Geometric Numeric Integration in Applications

In the GeNuIn project we consider simulation and modelling of multi-body dynamics, and problems arising in biomedical imaging. Our main scientific goal is to achieve more accurate and reliable computer simulations of the considered physical phenomena by exploiting the qualitative properties of the underlying models. In the first PhD project we are interested in the correct and accurate simulation of the pipe-laying process from ships on the bottom of the sea. The problem comprises the modelling of two interacting structures: a long and thin pipe (modelled as a rod) and a vessel (modelled as a rigid body). The system is subject to environmental forces (such as sea and wind effects). The control parameters for this problem are the vessel position and velocity, the pay-out speed and the pipe tension while the control objectives consist in determining the touchdown position of the pipe as well as ensuring the integrity of the pipe and to avoid critical deformations. We have considered numerical methods for the solution of these equations. With appropriate techniques, called splitting methods, we transform the system into several simpler problems which are solved in sequence. We use Jacobi elliptic functions and Lie group integrators to solve each sub-problem. An important feature of the numerical methods is to preserve the dissipative properties of the system of equations under numerical discretization. One of the post doscs has worked on numerical methods for similar mechanical systems and geometric methods for problems with holonomic and nonholonomic constraints. In the second PhD project we have considered volume preserving integrators. These are interesting in problems of image registration, shape matching and shape analysis. We have derived explicit, volume-preserving methods for polynomial divergence-free vector fields of arbitrary degree. The methods appear to be competitive with state of the art techniques. A different class of volume preserving integrators has been also derived by considering generating forms and functions for volume preserving mappings. The second post doc has worked on different, but related problems in medical imaging.

In this project we will consider simulation and modelling of multi-body dynamics, nonholonomic systems, and problems arising in biomedical imaging. Our main scientific goal is to achieve more accurate and reliable numerical simulations of the considered physical phenomena by exploiting the qualitative properties of the underlying models. Our starting point are models with an underlying geometric structure. Our research team has been in the international forefront in the development of geometric integ ration algorithms over the last 16 years. We are convinced that the exploitation of this knowledge in a more applied problem setting will lead to significant advance in the considered applications and give new insight for the design of superior numer ical strategies.