Dynamic analysis and optimum design for control of beams with time-dependent changes of length

Structural systems with longitudinal mass flow can be seen in belt-conveyers, running magnetic tapes and flexible manipulators, etc. Transverse vibration problems of flexible longitudinally extensible beams are considered as a special class of vibration problems of structural systems with longitudinal mass flow. The optimum design problem to control the dynamic response of such structural systems is very important from engineering and practical viewpoints, but until now it has not been the subject of much study.Based on the method of dynamic response analysis of extensible beams presented in the authors' previous studies, a method of sensitivity analysis of dynamic response is presented in this paper. Utilizing the obtained sensitivities of the response with respect to design parameters, such as mass flow velocity, tension and rigidity etc., a new optimum design method to control the responses of such structural systems is also presented.     

Optimization of a complex flexible multibody systems with composite materials

The paper presents a general optimization methodology for flexible multibody systems which is demonstrated to find optimal layouts of fiber composite structures components. The goal of the optimization process is to minimize the structural deformation and, simultaneously, to fulfill a set of multidisciplinary constraints, by finding the optimal values for the fiber orientation of composite structures. In this work, a general formulation for the computation of the first order analytical sensitivities based on the use of automatic differentiation tools is applied. A critical overview on the use of the sensitivities obtained by automatic differentiation against analytical sensitivities derived and implemented by hand is made with the purpose of identifying shortcomings and proposing solutions. The equations of motion and sensitivities of the flexible multibody system are solved simultaneously being the accelerations and velocities of the system and the sensitivities of the accelerations and of the velocities integrated in time using a multi-step multi-order integration algorithm. Then, the optimal design of the flexible multibody system is formulated to minimize the deformation energy of the system subjected to a set of technological and functional constraints. The methodologies proposed are first discussed for a simple demonstrative example and applied after to the optimization of a complex flexible multibody system, represented by a satellite antenna that is unfolded from its launching configuration to its functional state.     

First order sensitivity analysis of flexible multibody systems using absolute nodal coordinate formulation

Design sensitivity analysis of flexible multibody systems is important in optimizing the performance of mechanical systems. The choice of coordinates to describe the motion of multibody systems has a great influence on the efficiency and accuracy of both the dynamic and sensitivity analysis. In the flexible multibody system dynamics, both the floating frame of reference formulation (FFRF) and absolute nodal coordinate formulation (ANCF) are frequently utilized to describe flexibility, however, only the former has been used in design sensitivity analysis. In this article, ANCF, which has been recently developed and focuses on modeling of beams and plates in large deformation problems, is extended into design sensitivity analysis of flexible multibody systems. The Motion equations of a constrained flexible multibody system are expressed as a set of index-3 differential algebraic equations (DAEs), in which the element elastic forces are defined using nonlinear strain-displacement relations. Both the direct differentiation method and adjoint variable method are performed to do sensitivity analysis and the related dynamic and sensitivity equations are integrated with HHT-I3 algorithm. In this paper, a new method to deduce system sensitivity equations is proposed. With this approach, the system sensitivity equations are constructed by assembling the element sensitivity equations with the help of invariant matrices, which results in the advantage that the complex symbolic differentiation of the dynamic equations is avoided when the flexible multibody system model is changed. Besides that, the dynamic and sensitivity equations formed with the proposed method can be efficiently integrated using HHT-I3 method, which makes the efficiency of the direct differentiation method comparable to that of the adjoint variable method when the number of design variables is not extremely large. All these improvements greatly enhance the application value of the direct differentiation method in the engineering optimization of the ANCF-based flexible multibody systems.     

Design sensitivity analysis for transient response of non-viscously damped dynamic systems

A design sensitivity analysis for the transient response of the non-viscously damped dynamic systems is presented. The non-viscously (viscoelastically) damped system is widely used in structural vibration control. The damping forces in the system depend on the past history of motion via convolution integrals. The non-viscos damping is modeled by the generalized Maxwell model. The transient response is calculated with the implicit Newmark time integration scheme. The design sensitivity analysis method of the history dependent system is developed using the adjoint variable method. The discretize-then-differentiate approach is adopted for deriving discrete adjoint equations. The accuracy and the consistency of the proposed method are demonstrated through a single dof system. The proposed method is also applied to a multi-dof system. The validity and accuracy of the sensitivities from the proposed method are confirmed by finite difference results.     

Singular value decomposition for dynamic system design sensitivity analysis

A computer-based method for automatic generation and efficient numerical solution of mixed differential-algebraic equations for dynamic and design sensitivity analysis of dynamic systems is developed. The equations are written in terms of a maximal set of Cartesian coordinates to facilitate general formulation of kinematic and design constraints and forcing functions. Singular value decomposition of the system Jacobian matrix generates a set of composite generalized coordinates that are best suited to represent the system. The coordinates naturally partition into optimal independent and dependent sets, and integration of only the independent coordinates generates all of the system information. An adjoint variable method is used to compute design sensitivities of dynamic performance measures of the system. A general-purpose computer program incorporating these capabilities has been developed. A numerical example is presented to illustrate accuracy and properties of the method.     

Structural topology optimization of multibody systems

Flexible multibody dynamics (FMD) has found many applications in control, analysis and design of mechanical systems. FMD together with the theory of structural optimization can be used for designing multibody systems with bodies which are lighter, but stronger. Topology optimization of static structures is an active research topic in structural mechanics. However, the extension to the dynamic case is less investigated as one has to face serious numerical difficulties. One way of extending static structural topology optimization to topology optimization of dynamic flexible multibody system with large rotational and transitional motion is investigated in this paper. The optimization can be performed simultaneously on all flexible bodies. The simulation part of optimization is based on an FEM approach together with modal reduction. The resulting nonlinear differential-algebraic systems are solved with the error controlled integrator IDA (Sundials) wrapped into Python environment by Assimulo (Andersson et al. in Math. Comput. Simul. 116(0):26–43, 2015). A modified formulation of solid isotropic material with penalization (SIMP) method is suggested to avoid numerical instabilities and convergence failures of the optimizer. Sensitivity analysis is central in structural optimization. The sensitivities are approximated to circumvent the expensive calculations. The provided examples show that the method is indeed suitable for optimizing a wide range of multibody systems. Standard SIMP method in structural topology optimization suggests stiffness penalization. To overcome the problem of instabilities and mesh distortion in the dynamic case we consider here additionally element mass penalization.     

2-D and 3-D shape optimization using mesh velocities to integrate analytical sensitivities with associative CAD

The design sensitivities generated with the mesh velocity method, used by the authors, are compared with those obtained by the boundary layer and boundary displacement methods. The effect of adaptive mesh refinement and error control on the quality of the velocity fields is discussed, as well as their ability to yield accurate first-order predictions of constraint values. Two numerical shape optimization examples of a 2D and a 3D component are presented. These examples are used to illustrate the benefits of integrating analytical methods of design sensitivity analysis with parametric capabilities supported by state-of-the-art CAD systems.     

Erratum to: New efficient method for dynamic modeling and simulation of flexible multibody systems moving in plane

Efficient, precise dynamic analysis for general flexible multibody systems has become a research focus in the field of flexible multibody dynamics. In this paper, the finite element method and component mode synthesis are introduced to describe the deformations of the flexible components, and the dynamic equations of flexible bodies moving in plane are deduced. By combining the discrete time transfer matrix method of multibody system with these dynamic equations of flexible component, the transfer equations and transfer matrices of flexible bodies moving in plane are developed. Finally, a high-efficient dynamic modeling method and its algorithm are presented for high-speed computation of general flexible multibody dynamics. Compared with the ordinary dynamics methods, the proposed method combines the strengths of the transfer matrix method and finite element method. It does not need the global dynamic equations of system and has the low order of system matrix and high computational efficiency. This method can be applied to solve the dynamics problems of flexible multibody systems containing irregularly shaped flexible components. It has advantages for dynamic design of complex flexible multibody systems. Formulations as well as a numerical example of a multi-rigid-flexible-body system containing irregularly shaped flexible components are given to validate the method.     

New efficient method for dynamic modeling and simulation of flexible multibody systems moving in plane

Efficient, precise dynamic analysis for general flexible multibody systems has become a research focus in the field of flexible multibody dynamics. In this paper, the finite element method and component mode synthesis are introduced to describe the deformations of the flexible components, and the dynamic equations of flexible bodies moving in plane are deduced. By combining the discrete time transfer matrix method of multibody system with these dynamic equations of flexible component, the transfer equations and transfer matrices of flexible bodies moving in plane are developed. Finally, a high-efficient dynamic modeling method and its algorithm are presented for high-speed computation of general flexible multibody dynamics. Compared with the ordinary dynamics methods, the proposed method combines the strengths of the transfer matrix method and finite element method. It does not need the global dynamic equations of system and has the low order of system matrix and high computational efficiency. This method can be applied to solve the dynamics problems of flexible multibody systems containing irregularly shaped flexible components. It has advantages for dynamic design of complex flexible multibody systems. Formulations as well as a numerical example of a multi-rigid-flexible-body system containing irregularly shaped flexible components are given to validate the method.     

Structural topology optimization under harmonic base acceleration excitations

This work is focused on the structural topology optimization methods related to dynamic responses under harmonic base acceleration excitations. The uniform acceleration input model is chosen to be the input form of base excitations. In the dynamic response analysis, we propose using the large mass method (LMM) in which artificial large mass values are attributed to each driven nodal degree of freedom (DOF), which can thus transform the base acceleration excitations into force excitations. Mode displacement method (MDM) and mode acceleration method (MAM) are then used to calculate the harmonic responses and the design sensitivities due to their balances between computing efficiency and accuracy especially when frequency bands are taken into account. A density based topology optimization method of minimizing dynamic responses is then formulated based on the integration of LMM and MDM or MAM. Moreover, some particular appearances such as the precision of response analysis using MDM or MAM, and the duplicated frequencies are briefly discussed. Numerical examples are finally tested to verify the accuracy of the proposed schemes in dynamic response analysis and the quality of the optimized design in improving dynamic performances.     

Adjoint design sensitivity analysis of dynamic crack propagation using peridynamic theory

Based on the peridynamics of the reformulated continuum theory, an adjoint design sensitivity analysis (DSA) method is developed for the solution of dynamic crack propagation problems using the explicit scheme of time integration. Non-shape DSA problems are considered for the dynamic crack propagation including the successive branching of cracks. The adjoint variable method is generally suitable for path-independent problems but employed in this bond-based peridynamics since its path is readily available. Since both original and adjoint systems possess time-reversal symmetry, the trajectories of systems are symmetric about the u-axis. We take advantage of the time-reversal symmetry for the efficient and concurrent computation of original and adjoint systems. Also, to improve the numerical efficiency of large scale problems, a parallel computation scheme is employed using a binary space decomposition method. The accuracy of analytical design sensitivity is verified by comparing it with the finite difference one. The finite difference method is susceptible to the amount of design perturbations and could result in inaccurate design sensitivity for highly nonlinear peridynamics problems with respect to the design. It is demonstrated that the peridynamic adjoint sensitivity involving history-dependent variables can be accurate only if the path of the adjoint response analysis is identical to that of the original response.     

Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems

There are two commonly used analytical reliability analysis methods: linear approximation - first-order reliability method (FORM), and quadratic approximation - second-order reliability method (SORM), of the performance function. The reliability analysis using FORM could be acceptable in accuracy for mildly nonlinear performance functions, whereas the reliability analysis using SORM may be necessary for accuracy of nonlinear and multi-dimensional performance functions. Even though the reliability analysis using SORM may be accurate, it is not as much used for probability of failure calculation since SORM requires the second-order sensitivities. Moreover, the SORM-based inverse reliability analysis is rather difficult to develop.This paper proposes an inverse reliability analysis method that can be used to obtain accurate probability of failure calculation without requiring the second-order sensitivities for reliability-based design optimization (RBDO) of nonlinear and multi-dimensional systems. For the inverse reliability analysis, the most probable point (MPP)-based dimension reduction method (DRM) is developed. Since the FORM-based reliability index (β) is inaccurate for the MPP search of the nonlinear performance function, a three-step computational procedure is proposed to improve accuracy of the inverse reliability analysis: probability of failure calculation using constraint shift, reliability index update, and MPP update. Using the three steps, a new DRM-based MPP is obtained, which estimates the probability of failure of the performance function more accurately than FORM and more efficiently than SORM. The DRM-based MPP is then used for the next design iteration of RBDO to obtain an accurate optimum design even for nonlinear and/or multi-dimensional system. Since the DRM-based RBDO requires more function evaluations, the enriched performance measure approach (PMA+) with new tolerances for constraint activeness and reduced rotation matrix is used to reduce the number of function evaluations.     

A general optimization strategy for sound power minimization

A general approach for minimizing radiated acoustic power of a baffled plate excited by broad band harmonic excitation is given. The steps involve a finite element discretization for expressing acoustic power and vibration analysis, analytical design sensitivity analysis, and the use of gradient-based optimization algorithms. Acoustic power expressions are derived from the Rayleigh integral for plates. A general methodology is developed for computing design sensitivities using analytical expressions. Results show that analytical sensitivity analysis is important from both computational time and accuracy considerations. Applications of the optimization strategy to rectangular plates and an engine cover plate are presented. Thicknesses are chosen as design variables.     

Optimization of dynamic response using a monolithic-time formulation

The design of systems for dynamic response may involve constraints that need to be satisfied over an entire time interval or objective functions evaluated over the interval. Efficiently performing the constrained optimization is challenging, since the typical response is implicitly linked to the design variables through a numerical integration of the governing differential equations. Evaluating constraints is costly, as is the determination of sensitivities to variations in the design variables. In this paper, we investigate the application of a temporal spectral element method to the optimization of transient and time-periodic responses of fundamental engineering systems. Through the spectral discretization, the response is computed globally, thereby enabling a more explicit connection between the response and design variables and facilitating the efficient computation of response sensitivities. Furthermore, the response is captured in a higher order manner to increase analysis accuracy. Two applications of the coupling of dynamic response optimization with the temporal spectral element method are demonstrated. The first application, a one-degree-of-freedom, linear, impact absorber, is selected from the auto industry, and tests the ability of the method to treat transient constraints over a large-time interval. The second application, a related mass-spring-damper system, shows how the method can be used to obtain work and amplitude optimal time-periodic control force subject to constraints over a periodic time interval. This research was performed while the first author held a National Research Council Research Associateship Award at the Air Force Research Laboratory. An early version of this paper was presented at the 46th AIAA Aerospace Sciences Meeting and Exhibit, Jan 7–10, 2008, Reno, Nevada.     

Analytical expressions of sensitivities for shape variables in linear bending systems

Sensitivity analysis is a very interesting field in structural engineering because of its variety of uses. But the computational effort to obtain the analytical values of such sensitivities is a tough task that has been generally avoided when considering flexural systems. Instead some numerical approaches have been used to solve the problem. However, carrying out the sensitivity analysis by this method leads to considerable errors, especially with shape variables as many authors have pointed out. In this paper analytical expressions of sensitivities analysis with respect to shape variables are carried out for bending systems in linear theory. The development presented in this paper starts evaluating the sensitivity analysis of the nodal movements performing the loading vector and stiffness matrix sensitivity analysis. Then this research evaluates the sensitivity analysis of the maximum normal stresses. Finally, some structural examples where the previous analytical sensitivities are evaluated are exposed relating the results versus the corresponding results obtained by finite difference methods and some conclusions are drawn from the work presented.     

Semi-analytical sensitivity analysis for nonlinear transient problems

Efficient analytical sensitivity computations are essential elements of gradient-based optimization schemes; unfortunately, they can be difficult to implement. This implementation issue is often resolved by adopting the semi-analytical method which exhibits the efficiency of the analytical methods and the ease of implementation of the finite difference method. However, care must be taken as semi-analytical sensitivities may exhibit errors due to truncation and round-off. Additional errors are introduced if the convergence tolerance of the primal analysis is not sufficiently small. This paper gives a general overview and some new developments of the analytical and semi-analytical sensitivity analyses for nonlinear steady-state, transient, and dynamic problems. We discuss the restrictive assumptions, accuracy, and consistency of these methods. Both adjoint and direct differentiation methods are studied. Numerical examples are provided.     

带有超大型挠性网状天线航天器姿控系统的参数化多目标设计

本文研究了带有超大型挠性附件的卫星的姿态控制问题.关于挠性航天器振动抑制与姿态控制,绝大多数已有的控制方法都是针对某一单一指标提出的.然而工程上要同时兼顾精度、快速性、平稳性、挠性部件的振动抑制以及各种鲁棒性,因此挠性航天器的姿态控制系统设计实际上是一个典型的多目标设计问题.本文针对具有超大挠性网状天线卫星的俯仰通道姿态系统,提出了一种基于输出反馈的鲁棒极点配置的参数化多目标设计方法.首先给出了系统能控与能观的充分必要条件,然后给出了动态补偿器以及特征向量矩阵的参数化表达,并在此基础上进一步对自由参向量进行了多目标优化,使得控制系统具有:1)配置到期望区域的极点; 2)较低的特征值灵敏度;3)较强的高阶未建模动态抑制能力; 4)尽可能小的控制增益.最后,本文根据卫星工程参数进行了控制器设计与仿真验证,仿真结果表明本文提出的方法可以在动态响应、高阶未建模动态抑制能力、控制量峰值等方面优于传统的PID控制器.     

细长柔性空间结构几种动力学模型的比较

对于自由-自由边界的大型柔性梁式空间结构在轨搬运过程中,其大范围刚体运动和柔性振动会相互耦合,是一类典型的刚柔耦合动力学问题.建立相对准确的动力学模型是设计良好控制系统的前提,但现有文献在研究该问题时却采用了忽略刚柔耦合作用的动力学模型并依此设计控制器,因此有必要建立耦合模型,并探讨其与非耦合模型之间的区别和适用性.首先针对结构自身运动特点选择以瞬时质心为原点的浮动坐标系作为辅助坐标系,将结构两类不同的运动形式进行分解,并利用其产生的附加约束条件简化虚功表达式;其次选择Euler-Bernoulli梁变形形式描述结构变形并采用假设模态法对变形进行变量分离;基于虚功原理推导得到结构大范围运动的刚柔耦合动力学模型;通过仿真算例1对非耦合模型、零次近似模型和一次近似模型进行了对比,验证了非耦合模型的不合理性及零次近似简化模型的准确性和有效性;通过仿真算例2对零次近似简化模型和一次近似模型的对比,说明了二者的使用范围;仿真对比为后续的运动控制系统设计和振动抑制研究提供了依据.     

Mixed elements in the optimal design of plates

The theory of design sensitivity analysis of structures, based on mixed finite element models, is developed for static, dynamic and stability constraints. The theory is applied to the optimal design of plates with minimum weight, subject to displacement, stress, natural frequencies and buckling stresses constraints. The finite element model is based on an eight node mixed isoparametric quadratic plate element, whose degrees of freedom are the transversal displacement and three moments per node. The corresponding nonlinear programming problem is solved using the commercially available ADS (Automated Design Synthesis) program. The sensitivities are calculated by analytical, semi-analytical and finite difference techniques. The advantages and disadvantages of mixed elements in design optimization of plates are discussed with reference to applications.     

On shape sensitivities with heaviside-enriched XFEM

This paper investigates the behavior of shape sensitivities within the context of the eXtended Finite Element Method (XFEM) using a Heaviside enrichment strategy, wherein the shape derivative is computed by the adjoint method. The Heaviside function is discontinuous by construction. This feature of the enrichment function presents advantages as well as challenges in the computation of shape sensitivities, both of which are discussed in detail in this paper. Using continuum and discrete approaches, we present the derivation of analytical shape sensitivities with respect to the design variables which define the design geometry. We propose a robust semi-analytical approach to computing the shape sensitivities, which provides great ease of implementation as compared to fully analytical approaches. The behavior of the XFEM-based shape sensitivities is analyzed using linear heat diffusion examples in 2D, and an incompressible fluid flow example in 3D. We compare XFEM-based shape sensitivities against shape sensitivities obtained through the classical approach of using a body-fitted mesh. It is found that the former are not as smooth as those obtained using a comparable body-fitted mesh. This discrepancy is shown to be an outcome of the discretization error of the design geometry on a background mesh and is not a consequence of the approach by which the XFEM-based shape sensitivities are computed.