Spectral Equation Research Articles

Modelling the residually stressed magneto-electrically coupled soft elastic materials

Residual stresses may exist in different finitely deformed soft multifunctional materials such as in electro-active and magneto-active polymers. In order to develop accurate constitutive frameworks of these smart materials experiencing electro-magneto-mechanically coupled loads, the presence of residual stresses needs to be considered on the onset of the model development. In this contribution, a spectral constitutive equation for finite strain magneto-electric soft material bodies with residual stresses is developed using spectral invariants, where each spectral invariant has a clear physical meaning. A prototype total energy function comprising of single-variable functions is proposed; a single-variable function that depends on an invariant with a direct meaning is easily handled and is experimentally attractive. Results of some boundary value problems are given.

Transport synthetic acceleration for the solution of the one-speed nonclassical spectral S[formula omitted] equations in slab geometry

The nonclassical transport equation models particle transport processes in which the particle flux does not decrease as an exponential function of the particle’s free-path. Recently, a spectral approach was developed to generate nonclassical spectral SN equations, which can be numerically solved in a deterministic fashion using classical numerical techniques. This paper introduces a transport synthetic acceleration procedure to speed up the iteration scheme for the solution of the monoenergetic slab-geometry nonclassical spectral SN equations. We present numerical results that confirm the benefit of the acceleration procedure for this class of problems.

The generalized Wiener-Hopf equations for wave motion in angular regions: electromagnetic application.

In this work, we introduce a general method to deduce spectral functional equations and, thus, the generalized Wiener–Hopf equations (GWHEs) for wave motion in angular regions filled by arbitrary linear homogeneous media and illuminated by sources localized at infinity with application to electromagnetics. The functional equations are obtained by solving vector differential equations of first order that model the problem. The application of the boundary conditions to the functional equations yields GWHEs for practical problems. This paper shows the general theory and the validity of GWHEs in the context of electromagnetic applications with respect to the current literature. Extension to scattering problems by wedges in arbitrarily linear media in different physics will be presented in future works.

Mass Spectra and Decay Constants of Heavy-Light Axial Quarkonia in the Framework of Bethe-Salpeter Equation

In this work we calculate the mass spectrum and decay constants of ground and excited states of heavy-light P-wave mesons such as 1++ and 1+−, with quark composition, \(c\overline {u}, c\overline {s}, b\overline {u}, b\overline {s}\), and \(b\overline {c}\) in the framework of a QCD motivated Bethe-Salpeter equation (BSE) by making use of the exact treatment of the spin structure \((\gamma _{\mu }\bigotimes \gamma _{\mu })\) in the interaction kernel, In this 4 × 4 BSE framework, the coupled Salpeter equations for \(Q\overline {q}\) are first solved for the confining part of interaction, and are shown to decouple under heavy-quark approximation. Then the one-gluon-exchange interaction is perturbatively incorporated, leading to their mass spectral equations. The analytic forms of wave functions obtained from these equations are then used for calculation of leptonic decay constants of ground and excited states of 1++, and 1+− as a test of these wave functions and the over all framework.

Nonlinear rate-dependent spectral constitutive equation for viscoelastic solids with residual stresses

Nonlinear rate-dependent spectral constitutive equation for viscoelastic solids with residual stresses

Dynamic earthquake sequence simulation with a SBIEM without periodic boundaries

Dynamic earthquake sequence simulation is an important tool for investigating the behavior of a fault that hosts a series of earthquakes because it solves all interrelated stages in the earthquake cycle consistently, including nucleation, propagation and arrest of dynamic rupture, afterslip, locking, and interseismic stress accumulation. Numerically simulating and resolving these phenomena, which have different time and length scales, in a single framework is challenging. A spectral boundary integral equation method (SBIEM) that makes use of a fast Fourier transform is widely used because it reduces required computational costs, even though it can only be used for a planar fault. The conventional SBIEM has a periodic boundary condition as a result of the discretization of the wavenumber domain with a regular mesh; thus, to obtain an approximate solution for a fault in an infinite medium, it has been necessary to simulate a region much longer than the source distribution. Here, I propose a new SBIEM that is free from this artificial periodic boundary condition. In the proposed method, the periodic boundaries are removed by using a previously proposed method for the simulation of dynamic rupture. The integration kernel for the elastostatic effect, which reaches infinitely far from the source, is expressed analytically and replaces the one in the conventional SBIEM. The new method requires simulation of a region only twice as long as the source distribution, so the computational costs are significantly less than those required by the conventional SBIEM to simulate a fault in an infinite medium. The effect of the distance λ between the artificial periodic boundaries was investigated by comparing solutions for a typical problem setting between the conventional and proposed SBIEM. The result showed that the artificial periodic boundaries cause overestimation of the recurrence interval that is proportional to λ−2. If λ is four times the fault length, the interval is overestimated by less than 1%. Thus, the artificial periodic boundaries have only a modest effect on the conclusions of previous studies.

Reduced Multiplicative (BURA-MR) and Additive (BURA-AR) Best Uniform Rational Approximation Methods and Algorithms for Fractional Elliptic Equations

Numerical methods for spectral space-fractional elliptic equations are studied. The boundary value problem is defined in a bounded domain of general geometry, Ω⊂Rd, d∈{1,2,3}. Assuming that the finite difference method (FDM) or the finite element method (FEM) is applied for discretization in space, the approximate solution is described by the system of linear algebraic equations Aαu=f, α∈(0,1). Although matrix A∈RN×N is sparse, symmetric and positive definite (SPD), matrix Aα is dense. The recent achievements in the field are determined by methods that reduce the original non-local problem to solving k auxiliary linear systems with sparse SPD matrices that can be expressed as positive diagonal perturbations of A. The present study is in the spirit of the BURA method, based on the best uniform rational approximation rα,k(t) of degree k of tα in the interval [0,1]. The introduced additive BURA-AR and multiplicative BURA-MR methods follow the observation that the matrices of part of the auxiliary systems possess very different properties. As a result, solution methods with substantially improved computational complexity are developed. In this paper, we present new theoretical characterizations of the BURA parameters, which gives a theoretical justification for the new methods. The theoretical estimates are supported by a set of representative numerical tests. The new theoretical and experimental results raise the question of whether the almost optimal estimate of the computational complexity of the BURA method in the form O(Nlog2N) can be improved.

Damping effect on seismic input energy and its verification by shake table tests

Seismic input energy per unit mass ( EI/m) imparted into a structure is a function of earthquake (duration, frequency content, amplitude etc.), soil (shear velocity, dominant period etc.) and the structural (vibrational periods etc.) characteristics. Generally, the damping properties of the structure is assumed negligible for seismic input energy. Most of the existing spectral equations derived for SDOF systems generally use a constant damping ratio of 5%. In this study, the damping effect on EI/m is investigated experimentally and numerically on SDOF systems with distinct damping ratios. Experimental investigation and numerical computations proved that seismic input energy is very sensitive to variation of damping within the vicinity of fundamental frequencies. Specifically, up to 50% increment was observed in the plateau region of the input energy spectrum, where maximum EI/m values occur, by variation of damping from 2% to 10%. Hence, a novel damping modification factor ( DMF), which could be utilized for existing energy spectra, is proposed in this paper. Validation studies of the proposed DMF are achieved through the various energy spectra found in the literature.

Number of populated electronic configurations in a hot dense plasma.

In hot dense plasmas of intermediate or high-Z elements in the state of local thermodynamic equilibrium, the number of electronic configurations contributing to key macroscopic quantities such as the spectral opacity and equation of state can be enormous. In this work we present systematic methods for the analysis of the number of relativistic electronic configurations in a plasma. While the combinatoric number of configurations can be huge even for mid-Z elements, the number of configurations which have non-negligible population is much lower and depends strongly and nontrivially on temperature and density. We discuss two useful methods for the estimation of the number of populated configurations: (i) using an exact calculation of the total combinatoric number of configurations within superconfigurations in a converged super-transition-array (STA) calculation, and (ii) by using an estimate for the multidimensional width of the probability distribution for electronic population over bound shells, which is binomial if electron exchange and correlation effects are neglected. These methods are analyzed, and the mechanism which leads to the huge number of populated configurations is discussed in detail. Comprehensive average-atom finite-temperature density functional theory (DFT) calculations are performed in a wide range of temperature and density for several low-, mid-, and high-Z plasmas. The effects of temperature and density on the number of populated configurations are discussed and explained.

Investigation of a Heterogeneous RLC Lattice with Triangular Topology, Excited by a Lumped Voltage Source

Heterogeneous triangular network composed of resistors, capacitors and inductors gives a genuine electrical model for analyzing many engineering and physical problems. Here, we discuss various designs of RLC electrical circuits possessing a triangular topology. The theoretical formulation uses the WCIP technique. The proposed method rests on the definition of the incident and reflected waves from the voltage and current on each circuit branch. The use of the auxiliary source concept makes the execution of the employed technique simple and increments the capacity and effectiveness of the strategy for treating more intricate circuits. The mathematical formulation is subdivided into two definition domains: a spatial domain expresses the network architecture and imposes the boundary conditions, and a spectral one which characterizes the periodicity laws and describes the coupling of the electrical elements. The relationships formulated are resolved by an iterative process where the spectral and spatial equations are repeated up to the convergence. In numerical example, the results obtained by the WCIP method, for a resistors network of triangular topology, show good conformance with the analytical solutions. The method has also demonstrated its performances for investigating various designs of heterogeneous RLC lattice with triangular topology including the heterogeneous RL, RC and RLC circuits, perturbed RC circuits and a composite RLC circuit excited by vertical and horizontal lumped sources.

Analysis of the Propagation in High-Speed Interconnects for MIMICs by Means of the Method of Analytical Preconditioning: A New Highly Efficient Evaluation of the Coefficient Matrix

The method of analytical preconditioning combines the discretization and the analytical regularization of a singular integral equation in a single step. In a recent paper by the author, such a method has been applied to a spectral domain integral equation formulation devised to analyze the propagation in polygonal cross-section microstrip lines, which are widely used as high-speed interconnects in monolithic microwave and millimeter waves integrated circuits. By choosing analytically Fourier transformable expansion functions reconstructing the behavior of the fields on the wedges, fast convergence is achieved, and the convolution integrals are expressed in closed form. However, the coefficient matrix elements are one-dimensional improper integrals of oscillating and, in the worst cases, slowly decaying functions. In this paper, a novel technique for the efficient evaluation of such kind of integrals is proposed. By means of a procedure based on Cauchy integral theorem, the general coefficient matrix element is written as a linear combination of fast converging integrals. As shown in the numerical results section, the proposed technique always outperforms the analytical asymptotic acceleration technique, especially when highly accurate solutions are required.

A Linear Scaling in Accuracy Numerical Method for Computing the Electrostatic Forces in the $N$-Body Dielectric Spheres Problem

This article deals with the efficient and accurate computation of the electrostatic forces between charged, spherical dielectric particles undergoing mutual polarisation. We use the spectral Galerkin boundary integral equation framework developed by Lindgren et al. (J. Comput. Phys. 371 (2018): 712-731) and subsequently analysed in two earlier contributions of the authors to propose a linear scaling in cost algorithm for the computation of the approximate forces. We establish exponential convergence of the method and derive error estimates for the approximate forces that do not explicitly depend on the number of dielectric particles $N$. Consequently, the proposed method requires only $\mathcal{O}(N)$ operations to compute the electrostatic forces acting on $N$ dielectric particles up to any given and fixed relative error.

Spectral functions in the ϕ4 -theory from the spectral Dyson-Schwinger equations

We develop a non-perturbative functional framework for computing real-time correlation functions in strongly correlated systems. The framework is based on the spectral representation of correlation functions and dimensional regularisation. Therefore, the non-perturbative spectral renormalisation setup here respects all symmetries of the theories at hand. In particular this includes space-time symmetries as well as internal symmetries such as chiral symmetry, and gauge symmetries. Spectral renormalisation can be applied within general functional approaches such as the functional renormalisation group, Dyson-Schwinger equations, and two- or $n$-particle irreducible approaches. As an application we compute the full, non-perturbative, spectral function of the scalar field in the $\phi^4$-theory in $2+1$ dimensions from spectral Dyson-Schwinger equations. We also compute the $s$-channel spectral function of the full $\phi^4$-vertex in this theory.

Radiative decays of heavy-light quarkonia through M1 and E1 transitions in the framework of the Bethe-Salpeter equation

In this work we study the radiative decays of heavy-light quarkonia through M1 and E1 transitions, that involve quark-triangle diagrams with two hadron vertices, and are difficult to evaluate in BSE-CIA. We have expressed the transition amplitude, $M_{fi}$ as a linear superposition of terms involving all possible combinations of $++$, and $--$ components of Salpeter wave functions of final and initial hadron, with coefficients being related to results of pole integrals over complex $\sigma$- plane. We evaluate the decay widths for $M1$ transitions ($^3S_1 \rightarrow ^1S_0 +\gamma$), and $E1$ transitions ($^3S_1 \rightarrow ^1P_0 +\gamma$ and $^1P_0 \rightarrow ^3S_1 +\gamma$). We have used algebraic forms of Salpeter wave functions obtained through analytic solutions of mass spectral equations for ground and excited states of $0^{++},1^{--}$, and $0^{-+}$ heavy-light quarkonia in approximate harmonic oscillator basis, to calculate their decay widths. The input parameters used by us were obtained by fitting to their mass spectra. We have compared our results with experimental data and other models, and found reasonable agreements.

Accuracy aspects of conventional discretization methods for scalar transport with nonzero divergence velocity field arising from the energy balance equation

AbstractWe are concerned with the numerical solution of a linear transport problem with nonzero divergence velocity field that originates from the spectral energy balance equation describing the evolution of wind waves and swells in coastal seas. The discretization error of the commonly used first‐order upwind finite difference and first‐order vertex‐centered upwind finite volume schemes in one space dimension is analyzed. Smoothness of nondivergent velocity field plays a crucial role in this. No such analysis has been attempted to date for such problems. The two schemes studied differ in the manner in which they treat the scalar flux numerically. The finite difference variant is shock captured, whereas the vertex‐centered finite volume approximation employs an arithmetic mean of the velocity and appears not to be flux conservative. The methods are subsequently extended to two dimensions on triangular meshes. Numerical experiments are provided to verify the convergence analysis. The main finding is that the finite difference scheme displays optimal rates of convergence and offers higher accuracy over the finite volume scheme, regardless the regularity of the velocity field. The latter scheme notably yields convergence rates of 0.5 and 0 in L2‐norm and L∞‐norm, respectively, when the velocity field is not smooth. A test case illustrating wave shoaling and refraction over submerged shoals is also presented and demonstrates the practical importance of flux conservation.

RAP-method (random perturbation method) for minimax G-filter

Abstract The spectral equations for the minimax estimates of the parameters of some systems are obtained.

A Near-Shore Linear Wave Model with the Mixed Finite Volume and Finite Difference Unstructured Mesh Method

The near-shore and estuary environment is characterized by complex natural processes. A prominent feature is the wind-generated waves, which transfer energy and lead to various phenomena not observed where the hydrodynamics is dictated only by currents. Over the past several decades, numerical models have been developed to predict the wave and current state and their interactions. Most models, however, have relied on the two-model approach in which the wave model is developed independently of the current model and the two are coupled together through a separate steering module. In this study, a new wave model is developed and embedded in an existing two-dimensional (2D) depth-integrated current model, SRH-2D. The work leads to a new wave–current model based on the one-model approach. The physical processes of the new wave model are based on the latest third-generation formulation in which the spectral wave action balance equation is solved so that the spectrum shape is not pre-imposed and the non-linear effects are not parameterized. New contributions of the present study lie primarily in the numerical method adopted, which include: (a) a new operator-splitting method that allows an implicit solution of the wave action equation in the geographical space; (b) mixed finite volume and finite difference method; (c) unstructured polygonal mesh in the geographical space; and (d) a single mesh for both the wave and current models that paves the way for the use of the one-model approach. An advantage of the present model is that the propagation of waves from deep water to shallow water in near-shore and the interaction between waves and river inflows may be carried out seamlessly. Tedious interpolations and the so-called multi-model steering operation adopted by many existing models are avoided. As a result, the underlying interpolation errors and information loss due to matching between two meshes are avoided, leading to an increased computational efficiency and accuracy. The new wave model is developed and verified using a number of cases. The verified near-shore wave processes include wave shoaling, refraction, wave breaking and diffraction. The predicted model results compare well with the analytical solution or measured data for all cases.

A modified Lin equation for the energy balance in isotropic turbulence

• Resolution of the scale-invariance paradox is used to modify the Lin equation. • Exact symmetries justify the division of wavenumber space into different regions. • The roles of these different regions can be established by energy conservation. • Equivalent decompositions can be applied to the Navier–Stokes equations. • This could be the basis of new theoretical approaches to the turbulence problem. At sufficiently large Reynolds numbers, turbulence is expected to exhibit scale-invariance in an intermediate (“inertial”) range of wavenumbers, as shown by power law behavior of the energy spectrum and also by a constant rate of energy transfer through wavenumber. However, there is an apparent contradiction between the definition of the energy flux (i.e., the integral of the transfer spectrum) and the observed behavior of the transfer spectrum itself. This is because the transfer spectrum T ( k ) is invariably found to have a zero-crossing at a single point (at k = k * ), implying that the corresponding energy flux cannot have an extended plateau but must instead have a maximum value at k = k * . This behavior was formulated as a paradox and resolved by the introduction of filtered/partitioned transfer spectra, which exploited the symmetries of the triadic interactions (J. Phys. A: Math. Theor., 2008). In this paper we consider the more general implications of that procedure for the spectral energy balance equation, also known as the Lin equation. It is argued that the resulting modified Lin equations (and their corresponding Navier–Stokes equations) offer a new starting point for both numerical and theoretical methods, which may lead to a better understanding of the underlying energy transfer processes in turbulence. In particular the filtered partitioned transfer spectra could provide a basis for a hybrid approach to the statistical closure problem, with the different spectra being tackled using different methods.

Integrating evolution equations using Fredholm determinants

<p style='text-indent:20px;'>We outline the construction of special functions in terms of Fredholm determinants to solve boundary value problems of the string spectral problem. Our motivation is that the string spectral problem is related to the spectral equations in Lax pairs of at least three nonlinear evolution equations from mathematical physics.</p>

Spectral spherical harmonics discrete ordinate method

A new method for modeling the radiative transfer in inhomogeneous three-dimensional media illuminated by a Gaussian beam is described. This approach, called the Spectral Spherical Harmonics Discrete Ordinate Method (SSHDOM), uses the Fourier expansion method to transform the three-dimensional radiative transfer into an one-dimensional equation in the spectral domain, and the Spherical Harmonics Discrete Ordinate Method (SHDOM) for its solution. Specifically, (i) the source function is represented in the spectral domain through a spherical harmonic expansion, (ii) the spectral one-dimensional radiative transfer equation is integrated along discrete ordinates through a spatial grid, and (iii) the solution method is based on the Picard iteration. Both SSHDOM and SHDOM algorithms are implemented in a common computer code.