EFFICIENT METHOD OF CALCULATING LAYERED CONICAL SHELLS WITH LAGRANGE MULTIGRID ELEMENTS USE

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Introduction. The layered conical shells (LCS) are widely applied in the space-rocket and aircraft technol- ogy. Unlike cylindrical shells, LCS geometrical and stiff- ness properties depend on axial coordinate that creates great difficulties during analytical and numerical research of the stress strained state (SSS) of such shells.. Since during shells numerical calculations there is no unified approach suitable for the entire range of shell constructions in use, further research in the field of shells computing mechanics is being continued now. In recent years new numerical methods for LCS calculation have been developing and the existing methods have been improving. The method of differential quadratures developed for the solution of the linear and non-linear equations in partial derivatives [1] is applied to the LCS analysis with the equations of the three-dimensional elastic theory use in conic coordinates [2]. In works [3-5] the method of reference surfaces which is used for calculation of multilayer shells and plates is offered for homogeneous and layered shells in a spatial setting calculation. Movements in shell thickness are approximated by means of Lagrange polynomials, and movements in reference surfaces are set by functions which meet boundary conditions. For calculation of layered conical and cylindrical shells the method of discrete singular convolution is used [6]. At the same time only thin shells which deformation submits to Kirkhgofa- Lyava kinematic hypothesis are considered. The method of finite element (MFE) is the most widespread numerical method when calculating shells. The review of works on calculation of composite shells by means of MFE is provided in [7]. Various options of finite elements (FE) are applied to LCS calculation. For example, in work [8] a curvilinear 4-node FE with 20 degrees of freedom for the bearing layers of a 3-layer shell is used, and a filler interlayer is considered in the elastic theory three- dimensional setting. In work [9] the layered conical structures of a shell for bends with the use of isoparametric FE and high order displacement deformation models are analysed. The increased requirements to modern shell construc- tions cause the necessity of algorithms of approximate solutions tasks of the elastic theory with the given small error development and improvement. It is difficult to ap- ply the known approaches from the theory of materials strength, or the two-dimensional elastic theory based on the simplifying hypotheses that often leads to a bigger error of the achieved results to obtain such solutions. In this regard there is a need for the development of such FE in which the deformation of a composite shell is consid- ered in three-dimensional setting taking into account its structure without simplifying kinematic and static hy- potheses introduction. In works [10; 11] the computa- tional method of composite circular cylindrical shells in a three-dimensional setting with application of multigrid FE (MGFE) in which movements are approximated by means of Lagrange polynomials of various orders is offered. In this work the numerical computational method of LCS of various form and thickness at the arbitrary static loading in which three-dimensional curvilinear shell type Lagrange MGFE are used is described. The characteristic feature of the MGFE constructing offered procedure is that Lagrange polynomials nodes coincide in shell thick- ness with MGFE large grids nodes lying on the common borders of multi-module layers. The offered method pro- vides the uniform and fast convergence of approximate solutions that allows to make solutions with the specified (small) error. The effectiveness of the offered MGFE is that they generate discrete models which dimension is several orders less than dimensions of basic models. Ex- amples of calculations are given. Construction of single-grid FE for conical shell basic model. We will briefly consider the procedure of constructing curvilinear homogeneous single-grid FE (SGFE) construction which create a conic shell basic discrete model on the example of FE V (1) of the 1st order (fig. 1). The procedure of SGFE construction for cylindri- cal shells at approximation of fields of movements by degree polynomials is explicitly explained in work [12]. Let us consider that FE order is defined by order of a degree polynomial or a Lagrange polynomial constructed on its nodal grid, and the superscript in the symbol corresponds to the nodal grids quantity in an element. SGFE represents a part of the conical shell with the reference sizes h(1) (h(1) ) ´ h(1) ´ h(1) located in a local (1) (1) T (1) (1) T (1) x,1 x,2 y z P = ò (N ) F dV - ò (N ) q dS . (3) Cartesian coordinate system O1x1 y1z1 . In fig. 1 V S y designations are introduced: z1O1 y1 - a plane of symmetry, cd - a longitudinal axis of a conical shell, a - Let us note that the continuity of movements on FE curvilinear borders V (1) (fig. 1) is broken. However, as it FE V (1) corner angle, (1) h z - thickness, h(1) - length is well-known [14], realization of continuity of movements on borders of curvilinear FE is not a (height), (1) h = aR x,i i (i = 1, 2) , R1, R2 - radiuses of a shell necessary condition for convergence of numerical solubottom face at FE end faces, b - shell conicity angle, nodes in the drawing are noted by points. Movements, deformations and tension in SGFE V (1) satisfy to the equations of the three-dimensional elasticity theory, recorded in the local Cartesian coordinate system O1x1 y1z1 . tions to precise and is checked in each case. The carried out numerical experiments show that at curvilinear homogeneous FE V (1) reference sizes decrease numerical solutions converge to precise. In (3) we define integrals numerically. Let us present area V by elementary curvilinear subareas V 1,...,V N , Taking into account that FE reference sizes are small for minor basic splits, we use 1st order polynomials i. е. N V = ∪ n=1 V n , N - total number of subareas. For for approximation of movements functions u(1) , v(1) , w(1) V n area let us introduce designations: Dz = h(1) / m , z 1 of element V (1) Dy = h(1) / m , Da = a / m , Da - corner angle of area 1 2 1 3 1 4 1 5 1 1 u(1) , v(1) , w(1) = a + a x + a y + a z + a x y + y 2 V n ; m , m , m 1 2 3 3 - the given integral numbers; +a6 z1x1 + a7 z1 y1 + a8 x1 y 1 z1 . (1) N = m1m2m3 . The total potential energy of FE V (1) in a matrix form The form of area V n is a part of the truncated conical is the following [13] shell with thickness h ( Dz = h / cosb ), height Dy and П(1) (δ(1) ) = 1 ò (δ(1) )T (B(1) )T DB(1)δ(1) dV corner angle Da . Let us note that areas V n (irrespective 1 1 z 1 2 V of their sizes) geometrically precisely represent FE -ò (δ (1) )T (N (1) )T F(1) dV - ò (δ (1) )T (N(1) )T q(1) dS , (2) V (1) curvilinear area. Let xn , yn , n - area V n gravity V S centre coordinates in the local coordinate system where B(1) , D - matrixes of deformations and elastic O1x1 y1z1 . The volume DVn of area V n is defined by the modules V (1) ; F(1) , q(1) - vectors of volume and surface approximate formula DVn = DzDyDaRn , where Rn - forces; δ(1) , N(1) - a vector of nodal unknowns and a distance from a cone axis to an area V n gravity centre. matrix of form functions; V , S - FE V (1) area and sur- Matrix B(1) , which elements are calculated for values face; T - transposition. of coordinates xn , yn , zn , let us designate 1 1 1 1 1 1 From ¶П(1) (δ(1) ) / ¶δ(1) = 0 condition we find formulas B(1) (xn, yn , zn ) . We approximately find a stiffness matrix N for calculation of a stiffness matrix K(1) and a nodal K(1) by virtue to (3) on a formula 1 1 1 1 forces vector P(1) in the local coordinate frame O x y z K(1) = å (B(1) (xn , yn , zn ))T DB(1) (xn, yn, zn )DV . (4) K(1) = ò V (B(1) )T DB(1)dV , n=1 1 1 1 1 1 1 n а b Fig. 1. Homogeneous FE V (1) ( V (1) ) (a), the cross section of the FE plane z O y (b) n 1 1 1 Рис. 1. Однородный КЭ V (1) ( V (1) ) (а), сечение КЭ плоскостью z O y (б) n 1 1 1 The vector of nodal forces P(1) of element V (1) defined numerically. is also of calculating a k-layer conical shell it is necessary to use k- layer Lagrange TGFE of a k-order in thickness. The differences of the offered curvilinear SGFE V (1) construction procedure from isoparametric FE construc- tion [13] are as follows. Isoparametric FE use is proved by the necessity of FE stiffness matrix calculation simplification. Curvilinear coordinates are transformed to rectilinear (Cartesian) coordinates, and curvilinear FE is transformed in rectilinear (two - three-dimensional) by the equivalent transformations. Herewith stiffness matrix numerical calculation assumes the known quadrature formulas use [14]. Transformation of curvilinear coordinates demands calculation of a straight line and an inverse Jacobi matrix in each calculated point at a numerical integration. The offered option of SGFE stiffness matrix calculation (3), (4) is simpler and has the following advantages: m Fig. 2. Three-Layer TGFE V ( 2) ( V ( 2) ) Рис. 2. Трехслойный ДвКЭ V ( 2) ( V ( 2) ) - curvilinear FE V (1) is projected in the local three- m dimensional Cartesian coordinate system and therefore there is no need to define a straight line and an inverse Jacobi matrix [13; 14] that is required when using isoparametric FE; - when constructing approximating displacement SGFE nodes V (1) , n = 1,..., N , make a fine curvilinear n grid on which TGFE large grid is constructed. Let us note that large grid nodes on shell thickness lie on the common borders of TGFE multi-layers which generally have varifunctions u(1) , v(1) , w(1) FE V (1) we use the known degree ous thicknesses. Lagrange polynomials construction in the polynomials of the 1st, 2nd and 3rd orders [13] which are recorded in local Cartesian coordinate systems which do not contain FE rigid displacement. In case of local curvi- linear coordinate frames at constructing curvilinear shell FE application there is a need to construct such approxi- mating functions of movements in which FE rigid dis- placements are excluded that is connected with particular difficulties [15]; local curvilinear coordinate frame O2xhV on TGFE large grid for cylindrical shells is considered in [10; 11] and can be applied to LCS calculation. The basic function Nijk for node P(i, j, k ) (fig. 2) in curvilinear coordinates a, h, z is Nijk (a, h, z) = Li (a)Lj (h)Lk (z) , where Li (a) , Lj (h) , Lk (z) - Lagrange polynomials: - the numerical integration is performed according to the simplest formula when in each partial area V n the Li (a) = a - an , 1 n Õ n=1,n¹i ai - an Lj (h) = h - hn , 2 n Õ n=1,n¹ j h j - hn value of function is chosen constant and equal to the value of function in a gravity centre of this area. At decrease of (1) n 3 Lk (z) = Õ z - zn z - z . (5) the partial areas sizes the value of a FE V stiffness ma- trix in a limit converges to precise value. Procedures of the 2nd, 3rd order SGFE construction Using designations n=1,n¹k k n i i i i u(2) , v( 2) , w( 2) , N (2) for movewhich geometrically are similar to the FE (fig. 1) are similar to the above described. V (1) form ments and form functions of TGFE i node in the coordi- 2 2 2 2 nate frame O x y z , movements functions u(2) , v(2) , w(2) Further we will consider the construction of MGFE with ideal connections between the heterogeneous structure components in case of movements approximation by Lagrange polynomials on the example can be given as [13] n0 i i u( 2) = å N ( 2)u(2) i=1 n0 i i , v( 2) = å N ( 2) v( 2) , i=1 of three-grid FE (ТGFE) V (3) . Such element consists n 0 of M two-grid FE (TGFE) V (2) , ( m = 1,..., M ), each one w( 2) = å N ( 2) w(2) , n = n n n . (6) m is composed from N homogeneous SGFE i i i=1 0 1 2 3 n V (1) ( n = 1,..., N ). Construction of two-grid FE for a conical shell. Let We will record the functional of the total potential energy П(2) for basic TGFE V (2) split as follows us consider the procedure of multilayer TGFE for a coni- (2) N (1) 1 T (1) (1) (1) T (1) cal shell construction on the example of tree-layer TGFE V (2) of the 3rd order in its thickness which is used when П = å( (δn ) K n 2 n=1 δn - (δn ) Pn ) , (7) x,1 x,2 y calculating a 3-layer conical shell in thickness h with the where (1) K n - stiffness matrix, P(1) , (1) δ n - vectors of n reference sizes h( 2) (h(2) )´ h( 2) ´ h located in the local nodal forces and movements of SGFE V (1) which corren Cartesian coordinate system O2 x2 y2 z2 (fig. 2). In case spond to the coordinate frame O2 x2 y2 z2 . The use of small splits generates TGFE with a large (3) M (2) 1 T (2) (2) (2) T (2) number of nodal unknowns. For decrease in TGFE dimension the following procedure is used. By means of П = å( (δm ) K m 2 m=1 δm - (δm ) Pm ) , (10) functions (6) we present the vector of nodal movements where δ(2) - a nodal movements vector; K(2) , P(2) - δ(1) of SGFE V (1) , n = 1,..., N m m m through the vector of nodal n n a stiffness matrix and a nodal forces vector TGFE V (2) , movements δ(2) equality of TGFE V (2) . As a result, we receive which correspond to the coordinate frame m = 1,..., M . m O3 x3 y3 z3 , n n δ(1) = A( 2)δ(2) , (8) Movements functions u(3) , v(3) , w(3) ThrGFE V (3) , n where A(2) - a rectangular matrix, n = 1,..., N . constructed on a large grid by means of Lagrange polynomials, we will present as Substituting (8) in (7) and, following the principle of the total potential energy minimum, i. e. p0 u = å N u , (3) (3) (3) i i p0 v = å N v , (3) (3) (3) i i ¶П(2) (δ(2) ) / ¶δ(2) = 0 , for TGFE V ( 2) we get the ratio i=1 i=1 K (2)δ( 2) = P( 2) defining its an equilibrium state where (3) p0 (3) (3) w = å Ni wi , N N i=1 (11) K( 2) = å(A( 2) )T K(1) A( 2) , P( 2) = å (A(2) )T P(1) , (9) where u(2) , v(2) , w(2) , N (2) u(3) , v(3) , w(3) , N (3) - move- n=1 n n n n n n=1 i i i i i i i i K( 2) - stiffness matrix, P(2) - vector of nodal forces ments and an i node form function of a ThrGFE large grid in the coordinate frame O3 x3 y3 z3 ; p1, p2 , p3 - TGFE V (2) . Procedures of constructing composite Lagrange TGFE of p - order construction, geometrically similar to TGFE ThrGFE Lagrange polynomials orders on coordinates x3 , y3 , z3 , p0 = p1 p2 p3 . For decrease in number of ThrGFE nodal unknowns V (2) (fig. 2), with application of Lagrange polynoms the vector of FE V (2) nodal movements δ(2) by means of of p-order, are similar to the considered procedure. m (11) we present through the FE V (3) m vector of nodal m m The calculations show that at increase in dimensions of TGFE basic splits time expenditure on construction of movements δ(3) . As a result, we obtain equality matrixes K( 2) and P(2) according to formulas (9) δ(2) = A(3)δ(3) , (12) significantly increases. In this case it is expedient to apply ThrGFE which constraction requires less time expenditure and which generate shells discrete models of smaller dimension, than TGFE. Construction of three-grid FE for a conical shell. We will consider the procedure of multilayer ThrGFE for a conical shell construction we will consider on the where A(3) - a rectangular matrix, m = 1,..., M . Substituting (12) in (10) and, minimizing a functional m П(3) on movements δ(3), for ThrGFE V (3) we receive a matrix ratio K(3)δ(3) = P(3) which corresponds to its equilibrium state, where example of 3-layer ThrGFE V (3) of the 3rd order in its M (3) (3) T (2) (3) M (3) (3) T (2) thickness with the reference sizes (3) (3) (3) x,1 x,2 y ´ h , K = å(Am ) K m m=1 Am , P = å (Am ) Pm m=1 . (13) h (h )´ h disposed in the local Cartesian coordinate system where K(3) , P(3) - a stiffness matrix and a nodal forces O3 x3 y3 z3 . ThrGFE has the form similar to TGFE shown in fig. 2. For ThrGFE the order of Lagrange polynomials vector of ТhrGFE V (3) . Remark 1. The dimension of a vector δ(3) (i. e. on coordinates x3 , y3 can be arbitrary, different from the dimension of ThrGFE V (3) ) does not depend on TGFE polynomials order on these coordinates in TGFE. ThrGFE has the 3rd order in its thickness h (coordinate z3 ) which V (2) m n m total number included in ThrGFE. Therefore, is used when calculating 3-layer conical shells. In case of ThrGFE splitting into TGFE V (2) and SGFE V (1) can be a m-layer conical shell calculation it is necessary to use a m-layer Lagrange ThrGFE of m order in thickness. arbitrarily small that allows to consider a complex hetero- geneous structure and a form of ThrGFE V (3) . The ThrGFE area consists of M TGFE V (2) , Подпись: m Remark 2. The quantity of TGFE layers can be less m = 1,..., M which geometrically precisely represent the than the number of shell layers. For example, when con- ThrGFE area. The TGFE nodes, included in ThrGFE, generate a curvilinear grid on which a ThrGFE large grid is being constructed. Let us note that ThrGFE large grid nodes, as well as in case of TGFE lie on the common borders of multi-layers which generally have various structing a 6-layer ThrGFE it is possible to use 3-layer TGFE (fig. 2) or 2-layer TGFE. As the calculations show, it leads to decrease in time expenditure with an insignifi- cant change of solution error. The calculations show that the arrangement of thickness. The total potential energy V (3) is represented by П(3) of ThrGFE ThrGFE large grid nodes on borders of multi-layers pro- vides the uniform and fast convergence of approximate solutions. Using ThrGFE, according to the procedure similar h = 1 m; E = 1 h Pа; q = 1 MPа; n = 0, 3 ; b = 21,80 . n n 0 0 n n n s n 0 to p. 3, we construct 4-grid FE, and k -grid FE, k ³ 4 . 0 0 Let us note that k -grid FE generate discrete models of conical shells of smaller dimension, than (k -1) -grid FE. The proposed method can be used for calculation of multilayer conical shells with layers of various thickness. Designations are introduced in tab. 1: w* = w / (q h E-1) , s* = s / q , где w* , * - the di- mensionless normal movements and the equivalent stresses (for the model Rn reference points B and С ). We Results of numerical experiments determine stresses * according to the 4th theory of s n Example 1. Let us consider a 4-layer elastic console conical shell under the influence of external pressure q in the Cartesian coordinate system Oxyz , y -axial strenght. We get values las ds,n (%) , dw,n (%) by the formucoordinate, h - thickness, L - shell length. At y = 0 a ds (%) = 100 % ´ | s* - s* | /s* , ,n n n-1 shell is rigidly restrained. At shell end faces the radiuses of a median surface are equal to R at y = 0 and r at d (%) = 100 % ´ | w* - w* | /w* , n = 2,..., 5 . (16) w,n n n-1 n y = L , b - cone angle. Shell layers are isotropic The nature of sizes dw,n (%) , ds,n (%) change shows homogeneous bodies. Top and bottom layers have fast convergence of stress s* and movements w* . As for thickness h / 6 , two internal layers - h / 3 . Young’s model R the values d (%) = 0, 0049 , n n modules of 4 layers (starting with internal) are 5 w,5 B respectively equal: 10E; 3E; 5E; 20E , E - an elastic dw,5 (%)C = 0, 0232 and values ds,5 (%)B = 0, 0272 , module, n - Poisson’s ratio. The reference points B and ds,5 (%)C = 0, 007 are small, from the point of view of 5 5 C B С on the external surface of the shell lie on the crossing engineering practice it is possible to consider that of the plane Oyz and transverse sections y = L / 2; L . In movements (w*) = -0,82302 , (w*) = -0,3879 and calculations 1/4 part of the shell is used. Basic discrete stresses (s* ) = 17, 49340 , (s* ) = 11, 6266 in the conimodels of the 5 B 5 С 0 R shell consist of the 1st order n homogeneous SGFE h(1) (h(1) ) ´ h(1) ´ h(1) , V (1) with the reference sizes cal shell reference points B and C are calculated with a small error (less than 0,3 %). n The comparison of the results received by means of xn,1 xn,2 yn zn ThrGFE (grid1621´1621´ 61 ), by means of SGFE h(1) = h(1) / n , h(1) = h(1) / n , h(1) = h(1) / n , (grid163´163´13 ) received in the ANSYS program xn, j x1, j yn y1 zn z1 complex (PC) and by means of FE for a two-dimensional n = 1,..., 5 , j = 1, 2 , (14) task of the elasticity theory [13] is given in tab. 2. We will j = 1 corresponds to (1) V n size on the circumferential consider the numerical results received by means of two- dimensional axisymmetric task statement [13] the most coordinate at a larger FE end face, j = 2 - at a smaller precise within MFE. The smallest error (less than 0,04 %) n R n end face V (1) . The fine grid dimension of model 1/4 shell part is determined according to formulas m1 = 324n +1 , m2 = 324n +1 , 0 for for the field of movements in the reference points B and C is also provided by ThrGFE. For the equivalent stresses the error is less than 1,2 % for calculation in PC ANSYS, and less than 0,4 % when using ThrGFE. SGFE define n n n m3 = 12n +1, n = 1,..., 5 , (15) n where m1 - a grid dimension in the shell tangential direcmovements with an error less than 0,2 % and the stress with an error about 4 % on the free end of a conical shell. The grid size for SGFE exhausts the memory capacity used by electronic computing machine (ECM) that limits n tion, m2 - in axial, m3 - in radial. the possibility of constructing sequence of solutions by n n On basic models R0 , n = 1,..., 5 , we project multigrid means of SGFE. The basic discrete model R0 dimension (for 1/4 part of discrete models of the Rn shell which consist of Lagrange ThrGFE size 81h(1) (h(1) ) ´81h(1) ´ h s. ThrGFE consist 5 a shell) is 480364020 (approximately 0, 48´109 of nodal xn,1 xn,2 yn unknowns), MFE SLAE film width - 296710. The R5 of Lagrange TGFE with sizes 9h(1) (h(1) ) ´ 9h(1) ´ h . multigrid model has 54300 nodal unknowns, MFE SLAE xn,1 xn,2 yn Lagrange polynomials are used in ThrGFE, which film width is 2775. Realization of MFE for R5 multigrid are defined by formulas (5) which in local coordinates model reduces the order solved by MFE SLAE in n have the third order in the tangential and axial direction and the fourth order - in radial that corresponds to quan- tity of layers in the shell. In discrete models Rn TGFE and ThrGFE large grids nodes lie on the common borders of heterogeneous layers in shell thickness. The results of calculations for discrete models Rn at the following values of parameters are given in tab. 1: L = h0 ; R = h0 ; r = 0, 6h0 ; h = 0, 06h0 ; q = -0, 5q0 ; 8,8´103 times and demands in 0, 96 ´106 times less ECM memory capacity than for the basic model R0 in which only SGFE are used. The quantity of ThrGFE (400 5 ThrGFE) used for calculation in discrete model R5 is 14,6 times less than the quantity of FE in PC ANSYS (5850 FE). Thus, ThrGFE use when calculating SSS allows to save significantly ECM resources in comparison with PC ANSYS and when using SGFE. Example 2. Let us consider a conical shell with geo- metrical sizes and physical properties from example 1 in which two identical cutouts are located symmetrically relatively the planes Oyz and Oxy , with the length l and a cone angle a = p / 4 , 4l - the length of a frustum of a cone on the generatrix (fig. 3). and 0,11 % respectively) that is considered to be an acceptable result from the point of view of engineering practice. Comparison of these results with the results of task calculation is carried our in PC ANSYS. The dimensionless values of the equivalent stresses s0 and normal movements w0 in points B and C received in PC ANSYS are s0 = 9, 952 , s0 = 0, 638 and w0 = -1, 091 . B C B The relative accuracies of a deviation of movements and stresses values in points B and C received in R7 discrete model when using ThrGFE from the results received in PC ANSYS are less than 1,2 % for movements and less than 3 % for stresses. 5 In fig. 4 distributions on an external surface of a coni- cal shell of the dimensionless normal movements 5 Fig. 3. Shell design scheme ( w* = w* ) in sections y = L / 2; L and the equivalent Рис. 3. Расчетная схема оболочки stresses ( s* = s* ) in sections y = 0; L / 2; L depending Standard pressure of the distributed load q = -0, 5q0 , on the parameter s* = s / P , s - distance from an axis Oz to a point on an external surface of a shell, P - perimeter q0 = 1 MPa is enclosed on the area of the shell upper face 0, 5L £ y £ 0, 75L and a cone angle of a loading area g = p / 2 symmetrically concerning the plane Oyz . In of a shell cross section half are shown. Calculation of SSS is carried out by means of ThGFE for R7 model (solid line) and by means of PC ANSYS calculations we use a half of a shell. In calculation the same basic discrete models and Lagrange TGFE and ThrGFE as in example 1 are used. The results of calculations for discrete models Rn (dashed line). In all chosen sections of the composite shell construction it is possible to observe the coincidence of SSS, accepted in engineering calculations, received by means of ThGFE and PC ANSYS. Thr basic discrete model R0 dimension (for 1/2 of the ( n = 1,..., 7 ) are given in tab. 3. The nature of values 7 change d (%) , d (%) shows fast convergence of the shell) is 2460017130 (approximately 2, 46´109 nodal w,n s,n unknowns), the width of MFE SLAE film - 578601. The equivalent tension sn and normal deflections wn . multigrid model R7 has 199800 nodal unknowns, the As for R7 model deflections values width of MFE SLAE film is equal to 3840. Realization of dw,7 (%)B = 0, 025 , dw,7 (%)C = 0, 030 and values of stre- MFE for the multigrid model R7 reduces the order of the ses ds,7 (%)B = 0,1098 , ds,7 (%)C = 0, 0178 are small, it is 7 B possible to consider that movements (w*) = -1, 07661 , solved MFE SLAE by 12312 times and demands R 1,855´106 times less than CEM memory capacity than 7 (w*) = -1,13964 and stresses (s* ) = 10, 01830 , for the basic model 0 in which SGFE are used. The 7 C 7 B 7 B (s* ) = 0, 61903 in the reference points B and C of LCS are calculated with a small error (about 0,03 % ThGFE quantity used for calculation in discrete model R7 (240 FE) is 35 times less than FE quantity used when cal- culating in PC ANSYS (8436 FE). Тable 1 The sequence of solutions for a 4-layer conical shell Rn (w* ) n B (w* ) n C dw,n (%)B dw,n (%)C (s* ) n B (s* ) n C ds,n (%)B ds,n (%)C R1 -0.82538 - 17.41062 - -0.39056 11.69263 R2 -0.82341 0.2392 17.46738 0.3249 -0.38867 0.4863 11.64802 0.3830 R3 -0.82313 0.0340 17.48087 0.0772 -0.38817 0.1288 11.63100 0.1463 R4 -0.82306 0.0085 17.48864 0.0444 -0.38799 0.0464 11.62579 0.0448 R5 -0.82302 0.0049 17.49340 0.0272 -0.38790 0.0232 11.62660 0.0070 Тable2 Comparison of calculations results received in different variants of solution The method of task solutuion w* B dw (%)B w* C dw (%)B s* B ds (%)B s* C dC (%)С ThrGFE -0.82302 -0.38790 17.49340 11.62660 0.0255 0.0387 0.3787 0.0119 SGFE -0.82153 -0.38751 17.38464 12.04767 0.1556 0.0619 0.2454 3.6093 PC ANSYS -0.82329 -0.38871 17.447 11.760 0.0583 0.2476 0.1125 1.1354 [13] -0.82281 -0.38775 17.42740 11.62798 Table 3 The sequence of solutions for a 4-layer conical shell with cutouts Rn R1 R2 R3 R4 R5 R6 R7 (w* ) n B -1.07084 -1.07250 -1.07441 -1.07539 -1.07597 -1.07634 -1.07661 dw,n (%)B - 0.155 0.178 0.091 0.054 0.034 0.025 (w* ) n С -1.12031 -1.13299 -1.13632 -1.13782 -1.13865 -1.13918 -1.13954 dw,n (%)С - 1.119 0.293 0.132 0.073 0.047 0.030 (s* ) n B 9.65343 9.82601 9.92132 9.96536 9.99092 10.00730 10.01830 ds,n (%)B - 1.756 0.607 0.442 0.256 0.164 0.110 (s* ) n C 0.69330 0.63057 0.62157 0.61965 0.61926 0.61914 0.61903 ds,n (%)C - 9.948 1.448 0.310 0.063 0.019 0.018 а b Fig. 4. Distribution of deflections w* (a) and stresses s* (b) on the upper surface of the shell in cross sections: y = L; L / 2;0 . ThrGFE - solid line, ANSYS- dashed line Рис. 4. Распределение прогибов w* (а) и напряжений s* (б) по верхней поверхности оболочки в поперечных сечениях: y = L; L / 2;0 ; ТрКЭ - сплошная линия, ПК ANSYS - штриховая линия Thus, ThrFE use when calculating SSS allows to save significantly CEM resources in comparison with PC ANSYS and SGFE that considerably expands MFE possibilities in multigrid simulation option. Conclusion. In this work the numerical computational method of elastic layered conical shells of various form and thickness at arbitrary static loading is offered. In this method Lagrange MGFE, at construction of which Lagrange approximations are applied, are used. Lagrange polynomials allow to design large size three-dimensional MGFE. Realization of MFE for conical shells multigrid discrete models demands several orders less ECM mem- ory than when using SGFE, and allows to make calcula- tion of SSS with the given small error for movements and stresses. The given examples show high efficiency of the proposed method of conical shells calculation using MGFE which provide small error of solutions and save ECM resources.

About the authors

G. I. Rastorguev

Novosibirsk State Technical University

20, Karl Marx Av., Novosibirsk, 630073, Russian Federation

A. N. Grishanov

Novosibirsk State Technical University

20, Karl Marx Av., Novosibirsk, 630073, Russian Federation

A. D. Matveev

Institute of Computational Modeling SB RAS

Email: mtv241@mail.ru
50/44, Akademgorodok, Krasnoyarsk, 660036, Russian Federation

References

Supplementary files