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Journal articles on the topic 'High–Order Perturbation of Surfaces Methods'

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Author: Grafiati

Published: 10 December 2022

Last updated: 28 January 2023

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1

Nicholls, David P., Fernando Reitich, Timothy W. Johnson, and Sang-Hyun Oh. "Fast high-order perturbation of surfaces methods for simulation of multilayer plasmonic devices and metamaterials." Journal of the Optical Society of America A 31, no. 8 (July 24, 2014): 1820. http://dx.doi.org/10.1364/josaa.31.001820.

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2

Klahn, Mathias, Per A. Madsen, and David R. Fuhrman. "On the accuracy and applicability of a new implicit Taylor method and the high-order spectral method on steady nonlinear waves." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2243 (November 2020): 20200436. http://dx.doi.org/10.1098/rspa.2020.0436.

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This paper presents an investigation and discussion of the accuracy and applicability of an implicit Taylor (IT) method versus the classical higher-order spectral (HOS) method when used to simulate two-dimensional regular waves. This comparison is relevant, because the HOS method is in fact an explicit perturbation solution of the IT formulation. First, we consider the Dirichlet–Neumann problem of determining the vertical velocity at the free surface given the surface elevation and the surface potential. For this problem, we conclude that the IT method is significantly more accurate than the HOS method when using the same truncation order, M , and spatial resolution, N , and is capable of dealing with steeper waves than the HOS method. Second, we focus on the problem of integrating the two methods in time. In this connection, it turns out that the IT method is less robust than the HOS method for similar truncation orders. We conclude that the IT method should be restricted to M = 4, while the HOS method can be used with M ≤ 8. We systematically compare these two options and finally establish the best achievable accuracy of the two methods as a function of the wave steepness and the water depth.

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3

Yan, Jun Xi, Fang Peng, and Yu Feng Peng. "Influence of Thermal Distortion of Mirrors in Laser Resonators on Laser Modes." Applied Mechanics and Materials 214 (November 2012): 223–26. http://dx.doi.org/10.4028/www.scientific.net/amm.214.223.

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Taking into account the thermal absorption of laser light on laser mirror surface, the thermal deformation of the mirror substrate was analyzed. Influence of the thermal distortions of the mirror substrate on the laser resonant modes was calculated. The expressions for the field distributions of transverse laser modes with perturbation methods were derived. By comparing them with the elegant Hermite-Gaussian modes where there were no thermal distortions, the intensity profiles of the transverse laser modes versus the amount of the thermal deformation were obtained. The results show that the larger the thermal distortions of mirrors are, the wider the intensity profiles of the fundamental mode become, and the smaller the peak intensity becomes. Meanwhile, the high-order modes will be induced.

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4

Machado, M. R., and J. M. C. Dos Santos. "Reliability Analysis of Damaged Beam Spectral Element with Parameter Uncertainties." Shock and Vibration 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/574846.

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The paper examines the influence of uncertainty parameters on the wave propagation responses at high frequencies for a damaged beam structure in the structural reliability context. The reliability analyses were performed using the perturbation method, First-Order Reliability Method (FORM), and response surface method (RSM) which were compared with Monte Carlo simulation (MCS) under the spectral element method environment. The simulated results were performed to investigate the effects of material property and geometric uncertainties on the response at high frequency modes, such as the computational efficiency of reliability methods. For the first time, the spectral element method is used in the context of reliability analysis at medium and high frequency bands applied to damage detection. It has shown the effects of parameters uncertainty on the dynamic beam response due on an impulsive load and the robustness of each method. Numerical examples in a bending vibrating beam with random parameters are performed to verify the computational efficiency of the present study.

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5

Karton, Amir, and Laura K. McKemmish. "Can Popular DFT Approximations and Truncated Coupled Cluster Theory Describe the Potential Energy Surface of the Beryllium Dimer?" Australian Journal of Chemistry 71, no. 10 (2018): 804. http://dx.doi.org/10.1071/ch18269.

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The potential energy surface (PES) of the ground state of the beryllium dimer poses a significant challenge for high-level ab initio electronic structure methods. Here, we present a systematic study of basis set effects over the entire PES of Be2 calculated at the full configuration interaction (FCI) level. The reference PES is calculated at the valence FCI/cc-pV{5,6}Z level of theory. We find that the FCI/cc-pV{T,Q}Z basis set extrapolation reproduces the shape of the FCI/cc-pV{5,6}Z PES as well as the binding energy and vibrational transition frequencies to within ~10 cm−1. We also use the FCI/cc-pV{5,6}Z PES to evaluate the performance of truncated coupled cluster methods (CCSD, CCSD(T), CCSDT, and CCSDT(Q)) and contemporary density functional theory methods (DFT) methods for the entire PES of Be2. Of the truncated coupled cluster methods, CCSDT(Q)/cc-pV{5,6}Z provides a good representation of the FCI/cc-pV{5,6}Z PES. The GGA functionals, as well as the HGGA and HMGGA functionals with low percentages of exact exchange tend to severely overbind the Be2 dimer, whereas BH&HLYP and M06-HF tend to underbind it. Range-separated DFT functionals tend to underbind the dimer. Double-hybrid DFT functionals show surprisingly good performance, with DSD-PBEP86 being the best performer. Møller–Plesset perturbation theory converges smoothly up to fourth order; however, fifth-order corrections have practically no effect on the PES.

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6

Gerwe, Brian S., Ehsan Nasr Esfahani, Jiangyu Li, and Stuart B. Adler. "Detecting Dynamic Manipulation of the Space Charge Region in Doped Ceria with Scanning Thermo-Ionic Microscopy." ECS Meeting Abstracts MA2018-01, no. 32 (April 13, 2018): 1939. http://dx.doi.org/10.1149/ma2018-01/32/1939.

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A new generation of high performance energy storage and conversion devices is instrumental to commercial adoption of renewable, but intermittent, energy sources. Developing new materials for these devices often relies on bulk characterization techniques, such as electrochemical impedance spectroscopy, which are inherently unable to spatially resolve electrochemical phenomena and material properties. Conversely, scanning probe methods (SPM) have long been employed to probe local electrochemistry and material properties; however, relating these measurements to macroscopic performance can be challenging. We aim to bridge this gap by using SPM as a localized in operando witness during macroscopic measurements. Workers have recently employed Scanning Thermo-Ionic Microscopy (STIM), a strain-based SPM that leverages mechano-thermo-chemical coupled effects to detect local ionic concentrations. Briefly, STIM operates by inducing thermal stresses in the material from a periodic temperature perturbation at the scanning probe tip. The strain related to these thermal stresses drives ionic motion and is detected by deflection of the probe tip at a higher order harmonic of the temperature perturbation. This allows the ionic response to be distinguished from electrostatic and electromechanical responses, which occur at the temperature perturbation frequency. Previous ex-situ measurements demonstrated the potential of STIM to image grain boundary effects in doped ceria, possibly due to space charge regions. More recently, STIM was implemented point-wise to study manipulation of the space charge region under static polarization. Though promising, intensive hardware demands prevented imaging with STIM, thus ultimately reducing the impact of s-SPM implementation. Herein we demonstrate the use of data science enhanced SPM to overcome hardware limitations and extend our work to in operando, dynamic polarization studies. Furthermore, the effect of adsorbed surface species on STIM response is examined by controlling temperature, gas environment, and relative humidity.

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7

Hou, Thomas Y. "Numerical Solutions to Free Boundary Problems." Acta Numerica 4 (January 1995): 335–415. http://dx.doi.org/10.1017/s0962492900002567.

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Many physically interesting problems involve propagation of free surfaces. Vortex-sheet roll-up in hydrodynamic instability, wave interactions on the ocean's free surface, the solidification problem for crystal growth and Hele-Shaw cells for pattern formation are some of the significant examples. These problems present a great challenge to physicists and applied mathematicians because the underlying problem is very singular. The physical solution is sensitive to small perturbations. Naïve discretisations may lead to numerical instabilities. Other numerical difficulties include singularity formation and possible change of topology in the moving free surfaces, and the severe time-stepping stability constraint due to the stiffness of high-order regularisation effects, such as surface tension.This paper reviews some of the recent advances in developing stable and efficient numerical algorithms for solving free boundary-value problems arising from fluid dynamics and materials science. In particular, we will consider boundary integral methods and the level-set approach for water waves, general multi-fluid interfaces, Hele–Shaw cells, crystal growth and solidification. We will also consider the stabilising effect of surface tension and curvature regularisation. The issue of numerical stability and convergence will be discussed, and the related theoretical results for the continuum equations will be addressed. This paper is not intended to be a detailed survey and the discussion is limited by both the taste and expertise of the author.

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8

Stewart, James A., Joseph D. Olles, and Mitchell A. Wood. "Elucidating size effects on the yield strength of single-crystal Cu via the Richtmyer–Meshkov instability." Journal of Applied Physics 131, no. 11 (March 21, 2022): 114901. http://dx.doi.org/10.1063/5.0082495.

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Capturing the dynamic response of a material under high strain-rate deformation often demands challenging and time consuming experimental effort. While shock hydrodynamic simulation methods can aid in this area, a priori characterizations of the material strength under shock loading and spall failure are needed in order to parameterize constitutive models needed for these computational tools. Moreover, parameterizations of strain-rate-dependent strength models are needed to capture the full suite of Richtmyer–Meshkov instability (RMI) behavior of shock compressed metals, creating an unrealistic demand for these training data solely on experiments. Herein, we sweep a large range of geometric, crystallographic, and shock conditions within molecular dynamics (MD) simulations and demonstrate the breadth of RMI in Cu that can be captured from the atomic scale. Yield strength measurements from jetted and arrested material from a sinusoidal surface perturbation were quantified as [Formula: see text] GPa, higher than strain-rate-independent models used in experimentally matched hydrodynamic simulations. Defect-free, single-crystal Cu samples used in MD will overestimate [Formula: see text], but the drastic scale difference between experiment and MD is highlighted by high confidence neighborhood clustering predictions of RMI characterizations, yielding incorrect classifications.

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9

Veerappan, Anbu, and Vijayalakshmi Karattadipalayam Arumugam. "Quantum Chemical Studies on the Spectroscopic, Electronic Structural and Nonlinear Properties of an Organic N-Methyl-N- (2,4,6-Trinitrophenyl) Nitramide Energetic Molecule." Current Physical Chemistry 9, no. 1 (June 3, 2019): 5–21. http://dx.doi.org/10.2174/1877946809666190218154806.

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Background: Earlier studies on the energetic molecule MTNPN show a small HOMO-LUMO energy gap. In general, the material which acquires small energy gap exhibits NLO response and identical counterparts in both IR and Raman spectra. Hence, the combined experimental and theoretical studies were performed to explore the fundamental properties of the molecule. Objective: The objective of this study was to explore the fundamental structural properties of an energetic molecule MTNPN in addition to its application as a nonlinear optical material. Methods: FT-IR technique and quantum chemical methods were used to analyze the vibrational normal modes and structural properties of the molecule. Kurtz and Perry technique is used to find second harmonic generation efficiency in comparison to the standard NLO reference material. Results: The potential energy distribution was used to assign the vibrational normal modes of the molecule. The second order perturbation energies between the lone pair and anti-bonding species were predicted to understand the driving forces of molecular stability. The chemical reactivity of the molecule was determined from the molecular electrostatic potential surface and global reactivity descriptor results. The second-order hyperpolarizability of MTNPN and SHG efficiency of MTNPN were studied to find its NLO response and it was found from the results that MTNPN exhibits high NLO response than the standard NLO reference material. Conclusion: The vibrational degrees of freedom of MTNPN molecule were assigned and the experimental FT-IR spectra were compared with the scaled harmonic frequencies. The predicted second-order hyperpolarizability of MTNPN was about 6.46 times greater than the standard NLO reference urea. The interacting species between the lone pair orbitals and antibonding orbitals such as n3O8→ π*(N7-O9), n3O11→ π*(N10-O12) and n3O14→ π*(N13-O15) stabilized the molecule to a greater extent.

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10

Isabel B. LOBO, L., and David C. WILTON. "Effect of lipid composition on lipoprotein lipase activity measured by a continuous fluorescence assay: effect of cholesterol supports an interfacial surface penetration model." Biochemical Journal 321, no. 3 (February 1, 1997): 829–36. http://dx.doi.org/10.1042/bj3210829.

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The breakdown of normal substrates by lipases requires an interfacial binding step prior to hydrolysis. Interfacial binding and subsequent hydrolysis will be affected by the lipid components and hence physical properties of the substrate surface. In order to investigate in detail the effect of lipid structure on the activity of lipoprotein lipase (LPL), triolein-containing emulsion particles of defined composition have been used as substrates. In addition, lipase activity has been measured using a continuous fluorescence displacement assay that monitors the release of long-chain fatty acids as an alternative to normal radiochemical assays. Using this fluorescence assay, rates of hydrolysis of triolein were the same as when using a standard radiochemical assay under identical conditions. Activation by apolipoprotein CII was very similar by both methods; however, the extent of activation (2Ő3-fold) was less than has been reported previously using different assay conditions. In order to investigate the effect of cholesterol on LPL activity, emulsion particles were prepared in which the cholesterol/egg-phosphatidylcholine ratio was increased up to a 1:1 molar ratio. A pronounced stimulatory effect of cholesterol was observed under these assay conditions, with up to a 5-fold increase in rate compared with emulsion particles without cholesterol. Since high molar ratios of cholesterol are reported to exclude triacylglycerol from the phospholipid surface [Spooner and Small (1987) Biochemistry 26, 5820Ő5825], these results are not consistent with a mechanism involving LPL hydrolysis of surface triacylglycerol. Instead, they support an interfacial penetration model, allowing the enzyme's active site direct access to triacylglycerol in the lipoprotein core. Perturbation of the surface phospholipid monolayer of the emulsion particle as a result of hydrolysis by Naja naja phospholipase A2 resulted in a 10-fold activation of LPL, providing further support for an interfacial penetration model. The stimulatory effect of apolipoprotein CII was not modulated by modification of the interface with cholesterol.

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11

Nenov, Artur, Javier Segarra-Martí, Angelo Giussani, Irene Conti, Ivan Rivalta, Elise Dumont, Vishal K. Jaiswal, Salvatore Flavio Altavilla, Shaul Mukamel, and Marco Garavelli. "Probing deactivation pathways of DNA nucleobases by two-dimensional electronic spectroscopy: first principles simulations." Faraday Discussions 177 (2015): 345–62. http://dx.doi.org/10.1039/c4fd00175c.

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The SOS//QM/MM [Rivalta et al., Int. J. Quant. Chem., 2014, 114, 85] method consists of an arsenal of computational tools allowing accurate simulation of one-dimensional (1D) and bi-dimensional (2D) electronic spectra of monomeric and dimeric systems with unprecedented details and accuracy. Prominent features like doubly excited local and excimer states, accessible in multi-photon processes, as well as charge-transfer states arise naturally through the fully quantum-mechanical description of the aggregates. In this contribution the SOS//QM/MM approach is extended to simulate time-resolved 2D spectra that can be used to characterize ultrafast excited state relaxation dynamics with atomistic details. We demonstrate how critical structures on the excited state potential energy surface, obtained through state-of-the-art quantum chemical computations, can be used as snapshots of the excited state relaxation dynamics to generate spectral fingerprints for different de-excitation channels. The approach is based on high-level multi-configurational wavefunction methods combined with non-linear response theory and incorporates the effects of the solvent/environment through hybrid quantum mechanics/molecular mechanics (QM/MM) techniques. Specifically, the protocol makes use of the second-order Perturbation Theory (CASPT2) on top of Complete Active Space Self Consistent Field (CASSCF) strategy to compute the high-lying excited states that can be accessed in different 2D experimental setups. As an example, the photophysics of the stacked adenine–adenine dimer in a double-stranded DNA is modeled through 2D near-ultraviolet (NUV) spectroscopy.

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12

Chen, ZhongYing, ChongNan He, and Bin Wu. "High order finite volume methods for singular perturbation problems." Science in China Series A: Mathematics 51, no. 8 (July 24, 2008): 1391–400. http://dx.doi.org/10.1007/s11425-008-0120-1.

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13

Zhang, Dingkang, and Farzad Rahnema. "High-Order Perturbation Theory for Incident Flux Response Expansion Methods." Nuclear Science and Engineering 176, no. 1 (January 2014): 69–80. http://dx.doi.org/10.13182/nse13-1.

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14

Inogamov, N. A. "Many-dimensional periodicity and turbulence of Rayleigh–Taylor and Richtmyer–Meshkov instabilities." Laser and Particle Beams 15, no. 1 (March 1997): 53–64. http://dx.doi.org/10.1017/s0263034600010740.

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The work begins with a presentation of results concerning periodic Rayleigh-Taylor (RT)and Richtmyer–Meshkov (RM) dynamics. The periodic RT flows are considered in 2D and 3D cases. The periodic RM flows are considered in a 2D case. Nonstationary and steadystate statements of RT and RM problems are given. Nonstationary statements of RT and RM problems are given in the2D case. Steady-state statements of RT and RM problems are considered in the 2D and 3D cases (RT) and in the 2D case (RM). Questions about turbulence are given in the second part of the work. In the nonstationary statement, the beginning of a transition from an equilibrium close to hydrostatics and behavior of a dynamic system when it is close to the steady-state solution are considered. To study the smooth evolution of the dynamic system in the RT case (at last during an initial of order of unity t ≃ 1 stage) we have to cut of f an inverse turbulent cascade incoming from the small scales (inverse means that the typical k decreases in time during the cascade). The cascade starts from some small scale k ≫ 1 (here g = 1) and appears in the scales k ≃ 1 after time t ≃ 1 after start. We have to consider exponentially smoothed initial data to cut it off. This means that amplitudes of initial spectrum of perturbations have to be exponentially decreased in the direction of high k. The steadystates have been studied in two ways. In both of them the spectral decomposition of a velocity field and an elevation of the contact surface has been used. In one of them, corresponding to the nonstationary situation, the ordinary differential system for time dependent spectral components has been integrated. The other solutions of the algebraic equations for constant components have been studied. These results have been compared with a computer simulation that has been done by large particles and artificial compressibility methods. High-order nonlinear terms corresponding to the interaction of a wide spectrum of harmonics are considered in both ways. The spatial structure of the many dimensional periodic solutions is studied. It is known that there are two important features in the flows under consideration: the bubble envelope and the jet “phase.” Spatial structures of the 2D and 3D flows are compared in the report. It is shown that the apices of the bubbles are isolated points and the bubble envelopes in both flows are qualitatively similar; the difference has a quantitative character. On the contrary, the jet “phases” differ qualitatively in the 2D and 3D cases. In the 2D case in the plane picture, the jets are separated from each other. In this picture they are isolated fingers. In the 3D case, near the bubble envelope after some time of development following after cos kx cos qy type initial perturbation, they are not the isolated fingers. They form a structure of the intersecting walls. We see, thus, that the 2D and 3D cases are topologically different. Most powerful ejections of dense fluid are in the points where these wall jets intersect each other.

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15

Chen, Bilian, Yajun Xie, and Changfeng Ma. "SOME HIGH ORDER ITERATIVE METHODS FOR NONLINEAR EQUATIONS BASED ON THE MODIFIED HOMOTOPY PERTURBATION METHODS." Asian-European Journal of Mathematics 03, no. 03 (September 2010): 395–408. http://dx.doi.org/10.1142/s1793557110000349.

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In this paper, we present some efficient iterative methods for solving nonlinear equation (systems of nonlinear equations, respectively) by using modified homotopy perturbation methods. We also discuss the convergence criteria of the present methods. Some numerical examples are given to illustrate the performance and efficiency of the proposed methods.

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16

Sadlej, Andrzej J. "Infinite-Order Regular Approximation by the Metric Perturbation." Collection of Czechoslovak Chemical Communications 70, no. 5 (2005): 677–88. http://dx.doi.org/10.1135/cccc20050677.

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The regular approximation methods for the reduction of the Dirac equation to a fully equivalent two-component form are considered in the framework of the perturbation theory. The usual Dirac hamiltonian is first transformed with the change of metric. Then, the change of metric is considered as a perturbation to the zeroth-order (ZORA) problem. General formulae for perturbation corrections to the ZORA wave function and energy are expressed solely in terms of the two-component solutions. The method presented in this paper gives the energy- independent scheme for the step-by-step generation of the infinite-order results which are equivalent to solutions of the Dirac equations. Several formal and computational aspects of the infinite-order regular approximation are discussed. It is concluded that, because of the use of well-behaved operators, the high-order regular approximation methods can be considered as competitive to high-order Douglas-Kroll approaches.

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17

Zhang, Weiqun. "Improved High Order Methods Using Boundary Layer Detection for a Singular Perturbation Problem." Journal of Applied Mathematics and Computation 4, no. 4 (December 11, 2020): 189–94. http://dx.doi.org/10.26855/jamc.2020.12.010.

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18

Myllerup, C. M., and B. J. Hamrock. "Perturbation Approach to Hydrodynamic Lubrication Theory." Journal of Tribology 116, no. 1 (January 1, 1994): 110–18. http://dx.doi.org/10.1115/1.2927025.

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Three perturbation approaches that apply for regular hydrodynamic lubrication problems are discussed: a cross-film coordinate perturbation, an iterative scheme, and a regular perturbation in terms of the film aspect ratio. The methods are used to derive higher order terms for a driven corner flow with a Newtonian lubricant of constant properties. Reasons for preferring the regular perturbation scheme are presented, and this method is used to obtain the correct curvature correction in an infinitely long journal bearing. Criteria for identifying singular geometries in hydrodynamic lubrication are set fourth.

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19

KORCHEMSKY, G. P. "MATRIX MODEL PERTURBED BY HIGHER ORDER CURVATURE TERMS." Modern Physics Letters A 07, no. 33 (October 30, 1992): 3081–100. http://dx.doi.org/10.1142/s0217732392002470.

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The critical behavior of the D=0 matrix model with the potential perturbed by a nonlocal term generating touchings between random surfaces is studied. It is found that the phase diagram of the model has many features of the phase diagram of discretized Polyakov’s bosonic string with higher order curvature terms included. It contains the phase of smooth (Liouville) surfaces, the intermediate phase and the phase of branched polymers. The perturbation becomes irrelevant at the first phase and dominates at the third one.

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20

Chen, P. C., Danny D. Liu, and Darius Sarhaddi. "High-Order vs Low-Order Panel Methods for Unsteady Subsonic Lifting Surfaces." Journal of Aircraft 41, no. 4 (July 2004): 957–59. http://dx.doi.org/10.2514/1.c10304r.

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21

Nicholls, David P. "Numerical Solution of Diffraction Problems: A High-Order Perturbation of Surfaces and Asymptotic Waveform Evaluation Method." SIAM Journal on Numerical Analysis 55, no. 1 (January 2017): 144–67. http://dx.doi.org/10.1137/16m1059679.

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22

Nicholls, David P. "Numerical Simulation of Grating Structures Incorporating Two-Dimensional Materials: A High-Order Perturbation of Surfaces Framework." SIAM Journal on Applied Mathematics 78, no. 1 (January 2018): 19–44. http://dx.doi.org/10.1137/17m1123481.

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23

Saye, Robert. "High-order methods for computing distances to implicitly defined surfaces." Communications in Applied Mathematics and Computational Science 9, no. 1 (May 17, 2014): 107–41. http://dx.doi.org/10.2140/camcos.2014.9.107.

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24

Antonietti, Paola F., Andreas Dedner, Pravin Madhavan, Simone Stangalino, Björn Stinner, and Marco Verani. "High Order Discontinuous Galerkin Methods for Elliptic Problems on Surfaces." SIAM Journal on Numerical Analysis 53, no. 2 (January 2015): 1145–71. http://dx.doi.org/10.1137/140957172.

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25

Larson, Mats G., and Sara Zahedi. "Stabilization of high order cut finite element methods on surfaces." IMA Journal of Numerical Analysis 40, no. 3 (April 25, 2019): 1702–45. http://dx.doi.org/10.1093/imanum/drz021.

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Abstract We develop and analyse a stabilization term for cut finite element approximations of an elliptic second-order partial differential equation on a surface embedded in ${\mathbb{R}}^d$. The new stabilization term combines properly scaled normal derivatives at the surface together with control of the jump in the normal derivatives across faces, and provides control of the variation of the finite element solution on the active three-dimensional elements that intersect the surface. We show that the condition number of the stiffness matrix is $O(h^{-2})$, where $h$ is the mesh parameter. The stabilization term works for linear as well as for higher-order elements and the derivation of its stabilizing properties is quite straightforward, which we illustrate by discussing the extension of the analysis to general $n$-dimensional smooth manifolds embedded in ${\mathbb{R}}^d$, with codimension $d-n$. We also state the properties of a general stabilization term that are sufficient to prove optimal scaling of the condition number and optimal error estimates in energy- and $L^2$-norm. We finally present numerical studies confirming our theoretical results.

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26

Silverstone, Harris J., Jonathan G. Harris, Jirí Čížek, and Josef Paldus. "Asymptotics of high-order perturbation theory for the one-dimensional anharmonic oscillator by quasisemiclassical methods." Physical Review A 32, no. 4 (October 1, 1985): 1965–80. http://dx.doi.org/10.1103/physreva.32.1965.

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27

Nicholls, David P., and Xin Tong. "High-Order Perturbation of Surfaces Algorithms for the Simulation of Localized Surface Plasmon Resonances in Two Dimensions." Journal of Scientific Computing 76, no. 3 (February 14, 2018): 1370–95. http://dx.doi.org/10.1007/s10915-018-0665-2.

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28

Kaplan, Matthew, and David P. Nicholls. "A nonlinear least squares framework for periodic grating identification with a high–order perturbation of surfaces implementation." Applied Numerical Mathematics 143 (September 2019): 20–34. http://dx.doi.org/10.1016/j.apnum.2019.03.012.

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29

Nicholls, David P., and Xin Tong. "High–order perturbation of surfaces algorithms for the simulation of localized surface plasmon resonances in graphene nanotubes." Applied Numerical Mathematics 157 (November 2020): 544–62. http://dx.doi.org/10.1016/j.apnum.2020.07.012.

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30

Hong, Youngjoon, and David P. Nicholls. "A high-order perturbation of surfaces method for vector electromagnetic scattering by doubly layered periodic crossed gratings." Journal of Computational Physics 372 (November 2018): 748–72. http://dx.doi.org/10.1016/j.jcp.2018.06.041.

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31

Nicholls, David P. "A high–order perturbation of surfaces (HOPS) approach to Fokas integral equations: Three–dimensional layered–media scattering." Quarterly of Applied Mathematics 74, no. 1 (December 3, 2015): 61–87. http://dx.doi.org/10.1090/qam/1411.

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32

Nieto, Agustin. "Perturbative QCD at High Temperature." International Journal of Modern Physics A 12, no. 08 (March 30, 1997): 1431–64. http://dx.doi.org/10.1142/s0217751x97001043.

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Recent developments of perturbation theory at finite temperature based on effective field theory methods are reviewed. These methods allow the contributions from the different scales to be separated and the perturbative series to be reorganized. The construction of the effective field theory is shown in detail for ϕ4 theory and QCD. It is applied to the evaluation of the free energy of QCD at order g5 and the calculation of the g6 term is outlined. Implications for the application of perturbative QCD to the quark–gluon plasma are also discussed.

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Shen, Siyuan, Tianjia Shao, Kun Zhou, Chenfanfu Jiang, Feng Luo, and Yin Yang. "HoD-Net: High-Order Differentiable Deep Neural Networks and Applications." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 8 (June 28, 2022): 8249–58. http://dx.doi.org/10.1609/aaai.v36i8.20799.

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We introduce a deep architecture named HoD-Net to enable high-order differentiability for deep learning. HoD-Net is based on and generalizes the complex-step finite difference (CSFD) method. While similar to classic finite difference, CSFD approaches the derivative of a function from a higher-dimension complex domain, leading to highly accurate and robust differentiation computation without numerical stability issues. This method can be coupled with backpropagation and adjoint perturbation methods for an efficient calculation of high-order derivatives. We show how this numerical scheme can be leveraged in challenging deep learning problems, such as high-order network training, deep learning-based physics simulation, and neural differential equations.

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34

LI, WEN-ZUO, YU-WEI PEI, CAI-XIA SUN, QING-ZHONG LI, and JIAN-BO CHENG. "THEORETICAL STUDY ON HBeP- AND HPBe- ANIONS USING MULTICONFIGURATION SECOND-ORDER PERTURBATION THEORY." Journal of Theoretical and Computational Chemistry 11, no. 06 (December 2012): 1281–88. http://dx.doi.org/10.1142/s021963361250085x.

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Some low-lying states of the nine-valence-electron systems HBeP - and HPBe - anions have been studied for the first time using three methods CASSCF, CASPT2 and B3LYP with the contracted atomic natural orbital (ANO) and cc-pVTZ basis sets. The geometries of all stationary points along the potential energy surfaces were optimized at the CASSCF/ANO, CASPT2/ANO and B3LYP/cc-pVTZ levels. The potential energy curves of isomerization reactions between HBeP - and HPBe - were calculated as a function of HBeP bond angle. The ground and the first excited states of HBeP - are predicted to be X2Π and A2Σ+ states, respectively. The X2Σ+ and A2Π states of the linear HPBe - are both first-order saddle points because they have unique imaginary frequency. Two bent minima M1 and M2 were found along the 12A′ and 12A″ potential energy surfaces, respectively. The calculated results indicated that the ground-state HBeP - is linear, while the ground-state HPBe - is bent.

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35

Shields, Jonathan J., and William C. Webster. "On direct methods in water-wave theory." Journal of Fluid Mechanics 197 (December 1988): 171–99. http://dx.doi.org/10.1017/s0022112088003222.

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Model equations for three-dimensional, inviscid flow between two arbitrary, time-varying material surfaces are derived using a ‘direct’ or variational approach due to Kantorovich. This approach results in a hierarchy of approximate theories, each of a higher level of spatial approximation and complexity. It can be shown that the equations are equivalent in substance to ‘the theory of directed fluid sheets’ of Green & Naghdi (1974, 1976).The theory can be used to study the propagation of long waves in water of finite depth and, as such, competes with theories derived using the classical Rayleigh–Boussinesq perturbation methods. In order to demonstrate that there is an advantage to the present approach, we compare predictions for steady, two-dimensional waves over a horizontal bottom. Numerical solutions indicate that the direct theory converges more rapidly than the perturbation theories. Also, the equations of the higher-order direct theories contain singularities related to waves of limiting height, and indeed such waves can be predicted with relative accuracy. Finally, the range of applicability of the direct theory is far greater: waves as short as three times the water depth can be modelled. This is essentially a deep-water condition, well beyond the range of convergence of the Rayleigh–Boussinesq approach.

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Chaudhuri, Rajat K., and Karl F. Freed. "Comparison of high order perturbative convergence of multireference perturbation methods: Application to singlet states of CH2." Journal of Chemical Physics 107, no. 17 (November 1997): 6699–711. http://dx.doi.org/10.1063/1.474913.

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37

de la Luz Sosa, Jose, Daniel Olvera-Trejo, Gorka Urbikain, Oscar Martinez-Romero, Alex Elías-Zúñiga, and Luis Norberto López de Lacalle. "Uncharted Stable Peninsula for Multivariable Milling Tools by High-Order Homotopy Perturbation Method." Applied Sciences 10, no. 21 (November 6, 2020): 7869. http://dx.doi.org/10.3390/app10217869.

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In this work, a new method for solving a delay differential equation (DDE) with multiple delays is presented by using second- and third-order polynomials to approximate the delayed terms using the enhanced homotopy perturbation method (EMHPM). To study the proposed method performance in terms of convergency and computational cost in comparison with the first-order EMHPM, semi-discretization and full-discretization methods, a delay differential equation that model the cutting milling operation process was used. To further assess the accuracy of the proposed method, a milling process with a multivariable cutter is examined in order to find the stability boundaries. Then, theoretical predictions are computed from the corresponding DDE finding uncharted stable zones at high axial depths of cut. Time-domain simulations based on continuous wavelet transform (CWT) scalograms, power spectral density (PSD) charts and Poincaré maps (PM) were employed to validate the stability lobes found by using the third-order EMHPM for the multivariable tool.

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38

Hong, Youngjoon, and David P. Nicholls. "A stable high-order perturbation of surfaces method for numerical simulation of diffraction problems in triply layered media." Journal of Computational Physics 330 (February 2017): 1043–68. http://dx.doi.org/10.1016/j.jcp.2016.10.057.

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39

Saye, R. I. "High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles." SIAM Journal on Scientific Computing 37, no. 2 (January 2015): A993—A1019. http://dx.doi.org/10.1137/140966290.

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40

Wang, Xiangyu, Song Cen, and Chenfeng Li. "Generalized Neumann Expansion and Its Application in Stochastic Finite Element Methods." Mathematical Problems in Engineering 2013 (2013): 1–13. http://dx.doi.org/10.1155/2013/325025.

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An acceleration technique, termed generalized Neumann expansion (GNE), is presented for evaluating the responses of uncertain systems. The GNE method, which solves stochastic linear algebraic equations arising in stochastic finite element analysis, is easy to implement and is of high efficiency. The convergence condition of the new method is studied, and a rigorous error estimator is proposed to evaluate the upper bound of the relative error of a given GNE solution. It is found that the third-order GNE solution is sufficient to achieve a good accuracy even when the variation of the source stochastic field is relatively high. The relationship between the GNE method, the perturbation method, and the standard Neumann expansion method is also discussed. Based on the links between these three methods, quantitative error estimations for the perturbation method and the standard Neumann method are obtained for the first time in the probability context.

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Nicholls, David P. "Stable, high-order computation of impedance–impedance operators for three-dimensional layered medium simulations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2212 (April 2018): 20170704. http://dx.doi.org/10.1098/rspa.2017.0704.

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The faithful modelling of the propagation of linear waves in a layered, periodic structure is of paramount importance in many branches of the applied sciences. In this paper, we present a novel numerical algorithm for the simulation of such problems which is free of the artificial singularities present in related approaches. We advocate for a surface integral formulation which is phrased in terms of impedance–impedance operators that are immune to the Dirichlet eigenvalues which plague the Dirichlet–Neumann operators that appear in classical formulations. We demonstrate a high-order spectral algorithm to simulate these latter operators based upon a high-order perturbation of surfaces methodology which is rapid, robust and highly accurate. We demonstrate the validity and utility of our approach with a sequence of numerical simulations.

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Nicholls, David P., and Venu Tammali. "A high-order perturbation of surfaces (HOPS) approach to Fokas integral equations: Vector electromagnetic scattering by periodic crossed gratings." Applied Numerical Mathematics 101 (March 2016): 1–17. http://dx.doi.org/10.1016/j.apnum.2015.11.004.

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43

BONILLA, LUIS L., and JUAN S. SOLER. "HIGH-FIELD LIMIT OF THE VLASOV–POISSON–FOKKER–PLANCK SYSTEM: A COMPARISON OF DIFFERENT PERTURBATION METHODS." Mathematical Models and Methods in Applied Sciences 11, no. 08 (November 2001): 1457–68. http://dx.doi.org/10.1142/s0218202501001410.

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A reduced drift-diffusion (Smoluchowski–Poisson) equation is found for the electric charge in the high-field limit of the Vlasov–Poisson–Fokker–Planck system, both in one and three dimensions. The corresponding electric field satisfies a Burgers equation. Three methods are compared in the one-dimensional case: Hilbert expansion, Chapman–Enskog procedure and closure of the hierarchy of equations for the moments of the probability density. Of these methods, only the Chapman–Enskog method is able to systematically yield reduced equations containing terms of different order.

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44

Chang, Jing, Jin Zhang, and Ming Cai. "Series Solutions of High-Dimensional Fractional Differential Equations." Mathematics 9, no. 17 (August 24, 2021): 2021. http://dx.doi.org/10.3390/math9172021.

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In the present paper, the series solutions and the approximate solutions of the time–space fractional differential equations are obtained using two different analytical methods. One is the homotopy perturbation Sumudu transform method (HPSTM), and another is the variational iteration Laplace transform method (VILTM). It is observed that the approximate solutions are very close to the exact solutions. The solutions obtained are very useful and significant to analyze many phenomena, and the solutions have not been reported in previous literature. The salient feature of this work is the graphical presentations of the third approximate solutions for different values of order α.

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CINI, MICHELE. "SECOND-HARMONIC GENERATION AT SURFACES." Surface Review and Letters 01, no. 04 (December 1994): 443–48. http://dx.doi.org/10.1142/s0218625x94000412.

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A method for computing the intensity of electric dipole second-harmonic generation from semiconductor surfaces and interfaces is presented. It uses third-order perturbation theory and a Green's function technique to embody Fresnel's formulas. Calculations for As/Si(111) are compared to experimental data, and show fair agreement for the spectral and angular dependence of the second-harmonic signal; the calculated absolute intensity, however, is too high. To investigate the origin of the discrepancy, we generalize the theory to all orders in the electron-photon coupling. The new scheme, based on the method of excitation amplitudes, is outlined. It allows to set up a theory of the second-harmonic generation response and other inelastic light scattering phenomena, like the dynamical Stark effect. Many-photon effects can modify the response in a qualitative way, and the maximum intensity of the second-harmonic fields does not generally correspond exactly to twice the incident frequency. This formalism also leads to new exact sum rules for nonlinear optics, that complement those obtained recently by Bassani and Scandolo.

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VAN RITBERGEN, T., J. A. M. VERMASEREN, S. A. LARIN, and P. NOGUEIRA. "THE CALCULATION OF VARIOUS QUANTITIES WITHIN PERTURBATION THEORY AT THE 3- AND 4-LOOP ORDER." International Journal of Modern Physics C 06, no. 04 (August 1995): 513–18. http://dx.doi.org/10.1142/s0129183195000368.

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We discuss methods and techniques that were recently used in the following high order perturbative calculations: the large quark mass expansion of the decay rates Z0→ hadrons and τ→ν+ hadrons and the calculation of various quantities for deep inelastic lepton hadron scattering. Progress on the automatization of some general procedures is discussed.

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BATES, PETER W., and XIAOFENG REN. "Heteroclinic orbits for a higher order phase transition problem." European Journal of Applied Mathematics 8, no. 2 (April 1997): 149–63. http://dx.doi.org/10.1017/s0956792597002994.

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A standard model for one-dimensional phase transitions is the second-order semilinear equation with bistable nonlinearity, where one seeks a solution which connects the two stable values. From an Ising-like model but which includes long-range interaction, one is led to consider the equation where the second-order operator is replaced by one of arbitrarily high order. Others have found the desired heteroclinic solutions for such equations, under the assumption that the higher-order terms have small coefficients, by employing singular perturbation methods for dynamical systems. Here, without making any assumption on the sizes of the coefficients, we obtain such heteroclinic solutions by using variational methods under the assumption that the nonlinearity arises from a potential having two wells of equal depths.

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48

ARDALAN, F., H. ARFAEI, and A. SHAFII-DEHABAD. "BELAVIN-KNIZHNIK FORMULA FOR OPEN BOSONIC STRING." Modern Physics Letters A 04, no. 04 (February 1989): 375–83. http://dx.doi.org/10.1142/s0217732389000459.

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We derive the analog of Belavin-Knizhnik formula for both orientable and nonorientable open bosonic strings. At the order 2g of perturbation theory, the open string partition function is given for orientable and non-orientable topologies as an integral over a subvariety of the moduli space of genus g Riemann surfaces. The integrand is the modulus of the holomorphic function of Belavin and Knizhnik multipied by ( det Im T)−13 where T is obtained from the period matrix.

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Mosedale, Andrew, and Dimitris Drikakis. "Assessment of Very High Order of Accuracy in Implicit LES models." Journal of Fluids Engineering 129, no. 12 (July 27, 2007): 1497–503. http://dx.doi.org/10.1115/1.2801374.

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This paper looks at the use of high-resolution and very high-order methods for implicit large-eddy simulation (ILES), with the specific example of simulating the multicomponent two-dimensional single-mode Richtmyer–Meshkov instability for which experimental data is available. The two gases are air and SF6, making stringent demands on the models used in the code. The interface between the two gases is initialized with a simple sinusoidal perturbation over a wavelength of 59mm, and a shock of strength Mach 1.3 is passed through this interface. The main comparison is between the second-order monotone upwind-centered scheme for conservation law methods of van Leer (1979, “Towards the Ultimate Conservative Difference Scheme,” J. Comput. Phys. 32, pp. 101–136) and the current state-of-the-art weighted essentially nonoscillatory interpolation, which is presented to ninth order, concentrating on the effect on resolution of the instability on coarse grids. The higher-order methods as expected provide better resolved and more physical features than the second-order methods on the same grid resolution. While it is not possible to make a definitive statement, the simulations have indicated that the extra time required for the higher-order reconstruction is less than the time saved by being able to obtain the same or better accuracy at lower computational cost (fewer grid points). It should also be noted that all simulations give a good representation of the growth rate of the instability, comparing very favorably to the experimental results, and as such far better than the currently existing theoretical models. This serves to further indicate that the ILES approach is capable of providing accurately physical information despite the lack of any formal subgrid model.

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VANDENBOOMGAERDE, M., C. CHERFILS, D. GALMICHE, S. GAUTHIER, and P. A. RAVIART. "Efficient perturbation methods for Richtmyer–Meshkov and Rayleigh–Taylor instabilities: Weakly nonlinear stage and beyond." Laser and Particle Beams 21, no. 3 (July 2003): 321–25. http://dx.doi.org/10.1017/s0263034603213057.

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The simplified perturbation method of Vandenboomgaerdeet al.(2002) is applied to both the Richtmyer–Meshkov and the Rayleigh–Taylor instabilities. This theory is devoted to the calculus of the growth rate of the perturbation of the interface in the weakly nonlinear stage. In the standard approach, expansions appear to be series in time. We build accurate approximations by retaining only the terms with the highest power in time. This simplifies and accelerates the solution. High order expressions are then easily reachable. For the Richtmyer–Meshkov instability, multimode configurations become tractable and the selection mode process can be studied. Inferences for the intermediate nonlinear regime are also proposed. In particular, a class of homothetic configurations is inferred; its validity is verified with numerical simulations even as vortex structures appear at the interface. This kind of method can also be used for the Rayleigh–Taylor instability. Some examples are presented.

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