Rayleigh Ritz

corazón, por la sabiduría y bendición que nos ha dado para culminar nuestra tesis. A nuestro asesor Lic.Mat. Reupo Vallejos Raúl Eduardo por su permanente apoyo, logrando de esta manera la meta trazada y por la cual estamos comprometidas a cumplir sus enseñanzas para alcanzar el éxito profesional.

Flexible structures are increasingly prevalent in the commercial aviation industry, and the use of highly flexible structures is a prominent trend for the future. When analyzing those structures, it is crucial to consider geometric nonlinearities caused by large displacements. This means that the modeling of the structures must incorporate nonlinear structural models, which can lead to a reasonable increase in computational costs. To tackle this challenge, a framework has been developed for static and dynamic analyses of highly flexible structures. It is based on a linear structural model, utilizing the Rayleigh-Ritz method, coupled with multibody dynamics. The geometric nonlinearities are modeled through rigid connections between multiple flexible bodies that form the final structure. Two different approaches have been used for the multibody dynamics. The former considers all degrees of freedom of each body and solves only the kinematics of the constraint to maintain the connections between the bodies, which resulted in an augmented system with Lagrange multipliers that can be used to reconstruct forces and moments of constraint. The latter utilizes only the independent degrees of freedom whilst reconstructing the dependent ones through the equations that define the constraints between the bodies, directly solving the constraints. The results obtained show that proposed framework accurately describes the dynamics of highly flexible structures and can be used to simulate structures with various types of connections, showcasing its versatility for other applications like simulations of morphing structures such as wings with folding wingtips.

The problem in this paper is to construct accurate approximations from a subspace to eigenpairs for symmetric matrices using the harmonic Rayleigh-Ritz method. Morgan introduced this concept in 14 as an alternative f o r R a yleigh-Ritz in large scale iterative methods for computing interior eigenpairs. The focus rests on the choice and in uence of the shift and error estimation. We a l s o g i v e a discussion of the di erences and similarities with the re ned Ritz approach for symmetric matrices. Using some numerical experiments we compare di erent conditions for selecting appropriate harmonic Ritz vectors.

We derive error bounds for the Rayleigh-Ritz method for the approximation to extremal eigenpairs of a symmetric matrix. The bounds are expressed in terms of the eigenvalues of the matrix and the angle between the subspace and the eigenvector. We also present a sharp bound.

In this work, we investigate the system formed by the equations div w = g 0 and curl w = g in bounded star-shaped domains of R 3. A Helmholtz-type decomposition theorem is established based on a general solution of the above-mentioned div-curl system which was previously derived in the literature. When g 0 ≡ 0, we readily obtain a bounded right inverse of curl which is a divergence-free invariant. The restriction of this operator to the subspace of divergence-free vector fields with vanishing normal trace is the well-known Biot-Savart operator. In turn, this right inverse of curl will be modified to guarantee its compactness and satisfy suitable boundary-value problems. Applications to Beltrami fields, Vekua-type problems as well as Maxwell's equations in inhomogeneous media are included.

In this paper, a lumped-mass dynamic model of a single degree of freedom cable-actuated system is derived, and passivity-based control is considered. The dynamic model developed takes into consideration the changing cable stiffness and mass as the cable is wrapped around a winch. In addition, the change in the winch inertia as the cable is wrapped around the winch is modeled. It is assumed that the mass of the payload is much greater than the mass of the cables and the equivalent mass of the winches, which allows for an approximation where the rigid dynamics can be decoupled from the elastic dynamics of the system. This approximation enables the definition of a modified input torque and modified output rate, allowing the establishment of passive input-output mappings. Passivity-based controllers are investigated, shown to render the closed-loop system input-output stable, and tested in simulation. Index Terms-μ-tip rate, cable-actuated systems, lumpedmass model, parallel manipulators, passivity-based control, proportional-integral-derivative (PID) control. I. INTRODUCTION T HE study of cable-actuated systems is a rapidly expanding field of research that has shown promising potential. Cable robots, cranes, elevators, towing systems, and aerostats are examples of cable-actuated systems, which are characterized by the use of cables to transmit an actuating force to an object, and are typically operated with the goal of accurately positioning a payload. The advantages of using cables to transmit forces, rather than rigid links, include the possibility of high acceleration maneuvers and higher payload to weight ratios. Although the notion of maneuvering a payload by means of cables is conceptually simple, in practice, the control of the payload and cables is quite challenging. For example, cables are subjected to a unidirectional force constraint, as they only operate in tension, not compression, and the cables used in cable-actuated systems are often, to some degree, flexible. Cable dynamics can be neglected when used in cable robot applications, where light very stiff cables are used over Manuscript

corazón, por la sabiduría y bendición que nos ha dado para culminar nuestra tesis. A nuestro asesor Lic.Mat. Reupo Vallejos Raúl Eduardo por su permanente apoyo, logrando de esta manera la meta trazada y por la cual estamos comprometidas a cumplir sus enseñanzas para alcanzar el éxito profesional.

Microfabrication limitations are of concern especially for suspended Micro-Electro-Mechanical-Systems (MEMS) microstructures such as cantilevers. The static and dynamic qualities of such microscale devices are directly related to the invariant and variant properties of the microsystem. Among the invariant properties, microfabrication limitations can be quantified only after the fabrication of the device through testing. However, MEMS are batch fabricated in large numbers where individual testing is neither possible nor cost effective. Hence, a suitable test algorithm needs to be developed where the test results obtained for a few devices can be applied to the whole fabrication batch, and also to the foundry process in general. In this regard, this paper proposes a method to test MEMS cantilevers under variant electro-thermal influences in order to quantify the effective boundary support condition obtained for a foundry process. A non-contact optical sensing approach is employed for the dynamic testing. The Rayleigh-Ritz energy method using boundary characteristic orthogonal polynomials is employed for the modeling and theoretical analysis.

Camber morphing aerofoils have the potential to significantly improve the efficiency of fixed and rotary wing aircraft by providing significant lift control authority to a wing, at a lower drag penalty than traditional plain flaps. A rapid, mesh-independent and two-dimensional analytical model of the fish bone active camber concept is presented. Existing structural models of this concept are one-dimensional and isotropic and therefore unable to capture either material anisotropy or spanwise variations in loading/deformation. The proposed model addresses these shortcomings by being able to analyse composite laminates and solve for static two-dimensional displacement fields. Kirchhoff–Love plate theory, along with the Rayleigh–Ritz method, are used to capture the complex and variable stiffness nature of the fish bone active camber concept in a single system of linear equations. Results show errors between 0.5% and 8% for static deflections under representative uniform pressure loading...

This paper develops a least-squares functional that arises from recasting general second-order uniformly elliptic partial di erential equations in n = 2 or 3 dimensions as a system of rst-order equations. In part I 11] a similar functional was developed and shown to be elliptic in the H(div) H 1 norm and to yield optimal convergence for nite element subspaces of H(div) H 1. In this paper the functional is modi ed by adding a compatible constraint and imposing additional boundary conditions on the rst-order system. The resulting functional is proved to be elliptic in the (H 1) n+1 norm. This immediately implies optimal error estimates for nite element approximation by standard subspaces of (H 1) n+1. Another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent. As an alternative to perturbation based approaches, the least-squares approach developed here applies directly to convection-di usion-reaction equations in a uni ed way and also admits a fast multigrid solver, historically a missing ingredient in least-squares methodology.

We prove convergence for a meshfree first-order system least squares (FOSLS) partition of unity finite element method (PUFEM). Essentially, by virtue of the partition of unity, local approximation gives rise to global approximation in H(div)∩ H(curl). The FOSLS formulation yields local a posteriori error estimates to guide the judicious allotment of new degrees of freedom to enrich the initial point set in a meshfree dis- cretization. Preliminary numerical results are provided and remaining challenges are discussed. AMS subject classifications: 65N30, 65N50

In this paper, we develop several related finite dimensional variational principles for discrete optimal transport (DOT), Minkowski type problems for convex polytopes and discrete Monge-Ampere equation (DMAE). A link between the discrete optimal transport, discrete Monge-Ampere equation and the power diagram in computational geometry is established.

In this paper, we develop several related finite dimensional variational principles for discrete optimal transport (DOT), Minkowski type problems for convex polytopes and discrete Monge-Ampere equation (DMAE). A link between the discrete optimal transport, discrete Monge-Ampere equation and the power diagram in computational geometry is established.

We prove convergence for a meshfree first-order system least squares (FOSLS) partition of unity finite element method (PUFEM). Essentially, by virtue of the partition of unity, local approximation gives rise to global approximation in H(div)∩ H(curl). The FOSLS formulation yields local a posteriori error estimates to guide the judicious allotment of new degrees of freedom to enrich the initial point set in a meshfree dis- cretization. Preliminary numerical results are provided and remaining challenges are discussed. AMS subject classifications: 65N30, 65N50

We prove convergence for a meshfree first-order system least squares (FOSLS) partition of unity finite element method (PUFEM). Essentially, by virtue of the partition of unity, local approximation gives rise to global approximation in H(d i v) ∩ H(cur l). The FOSLS formulation yields local a posteriori error estimates to guide the judicious allotment of new degrees of freedom to enrich the initial point set in a meshfree discretization. Preliminary numerical results are provided and remaining challenges are discussed.

This paper develops a least-squares functional that arises from recasting general second-order uniformly elliptic partial differential equations in n = 2 or 3 dimensions as a system of first-order equations. ] a similar functional was developed and shown to be elliptic in the H(div) × H 1 norm and to yield optimal convergence for finite element subspaces of H(div)×H 1 . In this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first-order system. The resulting functional is proved to be elliptic in the (H 1 ) n+1 norm. This immediately implies optimal error estimates for finite element approximation by standard subspaces of (H 1 ) n+1 . Another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent. As an alternative to perturbation-based approaches, the least-squares approach developed here applies directly to convection-diffusion-reaction equations in a unified way and also admits a fast multigrid solver, historically a missing ingredient in least-squares methodology.

This paper develops a least-squares functional that arises from recasting general second-order uniformly elliptic partial differential equations in n = 2 or 3 dimensions as a system of first-order equations. ] a similar functional was developed and shown to be elliptic in the H(div) × H 1 norm and to yield optimal convergence for finite element subspaces of H(div)×H 1 . In this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first-order system. The resulting functional is proved to be elliptic in the (H 1 ) n+1 norm. This immediately implies optimal error estimates for finite element approximation by standard subspaces of (H 1 ) n+1 . Another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent. As an alternative to perturbation-based approaches, the least-squares approach developed here applies directly to convection-diffusion-reaction equations in a unified way and also admits a fast multigrid solver, historically a missing ingredient in least-squares methodology.