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A Simultaneous Numerical Integration Routine for the Fast Calculation of Similar Integrations

Year 2023, Volume: 7 Issue: 2, 133 - 141, 20.12.2023

Aytaç Alparslan

https://doi.org/10.38088/jise.1246719

Abstract

In this paper, a fast and simultaneous integration routine tailored for obtaining results of multiple numerical integrations is introduced. In the routine, the same nodes are used when integrating different functions along the same integration path. In the paper it is demonstrated by several examples that if the integrands of interest are similar on the integration path, then using the same nodes decreases the computational costs dramatically. While the method is introduced by updating the popular Gauss-Kronrod quadrature rule, the same steps given in the paper can be applied to any other numerical integration rule.

Keywords

Electromagnetic Line Sources Gauss-Kronrod Quadrature Layered Media Green’s Functions Numerical Integration Simultaneous Integration

References

  • Palacio-Betancur, A. and Gutierrez Soto, M. (2023). Recent Advances in Computational Methodologies for Real-Time Hybrid Simulation of Engineering Structures. Archives of Computational Methods in Engineering, 30(3), 1637–1662.
  • Jiang, M., Li, Y., Lei, L. and Hu, J. (2022). A Review on Fast Direct Methods of Surface Integral Equations for Analysis of Electromagnetic Scattering from 3-D PEC Objects. Electronics, 11(22), 3753–3764.
  • Botha, M. M. (2015). Numerical Integration Scheme for the Near-Singular Green Function Gradient on General Triangles. IEEE Transactions on Antennas and Propagation, 63(10), 4435–4445.
  • Davis, P. J., and Rabinowitz, P. (2007). Methods of Numerical Integration, Dover Publications Inc., New York, USA, 2nd Edition, 612 p. ISBN: 0-486-45339-1.
  • Kreyszig, E. (2006). Advanced Engineering Mathematics, John Wiley and Sons, New Jersey, USA, 9th Edition, 1248 p. ISBN: 0-471-72897-7.
  • Aimi, A., Diligenti, M., & Monegato, G. (1997). New Numerical Integration Schemes for Applications of Galerkin Bem to 2-D Problems. International Journal for Numerical Methods in Engineering, 40(11), 1977–1999.
  • Felsen, N., & Marcuvitz, L. B. (1994). Radiation and Scattering of Waves, John Wiley and Sons, New Jersey, USA, 1st Edition, 924 p. ISBN: 0-7803-1088-8.
  • Chew, W. C. (1990). Waves and Fields in Inhomogeneous Media, Van Nostrand Reinhold, New York, USA, 1st Edition, 611 p. ISBN-10: 0442238169.
  • Cheng, D. K. (1989). Field and Wave Electromagnetics, Addison Wesley, Massachusetts, USA, 2nd Edition, 703 p. ISBN-10: 0201528207.
  • Jin, J.-M. (2015). The Finite Element Method in Electromagnetics, John Wiley and Sons, New Jersey, USA, 3rd Edition, 876 p. ISBN: 1-118-57136-1.
  • Gibson, W. C. (2022). The Method of Moments in Electromagnetics, CRC Press, Florida, USA, 3rd Edition, 510 p. ISBN: 9780367365066.
  • Harrington, R. F. (1982). Field Computation by Moment Methods, Robert E. Krieger Publishing Company, Florida, USA, 1st Edition, 229 p. ISBN: 0-89874-465-1.
  • Hafner, C. (1999). Post-Modern Electromagnetics: Using Intelligent Maxwell Solvers, John Wiley and Sons, New Jersey, USA, 1st Edition, 320 p. ISBN: 0-471-98711-6.
  • Kahaner, D., Moler, C., & Nash, S. (1989). Numerical Methods and Software, Prentice-Hall, Inc., New Jersey, USA, 1st Edition, 495 p. ISBN: 0-13-626672-3.
  • Kronrod, A. S. (1965). Nodes and Weights of Quadrature Formulas: Sixteen-Place Tables, Consultants Bureau New York, USA, 1st Edition, 143 p. ISBN: 9780306651113.
  • Laurie, D. (1997). Calculation of Gauss-Kronrod Quadrature Rules. Mathematics of Computation, 66(219), 1133–1145.
  • MathWorks (2023). MATLAB & Simulink https://www.mathworks.com/products/matlab.html [Accessed: 1 February 2023]
  • Shampine, L. F. (2008). Vectorized adaptive quadrature in MATLAB. Journal of Computational and Applied Mathematics, 211(2), 131–140.
  • GNU Octave. (2023). GNU Octave: Scientific Programming Language https://octave.org/index [Accessed: 1 February 2023]
  • Schlömer, N. (2023). quadpy: Numerical Integration, Quadrature for Various Domains (0.16.22) https://github.com/sigma-py/quadpy [Accessed: 1 February 2023]
  • Gonnet, P. (2012). A Review of Error Estimation in Adaptive Quadrature. ACM Computing Surveys, 44(4), 22:1-22:36.
  • Boas, M. (1983). Mathematical Methods in the Physical Sciences, John Wiley and Sons, New Jersey, USA, 2nd Edition, 743 p. ISBN: 0-471-19826-0.
  • Alparslan, A. (2023). Constituents of Electromagnetic 2-D Layered Media Green’s Functions for All Material Types and Radiation Conditions. Waves in Random and Complex Media, 1–26.
  • Michalski, K. A. and Mosig, J. R. (2015). The Sommerfeld Half-Space Problem Redux: Alternative Field Representations, Role of Zenneck and Surface Plasmon Waves. IEEE Transactions on Antennas and Propagation, 63(12), 5777–5790.
  • Alparslan, A. (2013). Numerical analysis of photonic nano structures in layered geometries, PhD Thesis. ETH Zurich, ZH, Switzerland. 126p.
  • Aksun, M. I. (1996). A Robust Approach for the Derivation of Closed-Form Green’s Functions. IEEE Transactions on Microwave Theory and Techniques, 44(5), 651–658.

Year 2023, Volume: 7 Issue: 2, 133 - 141, 20.12.2023

Aytaç Alparslan

https://doi.org/10.38088/jise.1246719

Abstract

References

  • Palacio-Betancur, A. and Gutierrez Soto, M. (2023). Recent Advances in Computational Methodologies for Real-Time Hybrid Simulation of Engineering Structures. Archives of Computational Methods in Engineering, 30(3), 1637–1662.
  • Jiang, M., Li, Y., Lei, L. and Hu, J. (2022). A Review on Fast Direct Methods of Surface Integral Equations for Analysis of Electromagnetic Scattering from 3-D PEC Objects. Electronics, 11(22), 3753–3764.
  • Botha, M. M. (2015). Numerical Integration Scheme for the Near-Singular Green Function Gradient on General Triangles. IEEE Transactions on Antennas and Propagation, 63(10), 4435–4445.
  • Davis, P. J., and Rabinowitz, P. (2007). Methods of Numerical Integration, Dover Publications Inc., New York, USA, 2nd Edition, 612 p. ISBN: 0-486-45339-1.
  • Kreyszig, E. (2006). Advanced Engineering Mathematics, John Wiley and Sons, New Jersey, USA, 9th Edition, 1248 p. ISBN: 0-471-72897-7.
  • Aimi, A., Diligenti, M., & Monegato, G. (1997). New Numerical Integration Schemes for Applications of Galerkin Bem to 2-D Problems. International Journal for Numerical Methods in Engineering, 40(11), 1977–1999.
  • Felsen, N., & Marcuvitz, L. B. (1994). Radiation and Scattering of Waves, John Wiley and Sons, New Jersey, USA, 1st Edition, 924 p. ISBN: 0-7803-1088-8.
  • Chew, W. C. (1990). Waves and Fields in Inhomogeneous Media, Van Nostrand Reinhold, New York, USA, 1st Edition, 611 p. ISBN-10: 0442238169.
  • Cheng, D. K. (1989). Field and Wave Electromagnetics, Addison Wesley, Massachusetts, USA, 2nd Edition, 703 p. ISBN-10: 0201528207.
  • Jin, J.-M. (2015). The Finite Element Method in Electromagnetics, John Wiley and Sons, New Jersey, USA, 3rd Edition, 876 p. ISBN: 1-118-57136-1.
  • Gibson, W. C. (2022). The Method of Moments in Electromagnetics, CRC Press, Florida, USA, 3rd Edition, 510 p. ISBN: 9780367365066.
  • Harrington, R. F. (1982). Field Computation by Moment Methods, Robert E. Krieger Publishing Company, Florida, USA, 1st Edition, 229 p. ISBN: 0-89874-465-1.
  • Hafner, C. (1999). Post-Modern Electromagnetics: Using Intelligent Maxwell Solvers, John Wiley and Sons, New Jersey, USA, 1st Edition, 320 p. ISBN: 0-471-98711-6.
  • Kahaner, D., Moler, C., & Nash, S. (1989). Numerical Methods and Software, Prentice-Hall, Inc., New Jersey, USA, 1st Edition, 495 p. ISBN: 0-13-626672-3.
  • Kronrod, A. S. (1965). Nodes and Weights of Quadrature Formulas: Sixteen-Place Tables, Consultants Bureau New York, USA, 1st Edition, 143 p. ISBN: 9780306651113.
  • Laurie, D. (1997). Calculation of Gauss-Kronrod Quadrature Rules. Mathematics of Computation, 66(219), 1133–1145.
  • MathWorks (2023). MATLAB & Simulink https://www.mathworks.com/products/matlab.html [Accessed: 1 February 2023]
  • Shampine, L. F. (2008). Vectorized adaptive quadrature in MATLAB. Journal of Computational and Applied Mathematics, 211(2), 131–140.
  • GNU Octave. (2023). GNU Octave: Scientific Programming Language https://octave.org/index [Accessed: 1 February 2023]
  • Schlömer, N. (2023). quadpy: Numerical Integration, Quadrature for Various Domains (0.16.22) https://github.com/sigma-py/quadpy [Accessed: 1 February 2023]
  • Gonnet, P. (2012). A Review of Error Estimation in Adaptive Quadrature. ACM Computing Surveys, 44(4), 22:1-22:36.
  • Boas, M. (1983). Mathematical Methods in the Physical Sciences, John Wiley and Sons, New Jersey, USA, 2nd Edition, 743 p. ISBN: 0-471-19826-0.
  • Alparslan, A. (2023). Constituents of Electromagnetic 2-D Layered Media Green’s Functions for All Material Types and Radiation Conditions. Waves in Random and Complex Media, 1–26.
  • Michalski, K. A. and Mosig, J. R. (2015). The Sommerfeld Half-Space Problem Redux: Alternative Field Representations, Role of Zenneck and Surface Plasmon Waves. IEEE Transactions on Antennas and Propagation, 63(12), 5777–5790.
  • Alparslan, A. (2013). Numerical analysis of photonic nano structures in layered geometries, PhD Thesis. ETH Zurich, ZH, Switzerland. 126p.
  • Aksun, M. I. (1996). A Robust Approach for the Derivation of Closed-Form Green’s Functions. IEEE Transactions on Microwave Theory and Techniques, 44(5), 651–658.
There are 26 citations in total.

Details

Primary Language English
Subjects Electrical Engineering
Journal Section Research Articles
Authors

Aytaç Alparslan 0000-0002-0232-1479

Early Pub Date October 15, 2023
Publication Date December 20, 2023
Published in Issue Year 2023Volume: 7 Issue: 2

Cite

APA Alparslan, A. (2023). A Simultaneous Numerical Integration Routine for the Fast Calculation of Similar Integrations. Journal of Innovative Science and Engineering, 7(2), 133-141. https://doi.org/10.38088/jise.1246719
AMA Alparslan A. A Simultaneous Numerical Integration Routine for the Fast Calculation of Similar Integrations. JISE. December 2023;7(2):133-141. doi:10.38088/jise.1246719
Chicago Alparslan, Aytaç. “A Simultaneous Numerical Integration Routine for the Fast Calculation of Similar Integrations”. Journal of Innovative Science and Engineering 7, no. 2 (December 2023): 133-41. https://doi.org/10.38088/jise.1246719.
EndNote Alparslan A (December 1, 2023) A Simultaneous Numerical Integration Routine for the Fast Calculation of Similar Integrations. Journal of Innovative Science and Engineering 7 2 133–141.
IEEE A. Alparslan, “A Simultaneous Numerical Integration Routine for the Fast Calculation of Similar Integrations”, JISE, vol. 7, no. 2, pp. 133–141, 2023, doi: 10.38088/jise.1246719.
ISNAD Alparslan, Aytaç. “A Simultaneous Numerical Integration Routine for the Fast Calculation of Similar Integrations”. Journal of Innovative Science and Engineering 7/2 (December 2023), 133-141. https://doi.org/10.38088/jise.1246719.
JAMA Alparslan A. A Simultaneous Numerical Integration Routine for the Fast Calculation of Similar Integrations. JISE. 2023;7:133–141.
MLA Alparslan, Aytaç. “A Simultaneous Numerical Integration Routine for the Fast Calculation of Similar Integrations”. Journal of Innovative Science and Engineering, vol. 7, no. 2, 2023, pp. 133-41, doi:10.38088/jise.1246719.
Vancouver Alparslan A. A Simultaneous Numerical Integration Routine for the Fast Calculation of Similar Integrations. JISE. 2023;7(2):133-41.
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