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Algebraic Construction of Smooth Interpolants on Polygonal Domains
Elisabeth Anna Malsch
Gautam Dasgupta

A smooth and bounded interpolant can be constructed in explicit algebraic form within any polygon, convex or concave. The resulting function is not unique and accordingly can be adjusted to satisfy desired global conditions, such as linear fields. The closed-form representation is obtained by combining simple geometric descriptions, such as the side lengths and areas. The interpolant distributes values given at discrete nodes smoothly over the interior of the domain. On a convex polygon, the interpolant is a rational function of the product of areas. On a concave or multiply connected polygon, the interpolant is a function of areas and edge lengths, which introduces a square root term.

*Notebook


*PDF


*HTML

*Introduction

*Input Data

*Geometric Measures and Support Functions

*Minimum Functions

*Functions for Displaying Interpolants

*Interpolants for Convex Polygons

*Interior Points

*Concave Polygons

*Concave Polygons with Linearity on the Sides

*Combining Parts

*Enforcing Global Linearity

*Conclusion

*Acknowledgment

*References

About the Authors
Elisabeth Anna Malsch completed her B.S. (1999), M.S. (2000), and Ph.D. (2003) from Columbia University, New York. Currently she is an Alexander von Humboldt fellow, Bonn, Germany. She has been using symbolic computation in her undergraduate, graduate and Ph.D. courses and research focused on engineering mechanics. For details about her recent publications on concave finite elements and boundary elements, see www.civil.columbia.edu/~malsch.

Gautam Dasgupta is a professor of civil engineering and engineering mechanics at Columbia University, New York. His areas of research include computer mathematics, waves in random media, earthquake engineering, and finite and boundary element analysis with Mathematica. For details on his recent work, see www.columbia.edu/~gd18.

Elisabeth Anna Malsch
Postdoctoral Researcher
Department of Civil Engineering and Engineering Mechanics
Columbia University
500 West 120th Street
New York, NY 10027
malsch@civil.columbia.edu

Gautam Dasgupta
Professor
Department of Civil Engineering and Engineering Mechanics
Columbia University
500 West 120th Street
New York, NY 10027
dasgupta@columbia.edu


     
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