Extending fields in a level set method by solving a biharmonic equation
This work provides a simpler and more implicit approach for velocity extension in level set methods, benefiting computational scientists using these methods for interface tracking.
The authors propose a new method for extending fields in level set methods by solving a biharmonic equation, which avoids explicit interface properties and yields smooth extensions. The method is demonstrated on test problems showing effectiveness and natural handling of symmetry and periodicity.
We present an approach for computing extensions of velocities or other fields in level set methods by solving a biharmonic equation. The approach differs from other commonly used approaches to velocity extension because it deals with the interface fully implicitly through the level set function. No explicit properties of the interface, such as its location or the velocity on the interface, are required in computing the extension. These features lead to a particularly simple implementation using either a sparse direct solver or a matrix-free conjugate gradient solver. Furthermore, we propose a fast Poisson preconditioner that can be used to accelerate the convergence of the latter. We demonstrate the biharmonic extension on a number of test problems that serve to illustrate its effectiveness at producing smooth and accurate extensions near interfaces. A further feature of the method is the natural way in which it deals with symmetry and periodicity, ensuring through its construction that the extension field also respects these symmetries.