Higher-order Case Research Articles

Variational-based energy–momentum schemes of higher-order for elastic fiber-reinforced continua

This paper contributes to the improvement of recently published higher-order accurate energy–momentum schemes for an anisotropic hyperelastic material formulation based on the concept of structural tensors. These Galerkin-based energy–momentum schemes are developed by means of time finite elements with the focus on improving numerical stability and robustness in the presence of stiffness and large rotations. They show advantages over conventional time stepping schemes in combination with anisotropic materials formulated with polyconvex strain energy density functions. In this paper, higher-order energy–momentum schemes are developed by using a differential variational principle in order to remedy the drawbacks of these Galerkin-based schemes, namely a variationally inconsistent time approximation of the strains in the higher-order case, no coordinate free superimposed stress tensor and an unseparated time approximation of the fibers and the matrix material. These new structure-preserving time integrators preserve all conservation laws of a free flying hyperelastic anisotropic continuum, namely the total linear and the total angular momentum conservation as well as the total energy conservation. Numerical examples demonstrate dynamic behavior of anisotropic continua, discrete conservation properties of flying continua and the effect of higher-order accuracy subject to transient loads.

Global asymptotic output tracking of nonlinear second-order systems with power integrators

This brief considers the problem of designing controllers to make the output of a power-integrator planar system globally asymptotically track the reference signal generated by a power-integrator exosystem. A necessary and sufficient condition on the powers of the integrators is identified for the solvability of the problem. In the higher order case, the tracking controllers are explicitly constructed by employing the tool of adding a power integrator. In the lower order case, a weak version of adding a power integrator technique is developed to enable us to construct tracking controllers.

Generalized James' effective Hamiltonian method

James' effective Hamiltonian method has been extensively adopted to investigate largely detuned interacting quantum systems. This method is just corresponding to the second-order perturbation theory, and cannot be exploited to treat the problems which should be solved by using the third or higher-order perturbation theory. In this paper, we generalize James' effective Hamiltonian method to the higher-order case. Using the method developed here, we reexamine two examples published recently [Phys. Rev.Lett. 117, 043601 (2016), Phys. Rev A 92, 023842 (2015)], our results turn out to be the same as the original ones derived from the third-order perturbation theory and adiabatic elimination method respectively. For some specific problems, this method can simplify the calculating procedure, and the resultant effective Hamiltonian is more general.

Improved pedagogy for linear differential equations by reconsidering how we measure the size of solutions

ABSTRACTFor over 50 years, the learning of teaching of a priori bounds on solutions to linear differential equations has involved a Euclidean approach to measuring the size of a solution. While the Euclidean approach to a priori bounds on solutions is somewhat manageable in the learning and teaching of the proofs involving second-order, linear problems with constant co-efficients, we believe it is not pedagogically optimal. Moreover, the Euclidean method becomes pedagogically unwieldy in the proofs involving higher-order cases. The purpose of this work is to propose a simpler pedagogical approach to establish a priori bounds on solutions by considering a different way of measuring the size of a solution to linear problems, which we refer to as the Uber size. The Uber form enables a simplification of pedagogy from the literature and the ideas are accessible to learners who have an understanding of the Fundamental Theorem of Calculus and the exponential function, both usually seen in a first course in calculus. We believe that this work will be of mathematical and pedagogical interest to those who are learning and teaching in the area of differential equations or in any of the numerous disciplines where linear differential equations are used.

Synchronization for memristive chaotic neural networks using Wirtinger-based multiple integral inequality

This paper investigates the synchronization problem for a class of memristive chaotic neural networks with time-varying delays. Based on te Wirtinger-based double integral inequality, two novel inequalities are proposed, which are multiple integral forms of the Wirtinger-based integral inequality. Next, by applying the reciprocally convex combination approach for high order case and a free-matrix-based inequality, novel delay-dependent conditions are established to achieve the synchronization for the memristive chaotic neural networks. The results are based on dividing the bounding of activation function into two subintervals with equal length. Finally, a numerical example is provided to demonstrate the effectiveness of the theoretical results.

A few remarks on Poincaré-Perron solutions and regularly varying solutions

AbstractWe establish conditions guaranteeing the existence of (generalized) regularly varying solutions tonth order linear differential andq-difference equations. The proofs are based mainly on classical Poincaré’s and Perron’s theorems and certain transformations. In some special cases, our results reduce to the existing ones, thus actually we offer an alternative approach to some parts of the asymptotic theory of differential equations made in the framework of regular variation. For higher order cases, our statements are essentially new. Another important feature of this paper is that it reveals connections among various results and somehow revises some of them.

The Holmes–Wick theorem on two-weight bounds for higher order commutators revisited

A sufficient condition for the two-weight boundedness of higher order commutators was recently obtained by Holmes and Wick in terms of an intersection of two BMO spaces. We provide an alternative proof, showing that the higher order case can be deduced by a classical Cauchy integral argument from the corresponding first order result of Holmes, Lacey and Wick.

Wirtinger-based multiple integral inequality approach to synchronization of stochastic neural networks

This paper investigates the synchronization problem for a class of chaotic neural networks with discrete and unbounded distributed time delays under stochastic perturbations. Firstly, based on the Wirtinger-based double integral inequality, two novel inequalities are proposed, which are multiple integral forms of the Wirtinger-based integral inequality. Next, by applying the Jensen-type integral inequality for stochastic case and combining the Jensen integral inequality with the reciprocally convex combination approach, a delay-dependent criterion is developed to achieve the synchronization for the stochastic chaotic neural networks in the sense of mean square. In the case of no stochastic perturbations, by applying the reciprocally convex combination approach for high order case and a free-matrix-based inequality, novel delay-dependent conditions are established to achieve the synchronization for the chaotic neural networks. All the results are based on dividing the bounding of activation function into two subintervals with equal length. Finally, two numerical examples are provided to demonstrate the effectiveness of the theoretical results.

Finite element approximations of symmetric tensors on simplicial grids in ℝn: The lower order case

In this paper, we construct, in a unified fashion, lower order finite element subspaces of spaces of symmetric tensors with square-integrable divergence on a domain in any dimension. These subspaces are essentially the symmetric tensor finite element spaces of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296], enriched, for each [Formula: see text]-dimensional simplex, by [Formula: see text] face bubble functions in the symmetric tensor finite element space of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296] when [Formula: see text], and by [Formula: see text] face bubble functions in the symmetric tensor finite element space of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296] when [Formula: see text]. These spaces can be used to approximate the symmetric matrix field in a mixed formulation problem where the other variable is approximated by discontinuous piecewise [Formula: see text] polynomials. This in particular leads to first-order mixed elements on simplicial grids with total degrees of freedom per element [Formula: see text] plus [Formula: see text] in 2D, 48 plus 6 in 3D. The previous record of the degrees of freedom of first-order mixed elements is, 21 plus 3 in 2D, and 156 plus 6 in 3D, on simplicial grids. We also derive, in a unified way which is completely different from those used in [D. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions, Math. Comput. 77 (2008) 1229–1251; D. N. Arnold and R. Winther, Mixed finite element for elasticity, Number Math. 92 (2002) 401–419], a family of Arnold–Winther mixed finite elements in any space dimension. One example in this family is the Raviart–Thomas elements in one dimension, the second example is the mixed finite elements for linear elasticity in two dimensions due to Arnold and Winther, the third example is the mixed finite elements for linear elasticity in three dimensions due to Arnold, Awanou and Winther.

Time and frequency domain characteristics of detrending-operation-based scaling analysis: Exact DFA and DMA frequency responses

We develop a general framework to study the time and frequency domain characteristics of detrending-operation-based scaling analysis methods, such as detrended fluctuation analysis (DFA) and detrending moving average (DMA) analysis. In this framework, using either the time or frequency domain approach, the frequency responses of detrending operations are calculated analytically. Although the frequency domain approach based on conventional linear analysis techniques is only applicable to linear detrending operations, the time domain approach presented here is applicable to both linear and nonlinear detrending operations. Furthermore, using the relationship between the time and frequency domain representations of the frequency responses, the frequency domain characteristics of nonlinear detrending operations can be obtained. Based on the calculated frequency responses, it is possible to establish a direct connection between the root-mean-square deviation of the detrending-operation-based scaling analysis and the power spectrum for linear stochastic processes. Here, by applying our methods to DFA and DMA, including higher-order cases, exact frequency responses are calculated. In addition, we analytically investigate the cutoff frequencies of DFA and DMA detrending operations and show that these frequencies are not optimally adjusted to coincide with the corresponding time scale.

Commutator representations and roots of pseudo differential operators

Based on the fundamental commutator representation proposed by Cao [4] we established two explicit expressions for roots of a third order differential operator. By using those expressions we succeeded in clarifying the relationship between two major approaches in theory of integrable systems: the zero curvature and the Lax representations for the KdV and the Boussinesq hierarchies. The proposed procedure could be extended to the general case of higher order of differential operators that leads to the Gel’fand-Dickey hierarchy.

High-order finite-volume methods on locally-structured grids

We present an approach to designing arbitrarily high-order finite-volume spatial discretizations on locally-rectangular grids. It is based on the use of a simple class of high-order quadratures for computing the average of fluxes over faces. This approach has the advantage of being a variation on widely-used second-order methods, so that the prior experience in engineering those methods carries over in the higher-order case. Among the issues discussed are the basic design principles for uniform grids, the extension to locally-refined nest grid hierarchies, and the treatment of complex geometries using mapped grids, multiblock grids, and cut-cell representations.

On stability criteria for neural networks with time-varying delay using Wirtinger-based multiple integral inequality

This paper investigates the problem of delay-dependent stability analysis of neural networks with time-varying delay. Based on Wirtinger-based integral inequality which suggests very closed lower bound of Jensen's inequality, a new Wirtinger-based multiple integral inequality is presented and it is applied to time-varying delayed neural networks by using reciprocally convex combination approach of high order cases. Three numerical examples are given to describe the less conservatism of the proposed methods.

Derivation and inference of higher-order strictness types

We extend an existing first-order typing system for strictness analysis to the fully higher-order case, covering both the derivation system and the inference algorithm. The resulting strictness typing system has expressive capabilities far beyond that of traditional strictness analysis systems. This extension is developed with the explicit aim of formally proving soundness of higher-order strictness typing with respect to a natural operational semantics. A key aspect of our approach is the introduction of a proof assistant at an early stage, namely during development of the proof. As such, the theorem prover aids the design of the language theoretic concepts. The new results in combination with their formal proof can be seen as a case study towards the achievement of the long term PoplMark Challenge. The proof framework developed for this case study can furthermore be used in other typing system case studies.

NEW MIXED FINITE VOLUME SPACES FOR ELLIPTIC PROBLEMS

Mixed finite element methods for elliptic problems have been the subject of extensive research(see [1], [2], [3], [4], [5], [6]). The main idea of mixed finite element methods is to introduce the Darcy velocity as a new variable and write the equation into a system of differential equations. By discretizing this system, we can compute velocity and pressure together. However, the resulting mixed system has many more variables and gives rise to a saddle point problem. Thus there were some restrictions to use in industry. For the lowest order mixed methods, it is well known that there exist equivalent nonconforming finite element methods[7], [8]. For higher order cases, there exist some equivalent nonconforming finite element spaces. But this methods involve projections into

Assessment of shock capturing schemes for discontinuous Galerkin method

This paper carries out systematical investigations on the performance of several typical shock-capturing schemes for the discontinuous Galerkin (DG) method, including the total variation bounded (TVB) limiter and three artificial diffusivity schemes (the basis function-based (BF) scheme, the face residual-based (FR) scheme, and the element residual-based (ER) scheme). Shock-dominated flows (the Sod problem, the Shu-Osher problem, the double Mach reflection problem, and the transonic NACA0012 flow) are considered, addressing the issues of accuracy, non-oscillatory property, dependence on user-specified constants, resolution of discontinuities, and capability for steady solutions. Numerical results indicate that the TVB limiter is more efficient and robust, while the artificial diffusivity schemes are able to preserve small-scale flow structures better. In high order cases, the artificial diffusivity schemes have demonstrated superior performance over the TVB limiter.

Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients

The aim of this paper, which deals with a class of singular functionals involving difference quotients, is twofold: deriving suitable integral conditions under which a measurable function is polynomial and stating necessary and sufficient criteria for an integrable function to belong to a kth-order Sobolev space. One of the main theorems is a new characterization of Wk,p(Ω), k∈N and p∈(1,+∞), for arbitrary open sets Ω⊂Rn. In particular, we provide natural generalizations of the results regarding Sobolev spaces summarized in Brézis’ overview article [Brézis (2002)] to the higher-order case, and extend the work [Borghol (2007)] to a more general setting.

Asymptotic behavior of large radial solutions of a polyharmonic equation with superlinear growth

This paper concerns the blow-up behavior of large radial solutions of polyharmonic equations with power nonlinearities and positive radial weights. Specifically, we consider radially symmetric solutions of Δmu=c(|x|)|u|p on an annulus {x∈Rn|σ≤|x|<ρ}, with ρ∈(0,∞) and σ∈[0,ρ), that diverge to infinity as |x|→ρ. Here n,m∈N, p∈(1,∞), and c is a positive continuous function on the interval [σ,ρ]. Letting ϕρ(r):=QCρ/(ρ−r)q for r∈[σ,ρ), with q:=2m/(p−1), Q:=(q(q+1)⋯(q+2m−1))1/(p−1), and Cρ:=c(ρ)−1/(p−1), we show that, as |x|→ρ, the ratio u(x)/ϕρ(|x|) remains between positive constants that depend only on m and p. Extending well-known results for the second-order problem, we prove in the fourth-order case that u(x)/ϕρ(|x|)→1 as |x|→ρ and obtain precise asymptotic expansions if c is sufficiently smooth at ρ. In certain higher-order cases, we find solutions for which the ratio u(x)/ϕρ(|x|) does not converge, but oscillates about 1 with non-vanishing amplitude.

A two-level higher order local projection stabilization on hexahedral meshes

The two-level local projection stabilization with the pair (Qr,h,Qr−1,2hdisc), r≥1, of spaces of continuous, piecewise (mapped) polynomials of degree r on the mesh Th in each variable and discontinuous, piecewise (mapped) polynomials of degree r−1 on the macro mesh Mh in each variable satisfy a local inf–sup condition leading to optimal error estimates. In this note, we show that even the pair of spaces (Qr,h,Qr,2hdisc), r≥2, with the enriched projection space Qr,2hdisc satisfies the local inf–sup condition and can be used in this framework. This gives a new, alternative proof of the inf–sup condition for the pair (Qr,h,Qr−1,2hdisc) in higher order cases r≥2.

Phononic Frequency Combs through Nonlinear Resonances

We explore an analogue of optical frequency combs in driven nonlinear phononic systems, and present a mechanism for generating phononic frequency combs through nonlinear resonances. In the underlying process, a set of phonon modes is simultaneously excited by the external driving which yields frequency combs with an array of discrete and equidistant spectral lines of each nonlinearly excited phonon mode. Frequency combs through nonlinear resonance of different orders are investigated, and in particular the possibility of correlation tailoring in higher-order cases is revealed. We suggest that our results can be applied in various nonlinear acoustic processes, such as phonon harvesting, and can also be generalized to other nonlinear systems.