Computational method for solving boundary value problems of mechanics deformable body using non-orthogonal functions

. In this paper a complete system of non-orthogonal functions was built on the basis of orthogonal sines and cosines. It is shown that the known orthogonal systems of functions are a degenerate case of non-orthogonal systems of functions. It has been proven that the continuous function can be approximated non-orthogonal functions in such a way that one selected non-orthogonal function will not included in this amount. The boundary value problem of the elasticity theory has been considered for an inhomogeneous plate. A new method for solving the boundary value problem is developed for the fourth-order equation with variable coe ﬃ cients. The proposed method is based on separating of the stress state of the plate, use of complete systems of non-orthogonal functions and a generalized quadratic form. Criteria have been established under which the constructed approximate solution coincides with the exact solution. This method has been adapted to solving a boundary value problem for high-order di ﬀ erential equations. The high accuracy of the method has been conﬁrmed by numerical calculations.


Introduction
Currently, both continuous [1] and piecewise continuous [2,3] systems of orthogonal functions are widely used. The use of modern computers allows us to develop new techniques for modeling functions and various processes in engineering. In [4], it is proposed to use non-orthogonal systems of functions to solve the boundary value problems of the elasticity theory. The development of computer techniques for non-orthogonal functions made it possible to propose a new analytical-numerical approach to solve the boundary value problems of the elasticity theory [5][6][7], as well as differential equations [8] using the least squares method [9,10]. The computational method of solving boundary value problems of the mechanics of a deformable solids with variable elastic characteristics is considered in [11], where methods with use of non-orthogonal functions were proposed.
The non-orthogonal systems of functions naturally appear when constructing eigenfunctions of boundary value problems for differential equations [12]. They are also used to solve the boundary value problems of the elasticity theory [4]. However, up to date computational methods based on the non-orthogonal systems are not yet well developed for functions modelling, optimization of various processes, solving boundary value problems of mathematical physics, etc.
The purpose of research is to develop a new computer method to solve the boundary value problems involving both partial differential equations and ordinary differential equations and to devise analytical and computer tools for operating over the basis of the orthogonal sinecosine families.

Approximation of continuous functions
Consider a complete non-orthogonal system of functions on the interval [l 1 , l 2 ]. In this work, for its construction, we use the system of orthogonal functions (sinus and cosine), which will be considered on a smaller segment [l 1 , l 2 ] than the interval of their orthogonality [A, B]. An approximation of a continuous function f (x) on the interval [l 1 , l 2 ] will be will be realized as a finite sum of the following form where c k are unknown coefficients, . It is not difficult to verify that the selected system of functions {ϕ k (x)} will be complete and orthogonal in the interval [A, B], and and complete but not orthogonal in the segment x ∈ [l 1 , l 2 ]. Increasing the value N without bound in (1) leads to the conversion of the right hand side of equation (1) into a row. The use of decomposition (1) by non-orthogonal functions allows us to satisfy the different values of the function at the edges of the segment f (l 1 ) f (l 2 ) without disrupting of the convergence conditions at points l j , j = 1, 2. It requires a smaller quantity of number of the sum of the row (1) during approximation of functions with a given accuracy.
We formulate the important difference between non-orthogonal and orthogonal systems of functions. Corollary. When satisfying the boundary conditions, depending on the physical nature of the problem being solved, we have an opportunity not to use one basic non-orthogonal function.
The coefficients c k are determined from the condition of the minimum functional, which characterizes the deviation between the approximation of the function (1) and its value [6,7].

Solution of two-dimensional boundary value problems of the mechanics of the deformable solid body
Consider a two-dimensional boundary problem of the elasticity theory for a plate with variables Young's modulus and Poisson's ratio [13]. The median surface of the plate occupies a rectangular region with a contour L on which the load is given. Stresses in the region Π are expressed through stresses function F(x, y) The stresses function (2) satisfies the partial differential equation [13] T is the Poisson's ratio. We assume that the elastic constant of material E, ν are even with respect to the coordinate y and twice continuously differentiable functions in the region Π. Let us consider in detail when a ≫ b and normal loads are given only on the transverse sides, so that the tangent loads in the angular points of the plate are zero. Consider even normal stresses, odd tangent stresses σ x (a j , y) = σ j (y), τ xy (a j , y) = τ j (y), τ xy (a j , ±b) = 0, j = 1, 2, where σ j (y) are even normal, τ j (y) are odd tangent loads, a 1 = 0, a 2 = a. For problem (4) we will find the basic stress state of plate Π [5] so only one stress component σ 0 x (x, y) = σ 0 is not zero. Thus, we separate of the stressstrain state (SSS) of the plate by the sum of the basic SSS (σ 0 = σ 0 x ) with one component of the forces, and the perturbed self-balanced state with zero basic vectors of forces and moments [5].
Given the condition a ≫ b the symmetric self-balanced problem will be divided into two tasks. Consider the first of these tasks in which only one side of the rectangle is loaded. We have wrote boundary conditions where σ 1 (y) = σ 1 (y) − σ 0 is self-balanced load. On two longitudinal sides of the plate, zero boundary conditions (7) are presented. The Saint-Venant's principle implies that the selfbalanced stress state of the plate will be rapidly decreased by removing from the transverse side x = 0. Therefore, the function of stresses that satisfies equations (3), (5)-(7) we can search with the symmetry of the stress state in such form: where a k , d k are unknown coefficients; N is natural number; ϕ k (x) = cos(kωy), ϕ k+N (x) = = sin(kωy), k = 1, N, ω = 0.9π/b. We substitute the function of stresses (8) in relations (2) and we obtain an obvious type of components stresses The expression of the stress function (8) has been chosen so that it had a part that coincides with the expression of the stress function for a homogeneous isotropic material [5].
We introduce the norm [14] in the space of two-dimensional of two-dimensional functions F(x, y)specified in the rectangular region Π It is known [14] that if the function F(x, y) is a continuous function and �F(x, y)� = 0, then F(x, y) = 0. Note that unknown coefficients a k , d k of stress function (8) must satisfy the equation (3) in the region Π, which will be present in the following form: where Equation (11) defines the continuous function in the region Π. We minimize the deviations of the function (11) from zero. To do this, we substitute expressions (12) and after using the norm (10) in the region Π we get a non-negative quadratic form where M = 4N, T k+2N = S k ; c k = a k , c k+2N = d k , k = 0, 2N are variables and B k j = = � a 0 � b 0 T k T j dx dy, k = 1, M are the coefficients of the quadratic form. The given method allows us to reduce of numerical solution of the equation (3) to an estimation of the minimum of the quadratic form (13).
By means of selecting the exponentially reduced presentation of the stress state, the conditions (6) will be approximately satisfied. It remains to satisfy the boundary conditions (5), (7). After substitution the stresses (9) in them they are reduced to a compact form where A 1,k = � 6y sin(kωy) + 6y 2 kω cos(kωy) − k 2 ω 2 sin(kωy) � , A 2,k = kω � 3y 2 sin(kωy) + kωy 3 cos(kωy) � , A 3,k = k 2 ω 2 b 3 sin(kωb) e −kωx , We denote the left-hand sides of equation (14) by To solve equations (14), we had used the analytical-numerical method developed in [5,6]. The method is based on the convergence of all expressions in the left parts of equations (14), (15) to the loads P m (γ m ). We have written these expressions An effective analytic and numerical methodology has been developed, which allowed us to simultaneously minimize all K expressions (16) in the norms L 2 [0, α m ] and quadratic

Convergence criteria for the constructed solution
Lemma [7]. The function Λ(N) is nonnegative and not increasing.
If we assume that Λ(N) = 0, then all separate terms that are included in the left part of the expression (17) will be zero. Consequently, all equations (13), (16) will be satisfied and the initial task is solved.
We assume that the proposed computational process is stable [6]. The following theorem is faithful. Theorem 2. If there exists N such that Λ(N) < ε 2 /4 for any ε > 0, then the stresses written through function F(x, y) = lim N→∞ F N (x, y) exactly satisfy the conditions (5), (7), as well as equation (3).
Proof. Consider the sequence of the small positive numbers ε N convergent to zero. This sequence corresponds to the natural number sequence N → ∞. Now due to Lemma and theorem 2 assumption there exist N and ε N ≤ ε for any ε > 0 so that the relation