We derive a combined analytical and numerical scheme to solve the (1+1)-dimensional differential Kirchhoff system. Here the object is to obtain an accurate as well as an efficient solution process. Purely numerical algorithms typically have the disadvantage that the quality of solutions decreases enormously with increasing temporal step sizes, which results from the numerical stiffness of the underlying partial differential equations. To prevent that, we apply a differential Thomas decomposition and a Lie symmetry analysis to derive explicit analytical solutions to specific parts of the Kirchhoff system. These solutions are general and depend on arbitrary functions, which we set up according to the numerical solution of the remaining parts. In contrast to a purely numerical handling, this reduces the numerical solution space and prevents the system from becoming unstable. The differential Kirchhoff equation describes the dynamic equilibrium of one-dimensional continua, i.e. slender structures like fibers. We evaluate the advantage of our method by simulating a cilia carpet.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledgements: This work was partially supported by the Max Planck Center for Visual Computing and Communication (D.L.M.) as well as by the grant No. 13-01-00668 from the Russian Foundation for Basic Research (V.P.G.). The authors thank Robert Bryant for useful comments on the solution of PDE system (23)-(25) and the anonymous referees for their remarks and suggestions.