An analysis of the stress formula for energy-momentum methods in nonlinear elastodynamics

Romero, Ignacio

doi:10.1007/s00466-012-0693-y

Cited by 43 publications

(41 citation statements)

References 21 publications

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“…For a more general form of the internal energy function, the energy conservation property can be satisfied by introducing a higher‐order modification of the internal force such that the finite energy increment condition is satisfied. This procedure, introduced by Gonzalez , is typically based on an interior point evaluation of the internal force, . In the present formulation, the secant form of the internal force is obtained as an extension of the explicit global end‐point representation for a quartic potential function,

{boldg}_{normalq} = \frac{1}{2} ([], {boldg}_{n + 1} + {boldg}_{n}) - \frac{1}{12} normalΔ boldK normalΔ boldu .

The secant form is obtained by addition of a higher‐order term that modifies the energy increment associated with the actual displacement increment Δ u .…”

Section: Consistent Internal Forcementioning

confidence: 99%

“…In the discrete energy balance equation (13), the viscous dissipation is represented via a quadratic expression in the displacement increment u. The discretization leads to diminishing damping for the high frequencies beyond the aliasing limit.…”

Section: Consistent Algorithmic Dampingmentioning

confidence: 99%

“…Important early contributions to momentum energy‐based algorithms were made by Simo and Wong for rigid body motion and Simo and Tarnow for linear elasticity in terms of the nonlinear quadratic Green strain, demonstrating that energy conservation can be obtained by evaluating the effective internal force at a suitably defined mean state within the integration interval. The formation of the internal force at an internal mean state has since become a standard procedure, and for continuum mechanics, various choices of mean state have recently been discussed by Romero . The mean state is formed at the particle level, and in the context of models built by finite elements, this implies that the computation must revert to the element level to calculate the representative internal force at suitable Gauss points in this mean state.…”

Section: Introductionmentioning

confidence: 99%

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Global format for energy–momentum based time integration in nonlinear dynamics

Krenk

2014

Numerical Meth Engineering

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SUMMARYA global format is developed for momentum and energy consistent time integration of second order dynamic systems with general non-linear stiffness. The algorithm is formulated by integrating the state space equations of motion over the time increment. The internal force is first represented in fourth order form consisting of the end-point mean value plus a term containing the stiffness matrix increment. This form gives energy conservation for systems with internal energy as a quartic function of the displacement components. This representation is then extended to general energy conservation via a discrete gradient representation. The present procedure works directly with the internal force and the stiffness matrix at the time integration interval end-points, and in contrast to previous energy conserving algorithms does not require any special form of the energy function nor use of mean-value products at the element level or explicit use of a geometric stiffness matrix. An optional monotonic algorithmic damping, increasing with response frequency, is developed in terms of a single damping parameter. In the solution procedure the velocity is eliminated and the non-linear iterations are based on the displacement components alone. The procedure represents an energy consistent alternative to available collocation methods, with an equally simple implementation.

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{boldg}_{normalq} = \frac{1}{2} ([], {boldg}_{n + 1} + {boldg}_{n}) - \frac{1}{12} normalΔ boldK normalΔ boldu .

The secant form is obtained by addition of a higher‐order term that modifies the energy increment associated with the actual displacement increment Δ u .…”

Section: Consistent Internal Forcementioning

confidence: 99%

Section: Consistent Algorithmic Dampingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Global format for energy–momentum based time integration in nonlinear dynamics

Krenk

2014

Numerical Meth Engineering

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“…A hypoelastic formulation for applications in nonlinear elastodynamics using the eight‐node hexahedral element with one‐point quadrature techniques was used by Braun and Awruch . According to Romero , there are infinite ways to obtain second‐order accuracy as well as energy and momentum conservation algorithms, whereas the characterization of the conserving stress as a minimization problem leads to that conclusion.…”

Section: Introductionmentioning

confidence: 99%

A NURBS‐based finite element model applied to geometrically nonlinear elastodynamics using a corotational approach

Espath

Braun

Awruch

et al. 2015

Numerical Meth Engineering

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SUMMARYA numerical model to deal with nonlinear elastodynamics involving large rotations within the framework of the finite element based on NURBS (Non-Uniform Rational B-Spline) basis is presented. A comprehensive kinematical description using a corotational approach and an orthogonal tensor given by the exact polar decomposition is adopted. The state equation is written in terms of corotational variables according to the hypoelastic theory, relating the Jaumann derivative of the Cauchy stress to the Eulerian strain rate.The generalized-˛method (G˛) method and Generalized Energy-Momentum Method with an additional parameter (GEMM+ ) are employed in order to obtain a stable and controllable dissipative time-stepping scheme with algorithmic conservative properties for nonlinear dynamic analyses.The main contribution is to show that the energy-momentum conservation properties and numerical stability may be improved once a NURBS-based FEM in the spatial discretization is used. Also it is shown that high continuity can postpone the numerical instability when GEMM+ with consistent mass is employed; likewise, increasing the continuity class yields a decrease in the numerical dissipation. A parametric study is carried out in order to show the stability and energy budget in terms of several properties such as continuity class, spectral radius and lumped as well as consistent mass matrices.

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“…An essential point in these methods is the identification of a particular representation of the internal forces such that exact conservation properties are satisfied. In the context of continuum mechanics various forms of stress averages have recently been discussed by Romero [6].…”

Section: Introductionmentioning

confidence: 99%