OPTIMIZATION ALGORITHM
tion functions that required more basis functions. The reduction in the number of basis functions required to create an approximation function with error below tolerance reduces the likelihood of an ill-conditioned matrix when solving for the coefficients in the approximation function.
Recommendations for Future Research
Figure 15: Computational results on y-z plane where x=9. Discussion of Computational Results
For Example 1, computational error decreases as the number of nodes used to approximate the exact solution increas- es. Remember that the analytic solution to this boundary value problem is
-
The algorithm explores a set of pos- sible node locations, collocation point locations, and node, collocation order pairs. Because the algorithm is greedy the time to run the algorithm is expen- sive resulting in a restriction to the number of nodes and collocation points in the set of possible locations. Different problems will inherently result in differ- ent node locations. Future research into an algorithm that can reduce the set to only the most probable locations to offer the best approximation. This type of algorithm would use the gradient as criteria for deciding where to place more or less nodes.
The basis functions used in the approxi- mation are of the form
Figure 17: RMS error for twenty approximation functions.
Thus, the one node approximation selects the node closest to the origin of the source function it attempts to model. When additional nodes are added in an attempt to better the approximation, error decreases, but the change in error between each additional error decreases. Figure 16 depicts the reduction in error that occurs when a higher n approxi- mation function is used to approximate pressure.
Example problem 2 had similar error reduction patterns when more basis functions were introduced in the approx- imation function. Figure 17 depicts the
problem. Because of the computational difficulty of this problem more nodes are necessary to gain a better approxi- mation.
Conclusions
Figure 16: Error reduction depiction with increasing node use.
RMS error for 20 approximation basis functions with models developed for 1 to 20 basis functions for example problem 2.
12 TPG •
Jan.Feb.Mar 2019
Although there are some observa- tions stated in the literature as to computational accuracy improvement by use of different nodal point location strategies, there is not a formalized procedure for identifying the optimum location of modeling nodes that mini- mize computational error goals. Such a formal procedure is presented in this paper in the form a new algorithm that enables such optimum node locations to nodes are treated as additional degrees of freedom in the computational model- ing effort to reduce computational error in achieving problem boundary condi- tions. As expected, the use of the present- ed algorithm improved computational modeling accuracy. The over-arching conclusion can be made that the asso- ciated increase in available degrees of opportunities in reducing computational error. Additionally, the ability to opti- mize node locations enables the reduc- tion in the number of basis functions required to create an approximation function with the same amount of com- putation error as previous approxima-
Acknowledgements
The authors would like to thank the faculty of the United States Military Academy for the invaluable feedback they provided in the development of this paper, as well as their continued dedica- tion to our academic careers after our graduation.
References
1. C.S. Chen, A. Karageorghis, Y. Li, Numerical Algorithms 72(1), 107 (2016)
2. E. Knowles (ed.), The Oxford diction- ary of Phrase and Fable, 2nd edn. (Oxford University Press, Oxford, 2005)
3. R. Haberman, Partial Differential Equations with Fourier Series and - tion edn. (Pearson, New York, 2013)
4. B.D. Wilkins, J. Greenberg, B. Redmond, A. Baily, N. Flowerday, A. Kratch, T.V. Hromadka, R. Boucher, H.D. McInvale, S. Horton, Applied Mathematics 8(6), 878 (2017)
5. N.J. DeMoes, G. Bann, T.V. Hromadka, R. Boucher, International Journal for computational methods and experi- mental measurements (2018, accept- ed)
6. T.V. Hromadka, D. Zillmer, Advances in Engineering Software 43(1), 96 (2012)
7. W. Hui, Q. Qinghua, Acta Mechanica Solida Sinica 20(1), 21 (2007)
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