# A Locally Gradient-Preserving Reinitialization for Level Set Functions

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## Abstract

The level set method commonly requires a reinitialization of the level set function due to interface motion and deformation. We extend the traditional technique for reinitializing the level set function to a method that preserves the interface gradient. The gradient of the level set function represents the stretching of the interface, which is of critical importance in many physical applications. The proposed locally gradient-preserving reinitialization (LGPR) method involves the solution of three PDEs of Hamilton–Jacobi type in succession; first the signed distance function is found using a traditional reinitialization technique, then the interface gradient is extended into the domain by a transport equation, and finally the new level set function is found by solving a generalized reinitialization equation. We prove the well-posedness of the Hamilton–Jacobi equations, with possibly discontinuous Hamiltonians, and propose numerical schemes for their solutions. A subcell resolution technique is used in the numerical solution of the transport equation to extend data away from the interface directly with high accuracy. The reinitialization technique is computationally inexpensive if the PDEs are solved only in a small band surrounding the interface. As an important application, we show how the LGPR procedure can be used to make possible the local level set approach to the Eulerian Immersed boundary method.

## Keywords

Level-set method Eulerian immersed boundary method Fluid–structure interaction Reinitialization## Notes

### Acknowledgments

X. Xu acknowledges support by NSF-DMS Grant 1159133.

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