Hopf Equation Research Articles

Hopf Bifurcation Calculations for a Symmetric Airfoil in Transonic Flow

The application of a sparse matrix solver for the direct calculation of Hopf bifurcation points arising for an airfoil moving in pitch and plunge in a transonic flow is considered. The iteration scheme for solving the Hopf equations is based on a modified Newton’s method. Direct solution of the linear system for the updates has previously been restrictive for application of the method, and the sparse solver overcomes this limitation. Results of experiments with the approximation to the Jacobian matrix driving the iteration to convergence are presented. Finally, it is shown that an entire flutter boundary for the NACA0012 airfoil can be traced out in a time comparable to that required for a small number of time-response calculations. I. Introduction C OMPUTATIONAL fluid dynamics (CFD) has matured to the point where it is being applied to complex problems in external aerodynamics. Aeroelastic analysis relies on high-fidelity predictions of aerodynamics, particularly for phenomena associated with shock motions or separation. These two observations have motivated the development of CFD-based aeroelastic simulation, a field now being called computational aeroelasticity. Developments in computational aeroelasticity have mainly been focused on time-marching calculations, where the temporal response of a system to an initial perturbation is calculated to determine growth or decay, and from this to infer stability. This type of simulation has developed significantly in the past decade, with efforts concentrating on mesh movement, load and displacement transfer between the aerodynamic and structural grids, 1 and sequencing of solutions. 2,3 Recent and impressive example calculations have been made for complete aircraft configurations. 4,5 The time-marching method will remain a powerful tool in computational aeroelasticity because of its generality. However, the cost of these calculations motivates attempts to find quicker ways of evaluating stability while still retaining the detailed aerodynamic predictions given by CFD. One way of doing this is to boil down the CFD into a reduced-order model that still retains the essence of the aerodynamics. Various approaches have been proposed, with an expansion of the flowfield in a truncated series of modes derived from proper orthogonal decomposition currently receiving much attention. 6 A second approach proposed by Morton and Beran from the U.S. Air Force is to use dynamic systems theory to characterize the nature of the aeroelastic instability and then to use this additional information to concentrate the use of the CFD. Aeroelastic instabilities that are commonly termed flutter are of the Hopf type, where an eigenvalue of the system Jacobian matrix crosses the imaginary axis at the flutter point. A model problem was used to evaluate the approach 7 in which the main difficulties associated with the method (calculation of the Jacobian matrix, solution of the augmented system by Newton’s method, solution of a large sparse linear system) were considered. The method was applied to an aeroelastic system consisting of an airfoil moving in pitch and plunge in Ref. 8. The

Quantifying Uncertainty for Non-Gaussian Ensembles in Complex Systems

Many situations in complex systems require quantitative estimates of the lack of information in one probability distribution relative to another. In short-term climate and weather prediction, examples of these issues might involve a lack of information in the historical climate record compared with an ensemble prediction, or a lack of information in a particular Gaussian ensemble prediction strategy involving the first and second moments compared with the non-Gaussian ensemble itself. The relative entropy is a natural way to quantify this information. Here a recently developed mathematical theory for quantifying this lack of information is converted into a practical algorithmic tool. The theory involves explicit estimators obtained through convex optimization, principal predictability components, a signal/dispersion decomposition, etc. An explicit computationally feasible family of estimators is developed here for estimating the relative entropy over a large dimensional family of variables through a simple hierarchical strategy. Many facets of this computational strategy for estimating uncertainty are applied here for ensemble predictions for two ``toy'' climate models developed recently: the Galerkin truncation of the Burgers--Hopf equation and the Lorenz '96 model.

Plane Wave Diffraction by Impedance Planar Surfaces with a Step Discontinuity

The diffraction of electromagnetic plane waves by a thick impedance half-plane located over a nonhomogeneous half-space backed by an impedance plane with a step discontinuity is investigated rigorously by using the Fourier transform technique in conjunction with the mode matching method. The hybrid method of formulation gives rise to a scalar Wiener–Hopf equation of the second kind, the solution of which contains infinitely many constants satisfying an infinite system of linear algebraic equations. A numerical solution of this system is obtained for various values of the plate impedance, plate thickness, permittivity of the dielectric region, and the incidence angle wherefrom the effect of these parameters on the diffraction phenomenon is studied.

Towards a sufficient criterion for collapse in 3D Euler equations

A sufficient integral criterion for a blow-up solution of the Hopf equations (the Euler equations with zero pressure) is found. This criterion shows that a certain positive integral quantity blows up in a finite time under specific initial conditions. Blow up of this quantity means that solution of the Hopf equation in 3D cannot be continued in the Sobolev space H 2( R 3) for infinite time.

Transmission of sound waves in a cylindrical duct with an acoustically lined muffler

The propagation of sound in an infinite rigid cylindrical duct with an inserted expansion chamber whose walls are treated with an acoustically absorbent material is investigated rigorously through the Wiener–Hopf technique. By introducing the Fourier transform for the scattered field and applying the boundary conditions in the transform domain, the problem is reduced into a modified Wiener–Hopf equation. The solution involves four sets of infinitely many constants satisfying four infinite systems of linear algebraic equations. An approximate solution of these systems is obtained by means of numerical procedures.

Some developments in general variational inequalities

General variational inequalities provide us with a unified, natural, novel and simple framework to study a wide class of equilibrium problems arising in pure and applied sciences. In this paper, we present a number of new and known numerical techniques for solving general variational inequalities using various techniques including projection, Wiener–Hopf equations, updating the solution, auxiliary principle, inertial proximal, penalty function, dynamical system and well-posedness. We also consider the local and global uniqueness of the solution and sensitivity analysis of the general variational inequalities as well as the finite convergence of the projection-type algorithms. Our proofs of convergence are very simple as compared with other methods. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. Since the general variational inequalities include (quasi) variational inequalities and (quasi) implicit complementarity problems as special cases, results presented here continue to hold for these problems. Several open problems have been suggested for further research in these areas.

Passive Advection and the Degenerate Elliptic Operators M n

We prove estimates for the stationary state $n$-point functions at zero molecular diffusivity in the Kraichnan model. This is done by proving upper bounds for the heat kernels and Green's functions of the degenerate elliptic operators $M_n$ that occur in the Hopf equations for the $n$-point functions.

FUNCTIONALS WITH VALUES IN THE NON-ARCHIMEDEAN FIELD OF LAURENT SERIES AND THEIR APPLICATIONS TO THE EQUATIONS OF ELASTICITY THEORY

Functionals with values in Non‐Archimedean field of Laurent series applied to the definition of generalized solution (in the form of shock wave) of the Hopf equation and equations of elasticity theory. Calculation method for the profile of shock wave is proposed. It is shown that there is a possibility to find out some of the solutions of this system using the Newton iteration method. Examples and numerical tests are considered.

Asymptotic solutions of the nonlocal nonlinear Schrödinger equation in the limit of small dispersion

A method is developed for solving the initial value problem of the nonlocal nonlinear Schrödinger (NLS) equation in the limit of small dispersion. Whitham's modulation theory is used to characterize the main feature of the solution in terms of the single-phase periodic solution of the nonlocal NLS equation with the slowly varying wave parameters. The modulation equations for these parameters are derived by averaging the local conservation laws. A novel feature of the modulation equations is that they can be decoupled into the form of the integrable Hopf equation. An explicit example of the solution is exhibited for a step initial condition.

The Wiener--Hopf Technique for Impenetrable Wedges Having Arbitrary Aperture Angle

Diffraction by impenetrable wedges having arbitrary aperture angle is studied by means of the Wiener--Hopf (W-H) technique. A system of functional equations called generalized Wiener--Hopf equations (GWHE) is obtained. Only for certain values of the aperture angle are these equations recognizable as standard or classical Wiener--Hopf equations (CWHE). However, in all cases a mapping is found that reduces the GWHE to CWHE. It means that the diffraction by an impenetrable wedge always reduces to a standard W-H factorization. The solution for the diffraction by a wedge with given face impedances (the Malyuzhinets problem) is obtained in closed form by an explicit factorization of the kernel.

A Two-Dimensional Polynomial Filter

The criterion of minimal variance of transformation error is applied to design an nth-order nonlinear polynomial filter for image processing under given moments of order up to 2n of the two-dimensional input signal and additive disturbance. Structurally, the filter is represented by Hammerstein kernels (weight functions) determined from the solution of a system of two-dimensional linear integral equations. For a linear filter, this system is reduced to a two-dimensional Wiener–Hopf equation. The filter accuracy is shown to increase with the filter order n (in any case, it does not decrease). The filter performance is illustrated for practical operation conditions of information transmission channels.

Functionals with values in the non‐archimedean field of laurent series and their applications to the equations of elasticity theory. I 1

Abstract Functionals with values in Non‐Archimedean field of Laurent series applied to the definition of generalized solution (in the form of soliton) of the Hopf equation. Calculation method for the profile of infinitely narrow soliton is proposed. Applying this method, calculations of profiles are reduced to the nonlinear system of algebraic equations in R n+1, n > 1. It is shown that there is a possibility to find out some of the solutions of this system using the Newton iteration method. Example and numerical test are considered.

FUNCTIONALS WITH VALUES IN THE NON‐ARCHIMEDEAN FIELD OF LAURENT SERIES AND THEIR APPLICATIONS TO THE EQUATIONS OF ELASTICITY THEORY. I

Functionals with values in Non‐Archimedean field of Laurent series applied to the definition of generalized solution (in the form of soliton) of the Hopf equation. Calculation method for the profile of infinitely narrow soliton is proposed. Applying this method, calculations of profiles are reduced to the nonlinear system of algebraic equations in R n+1, n > 1. It is shown that there is a possibility to find out some of the solutions of this system using the Newton iteration method. Example and numerical test are considered.

Some new projection methods for variational inequalities

In this paper, we propose some new double-projection methods for solving variational inequalities by using the Wiener–Hopf equations technique. It is shown that these methods converge linearly under mild conditions and include some existing projection methods as special cases. Some examples are given to illustrate the efficiency of the proposed methods.

Diffraction by systems of finite strips

The talk is devoted to wave diffraction problems for the two-dimensional Helmholtz equation. Boundary-transmission problems for a union of intervals (i.e., strips of a negligible thickness) which lie on a system of one or several parallel lines are considered in a Sobolev space setting. These problems include the mixed Dirichlet/Neumann and impedance boundary and transmission problems. Such problems for a union of intervals are reduced to equivalent systems of Wiener–Hopf equations on Lebesgue spaces over the positive real semiaxis. The matrix symbols of corresponding Wiener–Hopf operators are semialmost periodic, that is, they are continuous on the real line and have, in general, different almost periodic behavior at semineighborhoods of infinity. The solvability of obtained systems of Wiener–Hopf equations is studied on the basis of factorization techniques for oscillating and sectorial matrix functions and a corona problem approach developed in M. A. Bastos, Yu. I. Karlovich, and A. F. dos Santos [Integr. Eq. Oper. Th. 42, 22–56 (2002)], A. Bottcher, Yu. I. Karlovich, and I. M. Spitkovsky [Convolution Operators and Factorization of Almost Periodic Matrix Functions (Birkhauser-Verlag, Basel, 2002)], and L. P. Castro and F.-O. Speck [Integr. Equat. Oper. Th. 37, 169–207 (2000)].

On the application of the Wiener–Hopf technique to problems in dynamic elasticity

Many problems in linear elastodynamics, or dynamic fracture mechanics, can be reduced to Wiener–Hopf functional equations defined in a strip in a complex transform plane. Apart from a few special cases, the inherent coupling between shear and compressional body motions gives rise to coupled systems of equations, and so the resulting Wiener–Hopf kernels are of matrix form. The key step in the solution of a Wiener–Hopf equation, which is to decompose the kernel into a product of two factors with particular analyticity properties, can be accomplished explicitly for scalar kernels. However, apart from special matrices which yield commutative factorizations, no procedure has yet been devised to factorize exactly general matrix kernels. This paper shall demonstrate, by way of example, that the Wiener–Hopf approximant matrix (WHAM) procedure for obtaining approximate factors of matrix kernels (recently introduced by the author in [SIAM J. Appl. Math. 57 (2) (1997) 541]) is applicable to the class of matrix kernels found in elasticity, and in particular to problems in QNDE. First, as a motivating example, the kernel arising in the model of diffraction of skew incident elastic waves on a semi-infinite crack in an isotropic elastic space is studied. This was first examined in a seminal work by Achenbach and Gautesen [J. Acoust. Soc. Am. 61 (2) (1977) 413] and here three methods are offered for deriving distinct non-commutative factorizations of the kernel. Second, the WHAM method is employed to factorize the matrix kernel arising in the problem of radiation into an elastic half-space with mixed boundary conditions on its face. Third, brief mention is made of kernel factorization related to the problems of flexural wave diffraction by a crack in a thin (Mindlin) plate, and body wave scattering by an interfacial crack.

Exact Solution of a System of Generalized Hopf Equations

In this paper we construct explicit solutions for initial value problem for a system of first order equations. When n = 1 , this system is just the standard Hopf equation in conservative form. When n > 1 , the system is non-conservative. We use the vanishing viscosity method to construct solutions. As the system is non-conservative we use Volpert product and the algebra of generalized Colombeau functions to make sense of the products which appear in the equations.

Implicit dynamical systems and quasi variational inequalities

In this paper, we propose and analyze some implicit projected dynamical systems associated with quasi variational inequalities by using the techniques of the projection and the Wiener–Hopf equations. We prove that the globally asymptotic stability of these dynamical systems requires only the pseudomonotonicity of the operator, which is a weaker condition than monotonicity. We also discuss some special cases, which can be obtained from our main results. The results obtained in this paper represent a significant improvement of the previously known results.

A priori tests of a stochastic mode reduction strategy

Several a priori tests of a systematic stochastic mode reduction procedure recently devised by the authors [Proc. Natl. Acad. Sci. 96 (1999) 14687; Commun. Pure Appl. Math. 54 (2001) 891] are developed here. In this procedure, reduced stochastic equations for a smaller collections of resolved variables are derived systematically for complex nonlinear systems with many degrees of freedom and a large collection of unresolved variables. While the above approach is mathematically rigorous in the limit when the ratio of correlation times between the resolved and the unresolved variables is arbitrary small, it is shown here on a systematic hierarchy of models that this ratio can be surprisingly big. Typically, the systematic reduced stochastic modeling yields quantitatively realistic dynamics for ratios as large as 1/2. The examples studied here vary from instructive stochastic triad models to prototype complex systems with many degrees of freedom utilizing the truncated Burgers–Hopf equations as a nonlinear heat bath. Systematic quantitative tests for the stochastic modeling procedure are developed here which involve the stationary distribution and the two-time correlations for the second and fourth moments including the resolved variables and the energy in the resolved variables. In an important illustrative example presented here, the nonlinear original system involves 102 degrees of freedom and the reduced stochastic model predicted by the theory for two resolved variables involves both nonlinear interaction and multiplicative noises. Even for large value of the correlation time ratio of the order of 1/2, the reduced stochastic model with two degrees of freedom captures the essentially nonlinear and non-Gaussian statistics of the original nonlinear systems with 102 modes extremely well. Furthermore, it is shown here that the standard regression fitting of the second-order correlations alone fails to reproduce the nonlinear stochastic dynamics in this example.

Exact solutions for some non-conservative hyperbolic systems

In this paper, we study initial value problem for some non-conservative hyperbolic systems of partial differential equations of first order. The first one is the Riemann problem for a model in elastodynamics and the second one the initial value problem for a system which is a generalization of the Hopf equation. The non-conservative products which appear in the equations do not make sense in the classical theory of distributions and are understood in the sense of Volpert (Math. USSR Sb. 2 (1967) 225). Following Lax (Comm. Pure Appl. Math. 10 (1957) 537) and Dal Maso et al. (J. Math. Pures Appl. 74 (1995) 483), we give an explicit solution for the Riemann problem for the elastodynamics equation. The coupled Hopf equation is studied using a generalization of the method of Hopf (Comm. Pure Appl. Math. 3 (1950) 201).