Efficient Implicit Non-linear LU-SGS Approach for Compressible Flow Computation Using High-Order Spectral Difference Method

An implicit non-linear lower-upper symmetric Gauss-Seidel(LU-SGS)solution algorithm has been developed for a high-order spectral difference Navier-Stokes solver on unstructured hexahedral grids.The non-linear LU-SGS solver is preconditioned by a block element matrix,and the system of equations is then solved with the LU decomposition approach.The large sparse Jacobian matrix is computed numerically,resulting in extremely simple operations for arbitrarily complex residual operators.Several inviscid and viscous test cases were performed to evaluate the performance.The implicit solver has shown speedup of 1 to 2 orders of magnitude over the multi-stage Runge-Kutta time integration scheme.

High-Order Multidomain Spectral Difference Method for the Navier-Stokes Equations on Unstructured Hexahedral Grids

A high order multidomain spectral difference method has been developed for the three dimensional Navier-Stokes equations on unstructured hexahedral grids.The method is easy to implement since it involves one-dimensional operations only,and does not involve surface or volume integrals.Universal reconstructions are obtained by distributing solution and flux points in a geometrically similar manner in a unit cube.The concepts of the Riemann solver and high-order local representations are applied to achieve conservation and high order accuracy.In this paper,accuracy studies are performed to numerically verify the order of accuracy using flow problems with analytical solutions.High order of accuracy and spectral convergence are obtained for the propagation of an isotropic vortex and Couette flow.The capability of the method for both inviscid and viscous flow problems with curved boundaries is also demonstrated.

Spectral difference analysis and identification of different maturity blueberry fruit based on hyperspectral imaging using spectral index

Hyperspectral imaging,with many narrow bands of spectra,is strongly capable to detect or classify objects.It has been become one research hotspot in the field of near-ground remote sensing.However,the higher demands for computing and complex operating of instrument are still the bottleneck for hyperspectral imaging technology applied in field.Band selection is a common way to reduce the dimensionality of hyperspectral imaging cube and simplify the design of spectral imaging instrument.In this research,hyperspectral images of blueberry fruit were collected both in the laboratory and in field.A set of spectral bands were selected by analyzing the differences among blueberry fruits at different growth stages and backgrounds.Furthermore,a normalized spectral index was set up using the bands selected to identify the three growth stages of blueberry fruits,aiming to eliminate the impact of background included leaf,branch,soil,illumination variation and so on.Two classifiers of spectral angle mapping(SAM),multinomial logistic regression(MLR)and classification tree were used to verify the results of identification of blueberry fruit.The detection accuracy was 82.1%for SAM classifier using all spectral bands,88.5%for MLR classifier using selected bands and 89.8%for decision tree using the spectral index.The results indicated that the normalization spectral index can both lower the complexity of computing and reduce the impact of noisy background in field.

Large Eddy Simulation of Flow over a Cylinder Using High-Order Spectral Difference Method

Large eddy simulation of the flow over a circular cylinder at Reynolds number ReD=2580 has been studied with a high-order unstructured spectral difference method.Grid and polynomial refinement studies were carried out to assess numerical errors.The mean and fluctuating velocity fields in the wake of a circular cylinder were compared with PIV experimental measurements.The numerical results are in excellent agreement with the experimental data for both the mean velocity and Reynolds stresses using the high-order SD scheme.Other wake characteristics such as the recirculation bubble length,vortex formation length and maximum intensity of the velocity fluctuations have also been predicted accurately.The numerical simulations demonstrated the potential of the high-order SD method in accurate large eddy simulation of physically complex problems.

On the Connection Between the Spectral Difference Method and the Discontinuous Galerkin Method

In this short note we present a derivation of the Spectral Difference Scheme from a Discontinuous Galerkin(DG)discretization of a nonlinear conservation law.This allows interpretation of the Spectral Difference Scheme as a particular discretization under the quadrature-free nodal DG paradigm.Moreover,it enables identification of the key differences between the Spectral Difference Scheme and standard nodal DG schemes.

Porosity Evaluation and the Power Spectral Densities Analyses of Carbon-Nickel Composite Films Annealed at Different Temperatures

The densification and the fractal dimensions of carbon-nickel films annealed at different temperatures 300, 500, 800, and 1000℃ with emphasis on porosity evaluation are investigated. For this purpose, the refractive index of films is determined from transmittance spectra. Three different regimes are identified, T 〈 500℃, 500℃ 〈 T 〈 800℃ and T 〉 800℃. The Rutherford baekscattering spectra show that with increasing the annealing temperature, the concentration of nickel atoms into films decreases. It is shown that the effect of annealing temperatures for increasing films densification at T 〈 500℃ and T 〉 800℃ is greater than the effect of nickel concentrations. It is observed that the effect of decreasing nickel atoms into films at 500℃ 〈 T 〈 800℃ strongly causes improving porosity and decreasing densification. The fractal dimensions of carbon-nickel films annealed from 300 to 500℃ are increased, while from 500 to 1000℃ these characteristics are decreased. It can be seen that at 800℃, films have maximum values of porosity and roughness.

Truncated Gaussian RBF Differences are Always Inferior to Finite Differences of the Same Stencil Width

Radial basis functions(RBFs)can be used to approximate derivatives and solve differential equations in several ways.Here,we compare one important scheme to ordinary finite differences by a mixture of numerical experiments and theoretical Fourier analysis,that is,by deriving and discussing analytical formulas for the error in differentiating exp(ikx)for arbitrary k.‘Truncated RBF differences”are derived from the same strategy as Fourier and Chebyshev pseudospectral methods:Differentiation of the Fourier,Chebyshev or RBF interpolant generates a differentiation matrix that maps the grid point values or samples of a function u(x)into the values of its derivative on the grid.For Fourier and Chebyshev interpolants,the action of the differentiation matrix can be computed indirectly but efficiently by the Fast Fourier Transform(FFT).For RBF functions,alas,the FFT is inapplicable and direct use of the dense differentiation matrix on a grid of N points is prohibitively expensive(O(N2))unless N is tiny.However,for Gaussian RBFs,which are exponentially localized,there is another option,which is to truncate the dense matrix to a banded matrix,yielding“truncated RBF differences”.The resulting formulas are identical in form to finite differences except for the difference weights.On a grid of spacing h with the RBF asφ(x)=exp(−α^(2)(x/h)^(2)),d f dx(0)≈∑^(∞)_(m)=1 wm{f(mh)−f(−mh)},where without approximation wm=(−1)m+12α^(2)/sinh(mα^(2)).We derive explicit formula for the differentiation of the linear function,f(X)≡X,and the errors therein.We show that Gaussian radial basis functions(GARBF),when truncated to give differentiation formulas of stencil width(2M+1),are significantly less accurate than(2M)-th order finite differences of the same stencil width.The error of the infinite series(M=∞)decreases exponentially asα→0.However,truncated GARBF series have a second error(truncation error)that grows exponentially asα→0.Even forα∼O(1)where the sum of these two errors is minimized,it is shown that the finite difference formulas are always superior.We explain,less rigorously,why these arguments extend to more general species of RBFs and to an irregular grid.There are,however,a variety of alternative differentiation strategies which will be analyzed in future work,so it is far too soon to dismiss RBFs as a tool for solving differential equations.