Conservation properties of a time FE method—part III: Mechanical systems with holonomic constraints

Betsch, Peter; Steinmann, Paul

doi:10.1002/nme.347

Cited by 72 publications

(52 citation statements)

References 30 publications

(38 reference statements)

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“…In order to guarantee the symplecticity construction for our time step scheme, the following condition has to be implemented to fulfill the constraints on velocity level at the last micro time step in every macro time step [2], [7].…”

Section: The Discrete Lagrangiañmentioning

confidence: 99%

Energy Conserving Time Integration Based on Galerkin-Variational Integrators With Constraints

Bartelt¹,

Groß²

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. Often, Lagrange's formalism based on the Lagrangian is the preferred way in order to derive equations of motion for a mechanical system. Therefore, the first step is the formulation of the total kinetic and total potential energy. In this formalism, holonomic constraints can be taken into account by using the Lagrange multiplier method, and external and dissipative forces can be included variationally consistently by means of the Lagrange-D'Alembert principle. In this way, we are directly arrive at a weak formulation of the equations of motion without using a strong form and integration by parts. The variational time integration method is now based on the discretization of the action functional and the discrete variational calculus. This method leads to time stepping schemes called variational integrators.In this paper, we show that variational integrators based on higher-order shape functions and quadrature rules leads to a higher-order time approximation, which preserve the total linear and total angular momentum of the mechanical system. A further topic of the paper is a new implementation of the Lagrange multiplier method for holonomic constraints in the higher-order variational integrator, in order to compute bearing forces by means of Lagrange multipliers. Usually, variational integrators show an excellent long time behavior and bind the total energy error per time step, but are not able to preserve the total energy exactly. Therefore, we introduce a discrete gradient of the total potential energy in the variational integrator to preserve the total energy. We show different discrete gradients with different results for numerical stability. We can show that these time stepping schemes preserve the total linear momentum, total angular momentum as well as the total energy for every time step size. As numerical examples, we show motions of three-dimensional continua, which are discretized in space by the finite element method. We start with a free flying hyper-elastic rotor to show the preservation of total linear and total angular momentum in combination with higher-order accuracy in time. In the second example, we consider a hyper-elastic beam and apply the Lagrange multiplier method in order to calculate the bearing forces at fixed nodes. In the last example, the energy conservation is shown for different discrete gradients.

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Section: The Discrete Lagrangiañmentioning

confidence: 99%

Energy Conserving Time Integration Based on Galerkin-Variational Integrators With Constraints

Bartelt¹,

Groß²

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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“…The optimal torques, which are constant in each time interval, are depicted in Figure 10. They yield a control effort of J d = 2.8242×10 6 . Finally, Figure 10 illustrates the evolution of the kinetic energy and a special attribute of the system under consideration.…”

Section: Optimal Control Of a Rigid Body With Rotorsmentioning

confidence: 99%

“…Because of the relatively simple structure of the evolution equations derived from the Lagrange multiplier method, their temporal discrete form can be derived easily using mechanical integrators as demonstrated among others in [6][7][8]. However, the presence of Lagrange multipliers among the set of unknowns enlarges the number of equations and causes the discrete system to be ill-conditioned for small time steps as reported (among others) by [9,10].…”

Section: Introductionmentioning

confidence: 99%

“…In the first step, the discrete form of the simple structured DAEs resulting from the use of the Lagrange multiplier method is derived using a mechanical integrator, for example, an energy-momentum conserving integrator [6,7] or a variational integrator leading to a symplectic-momentum conserving scheme [8]. For forced systems, both methods correctly compute the change in momentum maps.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Discrete mechanics and optimal control for constrained systems

Leyendecker

Ober‐Blöbaum

Marsden

et al. 2010

Optim. Control Appl. Meth.

127

108

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SUMMARYThe equations of motion of a controlled mechanical system subject to holonomic constraints may be formulated in terms of the states and controls by applying a constrained version of the Lagrange-d'Alembert principle. This paper derives a structure-preserving scheme for the optimal control of such systems using, as one of the key ingredients, a discrete analogue of that principle. This property is inherited when the system is reduced to its minimal dimension by the discrete null space method. Together with initial and final conditions on the configuration and conjugate momentum, the reduced discrete equations serve as nonlinear equality constraints for the minimization of a given objective functional. The algorithm yields a sequence of discrete configurations together with a sequence of actuating forces, optimally guiding the system from the initial to the desired final state. In particular, for the optimal control of multibody systems, a force formulation consistent with the joint constraints is introduced. This enables one to prove the consistency of the evolution of momentum maps. Using a two-link pendulum, the method is compared with existing methods. Further, it is applied to a satellite reorientation maneuver and a biomotion problem.

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“…To achieve conservation of energy and angular momentum the kinetic energy has to be reformulated with quadratic invariants to apply the discrete derivative developed by Gonzalez [5] and Betsch & Steinmann [6]. Using the quadratic invariants, given by the quaternion product π =ȬÔ , the kinetic energy writes…”

Section: Mass Matricesmentioning

confidence: 99%

Numerical integration of rigid body dynamics in terms of quaternions

Siebert

Betsch

2008

Proc Appl Math and Mech

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Unit-quaternions (or Euler parameter) are known to be well-suited for the singularity-free parametrization of finite rotations. Despite of this advantage, unit quaternions were rarely used to formulate the equations of motion (exceptions are the works by Nikravesh [1] and Haug [2]). This might be related to the fact, that the unit-quaternions are redundant, which requires the use of algebraic constraints in the equations of motion. Nowadays robust energy consistent integrators are available for the numerical solution of these differential-algebraic equations (DAEs). In the present work a mechanical integrator for the quaternions will be derived. This will be done by a size-reduction from the director formulation of the equations of motion, which also has the form of DAEs. Finite rotations described with directors and quaternionsThe directors are the columns of the rotation tensor. They may be arranged in the configuration vector q ∈ R 9 . The three directors are highly redundant due to the six independent orthogonality constraints of the rotation tensorIntroducing rotations described with unit quaternions, one may start with the Euler-Theorem, which states, that a rotation can be described by a rotation axis given by an unitvector n ∈ R 3 and a rotation angle θ ∈ R . The related unit-quaternion can be calculated by the formulaThe unit-quaternions must be of length one, which can be enforced through the algebraic unit-length constraintTo show the connection between unitquaternions and the rotation matrix, one has to use the well-known Euler-Rodrigues Parametrization. Relations between the different formulationsDifferent ways to formulate the equations of motion exist, such as the quaternion and the director formulation or the euler equations. The different possible descriptions of finite rotations can be connected with three matrices P 1 ∈ R 9×4 , P 2 ∈ R 4×3 and P 3 ∈ R 9×3 of a special form. The three diagrams below show the relations between the velocities, the jacobian of the constraints and the mass matrices in the different formulations. Jacobian of the constraints Mass matricesThe constant mass matrix in the director formulation is written as M 9 = diag(E 1 I 1 , E 2 I 2 , E 3 I 3 ) ∈ R 9×9 . The E i 's are the principal values of the Euler tensor, which is related to the classical inertia tensor through J = (trE) I 3 − E . The size-reduction automatically yields to the non-singular mass matrix used in the quaternion formulation, which is required for the transition to the Hamilton equations of motion. In previous works by Maciejewski [3] and Morton [4] the Euler equations were the starting point for the transition to the quaternion formulation. An introduction of an undetermined inertia term was required to set up the Hamiltonian formulation with quaternions. Clearly the mass matrix in the quaternion formulation becomes configuration dependent, which yields to higher nonlinearity of the quaternion formulation in comparison to the one with directors.

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Conservation properties of a time FE method—part III: Mechanical systems with holonomic constraints

Cited by 72 publications

References 30 publications

Energy Conserving Time Integration Based on Galerkin-Variational Integrators With Constraints

Energy Conserving Time Integration Based on Galerkin-Variational Integrators With Constraints

Discrete mechanics and optimal control for constrained systems

Numerical integration of rigid body dynamics in terms of quaternions

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