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.     

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。     

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.     

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     

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.     

Analytical derivatives technology for structural shape design

The ability to perform and evaluate the effect of shape changes on the stress and modal responses of components is an important ingredient in the “design” of aircraft engine components. The classical design of experiments (DOE)-based approach that is motivated from statistics (for physical experiments) is one of the possible approaches for the evaluation of the component response with respect to design parameters [Myers, Montgomery. Response surface methodology, process and product optimization using design of experiments. John Wiley and Sons, NY (1995)]. As the underlying physical model used for the component response is deterministic and understood through a computer simulation model, one needs to re-think the use of the classical DOE techniques for this class of problems. In this paper, we explore an alternate sensitivity-analysis-based technique where a deterministic parametric response is constructed using exact derivatives of the complex finite-element (FE)-based computer models to design parameters. The method is based on a discrete sensitivity analysis formulation using semi-automatic differentiation (Griewank, SIAM (2000), ADIFOR, Automatic Differentiation of FORTRAN codes ) to compute the Taylor series or its Pade equivalent for finite-element-based responses. Shape design or optimization in the context of finite element modeling is challenging because the evaluation of the response for different shape requires the need for a meshing consistent with the new geometry. This paper examines the differences in the nature and performance (accuracy and efficiency) of the analytical derivatives approach against other existing approaches with validation on several benchmark structural applications. The use of analytical derivatives for parametric analysis is demonstrated to have accuracy benefits on certain classes of shape applications.     

Optimization of hyperelastic materials with isotropic damage

In many practical problems, engineering structures under repeated loading exhibit softening material behaviour. The complex micromechanical processes yielding the observed loss of stiffness are often described phenomenologically on the macroscopic level by damage mechanics. A finite strain elastic constitutive model incorporating an isotropic damage mechanism was developed by Simo (1987). The additional theoretical and computational enhancements for utilizing this damage model and the associated finite element formulation for optimization purposes are outlined in this paper.?The structural response and its sensitivity expressions at a given time and position depend on the response and the response sensitivities of all previous locations and times. The expressions for variational design sensitivity analysis within damage mechanics are fully stated and related to prior work on history dependent material behaviour such as Prandtl-Reuss elastoplasticity, see Barthold and Wiechmann (1997) and Wiechmann et al. (1997). The theoretical details and the corresponding finite element formulation were described in the paper by Firuziaan (1998).?New problem functions based on the internal variables are shown to be adequate for controlling and optimizing the damage process.?Numerical experiments illustrate the method proposed and the efficiency of the overall optimization procedure. Received April 29, 1999     

Optimization of nonlinear trusses using a displacement-based approach

A displacement-based optimization strategy is extended to the design of truss structures with geometric and material nonlinear responses. Unlike the traditional optimization approach that uses iterative finite element analyses to determine the structural response as the sizing variables are varied by the optimizer, the proposed method searches for an optimal solution by using the displacement degrees of freedom as design variables. Hence, the method is composed of two levels: an outer level problem where the optimal displacement field is searched using general nonlinear programming algorithms, and an inner problem where a set of optimal cross-sectional dimensions are computed for a given displacement field. For truss structures, the inner problem is a linear programming problem in terms of the sizing variables regardless of the nature of the governing equilibrium equations, which can be linear or nonlinear in displacements. The method has been applied to three test examples, which include material and geometric nonlinearities, for which it appears to be efficient and robust. Received December 4, 2000     

Efficient reliability-based design optimization using a hybrid space with application to finite element analysis

The design of high technology structures aims to define the best compromise between cost and safety. The Reliability-Based Design Optimization (RBDO) allows us to design structures which satisfy economical and safety requirements. However, in practical applications, the coupling between the mechanical modelling, the reliability analyses and the optimization methods leads to very high computational time and weak convergence stability. Traditionally, the solution of the RBDO model is achieved by alternating reliability and optimization iterations. This approach leads to low numerical efficiency, which is disadvantageous for engineering applications on real structures. In order to avoid this difficulty, we propose herein a very efficient method based on the simultaneous solution of the reliability and optimization problems. The procedure leads to parallel convergence for both problems in a Hybrid Design Space (HDS). The efficiency of the proposed methodology is demonstrated on the design of a steel hook, where the RBDO is combined with Finite Element Analysis (FEA).     

Product platform design through sensitivity analysis and cluster analysis

Scale-based product platform design consists of platform configuration to decide which variables are shared among which product variants, and selection of the optimal values for platform (shared) and non-platform variables for all product variants. The configuration step plays a vital role in determining two important aspects of a product family: efficiency (cost savings due to commonality) and effectiveness (capability to satisfy performance requirements). Many existing product platform design methods ignore it, assuming a given platform configuration. Most approaches, whether or not they consider the configuration step, are single-platform methods, in which design variables are either shared across all product variants or not shared at all. In multiple-platform design, design variables may be shared among variants in any possible combination of subsets, offering opportunities for superior overall design but presenting a more difficult computational problem. In this work, sensitivity analysis and cluster analysis are used to improve both efficiency and effectiveness of a scale-based product family through multiple-platform product family design. Sensitivity analysis is performed on each design variable to help select candidate platform design variables and to provide guidance for cluster analysis. Cluster analysis, using performance loss due to commonization as the clustering criterion, is employed to determine platform configuration. An illustrative example is used to demonstrate the merits of the proposed method, and the results are compared with existing results from the literature.     

Multi-objective robust optimization using a sensitivity region concept

In multi-objective design optimization, it is quite desirable to obtain solutions that are multi-objectively optimum and insensitive to uncontrollable (noisy) parameter variations. We call such solutions robust Pareto solutions. In this paper we present a method to measure the multi-objective sensitivity of a design alternative, and an approach to use such a measure to obtain multi-objectively robust Pareto optimum solutions. Our sensitivity measure does not require a presumed probability distribution of uncontrollable parameters and does not utilize gradient information; therefore, it is applicable to multi-objective optimization problems that have non-differentiable and/or discontinuous objective functions, and also to problems with large parameter variations. As a demonstration, we apply our robust optimization method to an engineering example, the design of a vibrating platform. We show that the solutions obtained for this example are indeed robust.