Optimal spatiotemporal reduced order modeling, Part II: application to a nonlinear beam

A geometrically nonlinear, simply supported beam under the influence of time-dependent external forcing serves as a testbed to demonstrate application of the optimal spatiotemporal reduced order modeling (OPSTROM) framework proposed in Part I of this work. Fully resolved simulations, which are relatively expensive to perform, are used to accurately predict the beam response for a few forcing parameters. More affordable simulations are achieved with a conventional finite-difference scheme by coarsening the computational grid in space and time. Discretization errors are reduced with OPSTROM as subgrid-scale models are designed to account for the underlying space-time statistical structure using principles of mean-square error minimization, conditional expectations and stochastic estimation. When included in the under-resolved simulations, these optimal subgrid-scale models are shown to significantly improve the accuracy of predictions for both periodic and chaotic response types. This improved accuracy is further demonstrated through a set of numerical experiments designed to capture the complex bifurcation behavior of the beam response.     

Solving initial and boundary value problems using learning automata particle swarm optimization

In this article, the particle swarm optimization (PSO) algorithm is modified to use the learning automata (LA) technique for solving initial and boundary value problems. A constrained problem is converted into an unconstrained problem using a penalty method to define an appropriate fitness function, which is optimized using the LA-PSO method. This method analyses a large number of candidate solutions of the unconstrained problem with the LA-PSO algorithm to minimize an error measure, which quantifies how well a candidate solution satisfies the governing ordinary differential equations (ODEs) or partial differential equations (PDEs) and the boundary conditions. This approach is very capable of solving linear and nonlinear ODEs, systems of ordinary differential equations, and linear and nonlinear PDEs. The computational efficiency and accuracy of the PSO algorithm combined with the LA technique for solving initial and boundary value problems were improved. Numerical results demonstrate the high accuracy and efficiency of the proposed method.     

Lattice Discrete Particle Model (LDPM) for failure behavior of concrete. I: Theory

This paper deals with the formulation, calibration, and validation of the Lattice Discrete Particle Model (LDPM) suitable for the simulation of the failure behavior of concrete. LDPM simulates concrete at the meso-scale considered to be the length scale of coarse aggregate pieces. LDPM is formulated in the framework of discrete models for which the unknown displacement field is not continuous but only defined at a finite number of points representing the center of aggregate particles. Size and distribution of the particles are obtained according to the actual aggregate size distribution of concrete. Discrete compatibility and equilibrium equations are used to formulate the governing equations of the LDPM computational framework. Particle contact behavior represents the mechanical interaction among adjacent aggregate particles through the embedding mortar. Such interaction is governed by meso-scale constitutive equations simulating meso-scale tensile fracturing with strain-softening, cohesive and frictional shearing, and nonlinear compressive behavior with strain-hardening. The present, Part I, of this two-part study deals with model formulation leaving model calibration and validation to the subsequent Part II.     

Adaptive reduced order modeling for nonlinear dynamical systems through a new a posteriori error estimator: Application to uncertainty quantification

Multiquery problems such as uncertainty quantification (UQ), optimization of a dynamical system require solving a differential equation at multiple parameter values. Therefore, for large systems, the computational cost becomes prohibitive. This issue can be addressed by using a cheaper reduced order model (ROM) instead. However, the ROM entails error in the solution due to approximation in a lower dimensional subspace. Moreover, the ROM lacks robustness over a wide range of parameter values. To address these issues, first, an upper bound on the norm of the state transition matrix is derived. This bound, along with the residual in the governing equation, are then used to develop an error estimator for general nonlinear dynamical systems. Furthermore, this error estimator is used in conjunction with the modified greedy search algorithm proposed by Hossain and Ghosh (Int J Numer Methods Eng, 2018;116(12-13): 741-758) to adaptively construct a robust proper orthogonal decomposition-based ROM. This adaptive ROM is subsequently deployed for UQ by invoking it in a statistical simulation. Two numerical studies: (i) viscous Burgers' equation and (ii) beam on nonlinear Winkler foundation, showed an improved accuracy of the error estimator compared to the current literature. A significant computational speed-up in UQ is achieved.     

On the contact mechanics of a rolling cylinder on a graded coating. Part 1: Analytical formulation

In this paper, the fully coupled rolling contact problem of a graded coating/substrate system under the action of a rigid cylinder is investigated. Using the singular integral equation approach, the governing equations of the rolling contact problem are constructed for all possible stick/slip regimes. Applying the Gauss–Chebyshev numerical integration method, the governing equations are converted to systems of algebraic equations. A new numerical algorithm is proposed to solve these systems of equations. Both the coupled and the uncoupled solutions to the problem are found through an implemented iterative procedure. In Part I of this paper, the analytical formulation of the rolling contact problem and the discretization of the governing equations are introduced for all assumed stick/slip regimes. A detailed discussion of the proposed numerical algorithm, the iteration procedure and the numerical results, obtained using the analytical formulation, are given in Part II.     

Dynamics and control of multi-connected satellites aligned along local horizontal

The paper presents dynamics and control of multi-connected satellites aligned along an unstable local horizontal configuration. The satellites are modeled as point masses and they are connected through cables. The cables are assumed to be flexible but massless. The governing equations of motions of the system moving in an elliptic orbit are obtained through a Lagrangian approach. The control laws are developed and the cable deployment and retrieval laws are stated for an N-body system. Results of numerical simulations of the nonlinear governing equations of motion of two- and three-body systems and approximate closed-form solutions indicate the feasibility of controlling multi-connected satellites along the local horizontal. Finally, the effects of various system parameters as well as the cable deployment and retrieval on the system response have been investigated. It is important to mention that the multi-connected satellites system outperforms the free flyer multi-satellite system and the controller only requires force in the order of a Millinewton for multi-satellites in geostationary orbit with a desired inter-satellite separation distance of 100 m.     

A mixed-variable continuously deforming finite element method for parabolic evolution problems. Part II: The coupled problem of phase-change in porous media

The conceptual framework of a least squares rate variational approach to the formulation of continuously deforming mixed-variable finite element computational scheme for a single evolution equation was presented in Part I.¹ In this paper (Part II), we extend these concepts and present an adaptively deforming mixed variable finite element method for solving general two-dimensional transport problems governed by a system of coupled non-linear partial differential evolution equations. In particular, we consider porous media problems that involve coupled heat and mass transport processes that yield steep continuous moving fronts, and abrupt, discontinuous, moving phase-change interfaces. In this method, the potentials, such as the temperature, pressure and species concentration, and the corresponding fluxes, are permitted to jump in value across the phase-change interfaces. The equations, and the jump conditions, governing the physical phenomena, which were specialized from a general multiphase, multiconstituent mixture theory, provided the basis for the development and implementation of a two-dimensional numerical simulator. This simulator can effectively resolve steep continuous fronts (i.e. shock capturing) without oscillations or numerical dispersion, and can accurately represent and track discontinuous fronts (i.e. shock fitting) through adaptive grid deformation and redistribution. The numerical implementation of this simulator and numerical examples that demonstrate the performance of the computational method are presented in Part III² of this paper.     

Coupled longitudinal-transverse behaviour of a geometrically imperfect microbeam

Based on the modified couple stress theory, the coupled longitudinal-transverse nonlinear behaviour of an imperfect microbeam is investigated numerically. The equations governing the longitudinal and transverse motions are obtained using Hamilton’s principle for the system with an initial geometric imperfection. The Galerkin scheme is employed to discretize the two partial differential equations of motion, yielding a set of second-order nonlinear ordinary differential equations with coupled terms. This set is cast into new set of first-order nonlinear ordinary differential equations and solved by means of the pseudo-arclength continuation technique. The nonlinear resonant response of the system along with bifurcations are presented via frequency–response curves. Moreover, the effect of different system parameter on the frequency–response curves is highlighted.     

Coupled longitudinal-transverse-rotational behaviour of shear deformable microbeams

In this paper, for the first time, the nonlinear motion characteristics of a hinged-hinged third-order shear deformable microbeam are examined, based on the modified couple stress theory and the third-order shear deformation theory. The extensibility of the microbeam is modelled by taking into account the longitudinal displacement. The nonlinear equations governing the longitudinal, transverse, and rotational motions are derived by means of Hamilton's principle in conjunction with the modified couple stress theory (to take into account small-scale effects). The three coupled nonlinear partial differential equations are discretized via the Galerkin method and the resulting set of ordinary differential equations is solved by means of the pseudo-arclength continuation technique and via direct time-integration. The effects of the system parameters on the behaviour of the microbeam are studied. Results are presented in the form of frequency-responses and force-responses. Points of interest in the parameter space are also highlighted in the form of time histories, phase-plane portraits, and fast Fourier transforms (FFTs). Moreover, the similarities and differences in the response of the system obtained via the modified couple stress and classical continuum mechanics theories are discussed.     

Non-orthogonal stagnation-point flow of a micropolar fluid

This paper considers the problem of steady two-dimensional flow of a micropolar fluid impinging obliquely on a flat plate. The flow under consideration is a generalization of the classical modified Hiemenz flow for a micropolar fluid which occurs in the boundary layer near an orthogonal stagnation point. A coordinate decomposition transforms the full governing equations into a primary equation describing the modified Hiemenz flow for a micropolar fluid and an equation for the tangential flow coupled to the primary solution. The solution to the boundary-value problem is governed by two non-dimensional parameters: the material parameter K and the ratio of the microrotation to skin friction parameter n. The obtained ordinary differential equations are solved numerically for some values of the governing parameters. The primary consequence of the free stream obliqueness is the shift of the stagnation point toward the incoming flow.     

Size-dependent behaviour of electrically actuated microcantilever-based MEMS

In this paper, the nonlinear size-dependent static and dynamic behaviours of a microelectromechanical system under an electric excitation are investigated. A microcantilever is considered for the modelling of the deformable electrode of the MEMS. The governing equation of motion is derived based on the modified couple stress theory (MCST), a non-classical model capable of capturing small-size effects. With the aid of a high-dimensional Galerkin scheme, the nonlinear partial differential equation governing the motion of the deformable electrode is converted into a reduced-order model of the system. Then, the pseudo-arclength continuation technique is used to solve the governing equations. In order to investigate the static behaviour and static pull-in instabilities, the system is excited only by the electrostatic actuation (i.e., a DC voltage). The results obtained for the static pull-in instability predicted by both the classical theory and MCST are compared. In the second stage of analysis, the nonlinear dynamic behaviour of the deformable electrode due to the AC harmonic actuation is investigated around the deflected configuration, incorporating size dependence.     

Nonlinear dynamics of FGM circular cylindrical shell with clamped-clamped edges

An analysis on the nonlinear dynamics of a clamped-clamped FGM circular cylindrical shell subjected to an external excitation and uniform temperature change is presented in this paper. Material properties of the constituents are assumed to be temperature-independent and the effective properties of FGM cylindrical shell are graded in thickness direction according to a simple power law function in terms of the volume fractions. Based on the first-order shear deformation shell theory and von Karman type nonlinear strain-displacement relationship, the nonlinear governing equations of motion are derived by using Hamilton’s principle. Galerkin’s method is then utilized to discretize the governing partial equations to a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms under combined external excitations. Numerical results including the bifurcations, waveform, phase plots and Poincare maps are presented for clamped-clamped FGM cylindrical shells showing the influences of material gradient index, the thickness and the external loading on the nonlinear dynamics.     

Adaptive methodology for the meshless Galerkin boundary node method

The Galerkin boundary node method (GBNM) is a boundary-type meshless method that combines a variational form of boundary integral formulations for governing equations with the moving least-squares approximations for generation of the trial and test functions. In this paper, a posteriori error estimate and an effective adaptive h-refinement procedure are developed in conjunction with the GBNM. The error estimator is based on the difference between numerical solutions obtained using two successive nodal arrangements. The reliability and efficiency of this error estimator and the convergence of this adaptive meshless scheme are verified theoretically. Numerical examples are also given to show the efficiency of the adaptive methodology.     

Local integral equations implemented by MLS-approximation and analytical integrations

This paper is devoted to the development of advanced mesh free implementations of the governing equations and the boundary conditions for boundary value problems in elasticity. Both the strong and weak formulations are discretized by using the Moving Least Squares approximations. The weak formulation is represented by local integral equations considered on sub-domains around interior nodal points. The awkward evaluation of the shape functions and their derivatives is reduced by focusing to nodal points because of the development of analytical integrations. That results in significant saving of the computational time needed for creation of the system matrix. Furthermore, a modified differentiation scheme is developed for approximation of higher order derivatives of displacements appearing in the discretized formulations. The accuracy, convergence and computational efficiency are studied in simple numerical example.     

A set of decentralized PID controllers for an n – link robot manipulator

A class of stabilizing decentralized proportional integral derivative (PID) controllers for an n-link robot manipulator system is proposed. The range of decentralized PID controller parameters for an n-link robot manipulator is obtained using Kharitonov theorem and stability boundary equations. Basically, the proposed design technique is based on the gain-phase margin tester and Kharitonov??s theorem that synthesizes a set of PID controllers for the linear model while nonlinear interaction terms involve in system dynamics are treated as zero. The stability analysis of the composite system with the designed set of decentralized PID controllers is investigated by incorporating bounding parameters of interconnection terms in LMI formulation. From the range of controller gains obtained by the proposed method, a genetic algorithm is adopted to get an optimal controller gains so that the tracking error is minimum. Simulation results are shown to demonstrate the applicability of the proposed control scheme for solution of fixed as well as time-varying trajectory tracking control problems.     

The Global Nonlinear Galerkin Method for the Analysis of Elastic Large Deflections of Plates under Combined Loads: A Scalar Homotopy Method for the Direct Solution of Nonlinear Algebraic Equations

In this paper, the global nonlinear Galerkin method is used to perform an accurate and efficient analysis of the large deflection behavior of a simply-supported rectangular plate under combined loads. Through applying the Galerkin method to the governing nonlinear partial differential equations (PDEs) of the plate, we derive a system of coupled third order nonlinear algebraic equations (NAEs). However, the resultant system of NAEs is thought to be hard to tackle because one has to find the one physical solution from among the possible multiple solutions. Therefore, a suitable initial guess is required to lead to the real solution for given load conditions. The feature of this paper is that we apply the global nonlinear Galerkin method to the governing PDEs and solve the resultant NAEs directly in each load step. To keep track of the physical solution, the initial guess for the current load step is provided by taking the solution of the NAEs for the last step as the initial guess. Besides, the size of the NAEs grows dramatically larger, with the increase of the number of terms of the trial functions, which will cost much more computational efforts. An exponentially convergent scalar homotopy algorithm (ECSHA) is introduced to solve the large set of NAEs. The approach in the present paper is more direct and simpler than either the incremental global Galerkin method, or the incremental local Galerkin method (finite element method) based on a symmetric incremental weak-form; both of which methods lead to the inversion of tangent stiffness matrices and Newton-Raphson iterations in each load step. The present method of exponentially convergent scalar homotopy of directly solving the NAEs is much better than the quadratically convergent Newton-Raphson method. Several numerical examples are provided to validate the feasibility and efficiency of the proposed scheme.     

Three dimensional boundary formulations for nonlinear thermal shape sensitivities

Implicit differentiation of the discretized boundary integral equations governing the conduction of heat in three dimensional (3D) solid objects, subjected to nonlinear boundary conditions, and with temperature dependent material properties, is shown to generate an accurate and economical approach for the computation of shape sensitivities. The theoretical formulation for primary response (surface temperature and normal heat flux) sensitivities and secondary response (surface tangential heat flux components and internal temperature and heat flux components) sensitivities is given. Iterative strategies are described for the solution of the resulting sets of nonlinear equations and computational performances examined. Multi-zone analysis and zone condensation strategies are demonstrated to provide substantial computational economies in this process for models with either localized nonlinear boundary conditions or regions of geometric insensitivity to design variables. A series of nonlinear sensitivity example problems are presented that have closed form solutions. Sensitivities computed using the boundary formulation are shown to be in excellent agreement with these exact expressions.     

Unsteady mixed-convection boundary layer flow along a symmetric wedge with variable surface temperature

Laminar two-dimensional unsteady mixed-convection boundary-layer flow of a viscous incompressible fluid past a sharp wedge has been studied. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear system of partial differential equations is reduced to local non-similarity boundary layer equations, which are solved analytically for small time. Perturbation solutions are also obtained for small and large dimensionless time, τ. Solutions of the governing equations for all time are obtained employing the implicit finite difference method. Here we have focused our attention on the evolution of skin-friction coefficient (C_f) and local Nusselt number (Nu) (heat transfer rate), fluid velocity and fluid temperature with the effects of different governing parameters such as different time, τ, the exponent, m (=0.2, 0.4, 0.6, 0.8, 1.0), mixed convection parameter, λ (= 0.0, 0.5, 1.0) for fluids having Prandtl number, Pr = 0.1, 0.7, and 7.0.     

隔震结构非线性随机地震响应分析的复模态法

本对多自由度双线性滞变隔震结构在过滤白噪声地震激励下的随机响应问题进行了系统研究。首先建立了结构非线性运动方程；然后根据非线性随机振动理论对运动方程进行等效线性化；最后用复模态法获得了等效线性方程的解析解，将复杂的非线性随机响应问题简化为求解一元非线性代数方程问题，并给出了算例。从而建立了多自由度隔震结构非线性随机响应复模态分析的一整套计算方法。