A new energy and momentum conserving algorithm for the non‐linear dynamics of shells

Simo, J. C.; Tarnow, N.

doi:10.1002/nme.1620371503

Cited by 190 publications

(139 citation statements)

References 17 publications

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“…This benchmark problem, originally proposed by [78] and subsequently presented in [71,[78][79][80], is included in order to assess the ability of the algorithm to preserve angular momentum. We consider the motion of a threedimensional L-shaped block subjected to initial impulse traction boundary conditions at two of its sides described as follows (see Figure 11)…”

Section: L-shaped Blockmentioning

confidence: 99%

A stabilised Petrov–Galerkin formulation for linear tetrahedral elements in compressible, nearly incompressible and truly incompressible fast dynamics

Gil

Lee

Bonet

et al. 2014

Computer Methods in Applied Mechanics and Engineering

198

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A mixed second order stabilised Petrov-Galerkin finite element framework was recently introduced by the authors (C.H.Lee, A.J.Gil and J.Bonet. "Development of a stabilised Petrov-Galerkin formulation for conservation laws in Lagrangian fast solid dynamics", CMAME, 268:40-64, 2014). The new mixed formulation, written as a system of conservation laws for the linear momentum and the deformation gradient, performs extremely well in bending dominated scenarios (even when linear tetrahedral elements are used) yielding equal order of convergence for displacements and stresses. In this paper, this formulation is further enhanced for nearly and truly incompressible deformations with three key novelties. First, a new conservation law for the Jacobian of the deformation is added into the system providing extra flexibility to the scheme. Second, a variationally consistent PetrovGalerkin stabilisation methodology is derived. Third, an adapted fractional step method is presented for both incompressible and nearly incompressible materials in the context of nonlinear elastodynamics. For completeness and ease of understanding, these three improvements are presented both in small and large strain regimes, studying the eigenstructure of the resulting systems. A series of numerical examples are presented in order to demonstrate the robustness of the enhanced methodology with respect to the work previously published by the authors.

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Section: L-shaped Blockmentioning

confidence: 99%

A stabilised Petrov–Galerkin formulation for linear tetrahedral elements in compressible, nearly incompressible and truly incompressible fast dynamics

Gil

Lee

Bonet

et al. 2014

Computer Methods in Applied Mechanics and Engineering

198

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show abstract

“…Moreover, the DAEs turn out to be beneficial to the design of energy-momentum schemes [3][4][5]. Indeed our shell formulation can be shown to be equivalent to the previous work by Simo & Tarnow [6]. The newly established DAE framework facilitates the use of geometrically exact shells in flexible multibody dynamics.…”

mentioning

confidence: 70%

Geometrically exact shells in a flexible multibody framework

Sänger

Betsch

2008

Proc Appl Math and Mech

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The present work deals with the extension of contemporary finite element methods for nonlinear structural dynamics to the realm of flexible multibody dynamics. In particular, we aim at the incorporation of geometrically exact shell formulations into a multibody framework. Geometrically exact shell formulations are capable of dealing with finite deformation problems. In particular we focus on (i) a finite element discretization in space which inherits the frameindifference of the nonlinear strain measures from the underlying continuous shell formulation, and (ii) a discretization in time which leads to an energy and momentum conserving scheme or an energy decaying variant thereof. In nonlinear structural dynamics energy-momentum schemes have proven to be well-suited for the stable numerical integration of large deformation problems.We show that semi-discrete formulations emanating from the finite element discretization of geometrically exact shell theories can be regarded as discrete mechanical systems subject to holonomic constraints. Accordingly, the corresponding equations of motion can be written as differential-algebraic equations (DAEs) with index three. In particular, this viewpoint fits perfectly well into the framework of a rotationless description of rigid bodies [1] and geometrically exact beams [2]. Thus the DAEs provide a uniform framework for flexible multibody systems. Moreover, the DAEs turn out to be beneficial to the design of energy-momentum schemes [3][4][5]. Indeed our shell formulation can be shown to be equivalent to the previous work by Simo & Tarnow [6]. The newly established DAE framework facilitates the use of geometrically exact shells in flexible multibody dynamics. Representative numerical examples will demonstrate the capability of the proposed methodology. KeywordsNonlinear finite element methods, nonlinear structural dynamics, differentialalgebraic equations References[1] P. Betsch and P. Steinmann. Constrained integration of rigid body dynamics.

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“…The GEMM from Kuhl and Crisfield [23] is based on the energymomentum method developed by Simo and colleagues (see Refs. [31][32][33] for examples), which preserves energy and momentum exactly within each time step. However, due to convergence problems resulting from higher frequencies in stiff structural dynamics, controllable numerical dissipation is often desirable to prevent non-physical responses in these analyses.…”

Section: Discretisation Of Elastic Body Mechanicsmentioning

confidence: 99%

A constrained generalised- $$\alpha $$ α method for coupling rigid parallel chain kinematics and elastic bodies

Gransden

Bornemann²,

Rose

et al. 2015

Comput Mech

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A problem arises from combining flexible rotorcraft blades with stiffer mechanical links, which form a parallel kinematic chain. This paper introduces a method for solving index-3 differential algebraic equations for coupled stiff and elastic body systems with closed-loop kinematics. Rigid body dynamics and elastic body mechanics are independently described according to convenient mathematical measures. Holonomic constraint equations couple both the parallel chain kinematics and describe the coupling between the rigid and continuum bodies. Lagrange multipliers enforce the kinetic conditions for both sets of constraints. Additionally, to prevent numerical inaccuracy from inverting stiff mechanical matrices, a scaling factor normalises the dynamic tangential stiffness matrix. Finally, example tests show the verification of the algorithm with respect to existing computational tests and the accuracy of the model for cases relevant to the problem definition.

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A new energy and momentum conserving algorithm for the non‐linear dynamics of shells

Cited by 190 publications

References 17 publications

A stabilised Petrov–Galerkin formulation for linear tetrahedral elements in compressible, nearly incompressible and truly incompressible fast dynamics

A stabilised Petrov–Galerkin formulation for linear tetrahedral elements in compressible, nearly incompressible and truly incompressible fast dynamics

Geometrically exact shells in a flexible multibody framework

A constrained generalised- $$\alpha $$ α method for coupling rigid parallel chain kinematics and elastic bodies

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