An Introduction to the Level Set Method

1. Generalities about domain, or surface evolution problems 2 2. The Level Set Method for describing domain or interface evolution 3 2.1. Level Set framework and geometry 3 2.2. Surface evolution in the level set framework 4 2.3. A glimpse at the mathematical framework 5 2.4. Domain evolution as a boundary value problem: Eikonal equations 8 3. A model application of the level set framework 10 4. Solving the Level set Hamilton-Jacobi equation 12 4.1. Solving the Level Set Hamilton-Jacobi equation on Cartesian grids 12 4.2. Solving the Level Set Hamilton-Jacobi equation on triangular meshes 14 4.3. Semi-Lagrangian schemes 15 5. Initializing level set functions 16 5.1. The Fast Marching Method on Cartesian grids 17 5.2. Extension of the Fast Marching Method to triangular meshes 18 6. Operations within the Level Set framework 19 6.1. Evaluation of the normal vector or the mean curvature of Γ 19 6.2. Calculating integrals on Ω and Γ. 19 6.3. Algebraic operations over sets 20 6.4. Solving partial differential operations posed on Ω 20 7. Additional features 21 7.1. Redistancing 21 7.2. Velocity extension 22 7.3. Towards enhancing the efficiency of the Level Set method: the Narrow Band approach 23 7.4. Handling multiple phases with the Level Set method 24 8. Other domain evolution methods 25 8.1. Lagrangian methods 25 8.2. Eulerian methods 26 8.3. Hybrid methods 28 9. Applications of the Level Set Method 30 9.1. Applications in Computational Fluid Dynamics 30 9.2. Applications in shape optimization 30 9.3. Applications in image processing 30 References 31