A Highly Accurate and Efficient Method for Numerical Simulation of Reinforced Concrete Slab Bending
DOI:
https://doi.org/10.71310/pcam.5_69.2025.01Keywords:
reinforced concrete slab, bending, load, biharmonic equation, Chebyshev polynomials, discrete version of the pre-integration methodAbstract
Many practical problems, such as a solar panel, a microplate inside an electronic device, a thin layer of metal exposed to a laser, or the bending of a reinforced concrete slab, are described by various boundary value problems for the biharmonic equation. Solving biharmonic equations using iterative methods is extremely inconvenient due to the requirement to perform a large number of arithmetic operations, in addition, the number of iterations in which often turns out to be very large. The consideration of biharmonic equations with Dirichlet and Neumann boundary conditions limits the use of difference methods for their numerical solution. Therefore, the development of highly accurate and efficient direct numerical methods for solving such equations is of particular scientific interest. For this purpose, in this article, for the numerical solution of boundary value problems for the biharmonic equation, it is proposed to use a direct numerical method - a discrete version of the preliminary integration method with high accuracy and efficiency.
References
Gurlebeck K. On Some Applications of the Biharmonic Equation // Fundamental Theories of Physics. – 2011. – Vol. 94. – P. 109-128. – doi: http://dx.doi.org/10.1007/978-94-011-5036-1_10.
Zhuang Q., Chen L. Legendre-Galerkin spectral-element method for the biharmonic equations and its applications // Computers and Mathematics with Applications. – 2017. – Vol. 74, No. 12. – P. 1576-1592. – doi: http://dx.doi.org/10.1016/j.camwa.2017.07.039.
Ghasemi M. Spline-based DQM for multi-dimensional PDEs: Application to biharmonic and Poisson equations in 2D and 3D // Computers and Mathematics with Applications. – 2017. – Vol. 73. – P. 1576-1592. – doi: http://dx.doi.org/10.1016/j.camwa.2017.02.006.
Garnadi A.D. Mixed finite element formulation of the biharmonic equation // Journal of Mathematics and Its Applications. – 2018. – Vol. 4, No. 1. – P. 1-12. – doi: http://dx.doi.org/10.29244/jmap.4.1.1-12.
Liu X., Li H., Shi X., Fu J. Application of biharmonic equation in impeller profile optimization design of an aero-centrifugal pump // Engineering Computations. – 2019. – Vol. 36, No. 11. – P. 1764-1795. – doi: http://dx.doi.org/10.1108/EC-08-2018-0378.
Kim S., Palta B., Jeong J., Oh H.-S. Extraction of stress intensity factors of biharmonic equations with corner singularities corresponding to mixed boundary conditions of clamped, simply supported, and free (II) // Computers and Mathematics with Applications. – 2022. – Vol. 109. – P. 235-259. – doi: http://dx.doi.org/10.1016/j.camwa.2022.01.028.
Kov’aˇr’ık K., Muˇz’ık J., Gago F., Sit’anyiov’a D. The local boundary knots method for solution of Stokes and the biharmonic equation // Engineering Analysis with Boundary Elements. – 2023. – Vol. 155. – P. 1149-1159. – doi: http://dx.doi.org/10.1016/j.enganabound.2023.07.031.
Tang J., Li Z., Gao Y., Mao K. Bivariate Chebyshev polynomial-based reconstruction of surface residual stress fields with adaptive sampling // Measurement. – 2025. – Vol. 256. – P. 1-19. – doi: http://dx.doi.org/10.1016/j.measurement.2025.117992.
Gu S., Huo F., Liu S. A stabilizer-free weak Galerkin mixed finite element method for the biharmonic equation // Computers and Mathematics with Applications. – 2024. – Vol. 176. – P. 109-121. – doi: http://dx.doi.org/10.1016/j.camwa.2024.09.011.
Antonietti P.F., Matalon P., Verani M. Iterative solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order method // Computers and Mathematics with Applications. – 2024. – Vol. 171. – P. 154-163. – doi: http://dx.doi.org/10.1016/j.camwa.2024.07.018.
Hwang G., Kar M. Reconstructing unknown inclusions for the biharmonic equation // Journal of Mathematical Analysis and Applications. – 2024. – Vol. 530. – doi: http://dx.doi.org/10.1016/j.jmaa.2023.127745.
Cui X., Huang X. A decoupled nonconforming finite element method for biharmonic equation in three dimensions // Applied Numerical Mathematics. – 2025. – Vol. 212. – P. 300-311. – doi: http://dx.doi.org/10.1016/j.apnum.2025.02.012.
Li X., Wu J., Huang Y., Ding Z., Tai X., Liu L., Wang Y.-G. Fourier-feature induced physics informed randomized neural network method to solve the biharmonic equation // Journal of Computational and Applied Mathematics. – 2025. – Vol. 468. – doi: http://dx.doi.org/10.1016/j.cam.2025.116635.
Boukraa M.A., Caill’e L., Delvare F. Fading regularization method for an inverse boundary value problem associated with the biharmonic equation // Journal of Computational and Applied Mathematics. – 2025. – Vol. 457. – P. 1-19. – doi: http://dx.doi.org/10.1016/j.cam.2024.116285.
Haghighi D., Abbasbandy S., Guan Y., Shivanian E. Application of the fragile points method for two-dimensional generalized biharmonic equation on arbitrary domains // Computational and Applied Mathematics. – 2025. – Vol. 44. – P. 1-20. – doi: http://dx.doi.org/10.1007/s40314-025-03188-w.
Gu S., Huo F., Peng H., Wang X. A new stabilizer-free weak Galerkin mixed finite element method for the biharmonic equation on polygonal meshes // Journal Pre-proof. – 2025. – doi: http://dx.doi.org/10.1016/j.cnsns.2025.109084.
Dressler M., Foucart S., Joldes M., de Klerk E., Lasserre J.B., Xu Y. Optimization-aided construction of multivariate Chebyshev polynomials // Journal of Approximation Theory. – 2025. – Vol. 305. – P. 1-19. – doi: http://dx.doi.org/10.1016/j.jat.2024.106116.
Normurodov Ch.B. Mathematical modeling of hydrodynamic problems for two-phase planeparallel flows // Mathematical modeling. – 2007. – №19(6). – P. 53-60.
Normurodov CH.B. Ob odnom effektivnom metode resheniya uravneniya Orra–Zommerfel'da // Matematicheskoye modelirovaniye. – 2005. – №9(17). – S. 35-42.
Normurodov CH.B., Ziyakulova SH.A. Chislennoye modelirovaniye uravneniy ellipticheskogo tipa diskretnym variantom metoda predvaritel'nogo integrirovaniya // Problemy vychislitel'noy i prikladnoy matematiki. – 2024. – №5(61). – S. 59-68.
Normuradov Ch.B., Djurayeva N.T., Anuar M.S., Deraman F., Asi S.M. One Effective Method for Solving Singularly Perturbed Equations // Malaysian Journal of Science. – 2025. – Vol. 44, No. 1. – P. 62-68. – doi: http://dx.doi.org/10.22452/mjs.vol44no1.8.
Normurodov Ch.B., Tursunova B.A. Numerical modeling of the boundary value problem of an ordinary differential equation with a small parameter at the highest derivative by Chebyshev polynomials of the second kind // Results in Applied Mathematics. – 2023. – P. 1-6. – doi: http://dx.doi.org/10.1016/j.rinam.2023.100388.
Normurodov Ch.B., Ziyakulova Sh.A., Murodov S.K. On one highly accurate and efficient method for solving the biharmonic equation // International Journal of Applied Mathematics. – 2025. – Vol. 38, No. 4. – P. 437-453. – doi: http://dx.doi.org/10.12732/ijam.v38i4.1.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Ch.B. Normurodov
This work is licensed under a Creative Commons Attribution 4.0 International License.
