Updated Lagrangian/Arbitrary Lagrangian–Eulerian framework for interaction between a compressible neo‐Hookean structure and an incompressible fluid

Abstract

Summary We propose a numerical method for a fluid–structure interaction problem. The material of the structure is homogeneous, isotropic, and it can be described by the compressible neo‐Hookean constitutive equation, while the fluid is governed by the Navier–Stokes equations. Our study does not use turbulence model. Updated Lagrangian method is used for the structure and fluid equations are written in Arbitrary Lagrangian–Eulerian coordinates. One global moving mesh is employed for the fluid–structure domain, where the fluid–structure interface is an ‘interior boundary’ of the global mesh. At each time step, we solve a monolithic system of unknown velocity and pressure defined on the global mesh. The continuity of velocity at the interface is automatically satisfied, while the continuity of stress does not appear explicitly in the monolithic fluid–structure system. This method is very fast because at each time step, we solve only one linear system. This linear system was obtained by the linearization of the structure around the previous position in the updated Lagrangian formulation and by the employment of a linear convection term for the fluid. Numerical results are presented. Copyright © 2016 John Wiley & Sons, Ltd.

Three-Dimensional Simulation of Fluid–Structure Interaction Problems Using Monolithic Semi-Implicit Algorithm

“…Now, we present the updated Lagrangian framework, where the equations are written in the domain Ω S n . We follow a similar method as in [36], where the structure is a bi-dimensional compressible neo-Hookean material, or in [37], where the bi-dimensional linear elasticity model is used.…”

“…For the structure modeled by the linear elasticity equations, we use the updated Lagrangian framework and, for the fluid governed by the Navier-Stokes equations, we employ the ALE method. A similar strategy is used in [36] for a bi-dimensional compressible neo-Hookean model or in [37] for a bi-dimensional linear elasticity model for the structure. As in [38], we employ a global mesh for the fluid-structure domain where the fluid-structure interface is an interior boundary.…”

“…In [36], by using the Updated Lagrangian framework for a compressible Neo-Hookean structure, the weak formulation is written in the known configuration obtained at the precedent time step. By linearization around this configuration, at each time step, only a linear system is solved for the fluid-structure coupled equations and consequently the computing time is reduced.…”

Stable Semi-Implicit Monolithic Scheme for Interaction Between Incompressible Neo-hookean Structure and Navier-Stokes Fluid

“…In [11,16,[18][19][20]24] one global mesh for fluid-structure domain which fits to the interface is used. In [11,20], an Eulerian formulation derived from Cayley-Hamilton theorem is used for the incompressible Mooney-Rivlin structure.…”

“…In [16] where the structure is linear elastic and in [18] where the structure is compressible neo-hookean, the Updated Lagrangian coordinates are used for the structure combined with the Arbitrary Lagrangian Eulerian framework for the fluid equations. Using one velocity field defined over the fluid-structure mesh, and globally continuous finite elements, the continuity of the velocity at the interface is automatically verified.…”