Ritz Approximations Research Articles

Recycling of solution spaces in multipreconditioned FETI methods applied to structural dynamics

SummaryThis article presents a new method to recycle the solution space of an adaptive multipreconditioned finite element tearing and interconnecting algorithm in the case where the same operator is solved for multiple right‐hand sides like in linear structural dynamics. It accelerates the computation from the second time step on by applying a coarse space that is generated from Ritz approximations of local eigenproblems, using the solution space of the first time step. These eigenproblems are known to provide very efficient coarse spaces but must usually be solved a priori at high computational cost. Their Ritz approximations are much smaller and less expensive to solve. Recycling methods based on Ritz approximations of global eigenproblems have been published for classical finite element tearing and interconnecting algorithms, but their efficient application to multipreconditioned variants is not possible. This article also presents the application of a simpler recycling procedure, which reuses plain solution spaces, to adaptive multipreconditioned finite element tearing and interconnecting. Numerical results of the application of the presented methods to four test cases are shown. The new Ritz approximation method leads to coarse spaces, which turn out to be as efficient as those obtained from solving the unreduced eigenproblems. It is the most efficient recycling method currently available for multipreconditioned dual domain decomposition techniques.

An efficient 3D numerical beam model based on cross sectional analysis and Ritz approximations

A 3D beam model, i.e. a beam that may deform in space and experience longitudinal and torsional deformations, is developed considering Timoshenko's theory for bending and assuming that the cross section rotates as a rigid body but may deform in longitudinal direction due to warping. The cross sectional properties are firstly calculated and then inserted at the equation of motion. The beam is assumed to be with an arbitrary cross section, with linearly varying thickness and width, and with an initial twist. The model is appropriate for open and closed thin-walled cross sections, and also for solid cross sections. The objective of the current research is to demonstrate that complex beam structures can be modeled accurately with reduced number of degrees of freedom.

Computation of Creep in an Axially Moving Beam

AbstractTwo mechanical models for an axially moving beam supported by discrete elastic springs are derived, analysed and compared. It is considered that the beam has a constant thickness, a predefined temperature distribution and a given transversal load. The first beam model applies global Ritz approximations for the spans with springs. Gauss integration points are used in longitudinal and transverse direction. The second beam model uses a conventional FE description. The thermal and pressure loads cause stresses, which lead to strain rates due to creep of the material so that a time dependent deformation between the elastic supports results. Here the axial motion is considered to be so slow so that inertia effects can be neglected. The axial transport of the inelastic creep‐strains has been considered in the computation. The temperature field is constant and fixed in space. The deflection of the axially moving beam model is computed considering a variable number of spans separated by elastic springs. The computed results of the two mechanical models are compared for a beam with two spans. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the thermally induced bistability of composite cylindrical shells for morphing structures

The multistability of composite thin structures has shown potential for morphing applications. The present work combines a Ritz model with path-following algorithms to study bistable cylindrical panels. Polynomial discretisations of the displacements field are used to predict stable shapes’ geometry and other aspects of the nonlinear structural behaviour. In order to improve the inherently poor conditioning properties of Ritz approximations of slender structures, a non-dimensional Shell Lamination Theory with Sanders nonlinear strains is developed and presented. An investigation on the relative importance of different nonlinear strain terms is shown to provide useful insight into the applicability of common assumptions about shell kinematics. In the current approach, we continue numerical solutions in parameter space, that is, we path-follow equilibrium configurations as the control parameter varies, find stable and unstable configurations and identify bifurcations. The numerics is carried out using a set of in-house M atlab® routines for numerical continuation. Results are compared with detailed finite elements analysis throughout the course of the paper.

The Rayleigh–Ritz method, refinement and Arnoldi process for periodic matrix pairs

We extend the Rayleigh–Ritz method to the eigen-problem of periodic matrix pairs. Assuming that the deviations of the desired periodic eigenvectors from the corresponding periodic subspaces tend to zero, we show that there exist periodic Ritz values that converge to the desired periodic eigenvalues unconditionally, yet the periodic Ritz vectors may fail to converge. To overcome this potential problem, we minimize residuals formed with periodic Ritz values to produce the refined periodic Ritz vectors, which converge under the same assumption. These results generalize the corresponding well-known ones for Rayleigh–Ritz approximations and their refinement for non-periodic eigen-problems. In addition, we consider a periodic Arnoldi process which is particularly efficient when coupled with the Rayleigh–Ritz method with refinement. The numerical results illustrate that the refinement procedure produces excellent approximations to the original periodic eigenvectors.

Bistable plates for morphing structures: A refined analytical approach with high-order polynomials

The multistability of composite thin structures has shown potential for morphing applications. The present work combines a Ritz model with path-following algorithms to study bistable plates’ behaviour. Classic low-order Ritz models predict stable shapes’ geometry with reasonable accuracy. However, they may fail when modelling other aspects of the elastic structural behaviour. A refined higher-order model is here presented. In order to improve the inherently poor conditioning properties of Ritz approximations of slender structures, a non-dimensional version of Classical Plate Lamination Theory with von Kármán nonlinear strains is developed and presented. In the current approach, we continue numerical solutions in parameter space, that is, we path-follow equilibrium configurations as the control parameter varies, find stable and unstable configurations and identify bifurcations. The numerics are carried out using a set of in-house Matlab® routines for numerical continuation. The increased degrees of freedom within the model are shown to accurately reflect buckling loads and provide useful insight into the relative importance of different aspects of nonlinear behaviour. Finally, the complex, experimentally observed snap-through geometry is captured analytically for the first time. Results are validated against finite elements analysis throughout the course of the paper.

Improving the performance of a piezoelectric energy harvester using a variable thicknessbeam

In recent years, researchers have shown a growing interest in the possibility of harvestingmechanical energy from vibrating structures. A common way to proceed consists ofusing the direct piezoelectric effect of a bimorph cantilever beam with integratedpiezoelectric elements. Several studies focused on the development of analytical modelsdescribing the electromechanical coupling. Historically, most of these models havebeen limited to simple structures such as a constant cross-section cantilever beamharvester. This paper studies the effect of a variable thickness beam harvester on itselectromechanical performance. A semi-analytical mechanical model was developed usingRayleigh–Ritz approximations with a trigonometric functions set. The model was nextvalidated by a finite element (FE) modeling. Numerical simulations were thenperformed for different beam slope angles in order to find the optimum for a givenmaximal strain across the piezoelectric elements. For the case under study, it isshown that tapered beams lead to a more uniform strain distribution across thepiezoelectric material and increase the harvesting performance by a factor of 3.6.

A Refined Harmonic Lanczos Bidiagonalization Method and an Implicitly Restarted Algorithm for Computing the Smallest Singular Triplets of Large Matrices

The harmonic Lanczos bidiagonalization method can be used to compute the smallest singular triplets of a large matrix $A$. We prove that for good enough projection subspaces harmonic Ritz values converge if the columns of $A$ are strongly linearly independent. On the other hand, harmonic Ritz values may miss some desired singular values when the columns of $A$ almost linearly dependent. Furthermore, harmonic Ritz vectors may converge irregularly and even may fail to converge. Based on the refined projection principle for large matrix eigenproblems due to the first author, we propose a refined harmonic Lanczos bidiagonalization method that takes the Rayleigh quotients of the harmonic Ritz vectors as approximate singular values and extracts the best approximate singular vectors, called the refined harmonic Ritz approximations, from the given subspaces in the sense of residual minimizations. The refined approximations are shown to converge to the desired singular vectors once the subspaces are sufficiently good and the Rayleigh quotients converge. An implicitly restarted refined harmonic Lanczos bidiagonalization algorithm (IRRHLB) is developed. We study how to select the best possible shifts, and suggest refined harmonic shifts that are theoretically better than the harmonic shifts used within the implicitly restarted Lanczos bidiagonalization algorithm (IRHLB). We propose a novel procedure that can numerically compute the refined harmonic shifts efficiently and accurately. Numerical experiments are reported that compare IRRHLB with five other algorithms based on the Lanczos bidiagonalization process. It appears that IRRHLB is at least competitive with them and can be considerably more efficient when computing the smallest singular triplets.

Optimum design of composite laminates for frequency constraints

Abstract A method for minimum thickness multilayer rectangular composite laminates is presented which is based on a layerwise optimization procedure under natural frequency limitations. In this method, all types of boundary conditions and their combinations can be taken into account. Analysis is based on the Ritz approximations, and no limitations exist on the number of layers that can be considered or the plate’s aspect ratio. Coupling of flexural and torsional stiffness terms need not be zero in any of the cases. Optimization constraints can be either limits on the fundamental frequency or separation between two natural frequencies. Hybrid combination of layers is also made possible in order to enhance the laminate economy, and a short cut procedure is recommended to save total computational efforts. Examples are offered that illustrate the method’s capabilities, and the results are verified by comparison to some published ones.

A posteriori estimates for eigenvalue/vector approximations

AbstractWe combine abstract eigenvalue/eigenvector estimates (from our earlier work) with a saturation assumption for finite element solution of associated stationary problem to obtain a posteriori estimates of the accuracy of finite element Rayleigh–Ritz approximations. Attention will be payed to the interplay between the accuracy estimate for the finite element method and a strategy for generating an adapted mesh. The obtained results use a preconditioned residuum of Neymeyr and extend his study of eigenvalue approximations with eigenvector estimates. We also prove that this eigenvalue estimator is equivalent to the global error. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On Eigenvalue and Eigenvector Estimates for Nonnegative Definite Operators

We present a perturbation approach to the Rayleigh–Ritz approximations in the sense of Davis, Kahan, and Weinberger. We restrict ourselves to nonnegative definite self‐adjoint operators and obtain sharp bounds of relative type for both eigenvalues and eigenvectors. The operators are allowed to have nontrivial null‐spaces, and the test spaces need not be contained in the domain of the considered operator.

Cluster robust error estimates for the Rayleigh–Ritz approximation II: Estimates for eigenvalues

This is the second part of a paper that deals with error estimates for the Rayleigh–Ritz approximations of the spectrum and invariant subspaces of a bounded Hermitian operator in a Hilbert or Euclidean space. This part addresses the approximation of eigenvalues. Two kinds of estimates are considered: (i) estimates for the eigenvalue errors via the best approximation errors for the corresponding invariant subspaces, and (ii) estimates for the same via the corresponding residuals. Estimates of these two kinds are needed for, respectively, the a priori and a posteriory error analysis of numerical methods for computing eigenvalues. The paper’s major concern is to ensure that the estimates in question are accurate and ‘cluster robust’, i.e. are not adversely affected by the presence of clustered, i.e. closely situated eigenvalues among those of interest. The paper’s main new results introduce estimates for clustered eigenvalues whereby not only the distances between eigenvalues in the cluster are not present but also the distances between the cluster and the rest of the spectrum appear in asymptotically insignificant terms only.

Cluster robust error estimates for the Rayleigh–Ritz approximation I: Estimates for invariant subspaces

This is the first part of a paper that deals with error estimates for the Rayleigh–Ritz approximations to the spectrum and invariant subspaces of a bounded Hermitian operator in a Hilbert or Euclidean space. This part addresses estimates for the angles between the invariant subspaces and their approximations via the corresponding best approximation errors and residuals and, for invariant subspaces corresponding to parts of the discrete spectrum, via eigenvalue errors. The paper’s major concern is to ensure that the estimates in question are accurate and ‘cluster robust’, i.e. are not adversely affected by the presence of clustered, i.e. closely situated eigenvalues in the spectrum. Available estimates of such kind are reviewed and new estimates are derived. The paper’s main new results introduce estimates for invariant subspaces in which the operator may have clustered eigenvalues whereby not only the distances between eigenvalues in the cluster are not present but also the distances between the cluster and the rest of the spectrum appear in asymptotically insignificant terms only.

On Stabilization and Convergence of Clustered Ritz Values in the Lanczos Method

The Lanczos algorithm can be considered as an iterative method for finding a few eigenvalues and eigenvectors of large sparse symmetric matrices. In the present paper we solve a conjecture posed by Zdenek Strakos and Anne Greenbaum in [Open Questions in the Convergence Analysis of the Lanczos Process for the Real Symmetric Eigenvalue Problem, IMA Research Report, 1992] on the clustering of Ritz values, which occurs in finite precision computations. In particular, we prove that the conjecture is valid in most cases and describe the rare case when it is not. The established upper bounds measuring the quality of Ritz approximations imply also that Ritz values cluster only close to an eigenvalue.

On Ritz approximations for positive definite operators I (theory)

We give new lower bounds on the Rayleigh–Ritz approximations of a part of the spectrum of an elliptic operator. Furthermore, we present bounds for the accompanying Ritz vectors. The bounds include a form of a relative gap between the Ritz values and the rest of the spectrum of the operator. A model example shows that the obtained bounds may be very sharp.

A generalization of Saad's theorem on Rayleigh–Ritz approximations

Let (λ,x) be an eigenpair of the Hermitian matrix A of order n and let (μ,u) be a Ritz pair from a subspace K of C 2 . Y. Saad (Numerical Methods for Large Eigenvalue Problems: Theory and Algorithm, Wiley, New York, 1992) has given a simple inequality bounding sin ∠(x,u) in terms of sin ∠(x, K) . In this paper, we show that this inequality can be extended to an equally simple inequality for eigenspaces of non-Hermitian matrices.

Energy approximations of the solution of the first problem of the linear theory of elasticity

The condition for which the Ritz approximations and the Trefftz approximations lead to a single energy approximation of the solution of the first problem of the theory of elasticity is established.

Convergent Ritz approximations of the set of stabilizing controllers

A necessary and sufficient condition for linear exponential sums to be dense in L p is derived. Under this condition a Ritz approximation converges, which has been proposed to solve linear control problems. State-space representations of orthonormal basis functions for multivariable systems are given.

The rational Krylov algorithm for nonsymmetric eigenvalue problems. III: Complex shifts for real matrices

A new algorithm for the computation of eigenvalues of a nonsymmetric matrix pencil is described. It is a generalization of the shifted and inverted Lanczos (or Arnoldi) algorithm, in which several shifts are used in one run. It computes an orthogonal basis and a small Hessenberg pencil. The eigensolution of the Hessenberg pencil, gives Ritz approximations to the solution of the original pencil. It is shown how complex shifts can be used to compute a real block Hessenberg pencil to a real matrix pair. Two applicationx, one coming from an aircraft stability problem and the other from a hydrodynamic bifurcation, have been tested and results are reported.

Rational Krylov algorithms for nonsymmetric eigenvalue problems. II. matrix pairs

A new algorithm for the computation of eigenvalues of a nonsymmetric matrix pencil is described. It is a generalization of the shifted and inverted Lanczos (or Arnoldi) algorithm, in which several shifts are used in one run. It computes an orthogonal basis and a small Hessenberg pencil. The eigensolution of the Hessenberg pencil gives Ritz approximations to the solution of the original pencil. It is shown that the computed approximate solution is the exact solution of a perturbed pencil, and bounds and estimates of the perturbations are given. Results for a numerical example coming from a bifurcation problem arising from a hydrodynamical application are demonstrated.