Abstract: Grid generation and adaptation is an important task for numerical solution of partial differential equations. Over the past decades, effective moving grid approaches for generating adaptive structured and unstructured grids have been developed. This talk will focus on the deformation method and its applications. The method has its origin in Jurgen Moser’s work on Riemannian volume elements. It has been adopted to generate injective, smooth coordinate transformations with prescribed Jacobian determinant.
In the first part of this talk, we will discuss its theoretical foundation and indicate its applications to field simulation problems such as computational fluid dynamics and structure analysis. In the second part, we will discuss a new formulation of image registration as an optimal control problem. The deformation method provides the constraint in this approach.
References
1. J. Moser, Volume elements of a Riemannian manifold, Trans. AMS, 120 (1965)
2. A. D. Melas, An Example of a Harmonic Map Between Euclidean Balls, Proceedings of the American Mathematical Society, Vol. 117, No. 3. (1993), pp. 857-859.
4. T-W. Pan, J. Su and G. Liao, A numerical grid generator based on Moser's deformation method, Numerical Methods for Partial Differential Equations, Vol. 10, p. 20-31, 1994
5. F. Liu, S. Ji, and G. Liao, An adaptive grid method and its application to steady Euler flow calculations, SIAM J. Sci. Comput. 20, 811-825, 1998
6. G. Liao, F. Liu, C. de la Pena, D. Pang, and S. Osher, Level set based deformation methods for adaptive grids, Journal of Computational Physics, vol. 159, p. 103-122, 2000
7. Z. Lei and G. Liao, Adaptive grids for resolution enhancement, Shock Waves, vol. 12, p.153-156, (2002)
8. X. Han, C. Xu, and J. L. Prince, A 2D Moving Grid Geometric Deformable Model, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003
9. D. Fleitas, X. Cai, G. Liao, B. Jiang, The Least-Squares Finite Element Method on Overlapping Elements, Journal of Computational Information Systems 1:2 (2005) 203-213
10. S. Turek and D. Wan, Fictitious Boundary and Moving Mesh Methods for the Numerical Simulation of Particulate Flow, D. Anca, European Conference on Computational Fluid Dynamics, ECCOMAS CFD 2006, P. Wesseling, E. O˜nate and J. P´eriaux (Eds), TU Delft, The Netherlands, 2006
11. G. Liao et. al. Optimal Control Approach to Data Set Alignment, to appear in Applied Math. Letters
This seminar is easily accessible to persons with disabilities. For more information or for assistance, please contact the Mathematics Department at 743-3500.