Application of wavelets and conformal reflections to finding optimal scheme of fiber placement at 3d printing constructions from composition materials

The article is devoted to finding the optimal schemes of fiber placement at the production of constructions, reinforced with continuous fibers by 3D printing method. As the optimization of the objective function one of the criteria for the destruction of the composite was chosen. For the process acceleration of multiple solution of the system of partial differential equations describing the stress-strain state of the structure, a computational algorithm based on wavelets built through subdivision schemes is proposed. To set the local coordinate system, it is proposed to use analytical functions, which will be constructed using the well-known Dini and Cisotti formulas, just by specifying the direction of laying the fiber at the product boundary. The article also presents a lifting scheme (lifting scheme) allowing to construct biorthogonal wavelet systems with specified properties using some initial biorthogonal wavelet systems with filters.


Introduction
At the present time in the high-tech areas of industries composite materials (CM) are widely spread, which consist of reinforcing material and a binder. In the quality of reinforcing material, the carbon fibers are used, which have high specific strength. In this case, the mechanical properties of CM products depend on the direction of the fibers. 3D printing is a perspective technology for manufacturing structures of complex shapes by sequential placement of composite materials. With the usage of 3D printing, in principle, it is possible to obtain the structure with spatial reinforcement along the given paths. The total control over the placement of fibers during the printing (100% of the fibers are at the right direction) allows them to be stacked according to the required operating conditions. The geometry of the fiber placement is determined by the equations of CM mechanics themselves in the form of some (unknown) local orthogonal coordinate system [1]. To find the geometry of placement it is possible from these equations by solving them multiple times with different local coordinate systems. That is why it is necessary to work out rather fast algorithm for solving partial differential equations. To achieve this goal, it is proposed to use biorthogonal wavelets constructed by using subdivision schemes [2,3,4]. The choice of such scheme of decision is explained by the following: -firstly, when using the lifting scheme, you can control the properties of the wavelets themselves [5], -secondly, for a sufficiently accurate solution of the equations themselves, a small number of iterations is required.
To set the local coordinate system, it is proposed to use analytical functions that will be built at the usage of well-known Dini and Chizotti formulas just by specifying the direction of fiber laying at the product boundary [6]. Any of the CM fracture criteria can be selected as an optimization criterion for choosing fiber placement paths [7]. So, the goal is to develop methods and algorithms for finding the optimal fiber placement when 3D printing structures are made of composite materials reinforced with continuous fibers.

Experimental
Wavelets have a number of advantages over other basic functions. Firstly, the usage of the lifting scheme allows one to construct wavelets with the given properties: smoothness, compact support, symmetry, the required number of zero moments, vanishing on the boundary of the domain of functions corresponding to non-boundary mesh vertices [8]. Secondly, the high rate of decay of the wavelet coefficients, which allows, limiting themselves to a small number of terms in the expansion, to obtain sufficiently accurate approximations of the function. Thirdly, the presence of fast cascade algorithms for finding the coefficients of the wavelet expansion of the function. This section is devoted to the development of the technique for using biorthogonal wavelets in the approximate solution of partial differential equations.
Multi-scale analysis In describing the basis of wavelet analysis, including notations, we will follow the works [2][3]. Let it be ( , , ) X   -a measurable measure space [7]. We will consider the real space 2 L ( ) , ; The family of functions , , , , , are called biorthogonal wavelet systems. As , The sequences , , , , , , As 1 1, , , Consequently, From equalities (2) and (5) we obtain From the formula similar to (5), we obtain The formulas (6) are wavelet decomposition formulas or analysis formulas, and formula (7) is wavelet reduction or synthesis formula. Lifting scheme Lifting scheme allows to construct biorthogonal wavelet systems with the specified properties using some initial biorthogonal wavelet systems with filters  Note that you don't need to change the function , j k  , but raise , j k  . This mechanism is exactly the same and is called the dual lifting scheme [5,3]. It allows to improve the properties of the wavelet , j m  . At the dual lifting scheme, the new filters are determined by the formulas: Let us dwell briefly at the method for constructing of biorthogonal wavelets on triangulated spaces with the finite set of simplices, presented in the work [10].
Let it be ( , , ) T g X -triangulated space with a finite set of simplices [11].
-scaling functions and wavelets on Т. Let's define the scaling functions and wavelets on X by the following equalities: Then, if 2 : and , then [2] Application of wavelets to the approximate solution of partial differential equations One of the methods for the approximate solution of partial differential equations that we will use is the method of least squares. It is widely used in solving boundary value problems in mathematical physics. [12]. Let's consider the differential equation and The second used approximate method for solving goal (8) is the collocation method, in which it is required that the equation and boundary conditions are fulfilled at grid nodes (so-called collocation nodes).
In this work, the example was calculated using spline wavelets. Let's briefly dwell on their construction.
Let it be 2 j n of the required smoothness class, and put ( ) For example, the sequence 1 4 6 4 1 ( 0 0 ) , as it is known leads to cubic Вsplines [13]. Choosing the filter, we thereby constructed the scaling functions [14], which are determined by the scheme (7). Let it be e The scaling functions won't be changed. The tangent vectors to the curves along which the fibers are placed in 3D printing form a vector field r in X, which will be characterized by a complex number 1   2  1  1  1  2  2  2  1  2   r , where ( , ), ( , ) r ir r r x x r r x x = + = = . We will consider this field to be harmonic, i.e. solenoidal and potential [15]. Such field has no sources or vortices. Moreover, let it be X X  , where the area X will be considered simply connected, and the field is considered in this simply connected region. It means that expression 2 1 1 2 r dx r dx − + is the total differential of some function 2  , determined for X. This function is called the current function [16]. Moreover, the expression 1 1 2 2 r dx r dx + there is also the total differential of some function 1 1 2 ( , ) v x x , which is called the field potential [16]. The current function 2 1 2 ( , ) v x x and field potential 1 1 2 ( , ) v x x are the conjugate harmonic functions [16]. The current lines and the lines of equal potential form the orthogonal family. The analytical function is called the complex field potential [17]. Thus, any analytic function in the domain X also gives the scheme of fiber placement, and local curvilinear coordinate system in X X  . Let's agree the points 1 2 x ( ) T x x = depict at one complex plane, and the points at the other. Then the transformation 1 1 1 2 2 represent the transformation of some area X plane x at the set  plane v. Level line network 1

Conclusions
The theoretical foundations of mathematical modeling of the process of manufacturing structures from composite materials reinforced with continuous fibers by the method of 3D printing have been developed. To set the local coordinate system, analytical functions were used in the work, built using the well-known Dini and Cisotti formulas by specifying the direction of laying the fiber at the boundary of the product. As an optimization criterion for the choice of fiber placement trajectories, the criteria for the destruction of a structural material were chosen. The proposed techniques are implemented in a CAD / CAE system for constructing such structures, written using the Python © programming language.