Adjoint sensitivity index-3 augmented Lagrangian formulation with projections

Abstract The interest on closed-form analytical sensitivity equations based on the classical forward dynamics formulations, started shortly after the equations were ready and became popular. A vast effort was devoted by the authors in the last years to derive the forward and adjoint sensitivity equations for some state-of-the-art formulations of practical interest. Recently, the forward sensitivity equations for the index-3 augmented Lagrangian formulation with projections (the ALI3-P formulation) have been derived. In this article, the ALI3-P adjoint sensitivity equations are derived and implemented in the MBSLIM library as a general code and tested in one academic example and one real-life multibody system.

[1]  Juan C. García Orden.  Energy Considerations for the Stabilization of Constrained Mechanical Systems with Velocity Projection , 2010 .

[2]  Daniel Dopico,et al.  Determination of Holonomic and Nonholonomic Constraint Reactions in an Index-3 Augmented Lagrangian Formulation With Velocity and Acceleration Projections , 2014 .

[3]  Adrian Sandu,et al.  Adjoint sensitivity analysis of hybrid multibody dynamical systems , 2018, Multibody System Dynamics.

[4]  R. Ledesma,et al.  Augmented lagrangian and mass-orthogonal projection methods for constrained multibody dynamics , 1996 .

[5]  M. A. Serna,et al.  A modified Lagrangian formulation for the dynamic analysis of constrained mechanical systems , 1988 .

[6]  Daniel Dopico,et al.  On the Stabilizing Properties of Energy-Momentum Integrators and Coordinate Projections for Constrained Mechanical Systems , 2007 .

[7]  Antonio M. Recuero,et al.  Sensitivity Analysis for Hybrid Systems and Systems with Memory , 2019, Journal of Computational and Nonlinear Dynamics.

[8]  A Schaffer,et al.  Stabilized index-1 differential-algebraic formulations for sensitivity analysis of multi-body dynamics , 2006 .

[9]  Daniel Dopico,et al.  On the optimal scaling of index three DAEs in multibody dynamics , 2008 .

[10]  Juan C. García Orden,et al.  Controllable velocity projection for constraint stabilization in multibody dynamics , 2012 .

[12]  Shengtai Li,et al.  Adjoint Sensitivity Analysis for Differential-Algebraic Equations: The Adjoint DAE System and Its Numerical Solution , 2002, SIAM J. Sci. Comput..

[13]  Adrian Sandu,et al.  Direct and Adjoint Sensitivity Analysis of Multibody Systems Using Maggi’s Equations , 2013 .

[14]  Andrei Schaffer,et al.  Stability of the Adjoint Differential-Algebraic Equation of the Index-3 Multibody System Equation of Motion , 2005, SIAM J. Sci. Comput..

[15]  Adrian Sandu,et al.  Direct and Adjoint Sensitivity Analysis of Ordinary Differential Equation Multibody Formulations , 2014, Journal of Computational and Nonlinear Dynamics.

[16]  Javier García de Jalón,et al.  Kinematic and Dynamic Simulation of Multibody Systems , 1994 .

[17]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[18]  Daniel Dopico,et al.  Direct sensitivity analysis of multibody systems with holonomic and nonholonomic constraints via an index-3 augmented Lagrangian formulation with projections , 2018 .

[19]  Shengtai Li,et al.  Sensitivity analysis of differential-algebraic equations and partial differential equations , 2005, Comput. Chem. Eng..

[20]  Jintai Chung,et al.  A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method , 1993 .

[21]  E. Haug.  Well posed formulations of holonomic mechanical system dynamics and design sensitivity analysis , 2020, Mechanics Based Design of Structures and Machines.

[22]  J. G. Jalón,et al.  Multibody dynamics with redundant constraints and singular mass matrix: existence, uniqueness, and determination of solutions for accelerations and constraint forces , 2013 .

[23]  Emmanuel D. Blanchard,et al.  Polynomial chaos-based parameter estimation methods applied to a vehicle system , 2010 .

[24]  Benjamin Boudon,et al.  Design methodology of a complex CKC mechanical joint with a representation energetic tool multi-Bond graph: application to the helicopter , 2014 .

[25]  Adrian Sandu,et al.  A Polynomial Chaos-Based Kalman Filter Approach for Parameter Estimation of Mechanical Systems , 2010 .

[26]  Javier Cuadrado,et al.  A comparison in terms of accuracy and efficiency between a MBS dynamic formulation with stress analysis and a non‐linear FEA code , 2001 .

[27]  Emmanuel D. Blanchard,et al.  Parameter estimation for mechanical systems via an explicit representation of uncertainty , 2009 .

[28]  Emmanuel D. Blanchard,et al.  Polynomial-chaos-based numerical method for the LQR problem with uncertain parameters in the formulation , 2010 .

[29]  Adrian Sandu,et al.  Benchmarking of adjoint sensitivity-based optimization techniques using a vehicle ride case study , 2018 .

[30]  Adrian Sandu,et al.  Modeling and sensitivity analysis methodology for hybrid dynamical system , 2017, Nonlinear Analysis: Hybrid Systems.

[31]  Nathan M. Newmark,et al.  A Method of Computation for Structural Dynamics , 1959 .

[32]  Zhuang Fengqing,et al.  Patients’ Responsibilities in Medical Ethics , 2016 .

[33]  Weijie Wang,et al.  Alternating Direction Method of Multipliers for Linear Inverse Problems , 2016, SIAM J. Numer. Anal..