Skip to main content

Analytical Solution of Laplace and Poisson Equations Using Conformal Mapping and Kronecker Products

Abstract

In this paper, using the eigenvalues and eigenvectors of symmetric block diagonal matrices with infinite dimension and numerical method of finite difference, a closed-form solution for exact solution of Laplace equation is presented. The method of this paper has applications in different states of boundary conditions like Neumann, Dirichlet, and other mixed boundary conditions. Using the method of this paper, a mathematical model for the exact solution of the Poisson equation is derived. Since these equations have many applications in engineering problems, in each part of this paper, examples, like water seepage problem through the soil and torsion of prismatic bars, are presented. Finally, a method is provided for torsion problem of prismatic bars with non-circular and non-rectangular cross-sections utilizing conformal mapping.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

References

  1. 1.

    Francu J, Novackova P, Janicek P (2012) Torsion of a non-circular bar. Eng Mech 19(1):45–60

    Google Scholar 

  2. 2.

    Greenberg MD (1998) Advanced Engineering Mathematics. Prentice-Hall, Upper Saddle River

    MATH  Google Scholar 

  3. 3.

    Kreyszig E (2007) Advanced engineering mathematics. Wiley, New York

    MATH  Google Scholar 

  4. 4.

    Zill D, Wright WS, Cullen MR (2011) Advanced Engineering Mathematics. Jones & Bartlett Learning, Burlington

    MATH  Google Scholar 

  5. 5.

    Ugural AC, Fenster SK (2011) Advanced mechanics of materials and applied elasticity. Pearson Education, Upper Saddle River

    MATH  Google Scholar 

  6. 6.

    West DB (2001) Introduction to graph theory, 2nd edn. Prentice Hall, Upper Saddle River

    Google Scholar 

  7. 7.

    Imrich W, Klavžar S (2000) Product graphs. Structure and recognition. John Wiley, New York

    MATH  Google Scholar 

  8. 8.

    Kaveh A, Rahimi Bondarabady HA, Shahryari L (2006) Buckling load of symmetric planar frames with semi-rigid joints using graph theory. Int J Civil Eng IUST 4:157–175

    Google Scholar 

  9. 9.

    Rahami H, Kaveh A, Ardalan Asl M, Mirghaderi SR (2013) Analysis of near-regular structures with node irregularity using SVD of equilibrium matrix. Int J Civil Eng, IUST 11:227–243

    Google Scholar 

  10. 10.

    Kaveh A, Massoudi MS (2014) Efficient finite element analysis using graph-theoretical force method; tetrahedron elements. Int J Civil Eng IUST 12:347–367

    Google Scholar 

  11. 11.

    Kaveh A, Rahami H (2004) An efficient method for decomposition of regular structures using graph products. Int J Numer Methods Eng 61:1797–1808

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Kaveh A, Rahami H (2011) Block circulant matrices and applications in free vibration analysis of cyclically repetitive structures. Acta Mech 217:51–62

    Article  MATH  Google Scholar 

  13. 13.

    Kaveh A (2013) Optimal analysis of structures by concepts of symmetry and regularity. Springer, Wien-New York

    Book  MATH  Google Scholar 

  14. 14.

    Kaveh A, Rahami H, Mirhosseini SM (2013) Analysis of structures transformable to circulant form using U-transformation and Kronecker products. Acta Mech 224:1625–1642

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Rahami H, Kaveh A, Mirhosseini SM (2013) Efficient solution of differential equations for linear and non-linear analysis of structures. Asian J Civil Eng 14(6):831–847

    MATH  Google Scholar 

  16. 16.

    Amir MJ, Yaseen M, Iqbal R (2013) Exact solutions of Laplace equation by differential transform method. arXiv:1312.7277

  17. 17.

    Yaseen M, Samraiz M, Naheed S (2013) Exact solutions of Laplace equation by DJ method. Results Phys 3:38–40

    Article  Google Scholar 

  18. 18.

    He JH, Wu XH (2007) Variational iteration method: new development and applications. Comput Math Appl 54:881–894

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Peaceman DW, Rachford HHJR (1955) The numerical solution of parabolic and elliptic differential equations. J Soc Ind Appl Math 3:28–41

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Sadighi A, Ganji DD (2007) Exact solutions of Laplace equation by homotopy-perturbation and Adomian decomposition methods. Phys Lett A 367:83–87

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Wazwaz A-M (2007) The variational iteration method for exact solutions of Laplace equation. Phys Lett A 363:260–262

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Ely JF, Zienkiewicz OC (1960) Torsion of compound bars—a relaxation solution. Int J Mech Sci 1:356–365

    Article  Google Scholar 

  23. 23.

    Zongo OOM, Kam S, Kieno PF, Ouedraogo A, Kumar A (2012) Elastic torsion of bars with “pound” and “yen” cross sections using large singular finite element Method. Phys Rev Res Int 2:133–143

    Google Scholar 

  24. 24.

    Yovanovich MM, Muzychka YS (1997) Solutions of Poisson equation within singly and doubly connected prismatic domains. AAA J 97:1–11

    Google Scholar 

  25. 25.

    Ting TWU (1966) Elastic-plastic torsion of a square bar. Trans Am Math Soc 123:369–401

    MATH  Google Scholar 

  26. 26.

    Yueh W-C (2005) Eigenvalues of several tridiagonal matrices. Appl Math E-Notes 5:210–230

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Das BM, Mechanics AS (2013) Advanced soil mechanics. CRC Press, Boca Raton

    Google Scholar 

Download references

Acknowledgments

The second author is grateful to the University of Tehran for financial support under grant number 27938/1/15.

Author information

Affiliations

Authors

Corresponding author

Correspondence to H. Rahami.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mirhosseini, S.M., Rahami, H. & Kaveh, A. Analytical Solution of Laplace and Poisson Equations Using Conformal Mapping and Kronecker Products. Int J Civ Eng 14, 369–377 (2016). https://doi.org/10.1007/s40999-016-0037-y

Download citation

Keywords

  • Analytical solution
  • Laplace equation and Poisson equation
  • Block diagonal matrices
  • Water seepage through soil
  • Torsion of non-circular and non-rectangular cross-sections