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.
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Chin, C.H.K., May, R.L., Connell, H.J.: A numerical model of a towed cable-body system. J. Aust. Math. Soc. 42(B), 362–384 (2000)
Swigon, D., Coleman, B.D., Tobias, I.: The elastic rod model for DNA and its application to the tertiary structure of DNA minicircles in mononucleosomes. Biop. J. 74, 2515–2530 (1998)
Colemana, B.D., Olsonb, W.K., Swigonc, D.: Theory of sequence-dependent DNA elasticity. J. Chem. Phys. 118, 7127–7140 (2003)
Moakher, M., Maddocks, J.H.: A double-strand elastic rod theory. In: Workshop on Atomistic to Continuum Models for Long Molecules and Thin Films (2001)
Coleman, B.D., Swigon, D.: Theory of self-contact in Kirchhoff rods with applications to supercoiling of knotted and unknotted DNA plasmids. Philosophical Transactions: Mathematical, Physical and Engineering Sciences 362, 1281–1299 (2004)
Goriely, A., Tabor, M.: Spontaneous helix-hand reversal and tendril perversion in climbing plants. Phys. Rev. Lett. 80, 1564–1567 (1998)
Nizette, M., Goriely, A.: Towards a classification of Euler-Kirchhoff filaments. Journal of Mathematical Physics 40, 2830–2866 (1999)
Goriely, A., Nizette, M., Tabor, M.: On the dynamics of elastic strips. Journal of Nonlinear Science 11, 3–45 (2001)
da Fonseca, A.F., de Aguiar, M.A.M.: Solving the boundary value problem for finite Kirchhoff rods. Physica D 181, 53–69 (2003)
Thomas, Y.H., Klapper, I., Helen, S.: Romoving the stiffness of curvature in computing 3-d filaments. J. Comp. Phys. 143, 628–664 (1998)
Rappaport, K.D.: S. Kovalevsky: A mathematical lesson. American Mathematical Monthly 88, 564–573 (1981)
Goriely, A., Nizettey, M.: Kovalevskaya rods and Kovalevskaya waves. Regu. Chao. Dyna. 5(1), 95–106 (2000)
Shi, Y., Hearst, J.E.: The Kirchhoff elastic rod, the nonlinear Schroedinger equation and DNA supercoiling. J. Chem. Phys. 101, 5186–5200 (1994)
Pai, D.K.: STRANDS: Interactive simulation of thin solids using Cosserat models. Computer Graphics Forum 21, 347–352 (2002); Eurographics 2002
Robbins, C.R.: Chemical and physical behavior of human hair. Springer, Heidelberg (2002)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C++, 2nd edn. Cambridge University Press, Cambridge (2002)
Aluffi-Pentini, F., Parisi, V., Zirilli, F.: Sigma — a stochastic-integration global minimization algorithm. ACM Tran. Math. Soft. 14, 366–380 (1988)
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Shu, L., Weber, A. (2005). A Symbolic-Numeric Method for Solving Boundary Value Problems of Kirchhoff Rods. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2005. Lecture Notes in Computer Science, vol 3718. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11555964_33
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DOI: https://doi.org/10.1007/11555964_33
Publisher Name: Springer, Berlin, Heidelberg
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