Optimal spatiotemporal reduced order modeling, Part I: proposed framework

An optimal spatiotemporal reduced order modeling framework is proposed for nonlinear dynamical systems in continuum mechanics. In this paper, Part I, the governing equations for a general system are modified for an under-resolved simulation in space and time with an arbitrary discretization scheme. Basic filtering concepts are used to demonstrate the manner in which subgrid-scale dynamics arise with a coarse computational grid. Models are then developed to account for the underlying spatiotemporal structure via inclusion of statistical information into the governing equations on a multi-point stencil. These subgrid-scale models are designed to provide closure by accounting for the interactions between spatiotemporal microscales and macroscales as the system evolves. Predictions for the modified system are based upon principles of mean-square error minimization, conditional expectations and stochastic estimation, thus rendering the optimal solution with respect to the chosen resolution. Practical methods are suggested for model construction, appraisal, error measure and implementation. The companion paper, Part II, is devoted to demonstrating the methodology through a computational study of a nonlinear beam.     

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.     

Structural uncertainty modeling for nonlinear geometric response using nonintrusive reduced order models

The focus of the present investigation is on the introduction of uncertainty directly in reduced order models of the nonlinear geometric response of structures following maximum entropy concepts. While the approach was formulated and preliminary validated in an earlier paper, its broad application to a variety of structures based on their finite element models from commercial software was impeded by two key challenges. The first of these involves an indeterminacy in the mapping of the nonlinear stiffness coefficients identified from the finite element model to those of the reduced order model form that is suitable for the uncertainty analysis. The second challenge is that a key matrix in the uncertainty modeling was expected to be positive definite but was numerically observed not to be. This latter issue is shown here to be rooted in differences in nonlinear finite element modeling between the commercial software and the theoretical developments. Both of these challenges are successfully resolved and applications examples are presented that confirm the broad applicability of the methodology.     

几何非线性空间梁的动力学建模与实验研究

建立了大变形细长空间梁的几何非线性动力学模型,并通过动力学实验验证建模理论的正确性。首先用曲梁中线上任意一点的3个绝对位置坐标和横截面的3个姿态角描述横截面的位置和姿态,建立了应变和曲率与位置坐标、姿态角的关系,在此基础上基于中线切线与横截面法线重合的假设,对节点广义坐标进行缩减,简化了动力学模型。用虚功原理建立了大变形细长空间梁的动力学方程,将该方法的计算时间与现有大型工程软件(LS-DYNA)进行比较,验证了方法的有效性。引入运动学约束方程,建立了气浮台和柔性空间梁系统的多体系统动力学方程。在大变形情况下,开展了气浮台和柔性空间梁系统的刚-柔耦合动力学实验,验证了几何非线性空间梁动力学模型的准确性。     

Contact-friction modeling within elastic beam assemblies: an application to knot tightening

In this paper we propose a finite element approach which simulates the mechanical behaviour of beam assemblies that are subject to large deformations and that develop contact-friction interactions. We focus on detecting and modeling contact-friction interactions within the assembly of beams. Contact between beams—or between parts of the same beam in the case of self-contact, is detected from intermediate geometries defined within proximity zones associating close parts of beam axes. The discretization of contact-friction interactions is performed on these intermediate geometries by means of contact elements, constituted of pairs of material particles which are predicted to enter into contact. A 3D finite strain beam model is used to represent the behaviour of each beam. This model describes the kinematics of each beam cross-section using nine degrees of freedom, and is therefore able to represent plane deformations of these cross-sections. Algorithms are proposed to solve the global nonlinear problem using an implicit scheme, under quasi-static assumptions. Simulation results of the tightening and releasing of knots made on monofilament and multifilament yarns are shown as an application. Straight fibers are first twisted together to make a yarn, before suitable conditions are applied to their ends to form and tighten the knot. Tightening forces are finally released to obtain an equilibrium configuration of the knot without external forces.     

Fixed-point iteration to nonlinear finite element analysis. Part II: Formulation and implementation

The finite element formulation and implementation of the Fixed-Point Iteration (FPI) to linear/nonlinear structural static or dynamic analysis are developed. The direct and tangent formulations are presented and compared with the Newton–Raphson method (NRM). ‘Modified’ FPI algorithms have also been derived. A graphical interpretation of the method is introduced and suggested to call the FPI ‘the Saw method’. Mixing both the FPI and NRM is shown to be possible and may be useful in some applications. The overall strategies, iterative algorithms, and the appropriate norm convergence criteria necessary to implant the FPI into existing finite element programs are also included in the development. The superiority of the FPI over the NRM as seen from the development and the formulation lies in three major factors. First, the existing assembly process of element matrices is eliminated completely from the nonlinear finite element analysis. Secondly, the Gauss elimination or Crout's method is also eliminated. In the finite element terminology, this means that nonlinear structural static or dynamic responses can he obtained without recourse to the inverse of the structural stiffness matrix. Thirdly, the FPI can also be applied equally to linear structural analysis. Hence, the assembly process and the programming and storage associated with it can be removed from the existing finite element programs. While the FPI can solve problems that the NRM can, it will also be able to handle some engineering problems where the latter cannot. Buckling problems and problems where the force–displacement curve changes curvature are examples where the FPI is expected to be efficient.     

Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle. Part II: parametric study

The capabilities of the proposed nonlocal beam models in the companion paper in capturing the critical velocity of a moving nanoparticle as well as the dynamic response of double-walled carbon nanotubes (DWCNTs) under a moving nanoparticle are scrutinized in some detail. The role of the small-scale effect parameter, slenderness of DWCNTs and velocity of the moving nanoparticle on dynamic deflections and nonlocal bending moments of the innermost and outermost tubes as well as their maximum values are then investigated. The results reveal that the critical velocity increases with the slenderness of DWCNTs and the magnitude of the van der Waals interaction force. Nevertheless, the critical velocity generally decreases with the small-scale effect as well as the ratio of the mean diameter to the thickness of the innermost tube. Additionally, the predicted maximum dynamic deflections and nonlocal bending moments of the innermost and outermost tubes by using the nonlocal Euler?CBernoulli and Timoshenko beam theories are generally the lower and upper bounds of those obtained by the nonlocal higher-order beam theory (NHOBT). In the case of ??₁?<?20, the use of the NHOBT is highly recommended for more realistic prediction of dynamic response of DWCNTs under a moving nanoparticle.     

Optimal hybrid push/pull control strategies for a parallel multistage system: Part II

The strategies developed in part I of this paper are extended to a more general case in part II. Based on the optimal policy structure of a Markov Decision Process model and by control and information structure analysis, we propose that the ‘best’ strategy for a general series/parallel multistage (assembly) production/inventory system is to use a push (MRP) strategy at all initial stages of the system and a pull (JIT) strategy at all the other downstream stages. As viewed from the information structure, this recommended strategy is a group of decentralized controllers with a centralized coordinator.     

Progress in life time modeling of APS-TBC Part II: critical strains,macro-cracking,and thermal fatigue

Abstract

Here TBC top coat failure and life time are considered as a two-step process, where step 1 is time to delamination macro-cracking and step 2 is time to through macro-cracking. This two-step failure mechanism comes directly from critical strain measurement data and traditional coating failure theory. The critical strain data was measured by using a four point bend test on preoxidized samples. Included in the model is damage accumulation due to bond coat oxidation and thermal fatigue. The damage terms in the model have their origins in acoustic emission data from cyclic oxidation samples. The life time model is then used in concert with measured thermally grown oxide micro-crack lengths from isothermal oxidation samples. Finally a full model is presented for isothermal and cyclic oxidation of APS–TBC.     

Optimal placement of piezo actuators on a beam: a dynamic problem from stochastic to deterministic

In order to deal with uncertainties in optimization problems, this paper addresses a novel methodology which consists of the nonlinear interval number programming (NINP), particle swarm optimization (PSO) and finite element method (FEM). The placement of piezo actuators on a beam is optimized to obtain the dynamic maximum tip deflection when it is subjected to applied voltages varying in time by the use of the methodology; in the meantime, two constraints are considered, i.e. the deflection of midpoint of the beam should change in a specific range and the maximum bond shear stress should be below a specific value (interval number). To provide a valuable guide to designers, the influence degree of every stochastic parameter is studied, e.g. the dielectric permittivity and thickness of patches. In the optimization process, all stochastic parameters are eliminated through a NINP theory, and so a determined design result is finally obtained.     

Optimal platform design using non-dominated sorting genetic algorithm II and technique for order of preference by similarity to ideal solution; application to automotive suspension system

Unlike conventional approaches where optimization is performed on a unique component of a specific product, optimum design of a set of components for employing in a product family can cause significant reduction in costs. Increasing commonality and performance of the product platform simultaneously is a multi-objective optimization problem (MOP). Several optimization methods are reported to solve these MOPs. However, what is less discussed is how to find the trade-off points among the obtained non-dominated optimum points. This article investigates the optimal design of a product family using non-dominated sorting genetic algorithm II (NSGA-II) and proposes the employment of technique for order of preference by similarity to ideal solution (TOPSIS) method to find the trade-off points among the obtained non-dominated results while compromising all objective functions together. A case study for a family of suspension systems is presented, considering performance and commonality. The results indicate the effectiveness of the proposed method to obtain the trade-off points with the best possible performance while maximizing the common parts.     

Sliding beams,Part II: time integration

In this paper we obtain solutions for the discretized incremental system equations, as obtained in Part I, under certain initial and boundary conditions and/or specified applied loads, using the variable domain beam element. As a check on the validity of implementation, we first limit ourselves to linear analysis and obtain results for the axially inextensible sliding beams which we compare with the results reported in the literature. Second we set the axial velocity to zero and solve some special cases when the length of the beam is constant. In this case, we check the formulation and its implementation for non-linearities in the system due to large displacements. Finally, we solve the sliding beam problem for small amplitude oscillations, with a non-linear solver and compare the results with those obtained by the linear solver used for inextensible sliding beams. With these preliminary tests completed, we obtain the transient response of the free and forced large amplitude vibrations of the flexible sliding beam and demonstrate the need for using a non-linear analysis for this complex system. Finally, we consider the stability of the motion of periodically time varying flexible sliding beams and show that in the case of parametric resonance, the unstable regions obtained in the linear analysis, which imply unbounded amplitudes, are indeed stable and bounded when non-linear terms are taken into account. © 1998 John Wiley & Sons, Ltd.     

应用优化的同伦分析法求解非线性Jerk方程

应用优化的同伦分析法计算了具有三次非线性项的三阶微分方程(Jerk)的近似周期和近似解析周期解。文中给出一个算例说明由优化的同伦分析法可以容易得到精确的二阶近似周期解。当初速度 比较大时,一阶近似周期与精确周期的百分比误差为-0.415%,而二阶近似周期与精确周期的百分比误差为-0.0298%。与数值方法给出的“精确”周期解比较,一阶近似解析周期解和二阶近似周期解的精度很高。这个说明同伦分析法对求解非线性Jerk方程非常有效     

An intrinsic beam model based on a helicoidal approximation—Part II: Linearization and finite element implementation

The helicoidal beam model developed in the first part of this work is applied here to the development of a mixed finite element for space-curved and twisted beams undergoing large displacements and finite rotations. Starting from the governing weak form expressed by the principle of virtual work, a consistent linearization is obtained in the following and a novel updated Lagrangian finite element implementation is thoroughly discussed. The unique features and the distinguishing properties previously claimed for the helicoidal model are shown here to imply remarkable numerical consequences. For this purpose, meaningful example problems regarding the non-linear static response of beams are addressed in the following and the results are compared with those available from the literature. Furthermore, a finite element in time for the rigid body dynamic problem is developed within the framework of the helicoidal geometry. The underlying philosophy of this novel finite element for dynamics is the realization of the helicoidal decomposition of the rigid body motion within a time step.     

Stochastic reduced order models for random vectors: Application to random eigenvalue problems

An improved optimization algorithm is presented to construct accurate reduced order models for random vectors. The stochastic reduced order models (SROMs) are simple random elements that have a finite number of outcomes of unequal probabilities. The defining SROM parameters, samples and corresponding probabilities, are chosen through an optimization problem where the objective function quantifies the discrepancy between the statistics of the SROM and the random vector being modeled. The optimization algorithm proposed shows a substantial improvement in model accuracy and significantly reduces the computational time needed to form SROMs, as verified through numerical comparisons with the existing approach. SROMs formed using the new approach are applied to efficiently solve random eigenvalue problems, which arise in the modal analysis of structural systems with uncertain properties. Analytical bounds are established on the discrepancy between exact and SROM-based solutions for these problems. The ability of SROMs to approximate the natural frequencies and modes of uncertain systems as well as to estimate their dynamics in time is illustrated through comparison with Monte Carlo simulation in numerical examples.     

Development of a HTS magnet and application to a beam scanner

A solenoid magnet using high-temperature superconductor tape was designed, fabricated, and tested for its suitability as beam scanner. After successful cooling tests, the magnet performance was studied using DC and AC currents. With DC current the magnet was successfully operated as mirror coils for a 2.45 GHz ECR ion source. The coils could be operated at 100 A and frequencies above 1 Hz. The installation of iron pole pieces and return yokes, enabled us to generate fields in excess of 2 T at 197 A DC. In AC mode this magnet can be operated in the ranges of 0.14–1.73 and 1.22–1.67 T at frequencies of 0.05 and 0.25 Hz, respectively.     

Electron beam surface treatment. Part II: microstructure evolution of stainless steel and aluminum alloy during electron beam rapid solidification

Surface rapid solidification microstructures of AISI 321 austenitic stainless steel and 2024 aluminum alloy have been investigated by electron beam remelting process and optical microscopy observation. It is indicated that the morphologies of the melted layer of both stainless steel and aluminum alloy change dramatically compared to the original materials. Also, the microstructures were greatly refined after the electron beam irradiation.     

A methodology for fault diagnosis in large chemical processes and an application to a multistage flash desalination process: Part II

In Part I, an efficient method for identifying faults in large processes was presented. The whole plant is divided into sectors by using structural, functional, or causal decomposition. A signed directed graph (SDG) is the model used for each sector. The SDG represents interactions among process variables. This qualitative model is used to carry out qualitative simulation for all possible faults. The output of this step is information about the process behaviour. This information is used to build rules. When a symptom is detected in one sector, its rules are evaluated using on-line data and fuzzy logic to yield the diagnosis. In this paper the proposed methodology is applied to a multiple stage flash (MSF) desalination process. This process is composed of sequential flash chambers. It was designed for a pilot plant that produces drinkable water for a community in Argentina; that is, it is a real case. Due to the large number of variables, recycles, phase changes, etc., this process is a good challenge for the proposed diagnosis method.