Approximate Conformal Mappings and Elasticity Theory

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.


Introduction
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 [1].Muskhelishvili considered multiple solution methods of certain plane problems in the cited monograph.In particular, he gave the solution technique for basic plane elasticity theory problems and that for the rods torsion.In order to do this, the author made an extensive use of conformal mappings from the unit disk to a given domain.The easiest solution appears in the case of the polynomial mapping.
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., [2][3][4][5]).Note that if we want to map the disk to the given simply connected domain, then this problem solution turns out to be significantly harder and is traditionally reduced to nonlinear Theodorsen equation solution.The effective Wegmann method of this equation solution is based on iteration processes [6,7].
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 [8].We solve the integral Fredholm equation.This equation is well known and allows us to construct the conformal mapping of the given domain onto the unit disk [9].But we use this equation in order to find the monotone function that determines the necessary reparametrization of the given boundary.Moreover, our method provides us with the smooth solution in the form of Taylor polynomial.So it is possible to find the derivatives of the solutions.Also the method is connected with the domain boundary curve reparametrization described in [10].
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.

Approximate Conformal Mapping Construction by Means of the Boundary Reparametrization
The method was introduced and theoretically described in [10].Here, we give only the scheme of the function construction.Consider a finite simply connected domain  bounded by the smooth curve  = { = (),  ∈ [0, 2]} and (0) = (2).We trace the domain  counterclockwise along  as the parameter  increases.We only deal with the cases in which the boundary  representation is as follows: Note also that any smooth boundary may be approximated by a Fourier polynomial of this type.If Fourier polynomial representation (1) of the curve  possesses no summands with the negative degrees of   , then the function that maps the unit disk onto the domain  is immediately polynomial: Assume now that representation (1) contains nonzero coefficients  − ,  ∈ N.Then, it is possible to construct an approximate conformal mapping under reparametrization of (1) [10] leading to the coefficients  − ,  ∈ N, elimination.
Let () be the analytic function that gives the conformal mapping of  onto the unit domain so that (0) = 0.The necessary condition for the function log(()/) to be analytic in  is just as in [10]: where () =  0 + ∑ ∞ =1   cos  +   sin  = () − arg ().We consider the factor (  −  ) in the expression of ()− () in order to separate the improper VP integral in the last integral equation.Finally, the function () is the solution of the Fredholm integral equation of the second kind: where are continuous operators.
where () is 2 periodic and (, ) is 2 periodic with respect to both variables, can be reduced to solution of a finite linear system with error estimated by (1/ 2 ).Here,  is the finite linear system rank.
Proof.We search for the solution () in the form of Fourier series and present the kernel (, ) in the form of double Fourier series.Note that the restrictions on the kernel of integral equation and on the function () yield the following estimates of the kernel Fourier coefficients, | , | < /||  ||  , and of () Fourier coefficient |c  | < /|| 2 .We denote by  the vector of the function () Fourier coefficients and by  the vector of the function () Fourier coefficients.Now the integral equation reduces to the infinite linear system which can be presented as follows: Here,   is an  ×  matrix,  is an  × ∞ matrix,  is an ∞ ×  matrix,  is an ∞ × ∞ matrix, and   and  ∞ are the identity matrices of relative sizes.Each of the vectors  1 and  1 has  coordinates; the vectors  2 and  2 have the infinite number of coordinates.The Fourier coefficients of the smooth functions tend to zero as their numbers tend to infinity, so the coefficients of the matrices , , and  together with the coordinates of  2 decrease rapidly as  → ∞.Therefore, the matrix norm of  and the vector norm of  2 tend to zero as  → ∞.
Let us prove that there exists the number  ∈ N such that the matrix operator   +   is invertible ∀ >  since the limit for   integral operator  is compact and the operator  +  is invertible due to the lemma assumption.Note that we do not distinguish a finite dimensional vector and the Fourier polynomial with the corresponding finite coefficient set in our proof.Recall first that, due to chapter VI, paragraph 1 of [11], ‖ −   ‖ → 0 as  → ∞.Assume that ∀ ∈ N there exists   >  such that the spectrum of    contains −1.Then, there exists an infinite sequence (V   ) ∈N ⊂  2 such that ‖V   ‖ = 1 and    V   = −V   .Let us prove that then there should exist at least one limit point for the sequence {V   } ∈N .Since the operator  is compact, there exist both a subsequence Hence, V    → − 0 , ( → ∞).Note that since ‖V    ‖ = 1, ∀ ∈ N, the element  0 is nondegenerate.Let us show now that the relation  0 = − 0 holds true.Indeed, we have Hence, the spectrum of  contains −1, a contradiction with one of the assumptions.
We now take the number  so that ‖‖ < 1 and the matrix   +   possesses the inverse one.Now we have the relation Obviously, one can choose the value of  so large that ‖( ∞ + ) −1 ‖ 2,2 = (1/ 2 ) ≤ , where  < 1 is arbitrary small.Now we estimate the norm of the difference between the solution  1 and the solution x1 of the truncated system (  +   )x 1 =  1 : 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  2 can be estimated by D. Jackson's inequality by / 2 .The second summand (due to the series tail) also behaves no better than (1/ 2 ).So the error is (1/ 2 ).
We search for the integral equation solution in the form of the Fourier series: The approximate solution then is the finite sum here, the number  can be found from conditions that will be described later.Now the approximate solution of the integral equation reduces to solution of the linear equation system

Journal of Complex Analysis
The vectors on the right-hand side of the system consist of elements The block matrices of size  , , , then according to the Hilbert formula So there is a way to omit the polynomial roots calculation.The constructed function () must be monotone increasing.We construct the function inverse to it-the necessary reparametrization ()-with the help of FORTRAN spline function apparatus.According to Chapter 3 of [12], the algorithm computational complexity is ( 3 ).Assume now that we use -digit base  floating point arithmetic.Then, formula (3.5.2) of [12] gives us the calculation estimate: Here, ( α β ) is the computed solution.This reparametrization () allows us to construct the function (), || < 1, realizing the approximate conformal mapping on the given domain as a partial sum of the Taylor series: here, Existence of two different coefficient   and   forms for the mapping function () allows us to control the precision of the function () calculation by choosing the size  of the block matrices and the number of Fourier polynomial coefficients for this function ().
We can also find the function (),  ∈ [0, 2], by solving the integral equation with respect to its derivative: where Now the coefficients of the desired polynomial () must be calculated via the formulas This way of the mapping function () construction does not demand the inverse function ().So it is better than the initial one when we use differentiation technics.
The boundary point crowding in the example equals 75 to 1.
Again we construct the function  as a polynomial of degree  = 200.The array  then consists only of odd elements  2−1 ,  = 1, 5000.The array  elements simply vanish.The boundary point crowding in the example in Figure 2 equals 1.3 * 10 4 to 1.The approximation to the initial curve is (10 −3 ).Note that in this case we need a resulting polynomial of degree (); here,  is the greatest possible boundary point crowding.Also this example shows us that there is no strong connection between  and the border point crowding.The number  determines the precision with which we construct either the monotone function  or its derivative   that we need for reparametrization.
The following section gives the example of the approximate conformal mapping applicable to the plane elasticity theory.

The Example of Conformal Mapping onto the Interior of the Hypotrochoid Close to the Hexagon with the Smoothened Angles
Consider the complex parametric hypotrochoid equation: We fix  = 0.05; item (a) of Figure 3 shows this hypotrochoid.
Now we construct the mapping of the unit disc onto this hypotrochoid interior applying the procedure described above.We solve (4) and find the polynomial mapping function.The mapping of the unit circle relatively well visually approximates the boundary curve of the hypotrochoid even for 19 first coefficients ( 19 ()):  19 () = 0.985399 + 0.0441055 7 + 0.0116172 13 + 0.00429093 19 .

(29)
The corresponding domain is given by item (b) of Figure 3.
Item (c) of Figure 3 demonstrates the map of the unit disk by the polynomial of the 97th degree and the polar coordinate net transform under the mapping.
We also find the smooth solution after solving (24) and corresponding reparametrization of the given hypotrochoid.We name this polynomial function the smooth mapping.

Solution of the Torsion Problem for the Bar with the
Hypotrochoidal Section.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 [1], Chapter 7: the value of the shear stress is proportional to the expression where () is the polynomial mapping of the unit disk onto the hypotrochoid interior and () is the analytic in the unit disk function with the boundary condition The contour values of the tangent shear stress (), − <  < , were found for the functions of the preceding section: for the mapping  19 (), for the mapping  97 (), and for the smooth mapping.Figure 4 shows the graphs of the function () for these mappings.
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 solved this problem for the hypotrochoid interior in the case of () = 0.01 2 ,  = 2. Figure 5 shows us the displacements of points of the concentric curves being the images of || = const.

Spline-Interpolation Solution of the 3D Second Basic
Elasticity Problem.Consider now the 3D second basic elasticity theory problem for a bar  × [0, ℎ] parallel to axis in the space with coordinates (, , ℎ).We say that the spline-interpolation solution of this problem is the approximate solution which satisfies the elasticity equations and the boundary displacements at the finite number of levels ℎ = ℎ  ,  = 1, . . ., , and approximates the displacements at the end faces of the bar.Assume that we construct a linear spline following [13].Then, the solution is constructed for each element  × [ℎ  , ℎ +1 in the linear form on ℎ components of the displacement vector-the functions

Figure 1 :
Figure 1: Nonconvex domain and the image of the regular polar net.

Figure 2 :
Figure 2: The curve and the image of the regular polar net.

Figure 3 :
Figure 3: The mappings of the unit disk onto the hypotrochoidal interior: (a) hypotrochoid, (b)  19 () map of the unit disk, and (c)  97 () map of the unit disk.
Second Basic Elasticity Problem.We consider the second plane boundary value problem for a domain  that is the image of the unit disk under the conformal mapping ().Due to results of [1], we can reduce the problem to finding the analytic in the unit disk functions () and (), || < 1, via the boundary condition ([1, Chapter 2, formula (1)]) (− () +  ()   ()   () +  ())          =  =  () ;