A Hamiltonian state space approach to anisotropic elasticity and piezoelasticity

We present a Hamiltonian state space approach for problems of anisotropic elasticity and piezoelasticity. By means of Legendre’s transformation, the basic equations of piezoelasticity are formulated into a state equation and an output equation in terms of the state vector that comprises the generalized displacement vector and the conjugate generalized traction vector as the dual variables. The Hamiltonian features and symplectic orthogonality of the system, which are essential for the solution approach using eigenfunction expansion, are delineated at length. We show that the solution to 3D problems of a prismatic body hinges upon a 2D Hamiltonian eigensystem and the eigensolution associated with the zero eigenvalue leads to the solution to the generalized plane problem naturally. Based on the formalism, the solution to a problem of piezoelasticity is no more difficult than its elastic counterpart.     

Chaotic analyses of weakly damped parametrically excited cross waves with surface tension

Summary. The Wiggins-Holmes extension of the Generalized Melnikov Method (GMM) to higher dimensions and the extension of the Generalized Herglotz Algorithm (GHA) to non-autonomous systems are applied to weakly damped parametrically excited cross waves with surface tension in a long rectangular wave channel in order to demonstrate that cross waves are chaotic. The Luke Lagrangian density function for surface gravity waves with surface tension and dissipation is expressed in three generalized coordinates (or, equivalently, three degrees of freedom) that are the time-dependent components of three velocity potentials that represent three standing waves. The generalized momenta are computed from the Lagrangian, and the Hamiltonian is computed from a Legendre transform of the Lagrangian. This Hamiltonian contains both autonomous and non-autonomous components that must be suspended by applying an extension of the Herglotz algorithm for non-autonomous transformations in order to apply the Kolmogorov-Arnold-Moser (KAM) averaging operation and the GMM. Three canonical transformations are applied to (i) eliminate cross product terms by a rotation of axes; (ii) to transform to action-angle canonical variables and to eliminate two degrees of freedom; and (iii) to suspend the non-autonomous terms and to apply the Hamilton-Jacobi transformation. The system of nonlinear non-autonomous evolution equations determined from Hamiltons equations of motion of the second kind must be averaged in order to obtain an autonomous system that may be analyzed by the GMM. Hyperbolic saddle points that are connected by heteroclinic separatrices are computed from the unperturbed autonomous system. The non-dissipative perturbed Hamiltonian system with surface tension satisfies the KAM non-degeneracy requirements, and the Melnikov integral is calculated to demonstrate that the motion is chaotic. For the perturbed dissipative system with surface tension, the only hyperbolic fixed point that survives the averaged equations is a fixed point of weak chaos that is not connected by a homoclinic separatrix; consequently, the Melnikov integral is identically zero. The chaotic motion for the perturbed dissipative system with surface tension is demonstrated by numerical computation of positive Liapunov characteristic exponents.     

A perturbation solution for the steady heat flow around a thin circular insulating annulus on the surface of a conducting half-space

A constant temperature circular disc on the surface of a conducting half-space is surrounded by an insulating annulus. The remainder of the surface of the half space is maintained at zero temperature. The steady heat flow rate from the disc to the zero temperature surface is required. The mathematical problem, a three-part mixed boundary value problem can be reduced to integral equation form. Several alternative formulations are possible. The existing formulations do not readily yield solutions for the case in which the thickness-to-radius radio of the insulating annulus is small compared to unity. This case is considered here and a solution based on integral perturbation methods is obtained. An existing alternative integral equation formulation is also solved, by iteration, for cases in which the inner-to-outer radius ratio of the insulating annulus is small compared to unity. These two solutions are found to coalesce over an intermediate range of annulus thickness. Thus a composite solution is obtained which is valid for all cases. The solutions given here are also of practical interest in analogous problems in, e.g. the flow of fluids through porous media, and in elasticity theory.     

基于哈密顿体系求解空间粘性流体问题

本文通过变分原理,将哈密顿体系引入到小雷诺数空间粘性流体问题中,导出一套哈密顿算子矩阵的本征函数向量展开求解问题的方法.基于直接法求解流体力学基本方程,通过求零本征解及其约当型,得到几种常见的基本流动;求解非零本征值及本征向量的叠加,继可分析流场端部效应.从而在该领域用哈密顿体系辛几何空间中研究问题的方法代替了传统在拉格朗日体系欧氏空间分析问题的方法.     

Nonlinear semiclassical dynamics of open systems

A semiclassical approximation for an evolving density operator, driven by a 'closed' Hamiltonian and 'open' Markovian Lindblad operators, is reviewed. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the Hamiltonian is a quadratic function and the Lindblad operators are linear functions of positions and momenta. The semiclassical formulae are interpreted within a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra 'open' term in the double Hamiltonian is generated by the non-Hermitian part of the Lindblad operators in the general case of dissipative Markovian evolution. Decoherence narrows the relevant region of double phase space to the neighbourhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by the definition of a propagator, here developed in both representations. Generalized asymptotic equilibrium solutions are thus presented for the first time.     

陀螺系统辛子空间迭代法

转子系统的有限元分析可以导出陀螺系统的本征值问题．而陀螺本征值问题可在哈密顿体系下求解。基于辛子空间迭代法的思想，提出了一种求解陀螺系统本征值问题的算法。首先引入对偶变量，将陀螺动力系统导入哈密顿体系，将问题化为了哈密顿矩阵的本征值问题。由于稳定的陀螺系统其本征值必为纯虚数，利用这个特点。提出了对应陀螺系统的辛子空问迭代法，从而可以求出系统任意阶的本征值及其振型。算例证明了这种算法的有效性。     

Fourier series for computing the response of periodic structures with arbitrary stiffness distribution

A method based on Fourier series is presented, which allows to calculate the local stress–strain response of a three-dimensional periodic structure subjected to a spatial average of strain. The periodicity allows the reduction of the problem to that of a Representative Volume Element (RVE). The solution operator (which can easily be calculated in Fourier space) is defined for a simplified problem, and it is shown that this operator may also be used for the original problem. In order to illustrate the use of this procedure, an example problem is presented. A global error is defined and calculated for the example problem.     

On the Euler equation appearing in the synthesis of radiating systems

A problem appearing when the Euler equation is used in the theory of antenna synthesis is considered. Behavior of a solution to this equation at the ends of the synthesis interval contradicts the Meixner conditions, according to which this solution must tend to zero as a square root of the distance to the ends. It is shown that this contradiction arises if we seek solutions in the L ₂ space and disappears if an L ₂ subspace with limited energy norm is selected as the space of solutions. In this case, the Euler equation arises from a variational minimization problem for a functional involving the norm of current determined in the energy space.     

平面问题混合状态Hamiltonian动力元

本文提出一种对动力学问题的Ｈａｍｉｌｔｏｎ正则方程进行正则变换的方法，并给出一种沿时间方向迭代，沿空间方向半离散半解析求解动力问题Ｈａｍｉｌｔｏｎ正则方程的方法──混合状态Ｈａｍｉｌｔｏｎｉａｎ动力元．     

The linear-wave response of a periodic array of floating elastic plates

The problem of an infinite periodic array of identical floating elastic plates subject to forcing from plane incident waves is considered. This study is motivated by the problem of trying to model wave propagation in the marginal ice zone, a region of ocean consisting of an arbitrary packing of floating ice sheets. It is shown that the problem considered can be formulated exactly in terms of the solution to an integral equation in a manner similar to that used for the problem of wave scattering by a single elastic floating plate, the key difference here being the use of a modified periodic Green function. The convergence of this Green function in its original form is poor, but can be accelerated by a transformation. It is shown that the results from the method satisfy energy conservation and that in the particular case of a fixed rigid rectangular plate which spans the periodicity uniformly the solution reduces to that for a two-dimensional rigid dock. Solutions for a range of elastic-plate geometries are also presented.     

A SPACE–TIME FINITE ELEMENT METHOD FOR THE EXTERIOR STRUCTURAL ACOUSTICS PROBLEM: TIME-DEPENDENT RADIATION BOUNDARY CONDITIONS IN TWO SPACE DIMENSIONS

A time-discontinuous Galerkin space–time finite element method is formulated for the exterior structural acoustics problem in two space dimensions. The problem is posed over a bounded computational domain with local time-dependent radiation (absorbing) boundary conditions applied to the fluid truncation boundary. Absorbing boundary conditions are incorporated as ‘natural’ boundary conditions in the space–time variational equation, i.e. they are enforced weakly in both space and time. Following Bayliss and Turkel, time-dependent radiation boundary conditions for the two-dimensional wave equation are developed from an asymptotic approximation to the exact solution in the frequency domain expressed in negative powers of a non-dimensional wavenumber. In this paper, we undertake a brief development of the time-dependent radiation boundary conditions, establishing their relationship to the exact impedance (Dirichlet-to-Neumann map) for the acoustic fluid, and characterize their accuracy when implemented in our space–time finite element formulation for transient structural acoustics. Stability estimates are reported together with an analysis of the positive form of the matrix problem emanating from the space–time variational equations for the coupled fluid-structure system. Several numerical simulations of transient radiation and scattering in two space dimensions are presented to demonstrate the effectiveness of the space–time method.     

Cracked-rectangular sheet with linearly varying end displacements

It is shown that the stress intensity factors for the end rotation problem for a cracked-rectangular sheet can be determined from a knowledge of the solution for a uniform normal displacement without rotation. The argument depends on the extension of the “weight function” concept to configurations in which the applied forces vary with respect to both the space variables and the crack length parameter. Numerical illustrations are presented for the problem of an edge crack in a rectangular sheet with linearly varying end displacements.     

Rigorous near- to far-field transformation for vectorial diffraction calculations and its numerical implementation

A rigorous method for transforming an electromagnetic near-field distribution to the far field is presented. We start by deriving a set of self-consistent integral equations that can be used to represent the electromagnetic field rigorously everywhere in homogeneous space apart from the closed interior of a volume encompassing all charges and sinks. The representation is derived by imposing a condition analogous to Sommerfeld's radiation condition. We then examine the accuracy of our numerical implementation of the formula, also on a parallel computer cluster, by comparing the results with a case when the analytical solution is also available. Finally, an application example is shown for a nonanalytical case.     

The boundary element-linear complementarity method for the Signorini problem

The boundary element-linear complementarity method for solving the Laplacian Signorini problem is presented in this paper. Both Green's formula and the fundamental solution of the Laplace equation have been used to solve the boundary integral equation. By imposing the Signorini constraints of the potential and its normal derivative on the boundary, the discrete integral equation can be written into a standard linear complementarity problem (LCP). In the LCP, the unique variable to be affected by the Signorini boundary constraints is the boundary potential variable. A projected successive over-relaxation (PSOR) iterative method is employed to solve the LCP, and some numerical results are presented to illustrate the efficiency of this method.     

Measurement of Helmholtz wave fields

A simple formalism is found for the measurement of wave fields that satisfy the Helmholtz equation in free space. This formalism turns out to be analogous to the well-known theory of measurements for quantum-mechanical wave functions: A measurement corresponds to the squared magnitude of the inner product (in a suitable Hilbert space) of the wave field and a field that is associated with the detector. The measurement can also be expressed as an overlap in phase space of a special form of the Wigner function that is tailored for Helmholtz wave fields.     

Fractional generalized Hamiltonian mechanics

In this paper, we present a new fractional theory of dynamics, i.e., the dynamics of generalized Hamiltonian system with fractional derivatives (fractional generalized Hamiltonian mechanics). Based on the definition of Riemann–Liouville fractional derivatives, the fractional generalized Hamiltonian equations are obtained, the gradient representation and second-order gradient representation of the fractional generalized Hamiltonian system are studied, and then the conditions on which the system can be considered as a gradient system and a second-order gradient system are given, respectively. By using the method and results of this paper, the conditions under which a fractional generalized Hamiltonian equation can be reduced to a generalized Hamiltonian equation, a fractional Hamiltonian equation and a Hamiltonian equation are given, respectively, and then the existing conditions and their form of gradient equation and second-order gradient equation are investigated. Finally, an example of a fractional dynamical system is given to illustrate the method and results of the application.     

The spin oscillations of 3He-B in the presence of dipole forces

The problem of the spin oscillations of ³He-B is investigated in the presence of dipole forces under the assumption that only the zeroth parameter of the Fermi-liquid interaction is significant. The solution of the gap equation gives the new thermodynamical phases. The spin susceptibility tensor is obtained in the most general form. It is shown that in addition to the longitudinal and transverse components some extra circular components can appear. These components are analogous to the components of the dielectric tensor that are responsible for the circular dichroism of optically active substances.     

The Quantum Approximate Algorithm for Solving Traveling Salesman Problem

The Quantum Approximate Optimization Algorithm (QAOA) is an algorithmic framework for finding approximate solutions to combinatorial optimization problems. It consists of interleaved unitary transformations induced by two operators labelled the mixing and problem Hamiltonians. To fit this framework, one needs to transform the original problem into a suitable form and embed it into these two Hamiltonians. In this paper, for the well-known NP-hard Traveling Salesman Problem (TSP), we encode its constraints into the mixing Hamiltonian rather than the conventional approach of adding penalty terms to the problem Hamiltonian. Moreover, we map edges (routes) connecting each pair of cities to qubits, which decreases the search space significantly in comparison to other approaches. As a result, our method can achieve a higher probability for the shortest round-trip route with only half the number of qubits consumed compared to IBM Q’s approach. We argue the formalization approach presented in this paper would lead to a generalized framework for finding, in the context of QAOA, high-quality approximate solutions to NP optimization problems.     

Hamiltonian partial mixed finite element-state space symplectic semi-analytical approach for the piezoelectric smart composites and FGM analysis

A partial mixed finite element (FE)–state space method (SSM) semi-analytical approach is presented for the static analysis of piezoelectric smart laminate composite and functionally graded material (FGM) plates. Hence, using the Hamiltonian formalism, the three-dimensional piezoelectricity equations are first worked so that a partial mixed variational formulation, which retains the translational displacements, electric potential, transverse stresses, and transverse electric displacement as primary variables, is obtained; this allows, in particular, straightforward fulfillment of the electromechanical continuity constraints at the laminate interfaces. After an in-plane FE discretization only, the problem is first reduced, for a single layer, to a Hamiltonian eigenvalue problem that is solved using the symplectic approach; then, the multilayer solution is reached via the SSM propagator matrix. The proposed methodology is finally applied to the static analysis of piezoelectric-cross-ply hybrid laminated composite and FGM plates. In a comparison with open literature, available tabulated results show good agreements, thus validating the proposed approach.     

A numerical solution for the symmetric melting of spheres

A numerical method is presented for the solution of the radially symmetric heat conduction problem in a melting sphere. The method employs the embedding technique; this permits the solution to be written in the form of an ordinary integro-differential equation which is readily solved numerically by means of a forward integration scheme. The accuracy of the method is briefly discussed and numerical results for both constant and variable heat inputs are presented.