Exact energy and momentum conserving algorithms for general models in nonlinear elasticity

González, Óscar

doi:10.1016/s0045-7825(00)00189-4

Cited by 231 publications

(282 citation statements)

References 45 publications

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“…We implement a discrete gradient to correct energy error [8], [1]. The variational integrator loses its symplecticity, but the absolute value of the energy error in every time step is bounded by the NEWTON tolerance.…”

Section: Energy Conservingmentioning

confidence: 99%

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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: Energy Conservingmentioning

confidence: 99%

“…This knowledge can be derived from the generalization of the discrete derivative or gradient, respectively, in Reference [8] to higher order schemes [1]. For higher order schemes, the discrete gradient of the total energy is always split into two terms of a sum.…”

Section: Introductionmentioning

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 potential V (q) is introduced via of its finite derivative ∂V * /∂q, see e.g. [6], while the discrete form of the constraints follows from the increment of (3). This is a homogeneous quadratic form in q and can be represented by a combination of increments and mean values.…”

Section: State-space Time Integrationmentioning

confidence: 99%

Rigid Body Time Integration by Convected Base Vectors With Implicit Constraints

Krenk¹,

Nielsen²

2014

Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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“…An important extension of the range of conservative time integration methods was attained by the introduction of the so-called finite derivative introduced by Gonzalez [7,8], and further developed under the name of the 'discrete gradient' by McLachlan et al [9]. The basic idea is to replace the original expression of the internal force by an augmented form including a correction in terms of the increment of the quantity to be conserved.…”

Section: Introductionmentioning

confidence: 99%