Rayleigh-Ritz Principle Research Articles

Validation of Rayleigh-Ritz Deflection Equation for a Tapered Cantilever Beam using ANSYS Finite Element Software

Models are like our eyes and ears as they provide the basis for making very important design decisions in engineering. They give predictions that help engineers to foretell or foresee a system’s behaviour in real life. Validation ensures that these models give proper or accurate representation of the real system. In this study, a model equation derived for predicting the end deflection of a tapered width, constant height cantilever beam was adopted for validation. This equation was derived using the Finite Element Method's Rayleigh-Ritz principle and the virtual work concept. Using a finite element model developed in ANSYS as the representation of the real system and a basis for validation, this study aimed to validate the equation. Validation was achieved by the comparison of the load-deflection curve, the height-deflection curve and the end deflection of the model equation to that of the finite element model in ANSYS. Though the model equation was verified to really foretell deflection, the predictions of the equation was not in good agreement with that of the finite element model. The accuracy of the equation dwindled as the parameters of the tapered cantilever beam were varied. It gave an average percentage error of 30% higher than the allowable error of 5%. Though the equation was derived using the appropriate fundamental numerical principles as it basis, it’s accuracy can be exponentially increased by modifying it to include a larger number of discretized elements and a higher degree of shape/interpolation function.

Particle in a box inPT-symmetric quantum mechanics and an electromagnetic analog

In $\mathcal{PT}$-symmetric quantum mechanics a fundamental principle of quantum mechanics, that the Hamiltonian must be Hermitian, is replaced by another set of requirements, including notably symmetry under $\mathcal{PT}$, where $\mathcal{P}$ denotes parity and $\mathcal{T}$ denotes time reversal. Here we study the role of boundary conditions in $\mathcal{PT}$-symmetric quantum mechanics by constructing a simple model that is the $\mathcal{PT}$-symmetric analog of a particle in a box. The model has the usual particle-in-a-box Hamiltonian but boundary conditions that respect $\mathcal{PT}$ symmetry rather than Hermiticity. We find that for a broad class of $\mathcal{PT}$-symmetric boundary conditions the model respects the condition of unbroken $\mathcal{PT}$ symmetry, namely, that the Hamiltonian and the symmetry operator $\mathcal{PT}$ have simultaneous eigenfunctions, implying that the energy eigenvalues are real. We also find that the Hamiltonian is self-adjoint under the $\mathcal{PT}$-symmetric inner product. Thus we obtain a simple soluble model that fulfills all the requirements of $\mathcal{PT}$-symmetric quantum mechanics. In the second part of this paper we formulate a variational principle for $\mathcal{PT}$-symmetric quantum mechanics that is the analog of the textbook Rayleigh-Ritz principle. Finally we consider electromagnetic analogs of the $\mathcal{PT}$-symmetric particle in a box. We show that the isolated particle in a box may be realized as a Fabry-Perot cavity between an absorbing medium and its conjugate gain medium. Coupling the cavity to an external continuum of incoming and outgoing states turns the energy levels of the box into sharp resonances. Remarkably we find that the resonances have a Breit-Wigner line shape in transmission and a Fano line shape in reflection; by contrast, in the corresponding Hermitian case the line shapes always have a Breit-Wigner form in both transmission and reflection.

A theoretical study on shape control of arbitrary lay-up laminates using piezoelectric actuators

Abstract This paper investigates the application of a piezoelectric actuator to control deformation of thin arbitrary lay-up composite laminates. A theoretical model based on Rayleigh–Ritz principle is developed to predict laminate cured shape and the effect of piezoelectric actuator layer. Several stacking sequences are studied using the model. When applied to pure composite laminates it is observed that cured shape of near-unidirectional laminates is always saddle. When applied to arbitrary lay-up piezo-composite laminates, it is found that the principal direction of the laminates can change with electric field. This is one unique feature of such type of laminates. A new FEA technique is proposed to deal with the multiple solution issue and obtain the stable solution more precisely.

Theoretical study on shape control of thin cross-ply laminates using piezoelectric actuators

This paper investigates the application of a piezoelectric actuator to control deformation of thin unsymmetric cross-ply composite laminates . A theoretical model based on Rayleigh–Ritz principle is developed to predict laminate deformation and the effect of the piezoelectric actuator layer. Several stacking sequences are studied using the model. It is found that two stable shapes still exist for laminates with larger dimensions but the two shapes are not symmetric as in traditional cross-ply laminates. The bifurcation dimension increases when the thickness of piezoelectric layer increases. Laminate curvatures are in a perfectly linear relationship with an electric field applied to the piezoelectric actuator layer. Snap-through from one stable shape to the other can occur by applying an electric field. FEA results match model prediction fairly well. This model can be used as a guide to design piezocomposite laminates.

A Model for Shape Control of Cross-ply Laminated Shells using a Piezoelectric Actuator

This article investigates the technique of using a piezoelectric (P) material as the actuator to control the deformation of thin unsymmetric cross-ply laminated shells. A theoretical model is developed using the Rayleigh–Ritz principle to predict the effect of piezoelectric layer on laminated shell deformation. Several stacking sequences are investigated. Owing to the curvature of the shell, the position of the piezoelectric layer has a great effect on the deformation. The effect of the P-layer as the outer surface or inner surface is investigated. It is found that to achieve the same amount of deformation, different levels of electric fields have to be applied depending on the P-layer configuration. The efficiency of the P-layer on shells with various stacking sequences is also studied and it is observed that the efficiency is determined by the stacking sequence of the composite layers. This model can be used as a guide to design the piezocomposite laminated shell.

Boundary conditions for augmented plane-wave methods

The augmented plane wave method uses the Rayleigh-Ritz principle for basis functions that are continuous but with discontinuous derivatives and the kinetic energy is written as a pair of gradients rather than as a Laplacian. It is shown here that this procedure is fully justified from the mathematical point of view. The domain of the self-adjoint Hamiltonian, which does not contain functions with discontinuous derivatives, is extended to its form domain, which contains them, and this modifies the form of the kinetic energy. Moreover, it is argued that discontinuous basis functions should be avoided.

A unified approach to universal inequalities for eigenvalues of elliptic operators

We present an abstract approach to universal inequalities for the discrete spectrum of a self-adjoint operator, based on commutator algebra, the Rayleigh-Ritz principle, and one set of operators. The new proof uni.es classical inequalities of Payne-P?olya-Weinberger, Hile-Protter, and H.C. Yang and provides a Yang type strengthening of Hook's bounds for various elliptic operators with Dirichlet boundary conditions. The proof avoids the introduction of the free parameters of many previous authors and relies on earlier works of Ashbaugh and Benguria, and, especially, Harrell (alone and with Michel), in addition to those of the other authors listed above. The Yang type inequality is proved to be stronger under general conditions on the operator and the auxiliary operators. This approach provides an alternative route to recent results obtained by Harrell and Stubbe.

Alternatives to multireference methods for the molecular electronic structure problem

AbstractThe development of robust many‐body methods for the molecular electronic structure problem with respect to multireference functions has attracted much attention over the past two decades. In recent years, multireference methods based on the Brillouin–Wigner expansion have been shown to overcome the intruder state problems which have plagued similar approaches based on the Rayleigh–Schrödinger series. However, there are some problems in molecular electronic structure theory, which may be handled by means of multireference methods, but can be treated by methods that avoid the use of multireference functions. We consider two examples of alternatives to multireference methods for the molecular electronic structure problem. The study of excited states having the same symmetry as the ground state or some lower lying excited state often involves the use of a multireference function. We describe an alternative procedure based on the generalized Rayleigh–Ritz variation principle. Gidopoulos, Glushkov, and Wilson have termed this the optimized trace method. The generalized Rayleigh–Ritz principle for the relativistic formulation of the molecular electronic structure problem is briefly also considered. Molecular dissociative processes are frequently described by quantum chemical methods based on multireference functions. Single‐reference Hartree–Fock methods usually provide a qualitatively incorrect description of such processes even in the simplest of molecules, H2. However, approximate single configuration wave functions constructed from nonorthogonal orbitals can often afford a useful approximation to bond‐breaking processes. Each occupied nonorthogonal orbital is then an eigenfunction of a different Fock‐like operator. Each of these Fock‐like operators supports a spectrum of M single particle states, where M is the size of the basis set, and only the lowest of these is occupied. We briefly consider the relativistic formulation of the molecular electronic structure problem based on approximations involving products of nonorthogonal functions within the Furry bound state interaction picture of quantum electrodynamics. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Local variational principle

A generalization of the Gibbs-Bogoliubov-Feynman inequality for spinless particles is proven and then illustrated for the simple model of a symmetric double-well quartic potential. The method gives a pointwise lower bound for the finite-temperature density matrix and it can be systematically improved by the Trotter composition rule. It is also shown to produce ground state energies better than the ones given by the Rayleigh-Ritz principle as applied to the ground state eigenfunctions of the reference potentials. Based on this observation, it is argued that the local variational principle performs better than the equivalent methods based on the centroid path idea and on the Gibbs-Bogoliubov-Feynman variational principle, especially in the range of low temperatures.

Substructure synthesis method for simulating large molecular complexes.

This paper reports a computational method for describing the conformational flexibility of very large biomolecular complexes using a reduced number of degrees of freedom. It is called the substructure synthesis method, and the basic concept is to treat the motions of a given structure as a collection of those of an assemblage of substructures. The choice of substructures is arbitrary and sometimes quite natural, such as domains, subunits, or even large segments of biomolecular complexes. To start, a group of low-frequency substructure modes is determined, for instance by normal mode analysis, to represent the motions of the substructure. Next, a desired number of substructures are joined together by a set of constraints to enforce geometric compatibility at the interface of adjacent substructures, and the modes for the assembled structure can then be synthesized from the substructure modes by applying the Rayleigh-Ritz principle. Such a procedure is computationally much more desirable than solving the full eigenvalue problem for the whole assembled structure. Furthermore, to show the applicability to biomolecular complexes, the method is used to study F-actin, a large filamentous molecular complex involved in many cellular functions. The results demonstrate that the method is capable of studying the motions of very large molecular complexes that are otherwise completely beyond the reach of any conventional methods.

FREE OUT-OF-PLANE VIBRATIONS OF A ROTATING BEAM WITH NON-LINEAR ELASTOMERIC CONSTRAINTS

Free out-of-plane vibration of a rotating beam with a non-linear, elastomeric constraint has been investigated. The elastomer is modelled as a parallel combination of spring and damper elements. The stiffness and damping parameters are evaluated by using the previous experimental data. The linear analysis is performed by two techniques, one based on a power series expansion and the other based on the Rayleigh-Ritz principle. The results are found to be in excellent agreement. In the non-linear analysis, a numerical–perturbation technique is applied to determine the frequency-amplitude relationship. A parametric study revealing the influence of the non-linear constraint is presented.

Free vibration of composite cylindrical polygonal ducts

A simple numerical method using the Rayleigh-Ritz principle is presented for computing the natural frequencies and mode shapes of cylindrical polygonal ducts constructed from composite materials. The duct creases are assumed to be parallel to one of the principal axes of orthotropy. Numerical results are presented for triangular and square cylindrical ducts with various degrees of orthotropy.

Time-dependent variational approach for two-particle systems: photoionization of He (1s2)

An extension of the standard time-dependent Hartree-Fock (TDHF) approach to the problem of sequential multiple ionization of He by intense laser pulses is developed. The method is based on the TD Rayleigh-Ritz principle and employs a TD product wavefunction in coordinate space for He(1s, 1s'). While the method is comparable in effort to TDHF, it incorporates some additional two-particle correlation effects. Numerical results for high-frequency ionization of helium show deviations from TDHF results at moderate intensities, but, as in TDHF, simultaneous populations of He+ and He2+ according to binomial statistics are predicted in very strong fields. At optical laser frequencies and strong fields the results indicate a definite trend towards the sequential ionization mechanism suggested by experiment, but do not reproduce this effect fully.

Density-functional theory for ensembles of fractionally occupied states. I. Basic formalism.

A density-functional theory for ensembles of unequally weighted states is formulated on the basis of the generalized Rayleigh-Ritz principle of the preceding paper. From this formalism, two alternative approaches to the computation of excitation energies are derived, one equivalent to the equiensemble method proposed by Theophilou [J. Phys. C 12, 5419 (1979)], the other grounded on an expression relating the excitation energies to the Kohn-Sham single-particle eigenvalues.

Rayleigh-Ritz variational principle for ensembles of fractionally occupied states.

The Rayleigh-Ritz minimization principle is generalized to ensembles of unequally weighted states. Given the M lowest eigenvalues ${E}_{1}$\ensuremath{\le}${E}_{2}$\ensuremath{\le}...\ensuremath{\le}${E}_{M}$ of a Hamiltonian H, and given M real numbers ${w}_{1}$\ensuremath{\ge}${w}_{2}$\ensuremath{\ge}...\ensuremath{\ge}${w}_{M}$g0, an upper bound for the weighted sum ${w}_{1}$${E}_{1}$ +${w}_{2}$${E}_{2}$+...+${w}_{M}$${E}_{M}$ is established. Particular cases are the ground-state Rayleigh-Ritz principle (M=1) and the variational principle for equiensembles (${w}_{1}$=${w}_{2}$=...=${w}_{M}$). Applications of the generalized principle are discussed.

Time-dependent Hartree-Fock theory and the Rayleigh-Ritz principle

The trial function $\ensuremath{\Psi}$ in the Rayleigh-Ritz variational principle, $\ensuremath{\delta}〈\ensuremath{\Psi}|(H\ensuremath{-}E)|\ensuremath{\Psi}〉=0$, is restricted to a continuous superposition of Slater determinants $\ensuremath{\Phi}(t)$, $\ensuremath{\Psi}=\ensuremath{\int}\mathrm{dtf}(t)\ensuremath{\Phi}(t)$. The variation $\ensuremath{\delta}$ is performed upon the path {$\ensuremath{\Phi}(t)$}. It is found that time-dependent Hartree-Fock solutions $\ensuremath{\Phi}(t)$ are optimal choices for making the Rayleigh-Ritz functional stationary.NUCLEAR STRUCTURE TDHF paths best choices for generator coordinate method with Rayleigh-Ritz functional.

Complement to the Rayleigh-Ritz principle for the energy spectrum of a system composed of identical particles

A theorem is proved which states that the energies E/sub i/ of a general translation-invariant system of N identical particles interacting by pair potentials and obeying nonrelativistic quantum mechanics are bounded below one by one by the energies E/sup L//sub i/ of a related model system consisting of (N-1) noninteracting particles of the same symmetry type bound to a fixed center. The theorem is applied to a three-fermion harmonic oscillator test system whose exact eigenvalues are found. The ratios E/sup L//sub i//E/sub i/ for this problem satisfy E/sup L//sub i//E/sub i/ > or = E/sup L//sub 0//E/sub 0/ = ..sqrt..3/4 for at least the first twenty three-body states.

Subsidiary minimum principles for scattering parameters

We denote as a minimum one in which a quantity $B$ of physical interest is represented as the minimum value with respect to variations in a trial function ${\ensuremath{\psi}}_{t}$ of a functional $F({\ensuremath{\psi}}_{t})$; $F$ then provides a variational upper bound on $B$. (The Rayleigh-Ritz principle for the ground-state energy of a system is a familiar example.) If $F$ is quadratic in ${\ensuremath{\psi}}_{t}$, the variational property of $F$ enables one to determine the linear parameters relatively easily, but the minimum property is required if the nonlinear parameters are to be determined in a way which allows for systematic improvement of ${\ensuremath{\psi}}_{t}$. We show here that for a wide class of problems for which primary minimum do not exist, useful and rigorous secondary or minimum principles are available. That is, we construct a functional ${F}^{\ensuremath{'}}({\ensuremath{\psi}}_{t})$ whose minimum value is reached for ${\ensuremath{\psi}}_{t}$ equal to some function $\ensuremath{\psi}$ of dynamical interest. (The Rayleigh-Ritz method provides a subsidiary minimum principle for the approximate determination of the ground-state wave function of a system.) If $B=B(\ensuremath{\psi})$, then a study of ${F}^{\ensuremath{'}}({\ensuremath{\psi}}_{t})$ provides a powerful tool for the estimation of $\ensuremath{\psi}$ and therefore $B$, though $B({\ensuremath{\psi}}_{t})$ is not normally a variational bound on $B(\ensuremath{\psi})$. Subsidiary minimum have recently been obtained for the approximation of the auxiliary functions that appear in the variational principle for the matrix element (${\ensuremath{\chi}}_{n}$, $W{\ensuremath{\chi}}_{m}$), where ${\ensuremath{\chi}}_{n}$ and ${\ensuremath{\chi}}_{m}$ are bound-state wave functions and $W$ is an arbitrary operator. Here we extend the method to the estimation of matrix elements of the Green's function $g(\ensuremath{\epsilon})$ of a bound system with $\ensuremath{\epsilon}$ below the continuum threshold energy. The response of the system to an external perturbation can be represented by matrix elements of this type. While no new results on the bound-state problem are obtained, our formulation is a convenient starting point for the further extension of the method to continuum problems. The new result obtained here is the derivation of a subsidiary minimum principle for the problem of scattering of a projectile by a target whose bound-state wave function is only imprecisely known. The subsidiary minimum principle allows for systematic improvement of the closed-channel component of the trial scattering wave function that appears in a Kohn-type variational calculation of the scattering amplitude.

Studies in the accuracy of some lower-bound formulae-a simple one-dimensional problem

A comparative study is made of several lower-bound procedures for calculating the eigenvalues of a simple exactly soluble one-dimensional potential, when this potential V(x) is regarded as the sum of two harmonic-oscillator potentials 1/2(R-x)2 and 1/2(R+x)2 centred at x=+or-R. This potential resembles a molecular potential, the wavefunctions for which are to be represented as combinations of the 'atomic' wavefunctions of the two separate potentials 1/2(R+or-x)2. The resulting LCAO-type wavefunctions are chosen to give lower bounds according to the classical methods of Weinstein and Stevenson (1938) and the truncated hamiltonian method of Bazley and Fox (1963); upper bounds are also obtained according to the Rayleigh-Ritz principle. The calculations are made for different values of R. Acceptable energies can be calculated by all methods. Of the methods considered the Rayleigh-Ritz upper-bound method converges most rapidly; the Stevenson lower bound is best if the optimum value of a certain parameter is known; the Weinstein method gives the poorest convergence of all.

A six-dimensional quadrature procedure

An account is given of the use of Gaussian quadrature product formulae in the evaluation of certain six-dimensional, two-centre integrals involving one-electron Green's functions. These integrals occur in a new molecular variational principle recently proposed by Hall, Hyslop and Rees [1] from which an approximate energy may be derived which can be shown to be at least as good as that obtained from the Rayleigh-Ritz principle. Reductions in computing time are realized by removing certain singularities using a subtraction technique and also by using an empirically determined Richardson-type extrapolation formula.