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13. Example Problems

Several example problems are included with VBM which use a variety of visualization and computation modules.

13.1 Strut Problem

This example is a seven dimensional ODE BVP. It models a naturally twisted strut to which a force is being applied at the end. After discretization we will have a system with approximately 600 equations and two more unknowns than equations. This is the same problem that is used in the tutorial part of this manual.

The VBM file for this demo can be found in:


VBM/demos/planar_strut/strut.vbm

This file demos the VBM_1D_AUTO compute engine. Since it is the basis of the tutorial in Section it is a good place to start looking at VBM files.

13.2 Heavy Rigid Body Problem

This example, taken from the work of Maddocks in Bibliography, is a problem with six equations and eight unknowns. The system models the equilibrium configurations of a heavy rigid body.

The VBM file for this demo can be found in:


VBM/continuation_data/MC2_2par/mc2.data

This file demos the VBM_1D_AUTO and VBM_2D_Rheinboldt compute engines. Note, you must have Matlab installed on your computer to use the VBM_2D_Rheinboldt module.

13.3 AUTO BVP Problem

This demo is taken directly from the demo problems in AUTO94 (for more detail see Bibliography). It is a two dimensional ODE BVP. We provide it as an example of the changes that are necessary to take a problem originally for AUTO94 and use it in VBM.

The VBM file for this demo can be found in:


VBM/continuation_data/bvp/bvp.vbm

This file demos the VBM_1D_AUTO_FORT8, VBM_1D_AUTO, and VBM_1D_remote_AUTO compute engines. Note, the file must be modified to have the first line of the VBM_1D_remote_AUTO block be the address of a machine running a VBM compute server.

13.4 Ring Problem

This example is a fourteen dimensional Hamiltonian ODE BVP which has been used to model the equilibrium configurations of small DNA mini-circles. In this example VBM (or its predecessors) was used as part of the original research, and results have been published in Bibliography, Bibliography, Bibliography, and Bibliography.

After discretization we will have a system with approximately 1200 equations and six more unknowns then equations. We will not explicitly compute any six dimensional manifolds, but we can compute various one dimensional sub-bifurcation diagrams.

The VBM file for this demo can be found in:


VBM/continuation_data/dna_1par/dna.data

This file demos VBM_1D_AUTO and VBM_1D_remote_AUTO compute engines. Note, the file must be modified to have the first line of the VBM_1D_remote_AUTO block be the address of a machine running a VBM compute server. Also, this demo can use VBM's ability to transmit binary problem files. There is a Makefile in VBM/continuation_data/dna_1par which may be used to compile a parallel version of AUTO94 on SunOS machines with the Sun Workshop Pro compilers. This Makefile can be used as an example for the compilation on your specific machine.

13.5 Bibliography for Example Problems

  1. E. J. Doedel, Xianjun Wang, and Thomas Fairgrieve. AUTO 94, Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. Caltech, Dept. of Applied Mathematics, 1994.
  2. P. Furrer, R. Manning, and J. Maddocks. Multiple equilibria in DNA rings. 1999. In preparation.
  3. J. Maddocks, R. Manning, and J. Kahn. A continuum rod model of sequence-dependent DNA structure. J. Chem. Phys., 105:5626-5626, 1996
  4. J. Maddocks, R. Manning, R. Paffenroth, K. Rogers, and J. Warner. Interactive computation, parameter continuation, and visualization. International Journal of Bifurcation and Chaos, (8):1699-1715, 1997
  5. J. H. Maddocks and R. L. Sachs. Constrained Variational Principles and Stability in Hamiltonian Systems. In Meyer, Dumas, and Schmidt, editors, Hamiltonian Dynamical Systems, volume 63 of IMA Volume in Mathematics and Its Applications. Springer Verlag, 1995.
  6. K. Rogers. Stability of Elastic Rods. PhD Thesis, Unversity of Maryland, College Park, 1997.


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