# A Symbolic-Numeric Method for Solving Boundary Value Problems of Kirchhoff Rods

## Abstract

We study solution methods for boundary value problems associated with the static Kirchhoff rod equations. Using the well known Kirchhoff kinetic analogy between the equations describing the spinning top in a gravity field and spatial rods, the static Kirchhoff rod equations can be fully integrated. We first give an explicit form of a general solution of the static Kirchhoff equations in parametric form that is easy to use. Then by combining the explicit solution with a minimization scheme, we develop a unified method to match the parameters and integration constants needed by the explicit solutions and given boundary conditions. The method presented in the paper can be adapted to a variety of boundary conditions. We detail our method on two commonly used boundary conditions.

Keywords: computer algebra, hair modelling, Kirchhoff models

## Bibtex

@INPROCEEDINGS{shu-2005-symbolic, author = {Shu, Liu and Weber, Andreas}, editor = {Ganzha, V. G. and Mayr, E. W. and Vorozhtsov, E. V.}, pages = {387--398}, title = {A Symbolic-Numeric Method for Solving Boundary Value Problems of Kirchhoff Rods}, booktitle = {Computer Algebra in Scientific Computing (CASC '05)}, series = {Lecture Notes in Computer Science}, volume = {3718}, year = {2005}, month = sep, publisher = {Springer-Verlag}, keywords = {computer algebra, hair modelling, Kirchhoff models}, abstract = {We study solution methods for boundary value problems associated with the static Kirchhoff rod equations. Using the well known Kirchhoff kinetic analogy between the equations describing the spinning top in a gravity field and spatial rods, the static Kirchhoff rod equations can be fully integrated. We first give an explicit form of a general solution of the static Kirchhoff equations in parametric form that is easy to use. Then by combining the explicit solution with a minimization scheme, we develop a unified method to match the parameters and integration constants needed by the explicit solutions and given boundary conditions. The method presented in the paper can be adapted to a variety of boundary conditions. We detail our method on two commonly used boundary conditions.}, isbn = {3-540-28966-6}, conference = {Computer Algebra in Scientific Computing (CASC '05)} }