# Simulating slender structures with Lie group methods and data

We are Andrea Leone and Ergys Çokaj, two first year PhD students at the Department of Mathematical Sciences of the Norwegian University of Science and Technology (NTNU). Our positions as Early Stage Researchers ESR4 and ESR5 are part of the ETN THREAD project “Numerical Modelling of Highly Flexible Structures”.

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THREAD is a unique network of universities, research organisations and industrial partners in 11 European countries and addresses modelling problems related to essential parts of high-performance engineering systems such as highly flexible slender structures (yarns, cables, hoses or ropes). The complex response of these structures in real operational conditions is far beyond the capabilities of current modelling tools that are at the core of modern product development cycles. The THREAD project has brought together young mechanical engineers and mathematicians who will develop mechanical models and numerical methods for designing highly flexible slender structures and support the development of future virtual prototyping tools, in order to better meet the needs of different industries.

Why did we choose THREAD for our PhD studies?

Andrea: I come from Italy and I have a master’s degree in Mathematical Engineering and a bachelor’s degree in Physics, both from University of L’Aquila. I am interested in mathematical modelling and numerical analysis as well as machine learning and I believe that the ESR4 project on data driven modelling of beams will highly improve my expertise in these fields. I find it an exciting and challenging research topic, with significant industrial applications.

I also appreciate the interdisciplinary research environment of the THREAD network, based on the collaboration of mathematicians and engineers, and this is reflected in my academic background. Therefore, I chose THREAD as I think that it provides an invaluable opportunity to investigate fundamental modelling problems in a strong, international academic environment.

Ergys: I have a background in applied mathematics with a bachelor’s degree and a master’s degree in “Mathematics and applications” from the University of Camerino, Italy.

I applied for the ESR5 position in the THREAD project because I see it as an extension of my study interests and I considered my background strongly related to the prerequisites for the project. An exciting fact about THREAD is the direct opportunity to collaborate with researchers with different backgrounds, from all around the world.

I am very happy that I am accepted to be a part of this project and I am convinced that this experience will strengthen my skills, increase my knowledge and change the way I see the world around me.

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What will our contribute be on THREAD?

Andrea: My research project is about data driven modelling of cables and hoses. During the PhD I aim at developing numerical methods for PDE models of slender structures that respect fundamental, underlying geometric properties of the equations. These discretization methods should be able to cope with the high flexibility of the structure and resolve large rotations. I will consider suitable techniques for incorporating information from data (e.g. from laboratory experiments) and to this end I intend to combine machine learning techniques inspired by optimal control with the PDE models. I will also have the chance to pursue industrial applications with the Fraunhofer Institute in Germany and the industrial partner TechnipFMC in Norway.

Ergys: My individual research project is entitled: “Novel practical numerical methods for mechanical problems on Lie groups and their implementation”. The objectives of my individual research project are the study implementation issues for Lie group integrators and invariant preserving integrators. Expected results of my work are on variable step size Lie group integrators with error control: for Lie group integrators with stepwise updated local coordinates standard error control from integrators on linear spaces can be applied. I will also be working on invariant preserving schemes for mechanical systems evolving on Riemannian manifolds and model reduction applied to large dynamical systems on Lie groups and manifolds.

Visualization of a configuration of a geometrically exact beam as provided by the Cosserat rod theory. This is an appropriate model for the geometrically exact simulation of deformable rods in space (statics) and space-time (quasi-statics or dynamics). The rod is fully described by the curve of centroids $\textbf{u}(s)$ (center line) and the orientation of its cross sections, defined by the orthonormal frame $\textbf{d}_i=\textbf{R}\textbf{e}_i, \; i=1,2,3$, attached to each point of the center line, where $\textbf{d}_3(s)$ is normal to the plane cross section at $\textbf{u}(s)$ and $\textbf{R} \in SO(3)$.

We assume that the configurations and motions of the beam are such that sections do not deform. A measure of the change of orientation undergone by the section can then be obtained by comparing the vectors $\partial_s\textbf{u}(s)$ and $\textbf{d}_3(s)$. The relative change in length will measure the longitudinal strain, while the relative change in direction will measure the amount of shear. If we also keep track of the original orientation (before deformation) of the section with respect to the tangent to the center line, changes in direction of the vectors lying on the section will also provide a measure of torsion.

These deformation measures enter in the governing equations of geometrically exact beams and we consider models based on the pioneering work started by J.C. Simo and his collaborators in the 1980’s.

Picture from the Nwt1 course of THREAD.

Best regards,

Andrea & Ergys