Variational Principles for Non-Material Systems Within an Arbitrary Lagrangian Eulerian Description of Motion

2021 ◽  
Author(s):  
Giuseppe Pennisi ◽  
Olivier Bauchau
Keyword(s):  
Variational Principles ◽  
Arbitrary Lagrangian Eulerian ◽  
Eulerian Description ◽  
Material Systems

Variational Principles for Non-Material Systems Within an Arbitrary Lagrangian Eulerian Description of Motion

2020 ◽  
Author(s):  
Giuseppe Pennisi ◽  
Olivier Bauchau
Keyword(s):  
Finite Element ◽  
Dynamic Behavior ◽  
Variational Principles ◽  
Material System ◽  
Arbitrary Lagrangian Eulerian ◽  
Complex Dynamic ◽  
Eulerian Description ◽  
Material Systems ◽  
Axially Moving ◽  
Dynamic Description

Abstract Dynamics of axially moving continua, such as beams, cables and strings, can be modeled by use of an Arbitrary La-grangian Eulerian (ALE) approach. Within a Finite Element framework, an ALE element is indeed a non-material system, i.e. a mass flow occurs at its boundaries. This article presents the dynamic description of such systems and highlights the peculiarities that arise when applying standard mechanical principles to non-material systems. Starting from D’Alembert’s principle, Hamilton’s principle and Lagrange’s equations for a non-material system are derived and the significance of the additional transport terms discussed. Subsequently, the numerical example of a length-changing beam is illustrated. Energetic considerations show the complex dynamic behavior non-material systems might exhibit.

Optimization Using Arbitrary Lagrangian–Eulerian Formulation of the Navier–Stokes Equations

2015 ◽  
Vol 137 (6) ◽  
Author(s):  
Eysteinn Helgason ◽  
Siniša Krajnović
Keyword(s):  
Stokes Equations ◽  
Optimization Method ◽  
Navier Stokes ◽  
Lagrangian Description ◽  
Computational Domain ◽  
Arbitrary Lagrangian Eulerian ◽  
Navier Stokes Equations ◽  
Mesh Motion ◽  
Eulerian Description ◽  
Eulerian Formulation

In this paper, we present a new shape optimization method by using sensitivities obtained from the Arbitrary Lagrangian–Eulerian (ALE) form of the Navier–Stokes equations. In the ALE description, the nodes of the computational domain may be moved with the fluid as in the Lagrangian description, held fixed in space as in the Eulerian description, or moved in some arbitrary way in between. Applying the adjoint method with respect to mesh motion allows the whole sensitivity field for the shape changes to be calculated using only two solver calls, a primal solver call and an adjoint solver call. We show that the sensitivities with respect to the mesh motion can be calculated in a postprocessing step to the primal and adjoint flow simulations. The resulting ALE sensitivities are compared to sensitivities obtained using a finite difference approach. Finally, the sensitivities are coupled to a mesh motion smoothing algorithm, and a duct is optimized with respect to the total pressure drop using the proposed method.

A novel approach for simulation of soil-tool interaction based on an arbitrary Lagrangian–Eulerian description

2018 ◽  
Vol 178 ◽  
pp. 41-49 ◽  
Author(s):  
L.B. Zhang ◽  
Z.X. Cai ◽  
H.F. Liu
Keyword(s):  
Arbitrary Lagrangian Eulerian ◽  
Eulerian Description ◽  
Novel Approach

Interaction equations for Rossby waves derived from Hamilton’s principle in the Eulerian description

1995 ◽  
Vol 450 (1940) ◽  
pp. 667-676
Keyword(s):  
Variational Principle ◽  
Velocity Field ◽  
Variational Approach ◽  
Evolution Equations ◽  
Rossby Waves ◽  
Variational Principles ◽  
Lagrangian Description ◽  
Hydrodynamic Waves ◽  
Hamilton’S Principle ◽  
Eulerian Description

Variational principles provide a useful tool for deriving the evolution equations of a resonant triad of hydrodynamic waves, but they are often considered in the Lagrangian description, as Eulerian variational principles require the introduction of extra variables, related to the particle displacements. To show that a variational approach remains possible with these extra variables, the derivation of interaction equations is presented in detail for Rossby waves on a β -plane, using a variational principle based on the Clebsch representation of the velocity field.

Variational Principles in Mathematical Physics, Geometry, and Economics

2009 ◽  
Author(s):  
Alexandru Kristaly ◽  
Vicentiu D. Radulescu ◽  
Csaba Varga
Keyword(s):  
Variational Principles ◽  
Mathematical Physics

Invariant Variational Principles

1977 ◽  
Keyword(s):  
Variational Principles
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