Legendre Polynomial Expansion Method Research Articles

An efficient nonlinear interval uncertain optimization method using Legendre polynomial chaos expansion

Many specialized numerical techniques have been developed on interval uncertainty optimization. However, the conventional nested optimization strategy is time-consuming. The improved Taylor-based interval optimization avoids the laborious inner optimization, nonetheless, it is intrusive and presents significant computational challenges in the large-level uncertain optimization problems. To end these, this paper proposes a Legendre polynomial expansion method combined with the subinterval technique to evaluate the range enclosure of an interval function, where the expansion coefficients are computed through the collocation method. The detailed comparisons demonstrate that such Legendre-based polynomial expansion provides much more confidence in interval bounds evaluation than the Taylor expansion, and is capable of maintaining very sharp solution enclosures even when the uncertainty level is increased. More exciting, it is highly efficient. Subsequent application of the Legendre polynomial expansion in the interval bound estimation of the uncertain objective function and constraints, and combining an appropriate outer optimizer, result in a novel nonlinear interval uncertain optimization method for complicated engineering systems, whereby the time-consuming optimization nesting is avoided. A beam design example and an electromagnetic buffer optimization example illustrate that the proposed interval optimization method has fine computational precision and high computational efficiency, especially is capable of attacking the increase of uncertainty level. Moreover, the new method does not require the derivative information of the uncertain objective and constraints, such that applies to any engineering uncertain optimization problems, including the complex engineering problems which are usually referred to as “black-box”.

On computation of Hough functions

Abstract. Hough functions are the eigenfunctions of the Laplace tidal equation governing fluid motion on a rotating sphere with a resting basic state. Several numerical methods have been used in the past. In this paper, we compare two of those methods: normalized associated Legendre polynomial expansion and Chebyshev collocation. Both methods are not widely used, but both have some advantages over the commonly used unnormalized associated Legendre polynomial expansion method. Comparable results are obtained using both methods. For the first method we note some details on numerical implementation. The Chebyshev collocation method was first used for the Laplace tidal problem by Boyd (1976) and is relatively easy to use. A compact MATLAB code is provided for this method. We also illustrate the importance and effect of including a parity factor in Chebyshev polynomial expansions for modes with odd zonal wave numbers.

Inversion of Functional Graded Materials Elastic Properties from Ultrasonic Lamb Wave Phase Velocity Data Using Genetic Algorithm

An inverse method based on genetic algorithm (GA) is presented to determine the elastic properties of functional graded materials (FGMs) plate from lamb wave phase velocity data. The Legendre polynomial expansion method is used as a forward solver to calculate the phase velocity dispersion curves of lamb wave in FGMs plate, which has a distribution function to express the inhomogeneity of materials’ elastic properties. The properties of FGMs plate can be characterized by minimizing the standard deviations between the actual and calculated phase velocities of lamb waves. By using GA, the elastic parameters of FGMs plates with three different distribution functions are inversed, and the convergence and deviations of the inversion are also discussed. The investigation shows that this inverse method is efficient and accurate for determining the materials parameters of FGMs and the deviation can be controlled below 5 %.

一维衍射光栅和一维光子晶体组成的硅薄膜太阳能电池背反射器

We designed a type of reflector that consisted of one-dimensional diffraction gratings and one-dimensional photonic crystal.It can be used for thin-film silicon solar cells.The structure parameters were optimized by Legendre polynomial expansion method,and the reflectivity at the range of 400~1 200 nm was calculated.Under the high reflectivity one-dimensional photonic crystal,the propagation path of the photon can be extended and absorbed in the battery.The results show that the diffraction grating and a back reflector in the photonic crystal structure are efficient at the range of 750~1 100 nm.Back-reflection layer can greatly improve the light harvesting capacity and enhance the conversion efficiency of solar cells.

Artifact-free analysis of highly conducting binary gratings by using the Legendre polynomial expansion method

Analysis of highly conducting binary gratings in TM polarization has been problematic as the Fourier factorization fails and thus unwanted numerical artifacts appear. The Legendre polynomial expansion method (LPEM) is employed here, and the erroneous harsh variations attributed to the violation of the inverse rule validity in applying the Fourier factorization are filtered out. In this fashion, stable and artifact-free numerical results are obtained. The observed phenomenon is clearly demonstrated via several numerical examples and is explained by inspecting the transverse electromagnetic field profile.

Longitudinal Legendre polynomial expansion of electromagnetic fields for analysis of arbitrary-shaped gratings

The Legendre polynomial expansion method (LPEM), which has been successfully applied to homogenous and longitudinally inhomogeneous gratings [J. Opt. Soc. Am. B24, 2676 (2007)], is now generalized for the efficient analysis of arbitrary-shaped surface relief gratings. The modulated region is cut into a few sufficiently thin arbitrary-shaped subgratings of equal spatial period, where electromagnetic field dependence is now smooth enough to be approximated by keeping fewer Legendre basis functions. The R-matrix propagation algorithm is then employed to match the Legendre polynomial expansions of the transverse electric and magnetic fields across the upper and lower interfaces of every slice. The proposed strategy then enhances the overall computational efficiency, reduces the required memory size, and permits the efficient study of arbitrary-shaped gratings. Here the rigorous approach is followed, and analytical formulas of the involved matrices are given.

Ordering transition and demixing of a binary mixture of thick and thin rodlike molecules in the presence of an external field

The effect of an external field (electric/magnetic) on the phase behavior of the binary mixture of very long thick and thin rodlike particles is studied. Both the thick and thin particles possess positive but different susceptibility anisotropics (Delta alpha). The difference in the extent of interaction between the external field and the two species is varied by means of a coupling parameter (l = Delta alpha(thick)/Delta alpha(thin)). Isotropic-nematic phase transition and demixing phase transitions taking place both in the isotropic and nematic phases are examined as a function of field strength on the level of the second virial theory of Onsager in the range of 0 < l <1. The approximate sixth order Legendre polynomial expansion method is used to represent the excluded volume interaction between the rodlike particles. It is found that the isotropic phase becomes weakly nematic (paranematic) in the presence of external field and the field orients both components in the direction of the field even if the field does not have direct interaction with the thick component (l = 0). Analytical expressions are derived for the external field induced order parameters and birefringence. The increasing field destabilizes both types of demixing transitions (isotropic-isotropic and nematic-nematic) and the paranematic-nematic phase transition. Moreover it induces closed loop immiscibility, and upper and lower critical points terminating the paranematic-nematic phase coexistence may occur for low values of the coupling parameter. It is interesting that while the phase boundaries of the paranematic-paranematic demixing and the paranematic-nematic transitions are very sensitive to the value of the coupling parameter at low pressures, the paranematic-nematic and nematic-nematic phase boundaries are practically independent of the coupling parameter at high pressures.

The orbital precession around oblate spheroids

An exact series will be given for the gravitational potential generated by an oblate gravitating source. To this end the corresponding Epstein-Hubbell type elliptic integral is evaluated. The procedure is based on the Legendre polynomial expansion method and on combinatorial techniques. The result is of interest for gravitational models based on the linearity of the gravitational potential. The series approximation for such potentials is of use for the analysis of orbital motions around a nonspherical source. It can be considered advantageous that the analysis is purely algebraic. Numerical approximations are not required. As an important example, the expression for the orbital precession will be derived for an object orbiting around an oblate homogeneous spheroid.

A series approximation for disk galaxies by means of the Epstein-Hubbell integral

In this paper, an exact series will be derived for the calculation of rotational velocities in homogeneous disk galaxies with finite thickness. The procedure is based on the Legendre polynomial expansion method for the solution of the Epstein-Hubbell integral. A ‘scaling’ method will also be given in order to derive approximate values for rotational velocities in disk galaxies with a nonuniform distribution. An interesting identity, which shows up in the analysis, has been proved.

Legendre polynomial expansion method for evaluating multipole moments of current density in toroidal devices

It is found that multipole moments of toroidal current are given by the coefficients of the Legendre polynomial expansion of the magnetic field on a meridian contour. Using this fact and the orthogonality of the Legendre polynomials, a method is proposed for evaluating the moments from magnetic field measurements through an open-contour. As an application, exact expressions of the first few order moments and the current center position are formulated. Results show that this method is applicable to any aspect ratio tokamaks without the limitation of small displacement of the current center.

Deterministic modeling of impact ionization with a random-k approximation and the multiband Boltzmann equation

We present here an approach for determining impact ionization coefficients for the spherical multiband model in silicon. Using random-k approximation, the impact ionization rate is determined to reflect the multiband density of states in silicon. To account for the actual density of states, we have solved four coupled Boltzmann transport equations by combining a generalized Legendre polynomial expansion method with numerical techniques using finite differences and sparse matrices. Calculated values for the impact ionization coefficients agree with experiments for electrons in silicon, while being obtained in significantly less CPU time than required by analogous Monte Carlo calculations. Different multiband transport parameter sets are also compared.

О классификации сингулярности в конфигурации деформации в модели жидкой капли

The deformation shape of the liquid-drop model of nuclear fission can be excellently represented by the Legendre-polynomial expansion method. Unfortunately, in the past, it has appeared necessary to include a large number of higher-order terms in order to describe the more deformed shapes encountered in the later stages of fission. In this paper, we classify the degenerate singularity that occurs in the Businaro-Gallone-Swiatecki representation of the Legendre-polynomial expansion of the deformation shape for the slightly distorted ground state. In our analysis, we found the singularity to be double-degenerate and strongly 3-determinate. As a result, we have classified it as a hyperbolic-umbilic elementary catastrophe. In order to make the potential function of the singularity a universal unfolding, we introduced two new control parameters for the liquid-drop model.