High-order Perturbation Method Research Articles

Complex Multi-nonlinearity for piezoelectric energy harvesting systems based on an accurate higher-order perturbation method

A theoretical model of a complex nonlinear coupling energy harvester under hybrid excitations is established with analytical and numerical solutions. Different orders of the method of multiple scales are derived to approximate and analyze the equation of motion for the electromechanical coupling system, thus obtaining higher-order approximate analytical solutions for its operating status and output performance indicators. The significant effects of external excitation, linear and nonlinear damping coefficients, as well as impedance on the transverse displacement, voltage, and power of higher-order and lower-order methods are analyzed. The results show that higher-order solution methodology are more sensitive to parameter excitation. For a large damping, the higher-order analysis has more excellent performance in capturing energy. The higher order method shows more pronounced nonlinear softening phenomenon and it is more suitable for low frequency environments. The higher order methods improves solution accuracy of piezoelectric energy harvesting systems. Consequently, it is easier to derive the nonlinear performance of the harvesting systems, and thus improves the average output power.

EPR parameters and optical spectrum of α-Al2O3:Ti3+ with dynamic Jahn-Teller effect

The quenching phenomenon of trigonal crystal field (CF), spin–orbit (SO) coupling operators, and orbital operators reflected in electron paramagnetic resonance (EPR) parameters and energy splitting intervals observed in experiments of α-Al2O3:Ti3+ is theoretically investigated by the calculation of EPR g factors and the energy differences of the lowest three levers E3/2, 1E1/2 and 2E1/2 with the formulas derived with high-order perturbation method. In the calculation, the coupling of 2T2g and 2Eg with e vibrational model of dynamic Jahn-Teller effect are considered. The calculated results are in agreement with the experimental values. The local defect structure around Ti3+ ions (bond lengths and angles of [TiO6]9-) and trigonal CF parameters ν and v′ are determinedand the covalence effects on SO coupling parameters ζ, ζ’ and orbital reduction parameters k、k’ are determined also.

On the use of an high order perturbation method for numerical time integration in structural dynamics.

On the use of an high order perturbation method for numerical time integration in structural dynamics.

High-Order Perturbation Method for Updating Response Functions for Time-Dependent Transport Calculations

The hybrid stochastic deterministic continuous-energy coarse mesh transport method (COMET) has been recently extended for high-fidelity efficient kinetics calculations in highly heterogeneous reactor cores. The method discretizes the time variable as a series of time grids and solves the resulting set of steady-state neutron transport equations. In this work, a high-order perturbation method is developed to update the COMET unperturbed response function library on the fly for changes in the discretized time step size Δ t h . The unperturbed response functions are precomputed with 1 / v Δ t h = 0 . The perturbation expansion coefficients are also generated during the unperturbed response function library precomputation. The adjoint solution needed by the perturbation method is calculated using the reciprocity relation without solving the corresponding adjoint problems. As a result, the method can be easily implemented into any stochastic code to generate the perturbation expansion coefficients together with the unperturbed response functions. The high-order perturbation method is benchmarked by comparing both the response functions and the time-dependent COMET core solution (fission density) with the corresponding reference solutions. It is found that the response functions generated by the perturbation method at second order are in excellent agreement with those directly computed by the Monte Carlo method. When Δ t h changes from 1.0E-4 s to infinity, the average and maximum relative differences in the various response functions were found to be in the range of 0.0000106% to 0.00116% and 0.000011% to 0.00121%, respectively. The fission density as a function of time calculated by COMET using the perturbation method is in excellent agreement with the reference solutions, with an average relative difference of 0.0065% to 0.075%. These comparisons indicate that the perturbation method at second order is highly accurate.

Numerical convergence and error analysis for the truncated iterative generalized stochastic perturbation-based finite element method

The main aim of this work is to present a numerical analysis of convergence and accuracy of the selected generalized perturbation-based schemes in linear and nonlinear problems of solid mechanics. An algorithm for determining the basic probabilistic characteristics has been developed using the iterative generalized stochastic higher-order perturbation method adjacent to symmetrically truncated Gaussian random variables. It has been confirmed that usage of a sufficiently high order of the Truncated Iterative Stochastic Perturbation Technique (TISPT) allows for achieving any desired accuracy in determining up to the fourth-order probabilistic characteristics of static structural response. The semi-analytical probabilistic approach is the reference solution in this study, which is based upon the same composite response functions determined with the use of the Least Squares Method created using specific series of FEM experiments. The entire methodology has been provided for the given extreme value of the coefficient of variation αmax of the input uncertainty source. On the other hand, a selection procedure of the stochastic perturbation method order to achieve 1% numerical accuracy in all up to the fourth-order probabilistic moments has been proposed. Computational experiments include simply supported elastic Euler–Bernoulli beam, a set of steel diagrid structures, nonlinear tension of steel round bar as well as homogenization procedure of some particulate composite.

Localization of higher order exceptional points from finite element model and their applications to duct acoustics.

This work reviews the state of the art for high order perturbation method for parametric eigenvalue problems and propose some extensions for the multiparameters case. This approach allows to locate high order exceptional points (EP) arising in eigenvalue problems. EP correspond to a particular tuning of some complex-valued parameters which render the problem degenerate. These non-Hermitian degeneracies have raised considerable attention in the scientific community as these can have a great impact in a variety of physical problems (PT-symmetry, thermo-acoustic or fluid-structure instability, etc.) and their numerical solution. For applications dealing with dissipative acoustic waveguides, strong modal attenuation can be achieved close to EP and a maximum of attenuation occurs at EP of high order corresponding to the coalescence of more than two modes. The method is based on the automatic computation of the successive derivatives of some selected eigenpairs with respect to the parameters so that, after recombination, regular functions can be constructed. This algebraic manipulations permit to build a reduced order model allowing i) to quickly solve the eigenvalue problem for other parameters values, ii) to follow modal branches, iii) to locate higher order EPs. The method is applied to the particular case of a circular duct with a locally reacting liner at its wall which admittance varies with azimuthal position.

Fiscal policy and uncertainty

We investigate the impact of macroeconomic uncertainty on the stimulative effect of government spending. Using historical macroeconomic time series, we show that government spending multipliers are smaller in episodes characterized by high macroeconomic uncertainty. This state dependence is found to be statistically and economically significant and robust to several alternative specifications. To provide structure, we then build a DSGE model with an agency problem and a forecast-error based uncertainty index. We solve the model via a higher-order perturbation method and use the model to evaluate state-dependent government stimulus. In simulations, we find significant volatility in multipliers as well as co-movements that qualitatively match our empirical results. We find that these results are generated by a reallocation away from risky assets following the government spending shock which triggers a financial accelerator mechanism in our model; we then show that this channel is also active in the data. We conclude that uncertainty plays a significant role in determining the effectiveness of fiscal policy and that business cycle models which abstract from uncertainty can produce misleading policy recommendations.

Fine structure of energy levels and electron paramagnetic resonance parameters of Ni2+ doped in CsCdCl3

Complete diagonalization method (CDM) and high-order perturbation method (PTM) with forth-order formulae have been used to explain the optical spectra and ground term 3A2g electron paramagnetic resonance (EPR) parameters g//,g⊥ and zero-field splitting (ZFS) parameter D of Ni2+-doped CsCdCl3 within the framework of molecular orbital (MO) scheme, the calculated results are in good agreement with the experimental data. The trigonal parametersv and v′, Reach parameters B and C and the local defect structure are determined. The effect of vibronic interaction on optical spectra of 3T2g and 3A2g transition has been discussed with a1g and eg vibronic models, and the Huang–Phys parameters and vibronic coupling constant are evaluated.

Adjoint-Based Calculation of Parametric Thermoacoustic Maps of an Industrial Combustion Chamber

Abstract The aim of this study is to efficiently calculate parametric thermoacoustic maps of typical combustion chambers. Two configurations are considered: an academic configuration based on a Rijke tube, and an industrial combustion chamber, which is the core of a recently developed microturbine for power generation. Such maps can be understood as the collection of loci of thermoacoustic eigenfrequencies obtained under systematic variations of some defined parameters, while considering the Helmholtz equation as the thermoacoustic model of interest. In this study we consider variations on two parameters: the gain n and time-delay τ associated with a generic flame response model. We also show the feasibility of the proposed approach when considering more realistic flame responses. A straight-forward way to calculate such a thermoacoustic map is by solving the Helmholtz equation, and, thus, the corresponding nonlinear eigenvalue problem (NLEVP), one time per parameter combination. With that approach, the nonlinear eigenvalue problem needs to be solved hundreds or thousands of times if an adequate resolution of the thermoacoustic map is sought. Such a strategy may be computationally unaffordable. In order to overcome this difficulty, this work utilizes an adjoint-based, high-order perturbation method. The actual eigenvalue problem is only solved once at a baseline point. After applying the perturbation equations at that point, a polynomial rational function—the Padé approximant—is obtained to estimate the eigenfrequency drift that results for a small or large perturbation in the flame response. It is demonstrated, for both academic and industrial test cases, that the obtained maps are accurate. Additionally, it is shown that these maps reveal a large variety of thermoacoustic features, such as stability boundaries, intrinsic thermoacoustic modes, and exceptional points. The numerical costs for such calculations are negligible even for the industrial combustion chamber investigated.

A High-Order Perturbation Method for an Investigation into Relativistic Stimulated Raman Back Scattering of a High Power Laser in Plasma

The relativistic stimulated Raman back scattering (SRBS) of a high power linearly polarized laser pulse in the plasma is investigated. The nonlinear coupled equations of Raman instabilities in the presence of combined effects of relativistic and ponderomotive nonlinearities are derived. Using high-order perturbation method in relativistic fluid-Maxwell model, a set of coupled non-linear equations is obtained which describes the evolutions of two scattered electromagnetic waves and a modified plasma wave. The temporal evolution of SRBS process is then studied using numerical solutions of the set of coupled equations. The results showed that by using the linear approximation, the relativistic mass correction cannot be employed accurately and it leads to over-estimation of the perturbed current density and Raman Scattering. Our analysis shows that, in the relativistic regime, it is crucial to employ higher order perturbation method to obtain reliable results.

Effective non-Hermitian physics for degenerate ground states of a non-Hermitian Ising model with symmetry

In this paper, based on a one-dimensional non-Hermitian spin model with -invariant term, we study the non-Hermitian physics for the two (nearly) degenerate ground states. By using the high-order perturbation method, an effective pseudo-spin model is obtained to describe non-Hermitian physics for the two (nearly) degenerate ground states, which are precisely consistent with the numerical calculations. We found that there may exist effective (anti) symmetry for the effective pseudo-spin model of the two (nearly) degenerate ground states. In particular, there exists spontaneous (anti) -symmetry breaking for the topological degenerate ground states with tunable parameters in external fields. We also found that even a very tiny imaginary external field applied will drive phase transition.

Transformed Perturbation Solutions for Dynamic Stochastic General Equilibrium Models

Despite the recent introduction of novel solution methods for Dynamic Stochastic General Equilibrium (DSGE), perturbation methods are still among the most popular and widely used solution techniques for DSGE models. Unfortunately, nonlinear perturbation solutions produce paths with stochastic properties that invalidate the econometric analysis. This paper proposes a correction that renders the econometric analysis valid and sound. The proposed correction is simple to implement in existing software packages such as Dynare, it does not add any significant computational effort and, as a result, does not impact computational times. The corrected solution retains the same approximation properties as standard higher-order perturbation methods and, in contrast to those methods, generates stable sample paths that are stationary, geometrically ergodic and absolutely regular. Additionally, moments are shown to be bounded. Transformed perturbation solutions are an alternative to the pruning method as proposed in Kim et al. (2008). The advantages of our approach are that, unlike pruning, we do not need to sacrifice accuracy around the steady-state by ignoring higher-order effects, and furthermore, we also deliver a policy function. Moreover, the newly proposed solution is always more accurate globally than standard perturbation methods. We demonstrate the superior accuracy of our method in a range of simple examples.

Modelling Occasionally Binding Constraints Using Regime-Switching

Occasionally binding constraints are part of the economic landscape: for instance recent experience with the global financial crisis has highlighted the gravity of the lower bound constraint on interest rates; mortgagors are subject to more stringent borrowing conditions when credit growth has been excessive or there is a downturn in the economy. In this paper we take four common examples of occasionally binding constraints in economics and demonstrate how to use regime-switching to incorporate them into DSGE models. In particular we investigate the zero lower bound constraint on interest rates, occasionally binding collateral constraints, downward nominal wage rigidities and irreversible investment. We compare our approach against some well-known methods for solving occasionally-binding constraints. We demonstrate the versatility of our regime-switching approach by combining multiple occasionally binding constraints to a model solved using higher-order perturbation methods, a feat that is difficult to achieve using alternative methodologies.

A second-order perturbation method for fuzzy eigenvalue problems

Purpose For eigenvalue problems containing uncertain inputs characterized by fuzzy basic parameters, first-order perturbation methods have been developed to extract eigen-solutions, but either the result accuracy or the computational efficiency of these methods is less satisfactory. This paper presents an efficient method for estimation of fuzzy eigenvalues with high accuracy. Design/methodology/approach Based on the first order derivatives of eigenvalues and modes with respect to the fuzzy basic parameters, expressions of the second order derivatives of eigenvalues are formulated. Then a second-order perturbation method is introduced to provide more accurate fuzzy eigenvalue solutions. Only one eigenvalue solution is sought for the perturbed formulation, and quadratic programming is performed to simplify the alpha-level optimization. Findings Fuzzy natural frequencies and buckling loads of some structures are estimated with good accuracy, illustrating the high computational efficiency of the proposed method. Originality/value Up to the second order derivatives of the eigenvalues with respect to the basic parameters are represented in functional forms, which are used to introduce a second-order perturbation method for treatment of fuzzy eigenvalue problems. The corresponding alpha-level optimization is thus simplified into quadratic programming. The proposed method provides much more accurate interval solutions at alpha-cuts for the membership functions of fuzzy eigenvalues. Analogously, third- and higher-order perturbation methods can be developed for more stringent accuracy demands or for the treatment of stronger nonlinearity. The present work can be applied to realistic structural analysis in civil engineering, especially for those structures made of dispersed materials such as concrete and soil.

Theoretical Explanations of the Optical Band Positions and Local Structure for Cu2+Centers in ZnO-Bi2O3-B2O3-CuO Glasses

The local structure, two optical band positions and three electronic spin resonance parameters for Cu centers at the tetragonally-distorted octahedral sites in ZnO-Bi2O3-B2O3-CuO glasses are calculated from the high-order perturbation method based on the two-spin-orbit-parameter model, where the contributions from both the spin orbit coupling parameters of central d ion and ligand ion are included. The theoretical results are in good agreement with the experimental values. The calculations show that the high-order perturbation method based on the two-spin-orbit-parameter model is e ective in the explanations of optical spectra, electronic spin resonance parameters and local distortion structure for d ions in glasses.

Theoretical studies of the spin-Hamiltonian parameters of the oxycation MoO3+ in (NH4)2MoOCl5 and K2MoOF5 single crystals

Theoretical formulae for the spin-Hamiltonian (SH) parameters (g tensor g//, g⊥ and hyperfine structure A tensor A//, A⊥) of the ground state of oxycation MoO3+ in [MOX5]2− (M=Mo5+, Cr4+, V3+, X=F−, Cl− or Br− ) complexes are derived by the high-order perturbation method within the frame of molecular orbital (MO) scheme. In those formulae, the contributions to the SH parameters arising both from the crystal field (CF) and charge transfer (CT) excitations are taken into account, which are derived up to the third and second order, respectively. Those formulae are applied to calculate the SH of the oxycation MoO3+ in (NH4)2MoOCl5 and K2MoOF5 single crystals by the best match method, and the calculated results are in good agreement with the experimental data. The investigations show that the charge transfer mechanism play a decisive role in the understanding for the SH parameters for 4d1 ions in crystals with the strong coordinate covalence, especially for anomalously experimental results g//>g⊥ which cannot be explained by the traditional CF approximation alone. The MO coefficients are determined and the results are discussed.

Asset Prices in Affine Real Business Cycle Models

We develop a tractable way to solve for equilibrium quantities and asset prices in a class of real business cycle models featuring Epstein-Zin preferences and affine dynamics for productivity growth and volatility. The method relies on log-linearization and exploits the log-normality of all the quantities. It is an easy substitute for more involved numerical techniques, such as higher order perturbation methods, and allows for easy implementation and analytical results. We show explicitly the link with perturbation techniques and find that the quantitative difference between the two is insignificant for several models of interest.

Asset prices in affine real business cycle models

I describe a tractable way to study macroeconomic quantities and asset prices in a large class of dynamic stochastic general equilibrium models. The proposed approximate solution is analytical, log-linear, and adjusted for risk. Therefore, it is well suited to investigate economic mechanisms, describe the time series properties or estimate the model, and deal with stochastic volatility. I explain the pitfalls encountered by previous attempts to use simple approximation techniques, in particular with models featuring recursive preferences. Finally, I show the theoretical relationship between my solution and higher-order perturbation methods.

On the ground-state splitting, fine structure of multiplets and EPR spectrum of Ni2+ doped in MgF2 crystal with orthorhombic site symmetry

The 45×45 complete energy matrix for 3d8 ion at D2h site symmetry is used to calculate and assign the ground-state splitting and the fine structure of the multiplets of Ni2+(3d8) doped in MgF2 crystal with rutile type structure by the complete diagonalizaton method (CDM) in the frame of semi-empirical molecular orbital (MO) scheme in the strong crystal field (CF) approximation. In the calculation, all the configuration interactions though the cubic CF part, low-symmetry component (tetragonal and orthorhombic parts), Coulomb interaction and the spin–orbit coupling (SOC) interaction (both of the central ion and the liangds) are taken into account completely. The calculated results are in good agreement with the experimental data. In addition, the ground-state splitting is also calculated by the high-order perturbation method (PTM), together with the electron paramagnetic resonance (EPR) parameters D, E and g-factors (gx, gy and gz). The results of the spin–orbit splitting of the ground state calculated by CDM and PTM are not only close to each other, but also in good agreement with the observed data. The relationship between crystalline parameters and the fine structure of multiplets and EPR spectrum is established and the local defect structure is determined.

High-Order Perturbation Theory for Incident Flux Response Expansion Methods

In this work, a high-order perturbation method is developed to generate/compute response functions for the coarse mesh transport (COMET) method, which provides whole-core neutron transport solutions to various reactor core types. In this approach, the response functions are first generated at a reference eigenvalue point, and the response functions at an arbitrary eigenvalue are computed as a high-order perturbation from the reference point. The method has been tested at both the lattice level (response function generation) and the core level (whole-core transport calculations). At the lattice level, it is found that the response functions predicted by the perturbation method agree very well with those directly computed by the Monte Carlo method. The average relative difference in the surface-to-surface response functions is 0.29% to 0.46% when the eigenvalue k ranges from 0.6667 to 1.5. In whole-core transport calculations, the COMET calculations using the high-order perturbation method are almost identical to those using the interpolation method. The eigenvalue, assembly, and pin fission densities predicted by COMET agree very well with the MCNP reference solution. This indicates that the high-order perturbation response function generation method can achieve the same accuracy as the interpolation method while significantly improving the computational efficiency of the precomputation phase.