# JAMC

Article http://dx.doi.org/10.26855/jamc.2020.03.002

# Self-Adaptive Nonlinear Least Squares Optimization for Geometric Calculations

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Alpar A. Csendes

GRAPHISOFT SE, Záhony u. 7. (GRAPHISOFT Park 1.) 1031 Budapest, Hungary

*Corresponding author: Alpar A. Csendes. GRAPHISOFT SE, Záhony u. 7. (GRAPHISOFT Park 1.) 1031 Budapest, Hungary.

Published: March 6,2020

## Abstract

In this paper, two-dimensional parameterized geometries and related computational techniques are discussed. Simple and extendable parameterized computational model has been established, which provides the foundation for morphing the geometry. The well known Nonlinear Least Squares optimization technique has been applied in order to find the particular shape of the geometry satisfying a given set of criteria. This criteria-system is provided by the end-user, and so it has been fundamental to provide a meaningful technique as part of the setup. Our “dimension tool” was not only the right choice for our clients (Architects) but also the related mathematical model proved to be easily handled by the numerical solution process. The solution is found via minimization of the sum of properly weighted and squared residuals. This system of residuals has been extended by auxiliary residuals and related terms implementing an optimization-embedded continuation-technique, a new and promising approach to support stable and quick solution for stiff residual systems. At the end of the paper we show that the computational model and techniques are capable of supporting extraction of parametric sensitivity of the final geometry, which has great practical importance in design optimization.

## References

[1] Fletcher, R. Practical Methods of Optimization. 2nd ed.

[3] Beltrán, Carlos. (2011). A continuation method to solve polynomial systems and its complexity. Numerische Mathematik. 117. 89-113. 10.1007/s00211-010-0334-3.

[4] Automatic differentiation. https://en.wikipedia.org/wiki/Automatic_differentiation

## How to cite this paper

Self-Adaptive Nonlinear Least Squares Optimization for Geometric Calculations

How to cite this paper: Alpar A. Csendes. (2020) Self-Adaptive Nonlinear Least Squares Optimization for Geometric Calculations. Journal of Applied Mathematics and Computation, 4(1), 5-13.

DOI: http://dx.doi.org/10.26855/jamc.2020.03.002