Abstract
Because the complete set of Trefftz functions for the 3D biharmonic equation is not yet well established, a multiple-direction Trefftz method (MDTM) and an in-plane biharmonic functions method (IPBFM) are deduced in the paper. Inspired by the Trefftz method for the 2D biharmonic equation, a novel MDTM incorporates planar directors into the 2D like Trefftz functions to solve the 3D biharmonic equation. These functions being a series of biharmonic polynomials of different degree, automatically satisfying the 3D biharmonic equation, are taken as the bases to expand the solution. Then, we derive a quite large class solution of the 3D biharmonic equation in terms of 3D harmonic functions, and 2D biharmonic functions in three sub-planes. The 2D biharmonic functions are formulated as the Trefftz functions in terms of the polar coordinates for each sub-plane. Introducing a projective variable, we can obtain the projective type general solution for the 3D Laplace equation, which is used to generate the 3D Trefftz type harmonic functions. Several numerical examples confirm the efficiency and accuracy of the proposed MDTM and IPBFM.
Similar content being viewed by others
Data Availability
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
References
Lesnic D, Elliott L, Ingham DB (1998) The boundary element solution of the Laplace and biharmonic equations subjected to noisy boundary data. Int J Numer Meth Eng 43:479–492
Jin B (2004) A meshless method for the Laplace and biharmonic equations subjected to noisy boundary data. Comput Model Eng Sci 6:253–261
Karageorghis K, Fairweather G (1987) The method of fundamental solutions for the numerical solution of the biharmonic equation. J Comput Phys 69:434–459
Bialecki B (2003) A fast solver for the orthogonal spline collocation solution of the biharmonic Dirichlet problem on rectangles. J Comput Phys 191:601–621
Smyrlis YS, Karageorghis A (2003) Some aspects of the method of fundamental solutions for certain biharmonic problems. Comput Model Eng Sci 4:535–550
Tsangaris T, Smyrlis YS, Karageorghis A (2004) A matrix decomposition MFS algorithm for biharmonic problems in annular domains. Comput Mater Contin 1:245–258
Reutskiy SY (2005) The method of fundamental solutions for eigenproblems with Laplace and biharmonic operators. Comput Mater Contin 2:177–188
Melnikov YA, Melnikov MY (2001) Modified potentials as a tool for computing Green’s functions in continuum mechanics. Comput Model Eng Sci 2:291–306
Christodoulou E, Elliotis M, Georgiou G, Xenophontos C (2012) Analysis of the singular function boundary integral method for a biharmonic problem with one boundary singularity. Numer Meth Partial Diff Eqs 28:749–767
Diaz M, Herrera I (2005) TH-collocation for the biharmonic equation. Adv Eng Softw 36:243–251
Liu CS (2008) A highly accurate MCTM for direct and inverse problems of biharmonic equation in arbitrary plane domains. Comput Model Eng Sci 30:65–75
Dong Z, Ern A (2022) Hybrid high-order and weak Galerkin methods for the biharmonic problem. SIAM J Numer Anal 60:2626–2656
Dong Z, Mascotto L (2023) hp-Optimal interior penalty discontinuous Galerkin methods for the biharmonic problem. J Sci Comput 96:30
Altas I, Erhel J, Gupta MM (2002) High accuracy solution of three-dimensional biharmonic equations. Numer Algor 29:1–19
Gumerov NA, Duraiswami R (2006) Fast multipole method for the biharmonic equation in three dimensions. J Comput Phys 215:363–383
Ghasemi M (2017) Spline-based DQM for multi-dimensional PDEs: application to biharmonic and Poisson equations in 2D and 3D. Comput Math Appl 73:1576–1592
Shi Z, Cao Y, Chen Q (2012) Solving 2D and 3D Poisson equations and biharmonic equations by the Haar wavelet method. Appl Math Model 36:5143–5162
Atangana A, Kiliçman A (2013) Analytical solutions of boundary values problem of 2D and 3D Poisson and biharmonic equations by homotopy decomposition method. Abs Appl Anal 2013:380484
Jiang S, Ren B, Tsuji P, Ying L (2011) Second kind integral equations for the first kind Dirichlet problem of the biharmonic equation in three dimensions. J Comput Phys 230:7488–7501
Liu CS (2013) A multiple-scale Trefftz method for an incomplete Cauchy problem of biharmonic equation. Eng Anal Bound Elem 37:1445–1456
Liu CS (2016) A multiple/scale/direction polynomial Trefftz method for solving the BHCP in high-dimensional arbitrary simply-connected domains. Int J Heat Mass Transfer 92:970–978
Liu CS, Kuo CL (2016) A multiple-direction Trefftz method for solving the multi-dimensional wave equation in an arbitrary spatial domain. J Comput Phys 321:39–54
Liu CS (2016) A simple Trefftz method for solving the Cauchy problems of three-dimensional Helmholtz equation. Eng Anal Bound Elem 63:105–113
Liu CS, Qu W, Chen W, Lin J (2017) A novel Trefftz method of the inverse Cauchy problem for 3D modified Helmholtz equation. Inv Prob Sci Eng 25:1278–1298
Liu CS, Qu W, Chen W, Lin J (2018) Fast solving the Cauchy problems of Poisson equation in an arbitrary three-dimensional domain. Comput Model Eng Sci 114:351–380
Lin J, Liu CS, Chen W, Sun L (2018) A novel Trefftz method for solving the multi-dimensional direct and Cauchy problems of Laplace equation in an arbitrary domain. J Comput Sci 25:16–27
Mu L, Wang J, Wang Y, Ye X (2013) A weak Galerkin mixed finite element method for biharmonic equations. In: Iliev O, Margenov S, Minev P, Vassilevski P, Zikatanov L (eds) Numerical solution of partial differential equations: theory, algorithms, and their applications Springer Proceedings in Mathematics & Statistics, vol 45. Springer, New York
Zhang R, Zhai Q (2015) A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order. J Sci Comput 64:559–585
Gudi T, Nataraj N, Pani AK (2008) Mixed discontinuous Galerkin finite element method for the biharmonic equation. J Sci Comput 37:139–161
Li F, Shu CW (2006) A local-structure-preserving local discontinuous Galerkin method for the Laplace equation. Meth Appl Anal 13:215–234
Hiptmair R, Moiola A, Perugia I, Schwab C (2014) Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM. ESAIM: M2AN 48:727-752
Chernov A, Mascotto L (2019) The harmonic virtual element method: stabilization and exponential convergence for the Laplace problem on polygonal domains. IMA J Numer Anal 39:1787–1817
Mascotto L, Perugia I, Pichler A (2018) Non-conforming harmonic virtual element method: h- and p-versions. J Sci Comput 77:1874–1908
Mascotto L, Perugia I, Pichler A (2019) A nonconforming Trefftz virtual element method for the Helmholtz problem: Numerical aspects. Comput Meth Appl Mech Eng 347:445–476
Imbert-Gèrard LM (2021) Amplitude-based generalized plane waves: new quasi-Trefftz functions for scalar equations in two dimensions. SIAM J Numer Anal 59:1663–1686
Imbert-Gèrard LM, Desprès B (2014) A generalized plane-wave numerical method for smooth nonconstant coefficients. IMA J Numer Anal 34:1072–1103
Hiptmair R, Moiola A, Perugia I (2011) Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM J Numer Anal 49:264–284
Hiptmair R, Moiola A, Perugia I (2016) A survey of Trefftz methods for the Helmholtz equation. In Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lecture Notes in Computational Science and Engineering, 114, 237 - 279, Springer Science and Business Media LLC
Fung YC, Tong P (2001) Classical and computational solid mechanics. World Scientific, Singapore
Galerkin BG (1930) Computes rendus hebdomadaires des s\(\acute{\text{ e }}\)ances de l’acad\(\acute{\text{ e }}\)mie des sciences. Paris 190:1930
Little RW (1973) Elasticity. Prentice-Hall, New Jersey, p 1973
Bilotta A, Turco E (2009) A numerical study on the solution of the Cauchy problem in elasticity. Int J Solids Struct 46:4451–4477
Durand B, Delvare F, Bailly P (2011) Numerical solution of Cauchy problems in linear elasticity in axisymmetric situations. Int J Solids Struct 48:3041–3053
Marin L, Johansson BT (2011) A relaxation method of an alternating iterative algorithm for the Cauchy problem in linear isotropic elasticity. Comput Meth Appl Mech Eng 199:3179–3196
Marin L, Johansson BT (2011) Relaxation procedures for an iterative MFS algorithm for the stable reconstruction of elastic fields from Cauchy data in two-dimensional isotropic linear elasticity. Int J Solids Struct 47:3462–3479
Liu CS (2016) A fast multiple-scale polynomial solution for the inverse Cauchy problem of elasticity in an arbitrary plane domain. Comput Math Appl 72:1205–1224
Turco E (2017) Tools for the numerical solution of inverse problems in structural mechanics: review and research perspectives. Eur J Environ Civil Eng 21:509–554
Liu CS (2012) An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation. Eng Anal Bound Elem 36:1235–1245
Liu CS, Atluri SN (2013) Numerical solution of the Laplacian Cauchy problem by using a better postconditioning collocation Trefftz method. Eng Anal Bound Elem 37:74–83
Antunes PRS (2018) A numerical algorithm to reduce ill-conditioning in meshless methods for the Helmholtz equation. Numer Algor 79:879–897
Ku CY, Kuo CL, Fan CM, Liu CS, Guan PC (2015) Numerical solution of three-dimensional Laplacian problems using the multiple scale Trefftz method. Eng Anal Bound Elem 50:157–168
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. The authors declare no confict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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.
About this article
Cite this article
Liu, CS., Kuo, CL. The Trefftz methods for 3D biharmonic equation using directors and in-plane biharmonic functions. Engineering with Computers (2024). https://doi.org/10.1007/s00366-024-01977-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00366-024-01977-1