Nonlinear CG-like iterative methods

AbstractA nonlinear conjugate gradient method has been introduced and analyzed by J.W. Daniel. This method applies to nonlinear operators with symmetric Jacobians. Orthomin(1) is an iterative method which applies to nonsymmetric and definite linear systems. In this article we generalize Orthomin(1) to a method which applies directly to nonlinear operator equations. Each iteration of the new method requires the solution of a scalar nonlinear equation. Under conditions that the Hessian is uniformly bounded away from zero and the Jacobian is uniformly positive definite the new method is proved to converge to a globally unique solution. Error bounds and local convergence results are also obtained. Numerical experiments on solving nonlinear operator equations arising in the discretization of nonlinear elliptic partial differential equations are presented

Rainbow metamaterials for broadband multi-frequency vibration attenuation: Numerical analysis and experimental validation

In this study, we propose a ‘rainbow’ metamaterial to achieve broadband multi-frequency vibration attenuation. The rainbow metamaterial is constituted of a Π-shaped beam partitioned into substructures by parallel plates insertions with two attached cantilever-mass acting as local resonators. Both resonators inside each substructure can be non-symmetric such that the metamaterial can have multi-frequency bandgaps. Furthermore, these cantilever-mass resonators have a progressively variant design along the beam, namely rainbow-shaped, for the purpose of achieving broader energy stop bands. Π-shaped beams partitioned by parallel plate insertions can be extended to honeycomb sandwich composites, hence the proposed rainbow metamaterial can serve as a precursor for future honeycomb composites with superior vibration attenuation for more industrial applications. A mathematical model is first developed to estimate the frequency response functions of the metamaterial. Interaction forces between resonators and the backbone structure are calculated by solving the displacement of the cantilever-mass resonators. The plate insertions are modeled as attached masses with both their translational and rotational motion considered. Subsequently, the mathematical model is verified by comparison with experimental results. Metamaterials fabricated through an additive manufacturing technique are tested with a laser doppler receptance measuring system. After the validation of the mathematical model, a numerical study is conducted to explore the influences of the resonator spatial distributions on the frequency response functions of structures. Results show that for metamaterials with both symmetric and non-symmetric resonators, rainbow-shaped resonators can introduce inertial forces inside wider frequency range when compared to the periodic resonators of the same total mass, hence broader bandgaps. Meanwhile, the attenuation inside the bandgaps decreases when the bandgap become broader. Metamaterials with broadband multi-frequency range vibration attenuation can be achieved with non-symmetric sinusoidally varying resonators

On Nonlinear Generalized Conjugate Gradient Methods

this article (Sects. 2 and 3) we assume that the Jacobian of F (¸) has symmetric parts uniformly positive definite. In the final part (Sect. 4) a method is presented where this assumption is not required. The Newton method coupled with direct linear system solvers is an efficient way to solve such nonlinear systems when the dimension of the Jacobian is small. When the Jacobian is large and sparse some kind of iterative method may be used. This can be a nonlinear iteration (for example functional iteration for contractive operators), or an inexact Newton method. In an inexact Newton the solution of the resulting linear systems is approximated by a linear iterative method. The following are typical steps in an inexact Newton method for solving this nonlinear system

A Parallel Krylov-Type Method for Nonsymmetric Linear Systems

Abstract. Parallel Krylov (S-step and block) iterative methods for linear systems have been studied and implemented in the past. In this ar-ticle we present a parallel Krylov method based on block s-step method for nonsymmetric linear systems. We derive two new averaging algorithm to combine several approximations to the solution of a single linear sys-tem using the block method with multiple initial guesses. We implement the new methods with ILU preconditioners on a parallel computer. We test the accuracy and present performance results.

Source camera identification: a distributed computing approach using Hadoop

The widespread use of digital images has led to a new challenge in digital image forensics. These images can be used in court as evidence of criminal cases. However, digital images are easily manipulated which brings up the need of a method to verify the authenticity of the image. One of the methods is by identifying the source camera. In spite of that, it takes a large amount of time to be completed by using traditional desktop computers. To tackle the problem, we aim to increase the performance of the process by implementing it in a distributed computing environment. We evaluate the camera identification process using conditional probability features and Apache Hadoop. The evaluation process used 6000 images from six different mobile phones of the different models and classified them using Apache Mahout, a scalable machine learning tool which runs on Hadoop. We ran the source camera identification process in a cluster of up to 19 computing nodes. The experimental results demonstrate exponential decrease in processing times and slight decrease in accuracies as the processes are distributed across the cluster. Our prediction accuracies are recorded between 85 to 95% across varying number of mappers

Robust optimised design of 3D printed elastic metastructures: A trade-off between complexity and vibration attenuation

In this work, a strategy for optimal design of mechanical metastructure is proposed taken into account uncertainties arising from additive manufacturing. A locally resonant Π-shaped beam with parallel plate-like insertions and two cantilever mass resonators at each unit cell is manufactured through a selective laser sintering process. The variability of the material properties introduced by the additive manufacturing procedure is experimentally obtained. Given that such manufacturing approaches are predominantly employed for producing complex metastructure architectures, it can significantly compromise the optimality of the design. A transfer matrix approach is employed to propagate variability at a structural level and predict the structural receptance due to a point harmonic force in the finite length metastructure. Then, the mass ratio of the metastructure is optimised for maximising vibration attenuation considering different numbers of added resonators and relative masses. A cost function is introduced in the classical robust design approach in order to favour designs with least complexity, represented by the number of added resonators. It is exhibited in several cases that the robustly optimal design is away from the deterministic optimal one, emphasising the relevance of the proposed approach in the optimisation of complex and locally resonant structures. Moreover, it is shown that the frequency range of interest plays a major role on the derived optimal design for each number of implemented resonators. The presented results show that even small variability in the Young's modulus of up to 3% and in the mass density of up to 1% can still affect the robustness of the optimal design for locally resonant metastructure as due to the consequent mistuning of the added resonators. © 2022 Elsevier Lt