Horizontal decomposition of triangulated solids for the simulation of dip-coating processes

Abstract

In dip-coating processes a three-dimensional object, e.g. an entire car body, is dipped into a liquid bath. In order to simulate such processes, the space surrounding the object is decomposed into the so-called flow volumes, for which each intersection with a horizontal plane is connected. At any time the liquid's surface then has a unique level within such a flow volume, which greatly simplifies the simulation of the liquid. The decomposition into flow volumes corresponds to the Reeb graph of the object's exterior (considered as 3-manifold with boundary) with respect to the height function. This article presents an algorithm which computes this decomposition for an object represented as oriented triangular boundary mesh. First critical vertices of the surface are identified, which include the upper and lower ends of flow volumes. Using local information about horizontal intersection planes near the critical points, a sweep plane algorithm then constructs the volume decomposition in a second step. It is shown that the method can deal with realistic data.

Layered Reeb graphs for three-dimensional manifolds in boundary representation

“…On the one hand, the Morse functions may be determined by the application in mind. For example, the Reeb graph with respect to the height function is a valuable tool in the simulation of dip coating processes [29]. A similar situation may occur for specific applications of the Reeb space.…”

“…the connection to Jacobi sets. While the construction of the Reeb graph with respect to the height function finds an application in the simulation of dip-coating processes [29], the Reeb graph of a moving manifold could be applied in the simulation of the dipcoating process of an object that moves and rotates over time.…”

“…Other authors first identify all critical points and then find connections between them using monotone paths. For example, Doraiswamy and Natarajan [28] use paths of adjacent triangles of a simplex mesh of arbitrary dimension, and Strodthoff et al [29] use monotone edge paths to construct the Reeb graph of a three-dimensional manifold in boundary representation.…”

“…In contrast, we will consider three-dimensional manifolds in space which are given by an oriented triangular surface mesh that represents their boundary. The boundary-based construction algorithm for Reeb graphs of three-dimensional manifolds with respect to the height function was introduced in [29]. Here we will extend this approach to the computation of layered Reeb graphs.…”