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Meshfree Modeling and Analysis

R - Functions

"Theory of R-functions and Its Applications" lectures available online.


The theory of R-functions was developed in Ukraine by Rvachev and his students [1].

A real-valued function f(x1,x2,...,xn) is called an R-function if its sign is completely determined by the signs of its arguments xi. If the sign of a function is considered to be a logical property, negative values of a function can be considered to correspond with logical FALSE, and positive values can be considered to correspond with logical TRUE. In other words, an R-function f works as a Boolean switching function, changing its sign only when its arguments change their signs;  it can be regarded as ?on? or ?off? (or ?true? or ?false?) depending on the values of the input variables.  For example, the function xyz can be negative only when the number of its negative arguments is odd. As another example, min(x1, x2) is an R-function whose companion Boolean function is logical ?and? (/\) (logical conjunction), and max(x1, x2) is an R-function whose companion Boolean function is logical ?or? (\/) (logical disjunction).  This is seen in that function min(x1, x2) takes on positive values only when x1 AND x2 are positive;  similarly function max(x1, x2) takes on positive values when x1 OR x2 are positive. Except the functions min(x1, x2) and max(x1, x2) there are infinitely many R-functions with different differential properties [1, 2, 3]. Here is the most popular system of R-functions:

Similar to Boolean functions, R-functions are closed under composition, which means that a combination of several R-functions is another R-function which corresponds to a more complex logical expression.  Thus, just as any logical function can be written using only three operations (logical negation or NOT), \/ (logical disjunction or OR), and /\ (logical conjunction or AND), three corresponding R-functions can be combined into a corresponding R-function. Expressed another way, for every formal logical sentence (i.e., for every Boolean function), one may construct a corresponding R-function using R-conjunction, R-disjunction, and R-negation, whose sign is determined by the truth table of the logical sentence.  For nonzero arguments, the negation () operation is usually accomplished by changing the sign of the R-function.  The logical disjunction \/ and conjunction /\ operations can respectively be accomplished in the usual case by performing intersection and union operations.  Depending on the particular form of the R-conjunction, R-disjunction, and R-negation chosen to construct the corresponding R-function (i.e., depending on the ?family? from which the R-functions are chosen), a rich variety of differential properties may be obtained.

R-functions are a toolkit for construction of normalized implicit functions with desirable differential properties from any logical geometric representation.



[1] V. L. Rvachev. Theory of R-functions and Some Applications. Naukova Dumka, 1982. In Russian.

[2] Theory of R-functions and applications: A primer. V. Shapiro, Technical Report, Cornell University.

[3] V. Shapiro and I. Tsukanov, Implicit Functions with Guaranteed Differential Properties, In Proceedings of the Fifth ACM Symposium on Solid Modeling and Applications, June 1999, Ann Arbor, MI

[4] V. Shapiro and I. Tsukanov, Meshfree Simulation of Deforming Domains, Computer-Aided Design , Vol. 31, No. 7, 1999, pp. 459-471

[5] V.L.Rvachev, T.I.Sheiko, V.Shapiro and I.Tsukanov, On Completeness of RFM Solution Structures, Computational Mechanics, special issue on meshfree methods, Vol. 25, 2000, pp. 305-316

[6] V.L. Rvachev, T.I. Sheiko, V. Shapiro, I. Tsukanov, Transfinite Interpolation Over Implicitly Defined Sets, Computer Aided Geometric Design, Vol. 18, No. 4, 2001, pp.195-220

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