Direct Sensitivity Analysis of Multibody Systems: A Vehicle Dynamics Benchmark

2019 ◽  
Vol 14 (2) ◽  
Author(s):  
Alfonso Callejo ◽  
Daniel Dopico
Keyword(s):  
Sensitivity Analysis ◽  
Multibody Systems ◽  
Automatic Differentiation ◽  
Time Integration ◽  
General Purpose ◽  
Contact Forces ◽  
Design Parameters ◽  
Computational Techniques ◽  
Computationally Efficient ◽  
Integration Algorithm

Algorithms for the sensitivity analysis of multibody systems are quickly maturing as computational and software resources grow. Indeed, the area has made substantial progress since the first academic methods and examples were developed. Today, sensitivity analysis tools aimed at gradient-based design optimization are required to be as computationally efficient and scalable as possible. This paper presents extensive verification of one of the most popular sensitivity analysis techniques, namely the direct differentiation method (DDM). Usage of such method is recommended when the number of design parameters relative to the number of outputs is small and when the time integration algorithm is sensitive to accumulation errors. Verification is hereby accomplished through two radically different computational techniques, namely manual differentiation and automatic differentiation, which are used to compute the necessary partial derivatives. Experiments are conducted on an 18-degree-of-freedom, 366-dependent-coordinate bus model with realistic geometry and tire contact forces, which constitutes an unusually large system within general-purpose sensitivity analysis of multibody systems. The results are in good agreement; the manual technique provides shorter runtimes, whereas the automatic differentiation technique is easier to implement. The presented results highlight the potential of manual and automatic differentiation approaches within general-purpose simulation packages, and the importance of formulation benchmarking.

Direct Sensitivity Analysis of Spatial Multibody Systems With Joint Friction Using Index-1 Formulation

2021 ◽  
Author(s):  
Adwait Verulkar ◽  
Corina Sandu ◽  
Daniel Dopico ◽  
Adrian Sandu
Keyword(s):  
Sensitivity Analysis ◽  
Dynamic Systems ◽  
Multibody Systems ◽  
Friction Model ◽  
Design Parameters ◽  
Adjoint Sensitivity ◽  
Joint Friction ◽  
Multibody Dynamic ◽  
Model Dynamics ◽  
Direct Sensitivity

Abstract Sensitivity analysis is one of the most prominent gradient based optimization techniques for mechanical systems. Model sensitivities are the derivatives of the generalized coordinates defining the motion of the system in time with respect to the system design parameters. These sensitivities can be calculated using finite differences, but the accuracy and computational inefficiency of this method limits its use. Hence, the methodologies of direct and adjoint sensitivity analysis have gained prominence. Recent research has presented computationally efficient methodologies for both direct and adjoint sensitivity analysis of complex multibody dynamic systems. The contribution of this article is in the development of the mathematical framework for conducting the direct sensitivity analysis of multibody dynamic systems with joint friction using the index-1 formulation. For modeling friction in multibody systems, the Brown and McPhee friction model has been used. This model incorporates the effects of both static and dynamic friction on the model dynamics. A case study has been conducted on a spatial slider-crank mechanism to illustrate the application of this methodology to real-world systems. Using computer models, with and without joint friction, effect of friction on the dynamics and model sensitivities has been demonstrated. The sensitivities of slider velocity have been computed with respect to the design parameters of crank length, rod length, and the parameters defining the friction model. Due to the highly non-linear nature of friction, the model dynamics are more sensitive during the transition phases, where the friction coefficient changes from static to dynamic and vice versa.

Design Sensitivity Analysis of Nonlinear Multidisciplinary Multibody Systems in the MIXEDMODELS Platform

2007 ◽  
Author(s):  
Shilpa A. Vaze ◽  
Prakash Krishnaswami ◽  
James DeVault
Keyword(s):  
Sensitivity Analysis ◽  
Multibody Systems ◽  
Design Sensitivity ◽  
Differential Algebraic Equations ◽  
The State ◽  
Design Sensitivity Analysis ◽  
Design Parameters ◽  
State Variables ◽  
Sensitivity Coefficients ◽  
Design Variables

Most state-of-the-art multibody systems are multidisciplinary and encompass a wide range of components from various domains such as electrical, mechanical, hydraulic, pneumatic, etc. The design considerations and design parameters of the system can come from any of these domains or from a combination of these domains. In order to perform analytical design sensitivity analysis on a multidisciplinary system (MDS), we first need a uniform modeling approach for this class of systems to obtain a unified mathematical model of the system. Based on this model, we can derive a unified formulation for design sensitivity analysis. In this paper, we present a modeling and design sensitivity formulation for MDS that has been successfully implemented in the MIXEDMODELS (Multidisciplinary Integrated eXtensible Engine for Driving Metamodeling, Optimization and DEsign of Large-scale Systems) platform. MIXEDMODELS is a unified analysis and design tool for MDS that is based on a procedural, symbolic-numeric architecture. This architecture allows any engineer to add components in his/her domain of expertise to the platform in a modular fashion. The symbolic engine in the MIXEDMODELS platform synthesizes the system governing equations as a unified set of non-linear differential-algebraic equations (DAE’s). These equations can then be differentiated with respect to design to obtain an additional set of DAE’s in the sensitivity coefficients of the system state variables with respect to the system’s design variables. This combined set of DAE’s can be solved numerically to obtain the solution for the state variables and state sensitivity coefficients of the system. Finally, knowing the system performance functions, we can calculate the design sensitivity coefficients of these performance functions by using the values of the state variables and state sensitivity coefficients obtained from the DAE’s. In this work we use the direct differentiation approach for sensitivity analysis, as opposed to the adjoint variable approach, for ease in error control and software implementation. The capabilities and performance of the proposed design sensitivity analysis formulation are demonstrated through a numerical example consisting of an AC rectified DC power supply driving a slider crank mechanism. In this case, the performance functions and design variables come from both electrical and mechanical domains. The results obtained were verified by perturbation analysis, and the method was shown to be very accurate and computationally viable.

Analytical Derivatives Technology for Parametric Shape Design and Analysis in Structural Applications

2006 ◽  
Author(s):  
Srikanth Akkaram ◽  
Jean-Daniel Beley ◽  
Bob Maffeo ◽  
Gene Wiggs
Keyword(s):  
Sensitivity Analysis ◽  
Finite Element ◽  
Automatic Differentiation ◽  
Thermal Response ◽  
Aircraft Engine ◽  
Design Parameters ◽  
Shape Design ◽  
Computer Simulation Model ◽  
Structural Applications ◽  
Analytical Derivatives

The ability to perform and evaluate the effect of shape changes on the stress, modal and thermal response 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 [1]. Since 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 [2,3] 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.

scholarly journals Dynamic Response Optimization of Complex Multibody Systems in a Penalty Formulation Using Adjoint Sensitivity

2015 ◽  
Vol 10 (3) ◽  
Author(s):  
Yitao Zhu ◽  
Daniel Dopico ◽  
Corina Sandu ◽  
Adrian Sandu
Keyword(s):  
Sensitivity Analysis ◽  
Multibody Dynamics ◽  
Multibody Systems ◽  
Real Life ◽  
Design Parameters ◽  
Vehicle Model ◽  
Adjoint Sensitivity ◽  
Penalty Formulation ◽  
Sensitivity Approach ◽  
Dynamic Performance Analysis

Multibody dynamics simulations are currently widely accepted as valuable means for dynamic performance analysis of mechanical systems. The evolution of theoretical and computational aspects of the multibody dynamics discipline makes it conducive these days for other types of applications, in addition to pure simulations. One very important such application is design optimization for multibody systems. In this paper, we focus on gradient-based optimization in order to find local minima. Gradients are calculated efficiently via adjoint sensitivity analysis techniques. Current approaches have limitations in terms of efficiently performing sensitivity analysis for complex systems with respect to multiple design parameters. To improve the state of the art, the adjoint sensitivity approach of multibody systems in the context of the penalty formulation is developed in this study. The new theory developed is then demonstrated on one academic case study, a five-bar mechanism, and on one real-life system, a 14 degree of freedom (DOF) vehicle model. The five-bar mechanism is used to validate the sensitivity approach derived in this paper. The full vehicle model is used to demonstrate the capability of the new approach developed to perform sensitivity analysis and optimization for large and complex multibody systems with respect to multiple design parameters with high efficiency.

Automatic Differentiation of the General-Purpose Computational Fluid Dynamics Package FLUENT

2006 ◽  
Vol 129 (5) ◽  
pp. 652-658 ◽  
Author(s):  
Christian H. Bischof ◽  
H. Martin Bücker ◽  
Arno Rasch ◽  
Emil Slusanschi ◽  
Bruno Lang
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Automatic Differentiation ◽  
General Purpose ◽  
Computational Techniques ◽  
Software Packages ◽  
Broad Variety ◽  
Industrial Software ◽  
Crucial Ingredient ◽  
Numerical Approaches

Derivatives are a crucial ingredient to a broad variety of computational techniques in science and engineering. While numerical approaches for evaluating derivatives suffer from truncation error, automatic differentiation is accurate up to machine precision. The term automatic differentiation comprises a set of techniques for mechanically transforming a given computer program to another one capable of evaluating derivatives. A common misconception about automatic differentiation is that this technique only works on local pieces of fairly simple code. Here, it is shown that automatic differentiation is not only applicable to small academic codes, but scales to advanced industrial software packages. In particular, the general-purpose computational fluid dynamics software package FLUENT is transformed by automatic differentiation.

Discrete Adjoint Method for the Sensitivity Analysis of Flexible Multibody Systems

2019 ◽  
Vol 14 (2) ◽  
Author(s):  
Alfonso Callejo ◽  
Valentin Sonneville ◽  
Olivier A. Bauchau
Keyword(s):  
Sensitivity Analysis ◽  
Multibody Systems ◽  
Adjoint Method ◽  
Mechanical Systems ◽  
Rigid Bodies ◽  
Flexible Multibody Systems ◽  
Design Parameters ◽  
Integration Scheme ◽  
Discrete Version ◽  
Flexible Multibody

The gradient-based design optimization of mechanical systems requires robust and efficient sensitivity analysis tools. The adjoint method is regarded as the most efficient semi-analytical method to evaluate sensitivity derivatives for problems involving numerous design parameters and relatively few objective functions. This paper presents a discrete version of the adjoint method based on the generalized-alpha time integration scheme, which is applied to the dynamic simulation of flexible multibody systems. Rather than using an ad hoc backward integration solver, the proposed approach leads to a straightforward algebraic procedure that provides design sensitivities evaluated to machine accuracy. The approach is based on an intrinsic representation of motion that does not require a global parameterization of rotation. Design parameters associated with rigid bodies, kinematic joints, and beam sectional properties are considered. Rigid and flexible mechanical systems are investigated to validate the proposed approach and demonstrate its accuracy, efficiency, and robustness.

DynManto: A Matlab Toolbox for the Simulation and Analysis of Multibody Systems

2020 ◽  
Author(s):  
Alexander Held ◽  
Ali Moghadasi ◽  
Robert Seifried
Keyword(s):  
Sensitivity Analysis ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
The Body ◽  
Computationally Efficient ◽  
Adjoint Sensitivity ◽  
Modeling And Analysis ◽  
Gradient Based ◽  
Data Files ◽  
Parameter Identifications

Abstract The Dynamic Modeling and Analysis Toolbox DynManto is an acedemic Matlab code which allows the modeling, simulation and sensitivity analysis of spatial multibody systems. The kinematics of rigid and flexible bodies is described by the floating frame of reference formulation and the body properties are provided by standard input data files. In this way the evaluation of the equations of motion is computationally efficient and an arbitrary parameterization of the system can be achieved. The latter is important in the automated adjoint sensitivity analysis of multibody systems, which yields gradient information for system analyses, parameter identifications or gradient-based optimizations. The capabilities of DynManto are demonstrated by the application examples of a flexible two-arm manipulator and Chebyshev’s Lambda Mechanism.

scholarly journals Dynamic Sensitivity Analysis of Deployable Flexible Space Structures With a Large Number of Design Parameters

2021 ◽  
Author(s):  
Shuai Wang ◽  
Qiang Tian ◽  
Haiyan Hu ◽  
Junwei Shi ◽  
Lingbin Zeng
Keyword(s):  
Sensitivity Analysis ◽  
Automatic Differentiation ◽  
Differential Algebraic Equations ◽  
Design Parameters ◽  
Space Structures ◽  
Analytical Sensitivity ◽  
Flexible Space Structures ◽  
Dynamic Sensitivity ◽  
Computational Methodology ◽  
Dynamic Sensitivity Analysis

Abstract The reliability and dynamic performance of deployable flexible space structures significantly depend on their key design parameters. Sensitivity analysis of these design parameters in the frame of multibody dynamics can serve as a powerful tool to evaluate and improve the dynamic performances of deployable space structures. Nevertheless, previous studies on the dynamic sensitivity analysis are mainly confined to planar multibody systems with a few design parameters. In this study, an efficient computational methodology is proposed to perform the dynamic sensitivity analysis of the complex deployable space structures with a large number of design parameters. Firstly, the analytical sensitivity analysis formulations of objective functions with the parameter-dependent integration bounds are deduced via the direct differentiation method and adjoint variable method. A checkpointing scheme is further introduced to assist the backward integration of the high dimensional differential algebraic equations of the adjoint variables. The flexible beams in the deployable flexible space structure are described by the locking-free three-node spatial beam elements of absolute nodal coordinate formulation. Furthermore, a parallelized automatic differentiation algorithm is proposed to efficiently evaluate the complex partial derivatives in the sensitivity analysis formulations. Finally, four numerical examples are provided to validate the accuracy and efficiency of the proposed computational methodology.

scholarly journals Projected explicit and implicit Taylor series methods for DAEs

2021 ◽  
Author(s):  
Diana Estévez Schwarz ◽  
René Lamour
Keyword(s):  
Taylor Series ◽  
Multibody Systems ◽  
Optimization Problem ◽  
Automatic Differentiation ◽  
Taylor Coefficients ◽  
Integration Schemes ◽  
Consistent Initial Values ◽  
Taylor Series Methods ◽  
Systems Simulation ◽  
Derivative Array

AbstractThe recently developed new algorithm for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimization opens new possibilities to apply Taylor series integration methods. In this paper, we show how corresponding projected explicit and implicit Taylor series methods can be adapted to DAEs of arbitrary index. Owing to our formulation as a projected optimization problem constrained by the derivative array, no explicit description of the inherent dynamics is necessary, and various Taylor integration schemes can be defined in a general framework. In particular, we address higher-order Padé methods that stand out due to their stability. We further discuss several aspects of our prototype implemented in Python using Automatic Differentiation. The methods have been successfully tested on examples arising from multibody systems simulation and a higher-index DAE benchmark arising from servo-constraint problems.

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