on it, we deduce the explicit theta function representations of the Baker-Akhiezer function and the meromorphic function associated with the modified Boussinesq hierarchy. The modified Boussinesq flows are straightened using the Abel map and the Lagrange interpolation formula. The explicit theta function representations of solutions for the entire modified Boussinesq hierarchy are constructed with the aid of the asymptotic properties and the algebro-geometric characters of the meromorphic function.

Single scale cluster expansions with applications to many Boson and unbounded spin systems

We develop a cluster expansion to show exponential decay of correlations for quite general single scale spin systems, as they arise in lattice quantum field theory and discretized functional integral representations for observables of quantum statistical mechanics. We apply our results to the small field approximation to the coherent state correlation functions of the grand canonical Bose gas at negative chemical potential, constructed by Balaban et al. [Ann. Henri Poincaré 11, 151–350 (2010c)], and to N component unbounded spin systems with repulsive two body interaction and massive, possibly complex, covariance. Our cluster expansion is derived by a single application of the Brydges-Kennedy-Abdesselam-Rivasseau interpolation formula.

Smolyak type sparse grid collocation method for uncertainty quantification of nonlinear stochastic dynamic equations

This paper develops a Smolyak-type sparse-grid stochastic collocation method (SGSCM) for uncertainty quantification of nonlinear stochastic dynamic equations. The solution obtained by the method is a linear combination of tensor product formulas for multivariate polynomial interpolation. By choosing the collocation point sets to coincide with cubature point sets of quadrature rules, we derive quadrature formulas to estimate the expectations of the solution. The method does not suffer from the curse of dimensionality in the sense that the computational cost does not increase exponentially with the number of input random variables. Numerical analysis of a nonlinear elastic oscillator subjected to a discretized band-limited white noise process demonstrates the computational efficiency and accuracy of the developed method.

Finding the Real Surge Boundaries of Turbo-charged Automobiles Using Intelligent Fuzzy Reasoning Technique

Engine with turbo charger needs to be aware of the potential damage caused by surge phenomena. The focus of this paper is to propose a new intelligent reasoning to find the real surge boundary for a turbo-charged automobile without performing real complicated automobile experiment. Four physical fuzzy variables of a turbo-charged automobile, i.e., throttle size, pressure ratio, engine displacement, and total effective pipe length between the throttle and turbo charger, are chosen as the input to our intelligent reasoning algorithm system to produce the mass air flow for correction. The mass air flow for correction can then be applied on the compressor map of turbo charger to determine the real surge boundary for a certain turbo-charged automobile. In comparison with the previous approach using quadratic interpolation formula for exact real surge boundary, our new approach can indeed produce nearly the same answer with simplified computation load. Excellent benchmark tests are conducted on Saab NG 900 and 9000 with turbo chargers, Garrett T25-60.

Non-perturbative improvement of the axial current in [formula omitted] lattice QCD with Wilson fermions and tree-level improved gauge action

The coefficient cA required for O(a) improvement of the axial current in lattice QCD with Nf=3 flavors of Wilson fermions and the tree-level Symanzik-improved gauge action is determined non-perturbatively. The standard improvement condition using Schrödinger functional boundary conditions is employed at constant physics for a range of couplings relevant for simulations at lattice spacings of ≈0.09 fm and below. We define the improvement condition projected onto the zero topological charge sector of the theory, in order to avoid the problem of possibly insufficient tunneling between topological sectors in our simulations at the smallest bare coupling. An interpolation formula for cA(g02) is provided together with our final results.