Here, we present the new method of approximate conformal mapping of the unit disk to a one-connected domain with smooth boundary without auxiliary constructions and iterations. The mapping function is a Taylor polynomial. The method is applicable to elasticity problems solution.

Conformal mappings play an important part in solution of elasticity theory problems if we apply to them complex variable theory. These investigations were started by Muskhelishvili [

Computer progress stimulated appearance of many numerical conformal mapping construction methods. Many of these methods were connected with the integral equation solutions. If we want to map a given simply connected domain to the disk, then we solve a linear integral equation either analytically or numerically (see, e.g., [

The approximate conformal mapping of the unit disk to the given domain construction method presented here has the following advantages: it does not use any auxiliary constructions (triangulation, circle packing) or additional conformal mappings (zipper algorithm), it does not use accessory solutions of boundary value problems (conjugate function method, Wegmann method), and it even does not use iterations as Wegmann method or Fornberg method where the preimages of the given-on-the-unit circle points move along the given curve [

The auxiliary function involved in the function inverse to the reparametrizing one can be found by integral equation solution. This solution is reduced to solution of the infinite system whose truncated form is regulated by two different polynomial coefficients calculation methods. If the system size is reasonable, then both formulas lead to close values of the desired coefficients. We present the example of nonconvex domain and construct the approximate conformal mapping of the unit disk onto this domain with the help of the boundary curve reparametrization.

The example of the unit disk mapping onto the hypotrochoid interior leads to solution of plane elasticity problems. With the help of the constructed approximate mapping functions, we analytically find the boundary shear stresses and draw the corresponding graphs.

We also give the example of one boundary value problem solution for the corresponding domain which is reduced to finite linear equation system solution according to results of Muskhelishvili [

The method was introduced and theoretically described in [

Consider a finite simply connected domain

If Fourier polynomial representation (

Assume now that representation (

In order to find this reparametrization

Let

We consider the factor

Note that

Equation (

Let the numbers

We search for the solution

Here,

Let us prove that there exists the number

We now take the number

Obviously, one can choose the value of

Consider the first summand on the right-hand side of the last inequality. This is the summand that is determined by the operator approximation. The vector norm of

We search for the integral equation solution in the form of the Fourier series:

Now the approximate solution of the integral equation reduces to solution of the linear equation system

The vectors

The block matrices of size

The numbers

So there is a way to omit the polynomial roots calculation.

The constructed function

This reparametrization

Existence of two different coefficient

We can also find the function

Now the coefficients of the desired polynomial

This way of the mapping function

(

The corresponding domain is not convex and even not starlike with respect to the origin. The image of the unit disk

Nonconvex domain and the image of the regular polar net.

The boundary point crowding in the example equals

(

Again we construct the function

The curve and the image of the regular polar net.

The following section gives the example of the approximate conformal mapping applicable to the plane elasticity theory.

Consider the complex parametric hypotrochoid equation:

We fix

The mappings of the unit disk onto the hypotrochoidal interior: (a) hypotrochoid, (b)

Now we construct the mapping of the unit disc onto this hypotrochoid interior applying the procedure described above. We solve (

Item (c) of Figure

We also find the smooth solution after solving (

We examine the boundary shear stresses for the twisted bar with the hypotrochoidal interior as the cross section. We base the solution of the torsion problem on relation (13) of [

The contour values of the tangent shear stress

Shear stresses found for different functions: (a) for

Each of the graphs shows that the minimal possible stress values happen in the points that correspond to the “hexagon vertices” and the maximal values are in the edges centers.

We consider the second plane boundary value problem for a domain

We solved this problem for the hypotrochoid interior in the case of

Hypotrochoid interior section and concentric inner circles displacements.

Consider now the 3D second basic elasticity theory problem for a bar

We construct as an example the element

Top and bottom section deformations.

Spline

The spline-interpolation solution is particularly effective in comparison with the FEM for the bodies with singular boundary points, for example, cones [

The conformal mapping method suggested in the paper is computationally efficient (of

The authors declare that they have no competing interests.

The work was partly supported by the Russian Government Program of Competitive Growth of Kazan Federal University.