МНОГОСЕТОЧНЫЕ КОНЕЧНЫЕ ЭЛЕМЕНТЫ В РАСЧЕТАХ МНОГОСЛОЙНЫХ ЦИЛИНДРИЧЕСКИХ ОБОЛОЧЕК

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Introduction. Finite Element Method (FEM) [1; 2] is widely used in the study of stress strained state (SSS) of elastic shells [3-6]. In the calculation of shells, construct- ing the curvilinear finite elements (FE) causes various difficulties [3], in particular, related to the fulfillment of conformality conditions, which is necessary for the convergence of finite element solutions [7]. These dif- ficulties are largely due to the fact that to reduce the order of equations in the theory of shells, hypotheses are intro- duced, that impose certain restrictions on the fields of displacement, strain and stress [8-14], which generates irreducible errors in solutions and limits the applications of these theories. For example, in the work [15; 16] three- dimensional finite elements are considered with a given distribution of displacements through the thickness, given the compression of the shell. In the work [17] the review of the basic options of use of FEM for calculation of composite plates and covers in two-dimensional statement is presented. The attempts to calculate composite cylin- drical shells with application of FE in the formulation of the three-dimensional problem of elasticity theory with account of their structure leads to systems of linear alge- braic equations (SLAE) of the finite element method of high order (more 106). Application for such discrete shell models of calculation of ANSYS, NASTRAN etc. [3] is difficult. In addition, the solution obtained for the systems of high-order FEM equations contains a computational error, which is difficult to determine the exact value. In this regard, there is a need to develop such variants of FEM, in which the composite cylindrical shell is con- sidered in a three-dimensional formulation, but which lead to SLAE of a low order in compliance with the per- missible level of SSS error values. In the works [18-20] calculations of composite cylindrical panels and shells with the help of multigrid finite element (MFE) are car- ried out, that was constructed using power polynomials. In this paper, we propose an efficient numerical method of calculating linearly elastic multilayer cylindrical shells using a multilayer curvilinear Lagrangian MFE. Constructing n net finite element (FE), n ³ 2 , n of enclosed grid is used. Small grids are made by basic splitting of MFE, the other n -1 (larger) grids are used to reduce its dimension. The aim of this work is to develop Lagrangian curved multilayer shell-type MFE. A procedure for constructing Lagrange polynomials of different orders in local curvilinear coordinates is proposed. In constructing approximate solutions a multi-layer Lagrangian, MFE shell with a con- stant thickness, a multiple of the thickness of the shell is used. The order of the Lagrange polynomial in thickness is taken by a multiple to the number of shell layers. Cal- culations show that the arrangement of nodes of large MFE grids on the common boundaries of different- modular shell layers provides homogenous and fast con- vergence of sequences of finite-element solutions, which allows to construct approximate solutions with low error. The proposed MFE are effective in calculating the SSS of multilayer cylindrical shells of different thicknesses, especially in the calculation of thin shells having a com- plex shape, the complex nature of the fixations and loads. Multilayer shells are widely used in rocket-space and aviation technology. The advantages are as follows. Multilayer Lagrangian shell MFE: - take into account the heterogeneous structure of the shells; - describe the three-dimensional stress state in multi- layer shells; - form multigrid discrete shell models, the dimension of which is much smaller than the dimensions of the base models; - generate the numerical solution with fast conver- gence to accurate, which allows us to construct solutions with a small error. Calculations show that application of the FEM for multigrid discrete models requires 103-107 time less computer memory than the base models need. The im- plementation of the proposed method on single-processor computers requires a small amount of time. To analyze the convergence of approximate solutions constructed for the initial problem, we use the well-known numerical method [2]. The implementation of this method is per- formed by constructing a sequence of approximate solu- tions for a similar test problem using MFE, which are used in solving the original problem. An example of cal- culating a 4-layer shell of complex shape using 4-layer Lagrangian shell three-grid FE is given. The results of the calculations show the high efficiency of the application of the proposed three-grid FE. 1. Homogeneous curvilinear single-grid FE. The procedure for constructing curvilinear homogeneous sin- gle-grid FE, which form a basic discrete model of the shell, is briefly considered as the example of FE Ve of the the characteristic sizes of curved homogeneous FE Ve decrease, the numerical solutions converge to the exact ones. Procedures for the construction of homogeneous curvilinear single-grid FE of 2nd and 3rd order, which are 1st order, located in the local Cartesian coordinate system geometrically similar to the form of FE Ve (fig. 1), are O1x1 y1z1 (fig. 1). For FE Ve designations are given: x y z he ´ he ´ he - characteristic sizes, z1O1 y1 - a symmetry h h z - y 1 2 plane, cd - an axis of a shell, Re ( Re ) - radius of curva- ture of the bottom (top) surface, e - thickness, e analogous to the procedure in § 1. 2. Multilayer curvilinear Lagrangian two-grid FE The procedure of constructing multilayer curvilinear two-grid FE (TGFE) with the use of Lagrange polynomi- als is considered with the example of a three-layer TGFE V of the 3rd order with its thickness equal to h that is a length, he = a Re , a - an opening angle. The shape of x e 1 e y the FE Ve is a straight prism with height he . Deformation of FE Ve is described by the equations of the three- dimensional problem of the theory of elasticity [1], shown used in the calculation of 3-layer shells with the thickness h. In the calculation of m-layer shell m-layer Lagrangian TGFE of m-order in thickness are used. TGFE is located in a local Cartesian coordinate system O2 x2 y2 z2 (fig. 2), in coordinate system O1x1 y1 z1. Using a first order polyits dimensions are ha ´ ha ´ h , h - thickness, ha - length. x y y nomial (in the coordinate system O1 x1 y1 z1 ), for FE Ve Suppose that the bonds between the components of the e we define the stiffness matrix éëK1 ùû and the nodal force vector P1 with formulas [1; 2] inhomogeneous structure of TGFE are ideal. Basic parti- tioning of Ra TGFE, which consists of a homogeneous e curvilinear FE Ve of the 1st order (fig. 1), takes into e [K1] = ò [Be ]T [De ][Be ]dV , Ve (1) account in TGFE inhomogeneous structure, a complex type of loading and fastening, and generates a small curvilinear grid h , e = 1, ..., M , M is the total number of FE V . e P1 = ò [Ne ]T FedV + Ve ò [Ne ]T qedS , Se a On the grid e ha we define the large curvilinear grid where [B ] , [D ] are the matrix of differentiation and Ha Ì ha , TGFE, the nodes of this grid are marked with e e dots, 64 nodes in fig. 2. Note that the nodes of the large modules of elasticity of the FE Ve ; Fe , qe are the volume grid Ha lie on the common boundaries of differentand surface forces vectors FE Ve ; [Ne ] is the matrix of shape functions; Ve , Se are the area and the surface of the FE Ve . modular layers TGFE (fig. 2), in general they have differ- ent thickness. Suppose the axis O1 y1 (fig. 1) is parallel to the axis O2 y2 (fig. 2). Thus we can use a formula of relation between the nodal displacement vectors 1 FE Ve , δ δ , e e which correspond to the local Cartesian coordinate sys- tems O1x1 y1z1 and O2 x2 y2 z2 e e e δ1 = [T ]δ , (2) Fig. 1. Single-grid FE Ve Рис. 1. Односеточный КЭ Ve Note that the continuity of displacements is violated where [Te ] is a square matrix of rotations [2], e = 1, ..., M . on the curvilinear boundaries of the FE Ve (fig. 1). However, as it’s known [21], the implementation of continu- ous displacements at the boundaries of curvilinear FE is not a necessary condition for convergence of numerical solutions to the exact one. Calculations show that when Fig. 2. Three-Layer TGFE Vа Рис. 2. Трехслойный ДвКЭ Vа We consider the construction of Lagrange polynomi- als in the local curvilinear coordinate system O2xhV on Using small partitions Ra , the functional (7) has a high dimension and generates a multinodal FE with a a large grid Ha (fig. 2). Suppose that the node P(i, j, k) large number of nodal unknowns, which is not effective of grid H (dimensions n ´ n ´ n ) has coordinates x , for practice. To reduce the dimension of the functional (7), a 1 2 3 i we use the following procedure. Using (6), the vector h j , zk , in fig. 2 i = j = 3 , k = 4 . Note that y2 = h for of nodal displacements δe FE Ve is shown through the e a small opening angles aа , TGFE we can see that x2 »x , vector of nodal displacements δa of large grid Ha TGFE Va z2 » z . We have δe = [ Aa ]δ , (8) x2 = x, y2 = h, z2 = z. (3) where [ Aa ] is a rectangular matrix e = 1,..., M . The base function Nijk for a node P(i, j, k) in the Substituting (8) in (7) and following the principle of e Cartesian coordinate system O2 x2 y2 z2 using Lagrange the minimum of total potential energy for TGFE Va , polynomials Li (x2 ), Lj ( y2 ), Lk (z2 ) [2] is written in the ¶Пa (δa ) / ¶δa = 0 we obtain a ratio [Ka ]δa = Fa corre- M form of sponding to the equilibrium state of TGFE Va , where Nijk (x2 , y2 , z2 ) = Li (x2 )Lj ( y2 )Lk (z2 ), [K ] = å a T a , = å a T (9) n1 x2 - x2,n n2 y2 - y2,n a M e=1 [ Ae ] [Ke ][ Ae ] Fa e=1 [ Ae ] Pe . Li (x2 ) = Õ n=1,n¹i x2,i , - x2,n Lj ( y2 ) = Õ n=1,n¹ j y2, j , - y2,n (4) The matrix [Ka ] is called the stiffness matrix, Fa is n3 z2 - z2,n nodal forces vector of TGFE Va . Note that the functions z - z Lk (z2 ) = Õ , n=1,n¹k 2,k 2,n ua , va , wa are used only to reduce the dimension of the where x2,i , y2, j , z2,k are the coordinates of the node functional (7), the large grid Ha determines the dimension of the TGFE V , which is less than the dimension of P(i, j, k) in the coordinate system O x y z . a 2 2 2 2 For a point with a coordinate x lying on the cylindrithe base partition Ra . cal surface of the radius R , we have x= aR , a is the Note 1. By virtue of (8) the dimension of the vector δa angle for the coordinate x , fig. 3. Considering (3) the ratio of the form x= aR , xi = ai R in (4), we obtain (i. e. the dimension of the TGFE Va ) does not depend on the M which is the total number of FE Ve constituting Nijk (a, h, z) = Li (a)Lj (h)Lk (z) , where Li (a) , Lj (h) , L (z) are the Lagrange polynomials, having the form the TGFE Va . Consequently, it is possible to use arbitrarily small base partitions Ra , which allows to take into k n1 a- an n2 h - hn account the heterogeneous and micro-homogeneous struc- ture of the TGFE Va . Li (a) = Õ a - a , Lj (h) = Õ h - h , Note 2. In formula (9), matrices [K ] , P , [ Aa ] are n=1,n¹i i n n=1,n¹ j j n (5) e e e Lk (z) = n3 z- zn Õ . z - z constructed taking into account the curvilinear form of the base FE Ve (see formula (1)), which represent the region n=1,n¹k k n TGFE V geometrically accurately. Consequently, the It is convenient to use Lagrange polynomials (5) in a calculations. Displacement functions ua , va , wa TGFE, matrices [Ka ] , Fa are also determined taking into constructed on the grid Ha using Lagrange polynomials (5), are presented in the form of account the curvilinear form of the TGFE Va . Note 3. The determination of the stresses in TGFE Va b b n0 n0 n0 can be shown as follows. Let the vector δa be found. b b b b ua = å N qu , va = å N qv , wa = å N qw , (6) With the help of the formulas (8), (2) we find vectors δe , b=1 b=1 b=1 δ1 nodal displacements of FE V ( e = 1, ..., M ) respecb b b where qu , qv , qw , Nb are displacements and shape е tively, in coordinate systems e O x y z and O1x1 y1z1 . function of the b node of grid H , n = n n n , in the 2 2 2 2 present case n0 = 64 (fig. 2). a 0 1 2 3 Using vector δ1 we count the tension in the FE Ve with е e e Using (1), (2), the stiffness matrix [Ke ] and the nodal forces vector Pe of FE Ve in the coordinate system O2 x2 y2 z2 , we present [Ke ] = [Te ]T [K1 ][Te ] , Pe = [Te ]T P1 [1]. The functional of the full potential energy Пa of the basic partition of the Ra TGFE Va can be written in the form of algorithms of the finite element method [1; 2]. Note 4. Lagrange polynomials are used in Lagrangian TGFE polynomials, determined by formulas (5), which have the order of the polynomial multiple of the number of layers in the thickness of the shell on the coordinate z (i. e. z ). The calculations show that the location of the nodes of the large grid Ha TGFE at the boundaries of П = åM æ 1 δT [K ] δ - T ö (7) heterogeneous layers provides a homogenous and rapid a ç 2 e e e δe Pe ÷. convergence of sequences of approximate solutions. e=1 è ø The procedures of constructing composite Lagrangian TGFE of n-order, geometrically similar to TGFE Va (fig. 2), with the application of Lagrange polynomials of n -order, are similar to the procedure of § 2. Calculations show that by increasing the dimensions O2 x2 y2 z2 and O3 x3 y3 z3 , n = 1, ..., N respectively. According to the FEM [1] we define the following for- mula: δa = [T a ] qa , where [T a ] is the rotations matrix [2], n n n n n n n n n n n [M a ] = [T a ]T [Ka ][T a ] , Pa = [T a ]T Fa . Taking into of the basic partitions of TGFE (i. e., by increasing the account these relations, the total potential energy of the number M), the time spent on the construction of matrices Пb ThGFE Vb , i. e. the partition of Rb , is presented [Ka ] и Fa and formulas (9) significantly increase. in the form of In this case, it is advisable to apply 3-grid finite elements, N T 2 æ 1 П = a é a ù a a T a ö . (10) for the construction of which less time is required and b åç (qn ) ëMn û qn (qn ) Pn ÷ which generate the discrete shell model of lower dimenn=1 è ø sion than TGFE. 3. Multilayer curvilinear Lagrangian three-grid FE. Functions of the displacements up , vp , wp ThGFE The procedure of constructing curvilinear three-grid FE (ThGFE) with the use of Lagrange polynomials is Vb on the large grid Hb , using Lagrange polynomials are presented in the form of n b considered by the example of a six-layer ThGFE V of 0 n0 n0 z b b b b b b the 6-th order with its thickness hb , that is used in the up = å N qu , vp = å N qv , wp = å N qw , (11) calculation of 6-layer shells with thickness h , where h = hb . In the calculation of m-layer shell m-layer where b=1 qu , qv , b=1 qw , N b=1 are displacements and shape z b b b b Lagrangian ThGFE of m-order thickness are used. ThGFE V with the size hb ´ hb ´ hb is located in the local Cartefunction of the β node of grid Hb , n0 = n1n2 n3 , in this b x y z case n0 = 112 (fig. 3). sian coordinate system O3 x3 y3 z3 (fig. 3). To reduce the dimension of the functional (10) we use functions (11). Let’s denote: δb is the vector of nodal displacements of a large grid Hb . Expressing the nodal n displacements of vector qa TGFE V a through the nodal n displacement of vector δb of the grid Hb can see the equality ThGFE Vb , we n n b qa = éë Ab ùû δ , n where [ Ab ] is a rectangular matrix, n = 1, ..., N . (12) Using (12) in (10) and minimizing functional Пb in displacement of δb , we obtain the ratio for the ThGFE Vb [Kb ]δb = Fb that corresponds to its equilibrium state, Fig. 3. Six-Layer, ThGFE Vb where N N Рис. 3. Шестислойный ТрКЭ V [Kb ] = å[ Ab ]T [M a ][ Ab ] , Fb = å[ Ab ]T Pa . (13) b n=1 n n n n n n=1 The area of ThGFE consists of N curved 6-ply TGFE The matrix [Kb ] will be called the stiffness matrix, a V n with thickness h, n = 1, ..., N that geometrically accu- Fb is the vector of nodal forces ThGFE Vb . Note that the rately represent the area of ThGFE. TGFE V n make the large grid H determines the dimension of the ThGFE V , a partition Rb . The large grids Ha TGFE form a small b b n which is less than the partition dimension Rb consisting grid hb ThGFE. On the grid hb we define large grid of of the TGFE V a . Hb Ì hb ThGFE. The nodes of the large grid Hb marked Note 5. By virtue of (12) the dimension of the vector with points (112 nodes) lie on the common boundaries of different-modular layers of ThGFE (fig. 3). n n Suppose that the axis O2 y2 of ThGFE (fig. 2) is paral- lel to the axis O3 y3 (fig. 3). Suppose that δa , qa are the δb (i. e. the dimension of the ThGFE Vb ) does not depend on the total number of TGFE V a components of ThGFE. This means that the splitting of a ThGFE Vb into n a TGFE V a and, consequently, into single-grid FE V vectors of nodal displacements, [Kn ] , [M a ] are the n e a n (see § 2) can be arbitrarily small, which allows to describe stiffness matrices and Fa , Pa are the vectors of nodal with arbitrarily small error the three-dimensional stress n n n forces TGFE V a responsible for the coordinate systems state in the ThGFE taking into account its inhomogeneous structure. Note 6. Note that the number of layers of TGFE may m1 = 324n +1, m2 = 324n +1, n be less than the number of layers of the shell. For exam- n n (14) ple, constructing six-layered ThGFE you can use a threem3 = 12n +1, n = 1, ...,10, layered TGFE (fig. 2) or two-layered TGFE. As calcula- tions show, this leads to a decrease in time costs with 1 is the dimension of the circular coordinate; m2 - the m n n a minor change in the error of the solution. axis Oy, m3 - axis Oz . Characteristic sizes he , he , he In the formula (13), matrices [M a ] , Pa , [ Ab ] n are n xn yn zn n n n FE Ve are defined by the following formulas constructed taking into account the curvilinear form of TGFE V a (see § 2), which geometrically represent the he = he / n, he = he / n, n xn x1 yn y1 (15) area accurately, ThGFE Vb . Consequently, the matrices he = he / n, n = 1, ...,10, [Kb ] , Fb are also determined taking into account the curzn z1 vilinear form of the ThGFE Vb . where e e e h h , y1 z1 are characteristic dimensions of FE h , x1 The procedure of determining stresses in the ThGFE V e V is similar to the procedure for determining stresses 1 of the 1st order corresponding to the discrete model b R0 , where he = a R , he = L / 324 , he = h /12 , in the TGFE. Using ThGFE, according to the procedure similar to § 3, we construct four-grid FE, and the k grid of FE, k ³ 4 . Note that the k grid generate a discrete FE shell 1 x1 1 e y1 z1 e a1 =p / 324 , Re is the radius of the lower cylindrical surface FE V 1 . model of lower dimension than the k -1 FE grid. The described method can be used to calculate multilayer shells with layers of different thicknesses. Small enough partitions of composite shells are presented as homogeneous MFE, which are designed according to the procedures similar to § 1-3. 4. The results of numerical experiments. Consider the problem of deformation of a four-layered elastic cylindrical shell V0 of a complex shape with length 2L . The shell, clamped from two ends, is located in the Cartesian coordinate system Oxyz . When y = 0; 2L , displacement u = v = w = 0 . The radius of the shell on the median surface R = 2.0 m, the thickness of the shell Fig. 4. Left symmetric part of the shell V0 h = 0.03 m, length 2L = 12.0 m, i. e V0 is a thin shell with large geometric dimensions. The left symmetrical part of the shell is shown in fig. 4. Point A lies at the in- tersection of the planes Oyz and y = L on the top surface Рис. 4. Левая симметричная часть оболочки V0 n On base models R0 , n = 1, ...,10 we construct multiof the shell. Shell layers are isotropic homogeneous bodgrid discrete models Rn of shell V0 consisting of Laies. The upper and lower layers have h /12 thickness, grangian shell ThGFE with sizes 81he ´ 81he ´ h where the inner 2 layers have 5h / 12 . The Young’s modules of 4 layers (starting from the bottom) are equal to: 10, 3, 5, 20 GPA, respectively. Poisson’s ratio is 0.3. There is xn yn zn h = 12nhe . For all basic discrete models, ThGFE have a fixed size coordinate z which is equal to the thickness a uniformly distributed tensile radial load q = 0.05 MPa of the shell h . ThGFE are constructed on the procedure (fig. 4) on the outer surface of the shell 3L / 4 £ y £ L shown in § 3 and consist of Lagrangian TGFE with dimensions 9he ´ 9he ´ h , according to the procedure with the opening angle a = p /2 , which is symmetrical to the planes Oyz and y = L . In the area of the shell clamps xn yn shown in § 2. there are cutouts symmetrical to the plane Oyz , the opening angle of each cut is equal to the p /2 length is L /4 (fig. 4). As the shape, loading and fastening of the shell are symmetrical to the planes Oyz and y = L , we use 1/4 of the shell in the calculations. n The basic discrete model R0 of the shell consists of a The ThGFE uses Lagrange polynomials defined by the formulas (5), which have the third order of the poly- nomial by coordinates x, y, and the forth order by coordi- nate z , which corresponds to the number of layers in the thickness of the shell. As shown by numerical calcula- tions, if the nodes of large grids Ha and Hb of two-grid and three-grid FE lie on the common boundaries of multie curved homogeneous single grid FE of the 1st order V n , modulus layers, discrete models Rn provide even and fast geometrically similar to FE Ve (fig. 1). The model grid convergence of a sequence of finite element solutions. The results of the calculations for discrete models Rn R0 has a dimension of m1 ´ m2 ´ m3 , where n n n n are given in tab. 1, where we see: wn , sn are maximum radial displacement and equivalent stress for the model (test) problem with known exact solution u0 is solved. Rn , n = 6, ...,10 . We can find the stress sn with the 4th Suppose that || u0 - uh || ® 0 when h ® 0 , where uh strength theory. As you know, using the maximum equivalent stress the factors of safety of structures are determined. We find the values ds,n (%) , dw,n (%) with the formulas ds,n (%) = 100 % × | sn - sn-1 | /sn , is the solution of the test problem, constructed with the help of a family of new MFE, h is the characteristic size of MFE. Then we consider that the solutions constructed with the help of a family of new MFE and for the initial problem converge in the limit ( h ® 0 ) to the exact one. We consider the deformation of a 4-layer cylindrical dw,n (%) = 100 % × | wn - wn-1 | /wn , n = 2, ...,10 . shell V1 as a test problem, which is located in the Carte- The nature of changes in values d w,n (%) , d s,n (%) (tab. 1) sian coordinate system Oxyz , to have the same geometric dimensions, fastening conditions and elastic modules shows rapid convergence of the equivalent stresses sn as the shell V0 in § 4. However, the shell V1 has no cutand displacements wn . Since the values for the model outs. When 3L / 4 £ y £ 5L / 4 the radial tensile uniform R10 are small, dw,10 = 0.00116179 , ds,10 = 0.00719947 it can load of р = 0.1 MPa acts on the outer surface of the be considered from the point of view of engineering prac- tice that the displacement of w10 = 30.289362 mm and s10 = 31.371908 MPa are made with low error, i. e., w10 , s10 are little different from the exact (see § 5). R 10 The dimension of the underlying discrete model 0 shell V1 , i. e. axisymmetric three-dimensional stress state is realized in the shell V1 [1]. As you know [1], the sequence of approximate solutions of the axisymmetric problem, constructed by MFE with the use of standard FE, which are homogeneous rings with a rectangular cross-section, in the limit (when is 3722110998 (more than 3.7 billion), the width of the tape of the system equations (SE) FEM is 1176610 (over hm ® 0 hm is the characteristic size of the standard FE) converge to the exact solution. Calculations are carried 1.1 million). Multigrid model R10 has 203090 nodal out for discrete models Q , n = 1, ...,14 , shell V . The unknowns, the width of the tape SE FEM is equal n 1 s to 5445. Application of the FEM for the multigrid model results of calculations are given in tab. 2 for models Qn R10 requires 3960366 (approximately 3.96 million) less where, n = 7, ...,14 , w0 , 0 are the deflection and n 10 n times than the amount of computer memory of the base model R0 . equivalent voltage at the point A (fig. 4), dimensions of models Qn are given in the plane Oyz . The parameters 5. The study of the convergence of approximate of d0 (%) , d0 (%) are determined by the formulas s,n n n-1 n solutions. To study the convergence of approximate solu- tions constructed using the new MFE, we use the follow- ing numerical method, the brief essence of which is w,n s,n w,n n n-1 n d0 (%) = 100 % × | w0 - w0 | /w0 , (16) shown below. With the kind of new MFE that are used in the solution of the original problem (see § 4), the similar d0 (%) = 100 % × | s0 - s0 | /s0 , n = 2, ...,14. Rn R6 R7 R8 R9 R10 wn 30.032632 30.136577 30.205840 30.254172 30.289362 dw,n (%) 0.550568 0.344913 0.229303 0.159753 0.116179 sn 30.074687 30.544130 30.881125 31.146047 31.371908 ds,n (%) 2.374768 1.536934 1.091265 0.850580 0.719947 Displacements wn and equivalent stresses sn for models Rn Table 1 Displacements w0 and equivalent stresses s0 for models Q Table 2 n n n N Dimensions of models w0 ×103, м n d0 (%) w,n s0 , МPа n d0 (%) s,n 7 2269 ´ 43 2.24419205 0.0001359 21.9642981 0.0006719 8 2593 ´ 49 2.24419400 0.0000868 21.9641839 0.0005199 9 2917 ´ 55 2.24419529 0.0000574 21.9640928 0.0004147 10 3241´ 61 2.24419632 0.0000458 21.9640199 0.0003319 11 3565 ´ 67 2.24419706 0.0000329 21,.9639594 0.0002754 12 3889 ´ 73 2.24419754 0.0000213 21.9639074 0.0002367 End of table 2 N Dimensions of models w0 ×103, м n d0 (%) w,n s0 , МPа n d0 (%) s,n 13 4213 ´ 79 2.24419797 0.0000147 21.9638637 0.0001989 14 4537 ´ 85 2.24419832 0.0000155 21.9638261 0.0001711 Displacements wp and stresses s p for models R Table 3 n n n n wp ×103, м n d p (%) w,n s p , МPа n d p (%) s,n 7 2.24416383 0.0001038 21.9641016 0.0060703 8 2.24416590 0.0000922 21.9649716 0.0039608 9 2.24416869 0.0001243 21.9655875 0.0028039 10 2.24417150 0.0002810 21.9660517 0.0021132 11 2.24417898 0.0003333 21.9664194 0.0016739 12 2.24418121 0.0000993 21.9667143 0.0013424 13 2.24418263 0.0000632 21.9669569 0.0011043 14 2.24418427 0.0000730 21.9671598 0.0009236 The nature of the values change of d0 (%) , d0 (%) of view of engineering practice) to the exact solution of shows the rapid convergence of stresses w,n s0 s,n and displacethe axisymmetric problem. n The shell V0 considered in § 4 differs from the shell ments w0 to the exact solution w , s of the axisymmet- V considered in § 5 by the presence of cutouts and the n 0 0 1 ric problem [1]. As the sizes, 0 d w,14 = 0.000000155 method of applying the load, with full coincidence of the d 0 s,14 = 0.000001711 are sufficiently small, the displacedimensions, boundary conditions and physical character- istics of the shells. In addition, when constructing w 14 ment of 0 = 2.24419832 ×10-3 m and the equivalent sequences of approximate solutions for the initial and test problems, the same family of proposed ThGFE is used. 14 stress s0 = = 21.9638261 MPa can be considered as the Therefore, it can be assumed that the proposed shell 14 0 14 exact solution, i. e. we believe w0 = w0 , s = s0 . We consider the solution of this axisymmetric MFE problem with the use of FE, which were used in solving the problem in § 4. We construct approximate solutions of the axisymmetric problem using the laws of grinding (14), (15) of basic partitions. The results of calculations are ThGFE, which provide uniform convergence of approxi- mate solutions for the test problem (for the shell V1 ), gen- erate solutions wn , sn that in the limit (at n ®¥ ) will converge (from the point of view of engineering practice) to the exact values of displacement and equivalent stress for the original problem (for the shell V0 ), see § 4. given in the tab. 3, where, wp , s p is the deflection and Conclusion. In this work we propose a numerical n n w equivalent stress at the point A for a multigrid discrete model Rn , n = 7, ...,14 . The parameters d p,n (%) , method of calculation of multilayered linear elastic cylin- drical thin and medium-thickness shells with the use of curvilinear Lagrangian shell type MFE. Application of the d p s,n (%) are determined by formulas similar to formulas MFE for multigrid discrete shell models requires much (16). The nature of the change in values d p (%) , less computer memory than the base models, which d p s,n (%) w,n w demonstrates the rapid convergence of stresses allows to construct solutions with a small error and can explore SSS of shells of large geometric dimensions. The above calculations show the high efficiency of the s n p and displacements p to the limit values wp , proposed curvilinear Lagrangian shell MFE in the analys 0 0 n p . The errors for displacement p and stress sis of three-dimensional SSS multilayer shells. w s p dw (%) = 100 % × | w0 - wp | /w0 , d (%) = 100 % ´ References 14 14 14 14 14 s ´ | s0 - s p | /s0 , respectively, are equal to 0.00062828 % 14 14 14 0.0151749 %. In tab. 2, 3 values w0 , s0 , wp , s p , are 1. Zenkevich O. Metod konechnykh elementov v tekh- nike [The finite element method in engineering science]. 14 14 14 14 marked in bold. From the point of view of engineering practice, because of the smallness of the values dw (%) , ds (%) , we can assume that wp = w0 , s p = s0 . Then we Moscow, Mir Publ., 1975, 541 p. 2. Norri D., Zh. de Friz. Vvedenie v metod konechnykh elementov [An Introduction to Finite Element Analysis]. Moscow, Mir Publ., 1981, 304 p. 0 0 can conclude that the proposed ThGFE generate solutions s p , wp that in the limit (at n ®¥ ) tend (from the point 3. Golovanov A. I., Tyuleneva O. I., Shigabutdinov A. F. Metod konechnykh elementov v statike i dinamike tonkosn n tennykh konstruktsiy [Finite Element Method in statics

Об авторах

А. Д. Матвеев

Институт вычислительного моделирования СО РАН

Email: mtv241@mail.ru
Российская Федерация, 660036, г. Красноярск, Академгородок, 50/44

А. Н. Гришанов

Новосибирский государственный технический университет

Российская Федерация, 630073, г. Новосибирск, просп. К. Маркса, 20

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