CC BY
30
UDC 539.3
Siberian Journal of Science and Technology. 2018, Vol. 19, No. 1, P. 27-36
MULTIGRID FINITE ELEMENTS IN THE CALCULATIONS OF MULTILAYER CYLINDRICAL SHELLS
А. D. Matveev1*, A. N. Grishanov2
institute of Computational Modeling 50/44, Akademgorodok, Krasnoyarsk, 660036, Russian Federation Novosibirsk State Technical University 20, Karl Marx Av., Novosibirsk, 630073, Russian Federation E-mail: [email protected]
An effective numerical method for calculating linearly elastic multilayer cylindrical shells under static loading implemented on the basis of Finite Element Method (FEM) procedures using the multilayer curved Lagrangian multi-grid finite elements (MFE) of the shell type was proposed. Such shells are widely used in rocket-space and aircraft engineering. MFE are developed in local Cartesian coordinate systems based on small (basic) shell partitions that take into account their heterogeneous structure, irregular shape, combined loading and fixing. The stress strained state (SSS) in the MFE was described by the equations of the three-dimensional elasticity problem without using the additional kinematical and static hypotheses, which allow one to use MFE for the shells of various thicknesses to be calculated. The procedure of constructing the Langrage polynomials in local curvilinear coordinate systems used to develop the shell MFE is presented. The displacements in the MFE were approximated by the power and Lagrange polynomials of different orders. When constructing a n -grid finite element (FE), n > 2, n-nested grids were used. The fine grid was generated by the basic partition of the MFE; the other (coarse) grids were used to reduce its dimension. According to the method, the nodes of the coarse MFE grids are located on the common boundaries of the different modular layers of the shell. The proposed law of the expansion in the number of discrete models using MFE with a constant thickness, multiple of the shell thickness, provides a uniform and rapid convergence of approximate solutions, allowing one to frame solutions with a small error. Multigrid discrete models have 103...106 times less unknown MFE than the basic ones. The implementation of the MFE for multigrid models requires 104...107 times less computer storage space than for the reference models, which allows one using the proposed method to calculate some large shells. An example of calculating a multilayer cylindrical local loading shell of irregular shape was given. In the calculation, three-grid shell -type FE, developed on the basis of the reference models having from 2 million to 3.7 billion of the nodal MFE unknowns were used. To study the approximate solution convergence and error, a well-known numerical method was used.
Keywords: elasticity, cylindrical shells, composites, multigrid finite elements of shell type, Lagrange polynomials, small error.
Сибирский журнал науки и технологий. 2018. Т. 19, № 1. С. 27-36
МНОГОСЕТОЧНЫЕ КОНЕЧНЫЕ ЭЛЕМЕНТЫ В РАСЧЕТАХ МНОГОСЛОЙНЫХ ЦИЛИНДРИЧЕСКИХ ОБОЛОЧЕК
А. Д. Матвеев1 , А. Н. Гришанов2
1Институт вычислительного моделирования СО РАН Российская Федерация, 660036, г. Красноярск, Академгородок, 50/44
2Новосибирский государственный технический университет Российская Федерация, 630073, г. Новосибирск, просп. К. Маркса, 20 E-mail: [email protected]
Предложен эффективный численный метод расчета линейно-упругих многослойных цилиндрических оболочек при статическом нагружении с применением многослойных криволинейных лагранжевых многосеточных конечных элементов (МнКЭ) оболочечного типа. Такие оболочки широко используются в ракетно-космической и авиационной технике. МнКЭ проектируются в локальных декартовых системах координат на основе мелких (базовых) разбиений оболочек, которые учитывают их неоднородную структуру, сложную форму, сложное нагружение и закрепление. Напряженное деформированное состояние в МнКЭ описывается уравнениями трехмерной задачи теории упругости без использования дополнительных кинематических и статических гипотез, что позволяет применять МнКЭ для расчета многослойных оболочек различной толщины. Показана процедура построения в локальных криволинейных системах координат полиномов Лагранжа, которые применяются при проектировании оболочечных МнКЭ. Перемещения в МнКЭ аппроксимируются степенными
и лагранжевыми полиномами различных порядков. При построении п -сеточного конечного элемента (КЭ), п > 2, используют п вложенных сеток. Мелкая сетка порождена базовым разбиением МнКЭ, остальные п - 1 (крупные) сетки применяются для понижения его размерности. В предлагаемом методе узлы крупных сеток МнКЭ расположены на общих границах разномодульных слоев оболочки. Закон измельчения дискретных моделей, в которых используются МнКЭ с постоянной толщиной, кратной толщине оболочки, порождает равномерную и быструю сходимость приближенных решений, что дает возможность строить решения с малой погрешностью. Многосеточные дискретные модели имеют в раз меньше узловых неизвестных, чем
базовые. Реализация метода конечных элементов (МКЭ) для многосеточных моделей требует в 104-107 раз меньше объема памяти ЭВМ, чем для базовых, что позволяет использовать предложенный метод для расчета оболочек больших размеров. В приведенном расчете многослойной цилиндрической оболочки сложной формы, имеющей локальное нагружение, используются оболочечные трехсеточные КЭ, построенные на базовых моделях, которые имеют от 2 миллионов до 3,7 миллиарда неизвестных МКЭ. Для анализа сходимости приближенных решений используется известный численный метод.
Ключевые слова: упругость, цилиндрические оболочки, композиты, многосеточные конечные элементы оболочечного типа, полиномы Лагранжа, малая погрешность.
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, constructing 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 difficulties are largely due to the fact that to reduce the order of equations in the theory of shells, hypotheses are introduced, 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 cylindrical 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 algebraic 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 considered in a three-dimensional formulation, but which lead to SLAE of a low order in compliance with the permissible 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 carried 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 > 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 constant 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. Calculations show that the arrangement of nodes of large MFE grids on the common boundaries of different-modular shell layers provides homogenous and fast convergence 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 complex 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 multilayer 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 convergence 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 implementation 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 performed by constructing a sequence of approximate solutions for a similar test problem using MFE, which are used in solving the original problem. An example of calculating 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 single-grid FE, which form a basic discrete model of the shell, is briefly considered as the example of FE Ve of the 1st order, located in the local Cartesian coordinate system O1 x1 y1 z1 (fig. 1). For FE Ve designations are given:
hex x hey x hi - characteristic sizes, z1O1 y - a symmetry plane, cd - an axis of a shell, R (R-f) - radius of curvature of the bottom (top) surface, hf - thickness, hf -length, hf = aeRf, ae - an opening angle. The shape of the FE Ve is a straight prism with height hf. Deformation
of FE Ve is described by the equations of the three-dimensional problem of the theory of elasticity [1], shown in coordinate system O1x1y1 z1. Using a first order polynomial (in the coordinate system O1 x1 y1 z1), for FE Ve
we define the stiffness matrix [ Kj J and the nodal force vector Pf with formulas [1; 2]
[K]1 = J [Be]T[De][Be]dV,
Ve
P = J [N ]T FedV + J [Ne ]T qedS,
(1)
Ve Sf
where [Be ], [De ] are the matrix of differentiation and modules of elasticity of the FE Ve; Fe, qe are the volume and surface forces vectors FE Ve; [Ne ] is the matrix of shape functions; Ve , Se are the area and the surface of the FE Ve .
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 geometrically similar to the form of FE Ve (fig. 1), are 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 polynomials is considered with the example of a three-layer TGFE Va of the 3rd order with its thickness equal to h that is 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), its dimensions are ha x hay x h, h - thickness, hay - length.
Suppose that the bonds between the components of the inhomogeneous structure of TGFE are ideal. Basic partitioning of Ra TGFE, which consists of a homogeneous curvilinear FE Ve of the 1st order (fig. 1), takes into account in TGFE inhomogeneous structure, a complex type of loading and fastening, and generates a small curvilinear grid ha, e = 1, ...,M, M is the total number of FE Ve.
On the grid ha we define the large curvilinear grid Ha c ha, TGFE, the nodes of this grid are marked with dots, 64 nodes in fig. 2. Note that the nodes of the large grid Ha lie on the common boundaries of different-modular layers TGFE (fig. 2), in general they have different 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 sI 5 e, FE Ve, which correspond to the local Cartesian coordinate systems O1 x1 y1 z1 and O2 x2 y2 z2
«e = [Te ]8e
(2)
where [Te ] is a square matrix of rotations [2], e = 1, ..., M.
Fig. 1. Single-grid FE Ve Рис. 1. Односеточный КЭ Ve
Note that the continuity of displacements is violated on the curvilinear boundaries of the FE Ve (fig. 1). However, as it's known [21], the implementation of continuous 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 Va Рис. 2. Трехслойный ДвКЭ VQ
We consider the construction of Lagrange polynomials in the local curvilinear coordinate system O2£qq on
a large grid Ha (fig. 2). Suppose that the node P(i, j,k) of grid Ha (dimensions n1 x n2 x n3 ) has coordinates £, -q j, Qk, in fig. 2 i = j = 3 , k = 4 . Note that y2 = q for small opening angles aa, TGFE we can see that x2 «£, z2 « Q . We have
x2 =£ y2 =- z2 =C. (3)
The base function NiJ-k for a node P(i, j, k) in the Cartesian coordinate system O2x2y2z2 using Lagrange polynomials Li (x2), Lj (y2), Lk (z2) [2] is written in the form of
Nijk (X2 , y2 , z2) = Li (X2 )Lj (y2 )Lk (z2 ),
n1
L (X2) = П
X2 x2,n
n=1,n^i X2,i X2,n
Lk (Z2) = П
"2
Lj (У2) = П
У " У2.,
n=1, n^ j У 2, j - У2
(4)
2 2,n z и — Z",
L (a) = П
a — an
n2
Lj (л) = П
Л—Лп
n=1,n^j Л j Лп
Lk(C)= П
C—Cn
(5)
П = £ (2 [K ] Ö* — ÖeP )
(7)
Using small partitions , the functional (7) has a high dimension and generates a multinodal FE with a large number of nodal unknowns, which is not effective for practice. To reduce the dimension of the functional (7), we use the following procedure. Using (6), the vector of nodal displacements be FE Ve is shown through the vector of nodal displacements 8a of large grid Ha TGFE Va
Ö = [Л" ]8e
(8)
n=1,k z2,k z2,n
where x2 i, y2j-, z2k are the coordinates of the node P(i, j, k) in the coordinate system O2x2y2z2.
For a point with a coordinate £ lying on the cylindrical surface of the radius R , we have £ = aR , a is the angle for the coordinate £ , fig. 3. Considering (3) the ratio of the form £ = aR, £ = aiR in (4), we obtain Njk (a, q, Q = Lt (a) Lj (q) Lk (Q) , where Lt (a), Lj (q) , Lk (Q) are the Lagrange polynomials, having the form
n=1,n^k Qk Qn
It is convenient to use Lagrange polynomials (5) in calculations. Displacement functions ua, va, wa TGFE, constructed on the grid H a using Lagrange polynomials (5), are presented in the form of
n0 n0 n0 Ua = £ Nq, Va = £ Npq; , Wa = £ Npqpw , (6) p=1 p=1 p=1
where qU , qV;, qW, Np are displacements and shape function of the p node of grid Ha, n0 = n1n2n3, in the present case n0 = 64 (fig. 2).
Using (1), (2), the stiffness matrix [Ke ] and the nodal forces vector Pe of FE Ve in the coordinate system O2 x2y2z2, we present [Kg] = [Te]T[Ke][Te], Ve = [Te]TPj [1]. The functional of the full potential energy na of the basic partition of the Ra TGFE Va can be written in the form of
where [Aae ] is a rectangular matrix e = 1,...,M .
Substituting (8) in (7) and following the principle of the minimum of total potential energy for TGFE Va, 3na (6a)/ d6a = 0 we obtain a ratio [ Ka ]6a = Fa corresponding to the equilibrium state of TGFE Va, where
M M
[ Ka ] =2 A ]T [ Ke ]A ], Fa = £ A ]T Pe . (9)
e=1 e=1
The matrix [Ka ] is called the stiffness matrix, Fa is nodal forces vector of TGFE Va . Note that the functions ua, va, wa are used only to reduce the dimension of the functional (7), the large grid Ha determines the dimension of the TGFE Va , which is less than the dimension of the base partition Ra .
Note 1. By virtue of (8) the dimension of the vector 6a (i. e. the dimension of the TGFE Va) does not depend on the M which is the total number of FE Ve constituting the TGFE Va . Consequently, it is possible to use arbitrarily small base partitions Ra , which allows to take into account the heterogeneous and micro-homogeneous structure of the TGFE Va.
Note 2. In formula (9), matrices [Ke ], Pe, [Aae ] are constructed taking into account the curvilinear form of the base FE Ve (see formula (1)), which represent the region TGFE Va geometrically accurately. Consequently, the matrices [Ka ], Fa are also determined taking into account the curvilinear form of the TGFE Va .
Note 3. The determination of the stresses in TGFE Va can be shown as follows. Let the vector 6a be found. With the help of the formulas (8), (2) we find vectors 6e, 8g nodal displacements of FE Ve (e = 1, . ..,M ) respectively, in coordinate systems O2x2y2z2 and O1x1 y z1. Using vector we count the tension in the FE Ve with
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. Q ). The calculations show that the location of the nodes of the large grid Ha TGFE at the boundaries of heterogeneous layers provides a homogenous and rapid convergence of sequences of approximate solutions.
The procedures of constructing composite Lagrangian TGFE of «-order, geometrically similar to TGFE Va (fig. 2), with the application of Lagrange polynomials of « -order, are similar to the procedure of § 2.
Calculations show that by increasing the dimensions of the basic partitions of TGFE (i. e., by increasing the number M), the time spent on the construction of matrices [Ka ] h Fa and formulas (9) significantly increase.
In this case, it is advisable to apply 3-grid finite elements, for the construction of which less time is required and which generate the discrete shell model of lower dimension than TGFE.
3. Multilayer curvilinear Lagrangian three-grid FE.
The procedure of constructing curvilinear three-grid FE (ThGFE) with the use of Lagrange polynomials is considered by the example of a six-layer ThGFE Vb of
the 6-th order with its thickness hZ, , that is used in the calculation of 6-layer shells with thickness h, where h = hZ,. In the calculation of m-layer shell m-layer Lagrangian ThGFE of m-order thickness are used. ThGFE Vb with the size hX, x hby x h, is located in the local Cartesian coordinate system O3x3y3z3 (fig. 3).
Fig. 3. Six-Layer, ThGFE Vb
Рис. 3. Шестислойный ТрКЭ Vb
O2x2y2z2 and O3x3y3z3, n = 1, ..., N respectively. According to the FEM [1] we define the following formula: 5an = [T«a ] q« , where [T«a ] is the rotations matrix [2],
[M«a ] = [T«a ]T [K«a ][T«a ], P«a = [T«a ]T F«a. Taking into account these relations, the total potential energy of the ny ThGFE Vb, i. e. the partition of Rb, is presented in the form of
nb = £ |j (q« )T [M«a ] q«-(q a )T P«a ]. (10)
Functions of the displacements up , vp , wp ThGFE Vb on the large grid Hb, using Lagrange polynomials are presented in the form of
n0
n0
n0
= Z Nq , = Z Nqp, ; = Z Nq;, (11)
p=1
p=1
p=1
where qU, qp, qW, Np are displacements and shape function of the p node of grid Hb , «0 = «j«2«3, in this case «0 = 112 (fig. 3).
To reduce the dimension of the functional (10) we use functions (11). Let's denote: 5b is the vector of nodal displacements of a large grid Hb . Expressing the nodal displacements of vector qa« TGFE V«a through the nodal displacement of vector 5b of the grid Hb ThGFE Vb, we can see the equality
qa =[ Al ] 5b, (12)
where [A«b] is a rectangular matrix, « = 1, ...,N.
Using (12) in (10) and minimizing functional nb in displacement of 5b, we obtain the ratio for the ThGFE Vb [Kb ]5b = Fb that corresponds to its equilibrium state, where
[Kb] = £[A]T[M«a][A«], Fb = £[A«]TP«a . (13)
The area of ThGFE consists of N curved 6-ply TGFE Va with thickness h, « = 1, ...,N that geometrically accurately represent the area of ThGFE. TGFE V« make the partition Rb . The large grids Ha TGFE form a small grid hb ThGFE. On the grid hb we define large grid of Hb c hb ThGFE. The nodes of the large grid Hb marked
with points (112 nodes) lie on the common boundaries of different-modular layers of ThGFE (fig. 3).
Suppose that the axis O2 y2 of ThGFE (fig. 2) is parallel to the axis O3y3 (fig. 3). Suppose that 5an , qan are the vectors of nodal displacements, [K«], [Ma ] are the stiffness matrices and F«a , P«a are the vectors of nodal forces TGFE V«a responsible for the coordinate systems
The matrix [ Kb ] will be called the stiffness matrix, Fb is the vector of nodal forces ThGFE Vb. Note that the large grid Hb determines the dimension of the ThGFE Vb , which is less than the partition dimension Rb consisting of the TGFE Va .
Note 5. By virtue of (12) the dimension of the vector 5b (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 a TGFE V« and, consequently, into single-grid FE Ve (see § 2) can be arbitrarily small, which allows to describe with arbitrarily small error the three-dimensional stress state in the ThGFE taking into account its inhomogeneous structure.
n=1
n=1
Note 6. Note that the number of layers of TGFE may be less than the number of layers of the shell. For example, constructing six-layered ThGFE you can use a three-layered TGFE (fig. 2) or two-layered TGFE. As calculations show, this leads to a decrease in time costs with a minor change in the error of the solution.
In the formula (13), matrices [Mna ], Pna, [A] are constructed taking into account the curvilinear form of TGFE Va (see § 2), which geometrically represent the area accurately, ThGFE Vb . Consequently, the matrices [Kb ], Fb are also determined taking into account the curvilinear form of the ThGFE Vb.
The procedure of determining stresses in the ThGFE Vb is similar to the procedure for determining stresses in the TGFE.
Using ThGFE, according to the procedure similar to § 3, we construct four-grid FE, and the k grid of FE, k > 4. Note that the k grid generate a discrete FE shell 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 h = 0.03 m, length 2 L = 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 intersection of the planes Oyz and y = L on the top surface of the shell. Shell layers are isotropic homogeneous bodies. The upper and lower layers have h /12 thickness, 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 a uniformly distributed tensile radial load q = 0.05 MPa
(fig. 4) on the outer surface of the shell 3L /4 < y < L with the opening angle a = n /2, which is symmetrical to the planes Oyz and y = L . In the area of the shell clamps there are cutouts symmetrical to the plane Oyz, the opening angle of each cut is equal to the n /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.
The basic discrete model Rn0 of the shell consists of a curved homogeneous single grid FE of the 1st order VT, geometrically similar to FE Ve (fig. 1). The model grid R° has a dimension of m1n x m2 x m3, where
mln = 324n +1, m2 = 324n +1, m3n = 12n +1, n = 1, ...,10,
m1n is the dimension of the circular coordinate; axis Oy, m3n - axis Oz . Characteristic sizes hXcn , FE Ven are defined by the following formulas
hL = K\i Kn = Ki'n,
hezn = hezl/n, n = 1, ...,10,
(14)
- the
,, h'„
(15)
where hexl, hey1, hez1 are characteristic dimensions of FE Ve1 of the 1st order corresponding to the discrete model r1 , where hexl = a1Re a1 = n /324, Re is the radius of the lower cylindrical surface FE Ve1 .
hey1 = L /324 ;
hezl = h/12,
Fig. 4. Left symmetric part of the shell V0 Рис. 4. Левая симметричная часть оболочки V0
On base models R°, n = 1, ...,10 we construct multi-grid discrete models Rn of shell V0 consisting of La-grangian shell ThGFE with sizes 81hXn x 81hyn x h where
h = 12nhem. For all basic discrete models, ThGFE have a fixed size coordinate z which is equal to the thickness of the shell h . ThGFE are constructed on the procedure shown in § 3 and consist of Lagrangian TGFE with dimensions 9h^n x 9heyn x h , according to the procedure shown in § 2.
The ThGFE uses Lagrange polynomials defined by the formulas (5), which have the third order of the polynomial by coordinates x, y, and the forth order by coordinate z , which corresponds to the number of layers in the thickness of the shell. As shown by numerical calculations, if the nodes of large grids Ha and Hb of two-grid and three-grid FE lie on the common boundaries of multi-modulus layers, discrete models Rn provide even and fast convergence of a sequence of finite element solutions.
The results of the calculations for discrete models Rn are given in tab. 1, where we see: wn , an are maximum
radial displacement and equivalent stress for the model Rn, n = 6, ...,10 . We can find the stress an with the 4th strength theory. As you know, using the maximum equivalent stress the factors of safety of structures are determined. We find the values San (%), Swn (%) with the
formulas
Sa,n(%) = 100 % .|Gn -an_ii/an,
8w,n(%) = 100 % . | Wn - wn-i | /Wn , n = 2, ...,10 .
The nature of changes in values Swn (%), san (%) (tab. 1) shows rapid convergence of the equivalent stresses an and displacements wn. Since the values for the model R10 are small, 8w10 = 0.00116179, 8a10 = 0.00719947 it can be considered from the point of view of engineering practice that the displacement of w10 = 30.289362 mm and a10 = 31.371908 MPa are made with low error, i. e., w10, a10 are little different from the exact (see § 5).
The dimension of the underlying discrete model R100 is 3722110998 (more than 3.7 billion), the width of the tape of the system equations (SE) FEM is 1176610 (over 1.1 million). Multigrid model R10 has 203090 nodal
unknowns, the width of the tape SE FEM is equal to 5445. Application of the FEM for the multigrid model R10 requires 3960366 (approximately 3.96 million) less times than the amount of computer memory of the base model R100.
5. The study of the convergence of approximate solutions. To study the convergence of approximate solutions constructed using the new MFE, we use the following numerical method, the brief essence of which is shown below. With the kind of new MFE that are used in the solution of the original problem (see § 4), the similar
(test) problem with known exact solution u0 is solved. Suppose that || u0 - uh || ^ 0 when h ^ 0, where uh 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 shell V1 as a test problem, which is located in the Cartesian coordinate system Oxyz , to have the same geometric dimensions, fastening conditions and elastic modules as the shell V0 in § 4. However, the shell V1 has no cutouts. When 3L /4 < y < 5L /4 the radial tensile uniform load of p = 0.1 MPa acts on the outer surface of the 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 hm ^ 0 hm is the characteristic size of the standard FE) converge to the exact solution. Calculations are carried out for discrete models Qn , n = 1, ...,14 , shell V1. The results of calculations are given in tab. 2 for models Qn where, n = 7, ...,14, w°, an are the deflection and equivalent voltage at the point A (fig. 4), dimensions of models Qn are given in the plane Oyz. The parameters
of §w n (%), §a n (%) are determined by the formulas
(%) = 100 % •|w0 - w^i/w0, 8° (%) = 100 % • | a°„ -a^-i | , n = 2, ...,14.
(16)
Displacements wn and equivalent stresses an for models Rn
Table 1
Rn R6 R7 r8 R9 R10
w n 30.032632 30.136577 30.205840 30.254172 30.289362
8w,n (%) 0.550568 0.344913 0.229303 0.159753 0.116179
30.074687 30.544130 30.881125 31.146047 31.371908
8°,n (%) 2.374768 1.536934 1.091265 0.850580 0.719947
Table 2
Displacements w° and equivalent stresses an for models Qn
N Dimensions of models w°n-103, м 8W,n (%) стП , MPa 8°,n (%)
7 2269 x 43 2.24419205 0.0001359 21.9642981 0.0006719
8 2593x 49 2.24419400 0.0000868 21.9641839 0.0005199
9 2917x 55 2.24419529 0.0000574 21.9640928 0.0004147
10 3241x 61 2.24419632 0.0000458 21.9640199 0.0003319
11 3565x 67 2.24419706 0.0000329 21,.9639594 0.0002754
12 3889x 73 2.24419754 0.0000213 21.9639074 0.0002367
End of table 2
N Dimensions of models W0-103, м sW,„ (%) , MPa s°,B (%)
13 4213x 79 2.24419797 0.0000147 21.9638637 0.0001989
14 4537x 85 2.24419832 0.0000155 21.9638261 0.0001711
Table 3
Displacements wp and stresses ap for models Rn
n -103, м sw,n (%) <5pn , MPa 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 8W n (%), n (%) shows the rapid convergence of stresses a0 and displacements wn to the exact solution w0, a0 of the axisymmet-ric problem [1]. As the sizes, 8W,14 = 0.000000155 8a,14 = 0.000001711 are sufficiently small, the displacement of w04 = 2.24419832 10 3 m and the equivalent stress a04 = = 21.9638261 MPa can be considered as the exact solution, i. e. we believe w0 = w°4, a0 = a04.
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 given in the tab. 3, where, wp , ap is the deflection and equivalent stress at the point A for a multigrid discrete model Rn , n = 7, ...,14. The parameters 8Wn (%),
8a n (%) are determined by formulas similar to formulas
(16). The nature of the change in values 8Wn (%), 8a n (%) demonstrates the rapid convergence of stresses anp and displacements wnp to the limit values w0p , a0p . The errors for displacement w1p4 and stress af4 8w (%) = 100 % • | w04 - wp4 | /w04 , 8a (%) = 100 % x X | a04 -af4 | /a04, respectively, are equal to 0.00062828 % 0.0151749 %. In tab. 2, 3 values w04, a04, wf4, af4, are marked in bold. From the point of view of engineering practice, because of the smallness of the values 8w (%), 8a (%), we can assume that w0p = w0, ap = a0. Then we can conclude that the proposed ThGFE generate solutions ap , wp that in the limit (at n ^ro) tend (from the point
of view of engineering practice) to the exact solution of the axisymmetric problem.
The shell V0 considered in § 4 differs from the shell V considered in § 5 by the presence of cutouts and the method of applying the load, with full coincidence of the dimensions, boundary conditions and physical characteristics of the shells. In addition, when constructing sequences of approximate solutions for the initial and test problems, the same family of proposed ThGFE is used. Therefore, it can be assumed that the proposed shell ThGFE, which provide uniform convergence of approximate solutions for the test problem (for the shell V1), generate solutions wn , an that in the limit (at n ^-ro) 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.
Conclusion. In this work we propose a numerical method of calculation of multilayered linear elastic cylindrical thin and medium-thickness shells with the use of curvilinear Lagrangian shell type MFE. Application of the MFE for multigrid discrete shell models requires much less computer memory than the base models, which 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 proposed curvilinear Lagrangian shell MFE in the analysis of three-dimensional SSS multilayer shells.
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© Matveev А. D., Grishanov A. N., 2018
