Mathematical model

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Where do the equations of motion used by BhT come from?

Within BhT's mechanical model, articulated bodies' shape is assigned as a function of time. By virtue of Newton's third law (principle of action-reaction), these shape-changes generate hydrodynamic forces and torques by which the bodies propel and steer themself. Such a physical system based on both Solid Mechanics and Fluid Mechanics is called fluid-structure interaction system.

Fluid model

In Fluid Mechanics, there is a lot of fluid models depending on fluid's properties (viscosity, density...). The fluid model of BhT is the one for a  perfect fluid (i. e. inviscid and incompressible) and whose flow is irrotational (i. e. vortex free). It is well known that such a flow can be fully described by a so-called potential function. This potential function solves a Neumann boundary value problem (NBVP): it is harmonic in the fluid domain and satisfies Neumann boundary conditions.

Bodies' motion

All of the immersed bodies are subject to hydrodynamic forces and the gravity force. Newton's laws for linear and angular momenta apply and yield a system of ordinary differential equations (ODEs) whose unknowns are the degrees of freedom of the bodies.

Coupling fluid-bodies

Coupling between fluid and bodies (i.e. between the equations of Fluid Mechanics and the equations of Solid Mechanics) is realized not only through hydrodynamic forces but also through kinematic constraints. Indeed the fluid domain and hence also the potential function depend on the bodies' positions. Moreover, the boundary conditions of the NBVP depend also on the bodies' velocities.

More sophisticated approach: Lagrangian formalism

Based on energetic considerations, the Lagrangian formalism (of Analytic Mechanics) allows to handle the sytem fluid-bodies in its integrity. It is particularly well adapted to describe the inner liaisons of articulated bodies. Far for being trivial, it can be shown that the equations of motion obtained with Lagrangian formalism are equivalent to those obtained via Newtonian formalism. 

Explicit equations of motion and how they are derived is explained in the articles [17] and [18] in the references page.

Key points

BhT's design is based on the following master ideas:

• All we are interested in is the bodies' motion. The fluid model we choose and the Lagrangian formalism make that possible not to compute what happen inside the fluid (for example the pressure is never computed).
• All of the data required to compute the bodies' motion lie on the fluid's boundary. This is really important because all which is necessary to be discretized for the computations is the fluid's boundary. This point is achieved by turning the NBVP of the fluid potential into an integral equation.

No bodies hydrodynamically decoupled hypothesis

Within fluid-structure interaction theory, the equations of motion can be greatly simplified by assuming that the immersed bodies are hydrodynamically decoupled. It means that each body is managed as if being alone in the fluid. Although pertinent when the bodies are far one from the others, this hypothesis is no longer relevant for close bodies. The equations BhT deals with are not simplified. This allows to study the distant interactions between bodies in a fluid. 

Summarizing

The central equation of BhT is a system of ODEs whose unknowns are the degrees of freedom of the articulated bodies. Each evaluation of this system requires a set of integral equations set on the fluid's boundary to be solved.

See also

What is BhT?

Articulated body

References

2008 - A. Munnier and B. Pincon (Insitut Elie Cartan and INRIA Lorraine, Projet CORIDA, Nancy, France).