MATHEMATICAL MODEL AND ANALYSIS OF COMPUTATIONAL EXPERIMENTS OF THERMO-ELECTRO-MAGNETIC-ELASTIC NONLINEAR DEFORMATION PROCESSES OF ANISOTROPIC PLATES USING THE R-FUNCTION METHOD

SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 10 MATHEMATICAL MODEL AND ANALYSIS OF COMPUTATIONAL EXPERIMENTS OF THERMO-ELECTROMAGNETIC-ELASTIC NONLINEAR DEFORMATION PROCESSES OF ANISOTROPIC PLATES USING THE RFUNCTION METHOD F. Nuraliyev1, N. Tojiyev2 Tashkent University of Information Technologies named after Muhammad al-Khwarizmi, Tashkent, Uzbekistan1 The Science Research Institute for Development of Digital Technologies and Artificial Intelligence,Tashkent, Uzbekistan2 https://doi.org/10.5281/zenodo.17833385 Abstract. In this study, a mathematical model is developed based on the Kirchhoff–Lyav hypothesis, which enables the transformation of a three-dimensional coordinate system into a twodimensional one, thereby obtaining the initial expressions for displacements. The variational formulations of kinetic energy, potential energy, and the work done by external forces are derived. Using the heat-balance temperature equation, the Duhamel–Neumann equation, Cauchy’s relations, Hooke’s law, the Lorentz force, and the Maxwell electromagnetic tensor, the nonlinear stress–strain state of anisotropic thermo–electro–magnetoelastic plates with complex geometries under the influence of a thermo–electro–magnetic field is investigated. Keywords: anisotropic, isotropic, nonlinear model, Hamilton–Ostrogradsky principle, Fourier’s law, Duhamel–Neumann equation, Maxwell’s equations. Introduction Developing a mathematical model In this study, the problem of vibration (bending) of an anisotropic plate of complex shape, made of an electrically conductive material with constant h , located in a thermo–electro–magnetic field, is considered. The analysis takes into account the effects of heat conduction and deformation. In this problem, the intensities of the magnetic and electric fields are assumed to be prescribed in advance, and it is considered that external electric currents and charges do not act on the system. The plate of complex geometry is positioned in a Cartesian coordinate system such that its middle surface coincides with the ,xy -plane. The mathematical model for a plate with a complex shape located in a thermoelectromagnetic field is obtained based on the Hamilton-Ostrogradsky variational principle, Fourier's law of heat conduction, the Dugamel-Neumann equation, and the electrodynamic equations in the form of Maxwell's equations. General view of the Hamilton-Ostrogradsky variational principle: 0 t (δK δП+δА)dt = ;  (1) Here: δ variational energy; К kinetic energy; П potential energy; SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 11 А work done by external volume and surface forces. When deriving the equations of motion of a plate with a complex shape, the Kirchhoff-Lyew hypothesis is a simplified model for analyzing plates and layers, which facilitates the analysis of the strength of layers, the movement and deformation of plates under loads. The normal stresses (σ_z) on the upper and lower surfaces of the plate are not taken into account. Here, it is assumed that there is no deformation of the plate with a complex shape along the coordinate axis Oz, and the displacement projections of the plate midplane are written in the following form: 1 2 3 ( , , ) , ( , , ) , ( , , ) ww u u x y t z u v x y t z u w x y t xy        (2) Here: ,,u v w displacements. Based on the Hamilton-Ostrogradsky variational principle, the effects of the thermoelectromagnetic field on the nonlinear stress-strain state of anisotropic thermo-electromagnetic plates with complex shapes are investigated using the Cauchy relations, the heat-balance equation of temperature, Hooke's law, and the Maxwell electromagnetic tensor. As a result, a mathematical model in the form of a system of partial differential equations with initial and boundary conditions related to the displacement is obtained. Determining Variational Kinetic Energy To calculate the change in variational kinetic energy, we use the following relationship: 2 2 2 1 2 3 2 2 2 1. 2 t t V u u u Кdt dVdt t t t                 (3) Now we calculate the variation of kinetic energy. To do this, we apply the variation operation to the general formula for kinetic energy, and we obtain the following expression: As a result, the variation of kinetic energy (4) is obtained: 33 2 3 3 3 2 3 2 2 2 2 2 2 2 2 12 12 12 12 . t x y yx xy ttt t x y u v w h w K h u h v h w w t t t t x h w h w h w w dydx w dy w dx t y t x t y u v w h u h v h w dydxdt t t t                                                                      (4) Determining variational potential energy . (1) Based on the Hamilton-Ostrogradsky variational principle, we secondly calculate the general form of the variational potential energy. Here we perform the following sequence of operations. General form of variational potential energy:   11 11 22 22 12 12 , V П dV            (5) in this: 11 12 22 ,,     deformations, 11 12 22 ,,     voltages. SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 12 Using the tensor representation of nonlinear deformation elements according to the Cauchy relation and the Kirchhoff-Lyaw hypothesis (2), we obtain: From this, we vary the nonlinear deformation elements and obtain expression (6): As a result, the following general representation of the variational potential energy (6) was obtained: Here: 11 22 12 ,,M M M - bending and torsional moments; 11 22 12 ,,N N N - normal and shear forces. Determining the variation of work done by external forces In the third stage of developing a mathematical model based on the Hamilton-Ostragradsky variational principle in formula (1), we derive the variation of the work done by external forces, taking into account the thermo-electromagnetic field forces. For this, we use the following formula:                     1 2 3 1 2 3 1 2 3 1 (7) x y Ty z Tz t t v x zx Tzx y zy Tzy z zz Tzz t y x x xx Txx y xy Txy z xz Txz t y z x yx Tyx Аdt X K T u Y K u Z K u dvdt q T u q T u q T u dxdydt P T u P T u P T u dzdydt F T u F                                                                       23 . y yy Tyy z yz Tyz t x z T u F T u dzdxdt             (7) By replacing the positions in accordance with the variation of the work done by the external forces and performing the appropriate operations, in particular, integration, integration by parts, and also by condensing similar terms, we obtain the following expression: Here, we introduce definitions and substitutions to simplify the expression, as a result of which we obtain expression (8): 12 11 11 11 12 12 22 22 22 12 12 12 2 11 12 22 12 12 1 1 1 2 2 2 1 1 1 2 2 2 11 [ (6 22 y x xy M w w w П N u M N w N v w N w dy x x x y y w w w w N v M w N w N u M N w dx y y y x x N N N N M u u v v w x y y x x y                                                                            11 11 12 22 22 12 ) 1 1 1 1 ] 2 2 2 2 w w w w w w N M N w N M N w dxdy x x x y y y y x                                    SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 13               ] (8) Txx Txy Txz Tyx Ty x x x zx Tzx Tx y y y zy Tzy Ty t t y x z z z zz Tzz Tz Px Txx ty Py Txy Pz Txz Fx Tyx Fy Tyy tx Аdt R q T u R q T v R q T w dxdydt u v w dydt x u                                                                                      y Tyz Fz Tyz v w dxdt y           Temperature determination for anisotropic materials We obtain the Fourier law of heat conduction, which is essentially an anisotropic nonlinear differential equation. , 0 (9) ij ij ij ij T C T T        If we substitute the expression in the above formula here, our equation will look like the following (10). 00 0 2 2 2 2 2 11 12 21 22 22 2 3 3 2 3 3 11 22 2 2 2 2 2 2 3 3 12 11 ( ) ( ) (10) 22 1 ( ) 0. 2 C x y y x t xy u w w v w w zz x t y t x t x t y t y t u v w w z y t x t x y t x y t                                                                                   Equation of motion of thermo-electro-magnetoelastic anisotropic plates with complex shapes We substitute the variations of the work done by external forces, determined based on the Hamilton-Ostrogradsky variational principle, taking into account the changes in kinetic and potential energy and the thermo-electromagnetic field forces. As a result, a system of equations of motion with the following initial and boundary conditions for thermo-electromagnetically anisotropic plates of complex shape is obtained. SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 14 211 12 2 222 12 2 22 12 11 11 12 2 22 22 12 10, 2 10, 2 11 22 11 22 x x x zx Tzx x y y y zy Tzy y zz u N N h R q T xy t v N N h R q T yx t w M w w w h N M N x y x x x y t w w w N M N R y y y x                                                                                0 2 2 2 2 2 11 12 21 22 22 11 12 21 22 11 12 21 22 (11) 0. ( ) 0. z zz Tzz z qT C x y y x t xy t t t t                                                                           Initial conditions: 0 3 2 3 2 0, 0, 0, 0 0, 0; (12) 12 12 T t t t xy tt u v w h u h v h w T t t t t h w h w ww t x t y                             Boundary conditions:       11 12 22 12 12 11 11 12 22 22 12 12 11 0, 0, 0, 0, 22 11 0, 22 11 0, (13) 22 Txx Txy Txz x y y x Px Txx Py Txy Pz Txz N u N v N v N u w w M w M N w w N w x x x y y w w w w M w N w M N w y y y x x u v w                                                                               0 0, 0. ( ) 0 ; Tyx Tyy Tyz T x Fx Tyx Fy Tyy Fz Tyz y TT u v w T n                                    in this ,,u v w plate bends; h plate thickness; 11 22 12 ,,M M M  bending and torsional moments; 11 22 12 ,,N N N  normal and resultant forces, , , , , , x y z x y z R R R     electromagnetic field and bulk forces; , , , , , zx zy zz x y z T T T q q q  surface forces organizers; SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 15 , , , , , , Px Py Pz Fx Fy Fz       , , , , , Txx Txy Txz Tyx Tyy Tyz        organizers of the contour forces; , , , , , Txx Txy Txz Tyx Tyy Tyz              thermal conductivity. Thus, using the Kirchhoff-Lyaw hypothesis, Cauchy relations, Hooke's law, Fourier's law of heat conduction, and Maxwell's electromagnetic tensor expressions, the equation of motion (11), (12) initial and (13) boundary conditions of a plate with a thermo-electro-magneto-elastic complex shape were derived based on the Hamilton-Ostrogradsky variational principle. Finding the bending and torsional moments, as well as the normal and shear forces, of nonlinear anisotropic plates with complex shapes. We determine the bending and torsional moments 11 22 12 ,,M M M and the normal and shear forces 11 22 12 ,,N N N in the geometric nonlinear mathematical model of the thermo-electromagnetoelastic problem of nonlinear anisotropic plates with complex shapes. Considering that the plate we are studying is a nonlinear anisotropic material, Hooke's law and the Dugamel-Neumann equation are expressed as follows: 0 ( ). (14) ij ijkl kl ij C T T       From here 11 1111 11 1122 22 1112 12 11 0 22 2211 11 2222 22 2212 12 22 0 12 1211 11 1222 22 1212 12 12 0 () ( ) (15) () C C C T T C C C T T C C C T T                               Here 11 12 22 ,,     strain tensor components; 11  , 12 22 ,   stress tensor components;   , 1,2 ijkl C i j  constants. (2) Taking into account the Kirchhoff-Lyew hypothesis, expression (15) is defined as follows: 2 2 22 11 1111 1122 22 2 1112 11 0 2 22 22 2211 2222 2 11 22 1 1 1 ( ), 2 2 2 1 2 u w w v w w C z C z x x x y y y u v w w w C z T T y x x y x y u w w v w C z C z x x x y y                                                                               2 2 2 2212 22 0 2 2 22 12 1211 1222 22 2 1212 1 2 1 1 1 ( ), (16) 2 2 2 11 22 1 1 1 2 2 2 w y u v w w w C z T T y x x y x y u w w v w w C z C z x x x y y y u v w Cz y x x y                                                                              12 0 ( ). ww TT xy       SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 16 Based on the above, the moments and forces are defined by relations (17) and (18), respectively, as follows: 2 2 11 1111 1122 1112 11 0 2 2 22 2211 2222 2212 1 1 1 1 1 ( ) , 2 2 2 2 2 1 1 1 2 2 2 u w v w u v w w N h C C C T T x x y y y x x y u w v w N h C C C x x y y                                                                                  22 0 2 2 12 1211 1222 1212 12 0 11 ( ) , (17) 22 1 1 1 1 1 ( ) . 2 2 2 2 2 u v w w TT y x x y u w v w u v w w N h C C C T T x x y y y x x y                                                                         3 2 2 2 11 1111 1122 1112 22 3 2 2 2 22 2211 2222 2212 22 3 2 2 2 12 1211 1222 1212 22 , 12 , (18) 12 . 12 h w w w M C C C x y x y h w w w M C C C x y x y h w w w M C C C x y x y                                           Mathematical model of a thermo-electro-magnetoelastic anisotropic plate with a complex shape We substitute the determined values (17) and (18) of the bending and torsional moments and the normal and shear forces in the above equation of motion (11), (12) initial and (3) boundary conditions and summarize them. As a result, the mathematical model of the state of geometrically nonlinear deformation of thin complex-shaped anisotropic plates located in a thermoelectromagnetic field has the following complete form (19): SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 17 2 2 2 3 3 2 2 1111 1212 1111 1222 1112 1222 2 2 2 3 3 2 2 2 3 2 3 1122 1212 1122 1212 1211 1112 1112 1211 22 2 2 2 2 2 ( ) ( ) ( ) ( ) 2 2 2 2 2 2 ( u u h u h w h w h v h v h hC C C C C C t x y x y x y h v h h w h u h h w hC C C C hC C C C x y x y x y x y hx                                              11 0 12 0 2 2 2 2 2 3 3 2222 1212 2212 1211 2222 1211 2 2 2 2 2 3 3 22 2211 1212 2212 1222 221 ( )) ( ( )) 0, 4 2 2 2 4 ( ) ( ) ( 4 2 2 2 x x x zx Tzx T T h T T N R q T y v v h v h u h u h w h w h hC C C C C C t y x y x y x h u h h v h hC C C C C x y x y                                                   33 1 1212 2212 1222 22 22 0 12 0 2 2 4 2 4 1111 1122 1122 2211 1212 2 2 4 2 2 2 ) ( ) 4 2 4 ( ( )) ( ( )) 0, (19) 2 1 1 1 ( ) ( ) 2 2 2 2 2 2 4 y y y zy Tzy h w h h w C C C x y x y h h T T T T N R q T yx w h u w w h v w h w h h C C C C C xy t x x x x y                                                   2 1112 2 4 2 2 4 2 2 1112 1211 2211 2222 2212 3 2 2 4 2 2 4 2 2 2 2212 1222 1211 1222 1212 3 11 ( ) ( ) ( ) 2 2 2 2 2 4 1 ( ) ( 2 2 2 uw Cyx h w h u w h v w w h u w v w C C C C C x y y x x y y y y y y h w u w v w h u w C C hC hC C x x y y x y y xy                                                      2 2 2 2 4 4 11 0 22 0 12 0 1111 1122 2212 1211 2 2 4 2 2 4 4 4 1112 2211 2222 1212 1222 3 3 4 ) ( ) ( ) ( ) ( ) 2 2 12 12 ( ) ( ) 12 12 12 z z z zz T vw x y x x y h w h w w h w h w T T T T h T T C C C C xy x y x x y h w h w h w C C C C C N R q T x y x y y                                                   0 00 2 2 2 2 2 2 3 3 11 12 21 22 11 2 2 2 2 2 3 3 2 2 3 3 22 12 22 0, 1 () 2 11 ( ) ( ) 0. 22 zz u w w Cz x y y x t x t x y x t x t v w w u v w w zz y t y t x t x y t x y t y t y t                                                                                                                                  The volume forces, surface forces, and contour forces arising from the action of a thermoelectromagnetic field on an anisotropic plate with a complex shape are substituted into the system of equations (19), respectively. Computational algorithms for solving boundary value problems of thin magnetoelastic bodies with complex shapes To solve the problem numerically, a computational algorithm was developed using the analytical R-function method (RFM), the Newmark method, variational and several numerical methods to solve the magnetoelastic plates of complex structural shape. The structure of the solutions constructed using the R-function method can be expressed in the following form: 1,uф   2,vф   23 wф   , T Tф   (20) here   a normalized function expressing a complex configuration field. 1 2 3 , , , T ф ф ф ф  vague components of the solution structure. SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 18     12 34 12 11 311 ( ) , , ( ) ( , ), ( ) , , ( ) ( , ) nn i i j j ij nn k k T l l kl ф C t x y ф C t x y ф C t x y ф C t x y         (21) , , , i j k l C C C C - unknown coefficients that require determination, , , , i j k l     - complete systems of basic polynomials (for example, power, Chebyshev, trigonometric, splines, and others). He is still being searched u,v, w vа T(t,x,y) Functions can be written in the following form:     12 34 11 11 ( ) , , , ( ) ( , , ), ( ) , , , ( ) ( , , ) nn i i j j ij nn k k l l kl u C t u x y v C t v x y w C t w x y T C t T x y         (22) By substituting the structure of solutions (16) into equations (17), discretization is performed with respect to the x and y spatial variables, and discrete equations in vector-matrix form (discrete model) are constructed to calculate the unknown coefficients of the solution. The problem is considered in a dynamical case and the form of discrete equations is expressed as follows. MC NC J (23) Initial conditions: 00 •• 00 ;; t t t t C C C C   (24) Here 11 11 12 13 1 1 22 21 22 23 2 2 33 31 32 33 3 3 41 42 43 0 0 0 0 0 0 ; ; ; 0 0 0 0 0 0 nm nm nm nm nm n n M N N N N C f M N N N N C f M N C J M N N N N C f M N N N N C f                                                     n=1,2,3…. m=1,2,3…. 12 3 4 4 1 1 2 2 3 3 4 1, ; 1, ; 1, ; 1, , , ; , , ij kl M Lu d i M L v d M nj L w d k L Td n n lMn                  