35,493 research outputs found
A comparison of variational, differential quadrature, and approximate closed-form solution methods for buckling of highly flexurally anisotropic laminates
The buckling response of symmetric laminates that possess strong exural-twist coupling are studied using different methodologies. Such plates are dfficult to analyse due to localised gradients in the mode shape. Initially, the energy method (Rayleigh-Ritz) using Legendre polynomials is employed and the difficulty of achieving reliable solutions for some extreme cases is discussed. To overcome the convergence problems, the concept of Lagrangian multiplier is introduced into the Rayleigh-Ritz formulation. The Lagrangian multiplier approach is able to provide the upper and lower bounds of critical buckling load results. In addition, mixed variational principles are used to gain a better understanding of the mechanics behind the strong exural-twist anisotropy effect on buckling solutions. Specifically, the Hellinger-Reissner variational principle is used to study the effect of exural-twist coupling on buckling and also to explore the potential for developing closed form solutions for these problems. Finally, solutions using the differential quadrature method are obtained. Numerical results of buckling coefficients for highly anisotropic plates with different boundary conditions are studied using the proposed approaches and compared with finite element results. The advantages of both Lagrangian multiplier theory and variational principle in evaluating buckling loads are discussed. In addition, a new simple closed form solution is shown for the case of a exurally anisotropic plate with three sides simply supported and one long edge free
A Unified Approach to Variational Derivatives of Modified Gravitational Actions
Our main aim in this paper is to promote the coframe variational method as a
unified approach to derive field equations for any given gravitational action
containing the algebraic functions of the scalars constructed from the Riemann
curvature tensor and its contractions. We are able to derive a master equation
which expresses the variational derivatives of the generalized gravitational
actions in terms of the variational derivatives of its constituent curvature
scalars. Using the Lagrange multiplier method relative to an orthonormal
coframe, we investigate the variational procedures for modified gravitational
Lagrangian densities in spacetime dimensions . We study
well-known gravitational actions such as those involving the Gauss-Bonnet and
Ricci-squared, Kretchmann scalar, Weyl-squared terms and their algebraic
generalizations similar to generic theories and the algebraic
generalization of sixth order gravitational Lagrangians. We put forth a new
model involving the gravitational Chern-Simons term and also give three
dimensional New massive gravity equations in a new form in terms of the Cotton
2-form
A Distributed Asynchronous Method of Multipliers for Constrained Nonconvex Optimization
This paper presents a fully asynchronous and distributed approach for
tackling optimization problems in which both the objective function and the
constraints may be nonconvex. In the considered network setting each node is
active upon triggering of a local timer and has access only to a portion of the
objective function and to a subset of the constraints. In the proposed
technique, based on the method of multipliers, each node performs, when it
wakes up, either a descent step on a local augmented Lagrangian or an ascent
step on the local multiplier vector. Nodes realize when to switch from the
descent step to the ascent one through an asynchronous distributed logic-AND,
which detects when all the nodes have reached a predefined tolerance in the
minimization of the augmented Lagrangian. It is shown that the resulting
distributed algorithm is equivalent to a block coordinate descent for the
minimization of the global augmented Lagrangian. This allows one to extend the
properties of the centralized method of multipliers to the considered
distributed framework. Two application examples are presented to validate the
proposed approach: a distributed source localization problem and the parameter
estimation of a neural network.Comment: arXiv admin note: substantial text overlap with arXiv:1803.0648
A Primal-Dual Augmented Lagrangian
Nonlinearly constrained optimization problems can be solved by minimizing a sequence of simpler unconstrained or linearly constrained subproblems. In this paper, we discuss the formulation of subproblems in which the objective is a primal-dual generalization of the Hestenes-Powell augmented Lagrangian function. This generalization has the crucial feature that it is minimized with respect to both the primal and the dual variables simultaneously. A benefit of this approach is that the quality of the dual variables is monitored explicitly during the solution of the subproblem. Moreover, each subproblem may be regularized by imposing explicit bounds on the dual variables. Two primal-dual variants of conventional primal methods are proposed: a primal-dual bound constrained Lagrangian (pdBCL) method and a primal-dual 1 linearly constrained Lagrangian (pd1-LCL) method
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