Order-(n+m) direct differentiation determination of design sensitivity for constrained multibody dynamic systems

With the complexity and large dimensionality of many modern multibody dynamic applications, the efficiency of the sensitivity evaluation methods used can greatly impact the overall computation cost and as such can greatly limit the usefulness of the sensitivity information. Most current direct differentiation approaches suffer from prohibitive computational cost, which may be as great as O(n⁴+n²m²+nm³) (for systems with n generalized coordinates and m algebraic constraints). This paper presents a concise and computationally efficient sensitivity analysis scheme to facilitate such sensitivity calculations. A unique feature of this scheme is its use of recursive procedures to directly embed the algebraic constraint relations in forming and simultaneously solving a minimal set of equations. This results in far fewer operations than more traditional, or so-called O(n), counterparts. The algorithm determines the derivatives of generalized accelerations in O(n+m) operations overall. The resulting equations are exact to integration accuracy and enforce constraints exactly at both the velocity and acceleration levels.     

基于隐式微分/代数方程的多体系统动力学设计灵敏度分析方法

基于一般性的积分型目标函数、隐式相容初始条件及终止时刻表达式,系统建立了含设计参数的用隐式微分／代数方程表达的多体系统动力学设计灵敏度分析的直接微分方法和伴随变量方法．为降低目标函数及其对设计变量导数的计算复杂性。将其积分形式的计算转化为微分形式．所得到的结果可方便地应用于高效的间接最优化设计方法．最后通过采用绝对坐标建模的平面两连杆机械臂模型对该方法进行了验证．     

Structural optimization of finite deformation elastoplasticity using continuum-based shape design sensitivity formulation

A continuum-based shape design sensitivity formulation and optimization method is proposed for finite deformation elastoplasticity. In response analysis, the multiplicative decomposition of the deformation gradient into elastic and plastic parts is used for the hyperelasticity-based elastoplastic constitutive model with respect to the intermediate configuration. In design sensitivity analysis, the shape variation at the undeformed configuration is taken using a design velocity concept and then is transformed to the current configuration to recover the updated Lagrangian formulation. The design sensitivity equation of the direct differentiation method is solved at each time step without iteration. The effect of using different reference frames for response analysis and sensitivity analysis is discussed in detail. The path-dependency of the sensitivity is due to the evolutions of the intermediate configuration and the internal plastic variables. A numerical example is shown to confirm the accuracy and efficiency of the proposed computational method using a vehicle bumper optimization.     

Optimization of a contact surface

The paper introduces ideas from shape optimization to multibody system dynamics. A disk rolling down a given slope is taken as a simple example, for which it is the goal of the optimization to shape the rolling contour of the disk such that it takes a minimum time to cover a certain distance. The shape of the contour is described by its radius of curvature. The governing equations of motion result from the kinematics of relative motion and the Newton–Euler formalism. Three different kinds of spirals are defined and optimized.     

基于随机灵敏度分析的化工过程裕量设计

对化工过程进行最优设计时,由于过程中参数的不确定性,需要在既满足过程约束又保证经济效益最优的前提下对设计变量增加裕量。本文考虑过程不确定参数的随机分布,结合灵敏度分析,提出一种基于随机灵敏度的化工过程最优裕量设计方法。首先,取不确定参数的标称值,进行化工过程最优设计,得到最优设计点的标称值;其次,假设过程不确定参数服从正态分布,基于灵敏度分析确定约束变量的均值和方差,在保证低概率违反约束的条件下优化求解设计裕量,定量分析不确定参数对设计变量的影响;最后,通过对串联连续搅拌釜式反应器（Continuous stirred tank reactor,CSTR）系统进行仿真实验,说明该裕量设计方法的具体步骤,并得到合理的设计裕量值,验证所提方法的正确性。     

变拓扑柔性多体系统接触碰撞动力学研究

在实际工程领域中存在着大量接触碰撞等非连续动力学问题,现有的解决柔性多体系统连续动力学过程的建模理论与方法,已经无法解决或无法很好解决这些问题.本文基于变拓扑思想,提出了附加接触约束的柔性多体系统碰撞动力学建模理论；通过设计柔性圆柱杆接触碰撞实验,验证了所提出附加约束接触碰撞模型的有效性；针对柔性多体系统全局动力学仿真面临时间和空间的多尺度问题,提出多变量的离散方法,从而提高了柔性多体系统非连续动力学的仿真效率.     

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.     

Shape sensitivity analysis and design studies for CAD flume sections

This paper presents shape design sensitivity analysis (DSA) and design studies for recreational waterslides represented in computer-aided design (CAD) environment. The mathematical representations of a number of commonly used flume sections that serve as the building blocks for waterslide configurations are created in CAD tools. Geometric dimensions of the individual sections that affect not only their geometric shape but also the overall configurations are identified as design variables. These design variables can be varied to search for better design alternatives, for example, safer waterslides. A set of coupled differential equations based on Lagrange’s equations of motion that describe the motion of the riding object are derived. The equations of motion incorporate friction forces between the riding object and the surface of the flume sections. These second-order differential equations are then solved using Mathematica. Based on the equations of motion and design variables identified, a set of differential equations are derived for calculating shape DSA coefficients. These equations are solved numerically again using Mathematica. The major contribution of the paper are (1) extending waterslide design parameterization and shape DSA computation to true CAD-based flume sections, which greatly alleviates the design for manufacturing issues previously encountered, (2) incorporating friction forces into shape DSA computation, and (3) developing a design scenario that includes sensitivity display and what-if studies for a compromised design that is safer and with a larger acceleration, therefore, higher excitement levels. Incorporating friction forces into the computation supports design for rider’s excitement levels, which are related to accelerations. Waterslide design will not be realistic without including friction forces.     

Integration of topology and shape optimization for design of structural components

This paper presents an integrated approach that supports the topology optimization and CAD-based shape optimization. The main contribution of the paper is using the geometric reconstruction technique that is mathematically sound and error bounded for creating solid models of the topologically optimized structures with smooth geometric boundary. This geometric reconstruction method extends the integration to 3-D applications. In addition, commercial Computer-Aided Design (CAD), finite element analysis (FEA), optimization, and application software tools are incorporated to support the integrated optimization process. The integration is carried out by first converting the geometry of the topologically optimized structure into smooth and parametric B-spline curves and surfaces. The B-spline curves and surfaces are then imported into a parametric CAD environment to build solid models of the structure. The control point movements of the B-spline curves or surfaces are defined as design variables for shape optimization, in which CAD-based design velocity field computations, design sensitivity analysis (DSA), and nonlinear programming are performed. Both 2-D plane stress and 3-D solid examples are presented to demonstrate the proposed approach. Received January 27, 2000 Communicated by J. Sobieski     

On the enhancement of computation and exploration of discretization approaches for meshless shape design sensitivity analysis

A new implementation of Reproducing Kernel Particle Method (RKPM) is proposed to enhance the process of shape design sensitivity analysis (DSA). The acceleration process is accomplished by expressing RKPM shape functions and their derivatives explicitly in terms of kernel function moments. In addition, two different discretization approaches are explored elaborately, which emanate from discretizing design sensitivity equation using the direct differentiation method. Comparison of these two approaches is made, and the equivalence of these two superficially different approaches is demonstrated through two elastostatics problems. The effectiveness of the enhanced RKPM is also verified by comparison of consumption of computer time between the classical method and the improved method.     

翼伞系统最优归航轨迹设计的敏感度分析方法

本文对三自由度翼伞系统归航轨迹优化问题进行了研究,采用控制变量参数化与时间尺度变换相结合的优化算法对翼伞系统的最优控制问题进行数值求解.该方法是基于灵敏度分析的优化算法,将控制量以及控制量转换时间转化为一系列参数优化问题同时进行求解.仿真结果表明,相对于基于两端边值优化算法而言,灵敏度分析法只需要正向积分进行求解,因而具有计算简单、耗时短等优点,其控制效果良好,距离偏差和方向偏差均满足实际需求,有效地提高了翼伞系统的着陆精度,验证了该优化算法的可行性.     

Shape sensitivity analysis using a fixed basis function finite element approach

An approach is presented for the determination of solution sensitivity to changes in problem domain or shape. A finite element displacement formulation is adopted and the point of view is taken that the finite element basis functions and grid are fixed during the sensitivity analysis; therefore, the method is referred to as a “fixed basis function” finite element shape sensitivity analysis. This approach avoids the requirement of explicit or approximate differentiation of finite element matrices and vectors and the difficulty or errors resulting from such calculations. Effectively, the sensitivity to boundary shape change is determined exactly; thus, the accuracy of the solution sensitivity is dictated only by the finite element mesh used. The evaluation of sensitivity matrices and force vectors requires only modest calculations beyond those of the reference problem finite element analysis; that is, certain boundary integrals and reaction forces on the reference location of the moving boundary are required. In addition, the formulation provides the unique family of element domain changes which completely eliminates the inclusion of grid sensitivity from the shape sensitivity calculation. The work is illustrated for some one-dimensional beam problems and is outlined for a two-dimensional C⁰ problem; the extension to three-dimensional problems is straight-forward. Received December 5, 1999?Revised mansucript received July 6, 2000     

飞机结构总体优化设计方法

优化设计可以改善结构的应力分布,合理布置材料,从而提高材料的利用率.通过对全机结构逐层分解,确定优化设计变量、优化区域以及响应约束,并应用MSC Nastran中的可行方向法和敏度分析方法,对MA700飞机的机翼进行优化设计方法研究,考虑稳定性因素,初步解决长桁尺寸确定的问题,为后续优化设计工作积累经验.     

Eulerian shape design sensitivity analysis and optimization with a fixed grid

Conventional shape optimization based on the finite element method uses Lagrangian representation in which the finite element mesh moves according to shape change, while modern topology optimization uses Eulerian representation. In this paper, an approach to shape optimization using Eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed. A continuum geometric model is defined on the fixed grid of finite elements. An active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization, in which each element has a unique shape density. The shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements. Using this transformation, it has been shown that the shape design problem can be treated as a parameter design problem, which is a much easier method than the former. A detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation. Very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary. The accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement. Two design optimization problems are presented to show the feasibility of the proposed design approach.     

Improvement of ship design practice using a 3D CAD model of a hull structure

At the initial stage of ship design, a hull structural model, that is, a 3D CAD model of a hull structure is not generated by the existing shipbuilding CAD system because it is time-consuming and requires much effort. Without the hull structural model, a designer must manually calculate the production material information of a building block by using 2D drawings, parent ship data, and design experiences at the initial planning and scheduling stages. At the initial stage of hull structural analysis, the designer manually generates a structural analysis model, that is, a finite element model of the hull structure. Moreover, the piping model, that is a 3D CAD model of the pipes in the hull structure, is generated independently of the hull structural model at the detailed design stage. To lighten the burden imposed on the designer, we developed an initial hull structural modeling system in our previous study. Using this system, a designer can rapidly and easily generate the hull structural model at the initial stage of design. In this study, the generation methods of the production material information of a building block, the structural analysis model, and the piping model based on the hull structural model are developed. The applicability of the developed methods are demonstrated by applying them to a deadweight 300,000 ton very large crude oil carrier (VLCC). The results show that the developed methods can quickly generate the corresponding information or models at the initial design stage.     

Sensitivity Analysis of Rigid-Flexible Multibody Systems

An important step in the application of automated design techniques to rigid-flexible multibody systems is the calculation of the sensitivities with respect to design variables. Thispaper presents a general formulation for thecalculation of the first order analytical designsensitivities based on the direct differentiationmethod. The analytical sensitivities are comparedwith the numerical results obtained by the finitedifferences method and the accuracy and validity ofboth methods is discussed. Cartesian co-ordinates areused for the dynamic analysis of rigid-flexiblemultibody systems. To reduce the number ofco-ordinates associated with the flexible bodies, thecomponent mode synthesis method is used. Theequations of the sensitivities are obtainedsymbolically and integrated in time simultaneouslywith the dynamic equations. Examples of 2Dsensitivity analysis of the transient response of aslider-crank and of a vehicle with a flexible chassisare presented, and the accuracy and characteristics ofthe sensitivities are analyzed and discussed.     

Information requirements for multi-level-of-development BIM using sensitivity analysis for energy performance

The concept of multi-Level-of-Development (multi-LOD) modelling represents a flexible approach of information management and compilation in building information modelling (BIM) on a set of consistent levels. From an energy perspective during early architectural design, the refinement of design parameters by addition of information allows a more precise prediction of building performance. The need for energy-efficient buildings requires a designer to focus on the parameters in order of their ability to reduce uncertainty in energy performance to prioritise energy relevant decisions. However, there is no method for assigning and prioritising information for a particular level of multi-LOD. In this study, we performed a sensitivity analysis of energy models to estimate the uncertainty caused by the design parameters in energy prediction. This study allows to rank the design parameters in order of their influence on the energy prediction and determine the information required at each level of multi-LOD approach. We have studied the parametric energy model of different building shapes representing architectural design variation at the early design stage. A variance-based sensitivity analysis method is used to calculate the uncertainty contribution of each design parameter. The three levels in the uncertainty contribution by the group of parameters are identified which form the basis of information required at each level of multi-LOD BIM approach. The first level includes geometrical parameters, the second level includes technical specification and operational design parameters, and the third level includes window construction and system efficiency parameters. These findings will be specifically useful in the development of a multi-LOD approach to prioritise performance relevant decisions at early design phases.