Surfaces in Action 1: Ready to Move?

Garant předmětu:

Přednášející:

Cvičící:

Předmět zajišťuje:

katedra softwarového inženýrství

Anotace:

Have you ever wondered how Hollywood creates lifelike water and fire, how self-driving cars find the safest route, how doctors reconstruct the shape of a baby from MRI scans during pregnancy, how MRI data is turned into 3D models of the brain, how cardiologists simulate electrical signals propagating through the heart? These are only a glimpse of remarkable applications where evolving surfaces shapes that move, grow, or change over time are the invisible engine behind modern technologies.

This course is dedicated to answering the mentioned questions above by building a foundational understanding of surface evolution across physics, engineering, and applied mathematics. The course offers a deep, intuitive understanding of surface evolution, equipping you with essential tools for modeling motion, shape change, and geometry-driven dynamics. Youll explore these topics through real-world examples and hands-on formulations, before even hearing terms like level set method. More importantly, if you're excited about trendy neural network solvers in surface evolution, this course gives you the core mathematical foundations you'll need in the future because even machine learning must respect the physics of surfaces in motion.

Požadavky:

Basic linear algebra, Strong knowledge in Calculus, Basic understanding of Differential Geometry and Finite Difference Method.

Osnova přednášek:

1. Visual Introduction

2. Eulerian and Lagrangian formulations of evolving manifolds

3. Eikonal equation and reinitialization

4. Gedankenexperiment of the level set method

5. Extension of velocity off the interface

6. Connecting physical processes to evolving geometry

7. Curvature-driven flow and geometric shrinking

8. Anisotropy and directional motion

9. Surface evolution under spatial heterogeneity

10. Applications across science and engineering

11. Fast marching and travel-time modeling

12. Image-based surface motion

Osnova cvičení:

Cíle studia:

Studijní materiály:

Books:

[1] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences vol. 153, 2003.

[2] J. A. Sethian, Level Set Methods and Fast Marching Methods: evolving interfaces in computational geometry, uid mechanics, computer vision, and materials science, Cambridge University Press, 2nd edition, 1999

[3] J. Hahn, K. Mikula, P. Frolkovič, P. Priesching, M. Balažovjech, B. Basara, „Second-order accurate finite volume method for G-equation on polyhedral meshes,“ Japan Journal of Industrial and Applied Mathematics, vol. 40, pp. 1053-1082, 2023. DOI: 10.1007/s13160-023-00574-x

[4] J. Hahn, K. Mikula, P. Frolkovič, and B. Basara, „Finite volume method with the Soner boundary condition for computing the signed distance function on polyhedral meshes,“ International Journal for Numerical Methods in Engineering, vol. 123, no. 4, pp. 1057-1077, 2022. DOI: 10.1002/nme.6888

Additional materials:

https://link.springer.com/article/10.1007/s00158-024-03870-3

https://ww3.math.ucla.edu/camreport/cam04-02.pdf

https://www.sciencedirect.com/science/article/pii/S0898122123005655?via%3Dihub

https://link.springer.com/article/10.1007/s13160-023-00574-x

Poznámka:

Další informace:

Pro tento předmět se rozvrh nepřipravuje

Předmět je součástí následujících studijních plánů: