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.

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.

Energy-Momentum Conserving Time Integration of Modally Reduced Flexible Multibody Systems

2013 ◽  
Author(s):  
Alexander Humer ◽  
Johannes Gerstmayr
Keyword(s):  
Angular Momentum ◽  
Multibody Systems ◽  
Mass Matrix ◽  
Time Integration ◽  
Flexible Multibody Dynamics ◽  
Integration Scheme ◽  
Integration Schemes ◽  
Flexible Multibody ◽  
Floating Frame ◽  
Energy And Momentum

Many conventional time integration schemes frequently adopted in flexible multibody dynamics fail to retain the fundamental conservation laws of energy and momentum of the continuous time domain. Lack of conservation, however, in particular of angular momentum, may give rise to unexpected, unphysical results. To avoid such problems, a scheme for the consistent integration of modally reduced multibody systems subjected to holonomic constraints is developed in the present paper. As opposed to the conventional approach, in which the floating frame of reference formulation is combined with component mode synthesis for approximating the flexible deformation, an alternative, recently proposed formulation based on absolute coordinates is adopted in the analysis. Owing to the linear relationship between the generalized coordinates and the absolute displacement, the inertia terms in the equations of motion attain a very simple structure. The mass matrix remains independent of the current state of deformation and the velocity dependent term known from the floating frame approach vanishes due to the absence of relative coordinates. These advantageous properties facilitate the construction of an energy and momentum consistent integration scheme. By the mid-point rule, algorithmic conservation of both linear and angular momentum is achieved. In order to consistently integrate the total energy of the system, the discrete derivative needs to be adopted when evaluating the strain energy gradient and the derivative of the algebraic constraint equations.

scholarly journals Estimates in the Taylor series method for polynomial total systems of PDEs

2021 ◽  
Vol 17 (1) ◽  
pp. 27-39
Author(s):  
Levon K. Babadzanjanz ◽  
◽  
Irina Yu. Pototskaya ◽  
Yulia Yu. Pupysheva ◽  
◽  
...  
Keyword(s):  
Cauchy Problem ◽  
Taylor Series ◽  
Series Method ◽  
Initial Value ◽  
Taylor Coefficients ◽  
Recurrence Formulas ◽  
The Cauchy Problem ◽  
Taylor Series Method ◽  
Systems Of Pdes ◽  
Ode Systems

Many of total systems of PDEs can be reduced to the polynomial form. As was shown by various authors, one of the best methods for the numerical solution of the initial value problem for ODE systems is the Taylor Series Method (TSM). In the article, the authors consider the Cauchy problem for the total polynomial PDE system, obtain the recurrence formulas for Taylor coefficients, and then formulate and prove a theorem on the accuracy of its solutions by TSM.

scholarly journals Detailed wheel/rail geometry processing with the conformal contact approach

2020 ◽  
Author(s):  
Edwin Vollebregt
Keyword(s):  
Multibody Systems ◽  
Tangent Plane ◽  
Shear Stresses ◽  
Coordinate Transformations ◽  
Geometry Processing ◽  
Force Element ◽  
Systems Simulation ◽  
Planar Contact ◽  
Conformal Contact ◽  
Selection Of

Abstract This paper proposes a new way of considering wheel–rail contact in multibody systems simulation that goes beyond the traditional planar constraint and elastic approaches. In this approach, wheel–rail interaction is modelled as a force element with pressures and shear stresses distributed over a contact area that may be curved, supporting conformal contact situations. This by-passes the selection of the contact reference location and reference angle, which are delicate aspects of planar contact approaches. The idea is worked out introducing the curved reference surface as the new backbone for the computations, instead of the tangent plane used previously in planar contact approaches. The steps are described by which the curved reference is constructed in CONTACT, using generic facilities for markers, grids, and coordinate transformations, by which generic wheel/rail configurations can be analyzed in a fully automated way. Numerical results show the capabilities of the new method for measured, worn profiles, suppressing discontinuities in the forces when multiple contact patches split or merge. A further application concerns the evaluation of strategies used in planar contact approaches. There we find that the tangent plane’s inclination is of the biggest importance. This should be defined in an averaged way to achieve maximum correspondence to the more detailed curved contact approach.

Automatic Differentiation in Automatic Generation of the Linearized Equations of Motion

2021 ◽  
Author(s):  
Bruce Minaker ◽  
Francisco González
Keyword(s):  
Multibody Systems ◽  
Automatic Differentiation ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Order Reduction ◽  
Automatic Generation ◽  
Analysis Model ◽  
Direct Evaluation ◽  
Base Function ◽  
Linearized Equations

Abstract In the ongoing search for mathematically efficient methods of predicting the motion of vehicle and other multibody systems, and presenting the associated results, one of the avenues of continued interest is the linearization of the equations of motion. While linearization can potentially result in reduced fidelity in the model, the benefits in computational speed often make it the pragmatic choice. Linearization techniques are also useful in modal and stability analysis, model order reduction, and state and input estimation. This paper explores the application of automatic differentiation to the generation of the linearized equations of motion. Automatic differentiation allows one to numerically evaluate the derivative of any function, with no prior knowledge of the differential relationship to other functions. It exploits the fact that every computer program must evaluate every function using only elementary arithmetic operations. Using automatic differentiation, derivatives of arbitrary order can be computed, accurately to working precision, with minimal additional computational cost over the evaluation of the base function. There are several freely available software libraries that implement automatic differentiation in modern computing languages. In the paper, several example multibody systems are analyzed, and the computation times of the stiffness matrix are compared using direct evaluation and automatic differentiation. The results show that automatic differentiation can be surprisingly competitive in terms of computational efficiency.

scholarly journals Polynomial approach for the most general linear Fredholm integrodifferential-difference equations using Taylor matrix method

2006 ◽  
Vol 2006 ◽  
pp. 1-15 ◽  
Author(s):  
Mehmet Sezer ◽  
Mustafa Gülsu
Keyword(s):  
Approximate Solution ◽  
Difference Equations ◽  
Taylor Series ◽  
Matrix Method ◽  
Variable Coefficients ◽  
General Linear ◽  
Matrix Equations ◽  
Taylor Coefficients ◽  
Taylor Polynomials ◽  
The Given

A Taylor matrix method is developed to find an approximate solution of the most general linear Fredholm integrodifferential-difference equations with variable coefficients under the mixed conditions in terms of Taylor polynomials. This method transforms the given general linear Fredholm integrodifferential-difference equations and the mixed conditions to matrix equations with unknown Taylor coefficients. By means of the obtained matrix equations, the Taylor coefficients can be easily computed. Hence, the finite Taylor series approach is obtained. Also, examples are presented and the results are discussed.

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