Numerical Methods in Simulation and Optimal Control
Many phenomena occurring in real-life applications, e.g., in physics, finance, biology, biomechanics, are modeled by means of ordinary (ODEs) or partial (PDE) differential equations. The understanding of these differential equations is closely connected to the understanding of their physical meaning and the qualitative and quantitative behavior of their solutions. However, only the simplest ordinary and partial differential equations have closed form solutions, i.e., solutions, which can be expressed in terms of fundamental functions. Therefore, a numerical approximation is in most cases the only way to nd the solution. Due to the complexity of models or the accuracy needed, efficient algorithms have to be developed for numerical simulation. Moreover, the optimal control problem is even more computationally demanding. Thus, modern tools from numerical mathematics and scientific computing, along with careful investigation and exploitation of the problem structure, are required. In this project we deal with two real-life applications and develop the mathematical framework needed to efficiently simulate and solve the related control problem numerically, i.e., modeling, numerical analysis, implementation, etc. We focus on Musculoskeletal Modeling and Simulation in climbing and Crime Modeling.
Musculoskeletal Modeling and Simulation in climbing
In climbing, the whole musculoskeletal system is involved. However, in most of the movements, the upper body plays a central role. In this project we investigate how the components of the musculoskeletal system of the upper body segments interact during climbing.
As a first approach, we will consider a basic movement of the upper body which occurs in cycles. Since the movement is symmetric for both sides, we consider just one arm in the model. An isolated cycle for each arm can be separated in three phases: the grabbing, the pulling and the relaxing. The main goal of this project is to develop a 3D model for the shoulder and the arm to optimize the force and the energy during a three-phase cycle in climbing.
Technics how to optimize the performance in climbing have been proposed in the literature. Most of them are based on experimental results and do not provide a quantitative description of the movement, e.g., musles forces, besides the qualitative results out of statistics. However, a mathematical approach does. We think that a combination of the two approaches will lead us to a better understanding of the forces and the energy involved during climbing..
We assume that all systems are placed in equilibrium state and run simulations with real data. Moreover, we extend the ideas proposed by Chaturapruek et al. [2013] for running a two dimensional model based in Levy flights and the two models with real data.
Researchers:
Dr. Hermann Mena (Department of Mathematics, University of Innsbruck, Innsbruck, Austria)
Dr. Dieter Heinrich (Department of Sport Science, University of Innsbruck, Innsbruck, Austria)
BSc. Andreas Kofler (University of Innsbruck, Innsbruck, Austria)
BSc. Johannes Schwaighofer (University of Innsbruck, Innsbruck, Austria)
Martin Zarfl (University of Innsbruck, Innsbruck, Austria)
Posters related with the project:
Mathematics Day, University of Innsbruck, April 2014, Innsbruck, Austria
Tags: Simulation, Optimal control