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Süper Elipsler için bir Çevre Formülü Önerisi ve Fizikte Dikdörtgensel Sınır Değer Problemlerinde Kullanımı

Year 2022, Volume: 12 Issue: 2, 166 - 176, 24.12.2022

Kadir Can Erbaş

Abstract

Gabriel Lame tarafından tanıtılan süper elipsler veya Lame eğrileri, son yıllarda fizik ve mühendislikte sıkça çalışılan konular olmuştur. Süper elips alanlarının hesaplanması analitik olarak mümkün olsa da, çevre hesaplamaları için formüllerin eksikliği dikkat çekicidir. Bu eksikliği gidermek için bu çalışma, süper elipslerin çevrelerini sayısal olarak hesaplayan bir kod yazmayı ve sayısal sonuçlarla uyumlu yaklaşık bir çevre formülasyonu bulmayı amaçlamıştır. Ek olarak, elde edilen çevre formülasyonu, bir dikdörtgen sınır koşulunda, Laplace denkleminin süper elipslerle yaklaşık bir boyuta indirgenebileceğini ve zor bir fiziksel probleme pratik bir yaklaşık çözüm bulunabileceğini göstermektedir.

Keywords

Süper elips çevre Laplace denklemi dikdörtgensel sınırlar

References

  • Abbott, P. 2011. On the perimeter of an ellipse. Mathematica Journal, 11(2), 172. http://dx.doi.org/doi:10.3888/tmj.11.2-4
  • Aldaher, M. A. 2012. New Simpler Equations for Properties of Lame Curve (Hypoellipse, Ellipse, Superellipse and Asteroid Curves), ICMS 2012 : International Conference on Marketing Studies, Bangkok
  • Allen, J., Kundtz, N., Roberts, D. A., Cummer, S. A., Smith, D. R. 2009. Electromagnetic source transformations using superellipse equations. Appl. Phys. Lett., 94(19), 194101. https://doi.org/10.1063/1.3130182
  • Anishchenko, O. S., Kukhar, V. V., Grushko, A. V., Vishtak, I. V., Prysiazhnyi, A. H., Arifuzzaman, S. M., Dong, K., Hou, Q., Zhu, H., Zeng, Q. 2020. Explicit contact force model for superellipses by Fourier transform and application to superellipse packing. Powder Technol., 361, 112-123. https://doi.org/10.1016/j.powtec.2019.10.018
  • Balalayeva, E. Y. 2019. Analysis of the sheet shell's curvature with lame's superellipse method during superplastic forming. Mater. Sci. Forum. (Vol. 945, pp. 531-537). Trans Tech Publications Ltd. http://dx.doi.org/10.4028/www.scientific.net/MSF.945.531
  • Bar, M., Neta, M. 2006. Humans prefer curved visual objects. Psyc. Sci., 17(8), 645-648. https://doi.org/10.1111%2Fj.1467-9280.2006.01759.x
  • Bar, M., Neta, M. 2007. Visual elements of subjective preference modulate amygdala activation. Neuropsychologia, 45(10), 2191-2200. https://doi.org/10.1016/j.neuropsychologia.2007.03.008
  • Bremer, J., Rokhlin, V., Sammis, I. 2010. Universal quadratures for boundary integral equations on two-dimensional domains with corners. J. Comput. Phys., 229(22), 8259-8280. https://doi.org/10.1016/j.jcp.2010.06.040
  • Delaney, G. W., Cleary, P. W. 2010. The packing properties of superellipsoids. EPL (Europhysics Letters), 89(3), 34002. http://dx.doi.org/10.1209/0295-5075/89/34002
  • Dhia, A. S. B. B., Hazard, C., Monteghetti, F. 2021. Complex-scaling method for the complex plasmonic resonances of planar subwavelength particles with corners. J. Comput. Phys., 440, 110433. https://dx.doi.org/10.1016/j.jcp.2021.110433
  • Erbaş K.C. 2020. Calculation of the characteristic impedance of a square coaxial line by fitting the equipotential curves to the super circles, 1st international Ankara multidisciplinary studies congress, Ankara, Turkey, August 2020, pp. 115-125
  • Erbaş K.C. 2019. Reducing the Laplace Equation to a 1D Problem in a Square Shaped Boundary, ICCMAS2019 International Conference on Computational Methods in Applied Sciences, İstanbul, Turkey, July 2019, p. 284.
  • Gardner, M. 2020. Mathematical carnival. Am. Math. Soc.., 1989 Ed. Newyork, 297 s.
  • Gridgeman, N. T. 1970. Lamé ovals. The Mathematical Gazette, 54(387), 31-37.
  • Li, S., Boyd, J. P. 2015. Approximation on non-tensor domains including squircles, Part III: Polynomial hyperinterpolation and radial basis function interpolation on Chebyshev-like grids and truncated uniform grids. J. Comput.Phy, 281, 653-668. https://doi.org/10.1016/j.jcp.2014.10.035
  • Hallonborg, U. 1996. Super ellipse as tyre-ground contact area. J. Terramechanics, 33(3), 125-132. https://doi.org/10.1016/S0022-4898(96)00013-4
  • Kleev, A. I., Kyurkchan, A. G. 2015. Application of the pattern equation method in spheroidal coordinates to solving diffraction problems with highly prolate scatterers. Acoust. Phys., 61(1), 19-27. https://doi.org/10.1134/S1063771014060104
  • Krähenbühl, L., Buret, F., Perrussel, R., Voyer, D., Dular, P., Péron, V., Poignard, C. 2011. Numerical treatment of rounded and sharp corners in the modeling of 2D electrostatic fields. J. Microw. Optoelectron. Electromagn. Appl., 10(1), 66-81. http://dx.doi.org/10.1590/S2179-10742011000100008
  • Kyurkchan, A. G., Manenkov, S. A., Smirnova, N. I. 2019. Solution of Problems of Wave Scattering by Bodies Having Boundary Breaks and Fractal-Like Bodies of Rotation. Opt. and Spectrosc., 126(5), 466-472. http://dx.doi.org/10.1134/S0030400X19050175
  • Lamé Curve Calculator 2022. https://www.had2know.com/academics/lame-curve-area-perimeter-superellipse-calculator.html
  • Lin, Y. C., Chen𝐠, C. Y., Cheng, Y. W., Shih, C. T. 2016. Using Differential Evolution in Skull Prosthesis Modelling by Superellipse. 17th APIEMS conference, Taipei.
  • Mac Huang, J., Shelley, M. J., Stein, D. B. 2021. A stable and accurate scheme for solving the Stefan problem coupled with natural convection using the Immersed Boundary Smooth Extension method. J. Comput. Phys., 432, 110162. https://doi.org/10.1016/j.jcp.2021.110162
  • Manenkov, S. A. 2014. A new version of the modified method of discrete sources in application to the problem of diffraction by a body of revolution. Acoust. Phys., 60(2), 127-133. http://dx.doi.org/10.1134/S1063771014010102
  • Manenkov, S. A. 2019. Solution of the Problem of Diffraction by a Body of Revolution Located in a Dielectric Layer. Opt. Spectrosc., 127(6), 1032-1043. https://doi.org/10.1134/S0030400X19120142
  • Méndez, I., Casar, B. 2021. A novel approach for the definition of small-field sizes using the concept of superellipse. Radiat. Phys. Chem., 189, 109775. https://doi.org/10.1016/j.radphyschem.2021.109775
  • Nagornov, K. O., Kozhinov, A. N., Tsybin, Y. O. 2021. Spatially-distributed cyclotron oscillators approach to FT-ICR MS at the true cyclotron frequency: Computational evaluation of sensitivity. Int. J. Mass Spectrom., 466, 116604. http://dx.doi.org/10.1016/j.ijms.2021.116604
  • Natural Superellipse 2022, Super-ellipse Calculator and Plotter. http://www.procato.com/superellipse/
  • Osian, M., Tuytelaars, T., Van Gool, L. 2004, June. Fitting superellipses to incomplete contours. In 2004 Conference on Computer Vision and Pattern Recognition Workshop (pp. 49-49). IEEE. http://dx.doi.org/10.1109/CVPR.2004.73
  • Panda, S., Hazra, G. 2014. Boundary perturbations and the Helmholtz equation in three dimensions. Eur. Phys. J. Plus., 129(4), 1-20. http://dx.doi.org/10.1140/epjp/i2014-14053-y
  • Rosin, P. L. 2000. Fitting superellipses. IEEE Trans. Pattern Anal. Mach. Intell. , 22(7), 726-732. http://dx.doi.org/10.1109/34.865190
  • Sert, Z. 2021. Flow and Mixed Convection with Heat Transfer around a Square Cylinder. Karaelmas Fen ve Mühendislik Dergisi , 11 (2) , 145-153 . Retrieved from https://dergipark.org.tr/en/pub/karaelmasfen/issue/66240/889634
  • Silvia, P. J., Barona, C. M. 2009. Do people prefer curved objects? Angularity, expertise, and aesthetic preference. Empir. Stud. Arts, 27(1), 25-42. https://doi.org/10.2190%2FEM.27.1.b
  • Villarino, M. B. 2005. Ramanujan's Perimeter of an Ellipse. arXiv preprint math/0506384.
  • Weisstein, Eric W. 2021. "Superellipse." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Superellipse.html
  • Weisstein, Eric W. 2021. "Rectellipse." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Rectellipse.html
  • Westerman, S. J., Gardner, P. H., Sutherland, E. J., White, T., Jordan, K., Watts, D., Wells, S. 2012. Product design: Preference for rounded versus angular design elements. Psyc. and Market., 29(8), 595-605. https://doi.org/10.1002/mar.20546
  • Zhang, X., Rosin, P. L. 2003. Superellipse fitting to partial data. Pattern Recognit., 36(3), 743-752. http://dx.doi.org/10.1016/S0031-3203(02)00088-2

Suggestion of a Perimeter Formula for Super Ellipses and their Use in Rectangular Boundary Value Problems in Physics

Year 2022, Volume: 12 Issue: 2, 166 - 176, 24.12.2022

Kadir Can Erbaş

Abstract

Super ellipses or Lame curves introduced by Gabriel Lame have been frequently studied subjects in physics and engineering in recent years. While the calculation of the area of super-ellipses is analytically possible, the lack of formulations for circumference calculations is noteworthy. To overcome this deficiency, this study aimed to write a code that computes the circumferences of super ellipses numerically and to find an approximate circumference formulation compatible with the numerical results. Additionally, the perimeter formulation obtained shows that in a rectangular boundary condition, Laplace's equation can be reduced to approximately one dimension with super ellipses and a practical approximate solution to a difficult physical problem can be found.

Keywords

Super ellipse perimeter circumference Laplace equation rectangular boundary

References

  • Abbott, P. 2011. On the perimeter of an ellipse. Mathematica Journal, 11(2), 172. http://dx.doi.org/doi:10.3888/tmj.11.2-4
  • Aldaher, M. A. 2012. New Simpler Equations for Properties of Lame Curve (Hypoellipse, Ellipse, Superellipse and Asteroid Curves), ICMS 2012 : International Conference on Marketing Studies, Bangkok
  • Allen, J., Kundtz, N., Roberts, D. A., Cummer, S. A., Smith, D. R. 2009. Electromagnetic source transformations using superellipse equations. Appl. Phys. Lett., 94(19), 194101. https://doi.org/10.1063/1.3130182
  • Anishchenko, O. S., Kukhar, V. V., Grushko, A. V., Vishtak, I. V., Prysiazhnyi, A. H., Arifuzzaman, S. M., Dong, K., Hou, Q., Zhu, H., Zeng, Q. 2020. Explicit contact force model for superellipses by Fourier transform and application to superellipse packing. Powder Technol., 361, 112-123. https://doi.org/10.1016/j.powtec.2019.10.018
  • Balalayeva, E. Y. 2019. Analysis of the sheet shell's curvature with lame's superellipse method during superplastic forming. Mater. Sci. Forum. (Vol. 945, pp. 531-537). Trans Tech Publications Ltd. http://dx.doi.org/10.4028/www.scientific.net/MSF.945.531
  • Bar, M., Neta, M. 2006. Humans prefer curved visual objects. Psyc. Sci., 17(8), 645-648. https://doi.org/10.1111%2Fj.1467-9280.2006.01759.x
  • Bar, M., Neta, M. 2007. Visual elements of subjective preference modulate amygdala activation. Neuropsychologia, 45(10), 2191-2200. https://doi.org/10.1016/j.neuropsychologia.2007.03.008
  • Bremer, J., Rokhlin, V., Sammis, I. 2010. Universal quadratures for boundary integral equations on two-dimensional domains with corners. J. Comput. Phys., 229(22), 8259-8280. https://doi.org/10.1016/j.jcp.2010.06.040
  • Delaney, G. W., Cleary, P. W. 2010. The packing properties of superellipsoids. EPL (Europhysics Letters), 89(3), 34002. http://dx.doi.org/10.1209/0295-5075/89/34002
  • Dhia, A. S. B. B., Hazard, C., Monteghetti, F. 2021. Complex-scaling method for the complex plasmonic resonances of planar subwavelength particles with corners. J. Comput. Phys., 440, 110433. https://dx.doi.org/10.1016/j.jcp.2021.110433
  • Erbaş K.C. 2020. Calculation of the characteristic impedance of a square coaxial line by fitting the equipotential curves to the super circles, 1st international Ankara multidisciplinary studies congress, Ankara, Turkey, August 2020, pp. 115-125
  • Erbaş K.C. 2019. Reducing the Laplace Equation to a 1D Problem in a Square Shaped Boundary, ICCMAS2019 International Conference on Computational Methods in Applied Sciences, İstanbul, Turkey, July 2019, p. 284.
  • Gardner, M. 2020. Mathematical carnival. Am. Math. Soc.., 1989 Ed. Newyork, 297 s.
  • Gridgeman, N. T. 1970. Lamé ovals. The Mathematical Gazette, 54(387), 31-37.
  • Li, S., Boyd, J. P. 2015. Approximation on non-tensor domains including squircles, Part III: Polynomial hyperinterpolation and radial basis function interpolation on Chebyshev-like grids and truncated uniform grids. J. Comput.Phy, 281, 653-668. https://doi.org/10.1016/j.jcp.2014.10.035
  • Hallonborg, U. 1996. Super ellipse as tyre-ground contact area. J. Terramechanics, 33(3), 125-132. https://doi.org/10.1016/S0022-4898(96)00013-4
  • Kleev, A. I., Kyurkchan, A. G. 2015. Application of the pattern equation method in spheroidal coordinates to solving diffraction problems with highly prolate scatterers. Acoust. Phys., 61(1), 19-27. https://doi.org/10.1134/S1063771014060104
  • Krähenbühl, L., Buret, F., Perrussel, R., Voyer, D., Dular, P., Péron, V., Poignard, C. 2011. Numerical treatment of rounded and sharp corners in the modeling of 2D electrostatic fields. J. Microw. Optoelectron. Electromagn. Appl., 10(1), 66-81. http://dx.doi.org/10.1590/S2179-10742011000100008
  • Kyurkchan, A. G., Manenkov, S. A., Smirnova, N. I. 2019. Solution of Problems of Wave Scattering by Bodies Having Boundary Breaks and Fractal-Like Bodies of Rotation. Opt. and Spectrosc., 126(5), 466-472. http://dx.doi.org/10.1134/S0030400X19050175
  • Lamé Curve Calculator 2022. https://www.had2know.com/academics/lame-curve-area-perimeter-superellipse-calculator.html
  • Lin, Y. C., Chen𝐠, C. Y., Cheng, Y. W., Shih, C. T. 2016. Using Differential Evolution in Skull Prosthesis Modelling by Superellipse. 17th APIEMS conference, Taipei.
  • Mac Huang, J., Shelley, M. J., Stein, D. B. 2021. A stable and accurate scheme for solving the Stefan problem coupled with natural convection using the Immersed Boundary Smooth Extension method. J. Comput. Phys., 432, 110162. https://doi.org/10.1016/j.jcp.2021.110162
  • Manenkov, S. A. 2014. A new version of the modified method of discrete sources in application to the problem of diffraction by a body of revolution. Acoust. Phys., 60(2), 127-133. http://dx.doi.org/10.1134/S1063771014010102
  • Manenkov, S. A. 2019. Solution of the Problem of Diffraction by a Body of Revolution Located in a Dielectric Layer. Opt. Spectrosc., 127(6), 1032-1043. https://doi.org/10.1134/S0030400X19120142
  • Méndez, I., Casar, B. 2021. A novel approach for the definition of small-field sizes using the concept of superellipse. Radiat. Phys. Chem., 189, 109775. https://doi.org/10.1016/j.radphyschem.2021.109775
  • Nagornov, K. O., Kozhinov, A. N., Tsybin, Y. O. 2021. Spatially-distributed cyclotron oscillators approach to FT-ICR MS at the true cyclotron frequency: Computational evaluation of sensitivity. Int. J. Mass Spectrom., 466, 116604. http://dx.doi.org/10.1016/j.ijms.2021.116604
  • Natural Superellipse 2022, Super-ellipse Calculator and Plotter. http://www.procato.com/superellipse/
  • Osian, M., Tuytelaars, T., Van Gool, L. 2004, June. Fitting superellipses to incomplete contours. In 2004 Conference on Computer Vision and Pattern Recognition Workshop (pp. 49-49). IEEE. http://dx.doi.org/10.1109/CVPR.2004.73
  • Panda, S., Hazra, G. 2014. Boundary perturbations and the Helmholtz equation in three dimensions. Eur. Phys. J. Plus., 129(4), 1-20. http://dx.doi.org/10.1140/epjp/i2014-14053-y
  • Rosin, P. L. 2000. Fitting superellipses. IEEE Trans. Pattern Anal. Mach. Intell. , 22(7), 726-732. http://dx.doi.org/10.1109/34.865190
  • Sert, Z. 2021. Flow and Mixed Convection with Heat Transfer around a Square Cylinder. Karaelmas Fen ve Mühendislik Dergisi , 11 (2) , 145-153 . Retrieved from https://dergipark.org.tr/en/pub/karaelmasfen/issue/66240/889634
  • Silvia, P. J., Barona, C. M. 2009. Do people prefer curved objects? Angularity, expertise, and aesthetic preference. Empir. Stud. Arts, 27(1), 25-42. https://doi.org/10.2190%2FEM.27.1.b
  • Villarino, M. B. 2005. Ramanujan's Perimeter of an Ellipse. arXiv preprint math/0506384.
  • Weisstein, Eric W. 2021. "Superellipse." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Superellipse.html
  • Weisstein, Eric W. 2021. "Rectellipse." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Rectellipse.html
  • Westerman, S. J., Gardner, P. H., Sutherland, E. J., White, T., Jordan, K., Watts, D., Wells, S. 2012. Product design: Preference for rounded versus angular design elements. Psyc. and Market., 29(8), 595-605. https://doi.org/10.1002/mar.20546
  • Zhang, X., Rosin, P. L. 2003. Superellipse fitting to partial data. Pattern Recognit., 36(3), 743-752. http://dx.doi.org/10.1016/S0031-3203(02)00088-2
There are 37 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Research Articles
Authors

Kadir Can Erbaş 0000-0002-6446-829X

Publication Date December 24, 2022
Published in Issue Year 2022 Volume: 12 Issue: 2

Cite

APA Erbaş, K. C. (2022). Suggestion of a Perimeter Formula for Super Ellipses and their Use in Rectangular Boundary Value Problems in Physics. Karaelmas Fen Ve Mühendislik Dergisi, 12(2), 166-176. https://doi.org/10.7212/karaelmasfen.1052608
AMA Erbaş KC. Suggestion of a Perimeter Formula for Super Ellipses and their Use in Rectangular Boundary Value Problems in Physics. Karaelmas Fen ve Mühendislik Dergisi. December 2022;12(2):166-176. doi:10.7212/karaelmasfen.1052608
Chicago Erbaş, Kadir Can. “Suggestion of a Perimeter Formula for Super Ellipses and Their Use in Rectangular Boundary Value Problems in Physics”. Karaelmas Fen Ve Mühendislik Dergisi 12, no. 2 (December 2022): 166-76. https://doi.org/10.7212/karaelmasfen.1052608.
EndNote Erbaş KC (December 1, 2022) Suggestion of a Perimeter Formula for Super Ellipses and their Use in Rectangular Boundary Value Problems in Physics. Karaelmas Fen ve Mühendislik Dergisi 12 2 166–176.
IEEE K. C. Erbaş, “Suggestion of a Perimeter Formula for Super Ellipses and their Use in Rectangular Boundary Value Problems in Physics”, Karaelmas Fen ve Mühendislik Dergisi, vol. 12, no. 2, pp. 166–176, 2022, doi: 10.7212/karaelmasfen.1052608.
ISNAD Erbaş, Kadir Can. “Suggestion of a Perimeter Formula for Super Ellipses and Their Use in Rectangular Boundary Value Problems in Physics”. Karaelmas Fen ve Mühendislik Dergisi 12/2 (December 2022), 166-176. https://doi.org/10.7212/karaelmasfen.1052608.
JAMA Erbaş KC. Suggestion of a Perimeter Formula for Super Ellipses and their Use in Rectangular Boundary Value Problems in Physics. Karaelmas Fen ve Mühendislik Dergisi. 2022;12:166–176.
MLA Erbaş, Kadir Can. “Suggestion of a Perimeter Formula for Super Ellipses and Their Use in Rectangular Boundary Value Problems in Physics”. Karaelmas Fen Ve Mühendislik Dergisi, vol. 12, no. 2, 2022, pp. 166-7, doi:10.7212/karaelmasfen.1052608.
Vancouver Erbaş KC. Suggestion of a Perimeter Formula for Super Ellipses and their Use in Rectangular Boundary Value Problems in Physics. Karaelmas Fen ve Mühendislik Dergisi. 2022;12(2):166-7.
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