Skip to main content
SpringerLink
Log in

Hermite Interpolation Polynomials on Parallelepipeds and FEM Applications

  • Published:
Mathematics in Computer Science Aims and scope Submit manuscript

Abstract

We implement in Maple and Mathematica an algorithm for constructing multivariate Hermitian interpolation polynomials (HIPs) inside a d-dimensional hypercube as a product of d pieces of one-dimensional HIPs of degree \(p'\) in each variable, that are calculated analytically using the authors’ recurrence relations. The piecewise polynomial functions constructed from the HIPs have continuous derivatives and are used in implementations of the high-accuracy finite element method. The efficiency of our finite element schemes, algorithms and GCMFEM program implemented in Maple and Mathematica are demonstrated by solving reference boundary value problems (BVPs) for multidimensional harmonic and anharmonic oscillators used in the Geometric Collective Model (GCM) of atomic nuclei. The BVP for the GCM is reduced to the BVP for a system of ordinary differential equations, which is solved by the KANTBP 5 M program implemented in Maple.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Berezin, I.S., Zhidkov, N.P.: Computing Methods. Pergamon Press, Oxford (1965)

    MATH  Google Scholar 

  2. Lorentz, R.A.: Multivariate Birkhoff Interpolation. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  3. Lekien, F., Marsden, J.: Tricubic interpolation in three dimensions. Int. J. Num. Meth. Eng. 63, 455–471 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chuluunbaatar, G., Gusev, A.A., Chuluunbaatar, O., Gerdt, V.P., Vinitsky, S.I., Derbov, V.L., Góźdź, A., Krassovitskiy, P.M., Hai, L.L.: Construction of multivariate interpolation Hermite polynomials for finite element method. EPJ Web Conf. 226, 02007 (2020)

  5. Gusev, A.A., Chuluunbaatar, O., Vinitsky, S.I., Derbov, V.L., Góźdź, A., Hai, L.L., Rostovtsev, V.A.: Symbolic-numerical solution of boundary-value problems with self-adjoint second-order differential equation using the finite element method with interpolation Hermite polynomials. LNSC 8660, 138–154 (2014)

  6. Gusev, A.A., Chuluunbaatar, G., Chuluunbaatar, O., Gerdt, V.P., Vinitsky, S.I., Hai, L.L., Lua, T.T., Derbov, V.L., Góźdź, A.: Algorithm for calculating interpolation Hermite polynomials in \(d\)-dimensional hypercube in the analytical form. In “Computer algebra” Conference Materials, Moscow, June 17–21, 2019 / ed. S.A. Abramov, L.A. Sevastianov. - Peoples’ Friendship University of Russia, 119–128 http://www.ccas.ru/ca/_media/ca-2019.pdf

  7. Troltenier, D., Maruhn, J.A., Hess, P.O.: Numerical application of the geometric collective model. In: Langanke, K., Maruhn, J.A., Konin, S.E. (eds.) Computational Nuclear Physics, vol. 1, pp. 105–128. Springer-Verlag, Berlin (1991)

    Chapter  Google Scholar 

  8. Chuluunbaatar, G., Gusev, A., Derbov, V., Vinitsky, S., Chuluunbaatar, O., Hai, L.L., Gerdt, V.: A Maple implementation of the finite element method for solving boundary-value problems for systems of second-order ordinary differential equations. Commun. Comput. Inform. Sci. 1414, 152–166 (2021)

    Google Scholar 

  9. Gusev, A., Vinitsky, S., Chuluunbaatar, O., Chuluunbaatar, G., Gerdt, V., Derbov, V., Gozdz, A., Krassovitskiy, P.: Interpolation Hermit polinomials for finite element method. EPJ Web Conf. 173, 03009 (2018)

  10. Bathe, K.J.: Finite Element Procedures in Engineering Analysis. Eng. Cliffs, NY (1982)

    Google Scholar 

  11. Walker, P.: Quadcubic interpolation: a four-dimensional spline method, preprint (2019), available at http://arxiv.org/abs/1904.09869v1; Walker, P., Krohn, U. and Carty, D.: ARBTools: A tricubic spline interpolator for three-dimensional scalar or vector fields. Journal of Open Research Software, 7(1), p12. (2019)

  12. Schwarz H. R.: Methode der finiten Elemente. 2-nd edn. B.G. Teubner, Stuttgart (1984)

  13. Schwarz, H.R.: FORTRAN-Programme zur methode der finiten Elemente. Springer, Fachmedien Wiesbaden (1991)

    Book  MATH  Google Scholar 

  14. Troltenier, D., Maruhn, J.A., Greiner, W., Hess, P.O.: A general numerical solution of collective quadrupole surface motion applied to microscopically calculated potential energy surfaces. Z. Phys. A. Hadrons Nuclei 343, 25–34 (1992)

    Article  Google Scholar 

  15. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)

    MATH  Google Scholar 

  16. Deveikis, A., Gusev, A.A., Vinitsky, S.I., Blinkov, Y.A., Góźdź, A., Pȩdrak, A., Hess, P.O.: Symbolic-numeric algorithm for calculations in geometric collective model of atomic nuclei. Comput. Sci. 13366, 103–123 (2022)

    MathSciNet  MATH  Google Scholar 

  17. Deveikis, A., Gusev A.A., Vinitsky S.I., Góźdź, A., Pȩdrak, A., Burdik, Č., Pogosyan, G.S.: Symbolic-numeric algorithm for computing orthonormal basis of \(O(5)\times SU(1,1)\) group. CASC 2020. LNCS 12291, 206–227 (2020)

  18. Moshinsky, M.: The harmonic oscillator in modern physics and Smirnov. HAP, Y.F. (1996)

    MATH  Google Scholar 

  19. Yannouleas, C., Pacheco, J.M.: An algebraic program for the states associated with the \( U(5) \supset O(5)\supset O(3)\) chain of groups. Comput. Phys. Commun. 52, 85–92 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yannouleas, C., Pacheco, J.M.: Algebraic manipulation of the states associated with the \( U(5) \supset O(5)\supset O(3)\) chain of groups: orthonormalization and matrix elements. Comput. Phys. Commun. 54, 315–328 (1989)

    Article  MATH  Google Scholar 

  21. Varshalovitch, D.A., Moskalev, A.N., and Hersonsky, V.K.: Quantum theory of angular momentum Leningrad Nauka. (1975); Singapore: World Scientific (1988)

  22. Bohr, A. and Mottelson, B.R.: Nuclear Structure. N Y, Amsterdam: W A Bejamin Inc, Vol 2, (1970)

  23. Eisenberg, J.M., Greiner W.: Nuclear theory. Vol. 1: Nuclear models. Collective and single-particle phenomena. Amsterdam, London, North-Holland Publ. Co. (1970); Moscow, Atomizdat (1975)

  24. Dobrowolski, A., Mazurek, K., Góźdź, A.: Consistent quadrupole-octupole collective model. Phys. Rev. C 94, 054322 (2016)

    Article  Google Scholar 

  25. Dobrowolski, A., Mazurek, K., Góźdź, A.: Rotational bands in the quadrupole-octupole collective model. Phys. Rev. C 97, 024321 (2018)

    Article  Google Scholar 

  26. Ermamatov, M.J., Hess, Peter O.: Microscopically derived potential energy surfaces from mostly structural considerations. Ann. Phys. 37, 125–158 (2016)

    Article  Google Scholar 

  27. Rohoziński, S.G., Dobaczewski, J., Nerlo-Pomorska, B., Pomorski, K., Srebrny, J.: Microscopic dynamic calculations of collective states in xenon and barium isotopes. Nucl. Phys. A 292, 66–87 (1977)

    Article  Google Scholar 

  28. Mardyban, E.V., Kolganova, E.A., Shneidman, T.M., Jolos, R.V.: Evolution of the phenomenologically determined collective potential along the chain of Zr isotopes. Phys. Rev. C 105, 024321 (2022)

    Article  Google Scholar 

  29. Hess, P.O., Ermamatov, M.: In search of a broader microscopic underpinning of the potential energy surface in heavy deformed nuclei. J. Phys.: Conf. Ser. 876, 012012 (2017)

    Google Scholar 

Download references

Acknowledgements

The authors thank Prof. Andrzej Góźdź for the long-term collaboration, Profs. R.V. Jolos and T.M. Shneidman for fruitful discussions. This publication has been supported by the Russian Foundation for Basic Research and the Ministry of Education, Culture, Science and Sports of Mongolia (grant No. 20-51-44001) and the Peoples’ Friendship University of Russia (RUDN) Strategic Academic Leadership Program, project No. 021934-0-000. POH acknowledges financial support from DGAPA-UNAM (IN100421). This research is funded by Ho Chi Minh City University of Education Foundation for Science and Technology (grant No. CS.2021.19.47). OCH acknowledges financial support from the Ministry of Education and Science of Mongolia (grant No. ShuG 2021/137).

Author information

Authors and Affiliations

  1. Joint Institute for Nuclear Research, Dubna, Russia

    Alexander A. Gusev, Galmandakh Chuluunbaatar, Ochbadrakh Chuluunbaatar & Sergue I. Vinitsky

  2. Dubna State University, Dubna, Russia

    Alexander A. Gusev

  3. Peoples’ Friendship University of Russia (RUDN University), Moscow, Russia

    Galmandakh Chuluunbaatar, Sergue I. Vinitsky & Yuri A. Blinkov

  4. Chernyshevsky Saratov National Research State University, Saratov, Russia

    Yuri A. Blinkov

  5. Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences, Ulaanbaatar, Mongolia

    Ochbadrakh Chuluunbaatar

  6. Department of Applied Informatics, Vytautas Magnus University, Kaunas, Lithuania

    Algirdas Deveikis

  7. Instituto de Ciencias Nucleares, UNAM, Circuito Exterior, C.U., A.P. 70-543, 04510, Mexico, D.F., Mexico

    Peter O. Hess

  8. Frankfurt Institute for Advanced Studies, 60438, Frankfurt am Main, Germany

    Peter O. Hess

  9. Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam

    Luong Le Hai

Authors
  1. Alexander A. Gusev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Galmandakh Chuluunbaatar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ochbadrakh Chuluunbaatar
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sergue I. Vinitsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Yuri A. Blinkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Algirdas Deveikis
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Peter O. Hess
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Luong Le Hai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergue I. Vinitsky.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file 1 (zip 8 KB)

Appendix A. The GCMFEM Program

Appendix A. The GCMFEM Program

The GCMFEM program is intended to solve a self-adjoint BVP for the system of elliptic differential equations (4.10) with Neumann BC decribing collective nuclear model.

  • On INPUT

    • L is the angular momentum

    • b2 is the mass \(\bar{B}_2\) in (4.10)

    • c2,c3,c4,c5,c6,d6 are coefficients \(C_2\), \(C_3\), \(C_4\), \(C_5\), \(C_6\), \(D_6\) of potentials (4.12)

    • zmesh is the mesh in the form of nested list [[],[]], where values of nodes are given in angstroms;

    • EmaxMeV is the maximum energy of printed eigenvalues (in MeV)

    • filename is the part of names of working files (see OUTPUT)

  • On OUTPUT

    • EIGV; is the set of eigenvalues below EmaxMeV (in MeV)

    • EIGF; is the set of corresponding eigenfunctions of the algebraic eigenvalue problem

    • The set of global Gaussian nodes and weights is written to file filename.dat

    • The set of the eigenvalues is written to file filenameL*.dat, where asterisk means the value of L

    • The set of the eigenfunctions in the global Gaussian nodes is written to file filenameL*K*n*.dat, where asterisks means the value of L, K and n

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gusev, A.A., Chuluunbaatar, G., Chuluunbaatar, O. et al. Hermite Interpolation Polynomials on Parallelepipeds and FEM Applications. Math.Comput.Sci. 17, 18 (2023). https://doi.org/10.1007/s11786-023-00568-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11786-023-00568-5

Keywords

Search

Navigation