Time finite element based Moreau‐type integrators

Capobianco, Giuseppe; Eugster, Simon R.

doi:10.1002/nme.5741

Cited by 19 publications

(8 citation statements)

References 49 publications

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“…Finally, we can add that the variational formulation has several computational advantages (see for instance [46] for a general discussion and [47] for applications to beam problems).…”

Section: Euler-lagrange Equationsmentioning

confidence: 99%

Large deformations of Timoshenko and Euler beams under distributed load

Corte

Battista

Dell’Isola

et al. 2019

Z. Angew. Math. Phys.

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In this paper, the general equilibrium equations for a geometrically nonlinear version of the Timoshenko beam are derived from the energy functional. The particular case in which the shear and extensional stiffnesses are infinite, which correspond to the inextensible Euler beam model, is studied under a uniformly distributed load. All the global and local minimizers of the variational problem are characterized, and the relative monotonicity and regularity properties are established. MathematicsSubject Classification. 74B20, 34B15, 49J45. Keywords. Nonlinear elasticity, Timoshenko beam, Euler beam, Stability of solutions of nonlinear ODEs.the case of concentrated load. Afterward, some numerical results for the inextensible Euler beam under distributed load were published in [37,38]. The paper is organized as follows: in Sect. 2 a nonlinear version of the extensible Timoshenko beam model is introduced and the problem of a clamped-free beam is formulated. Euler-Lagrange equations are formally derived. In Sect. 3, some numerical results concerning curled equilibrium configurations are shown. These results motivate the analytical study of the properties of the equilibrium solutions done in Sect. 4, in which the Euler-Lagrange equation is studied in the particular case of an infinite shear stiffness, which leads to the nonlinear Euler beam.

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“…Finally, we can add that the variational formulation has several computational advantages (see for instance [46] for a general discussion and [47] for applications to beam problems).…”

Section: Euler-lagrange Equationsmentioning

confidence: 99%

Large deformations of Timoshenko and Euler beams under distributed load

Corte

Battista

Dell’Isola

et al. 2019

Z. Angew. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…Further background can be found in the texts of Glocker (2001) and Studer (2009). Recent work in this area includes extensions to continuum mechanics (Capobianco and Eugster, 2018) and higher-order integration schemes (Acary, 2012). Although we will exploit analogies to the simulation of physical systems, the focus of our theoretical analysis is in developing algorithms that efficiently compute approximate local minima of constrained nonlinear programming problems.…”

Section: Related Workmentioning

confidence: 99%

On Constraints in First-Order Optimization: A View from Non-Smooth Dynamical Systems

Muehlebach¹,

Jordan²

2021

Preprint

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We introduce a class of first-order methods for smooth constrained optimization that are based on an analogy to non-smooth dynamical systems. Two distinctive features of our approach are that (i) projections or optimizations over the entire feasible set are avoided, in stark contrast to projected gradient methods or the Frank-Wolfe method, and (ii) iterates are allowed to become infeasible, which differs from active set or feasible direction methods, where the descent motion stops as soon as a new constraint is encountered. The resulting algorithmic procedure is simple to implement even when constraints are nonlinear, and is suitable for large-scale constrained optimization problems in which the feasible set fails to have a simple structure. The key underlying idea is that constraints are expressed in terms of velocities instead of positions, which has the algorithmic consequence that optimizations over feasible sets at each iteration are replaced with optimizations over local, sparse convex approximations. The result is a simplified suite of algorithms and an expanded range of possible applications in machine learning.

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“…The DCP model formulates the intermittent contact between bodies in motion as a complementarity constraint [19,20,21,22,23,12,24,25,26,27,28]. DCP models are solved numerically with time-stepping schemes.…”

Section: Related Workmentioning

confidence: 99%

Modeling and Prediction of Rigid Body Motion with Planar Non-Convex Contact

Xie

Chakraborty

2020

Preprint

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We present a principled method for motion prediction via dynamic simulation for rigid bodies in intermittent contact with each other where the contact region is a planar nonconvex contact patch. Such methods are useful in planning and control for robotic manipulation. The planar non-convex contact patch can either be a topologically connected set or disconnected set. Most work in rigid body dynamic simulation assume that the contact between objects is a point contact, which may not be valid in many applications. In this paper, by using the convex hull of the contact patch, we build on our recent work on simulating rigid bodies with convex contact patches for simulating motion of objects with planar non-convex contact patches. We formulate a discrete-time mixed complementarity problem where we solve the contact detection and integration of the equations of motion simultaneously. We solve for the equivalent contact point (ECP) and contact impulse of each contact patch simultaneously along with the state, i.e., configuration and velocity of the objects. We prove that although we are representing a patch contact by an equivalent point, our model for enforcing nonpenetration constraints ensure that there is no artificial penetration between the contacting rigid bodies. We provide empirical evidence to show that our method can seamlessly capture transition among different contact modes like patch contact, multiple or single point contact.

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Time finite element based Moreau‐type integrators

Cited by 19 publications

References 49 publications

Large deformations of Timoshenko and Euler beams under distributed load

Large deformations of Timoshenko and Euler beams under distributed load

On Constraints in First-Order Optimization: A View from Non-Smooth Dynamical Systems

Modeling and Prediction of Rigid Body Motion with Planar Non-Convex Contact

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