The aim of the present chapter is to provide an in-depth survey of arbitrary Lagrangian–Eulerian (ALE) methods, including both conceptual aspects of the mixed kinematical description and numerical implementation details. Applications are discussed in fluid dynamics, nonlinear solid mechanics and coupled problems describing fluid–structure interaction. The need for an adequate mesh-update strategy is underlined, and various automatic mesh-displacement prescription algorithms are reviewed. This includes mesh-regularization methods essentially based on geometrical concepts, as well as mesh-adaptation techniques aimed at optimizing the computational mesh according to some error indicator. Emphasis is then placed on particular issues related to the modeling of compressible and incompressible flow and nonlinear solid mechanics problems. This includes the treatment of convective terms in the conservation equations for mass, momentum, and energy, as well as a discussion of stress-update procedures for materials with history-dependent constitutive behavior.


Keywords:

ALE description;
convective transport;
finite elements;
stabilization techniques;
mesh regularization and adaptation;
fluid dynamics;
nonlinear solid mechanics;
stress-update procedures;
fluid–structure interaction

https://onlinelibrary.wiley.com/doi/pdf/10.1002/0470091355.ecm009

Arbitrary Lagrangian–Eulerian Methods

https://hal.inria.fr/inria-00071499/document

Added-mass effect in the design of partitioned algorithms for fluid-structure problems

We present a fully-coupled monolithic formulation of the fluid-structure interaction of an incompressible fluid on a moving domain with a nonlinear hyperelastic solid. The arbitrary Lagrangian–Eulerian description is utilized for the fluid subdomain and the Lagrangian description is utilized for the solid subdomain. Particular attention is paid to the derivation of various forms of the conservation equations; the conservation properties of the semi-discrete and fully discretized systems; a unified presentation of the generalized-α time integration method for fluid-structure interaction; and the derivation of the tangent matrix, including the calculation of shape derivatives. A NURBS-based isogeometric analysis methodology is used for the spatial discretization and three numerical examples are presented which demonstrate the good behavior of the methodology.

Isogeometric fluid-structure interaction: theory, algorithms, and computations

A fixed-point fluid–structure interaction (FSI) solver with dynamic relaxation is revisited. New developments and insights gained in recent years motivated us to present an FSI solver with simplicity and robustness in a wide range of applications. Particular emphasis is placed on the calculation of the relaxation parameter by both Aitken’s \({\Delta^{2}}\) method and the method of steepest descent. These methods have shown to be crucial ingredients for efficient FSI simulations.

Fixed-point fluid-structure interaction solvers with dynamic relaxation

A NURBS (non-uniform rational B-splines)-based isogeometric fluid–structure interaction formulation, coupling incompressible fluids with non-linear elastic solids, and allowing for large structural displacements, is developed. This methodology, encompassing a very general class of applications, is applied to problems of arterial blood flow modeling and simulation. In addition, a set of procedures enabling the construction of analysis-suitable NURBS geometries directly from patient-specific imaging data is outlined. The approach is compared with representative benchmark problems, yielding very good results. Computation of a patient-specific abdominal aorta is also performed, giving qualitative agreement with computations by other researchers using similar models.

Isogeometric Fluid structure Interaction Analysis with Applications to Arterial Blood Flow

Fluid structure interaction with large structural displacements

We introduce a new numerical method to model the fluid–structure interaction between a microcapsule and an external flow. An explicit finite element method is used to model the large deformation of the capsule wall, which is treated as a bidimensional hyperelastic membrane. It is coupled with a boundary integral method to solve for the internal and external Stokes flows. Our results are compared with previous studies in two classical test cases: a capsule in a simple shear flow and in a planar hyperbolic flow. The method is found to be numerically stable, even when the membrane undergoes in-plane compression, which had been shown to be a destabilizing factor for other methods. The results are in very good agreement with the literature. When the viscous forces are increased with respect to the membrane elastic forces, three regimes are found for both flow cases. Our method allows a precise characterization of the critical parameters governing the transitions. Copyright © 2010 John Wiley & Sons, Ltd.

/pdf/coupling-of-finite-element-and-boundary-integral-methods-for-1qfbotvd34.pdf

Coupling of finite element and boundary integral methods for a capsule in a Stokes flow

A non-affine microsphere model for rubberlike materials is proposed, based on a local minimization of the network free energy under a maximal advance path constraint. It accounts for any chain weight distribution and for damage such as Mullins softening observed in filled rubber materials. The non-affine equal-force model is compared to the common affine model and a hybrid equal-force model from the literature, when considering the isotropic hyperelastic behavior without damage of rubber materials presenting chains of various lengths. The non-affine model shows an improved deformability compared to the affine model limited by the maximal extension of the shorter chains and a significantly softer behavior. Possible damage is introduced by increasing the chain lengths according to the submitted maximal chain traction force. Each chain are impacted independently resulting in a directional softening that introduces the evolution of the stress-free configuration that needs to be assessed over the loadings. The model was successfully tested on the cyclic uniaxial tension stretch-stress responses of carbon-black filled styrene butadiene rubbers that were well fitted with three parameters only.

https://hal.archives-ouvertes.fr/hal-01986861/document

A fully equilibrated microsphere model with damage for rubberlike materials

Strain-crystallising rubber exhibits interesting properties: for instance, fatigue lifetime is known to be modified by this microstructural evolution which dissipates energy and creates a strong anisotropic reinforcement. We develop herein a micro-sphere 3D constitutive model for such strain-crystallising rubber. It is based on a simplified 1D micromechanical model that we extend with a micro-sphere approach to a full thermodynamically consistent evolutive anisotropic model. A specific numerical strategy is then proposed. The model is assessed on several significative configurations and reproduces the main experimental features while predicting the evolution of anisotropy as a function of the loading history. We finally show that it can also predict the crystallised zone in front of a mode I crack.

Micro-sphere model for strain-induced crystallisation and three-dimensional applications

https://hal.inria.fr/hal-01166967/document

P. Le Tallec

Papers

Fluid structure interaction with large structural displacements

Coupling of finite element and boundary integral methods for a capsule in a Stokes flow

A fully equilibrated microsphere model with damage for rubberlike materials

Micro-sphere model for strain-induced crystallisation and three-dimensional applications

Dynamics of a spherical capsule in a planar hyperbolic flow: influence of bending resistance