A version of collocations and least residuals method (CLS) based on polynomial approximation of high degree (p - approach) was proposed and implemented. In rectangular domains collocation points are selected using the roots of Chebyshev polynomials and approximate solution is represented in the form of direct products of Chebyshev polynomials series. It was shown that the use of p - approach in the CLS method allows to obtain numerical solutions with high accuracy and to implement complex boundary conditions with no special techniques. The numerical method used to solve a problem of bending of laminated anisotropic rectangular plates within frameworks of classical laminated plate theory, first order shear deformation theory and Grigolyuk-Chulkov's broken line theory. Several specific example problems are solved, including fixed three-ply laminates with transversely isotropic layers under transverse uniform loading.
Introduction The collocation and least residuals method (CLS) is an efficient method for numerial solution of boundary value problems both for systems of ordinary and partial differential equations. It is based on the collocation method (CM) [1], with approximate solution is represented as a linear combination of basis functions in some functional space. To determine it unknown coefficients in CM residual of equations ( ) vanishes at given points (collocation points) ( ) = 0, { } – collocation points.
(
In CLS method the number of equations can exceed the number of unknown
coefficients in representation of the solution. The solutions of arising
overdetermined systems of linear algebraic equations (SLAE) are defined in the sense
of (
In this paper, an approximate solution is represented as a linear combination
of polynomials of high degrees (p - approach) that is typical for spectral methods.
This allows to obtain numerical solutions of high accuracy at low computational
cost. Term (
We will demonstrate the application of a method to solving problems of solid mechanics – bending of laminated anisotropic rectangular plates. On a practical level, for calculating the stress and displacement fields of such structures the theories of plates are used. They lead to a smaller computational efforts compared to three-dimentional elasticity formulation.
Boundary value problems arising in a plate theories have a number of features
that present difficulties for many well-known numerical methods. First,
governing equations of plate theories may contain derivatives of high orders. Second,
boundary conditions may be quite complicated, for example, in a form of linear
combination of functions and their higher order derivatives. Third, equations of
the plate theory may contain small parameters in the derivatives. These features
cause serious difficulties for widely used finite differences and finite element
methods. The use of both p - approach and term (
Formulation of Problem and Governing Equations Let us consider a static bending of laminate composed of 3 layers of constant thickness (Fig. 1). Layers are transversely isotropic with material symmetry axis in the plate’s plane. Layers orientation scheme is 1 = 3 = 0, 2 = / 2, where is an angle measured counterclockwise from the coordinate axis to the -th layer material symmetry axis. x b z a q(x,y) h y z h 2 3 2 1 z2 z 3 z1 x mensions in the directions , , respectively, — -th layer lower surface coordinate,
Engineering constants of transversely isotropic material are [5] = 0.2 Mpsi, = 2 1 (︂ 0 + −
= , − Here – strains; 0(, ), 0(, ), 0(, ) – central plane displacements.
Constitutive equations for -th layer are expressed by ⎛ ⎞
⎜ ⎟⎠ = ⎜ 12 ⎝ ⎝ 22 26⎠⎟ ⎜⎝ ⎟ ,
⎠ 16 Coefficients express in terms of the engineering constant as follows: It is convenient to define the following quantities 11 = 12 =
1 − 1 − , 22 =
1 −
, ,
66 = .
3 ∫︁ +1 ︁∑ =
︁∑ = =1 3 ∫︁ +1 =1
3 =1 1 ∑︁ 3
3 =1 ︁∑ =
( +1 − ),
2 =
( 3+1 − 3),
(
We consider two kinds of boundary conditions: – clamped – simply-supported normal about central surface: (, ) and (, ). The strains are obtained by = 0 +
, = ︂( 0 + 0︂) = 0 +
, + ︂( +
︂) , = 0 + ,
= 0 + , = 0.
Constitutive equations of FSDT are obtained by adding to (
44 45︃)(︃ 45 55 )︃ , , 2 = ︂( cos −sin cos , sin )︂ where stiffness coefficients are express by engineering constants
44 = , 55 = . Governing equations for FSDT can be written as erties of each layer are considered separately. For this purpose, in each layer the rotations of transverse normal about central surface (, ) and (, ) are defined. It can be assumed that the GCT is a generalization of the FSDT that takes into account transverse shear stresses in each layer separately.
Expressions for geometrical equations in GCT have the form where
︂) 0 ︁∑ = = 0 + 0 + + = 3 =1 + ︁∑ ︁∑ 3 =1 3 =1 ︃( + ( − −1)
, + ( − −1)
, + )︃
+ ( − −1) , = + ,
= 0, + )︃
, Boundary conditions for clamped edges in GCT are ( = 1, 2, 3) 0 = 0, 0 = 0, 0 = 0, = 0, = 0,
Because of awkward form, governing equations of GCT are not presented. Detailed description of the theory can be found in [7]. It is necessary to note, that plate theories are approximations to elasticity theory and bring about their own errors which are important to estimate. 3
hp - Version of CLS Method Implementation aspects of CLS method are similar to the CM. Consider a general boundary value problem for a linear elliptic system in a rectangular domain = [0, ] × [0, ]:
(, ) = (, ), (, ) = (, ), (, ) ∈ ,
Let us define a grid with non-overlapping rectangular cells ( = 1, . . . ). In each cell we introduce local variables ( 1 , 2 ), which are associated with global variables (, ) in the Cartesian coordinate system by 1 = − * 1 , 2 = − * 2 , further. where 2 1 , 2 2 — sizes of cell in and directions, ( * , * ) — the coordinates of the cell centers. Local variables are varying in canonical interval 1 ∈ [−1, 1], 2 ∈ [−1, 1]. The upper index , that indicates the cell’s number, will be omitted
In this version of CLS method approximate solution in the cell is represented in form of direct product of single variable basis functions : ( 1, 2) = ∑︁ 1−1 ︁∑ 2−1 1=0 2=0 1 2 1 ( 1) 2 ( 2).
(
In previous paper [8] we used cardinal functions in Lagrange-like form 1 ( 1) = ∏︁ 1−1 ̸= 1
1 − ( 1)
=0 ( 1) 1 − ( 1) , 2 ( 2) = ∏︁ 2−1 =0 ̸= 2
2 − ( 2)
( 2) 2 − ( 2)
,
(
︀) are local coordinates of collocation points. But in
practice polynomials (
To determine the unknown coefficients in representation (
– solution defined in neighbour cell .
Matching conditions
usually require the continuity of the solutions and the necessary number of its derivatives along the normal to the boundary of the cell.
In this version of CLS method the local coordinates of collocation points are roots of Chebyshev polynomials ( 1 = 1, . . . , 1, 2 = 1, . . . , 2) ︀( ( 1) 1 , ( 2) 2
︀) = ( 11 , 22 ), spectively. By the same way we define (︀ 1 boundaries ( 1 = 1, . . . , 1, 2 = 1, . . . , 2) where 11 and 22 – roots of Chebyshev polynomials of 1 and 2 degree re, 2 ︀) and ( 1 , 2 ) on cell (−1, 22 ), (1, 22 ), ( 11 , −1), ( 11 , 1).
Thus, for 1 2 unknown coefficients in cell we use 1 2 collocation
equations appended by equations on cell boundary. Thus, in hp - version of CLS
method corresponding SLAE becomes overdetermined. In this version of method
approximate solution form does not satisfy to boundary and matching conditions
identically, so residual ( ) (
To solve the overdetermined linear systems in the least squares sense (
factorization of its matrix, implemented by Householder method. In the case of large linear systems we use domain decomposition method [9]. This allows to reduce the solution of the problem in whole region to iterative process through subdomains with computational complexity is much smaller than the original problem for the region. For linear systems in a subdomain we use Householder method again. In this case special matching conditions between subdomains are used. For example, the continuity of function and its first derivative at the boundary of the cell can be written as + 1 = + 1 , where is solution in the cell at the current iteration; — solution in the neighbor cell;
— the outer normal to the boundary . For plate theories, that may contain derivatives up to the 4th order, we can additionally require the continuity of a linear combination of the second and third derivatives of the approximate solution:
The choice of weights 1, 2 can affect the properties of the numerical solutions and speed of convergence of the iterative process. 4
Numerical Experiments In all numerical experiments a single cell that coincides with the entire domain is used. Further we will consider only square plates = = 1 m.
To demonstrate the capabilities of hp - version of CLS method let us consider
the problem with the known exact solution. The last equation of the system (
Consider 3-ply simply-supported square laminate under uniform transverse
load 0 with stiffness coefficients defined by (
Deflections in the center of the plate are calculated by hp – version of CLS method are shown in Table 1.
Table 1 demonstrates high accuracy of numerical results, obtained by hp –
version of CLS method even for the differential equation with derivatives of
4-th order of the unknown functions. Thus, hp – version of CLS method has no
difficulties when working with differential equations containing derivatives of
high orders like (
Consider another formulation of the problem. Let us use the free edge conditions on the one of the edges ( = 1) in the previous problem.
These boundary conditions by the use of term (
Now let us consider the bending of clamped 3-ply laminates with different relative thickness ℎ/ , as described in Section 2. Stress and displacement fields calculation for such plates will be carried out within framework of three theories described above. Brief analysis of solution shows that prevails in the stress state of outer layers (Fig. 3). And maximum absolute values are observed in the vicinity of the clamped edges ( = 0, 1). Similar conclusion are true for the middle layer of the plate and component , which is associated with an orientation of transversely isotropic material.
Further we will use the following normalized quantities ¯ = 0 2 ,
, .
out in the framework of the CLPT, FSDT and GCT plate theories. Sign (%) is used for relative percentage deviation from GCT.
mum absolute values. We assume GCT as the most accurate among considered theory, because its hypothesis is the most suitable in given structure [8, 7]. Therefore, we will treat it as a reference.
GCT 0.975 Despite the simplicity, the CLPT may be used for very thin laminated plates. If the 5% deviations from GCL are admissible, CLPT can be applied for ℎ/ < 0.02 case. More accurate results are obtained within the framework of FSDT. It can be used for plates with aspect ratio ℎ/ < 0.05. For thicker plates GCT theory differs significantly.
In presented calculations all functions are approximated with 1 = 2 = 17 that allowed to get three guaranteed digits for the maximum deflection. In this case, the CLPT is required to determine 17 · 17 · 3 = 867 unknown coefficients, FSDT – 17 · 17 · 5 = 1445, and GCT – 2601. It is interesting that for ℎ/ < 0.02 CLPT results do not differ significantly from those of FSDT. In particular, it means that for plates with ℎ/ < 0.02 CLPT is preferable, because of lower computational efforts.
Fig. 4 shows the distribution of stresses and along the coordinate (thickness) for different points (, ): in the center of the plate and on the border where they reach the maximum values. The absolute maximum stresses values are observed at the outer sutfaces of layers. And the absolute values at the edges of the plate exceed values at the center of the plate a few times, that is true for both the stress tensor components.
Presented numerical results shows that hp - version of CLS method can be
successfully applied to problems of mechanics of laminates anisotropic
rectangular plates within framework the various plates theories. Term (