Implicit Function Modeling of Solidification in Metal Castings

Abstract

Solidification of metal castings can be modeled by an implicit real-valued function whose behavior is determined by physical parameters prescribed on the boundary of a casting. We show how to construct such functions using theory of R-functions for two-dimensional castings represented by their boundaries. The parameterized form of the constructed functions is convenient for studying, controlling, and optimizing their behavior in terms of the physical parameters specified on the boundary of the casting. The proposed approach can also be used for modeling multiple cavities in a same sand mold, generalizes to three-dimensional castings, and is applicable to other physical phenomena that may be suitable for analysis based on empirical knowledge.

Approximate distance fields with non-vanishing gradients

“…6A. Similar techniques were used experimentally in [36] for modeling of solidification in metal castings. These examples suggest that it may be possible to construct fields normalized at all points for arbitrary decompositions of curves and surfaces.…”

“…The discussion in this section is a straightforward application of the trimming procedures described and studied in [29,35,36] for arbitrary implicitly defined curves and surfaces. Here, the same procedures are used to construct a normalized function for a straight line segment.…”

A meshfree method for incompressible fluid dynamics problems

“…Informally, R-functions serve as a construction toolkit transforming a set-theoretic description of the boundary of a geometric object into a real valued function whose zero set coincides with the boundary. Detailed discussion on R-functions and construction techniques is outside of the scope of this paper, but it can be found in numerous references [23][24][25][26][27] and will be illustrated in Section 2.2. Functions constructed using R-functions are di erentiable everywhere except a ÿnite number of points [23,25] and behave as distances to the boundaries near the boundary points.…”

TheR-function method in boundary-value problems with geometric and physical symmetry

“…The first two of formulas (4) correspond to the known logical operations of intersection and union of sets, the third corresponds to complementation (negation), and the other two to equivalence (negation of the symmetric difference) [2,6]. Information on other R-functions, methods of constructing them, and their use, can be found in [2,5].…”

“…0V o'i=0 | According to the R-function method, to obtain the normalized equation of the boundary of the region defined by the formula (6), it suffices to eliminate the symbols "> 0" in (6) and replace the operation of intersection "n" by the symbol "ix=" , [2,5]:…”