Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear systems

The response of a structural system to white noise excitation (deltacorrelated) constitutes a Markov vector process whose transitional probability density function (TPDF) is governed by both the forward Fokker-Planck and backward Kolmogorov equations. Numerical solution of these equations by finite element and finite difference methods for dynamical systems of engineering interest has been hindered by the problem of dimensionality. In this paper numerical solution of the stationary and transient form of the Fokker-Planck (FP) equation corresponding to two state nonlinear systems is obtained by standard sequential finite element method (FEM) using C⁰ shape function and Crank-Nicholson time integration scheme. The method is applied to Van-der-Pol and Duffing oscillators providing good agreement between results obtained by it and exact results. An extension of the finite difference discretization scheme developed by Spencer, Bergman and Wojtkiewicz is also presented. This paper presents an extension of the finite difference method for the solution of FP equation up to four dimensions. The difficulties associated in extending these methods to higher dimensional systems are discussed. This paper is dedicated to Prof R N Iyengar of the Indian Institute of Science on the occasion of his formal retirement.     

Thermoelectrical properties and optical third harmonic generation of Gd-doped PbTe

Nanocrystalline TiO₂ porous films were prepared by a sol-gel dip coating method from a solution using poly(ethylene glycol) as a template, ethanol as a solvent and terpineol as a highly viscous solvent. The thickness of films increases greatly by increasing the amount of terpineol. However, the overmuch incorporation of terpineol hinders the formation of porous structure and accelerates the crack creation. The crack-free film having a three-dimensionally extended porous structure was obtained at the ethanol/terpineol ratio of 8:1, the film thickness with three depositions being 2.5 μm. Pretreatment of the precursor at an appropriate temperature also plays an important role in pore formation. Single-phase anatase TiO₂ porous films were obtained after calcinations at 550°C for 1 h, and the crystal size increases with an increase in the amount of terpineol. The pore formation mechanism is discussed primarily in relation to the phase separation in the system and self-assembly of PEG.     

Quasi-phase matching of third harmonic generation in corrugated discharge capillary

Quasi-phase matching third harmonic of Ti:sapphire laser was measured by propagation through corrugated discharge capillaries. The 1 cm long capillary with periodical varying inner radius was used with corrugation frequency of 200 μm. The longitudinal plasma density was periodically varied by a discharge current ablating the inner capillary walls, altering the refractive index accordingly. Average plasma density measured was 10¹⁸ cm⁻³ at minimum corrugation radius. Peak laser intensity was 10¹⁰ W/cm² allowing enough pondermotive energy to the electron for high harmonic generation.     

Universally applicable uniaxial perfect matched layer formulation for explicit and implicit finite difference time domain algorithms

An important advance in the finite difference time domain (FDTD) algorithms is Berenger's perfect matched layer (PML), along with its derivatives. In recent years several new explicit and implicit FDTD algorithms have appeared on the scene, of which each requires a specific PML. To simplify programming, the authors derive a universal uniaxial PML formulation, which is universally applicable to the Yee FDTD, FDTD with higher-order stencils and implicit FDTD algorithms. Numerical results show the efficiency and versatility of the new method.     

Dynamic response in the time domain by coupled boundary and finite elements

This paper reports on the development of a finite element — boundary element coupling procedure for the analysis of arbitrary shaped elastic bodies subjected to dynamic loads. The coupling is accomplished through equilibrium and compatibility considerations along the boundary element — finite element interface.Several numerical studies are performed where one part of a uniform body is treated by finite elements, whereas the remaining region is descretized by boundary elements. The examples demonstrate the influence of different finite element approaches and the applicability and the accuracy of the proposed procedure.     

The influence of the absolute phase of a short light pulse on the third harmonic generation in an optically thin nonlinear layer

The third harmonic generation dynamics and the response spectrum of an optically thin layer of a medium possessing cubic nonlinearity depend on the absolute phase of a short light pulse with a small (～2%) detuning of the carrier frequency from that of the linear oscillator in the medium. The results of computer simulation show that a relatively small change in the absolute phase can result in a two-fold increase in the amplitude of the response pulse generated at the triple frequency.     

非线性伺服气动弹性的时域数值模拟

运用非定常流场求解器 ,直接求解翼面作任意运动的时域非定常气动力。将拉氏域内的伺服传递函数转化成时域内的伺服状态方程。通过时域气动力、结构状态方程和伺服状态方程的耦合求解 ,实现伺服气动弹性的时域数值模拟。可以方便地模拟如跨音速、大迎角及高超音速等带有不同非线性的伺服气动弹性响应。通过数值模拟发现 ,伺服系统会降低原有气动弹性系统的稳定性 ,结构陷波器的引入可以减弱伺服系统和结构模态的耦合。应用算例给出了大迎角下翼面的极限环响应和 M=7时伺服颤振速度随迎角变化的非线性特性。     

Aberration correction by nonlinear beam mixing: generation of a pseudo point sound source

Nonlinear beam mixing with microbubbles was explored to create a pseudo point source for aberration correction of therapeutic ultrasound. A damping coefficient for a bubble driven by a dual frequency sound field was derived by revisiting Prosperetti's linearized damping model. As a result, the overall damping term for dual frequency was obtained by linear summation of two damping terms for each frequency. The numerical simulation based on the bubble model suggests that the most efficient size range to generate a 1 MHz frequency from 4 MHz and 5 MHz sound sources is 2.6 to 3.0 microm. Furthermore, this size range constitutes the primary distribution of a specific ultrasound contrast agent. When a chamber of 0.1% of the diluted agent is sonified by 4 MHz and 5 MHz sound beams with 80 degrees incident angle between them, an approximately 100 Pa, 1 MHz difference frequency signal can be measured approximately 10 cm away. In addition, the received 1 MHz difference frequency signal shows omni-directional characteristics, even though the overlap zone of the two sound beams is on the order of the difference frequency wavelength. Therefore, the induced sound source can be considered as a pseudo point source and is expected to be useful for aberration correction for therapeutic ultrasound.     

Numerical simulation of cascaded third harmonic generation for both phase matched and mismatched schemes

Numerical solutions to the nonlinear coupling-wave equations of second harmonic (SH) and third harmonic (TH) generators are investigated for both phase matched and phase mismatched configurations. For phase mismatched TH generation, several kinds of schemes (the phase mismatch either in second harmonic generation (SHG) or sum-frequency generation (SFG) process) are considered and analyzed. The physical nature corresponding to the different ratios of the coupling coefficients is discussed.     

Fabrication of gold micro- and nanostructures by photolithographic exposure of thiol-stabilized gold nanoparticles

Exposure of thiol-stabilized gold nanoparticles supported on silicon wafers to UV light leads to oxidation of the thiol molecules and coagulation of the nanoparticles, forming densified structures that are resistant to removal by solvent exposure. Unoxidized particles may, in contrast, readily be removed leaving gold structures behind at the surface. This process provides a convenient and simple route for the fabrication of gold structures with dimensions ranging from micrometers to nanometers. The use of masks enables micrometer-scale structures to be fabricated rapidly. Exposure of nanoparticles to light from a near-field scanning optical microscope (NSOM) leads to the formation of gold nanowires. The dimensions of these nanowires depend on the method of preparation of the film: for spin-cast films, a width of 200 nm was achieved. However, this was reduced significantly, to 60 nm, for Langmuir-Schaeffer films.     

IR operation by third harmonic generation of Tl4PbTe3 and Tl4SnS3 single crystals

We have discovered substantial influence of the 3 μs CO₂ laser treatment on the output third harmonic generation (THG) for Tl₄SnS₃ and Tl₄PbTe₃ single crystals. The treatment was performed in the regime of the pump-probing delay. The optimal pump-probe delaying time was 230 ps for the Tl₄SnS₃ crystals and 250 ps for the Tl₄PbTe₃ single crystals. It is crucial that for the Tl₄PbTe₃ the effect is substantially smaller. The effect may be explained within a framework of conception of excitation of the effective phonon subsystem effectively contributing to the THG.     

Facile fabrication of 3D dendritic gold nanostructures with an AuSn alloy by square wave potential pulse

Multifunctional three dimensional (3D) dendritic gold (DG) nanostructures have been synthesized facilely by square wave potential pulse (SWPP) with an Au₄₅Sn₅₅ (wt.%) alloy in an electrolyte of NaOH or H₂SO₄. The dendrite formation involves combined actions of selective removal of Sn, oxidation-reduction cycles (ORCs) of gold and intensive hydrogen evolution, wherein the nanopores from dealloying and the movement of hydrogen bubbles facilitate the dendrite formation by rearranging the newly produced Au atoms from the ORCs. An alkaline medium is preferable and the dendritic structure can be obtained as quickly as in 100 s. Some applications have been demonstrated with the prepared DG including electrocatalytic oxidation of glucose, surface-enhanced Raman scattering, and superhydrophobicity.     

Axisymmetric nonlinear analysis of shallow spherical shells by a combined boundary element and finite difference method

This paper is concerned with the development of the mixed boundary element method and finite element method for the analysis of spherical annular shells under axisymmetric loads. The boundary element techniques are used to solve the equilibrium equation of shells and the central difference operator is adopted to deal with the compatibility equations. Iterative techniques are used throughout the analysis procedure. A number of numerical examples are given in the paper to illustrate the validity of the present approach.     

时域有限差分法及其在室内声场模拟中的应用

时域有限差分法将声波基本方程表示为一个时间和空间的迭代方程组,实现对室内声场中低频段的模拟计算。在介绍时域肯限差分法基本原理的基础上,着重探讨采用时域有限差分法模拟室内声场时边界条件的处理方法及该方法在室内声场模拟中的应用。分析了该方法存在的一些问题及其应用前景。     

A high order hybrid finite element method applied to the solution of electromagnetic wave scattering problems in the time domain

The development of a hybrid high order time domain finite element solution procedure for the simulation of two dimensional problems in computational electromagnetics is considered. The chosen application area is that of electromagnetic scattering. The spatial approximation adopted incorporates both a continuous Galerkin spectral element method and a high order discontinuous Galerkin method. Temporal discretisation is achieved by means of a fourth order Runge–Kutta procedure. An exact analytical solution is employed initially to validate the procedure and the numerical performance is then demonstrated for a number of more challenging examples.     

Angular distribution of nonlinear harmonic generation in helical undulators: A comparison between experiments and theory

Using harmonic emission from circularly polarized undulator is a procedure that is normally employed on synchrotron beamlines in order to extend the covered spectral range. A similar capability is likewise beneficial for next-generation free-electron lasers. In this paper, we perform a first quantitative experimental analysis of the angular distribution of free-electron laser harmonic emission from helical undulators. Experimental results are compared to those obtained by means of a theoretical model based on the paraxial solution of Maxwell's equations.     

Numerical modelling of elastic wave scattering in frequency domain by the partition of unity finite element method

In this paper, we investigate a numerical approach based on the partition of unity finite element method, for the time‐harmonic elastic wave equations. The aim of the proposed work is to accurately model two‐dimensional elastic wave problems with fewer elements, capable of containing many wavelengths per nodal spacing, and without refining the mesh at each frequency. The approximation of the displacement field is performed via the standard finite element shape functions, enriched by superimposing pressure and shear plane wave basis, which incorporate knowledge of the wave propagation. A variational framework able to handle mixed boundary conditions is described. Numerical examples dealing with the radiation and the scattering of elastic waves by a circular body are presented. The results show the performance of the proposed method in both accuracy and efficiency. Copyright © 2008 John Wiley & Sons, Ltd.     

利用直达波和海面反射波到达时延反演深海声速剖面

研究了利用直达波与海面反射波到达时延反演深海声速剖面的方法,采用最严经验模态方法组织利用历史水文数据并进行反演,并利用 2016年南海中南部冬季调查的实验数据进行了验证。反演结果与实测声速剖面在深度小于 710 m处的平均绝对误差的最大值为 2.15 m·s^-1,反演结果与实测值基本相符。     

Dynamic finite element analysis of viscoelastic composite plates in the time domain

Numerical procedures for analysing transient responses of anisotropic viscoeiastic composite plates have been developed in the time domain using variational principles and a direct time integration method. Recursion relationships have been formulated in order to reduce computer storage substantially. The method developed herein requires only two previous lime solutions to evaluate the next time solution and the necessary storage is independent of the number of the time steps. Verification studies were conducted to evaluate the accuracy and convergence of the present numerical algorithm and they show that the numerical solutions agree well with analytical ones. As examples, dynamic transient responses of viscoeiastic composite plates subjected to unit step loads were calculated using the presently developed procedure.