The consideration of Finite Element (FE) structures in multibody simulation (MBS) has become an established method, especially when the number of force application points remains moderate. In recent years, a trend can be observed in which distributed loads are considered as well, such as those arising from the contact of two elastic bodies. For finely-meshed FE structures, this results in a large number of possible force application points. In such a case, conventional methods fail, leading to exorbitantly high computation times.
In the last decade, approaches for the reduced computation of deformations inside distributed load application areas were introduced. Special approach vectors are used, called "local modes" here. These local modes lead to a reduction in the involved equations by several orders of magnitude. However, for very large potential load application areas, a large number of local modes is still required—for example, several thousand. Since each local mode leads to a differential equation, a fast numerical time integration is not possible with common methods. In this work, two methodological improvements are proposed for a fast and accurate time integration of such systems.
Table of Contents
1 Introduction
1.1 Motivation
1.2 Goal
1.3 Outline
2 Brief review on model reduction of finite element structures via projection
2.1 Linear FE models
2.1.1 Introduction
2.1.2 General equations
2.1.3 Component Mode Synthesis
2.1.4 Moment matching
2.1.5 Balanced truncation
2.1.6 A brief comparison of CMS, MM and BT
2.2 Nonlinear FE models
2.2.1 Introduction
2.2.2 Literature on MOR for geometric nonlinearities
2.2.3 MOR for nonlinearities caused by contact and friction
2.3 Data based reduction methods
2.4 Challenge of many modes in case of distributed nonlinear surface forces
3 Detailed analysis of the equations of motion of a free flexible body in the floating frame of reference formulation
3.1 Introduction
3.2 Brief review of the equations of motion
3.3 Assumptions and simplifications
3.3.1 Euler parameter
3.3.2 Linearized mean-axis conditions
3.3.3 Use of (pseudo) free surface modes together with an FF origin fixed to the center of gravity of the undeformed body
3.3.4 Use of mass normalized modes
3.3.5 Use of central axis of inertia
3.4 Final equation of motion
3.4.1 Comments on the use of flexible body’s modeled via the FEM
3.5 Illustrative Examples used in the section
3.6 Assumption of small deformations
3.6.1 Small elastic deformations with respect to the body’s dimension
3.6.2 Magnitude of modes and modal coordinates
3.7 Magnitude of entries in invariants W1, W2 and W3
3.7.1 Magnitude of entries in matrix W1
3.7.2 Magnitude of entries in matrices W2 and W3
3.8 Investigations on the relevance of the single terms in the equations of motion
3.8.1 Rotational inertia of the deformed body
3.8.2 Inertia coupling
3.8.3 Quadratic velocity vector
3.9 “Set of guidlines” for practical use
3.10 Benefit
4 Separate time integration for flexible body’s with a large number of modes
4.1 Introduction
4.2 Theory
4.2.1 Brief review of equations of motions of a multibody system
4.2.2 Properties of local modes and its implications
4.2.3 Separate consideration of a flexible body’s non-stiff and stiff equations
4.2.4 Separate time integration of a flexible body’s stiff equations via a fixed point iteration based on the HHT method
4.3 Numerical examples
4.3.1 2 dof Oscillator
4.3.2 Planar crank drive with elastic piston and elastohydrodynamic oil film model
4.4 Discussion and Conclusion
5 Approximation of surface loads via stress trial vectors
5.1 General concept
5.2 Determination of a proper stress mode base
5.2.1 Simulation-free generation of potential stress space
5.2.2 Simulation-based generation of a potential stress space
5.2.3 Computation of the finally used stress mode base out of the potential stress space inside
5.3 Definitions of reduction, hyper-reduction and semi-hyper-reduction
5.3.1 Reduction
5.3.2 Hyper-reduction
5.3.3 Semi-hyper-reduction
6 Semi-hyper-reduction
6.1 Semi-hyper-reduction
6.2 Numerical Example: 2D crank drive with elastic piston
6.2.1 Model description
6.2.2 Deformation modes of the piston
6.2.3 Convergence analysis of the pressure distribution
6.2.4 Computation of stress modes
6.2.5 Remarks on the time integration
6.2.6 Non-reduced reference solution based on the Finite Difference Method
6.2.7 Application of semi-hyper-reduction
6.2.8 Results
6.3 Conclusion
7 Hyper-reduction
7.1 Introduction
7.2 Theoretical background
7.2.1 Calculus of variation for surface loads
7.2.2 Hyper reduced approximation of the cost function
7.2.3 Selection of FE nodes for the reduced summation
7.3 Numerical Example: 2D crank drive with elastic piston
7.3.1 Model description
7.3.2 Equations for dry contact
7.3.3 Reduced summation: Convergence analysis
7.3.4 Time saving because of hyper–reduction
7.4 Conclusion
8 Summary
Research Goal and Thematic Scope
This work aims to develop computationally efficient numerical methods for the simulation of flexible bodies in multibody systems, specifically addressing scenarios where a large number of modes is required to accurately model distributed nonlinear loads. The primary research goal is to achieve significant computational time savings without compromising the quality of the simulation results, particularly for problems involving contact mechanics and elastohydrodynamics.
- Model Order Reduction (MOR) for finite element structures via projection methods.
- Advanced numerical time integration techniques for systems involving a large number of stiff local modes.
- Optimized computation of distributed nonlinear forces using stress trial vectors and stress modes.
- Implementation of Semi-hyper-reduction (SHR) and Hyper-reduction (HR) to reduce the computational cost of nonlinear surface load evaluations.
Excerpt from the Book
2.4 Challenge of many modes in case of distributed nonlinear surface forces
The trial vectors briefly reviewed in section 2.2.3, allow the accurate reconstruction of deformations in contact regions with significant less dof than with the FE method. As documented in the publications cited there, problems become efficiently and accurately computable which have been previously considered as non-computable or associated with extremely long simulation times. Under certain circumstances, the number of necessary trial vectors may still be very high, despite a significant reduction. As an example, the roller - roller problem from Sherif, Witteveen, Holl, Irschik and Mayrhofer (2013) and Sherif and Witteveen (2014) can be mentioned. Each of the two rolls involved has several 10000 FE nodes on the surface which is the potential contact area. Finally, the deformations in the contact area could be represented with about 5000 trial vectors. This is already a remarkable reduction factor but 5000 equations for the numerical time integration is still a lot. Therefore, in section 4 a method is presented, which enables efficient time integration even with a large number of trial vectors. Section 3 is necessary to lay the theoretical foundation.
Summary of Chapters
1 Introduction: Provides an overview of multibody system simulation and the motivation for this work regarding reduced computation times for flexible bodies.
2 Brief review on model reduction of finite element structures via projection: Recapitulates standard reduction methods for linear and nonlinear structures, including CMS, MM, and BT.
3 Detailed analysis of the equations of motion of a free flexible body in the floating frame of reference formulation: Investigates inertia-related terms in the equations of motion to determine their relevance for different structural applications.
4 Separate time integration for flexible body’s with a large number of modes: Proposes an efficient integration scheme that decouples stiff local modes from non-stiff global modes to accelerate simulation.
5 Approximation of surface loads via stress trial vectors: Introduces the concept of approximating surface loads using stress modes instead of full nodal displacement computations.
6 Semi-hyper-reduction: Details the semi-hyper-reduction method as an intermediate step to reduce the number of unknowns in load computation.
7 Hyper-reduction: Describes the hyper-reduction approach, which enables the efficient calculation of nonlinear surface forces independently of the number of nodes in the full finite element model.
8 Summary: Reviews the proposed strategies for efficient numerical time integration and hyper-reduction of nonlinear surface loads.
Keywords
Flexible Multibody Dynamics, Floating Frame of Reference Formulation, Model Order Reduction, Numerical Time Integration, Hyper-Reduction, Semi-Hyper-Reduction, Contact Mechanics, Stress Modes, Local Modes, Finite Element Method, Substructuring, Computational Efficiency, Modal Analysis, Nonlinear Loads.
Frequently Asked Questions
What is the core problem addressed in this research?
The research addresses the high computational cost associated with multibody simulations of flexible bodies when a large number of modes are required to model distributed nonlinear loads, such as those arising from contact mechanics.
What are the primary themes of the work?
The core themes include model order reduction, efficient numerical time integration for stiff differential equations, and the hyper-reduction of nonlinear force computations.
What is the main objective of the proposed methods?
The goal is to enable fast and accurate simulations of problems previously considered economically non-computable, specifically by reducing the complexity of the equations to be solved during time integration and load evaluation.
Which scientific methods are utilized?
The work utilizes the Floating Frame of Reference Formulation (FFRF), Component Mode Synthesis (CMS), Proper Orthogonal Decomposition (POD), and the HHT (Hilber-Hughes-Taylor) time integration scheme.
What topics are discussed in the main body?
The main body covers the theoretical foundation of flexible body dynamics, the separation of stiff and non-stiff equations for time integration, and the development of semi-hyper-reduction and hyper-reduction for surface load calculations.
Which keywords characterize the work?
Key terms include Flexible Multibody Dynamics, Model Order Reduction, Hyper-Reduction, Contact Mechanics, and Numerical Time Integration.
How does the proposed separation of stiff and non-stiff equations improve performance?
By decoupling the equations, it allows for a main integrator to handle the non-stiff equations while using a specialized, faster iteration method for the highly stiff equations, avoiding the costly construction and decomposition of a large Jacobian matrix.
How do stress modes contribute to efficiency in this work?
Stress modes allow for the approximation of nonlinear loads in a reduced-dimensional space rather than in the full physical nodal space, effectively decoupling the computational effort from the mesh density of the finite element model.
- Citar trabajo
- Wolfgang Witteveen (Autor), 2021, Computational efficient flexible multibody dynamics with nonlinear loads and many modes, Múnich, GRIN Verlag, https://www.grin.com/document/1190639
