Truncation Order Research Articles

Comprehensive analysis of entropy conservation property of non-dissipative schemes for compressible flows: KEEP scheme redefined

A theoretical analysis of the entropy conservation properties is conducted to explain the different behaviors of the non-dissipative finite-difference spatial discretization schemes, such as the kinetic-energy and entropy preserving (KEEP) schemes. The analysis is conducted based on the spatially-discretized entropy-evolution equation derived from the Euler equations with retaining the discrete-level strictness. The present analysis shows that the analytical relations (ARs) employed by the KEEP schemes eliminate some terms of the discretized entropy-evolution equation and help simplify the equation, while the ARs are not sufficient to explain the entropy-conservation property. Therefore, the entropy error is decomposed into several terms, and the behaviors of those decomposed error terms are observed in the compressible inviscid Taylor–Green vortex test case. The results of the test case show that the terms containing the velocity difference between two grid points cause significant entropy conservation error and result in the different entropy conservation properties of the tested non-dissipative schemes. Furthermore, the KEEP schemes are redefined based on the present entropy-error analysis. In the redefined KEEP schemes, the formulation does not contain logarithmic means, and the strictness of entropy conservation can be adjusted easily by the truncation order of the Maclaurin expansions. Finally, the present entropy-error analysis and the redefined KEEP schemes are extended to the generalized curvilinear coordinates.

Two-dimensional sound field reproduction based on Mathieu function expansion.

In this study, a two-dimensional sound field reproduction method based on Mathieu function expansion (MFE) is proposed. Mathieu functions are orthogonal solutions of the Helmholtz equation in an elliptical coordinate system. A MFE, which is similar to the conventional circular harmonic expansion, is applied to the sound field. The MFE-based method introduces elliptical properties such that the listening area is an ellipse. Three methods are described: an analytical sound field reproduction method for elliptical loudspeaker arrays, a method applying transformation (i.e., stretch, rotation, and translation) to the listening area, and a numerical approach for arbitrarily shaped arrays. A suitable truncation order for the MFE is also derived. The performance of all methods is tested in computer simulations with examples.

Real-time methods for spectral functions

In this paper we develop and compare different real-time methods to calculate spectral functions. These include classical-statistical simulations, the Gaussian state approximation (GSA), and the functional renormalization group (FRG) formulated on the Keldysh closed-time path. Our test-bed system is the quartic anharmonic oscillator, a single self-interacting bosonic degree of freedom, coupled to an external heat bath providing dissipation analogous to the Caldeira-Leggett model. As our benchmark we use the spectral function from exact diagonalization with constant Ohmic damping. To extend the GSA for the open system, we solve the corresponding Heisenberg-Langevin equations in the Gaussian approximation. For the real-time FRG, we introduce a novel general prescription to construct causal regulators based on introducing scale-dependent fictitious heat baths. Our results explicitly demonstrate how the discrete transition lines of the quantum system gradually build up the broad continuous structures in the classical spectral function as temperature increases. At sufficiently high temperatures, classical, GSA, and exact-diagonalization results all coincide. The real-time FRG is able to reproduce the effective thermal mass, but it overestimates broadening and only qualitatively describes higher excitations, at the present order of our combined vertex and loop expansion. As temperature is lowered, the GSA follows the ensemble average of the exact solution better than the classical spectral function. In the low-temperature strong-coupling regime, the qualitative features of the exact result are best captured by our real-time FRG calculation, with quantitative improvements to be expected at higher truncation orders.

An improved spectral approach for solving the nonclassical neutral particle transport equation

An improved modification of the Spectral Approach (SA) used for approximating the nonclassical neutral particle transport equation is described in this work. The term “Spectral” is used to indicate that the nonclassical angular flux is approximated as an expansion in terms of spectral basis functions. In the SA the basis functions are the Laguerre polynomials. The main focus of the modified SA lies on a slight modification of the nonclassical angular flux representation as a truncated Laguerre series. This leads, in some cases, to a considerable decrease of the Laguerre truncation order required to generate accurate solutions, thus improving efficiency. Numerical results are given to illustrate this offered improvement.

The analytic structure of the fixed charge expansion

We investigate the analytic properties of the fixed charge expansion for a number of conformal field theories in different space-time dimensions. The models investigated here are O(N) and QED3. We show that in d = 3 − ϵ dimensions the contribution to the O(N) fixed charge Q conformal dimensions obtained in the double scaling limit of large charge and vanishing ϵ is non-Borel summable, doubly factorial divergent, and with order sqrt{Q} optimal truncation order. By using resurgence techniques we show that the singularities in the Borel plane are related to worldline instantons that were discovered in the other double scaling limit of large Q and N of ref. [1]. In d = 4 − ϵ dimensions the story changes since in the same large Q and small E regime the next order corrections to the scaling dimensions lead to a convergent series. The resummed series displays a new branch cut singularity which is relevant for the stability of the O(N) large charge sector for negative ϵ. Although the QED3 model shares the same large charge behaviour of the O(N) model, we discover that at leading order in the large number of matter field expansion the large charge scaling dimensions are Borel summable, single factorial divergent, and with order Q optimal truncation order.

A combined sound field reconstruction method for non-cylindrical rotative surfaces based on statistically optimal cylindrical near-field acoustic holography and multipoint Helmholtz equation least square

For the sound field reconstruction of non-cylindrical rotative surfaces, current near-field acoustic holography (NAH) methods have relatively poor applicability and low accuracy. To overcome the problem, a combined sound field reconstruction method is proposed. First, an improved multipoint Helmholtz equation least square (HELS) method is proposed, which is more applicable to reconstruct the sound field with an aspect ratio larger than 1. Second, based on single criterion traversing method (SCTM), an improved double criterion traversing method (DCTM) is proposed to determine the optimal truncation order, and reduce the reconstruction error of multipoint HELS. Third, with the sound pressure measured on the holographic surface, the sound pressure on the transitional surface is reconstructed using statistically optimal cylindrical near-field acoustic holography (SOCNAH), and the sound pressure on the reconstruction surface is reconstructed using multipoint HELS. Finally, typical numerical case studies and experimental studies on a test bed are carried out, which validate that the combined method can obviously improve the accuracy and robustness of sound field reconstruction for non-cylindrical rotative surfaces, and thus provide reliable evidences for noise monitoring and control of mechanical systems.

Finite volume based asymptotic homogenization of viscoelastic unidirectional composites

The previously developed finite volume based asymptotic homogenization theory designed for unidirectional composites with finite-sized microstructures is further extended to non-ageing linear viscoelastic region via correspondence principle. The theory is a unified framework able to accommodate general loadings, both out-of-plane and in-plane. It first transforms the problems in time domain to Laplace domain via elastic–viscoelastic correspondence principle, solves the problems in transformed region, and transforms the solutions back to time domain through numerical inversion technique. In Laplace domain, the microscale problems are solved up to the third term of expansion, and the homogenized stiffness tensors of all orders are passed to the macroscale, leading to the solutions of macroscale displacements. The method’s capability is assessed through comparison with direct numerical solution, showing increasing accuracy with increasing truncation order and decreasing scale ratio, and further enhancement by reconstructing boundary layer. The robustness of theory is verified through parametric studies. The possibility of reducing the highest order of spatial derivative of homogenized displacement is also investigated. The finite volume based methodology marks the theory’s uniqueness among the existing higher order asymptotic homogenization methods, providing an efficient alternative to solve the viscoelastic composite structural problems.

Numerical Analysis of the Monte-Carlo Noise for the Resolution of the Deterministic and Uncertain Linear Boltzmann Equation (Comparison of Non-Intrusive gPC and MC-gPC)

Monte Carlo-generalized Polynomial Chaos (MC-gPC) has already been thoroughly studied in the literature. MC-gPC both builds a gPC based reduced model of a partial differential equation (PDE) of interest and solves it with an intrusive MC scheme in order to propagate uncertainties. This reduced model captures the behavior of the solution of a set of PDEs subject to some uncertain parameters modeled by random variables. MC-gPC is an intrusive method, it needs modifications of a code in order to be applied. This may be considered a drawback. But, on another hand, important computational gains obtained with MC-gPC have been observed on many applications. The MC-gPC resolution of Boltzmann equation has been investigated in many different ways: the wellposedness of the gPC based reduced model has been proved, the convergence with respect to the truncation order P has been theoretically and numerically studied and the coupling to nonlinear physics has been performed. But the study of the MC noise remains, to our knowledge, to be done. This is the purpose of this paper. We are interested in understanding what can be expected in terms of error estimations with respect to NMC , the number of MC particles. For this, we estimate the variances of non-intrusive gPC and MC-gPC, theoretically and numerically, and compare them in several configurations for several MC schemes (the semi-analog and the non-analog ones). The results show that the MC schemes of the literature used to solve MC-gPC present an excess of variance with respect to the non-intrusive strategies for comparable particle numbers NMC (even if this excess of variance remains acceptable and competitive in many situations).

Gravity field forward modelling using tesseroids accelerated by Taylor series expansion and symmetry relations

SUMMARY In this study, we developed a new method that can significantly accelerate the forward modelling of gravity fields generated by large-scale tesseroids while keeping the computational accuracy as high as possible. The cost of the high efficiency is that the method only works under the assumptions that (1) all tesseroids in the same latitude band have the same horizontal dimension, (2) the computation points are located at the same surface level and aligned with the horizontal centres of tesseroids and (3) each tesseroid has a constant or linearly varying density. The new method first integrates the kernel function of the Newton’s volume integral analytically in the radial direction to eliminate its dependence on the vertical dimension of the tesseroid, and then expands the integrated kernel function into a Taylor series up to a certain order. Because the Taylor series expansion term of the integrated kernel function is an odd or even function of the difference between the longitudes of the tesseroid and computation point, there exist shifting or swapping symmetry relations among the gravity field of tesseroids. Consequently, the shifting or swapping symmetry is extended to the tesseroids with unequal vertical dimensions. Numerical experiments using the spherical shell model are conducted to verify the effectiveness of the new method. The results show that the computational speed of the new method is about 30 times faster than that of the traditional method, which employs the Gauss–Legendre quadrature rule and a 2-D adaptive subdivision approach, while keeping almost the same computational accuracy. When applying the new method to an ice shell with unequal thicknesses, the results reveal that the relative errors of calculating V, Vz and Vzz are smaller than 10−8, 10−6 and 10−4, respectively if the Taylor series expansion is truncated at order 4, while the computational time consumed by the new method is about 7 times less than that of the traditional method. Finally, the influence of the truncation order on the computational accuracy and the strategies for dividing the latitude band into several parts to further improve the accuracy are discussed.

Development of explicit formulations of G45-based gas kinetic scheme for simulation of continuum and rarefied flows.

In this work, the explicit formulations of the Grad's distribution function for the 45 moments (G45)-based gas kinetic scheme (GKS) are presented. Similar to the G13 function-based gas kinetic scheme (G13-GKS), G45-GKS simulates flows from the continuum regime to the rarefied regime by solving the macroscopic governing equations based on the conservation laws, which are widely used in conventional Navier-Stokes solver. These macroscopic governing equations are discretized by the finite volume method, where the numerical fluxes are evaluated by the local solution to the Boltzmann equation. The initial distribution function is reconstructed by the G45 distribution function, which is a higher order truncation of the Hermite expansion of distribution function compared with the G13 distribution function. Such high order truncation of Hermite expansion helps the present solver to achieve a better accuracy than G13-GKS. Moreover, the reconstruction of distribution function makes the development of explicit formulations of numerical fluxes feasible, and the evolution of the distribution function, which is the main reason why the discrete velocity method is expensive, is avoided. Several numerical experiments are performed to examine the accuracy of G45-GKS. Results show that the accuracy of the present solver for almost all flow problems is much better than G13-GKS. Moreover, some typical rarefied effects, such as the direction of heat flux without temperature gradients and thermal creep flow, can be well captured by the present solver.

Three-dimensional transient flow in heterogeneous reservoirs: Integral transform solution of a generalized point-source problem

Appraisal well testing plays a key role in optimizing a complex field’s exploitation strategy — and increasing complexity often calls for more elaborate reservoir models for the proper interpretation of each test’s results. The classical analytical methods usually require restrictive simplifying assumptions, which limit their usefulness when dealing with reservoir heterogeneity. We seek to solve this issue by applying the hybrid analytical–numerical Generalized Integral Transform Technique (GITT) to the pressure diffusivity equation — and specifically, to the point-source problem extended to a reservoir with arbitrary permeability variation irregularly distributed throughout the 3D domain. The technique is first demonstrated to produce an exact alternative form of the well-known solution to the classical point-source in a homogeneous reservoir. This equivalence is then used to derive a computationally efficient working expression for the generalized point-source in a heterogeneous reservoir. Finally, this building block is applied to construct a uniform flow solution for a limited entry vertical well through spatial superposition — thus demonstrating the usage of the GITT for other wellbore geometries. Synthetic examples are used to show that the obtained expressions are in good agreement with results from a commercial reservoir simulator. In all cases presented, the eigenfunction expansions for both the drawdown pressure and its logarithmic derivative converge to at least four and two significant digits, respectively, within the adopted practical ranges of the series’ truncation orders. It is clear that the pressure expansions have better convergence characteristics, which are also influenced by the time value and by the distance from the position under consideration to the point-source. The resulting novel solution to the point-source provided in the present work is the most general and least restrictive expression presented so far to this single problem in a heterogeneous domain, and it is suitable for pressure transient analysis. • Generalized Integral Transform Technique (GITT) for pressure transient analysis. • Generalization of the point-source solution for a heterogeneous reservoir. • Building block for any wellbore geometry — illustrated with the limited entry well.

On the selection of the number of beamformers in beamforming-based binaural reproduction

In recent years, spatial audio reproduction has been widely researched with many studies focusing on headphone-based spatial reproduction. A popular format for spatial audio is higher order Ambisonics (HOA), where a spherical microphone array is typically used to obtain the HOA signals. When a spherical array is not available, beamforming-based binaural reproduction (BFBR) can be used, where signals are captured with arrays of a general configuration. While shown to be useful, no comprehensive studies of BFBR have been presented and so its limitations and other design aspects are not well understood. This paper takes an initial step towards developing a theory for BFBR and develops guidelines for selecting the number of beamformers. In particular, the average directivity factor of the microphone array is proposed as a measure for supporting this selection. The effect of head-related transfer function (HRTF) order truncation that occurs when using too many beamformer directions is presented and studied. In addition, the relation between HOA-based binaural reproduction and BFBR is discussed through analysis based on a spherical array. A simulation study is then presented, based on both a spherical and a planar array, demonstrating the proposed guidelines. A listening test verifies the perceptual attributes of the methods presented in this study. These results can be used for more informed beamformer design for BFBR.

A robust and efficient stability analysis of periodic solutions based on harmonic balance method and Floquet-Hill formulation

Harmonic balance method (HBM) incorporated with arc-length continuation and Floquet-Hill formulation now becomes an essential and strong tool for computing periodic solution branches and performing stability analysis, which can provide great insight into the responses of nonlinear dynamic systems. However, due to the redundancy of eigenvalues of Hill’s matrix over the actual Floquet exponents of periodic solutions and the distortion on the redundancy relation of the eigenvalues induced by order truncation of harmonics, it is not routine work to correctly select the actual Floquet exponents from the eigenvalues of Hill’s matrix in order to achieve correct stability and bifurcation analysis, especially the order of harmonics in HBM is low. In this work, a new robust strategy for Floquet exponents filtering is introduced, which is chosen based on the character of the real part (FEF-RP) of the Hill’s eigenvalues. Furthermore, an adaptive arc-length control for the arc-length continuation (ALC) is proposed, which is continuously governed by the minimum parameterized eigenvalue of the Jacobian matrix (ALC-MPE) and can provide smooth and efficient continuation of the turning points of periodic solutions. The study on a 4DOFs rotor–stator rubbing system and a 3DOFs damped Duffing oscillator system shows the feasibility of the proposed methods for arc-length control (ALC-MPE) and Floquet exponents filtering (FEF-RP).

Quasi-3-D Nonlinear Analytical Modeling of Magnetic Field in Axial-Flux Switched Reluctance Motors

Analytical modeling (AM) of the magnetic field for axial-flux switched reluctance motors (AFSRMs), considering three-dimensional (3-D) features and nonlinear saturation effects, remains to be a knotty problem. This paper proposes a quasi-3-D nonlinear AM method to predict magnetic field distributions and electromagnetic performances of an AFSRM. Firstly, the harmonic modeling technique is applied to obtain the spatiotemporal distribution characteristics of the axial and tangential magnetic flux density at the mean radius. Secondly, the nonlinearity of the soft magnetic material in teeth is considered by an adaptive convergence rate algorithm. Finally, a radial dependence function is formulated to extend the solution of the annular section to any point in the 3-D space of the AFSRM. To confirm the accuracy of the proposed method, the analytical solutions of magnetic flux density and electromagnetic torque have been compared with FEM simulation results. Meanwhile, an additional validation has been done for the stability of this nonlinear AM method under severe saturation conditions. At last, the reasonable selection of the harmonic truncation order number has been discussed to reach a good balance between the accuracy and time cost. The quasi-3-D nonlinear AM method can consider the curvature and edge effect as well as the saturation effect effectively, which is suitable for performance predictions and parametric studies of AFSRMs.

A correctly scaling rigorously spin-adapted and spin-complete open-shell CCSD implementation for arbitrary high-spin states.

In this paper, we report on a correctly scaling novel coupled cluster singles and doubles (CCSD) implementation for arbitrary high-spin open-shell states. The chosen cluster operator is completely spin-free, i.e., employs spatial substitutions only. It is composed of our recently developed Löwdin-type operators [N. Herrmann and M. Hanrath, J. Chem. Phys. 153, 164114 (2020)], which ensure (1) spin completeness and (2) spin adaption, i.e., spin purity of the CC wave function. In contrast to the proof-of-concept matrix-representation-based implementation presented there, the present implementation relies on second quantization and factorized tensor contractions. The generated singles and doubles operators are embedded in an equation generation engine. In the latter, Wick's theorem is used to derive prefactors arising from spin integration directly from the spin-free full contraction patterns. The obtained Wick terms composed of products of Kronecker deltas are represented by special non-antisymmetrized Goldstone diagrams. Identical (redundant) diagrams are identified by solving the underlying graph isomorphism problem. All non-redundant graphs are then automatically translated to locally-one term at a time-factorized tensor contractions. Finally, the spin-adapted and spin-complete (SASC) CCS and CCSD variants are applied to a set of small molecular test systems. Both correlation energies and amplitude norms hint toward a reasonable convergence of the SASC-CCSD method for a Baker-Campbell-Hausdorff series truncation of order four. In comparison to spin orbital CCSD, SASC-CCSD leads to slightly improved correlation energies with differences of up to 1.292mEH (1.10% with respect to full configuration identification) for quintet CH2 in the cc-pVDZ basis set.

A new family of <i>L</i>-stable block methods with relative measure of stability

There are several nonlinear and stiff mathematical models in fields of science and engineering that have always remained a challenge for numerical analysts and applied mathematicians. Various numerical methods are proposed to deal with stiff models; however, it requires the model to have strong stability characteristics to handle the stiffness in the model. This paper develops a new family of L-stable block methods with a relative measure of stability for the solution of stiff differential equations with different characteristics. First, the theoretical properties of the proposed block method in terms of local truncation errors, absolute stability, consistency, convergence, and order stars have been analysed and investigated. Then, seven illustrative stiff differential models have been solved to measure the proposed method's performance, suitability, effectiveness, and efficiency. Finally, the error distributions and the precision factors are computed in the comparison of several existing methods having similar properties as that of the proposed L-stable block method.

Vector-Norm Based Truncation of Harmonic Transfer Functions in Black-Box Electronic Power Systems

Power systems equipped with power electronic converters can be modeled by harmonic transfer functions (HTFs) in a black-box manner for dynamic analysis. This paper studies the truncation of HTFs. It is proposed to define the gain function of an HTF as the norm of its central-column vector. Then, the error bound of the gain function in relation to the truncation order is explicitly derived, which can be used as an indicator for the HTF truncation. Compared with existing solutions, the proposed method is practical in truncating black-box systems with unknown internal parameters, since the truncation error bound can be estimated by the central-column elements of the HTF, which can be easily measured through frequency scan. The truncation approach is finally verified on a three-phase electronic power system by electromagnetic transient simulations.

Spectral Dyadic Green’s Function for Multilayer Periodic Bi-Anisotropic Media

A computational technique for evaluating the spectral dyadic Green’s function utilized in the analysis of multilayer structures composed of periodic bi-anisotropic layers is presented. The Green’s function is achieved with the help of a fully vectorial rigorous matrix formulation based on a generalized transmission line (TL) formulation of Maxwell’s equations. This matrix formulation simplifies the mathematics involved in the modeling of the coupled diffraction orders within a periodic bi-anisotropic layer. To examine the soundness of this modeling, the extracted Green’s function is used in an integral equation for the analysis of metallic gratings on homogeneous/inhomogeneous anisotropic and chiral materials. The resulting integral equation is then solved by the Method of Moments (MoM) formulated in terms of sub-domain basis and Galerkin’s test functions. As verification, several examples are analyzed and the results obtained by this method are compared with those available in the literature or obtained by EM-solvers. The efficiency assessment of this method is carried out by considering its convergence rate and cost-time in terms of truncation orders. It is revealed not only the metallic grating with inhomogeneous periodic bi-anisotropic layers can be analyzed by this method but also analysis time, memory, and CPU occupancies are decreased in comparison with commercial EM-solvers.

The valid regions of Gram–Charlier densities with high-order cumulants

Based on derivatives of a Gaussian density, the Gram–Charlier series is an infinite expansion. Its truncated series is often used in many fields to approximate probability density functions. Although the expansions are useful, there are constrained regions on the value of the cumulants (or moments) that admit a valid (nonnegative) probability density function. When the truncation order is low (just at fourth-order), the truncated Gram–Charlier density may be difficult to approximate an implied probability distribution as closely as possible, especially for distributions that are not sufficiently close to a normal distribution. One might increase the order after which the series is truncated until a perfect fit is achieved. However, the series expansion is usually truncated in the existing literature until the fourth-order term because it becomes difficult to find valid regions. This paper shows how the valid region of higher cumulants can be numerically implemented by the semi-definite algorithm, which ensures that a series truncated at a cumulant of arbitrary even order represents a valid probability density. We provide examples of two valid regions of the sixth and eighth Gram–Charlier densities (i.e., truncated at the sixth and eighth terms). Our analysis proves the fact that valid regions can be broadened with the higher-order expansions. Furthermore, the impact of higher cumulants on the valid regions has been shown.

A new family of 𝒜− acceptable nonlinear methods with fixed and variable stepsize approach

Solving stiff, singular, and singularly perturbed initial value problems (IVPs) has always been challenging for researchers working in different fields of science and engineering. In this research work, an attempt is made to devise a family of nonlinear methods amongst which second- to fourth-order methods are not only — stable but — acceptable as well under order stars’ conditions. These features make them more suitable for solving stiff and singular systems in ordinary differential equations. Methods with remaining orders are either zero- or conditionally stable. The theoretical analysis contains local truncation error, consistency, and order of accuracy of the proposed nonlinear methods. Furthermore, both fixed and variable stepsize approaches are introduced wherein the latter improves the performance of the devised methods. The applicability of the methods for solving the system of IVPs is also described. When used to solve problems from physical and real-life applications, including nonlinear logistic growth and stiff model for flame propagation, the proposed methods are found to have good results.