On the Use of Lie Group Time Integrators in Multibody Dynamics

2010 ◽  
Vol 5 (3) ◽  
Author(s):  
Olivier Brüls ◽  
Alberto Cardona
Keyword(s):  
Lie Group ◽  
Dynamic Systems ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Second Order ◽  
Order Accuracy ◽  
Time Integrators ◽  
Group Time ◽  
Constrained Equations ◽  
New Algorithms

This paper proposes a family of Lie group time integrators for the simulation of flexible multibody systems. The method provides an elegant solution to the rotation parametrization problem. As an extension of the classical generalized-α method for dynamic systems, it can deal with constrained equations of motion. Second-order accuracy is demonstrated in the unconstrained case. The performance is illustrated on several critical benchmarks of rigid body systems with high rotation speeds, and second-order accuracy is evidenced in all of them, even for constrained cases. The remarkable simplicity of the new algorithms opens some interesting perspectives for real-time applications, model-based control, and optimization of multibody systems.

scholarly journals A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

2021 ◽  
Author(s):  
Stefan Hante ◽  
Denise Tumiotto ◽  
Martin Arnold
Keyword(s):  
Lie Group ◽  
Group Structure ◽  
Equations Of Motion ◽  
Second Order ◽  
The Body ◽  
Benchmark Problems ◽  
Beam Model ◽  
Step Size ◽  
Time Step ◽  
Time Step Size

AbstractIn this paper, we will consider a geometrically exact Cosserat beam model taking into account the industrial challenges. The beam is represented by a framed curve, which we parametrize in the configuration space $\mathbb{S}^{3}\ltimes \mathbb{R}^{3}$ S 3 ⋉ R 3 with semi-direct product Lie group structure, where $\mathbb{S}^{3}$ S 3 is the set of unit quaternions. Velocities and angular velocities with respect to the body-fixed frame are given as the velocity vector of the configuration. We introduce internal constraints, where the rigid cross sections have to remain perpendicular to the center line to reduce the full Cosserat beam model to a Kirchhoff beam model. We derive the equations of motion by Hamilton’s principle with an augmented Lagrangian. In order to fully discretize the beam model in space and time, we only consider piecewise interpolated configurations in the variational principle. This leads, after approximating the action integral with second order, to the discrete equations of motion. Here, it is notable that we allow the Lagrange multipliers to be discontinuous in time in order to respect the derivatives of the constraint equations, also known as hidden constraints. In the last part, we will test our numerical scheme on two benchmark problems that show that there is no shear locking observable in the discretized beam model and that the errors are observed to decrease with second order with the spatial step size and the time step size.

Novel Linear Implicit Integration Method and its Application to Linear/Nonlinear Structural System

2014 ◽  
Vol 1065-1069 ◽  
pp. 1535-1539
Author(s):  
Chuan Guo Jia ◽  
Yan Xing Liu ◽  
Ying Min Li ◽  
Min Mao Liao
Keyword(s):  
Computational Efficiency ◽  
Unconditional Stability ◽  
Second Order ◽  
Order Accuracy ◽  
Dynamic Simulations ◽  
Order Of Accuracy ◽  
Engineering Research ◽  
Time Integrators ◽  
Shear Type ◽  
The Stability

Dynamic simulations of structures to determine their seismic performance is an essential part of civil engineering research. Time integrators of increasing sophistication has been elaborated over the last few decades to achieve higher order accuracy, unconditional stability, computational efficiency and high-frequency dissipation. This paper tries to extend 1-stage Rosenbrock-based integrator to an integrator of second-order accuracy without losing computational efficiency and unconditional stability. Initially, 1-stage Rosenbrock integrator is introduced and its order of accuracy is studied theoretically. According to accuracy analysis, a force term at the end of current step is considered, resulting a novel integrator of second-order accuracy. Moreover, the stability of the proposed method is studied by means of the energy method. To investigate its performance for nonlinear structures, numerical simulations are conducted on a shear-type structure including a pendulum.

Stabilized Time-Discontinuous Galerkin Methods With Applications to Structural Acoustics

2006 ◽  
Author(s):  
Lonny L. Thompson ◽  
Prapot Kunthong
Keyword(s):  
Discontinuous Galerkin ◽  
Hyperbolic Systems ◽  
Equations Of Motion ◽  
Iterative Solution ◽  
Single Step ◽  
Second Order ◽  
High Order ◽  
High Order Accuracy ◽  
Structural Acoustics ◽  
Order Accuracy

The time-discontinuous Galerkin (TDG) method possesses high-order accuracy and desirable C-and L-stability for second-order hyperbolic systems including structural acoustics. C- and L-stability provide asymptotic annihilation of high frequency response due to spurious resolution of small scales. These non-physical responses are due to limitations in spatial discretization level for large-complex systems. In order to retain the high-order accuracy of the parent TDG method for high temporal approximation orders within an efficient multi-pass iterative solution algorithm which maintains stability, generalized gradients of residuals of the equations of motion expressed in state-space form are added to the TDG variational formulation. The resultant algorithm is shown to belong to a family of Pade approximations for the exponential solution to the spatially discrete hyperbolic equation system. The final form of the algorithm uses only a few iteration passes to reach the order of accuracy of the parent solution. Analysis of the multi-pass algorithm shows that the first iteration pass belongs to the family of (p+1)-stage stiff accurate Singly-Diagonal-Implicit-Runge-Kutta (SDIRK) method. The methods developed can be viewed as a generalization to the SDIRK method, retaining the desirable features of efficiency and stability, now extended to high-order accuracy. An example of a transient solution to the scalar wave equation demonstrates the efficiency and accuracy of the multi-pass algorithms over standard second-order accurate single-step/single-solve (SS/SS) methods.

Two Lie Group Formulations for Dynamic Multibody Systems With Large Rotations

2011 ◽  
Author(s):  
Olivier Bru¨ls ◽  
Martin Arnold ◽  
Alberto Cardona
Keyword(s):  
Numerical Solution ◽  
Lie Group ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Time Integration ◽  
Computational Cost ◽  
Differential Algebraic Equations ◽  
Rigid Body Dynamics ◽  
Time Step ◽  
Iteration Matrix

This paper studies the formulation of the dynamics of multibody systems with large rotation variables and kinematic constraints as differential-algebraic equations on a matrix Lie group. Those equations can then be solved using a Lie group time integration method proposed in a previous work. The general structure of the equations of motion are derived from Hamilton principle in a general and unifying framework. Then, in the case of rigid body dynamics, two particular formulations are developed and compared from the viewpoint of the structure of the equations of motion, of the accuracy of the numerical solution obtained by time integration, and of the computational cost of the iteration matrix involved in the Newton iterations at each time step. In the first formulation, the equations of motion are described on a Lie group defined as the Cartesian product of the group of translations R3 (the Euclidean space) and the group of rotations SO(3) (the special group of 3 by 3 proper orthogonal transformations). In the second formulation, the equations of motion are described on the group of Euclidean transformations SE(3) (the group of 4 by 4 homogeneous transformations). Both formulations lead to a second-order accurate numerical solution. For an academic example, we show that the formulation on SE(3) offers the advantage of an almost constant iteration matrix.

Separated-Form Equations of Motion of Controlled Flexible Multibody Systems

1994 ◽  
Vol 116 (4) ◽  
pp. 702-712 ◽  
Author(s):  
Junghsen Lieh
Keyword(s):  
Multibody Systems ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Second Order ◽  
Active Suspensions ◽  
Physical Term ◽  
Moving Base ◽  
Matrix Expansion ◽  
Generalized Force ◽  
Coordinate Partitioning

This paper introduces a method leading to separated-form formulation of dynamic equations of multibody systems subject to control. The algorithm is derived from the virtual work principle and includes the moving base effects. The elastic members are treated as Euler-Bernoulli beams. Equations of motion are expanded using generalized coordinate partitioning and a Jacobian matrix expansion. The formulation of each physical term is separated, i.e., the inertia matrix, nonlinear coupling vector, generalized force vector and base motion-induced terms are established individually. The formulation is implemented on a workstation using MAPLE. Nonlinear and linearized equations with control are generated in FORTRAN format. The control design adopts second-order models directly. Several examples including a spin-up cantilever beam, an elastic vehicle with active suspensions and an elastic slider-crank mechanism are given. Numerical results for nonlinear and linear spin-up beam models are provided. Simulation for the active vehicle model using second-order control theory is presented.

Non-Linear Random Vibrations Using Second-Order Adjoint and Projected Differentiation Methods

2021 ◽  
Author(s):  
Dimitrios Papadimitriou ◽  
Zissimos P. Mourelatos ◽  
Zhen Hu
Keyword(s):  
Duffing Oscillator ◽  
Equations Of Motion ◽  
Random Variables ◽  
Random Excitation ◽  
Second Order ◽  
Output Process ◽  
Random Vibrations ◽  
Order Accuracy ◽  
Vibratory System ◽  
Non Linear

Abstract This paper proposes a new computationally efficient methodology for random vibrations of nonlinear vibratory systems using a time-dependent second-order adjoint variable (AV2) method, and a second-order projected differentiation (PD2) method. The proposed approach is called AV2-PD2. The vibratory system can be excited by stationary Gaussian or non-Gaussian random processes. A Karhunen-Loeve (KL) expansion expresses each input random process in terms of standard normal random variables. A second-order adjoint approach is used to obtain the required first and second-order output derivatives accurately by solving as many sets of equations of motion (EOMs) as the number of KL random variables. These derivatives are used to compute the marginal CDF of the output process with second-order accuracy. Then, a second-order projected differentiation method calculates the autocorrelation function of each output process with second-order accuracy, at an additional cost of solving as many sets of EOM as the number of outputs of interest, independently of the time horizon (simulation time). The total number of solutions of the EOM scales linearly with the number of input KL random variables and the number of output processes. The efficiency and accuracy of the proposed approach is demonstrated using a non-linear Duffing oscillator problem under a quadratic random excitation.

scholarly journals Higher order Hamiltonian Monte Carlo sampling for cosmological large-scale structure analysis

2021 ◽  
Vol 502 (3) ◽  
pp. 3976-3992
Author(s):  
Mónica Hernández-Sánchez ◽  
Francisco-Shu Kitaura ◽  
Metin Ata ◽  
Claudio Dalla Vecchia
Keyword(s):  
Fourth Order ◽  
Large Scale ◽  
Equations Of Motion ◽  
Large Scale Structure ◽  
Real Space ◽  
Higher Order ◽  
Second Order ◽  
Scale Structure ◽  
Integration Schemes ◽  
Reconstruction Methods

ABSTRACT We investigate higher order symplectic integration strategies within Bayesian cosmic density field reconstruction methods. In particular, we study the fourth-order discretization of Hamiltonian equations of motion (EoM). This is achieved by recursively applying the basic second-order leap-frog scheme (considering the single evaluation of the EoM) in a combination of even numbers of forward time integration steps with a single intermediate backward step. This largely reduces the number of evaluations and random gradient computations, as required in the usual second-order case for high-dimensional cases. We restrict this study to the lognormal-Poisson model, applied to a full volume halo catalogue in real space on a cubical mesh of 1250 h−1 Mpc side and 2563 cells. Hence, we neglect selection effects, redshift space distortions, and displacements. We note that those observational and cosmic evolution effects can be accounted for in subsequent Gibbs-sampling steps within the COSMIC BIRTH algorithm. We find that going from the usual second to fourth order in the leap-frog scheme shortens the burn-in phase by a factor of at least ∼30. This implies that 75–90 independent samples are obtained while the fastest second-order method converges. After convergence, the correlation lengths indicate an improvement factor of about 3.0 fewer gradient computations for meshes of 2563 cells. In the considered cosmological scenario, the traditional leap-frog scheme turns out to outperform higher order integration schemes only when considering lower dimensional problems, e.g. meshes with 643 cells. This gain in computational efficiency can help to go towards a full Bayesian analysis of the cosmological large-scale structure for upcoming galaxy surveys.

scholarly journals Closed-form time derivatives of the equations of motion of rigid body systems

2021 ◽  
Author(s):  
Andreas Müller ◽  
Shivesh Kumar
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Alternative Formulation ◽  
Dynamics Of Rigid Body ◽  
Control Forces ◽  
And Control ◽  
Derivatives Of ◽  
Insight Into

AbstractDerivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

A Second Order Lagrangian Model for Irregular Ocean Waves

2005 ◽  
Vol 128 (3) ◽  
pp. 177-183 ◽  
Author(s):  
Sébastien Fouques ◽  
Harald E. Krogstad ◽  
Dag Myrhaug
Keyword(s):  
Specular Reflection ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Wave Theory ◽  
Ocean Waves ◽  
Second Order ◽  
Lagrangian Model ◽  
Lagrangian Description ◽  
Linear Wave ◽  
Irregular Solution

Synthetic aperture radar (SAR) imaging of ocean waves involves both the geometry and the kinematics of the sea surface. However, the traditional linear wave theory fails to describe steep waves, which are likely to bring about specular reflection of the radar beam, and it may overestimate the surface fluid velocity that causes the so-called velocity bunching effect. Recently, the interest for a Lagrangian description of ocean gravity waves has increased. Such an approach considers the motion of individual labeled fluid particles and the free surface elevation is derived from the surface particles positions. The first order regular solution to the Lagrangian equations of motion for an inviscid and incompressible fluid is the so-called Gerstner wave. It shows realistic features such as sharper crests and broader troughs as the wave steepness increases. This paper proposes a second order irregular solution to these equations. The general features of the first and second order waves are described, and some statistical properties of various surface parameters such as the orbital velocity, slope, and mean curvature are studied.

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