A three-dimensional nonlinear finite element method implementation toward surgery simulation
Author
Gülümser, Emir
Advisor
Güdükbay, Uğur
Date
2011Publisher
Bilkent University
Language
English
Type
ThesisItem Usage Stats
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Abstract
Finite Element Method (FEM) is a widely used numerical technique for finding
approximate solutions to the complex problems of engineering and mathematical
physics that cannot be solved with analytical methods. In most of the applications
that require simulation to be fast, linear FEM is widely used. Linear FEM
works with a high degree of accuracy with small deformations. However, linear
FEM fails in accuracy when large deformations are used. Therefore, nonlinear
FEM is the suitable method for crucial applications like surgical simulators. In
this thesis, we propose a new formulation and finite element solution to the nonlinear
3D elasticity theory. Nonlinear stiffness matrices are constructed by using
the Green-Lagrange strains (large deformation), which are derived directly from
the infinitesimal strains (small deformation) by adding the nonlinear terms that
are discarded in infinitesimal strain theory. The proposed solution is a more
comprehensible nonlinear FEM for those who have knowledge about linear FEM
since the proposed method directly derived from the infinitesimal strains. We
implemented both linear and nonlinear FEM by using same material properties
with the same tetrahedral elements to examine the advantages of nonlinear FEM
over the linear FEM. In our experiments, it is shown that nonlinear FEM gives
more accurate results when compared to linear FEM when rotations and high
external forces are involved. Moreover, the proposed nonlinear solution achieved
significant speed-ups for the calculation of stiffness matrices and for the solution
of a system as a whole.
Keywords
Tetrahedral elementDeformation
Finite element method
GreenLagrange strain
Surgery simulation