Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics

Glowinski, Roland; Tallec, Patrick Le

doi:10.1137/1.9781611970838

Cited by 938 publications

(921 citation statements)

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“…In (8) κ denotes the bulk modulus, whereas μ refers to the shear modulus in (9). The anisotropic material parametersk 1 , k 2 in (10) and k 1 , k 2 in (11) represent a stress-like material parameter and a dimensionless parameter, respectively.…”

Section: A Particular Form Of the Model By Holzapfel Et Al [16]mentioning

confidence: 99%

“…One remedy for (ii) is to implement the computationally (rather) expensive augmented Lagrangian method to bring the analysis towards the incompressibility limit, see Glowinski and Le Tallec [8,9] and Simo and Taylor [31] among others. Alternatively, the multiplicative decomposition of the deformation gradient can be avoided for the anisotropic part, as suggested by Sansour [26] and Helfenstein et al [14], which solves the issue on the constitutive level without using any ad hoc algorithm.…”

Section: Introductionmentioning

confidence: 99%

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On the quasi-incompressible finite element analysis of anisotropic hyperelastic materials

2018

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Quasi-incompressible behavior is a desired feature in several constitutive models within the finite elasticity of solids, such as rubber-like materials and some fiber-reinforced soft biological tissues. The Q1P0 finite element formulation, derived from the three-field Hu-Washizu variational principle, has hitherto been exploited along with the augmented Lagrangian method to enforce incompressibility. This formulation typically uses the unimodular deformation gradient. However, contributions by Sansour (Eur J Mech A Solids 27: [28][29][30][31][32][33][34][35][36][37][38][39] 2007) and Helfenstein et al. (Int J Solids Struct 47:2056-2061 conspicuously demonstrate an alternative concept for analyzing fiber reinforced solids, namely the use of the (unsplit) deformation gradient for the anisotropic contribution, and these authors elaborate on their proposals with analytical evidence. The present study handles the alternative concept from a purely numerical point of view, and addresses systematic comparisons with respect to the classical treatment of the Q1P0 element and its coalescence with the augmented Lagrangian method by means of representative numerical examples. The results corroborate the new concept, show its numerical efficiency and reveal a direct physical interpretation of the fiber stretches.

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Section: A Particular Form Of the Model By Holzapfel Et Al [16]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the quasi-incompressible finite element analysis of anisotropic hyperelastic materials

2018

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“…Regularized models have first been proposed to replace the non-smooth viscoplastic constitutive law by a smooth purely viscous model [3,4]. Augmented Lagrangian (AL) approaches have then emerged as an interesting alternative to the use of regularized models to solve viscoplastic fluid flows, it is now one of the most popular method to solve such problems [5][6][7][8][9]. However, it still suffers from a slow convergence rate so that three-dimensional simulations are still extremely expensive.…”

Section: Introductionmentioning

confidence: 99%

Advances in the simulation of viscoplastic fluid flows using interior-point methods

Bleyer

2018

Computer Methods in Applied Mechanics and Engineering

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We present a primal-dual interior point algorithm for the resolution of steady-state viscoplastic fluid flows formulated as a conic optimization problem. We give a complete description of the algorithm including some advanced aspects such as a predictor-corrector and scaling scheme to improve its efficiency. Our interior-point approach is shown to be largely more efficient than Augmented Lagrangian (AL) approaches which are traditionally used to solve such problems. In particular, the interior-point approach is roughly 5 times faster than the modern accelerated version of AL algorithms. The yield surfaces are shown to be accurately predicted and various examples ranging from channel flows to three-dimensional flows through a porous medium demonstrate its efficiency.

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“…Discrete multisymplectic form formula. Consider the discrete action functional S ns d defined in (55). With the help of the discrete Cartan forms defined in (44), the derivative of…”

Section: Generalized Lagrange Multiplier Approachmentioning

confidence: 99%

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

Demoures

Gay–Balmaz

Raţiu

2016

Forum of Mathematics, Sigma

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This paper develops the theory of multisymplectic variational integrators for nonsmooth continuum mechanics with constraints. Typical problems are the impact of an elastic body on a rigid plate or the collision of two elastic bodies. The integrators are obtained by combining, at the continuous and discrete levels, the variational multisymplectic formulation of nonsmooth continuum mechanics with the generalized Lagrange multiplier approach for optimization problems with nonsmooth constraints. These integrators verify a spacetime multisymplectic formula that generalizes the symplectic property of time integrators. In addition, they preserve the energy during the impact. In the presence of symmetry, a discrete version of the Noether theorem is verified. All these properties are inherited from the variational character of the integrator. Numerical illustrations are presented.

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Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics

Cited by 938 publications

References 0 publications

On the quasi-incompressible finite element analysis of anisotropic hyperelastic materials

On the quasi-incompressible finite element analysis of anisotropic hyperelastic materials

Advances in the simulation of viscoplastic fluid flows using interior-point methods

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

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