Lowest Order Approximation Research Articles

Speed of sound in cosmological phase transitions and effect on gravitational waves

The energy budget for gravitational waves of a cosmological first order phase transitions depends on the speed of sound in the thermal plasma in both phases around the bubble wall. Working in the real-singlet augmented Standard Model, which admits a strong two-step electroweak phase transition, we compute higher order corrections to the pressure and sound speed. We compare our result to lower-order approximations to the sound speed and the energy budget and investigate the impact on the gravitational wave signal. We find that deviations in the speed of sound from {c}_s^2 = 1/3 are enhanced up to mathcal{O} (5%) in our higher-order computation. This results in a suppression in the energy budget of up to mathcal{O} (50%) compared to approximations assuming {c}_s^2 = 1/3. The effect is most significant for hybrid and detonation solutions. We generalise our discussion to the case of multiple inert scalars and the case of a reduced number of fermion families in order to mimic hypothetical dark sector phase transitions. In this sector with modified field content, the sound speed can receive significant suppression, with potential order-of-magnitude impact on the gravitational wave amplitude.

Structural aspects of FRG in quantum tunneling computations

We probe both the unidimensional quartic harmonic oscillator and the double well potential through a numerical analysis of the Functional Renormalization Group flow equations truncated at first order in the derivative expansion. The two partial differential equations for the potential Vk(φ) and the wave function renormalization Zk(φ), as obtained in different schemes and with distinct regulators, are studied down to k=0, and the energy gap between lowest and first excited state is computed, in order to test the reliability of the approach in a strongly non-perturbative regime. Our findings point out at least three ranges of the quartic coupling λ, one with higher λ where the lowest order approximation is already accurate, the intermediate one where the inclusion of the first correction produces a good agreement with the exact results and, finally, the one with smallest λ where presumably the higher order correction of the flow is needed. Some details of the specifics of the infrared regulator are also discussed.

Virtual neutrino propagation at short baselines

Within a covariant perturbative field-theoretical approach, the wave-packet modified neutrino propagator is expressed as an asymptotic expansion in powers of dimensionless Lorentz- and rotation-invariant variables. The expansion is valid at high energies and short but macroscopic space-time distances between the vertices of the proper Feynman macrodiagram. In terms of duality between the propagator and the effective neutrino wave packet, at short times and distances, neutrinos are deeply virtual and move quasi-classically. In the lowest-order approximation, this leads to the classical inverse-square dependence of the modulus squared flavor transition amplitude and related neutrino-induced event rate from distance L between the source and detector, and the above-mentioned asymptotics results in the corrections to the classical behavior represented by powers of L^2. This is very different from the long-baseline regime, where similar corrections are given by an asymptotic expansion in inverse powers of L^2. However, in both short- and long-baseline regimes, the main corrections lead to a decrease in number of neutrino events.

Epicyclic frequencies of spheroidal stars with non-uniform density

ABSTRACT We consider the gravitational potential of a rotating star with non-uniform density to derive the orbital and epicyclic frequencies of the particles orbiting the star. We assume that the star is composed of concentric spheroids of constant density, with a global power-law distribution of density inside the star. At the lowest order approximation, we recover the known result for the Maclaurin spheroid that the maximum in the radial epicyclic frequency occurs at $r=\sqrt{2}ae$, for eccentricities ${\ge} 1/\sqrt{2}$. We find that the nature of these characteristic frequencies differs based on the geometry of the rotating star. For an oblate spheroid, the orbits resemble retrograde Kerr orbits and the location of the radial epicyclic maximum approaches the stellar surface as the density variation inside the star becomes steeper. On the contrary, orbits around a prolate spheroid resemble prograde Kerr orbits, but the marginally stable orbit does not exist for prolate-shaped stars. The orbital frequency is larger (smaller) than the Keplerian value for an oblate (prolate) star with the equality attained as e → 0 or r → ∞. The radial profiles of the angular velocity and the angular momentum allow for a stable accreting disc around any nature of oblate/prolate spheroid.

DPG Methods for a Fourth-Order div Problem

Abstract We study a fourth-order div problem and its approximation by the discontinuous Petrov–Galerkin method with optimal test functions. We present two variants, based on first and second-order systems. In both cases, we prove well-posedness of the formulation and quasi-optimal convergence of the approximation. Our analysis includes the fully-discrete schemes with approximated test functions, for general dimension and polynomial degree in the first-order case, and for two dimensions and lowest-order approximation in the second-order case. Numerical results illustrate the performance for quasi-uniform and adaptively refined meshes.

A New Approach to Reduce Transverse Shear Locking of Reissner–Mindlin Plate Based on The Cell-Centered Finite Element Method

This study investigates a new approach to solve a shear locking free on a low-order cell-centered finite element method (FECC) of the Reissner–Mindlin plate. The proposed method pursues the Lagrange piecewise of linear polynomial functions to attain the transverse displacement and rotations on the dual sub-mesh derived from the general one. The suggested theory is employed in numerical computation to verify the effectiveness and accurate performance. Detailed studies are carried out on several mesh types to compare with the analytical solutions reported by the low-order mimetic finite difference method [Beir ao Da Veiga, L., Lovadina, C. and Mora, D. [2013] “Numerical results for mimetic discretization of Reissner–Mindlin plate problems,” Calcolo 50, 209–237] and high-order mixed finite element methods [Chinosi, C. and Lovadina, C. [1995] “Numerical analysis of some mixed finite element methods for Reissner–Mindlin plates,” Comput. Mech. 16, 36–44]. The numerical results indicated that the present scheme based on the low-order FECC approximation can effectively predict on even the distorted meshes, satisfy the optimized convergence and be uniformly stable to the thin plate thicknesses.

Robust optimal attitude determination using interval analysis

The attitude determination of the satellite leads to solving the Wahba problem. The Wahba problem utilizes a set of at least two independent sensor measurements and reference vectors for attitude determination. These input vectors are not accurate due to sensor noises, misalignment, and low-order approximations. But these uncertainties do not consider in the classic Wahba problem directly. This paper presents a robust multi-objective optimization method for solving the static attitude determination with bounded uncertainty. The bounded uncertainty due to modeling error, measurement noise, and biases have modeled using interval analysis. Then, the Wahba problem is reformulated to a robust multi-objective optimization problem that consists of two performance indexes. The effectiveness of the proposed approach is evaluating by solving several examples.

A Spline Dimensional Decomposition for Uncertainty Quantification in High Dimensions

This study debuts a new spline dimensional decomposition (SDD) for uncertainty quantification analysis of high-dimensional functions, including those endowed with high nonlinearity and nonsmoothness, if they exist, in a proficient manner. The decomposition creates a hierarchical expansion for an output random variable of interest with respect to measure-consistent orthonormalized basis splines (B-splines) in independent input random variables. A dimensionwise decomposition of a spline space into orthogonal subspaces, each spanned by a reduced set of such orthonormal splines, results in SDD. Exploiting the modulus of smoothness, the SDD approximation is shown to converge in mean-square to the correct limit. The computational complexity of the SDD method is polynomial, as opposed to exponential, thus alleviating the curse of dimensionality to the extent possible. Analytical formulae are proposed to calculate the second-moment properties of a truncated SDD approximation for a general output random variable in terms of the expansion coefficients involved. Numerical results indicate that a low-order SDD approximation of nonsmooth functions calculates the probabilistic characteristics of an output variable with an accuracy matching or surpassing those obtained by high-order approximations from several existing methods. Finally, a 34-dimensional random eigenvalue analysis demonstrates the utility of SDD in solving practical problems.

On Robustly Convergent and Efficient Iterative Methods for Anisotropic Radiative Transfer

This paper considers the iterative solution of linear systems arising from discretization of the anisotropic radiative transfer equation with discontinuous elements on the sphere. In order to achieve robust convergence behavior in the discretization parameters and the physical parameters we develop preconditioned Richardson iterations in Hilbert spaces. We prove convergence of the resulting scheme. The preconditioner is constructed in two steps. The first step borrows ideas from matrix splittings and ensures mesh independence. The second step uses a subspace correction technique to reduce the influence of the optical parameters. The correction spaces are build from low-order spherical harmonics approximations generalizing well-known diffusion approximations. We discuss in detail the efficient implementation and application of the discrete operators. In particular, for the considered discontinuous spherical elements, the scattering operator becomes dense and we show that mathcal {H}- or mathcal {H}^2-matrix compression can be applied in a black-box fashion to obtain almost linear or linear complexity when applying the corresponding approximations. The effectiveness of the proposed method is shown in numerical examples.

Low-frequency acoustic radiation from a flanged circular pipe at an inclined angle.

The generic problem of low-frequency acoustic radiation through quiescent air from a circular pipe that is inclined with respect to its exit flange is studied in this work. The exit flange is taken to extend as an infinite plane away from the pipe opening. The analysis implements a hybrid method that combines modal expansions with the boundary element method. The reflection coefficient and pipe end correction for Helmholtz numbers (based on the pipe radius) less than 2.5 are calculated for various inclination angles up to 75°. Calculations are validated using simulations from the finite-element solver of the commercial software package COMSOL. The reflection coefficient and end correction predictions agree closely with the validation simulations yet differ notably from the results available in the literature. The solution obtained from the hybrid method is subsequently used to analyse the acoustic field at the pipe exit and in the downstream space. The key aspects of the governing physics pertaining to practical engineering applications at low frequencies are captured in a low-order approximation, which significantly reduces the degrees of freedom of the problem and provides generally good estimates of the reflection coefficient and end correction, as well as the downstream acoustic field.

Provable randomized rounding for minimum-similarity diversification

When searching for information in a data collection, we are often interested not only in finding relevant items, but also in assembling a diverse set, so as to explore different concepts that are present in the data. This problem has been researched extensively. However, finding a set of items with minimal pairwise similarities can be computationally challenging, and most existing works striving for quality guarantees assume that item relatedness is measured by a distance function. Given the widespread use of similarity functions in many domains, we believe this to be an important gap in the literature. In this paper we study the problem of finding a diverse set of items, when item relatedness is measured by a similarity function. We formulate the diversification task using a flexible, broadly applicable minimization objective, consisting of the sum of pairwise similarities of the selected items and a relevance penalty term. To find good solutions we adopt a randomized rounding strategy, which is challenging to analyze because of the cardinality constraint present in our formulation. Even though this obstacle can be overcome using dependent rounding, we show that it is possible to obtain provably good solutions using an independent approach, which is faster, simpler to implement and completely parallelizable. Our analysis relies on a novel bound for the ratio of Poisson-Binomial densities, which is of independent interest and has potential implications for other combinatorial-optimization problems. We leverage this result to design an efficient randomized algorithm that provides a lower-order additive approximation guarantee. We validate our method using several benchmark datasets, and show that it consistently outperforms the greedy approaches that are commonly used in the literature.

Nonlinear Stall Flutter Suppression of Wind Turbine Blade Based on LMI Method

Aiming at the failure of stall flutter in aeroelastic system of wind turbine blade, the active control process of stall flutter based on linear matrix inequality (LMI) design is described. The structural model is a typical blade section model based on spring-mass-damper, and the aerodynamic force is the ONERA stall aerodynamic model suitable for pure pitch motion. Based on the state variables, the nonlinear aeroelastic equations are expanded by Taylor series and linearized by low order approximation. The state feedback gain is calculated through LMI, and the time domain response stability analysis and stall flutter suppression method based on linearization are studied. The simulation results show that the maximum amplitude is greatly reduced after flutter control, and the system can be stabilized in a short time.

ADG: automated generation and evaluation of many-body diagrams

The goal of the present paper is twofold. First, a novel expansion many-body method applicable to superfluid open-shell nuclei, the so-called Bogoliubov in-medium similarity renormalization group (BIMSRG) theory, is formulated. This generalization of standard single-reference IMSRG theory for closed-shell systems parallels the recent extensions of coupled cluster, self-consistent Green’s function or many-body perturbation theory. Within the realm of IMSRG theories, BIMSRG provides an interesting alternative to the already existing multi-reference IMSRG (MR-IMSRG) method applicable to open-shell nuclei. The algebraic equations for low-order approximations, i.e., BIMSRG(1) and BIMSRG(2), can be derived manually without much difficulty. However, such a methodology becomes already impractical and error prone for the derivation of the BIMSRG(3) equations, which are eventually needed to reach high accuracy. Based on a diagrammatic formulation of BIMSRG theory, the second objective of the present paper is thus to describe the third version (v3.0) of the ADG code that automatically (1) generates all valid BIMSRG(n) diagrams and (2) evaluates their algebraic expressions in a matter of seconds. This is achieved in such a way that equations can easily be retrieved for both the flow equation and the Magnus expansion formulations of BIMSRG. Expanding on this work, the first future objective is to numerically implement BIMSRG(2) (eventually BIMSRG(3)) equations and perform ab initio calculations of mid-mass open-shell nuclei.

Low-order divergence-free approximations for the Stokes problem on Worsey–Farin and Powell–Sabin splits

We derive low-order, inf–sup stable and divergence-free finite element approximations for the Stokes problem using Worsey–Farin splits in three dimensions and Powell–Sabin splits in two dimensions. The velocity space simply consists of continuous, piecewise linear polynomials, whereas the pressure space is a subspace of piecewise constants with weak continuity properties at singular edges (3D) and singular vertices (2D). We discuss implementation aspects that arise when coding the pressure space, and in particular, show that the pressure constraints can be enforced at an algebraic level.

On computing the effective multiplication factor using the ADO method

In this work, the Analytical Discrete Ordinates method, ADO method, is applied to evaluate the effective multiplication factor in criticality problems of a nuclear reactor. To address this class of problems, the multigroup ADO formulation in multislab geometry is here derived taking into account real and complex spectrum. Two approaches are presented to find the desired dominant k-eigenvalue.In the first one, a criticality (characteristic) equation is derived from the complete original equation defined in multiplying media and a combined scheme, using regula-falsi and secant root-finding methods, is used to extract a root of that equation. Such scheme is shown to be very efficient and provided high-quality benchmark results, from lower quadrature order approximations than other schemes available in the literature, as found by the application of the ADO formulation also in several other transport problems.As a second proposed approach, the ADO solution is used along with the usual power iteration method in a simpler procedure. This simpler procedure avoids complex eigenvalues, but it is shown to be less accurate due to required approximations in the source approximation procedure.

A discontinuous Galerkin method for shock capturing using a mixed high-order and sub-grid low-order approximation space

This article considers a new discretization scheme for conservation laws. The discretization setting is based on a discontinuous Galerkin scheme in combination with an approximation space that contains high-order polynomial modes as well as piece-wise constant modes on a sub-grid. The high-order modes can continuously be suppressed with a penalty function that is based on a sensor which is intertwined with the approximation space. Numerical tests finally illustrate the performance of this scheme.

Online DMDc Based Model Identification Approach for Transient Stability Enhancement Using Wide Area Measurements

This letter proposes an online control strategy to enhance the overall damping of low-frequency power system oscillations of large inter-connected electrical networks. To achieve this objective, the proposed approach employs wide area measured signals to estimate an accurate online model of power system dynamics using the dynamic mode decomposition with control (DMDc) method. The DMDc technique applies singular value decomposition (SVD) to the collected data matrices such that an accurate low-order approximation of system dynamics is obtained. The estimated model is thereafter utilized in designing the wide area damping controller (WADC) using discrete linear quadratic regulator (DLQR). Furthermore, kalman filtering is deployed to update the states of the designed controller and to remove the noise that is contained in the measured signals. The effectiveness of the proposed method is tested on the 10 machines, 39-bus New England test system.

Dynamic wind turbine wake reconstruction: A Koopman-linear flow estimator

A challenging topic arising in dynamic wind turbine wake is modeling, especially the low-order approximation. The central problem is the fact that it has high-dimensional and nonlinear wake characteristics. In this paper, a Koopman-linear flow estimator is designed according to the Koopman operator theory. Different from the conventional flow reconstruction with the linear stochastic estimation method, a dynamic state-space model with physical states is constructed. The wake dynamics are approximated using a limited number of measurable physical parameters by the dynamic part; then, the full wake flow is reconstructed from the low-order states by the estimation part. The flow estimator is designed into three different forms following Extended Dynamic Mode Decomposition (EDMD) method. Each form has its unique advantages. Precisely, probe sensors are placed in the studied space and provide direct information of the wake, and a few in-directly physical parameters are also included. Nonlinear integer programming is further adopted using a heuristic optimization algorithm, by which the sensor configurations are optimized. Comparisons with the standard Dynamic Mode Decomposition (DMD)-based wake model are adopted in time domain and frequency domain to verify the effectiveness of the proposed flow estimators. The results show acceptable accuracy in typical modeling cases and maintain good estimation accuracy when the measurement noises are involved. Finally, the proposed Koopman-linear flow estimator is compared with related stochastic estimation methods, in which the connections of the proposed estimator with stochastic ones are also discussed.

Discussion of “Accurate explicit analytical solution for Colebrook-White equation” by Zahreddine Hafsi, Mechanics Research Communications 111 (2021) 103646)

The discussed paper introduces the concept of approximating the Colebrook-White equation (CW) by Taylor series expansion and solving analytically the resulting polynomial equations. The proposed 3rd order expansion based approximation is overly accurate but shows rather poor computational performance. We show in this letter that lower-order approximations developed according to the concept provide sufficient accuracy and greatly improved computational speed for engineering and scientific applications.

The possible equation of state of dark matter in low surface brightness galaxies * *Supported by the National Natural Science Foundation (NSF) of China (11973081, 11573062, 11403092, 11390374, 11521303), the YIPACAS Foundation (2012048), the Chinese Academy of Sciences (CAS, KJZD-EW-M06-01), the NSF of Yunnan Province (2019FB006) and the Youth Project of Western Light of CAS

The observed rotation curves of low surface brightness (LSB) galaxies play an essential role in studying dark matter, and indicate the existence of a central constant density dark matter core. However, the cosmological N-body simulations of cold dark matter predict an inner cusped halo with a power-law mass density distribution, and cannot reproduce a central constant-density core. This phenomenon is called cusp-core problem. When dark matter is quiescent and satisfies the condition for hydrostatic equilibrium, the equation of state can be adopted to obtain the density profile in the static and spherically symmetric space-time. To address the cusp-core problem, we assume that the equation of state is independent of the scaling transformation. Its lower order approximation for this type of equation of state can naturally lead to a special case, i.e., , where p and represent the pressure and density, respectively, depicts the rotation velocity of galaxy, and and are positive constants. It can obtain a density profile that is similar to the pseudo-isothermal halo model when is approximately 0.15. To obtain a more universally used model, let the equation of state include the polytropic model, i.e. , from which we can obtain other types of density profiles, such as the profile that is nearly same as the Burkert profile, where s and are positive constants.