Workshop on Biomechanics and Chemotaxis, Thu, 13 Dec, 2007
Speaker: Oliver Sander
Abstract
We present a new way to couple linear elastic three-dimensional bodies
to slender objects modelled as one-dimensional nonlinear Cosserat rods.
Starting from a full 3d nonlinear elastic formulation we derive suitable coupling
conditions for a reduced model consisting of a 3d linear elastic body
and a 1d nonlinear elastic rod. These involve the total force and torque transmitted through the interface
as well as its averaged position and an average orientation.
The resulting domain decomposition problem is solved using a
Dirichlet-Neumann algorithm.
The configuration space of a discrete special Cosserat rod is
$\mathbb{R}^{3n} \times \mbox{SO}(3)^n$. We present an $\infty$-norm
Riemannian trust-region algorithm for the minimization of the rod
energy functional on this nonlinear space. In conjunction with a
nonsmooth Newton multigrid method as the inner solver this yields an
efficient method with provable global convergence.
We use this coupling approach to model a human knee joint, where
the bones are modelled using 3d linear elasticity and the ligaments
as 1d rods. The use of rods for the ligaments decreases the overall
number of degrees of freedom and avoids meshing problems.
The additional problem of modelling the contact between the bones
is treated using a mortar element
discretization and a nonsmooth Newton multigrid method for the solution of
the resulting discrete system.
URL: www.ricam.oeaw.ac.at/specsem/ssqbm/participants/abstracts/index.php
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