1 Conformally mapping the physical domain
Consider the unbounded multiply connected domain being exterior to the rectilinear slits (see Figure 2 (right)). We assume each slit make an angle with the positive -axis for . The iterative method presented in  computes an unbounded preimage domain in the exterior of ellipses (see Figure 2 (left)). The ellipses are parameterized by
where is the ratio between minor and major axes’ lengths of the ellipses. The parameters and , , are computed by the iterative method. The method provides also the boundary values of the unique conformal mapping from the domain in -plane onto the domain in -plane with the normalization
Putting , the domain is on the left of because the curves are clockwise oriented in view of (2). According to (3) the mapping function is analytic in with . Thus, the Cauchy integral formula implies that the values of the mapping function can be obtained for by
We can also use the Cauchy integral formula to determine the values of the inverse mapping function for (see e.g., ). The inverse mapping function satisfies also the normalization [23, p. 114, p. 127]
Therefore the values of for can be computed by
where and means for . In this way, we define a parameterization of the boundary on by
We will drop the index in the notation of as the value of such that will be clear from the context. Therefore we simply write
The iterative method in  provides us with the boundary values , which are used to get a parameterization of the boundary of
Thus, by making use of (4), we can determine the values of the inverse map for through
To compute the values of , we first approximate the real and imaginary parts of on each interval ,
, by trigonometric interpolating polynomials, and then differentiate.
Now we are ready to compute the preimage of the given domain (see Figure 3 (right)). The inverse mapping function maps the domain outside the slits onto the domain outside of the ellipses . Furthermore, maps the squares and onto piecewise smooth Jordan curves and , respectively, such that the ellipses are in the ring domain between the curves and (see Figure 3 (left)). Thus the whole boundary of the domain is .
For each , we parameterize the square by , , where is chosen using the same approach used in [8, pp. 696-697]. Hence for , the curve is parameterized by
such that the values of are computed through (1). The values of the derivative , for , are computed numerically again by using trigonometric interpolating polynomials as explained above. Recall that the parameterization of the ellipses are given by (2). Henceforth if we take as the disjoint union of the intervals , then we parameterize the whole boundary by
2 Computing the temperature distribution
such that is the unique solution of the following boundary value problem on the domain in -plane,
As is harmonic, we can write
where is a single-valued analytic function on the domain . Assuming the boundary values of the function takes the form
on the boundary of , it follows from (7b) that
Similarly, equation (7e) implies that the values of the function are known for with
where is an undetermined real constant.
Let us define the function on by
which are known as the Riemann-Hilbert problem. To solve the problem (12), we tale an auxiliary given point in the domain . Since we are only interested in determining the real function , we can assume that is real. We define an auxiliary function on the domain by
The function is clearly analytic in . We consider also a complex-valued function defined on by
where is the piecewise constant function given by
Therefore the boundary conditions in (12) implies that solves the Riemann-Hilbert problem
where for , , and . The piecewise constant function is an unknown and need to be determined together with the function . Both of these two functions can be found using a boundary integral equation method based on the generalized Neumann kernel. More precisely, following  the boundary values of the analytic function are determined through
where uniquely solves the integral equation
and the function is computed by
Here, is the identity operator and the integral operators and are respectively defined by
[mu,h] = fbie(et,etp,A,gamk,n,iprec,restart,tol,maxit).
This functions employs a discretization of the integral equation (14) by the Nyström method using the trapezoidal rule to obtain an algebraic linear system [4, 20]. Each boundary component is discretized by nodes yielding a linear system of size . This system is solved by the MATLAB function
in which the matrix-vector multiplication is computed using the MATLAB functionfrom toolbox . In our numerical calculations, we choose ; which means the tolerance in the FMM is . The GMRES performs a maximum number of iterations maxit=100 with an accuracy tolerance tol=1e-14, and the method is used without restart by choosing restart=[ ].
is an analytic function on the medium called complex distribution temperature. The derivative if this function is usefully related to the heat flux as we will see in the next section.
3 Computing the heat flux
According to Fourier’s law of conduction the heat flux vector related to the gradient temperature is given by the equation
where is the equivalent thermal conductivity which we assume to be normalized to unity.
From (16) and using the Cauchy-Riemann equations, it follows that the derivative of the complex temperature distribution on is given by
One the other hand (17) yields
where the denominator does not vanish in the domain since is a conformal mapping. Therefore the heat flux can be expressed in terms of by the formula
The values of the heat flux can be estimated on the domain by first computing the derivatives of the boundary values of the analytic functions and on each boundary components using trigonometric interpolating polynomials as explained above. Then, the values of and , in the right-hand side of (18), are computed for using the Cauchy integral formula.
We apply the method presented above to compute the temperature field and the heat flux for four different number of CNTs. In the first example, we consider the case of CNTs of different lengths with an inner square of side length (see Figure 4). In the second example, we have CNTs of different lengths and (see Figure 5). The third example involves CNTs of equal length with (see Figure 6). In the fourth example, we consider CNTs of random lengths between and with (see Figure 6). The centers and angles of the rigid line inclusions are chosen randomly so that all lines are non-overlapping. For these fourth all examples, we use .
Estimating the boundary values of the function , requires two steps:
Computing the domain . For this step, the total number of discretization points is .
Compute the boundary values of the function . The total number of discretization points is .
The total CPU time required by the two steps to compute is shown in Table 1.
To compute the values of and , we discretize the domain using a matrix
W with approximately million points in . Thus is a matrix of points that discretize the domain . Afterwards, we compute the values of the temperature distribution and the heat flux at the points of the matrix using the formulas (16) and (18), respectively. For the temperature distribution , we use the MATLAB function
contourf to plot the contours of the function on the domain . To visualize the heat flux , we present a phase portrait with modulus contour lines of . The colors represent the phase of where red means is in the direction of the positive real axis, green for the positive imaginary axis, cyan for the negative real axis, and violet for the negative imaginary . We show also the contour lines of the amplitude of .
The total CPU time required to compute the values of and are shown in Table 1.
For the four examples, we plot in Figure 8 the computed values of the real constants (sorted from the smallest to the largest). We see from the figures that the sorted values of these constants almost lie on a straight line for large .
These examples demonstrate the computational effectiveness of the developed method of integral equations. We select a particular type of 2D structures, namely, a square with a large square hole. Such a hole models a cavity or a defect. The slits model perfectly conducting nanotubes randomly distributed in the bulk medium. The number of slits increasing from to clearly display the local fields in the considered composites. Different inclinations of slits (nanotubes) in Figure 4 show that the slit perpendicular to the external flux does not essentially disturb it contrary to the slit parallel to the external flux. The flux intensity can increase in 2 times due to the nanotube. Such an irregularity of the flux was noted in [19, 26] for analogous problems. With increasing the number of slits the perturbations of the local fields become smaller and a heterogeneous structure can be homogenized. The same examples demonstrate the elastic stress perturbations in the anti-plane problem. For elasticity problem, the modulus of the flux has to be replaced by the stress intensity where
stands for the components of the stress tensor.
|Number of rigid||Side length of||and|
|line inclusions||the inner square|
We would like to thank Vladimir Mityushev for suggesting this research problem and for his helpful comments and discussions.
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