High-order methods for computing distances to implicitly defined surfaces

Abstract

Implicitly embedding a surface as a level set of a scalar function φ : ‫ޒ‬ d → ‫ޒ‬ is a powerful technique for computing and manipulating surface geometry. A variety of applications, e.g., level set methods for tracking evolving interfaces, require accurate approximations of minimum distances to or closest points on implicitly defined surfaces. In this paper, we present an efficient method for calculating high-order approximations of closest points on implicit surfaces, applicable to both structured and unstructured meshes in any number of spatial dimensions. In combination with a high-order approximation of φ, the algorithm uses a rapidly converging Newton's method initialised with a guess of the closest point determined by an automatically generated point cloud approximating the surface. In general, the order of accuracy of the algorithm increases with the approximation order of φ. We demonstrate orders of accuracy up to six for smooth problems, while nonsmooth problems reliably reduce to their expected order of accuracy. Accompanying this paper is C++ code that can be used to implement the algorithms in a variety of settings.

Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid–structure interaction, and free surface flow: Part I

“…Under-resolved cases, e.g., dispersal of liquid droplets with diameter comparable to or smaller than the typical element size, are not considered as part of the present work. In under-resolved cases, the construction of the implicit mesh and the transference of state may need to be augmented with other strategies to assist with any desired conservation properties; possibilities may include, for example, adopting ideas from volume of fluid methods [13], which can conserve fluid mass to machine precision, to develop a hybrid interface tracking algorithm, see, e.g., [66,67].…”

High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles

“…• Approximation of the level set function: In many applications of implicit interface methods, the level set function is known only at the grid points of a computational grid/mesh (see, e.g., [25,18,22,23,24]), and therefore must be interpolated in order to apply the quadrature scheme on each mesh element.…”

Numerical simulation of artificial microswimmers driven by Marangoni flow

“…After advecting the level set, a correction of Φ may still become necessary in order to enforce the signed-distance property. This manipulation, called reinitialization, must not displace the interface and it can be performed in different ways [64][65][66][67]. However, due to undesired interface displacements, the reinitialization step is known to cause mass loss.…”