Fast and Accurate Predictor-Corrector Methods Using Feedback-Accelerated Picard Iteration for Strongly Nonlinear Problems

Although predictor-corrector methods have been extensively applied,they might not meet the requirements of practical applications and engineering tasks,particularly when high accuracy and efficiency are necessary.A novel class of correctors based on feedback-accelerated Picard iteration(FAPI)is proposed to further enhance computational performance.With optimal feedback terms that do not require inversion of matrices,significantly faster convergence speed and higher numerical accuracy are achieved by these correctors compared with their counterparts;however,the computational complexities are comparably low.These advantages enable nonlinear engineering problems to be solved quickly and accurately,even with rough initial guesses from elementary predictors.The proposed method offers flexibility,enabling the use of the generated correctors for either bulk processing of collocation nodes in a domain or successive corrections of a single node in a finite difference approach.In our method,the functional formulas of FAPI are discretized into numerical forms using the collocation approach.These collocated iteration formulas can directly solve nonlinear problems,but they may require significant computational resources because of the manipulation of high-dimensionalmatrices.To address this,the collocated iteration formulas are further converted into finite difference forms,enabling the design of lightweight predictor-corrector algorithms for real-time computation.The generality of the proposed method is illustrated by deriving new correctors for three commonly employed finite-difference approaches:the modified Euler approach,the Adams-Bashforth-Moulton approach,and the implicit Runge-Kutta approach.Subsequently,the updated approaches are tested in solving strongly nonlinear problems,including the Matthieu equation,the Duffing equation,and the low-earth-orbit tracking problem.The numerical findings confirm the computational accuracy and efficiency of the derived predictor-corrector algorithms.

The Use of High-Performance Fatigue Mechanics and the Extended Kalman/Particle Filters,for Diagnostics and Prognostics of Aircraft Structures

In this paper,we propose an approach for diagnostics and prognostics of damaged aircraft structures,by combing high-performance fatigue mechanics with filtering theories.Fast&accurate deterministic analyses of fatigue crack propagations are carried out,by using the Finite Element Alternating Method(FEAM)for computing SIFs,and by using the newly developed Moving Least Squares(MLS)law for computing fatigue crack growth rates.Such algorithms for simulating fatigue crack propagations are embedded in the computer program Safe-Flaw,which is called upon as a subroutine within the probabilistic framework of filter theories.Both the extended Kalman as well as particle filters are applied in this study,to obtain the statistically optimal and semi-optimal estimates of crack lengths,from a series of noisy measurements of crack-lengths over time.For the specific problem,a simple modification to the particle filter,which can drastically reduce the computational burden,is also proposed.Based on the results of such diagnostic analyses,the prognostics of aerospace structures are thereafter achieved,to estimate the probabilistic distribution of the remaining useful life.By using a simple example of a single-crack near a fastener hole,we demonstrate the concept and effectiveness of the proposed framework.This paper thus forms the scientific foundation for the recently proposed concepts of VRAMS(Virtual Risk-Informed Agile Maneuver Sustainment)and Digital Twins of aerospace vehicles.

A Unification of the Concepts of the Variational Iteration, Adomian Decomposition and Picard Iteration Methods;and a Local Variational Iteration Method

This paper compares the variational iteration method(VIM),the Adomian decomposition method(ADM)and the Picard iteration method(PIM)for solving a system of first o rder n onlinear o rdinary d ifferential e quations(ODEs).A unification of the concepts underlying these three methods is attempted by considering a very general iterative algorithm for VIM.It is found that all the three methods can be regarded as special cases of using a very general matrix of Lagrange multipliers in the iterative algorithm of VIM.The global variational iteration method is briefly reviewed,and further recast into a Local VIM,which is much more convenient and capable of predicting long term complex dynamic responses of nonlinear systems even if they are chaotic.

Weakly Singular Symmetric Galerkin Boundary Element Method for Fracture Analysis of Three-Dimensional Structures Considering Rotational Inertia and Gravitational Forces

The Symmetric Galerkin Boundary Element Method is advantageous for the linear elastic fracture and crackgrowth analysis of solid structures,because only boundary and crack-surface elements are needed.However,for engineering structures subjected to body forces such as rotational inertia and gravitational loads,additional domain integral terms in the Galerkin boundary integral equation will necessitate meshing of the interior of the domain.In this study,weakly-singular SGBEM for fracture analysis of three-dimensional structures considering rotational inertia and gravitational forces are developed.By using divergence theorem or alternatively the radial integration method,the domain integral terms caused by body forces are transformed into boundary integrals.And due to the weak singularity of the formulated boundary integral equations,a simple Gauss-Legendre quadrature with a few integral points is sufficient for numerically evaluating the SGBEM equations.Some numerical examples are presented to verify this approach and results are compared with benchmark solutions.

Double Optimal Regularization Algorithms for Solving Ill-Posed Linear Problems under Large Noise

A double optimal solution of an n-dimensional system of linear equations Ax=b has been derived in an affine m-dimensional Krylov subspace with m <<n.We further develop a double optimal iterative algorithm(DOIA),with the descent direction z being solved from the residual equation Az=r0 by using its double optimal solution,to solve ill-posed linear problem under large noise.The DOIA is proven to be absolutely convergent step-by-step with the square residual error ||r||^2=||b-Ax||^2 being reduced by a positive quantity ||Azk||^2 at each iteration step,which is found to be better than those algorithms based on the minimization of the square residual error in an m-dimensional Krylov subspace.In order to tackle the ill-posed linear problem under a large noise,we also propose a novel double optimal regularization algorithm(DORA)to solve it,which is an improvement of the Tikhonov regularization method.Some numerical tests reveal the high performance of DOIA and DORA against large noise.These methods are of use in the ill-posed problems of structural health-monitoring.

Harmonic Balance Methods:A Review and Recent Developments

The harmonic balance(HB)method is one of the most commonly used methods for solving periodic solutions of both weakly and strongly nonlinear dynamical systems.However,it is confined to low-order approximations due to complex symbolic operations.Many variants have been developed to improve the HB method,among which the time domain HB-like methods are regarded as crucial improvements because of their fast computation and simple derivation.So far,there are two problems remaining to be addressed.i)A dozen of different versions of HB-like methods,in frequency domain or time domain or in hybrid,have been developed;unfortunately,misclassification pervades among them due to the unclear borderlines of different methods.ii)The time domain HB-like methods suffer from non-physical solutions,which have been shown to be caused by aliasing(mixture of the high-order into the low-order harmonics).Although a series of dealiasing techniques have been developed over the past two decades,the mechanism of aliasing and the final solution to dealiasing are still not well known to the academic community.This paper aims to provide a comprehensive review of the development of HB-like methods and enunciate their principal differences.In particular,the time domain methods are emphasized with the famous aliasing phenomenon clearly addressed.

基于三维纤维计算晶粒的纤维复合材料直接细观力学建模

面向纤维复合材料与结构的优化设计需要,建立高效准确的纤维复材细观力学仿真工具具有重要意义.本文首次提出了包含内嵌纤维的三维计算晶粒(computational grains),用于纤维增强复合材料的直接细观力学建模,基于所开发的纤维计算晶粒方法,可以直接构造含有多根随机分布纤维的代表性体积单元(RVE).此外,本文提出了一种基于缩放柱调和函数的Papkovich-Neuber解来表征纤维和基体中独立的Trefftz试函数位移场,并且开发了一种新的多场边界变分原理来计算纤维计算晶粒的刚度矩阵.数值算例表明,对于内嵌大量随机分布纤维的RVE,纤维计算晶粒不需要构造复杂的网格,即可快速计算RVE的精确应力场,并预测其有效力学性能.此外,这也是内嵌纤维的三维有限元单元的首次提出.