Classical Energy Method Research Articles

Numerical solution of convected wave equation in free field using artificial boundary method

AbstractIn this article, we propose two procedures focusing on the computation of the time‐dependent convected wave equation in a free field with a uniform background flow. Both procedures are based on a framework, expended from Du et al. (SIAM J. Sci. Comput. 40 (2018), A1430–A1445.), of constructing the Dirichlet‐to‐Dirichlet (DtD)‐type discrete absorbing boundary conditions (ABCs). The first procedure is dedicated to reducing the infinite problem into a finite problem by a direct application of the framework on the finite difference discretization of the convected wave equation. However, the presence of convection terms makes the stability analysis hard to implement, which motivates us to develop the second procedure. First, the convected wave equation is transformed into a standard wave equation by using the Prandtl‐Glauert‐Lorentz transformation. After that, we obtain the DtD‐type ABC by using the above framework, and on this basis, derive an equivalent Dirichlet‐to‐Neumann‐type ABCs, which makes stability and convergence analysis easy to be obtained by the classical energy method. The effectiveness and comparison of these two procedures are investigated through numerical experiments.

Variability of strength and stiffness assessment for non-profile beam structures

Introduction. Traditional ways to increase the structural strength and stiffness of beams are now almost exhausted, and optimization of production and operation technologies is mostly based on the use of new materials and increasing their reliability as bearing elements of a complex geometric profile. Bearing elements of building structures operate under high loads and, even despite their predominantly static nature, experience a complex volumetric stress and strain state, which can scarcely ever be confirmed empirically and statistically.Aim. To propose an approach to assessing the strength and stiffness of beam structures operating under conditions of transverse bending of a complex geometric profile.Materials and methods. Classical energy methods, including Castigliano, Maxwell–Mohr and Vereshchagin methods (moment area method), have been used to assess the strength and stiffness of non-profile beam structures, initial parameters, differential equations of the deflection curve.Results. The similarity parameters (fitting criteria) of Mises–Hencky and Zhurkov reflect the behavior of a brittle or plastic beam material reasonably well. These characteristics are proposed to be combined in the Arrhenius–Zhurkov durability functions.Conclusions. The dependencies given in the calculations can be recommended for improving the flexural static and fatigue strength, as well as the stiffness of the bearing elements of building structures.

Homogenization of the steady-state Navier-Stokes equations with prescribed flux rate or pressure drop in a perforated pipe

The steady motion of a viscous incompressible fluid in a pipe (perforated with a large number of small holes) is modeled through the Navier-Stokes equations with mixed boundary conditions involving the Bernoulli pressure and the tangential velocity on the inlet and outlet of the tube, while either the transversal flux rate or the pressure drop is prescribed along the pipe. Applying the classical energy method in homogenization theory, we study the asymptotic behavior of the solutions to these systems, without any restriction on the magnitude of the data, as the size of the perforations goes to zero and show that the effective equations remain unmodified in the limit. The main novelty of the present work lies in the obtainment of the required uniform bounds, which are achieved (in the case of the prescribed flux problem) by a contradiction argument based on Bernoulli's law for solutions of the stationary Euler equations.

Influence of rheological behavior of low-melting alloy on instability and control method for sheet metal

The wrinkling-free forming method for sheet metal forming based on the low-melting alloy (LMA) is a novel sheet forming method. However, the wrinkling phenomenon is affected by the transmission of loading force depending on the rheological behavior of material LMA. The effect of the rheological behavior of LMA medium on the wrinkling mechanism needs to be deeply studied. The blank holder force exerted by the LMA medium on the edge of the sheet is analyzed using the plane strain method. The critical stress of sheet deformation and wrinkling is studied by combining the classical energy method. The range of blank holder force was computed at the critical wrinkling stage. The flow resistance of the sheet material flowing into the plastic region is affected by the stress angle in the transition zone through numerical simulation and experimental methods. The influence of different forming process conditions on stress angle on the transition area is studied, and the evolution law of the stress angle and the material flowing status are revealed. The effect of the upper layer thickness of LMA, the temperature of LMA, and the dimension of LMA in the flange region on wrinkling occurrence are performed. Experiments are carried out to verify the results for analytical and FE simulation. The experimental results indicated that research results show that a decrease in the stress angle will increase the resistance of the sheet flowing to the plastic deformation region.

Estimation of decay rates to large-solutions of 3D compressible magnetohydrodynamic system

The aim of this paper is to get an estimation of decay rates to first-order and second-order derivatives of space for large-solutions to 3D compressible magnetohydrodynamic system. While the condition (σ0 − 1, u0, Q0) ∈ L1 ∩ H2 is satisfied via a classical energy method and Fourier splitting method, first-order and second-order derivatives of space for large-solutions tending to 0 by L2-rate (1+t)−54 are shown. It is a necessary supplement to the result of Gao, Wei, and Yao [Appl. Math. Lett. 102, 106100 (2020)] in which they only obtained an estimation of decay rates to magnetic fields. Meanwhile, compared with the work of Gao, Wei, and Yao [Physica D 406, 132506 (2020)], we find that the appearance of magnetic fields does not have any bad effect on the estimation of decay rates to both the velocity field and density.

Linear-Time Algorithms for RNA Structure Prediction.

RNA secondary structure prediction is widely used to understand RNA function. Existing dynamic programming-based algorithms, both the classical minimum free energy (MFE) methods and partition function methods, suffer from a major limitation: their runtimes scale cubically with the RNA length, and this slowness limits their use in genome-wide applications. Inspired by incremental parsing for context-free grammars in computational linguistics, we designed linear-time heuristic algorithms, LinearFold and LinearPartition, to approximate the MFE structure, partition function and base pairing probabilities. These programs are orders of magnitude faster than Vienna RNAfold and CONTRAfold on long sequences. More interestingly, LinearFold and LinearPartition lead to more accurate predictions on the longest sequence families for which the structures are well established (16S and 23S Ribosomal RNAs), as well as improved accuracies for long-range base pairs (500 + nucleotides apart). This chapter provides protocols for using LinearFold and LinearPartition for secondary structure prediction.

Local Well-Posedness and Incompressible Limit of the Free-Boundary Problem in Compressible Elastodynamics

We consider the three dimensional free-boundary compressible elastodynamic system under the Rayleigh–Taylor sign condition. This describes the motion of an isentropic inviscid elastic medium with moving boundary. The deformation tensor is assumed to satisfy the neo-Hookean linear elasticity. The local well-posedness was proved by Trakhinin (J Differ Eq 264(3):1661–1715, 2018) by Nash–Moser iteration. In this paper, we give a new proof of the local well-posedness by the combination of classical energy method and hyperbolic approach. In the proof, we apply the tangential smoothing method to define the approximation system. The key observation is that the structure of the wave equation of pressure together with Christodoulou–Lindblad (Commun Pure Appl Math 53(12):1536–1602, 2000) elliptic estimates reduces the energy estimates to the control of tangentially-differentiated wave equations despite a potential loss of derivative in the source term. To the best of our knowledge, we first establish the nonlinear energy estimate without loss of regularity for free-boundary compressible elastodynamics. The energy estimate is also uniform in sound speed which yields the incompressible limit, that is, the solutions of the free-boundary compressible elastodynamic equations converge to the incompressible counterpart provided the convergence of initial datum. It is worth emphasizing that our method is completely applicable to compressible Euler equations. Our observation also shows that it is not necessary to include the full time derivatives in the boundary energy and analyze higher order wave equations as in Lindblad–Luo (Commun Pure Appl Math 71(7):1273–1333, 2018) and Luo (Ann. PDE 4(2):2506–2576, 2018) even if we require the energy is uniform in sound speed. Moreover, the enhanced regularity for compressible Euler equations obtained in Lindblad–Luo (Commun Pure Appl Math 71(7):1273–1333, 2018) and Luo (Ann. PDE 4(2):2506–2576, 2018) can still be recovered for a slightly compressible elastic medium by further delicate analysis of the Alinhac good unknowns, which is completely different from Euler equations.

Exponential stabilization of the problem of transmission of wave equation with linear dynamical feedback control

<p style='text-indent:20px;'>In this paper, we address exponential stabilization of transmission problem of the wave equation with linear dynamical feedback control. Using the classical energy method and multiplier technique, we prove that the energy of system exponentially decays.</p>

Pointwise estimates of the solution to one dimensional compressible Naiver-Stokes equations in half space

<p style='text-indent:20px;'>In this paper, we study the global existence and pointwise behavior of classical solution to one dimensional isentropic Navier-Stokes equations with mixed type boundary condition in half space. Based on classical energy method for half space problem, the global existence of classical solution is established firstly. Through analyzing the quantitative relationships of Green's function between Cauchy problem and initial boundary value problem, we observe that the leading part of Green's function for the initial boundary value problem is composed of three items: delta function, diffusive heat kernel, and reflected term from the boundary. Then applying Duhamel's principle yields the explicit expression of solution. With the help of accurate estimates for nonlinear wave coupling and the elliptic structure of velocity, the pointwise behavior of the solution is obtained under some appropriate assumptions on the initial data. Our results prove that the solution converges to the equilibrium state at the optimal decay rate <inline-formula><tex-math id="M1">\begin{document}$ (1+t)^{-\frac{1}{2}} $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula> norm.</p>

Damage Detection in Double Layer Grids with Modal Strain Energy Method and Dempster-Shafer Theory

Change in modal strain energy is one of the indicators used to detect damage in structures. However, in structures with high degrees of freedom, such as double-layer grids, this method requires a relatively large number of mode shapes which in practice is difficult to determine. Therefore, it is necessary to reduce the number of required mode shapes. In this study, a damage detection technique based on modal strain energy and Dempster-Shafer evidence theory is presented for locating damage in double layer grids using only a few number of mode shapes. First, by calculating mode shapes of the grid in undamaged and damaged states, the modal strain energy based index for each mode shape is determined. Then, the results obtained from separate mode shapes are combined using Dempster-Shafer theory to achieve better results. In order to investigate the effect of noise on damage detection, 3% random noise is added to mode shapes. To demonstrate the performance of the proposed method, different single and multiple damage cases with different damage intensities are considered. Numerical results show that using 5 mode shapes, the presented technique can detect up to 3 damaged elements with different damage intensities in different parts of the grid with good accuracy (probability of 92.3%). Considering the fact that the classical modal strain energy method fails to distinguish even 1 damaged element in the double layer grid, the result shows significant improvement.

Asymptotic decay of bipolar isentropic/non-isentropic compressible Navier-Stokes-Maxwell systems

The initial value problems of bipolar isentropic/non-isentropic compressible Navier-Stokes-Maxwell (CNS-M) systems arising from plasmas in R3 are studied. The main difficulty of studying the bipolar isentropic/non-isentropic CNS-M systems lies in the appearance of the electromagnetic fields satisfying the hyperbolic Maxwell equations. The large time-decay rates of global smooth solutions with small amplitude in Lq(R3) for 2≤q≤∞ are established. For the bipolar non-isentropic CNS-M system, the difference of velocities of two charged carriers decay at the rate (1+t)−34+14q which is faster than the rate (1+t)−34+14q(ln⁡(3+t))1−2q of the bipolar isentropic CNS-M system, meanwhile, the magnetic field decay at the rate (1+t)−34+34q(ln⁡(3+t))1−2q which is slower than the rate (1+t)−34+34q for the bipolar isentropic CNS-M system. The approach adopted is the classical energy method but with some new developments, where the techniques of choosing symmetrizers and the spectrum analysis on the linearized homogeneous system play the crucial roles.

Weak solutions to the Cauchy problem of fractional time‐space Keller–Segel equation

This paper investigates the Cauchy problem of the time‐space fractional parabolic‐elliptic Keller–Segel equation in . The global existence and mass conservation of weak solutions are established. Furthermore, the decay estimates and hyper‐contractive estimates of the weak solutions in for any 1 < r < ∞ are constructed. The existence result is obtained by the classical energy method combined with the generalized strong compactness criteria to time fractional PDEs (partial differential equations). And the proofs of decay estimates and hyper‐contractive estimates are based on the generalized comparison principle with Caputo derivative and the decay behavior of the solutions to the nonlinear fractional differential equation with ν > 0 and γ > 1.

An energy method for averaging lemmas

This work introduces a new approach to velocity averaging lemmas in kinetic theory. This approach -- based upon the classical energy method -- provides a powerful duality principle in kinetic transport equations which allows for a natural extension of classical averaging lemmas to previously unknown cases where the density and the source term belong to dual spaces. More generally, this kinetic duality principle produces regularity results where one can trade a loss of regularity or integrability somewhere in the kinetic transport equation for a suitable opposite gain elsewhere. Also, it looks simpler and more robust to rely on proving inequalities instead of constructing exact parametrices. The results in this article are introduced from a functional analytic point of view. They are motivated by the abstract regularity theory of kinetic transport equations. However, we may recall that velocity averaging lemmas have profound implications in kinetic theory and its related physical models. In particular, the precise formulation of such results has the potential to lead to important applications to the regularity of renormalizations of Boltzmann-type equations, as well as kinetic formulations of gas dynamics, for instance.

Asymptotic behavior of nonlinear sound waves in inviscid media with thermal and molecular relaxation

Nonlinear sound propagation through media with thermal and molecular relaxation can be modeled by third-order in time wave-like equations with memory. We investigate the asymptotic behavior of a Cauchy problem for such a model, the nonlocal Jordan–Moore–Gibson–Thompson equation, in the so-called critical case, which corresponds to propagation through inviscid fluids or gases. The memory has an exponentially fading character and type I, meaning that involves only the acoustic velocity potential. A major challenge in studying global behavior is that the linearized equation’s decay estimates are of regularity-loss type. As a result, the classical energy methods fail to work for the nonlinear problem. To overcome this difficulty, we construct appropriate time-weighted norms, where weights can have negative exponents. These problem-tailored norms create artificial damping terms that help control the nonlinearity and the loss of derivatives, and ultimately allow us to discover the model’s asymptotic behavior.

A Review of Modeling Modification Methods for Mixed FE-SEA

Since the traditional deterministic method and classical statistical energy method have little room for improvement, more and more researches gradually turn to the combination of the two methods to solve the low and medium frequency “acoustic vibration” problem. However, in the application of mixed FE-SEA model, researchers found that when the coupling interface between FEA substructure and SEA statistical substructure is in the form of continuous interface, a very large finite element grid needs to be established at the boundary of the mixed model, which leads to huge time-consuming problem in the calculation of medium and low frequency of the mixed model. To solve this problem, the hybrid FE-SEA model needs to be improved and modified. In this paper, some common hybrid model correction methods, including deterministic subsystem edge gradient correction, are reviewed to provide theoretical support for the comprehensive study of hybrid model correction.

Coupled Mode Flutter of a Linear Compressor Cascade in Subsonic and Transonic Flow Conditions

Flutter onset in turbomachinery is typically investigated numerically via decoupled methods due to a high mass ratio of structure to air. The unsteady aerodynamic response of a forced motion vibration is evaluated for a positive or negative work entry. The forced motion simulations assume vacuum mode shape vibrations at certain amplitudes without modal coupling due to aerodynamic forces. This approach, also known as the energy method, is valid for a high blade mass ratio and small logarithmic decrement values. An aeroelastic study of a multi-passage linear compressor cascade was performed. In fluid-structure-coupled time-marching CFD, generic heave and pitch degrees of freedom are allowed to vibrate freely in reaction to any aerodynamic forces. For one subsonic and one transonic flow condition predicted to be stable by the classical energy method approach, aerodynamically coupled-mode flutter is observed. It is shown that variations in the starting conditions, i.e. the initial vibration and inter-blade phase angle of the system, can have a strong influence on the number of CFD iterations required until amplitudes grow. However, if coupled-mode flutter is present in the system, it will ultimately set in at a distinct inter-blade phase angle.

LinearPartition: linear-time approximation of RNA folding partition function and base-pairing probabilities.

MotivationRNA secondary structure prediction is widely used to understand RNA function. Recently, there has been a shift away from the classical minimum free energy methods to partition function-based methods that account for folding ensembles and can therefore estimate structure and base pair probabilities. However, the classical partition function algorithm scales cubically with sequence length, and is therefore prohibitively slow for long sequences. This slowness is even more severe than cubic-time free energy minimization due to a substantially larger constant factor in runtime.ResultsInspired by the success of our recent LinearFold algorithm that predicts the approximate minimum free energy structure in linear time, we design a similar linear-time heuristic algorithm, LinearPartition, to approximate the partition function and base-pairing probabilities, which is shown to be orders of magnitude faster than Vienna RNAfold and CONTRAfold (e.g. 2.5 days versus 1.3 min on a sequence with length 32 753 nt). More interestingly, the resulting base-pairing probabilities are even better correlated with the ground-truth structures. LinearPartition also leads to a small accuracy improvement when used for downstream structure prediction on families with the longest length sequences (16S and 23S rRNAs), as well as a substantial improvement on long-distance base pairs (500+ nt apart).Availability and implementationCode: http://github.com/LinearFold/LinearPartition; Server: http://linearfold.org/partition.Supplementary information Supplementary data are available at Bioinformatics online.

Thermodynamic analysis of the thermal-technological complex of sugar production: the energy and entropy characteristics of an enterprise

This paper reports the approbation of a procedure of thermodynamic analysis of the thermal-technological complex of sugar production using the analysis of an enterprise of typical configuration as an example. Currently, the thermodynamic analysis of sugar production systems is mainly performed on the basis of a classical energy method. Minor attempts to exploit the potential of the second law of thermodynamics in the form of the adaptation of an exergy method are not systemic. Underlying the applied procedure is a joint analysis of general synthetic and analytical balances of mass, energy, and entropy. Such a procedure makes it possible to quantify the level of perfection of the existing and proposed thermal circuits, as well as the impact exerted on their perfection by energy efficiency measures, and it could be applied both to optimize the energy characteristics of the existing ones and in designing new enterprises of the sugar industry. It has been shown that the thermodynamic analysis of the thermal-technological complex of sugar production as a single system makes it possible to analyze the main factors of influence on the energy efficiency of the complex disregarding the course of the processes implemented therein. Such an approach can also be effectively used for the quick evaluation of the thermodynamic perfection of an enterprise and for determining its «energy-saving potential». Based on the results of energy analysis, the relationship has been established between the fuel and energy resources, supplied to the system, and the sources of their losses; a set of measures has been proposed to reduce the impact of each of these factors on resource consumption. Entropy analysis has revealed the internal and external causes of the irreversibility of processes; the principle of «energy irreversibility compensation» has made it possible to compile a rating of major imperfections and determine the optimal sequence of resource-saving measures. The results of the analysis have confirmed the efficiency of the procedure, which allows a comprehensive study, while operating only with the fundamental laws and the principles of classical thermodynamics, as opposed to procedures, based on energy-exergy characteristics

The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case

In this paper, we study the Cauchy problem for the linear and semilinear Moore-Gibson-Thompson (MGT) equation in the dissipative case. Concerning the linear MGT model, by utilizing WKB analysis associated with Fourier analysis, we derive some $L^2$ estimates of solutions, which improve those in the previous research [46]. Furthermore, asymptotic profiles of the solution and an approximate relation in a framework of the weighted $L^1$ space are derived. Next, with the aid of the classical energy method and Hardy's inequality, we get singular limit results for an energy and the solution itself. Concerning the semilinear MGT model, basing on these sharp $L^2$ estimates and constructing time-weighted Sobolev spaces, we investigate global (in time) existence of Sobolev solutions with different regularities. Finally, under a sign assumption on initial data, nonexistence of global (in time) weak solutions is proved by applying a test function method.

Dirichlet-boundary value problem for one dimensional nonlinear Schrödinger equations with large initial and boundary data

We consider the inhomogeneous Dirichlet-boundary value problem with large initial and boundary data for nonlinear Schrodinger equations in one space dimension. Global existence and asymptotic behavior in time of solutions to the problem are obtained by using the classical energy method and factorization techniques.