ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

approximated by the so-called Ritz vectors. Remke and Rothert [4] and Kapania and
Byun [5] expand the modal basis reduction and the load dependent Ritz method to
geometrically nonlinear truss systems. Originating from the turbulence research field
the proper orthogonal decomposition (e.g. [6, 7]) method builds up the subspace by
using snapshots from a precomputation. This method is also used for various nonlinear
problems. Especially in biomechanical applications Niroomandi et al. [8] and Dogan
and Serdar [9] use proper orthogonal decomposition based methods to reduce nonlinear
mechanical problems. The second group, the Krylov-based methods, employs moment
matching of the transfer functions (e.g. [1]) to reduce the system and is recognized as
being not useful for nonlinear problems [10].
In the following we extend the modal basis reduction, the load dependent Ritz and the
proper orthogonal decomposition method to nonlinear elasticity and inelasticity
including large deformations. The methods are tested for different material laws, a
hyperelastic Neohookean and a viscohyperelastic model. The performance of these
methods in the nonlinear context is first studied at the example of a cube under
compression. Afterwards we investigate a more complex and realistic geometry of an
inferior turbinate.
3. MODEL REDUCTION VIA PROPER ORTHOGONAL DECOMPOSITION
Nonlinear solid mechanics is based on three equations (momentum equation, kinematic
relation and constitutive law). Inserting them into each other and using a displacement
based finite element formulation to solve this problem leads to the discrete nonlinear
vector equation