The Characteristic Matrix of Nonuniqueness for First-Kind Equations

Abstract

The matrix characterizing nonuniqueness of solution for Fredholm integral equations of the first kind is constructed and approximated numerically in the case of plane elastic deformations.

10.1 Introduction

Let S be a finite domain in $\mathbb{R}^{2}$, bounded by a simple, closed, C ² curve $\partial S$. We denote by x and y generic points in $S \cup \partial S$ and by | x − y | the distance between x and y in the Cartesian metric. Also, let C ^0, α(∂ S) and C ^1, α(∂ S), α ∈ (0, 1), be, respectively, the spaces of Hölder continuous and Hölder continuously differentiable functions on $\partial S$. In what follows, Greek and Latin indices take the values 1, 2 and 1, 2, 3, respectively, and a superscript T denotes matrix transposition.

For any function f continuous on $\partial S$, we define the ‘calibration’ functional p by

$$\displaystyle{ pf =\int \limits _{\partial S}f\,ds. }$$

Using the fundamental solution for the two-dimensional Laplacian

$$\displaystyle{ g(x,y) = -\frac{1} {2\pi }\,\ln \vert x - y\vert, }$$

we define the single-layer harmonic potential of density φ by

$$\displaystyle{ (V \varphi )(x) =\int \limits _{\partial S}g(x,y)\varphi (y)\,ds(y). }$$

The proof of the following assertion can be found, for example, in [Co94] or [Co00].

Theorem 1.

For any α ∈ (0,1), there are a unique nonzero function Φ ∈ C ^0,α (∂S) and a unique number ω such that

$$\displaystyle{ V \varPhi =\omega \quad \mbox{ on $\partial S$},\quad p\varPhi = 1. }$$

It is easy to see that Φ and ω depend on g and ∂ S.

The numbers 2π ω and e ^{−2π ω} are called Robin’s constant and the logarithmic capacity of ∂ S.

For a circle with the center at the origin and radius R, both Φ and ω can be determined explicitly:

$$\displaystyle{ \varPhi = \frac{1} {2\pi R},\quad \omega = -\frac{1} {2\pi }\,\ln R. }$$

For other boundary curves, Φ and ω are practically impossible to determine analytically and must be computed by numerical methods.

If the solution of the Dirichlet problem in S with data function f on ∂ S is sought as u = V φ, then φ is a solution of the (weakly singular) first-kind boundary integral equation

$$\displaystyle{ V \varphi = f\quad \mbox{ on $\partial S$}. }$$

This is a well-posed problem if and only if ω ≠ 0. If ω = 0, the above equation has infinitely many solutions, which are expressed in terms of Φ.

10.2 Plane Elastic Strain

Consider a plate made of a homogeneous and isotropic material with Lamé constants λ and μ, which undergoes deformations in the (x ₁, x ₂)-plane. If the body forces are negligible, then its (static) displacement vector u = (u ₁, u ₂)^T satisfies the equilibrium system of equations [Co00]

$$\displaystyle{ Au = 0\quad \mbox{ in $S$}, }$$

where

$$\displaystyle{ A(\partial _{1},\partial _{2}) = \left (\begin{array}{*{10}c} \mu \varDelta +(\lambda +\mu )\partial _{1}^{2} & (\lambda +\mu )\partial _{1}\partial _{2} \\ (\lambda +\mu )\partial _{1}\partial _{2} & \mu \varDelta +(\lambda +\mu )\partial _{2}^{2} \end{array} \right ). }$$

It is not difficult to show [Co00] that the columns F ⁽ⁱ⁾ of the matrix

$$\displaystyle{ F = \left (\begin{array}{rrr} 1&0& x_{2} \\ 0&1& - x_{1} \end{array} \right ) }$$

form a basis for the space of rigid displacements.

The ‘calibrating’ vector-valued functional p is defined for continuous 2 × 1 vector functions f by

$$\displaystyle{ pf =\int \limits _{\partial S}F^{\mathrm{T}}f\,ds. }$$

A matrix of fundamental solutions for A is [Co00]

$$\displaystyle\begin{array}{rcl} & & D(x,y) = - \frac{1} {4\pi \mu (\gamma +1)} {}\\ & & \quad \times \left (\begin{array}{*{10}c} 2\gamma \ln \vert x - y\vert + 2\gamma + 1 -\dfrac{2(x_{1} - y_{1})^{2}} {\vert x - y\vert ^{2}} & -\dfrac{2(x_{1} - y_{1})(x_{2} - y_{2})} {\vert x - y\vert ^{2}} \\ -\dfrac{2(x_{1} - y_{1})(x_{2} - y_{2})} {\vert x - y\vert ^{2}} & 2\gamma \ln \vert x - y\vert + 2\gamma + 1 -\dfrac{2(x_{2} - y_{2})^{2}} {\vert x - y\vert ^{2}} \end{array} \right ), {}\\ & & \qquad \qquad \qquad \qquad \qquad \quad \qquad \qquad \gamma = \frac{\lambda +3\mu } {\lambda +\mu }. {}\\ \end{array}$$

The single-layer potential of density φ is defined by

$$\displaystyle{ (V \varphi )(x) =\int \limits _{\partial S}D(x,y)\varphi (y)\,ds(y). }$$

The proof of the following assertion can be found in [Co00].

Theorem 2.

There is a unique 2 × 3 matrix function Φ ∈ C ^0,α (∂S) and a unique 3 × 3 constant symmetric matrix $\mathcal{C}$ such that the columns Φ ⁽ⁱ⁾ of Φ are linearly independent and

$$\displaystyle{ V \varPhi = F\mathcal{C}\quad \mbox{ on $\partial S$},\quad p\varPhi = I, }$$

where I is the identity matrix.

Clearly, Φ and $\mathcal{C}$ depend on A, D, and ∂ S.

In the so-called alternative indirect method [Co00], the solution of the Dirichlet problem in S with data function f on ∂ S is sought in the form

$$\displaystyle{ u = V \varphi. }$$

(10.1)

Then the problem reduces to the (weakly singular) boundary integral equation

$$\displaystyle{ V \varphi = f\quad \mbox{ on $\partial S$}. }$$

(10.2)

Theorem 3.

Equation (10.2) has a unique solution φ ∈ C ^0,α (∂S), α ∈ (0,1), for any f ∈ C ^1,α (∂S) if and only if $\det \mathcal{C}\not =0$ . In this case, (10.1) is the unique solution of the Dirichlet problem.

If $\det \mathcal{C} = 0$, then the unique solution of the Dirichlet problem is obtained by solving an ill-posed modified boundary integral equation whose infinitely many solutions are constructed with Φ and $\mathcal{C}$.

In the so-called refined indirect method [Co00], the solution of the Dirichlet problem is sought as a pair {φ, c} such that

$$\displaystyle{ u = V \varphi - Fc\quad \mbox{ in $S$},\quad p\varphi = s, }$$

where s a constant 3 × 1 vector chosen (arbitrarily) a priori and c is a constant 3 × 1 vector. This leads to the system of boundary integral equations

$$\displaystyle{ V \varphi - Fc = f\quad \mbox{ on $\partial S$},\quad p\varphi = s. }$$

(10.3)

Theorem 4.

System (10.3) has a unique solution {φ, c} with φ ∈ C ^0,α (∂S) for any f ∈ C ^1,α (∂S), α ∈ (0,1), and any s.

It is important to evaluate the arbitrariness in the representation of the solution with respect to the prescribed ‘calibration’ s.

Theorem 5.

If {φ ⁽¹⁾ , c ⁽¹⁾ }, {φ ⁽²⁾ , c ⁽²⁾ } are two solutions of (10.3) constructed with s ⁽¹⁾ and s ⁽²⁾ , respectively, then

$$\displaystyle\begin{array}{rcl} & & \varphi ^{(2)} =\varphi ^{(1)} +\varPhi (s^{(2)} - s^{(1)}), {}\\ & & c^{(2)} = c^{(1)} + \mathcal{C}(s^{(2)} - s^{(1)}). {}\\ \end{array}$$

It is not easy to compute Φ and $\mathcal{C}$ analytically, or even numerically, in arbitrary domains S, but this can be accomplished if S is a circular disk. Let ∂ S be the circle with center at the origin and radius R. In this case, Φ and $\mathcal{C}$ can be determined analytically as

$$\displaystyle{ \varPhi = \frac{1} {2\pi R}\,\left (\begin{array}{ccr} 1&0& R^{-2}x_{ 2} \\ 0&1& - R^{-2}x_{1} \end{array} \right ), }$$

$$\displaystyle\begin{array}{rcl} & & \mathcal{C} = - \frac{1} {4\pi \mu (\lambda +2\mu )R^{2}} {}\\ & & \qquad \qquad \times \left (\begin{array}{*{10}c} (\lambda +3\mu )R^{2}(\ln R + 1)& 0 & 0 \\ 0 &(\lambda +3\mu )R^{2}(\ln R + 1)& 0 \\ 0 & 0 &-(\lambda +\mu ) \end{array} \right ).\quad {}\\ \end{array}$$

Clearly, $\det \mathcal{C} = 0$ if and only if $R = e^{-1}$.

Analytic computation of Φ and $\mathcal{C}$ is practically impossible for non-circular domains, and must be performed numerically.

We choose four 3 × 1 constant vectors s ⁽⁰⁾, s ⁽ⁱ⁾ such that the set {s ⁽ⁱ⁾ − s ⁽⁰⁾} is linearly independent, and form the 3 × 3 matrix $\varSigma$ with columns s ⁽ⁱ⁾ − s ⁽⁰⁾. Also, we choose an arbitrary function f. Next, we compute the solutions {φ ⁽⁰⁾, c ⁽⁰⁾}, {φ ⁽ⁱ⁾, c ⁽ⁱ⁾} of (10.3) corresponding to s ⁽⁰⁾, s ⁽ⁱ⁾, respectively, and f, by the refined indirect method, then form the 2 × 3 matrix function Ψ with columns φ ⁽ⁱ⁾ −φ ⁽⁰⁾ and the constant 3 × 3 matrix Γ with columns c ⁽ⁱ⁾ − c ⁽⁰⁾.

From Theorem 4 it follows that

$$\displaystyle\begin{array}{rcl} \varphi ^{(i)} -\varphi ^{(0)}& =& \varPhi (s^{(i)} - s^{(0)}), {}\\ c^{(i)} - c^{(0)}& =& \mathcal{C}(s^{(i)} - s^{(0)}), {}\\ \end{array}$$

or, what is the same,

$$\displaystyle{ \varPhi \varSigma =\varPsi,\quad \mathcal{C}\varSigma =\varGamma; }$$

hence,

$$\displaystyle{ \varPhi =\varPsi \varSigma ^{-1},\quad \mathcal{C} =\varGamma \varSigma ^{-1}. }$$

A similar analysis can be performed for other two-dimensional linear elliptic systems with constant coefficients—for example, the system modeling bending of elastic plates with transverse shear deformation [Co00]. No apparent connection exists between the matrix $\mathcal{C}$ and the characteristic constant ω of ∂ S.

10.3 Numerical Examples

Consider a steel plate with scaled Lamé coefficients

$$\displaystyle{ \lambda = 11.5,\quad \mu = 7.69, }$$

and let ∂ S (see Figure 10.1) be the curve of parametric equations

$$\displaystyle\begin{array}{rcl} x_{1}(t)& =& 2\cos (\pi t) -\tfrac{4} {3}\,\cos (2\pi t) + \tfrac{10} {3}, {}\\ x_{2}(t)& =& 2\sin (\pi t) + 2,\quad 0 \leq t \leq 2. {}\\ \end{array}$$

Fig. 10.1

The boundary curve ∂ S.

We choose the vectors

$$\displaystyle\begin{array}{rcl} & s^{(0)} = \left (\begin{array}{*{10}c} 1\\ 1 \\ 1 \end{array} \right ),\quad s^{(1)} = \left (\begin{array}{*{10}c} 1\\ 0 \\ 0 \end{array} \right ),\quad s^{(2)} = \left (\begin{array}{*{10}c} 0\\ 1 \\ 0 \end{array} \right ),\quad s^{(3)} = \left (\begin{array}{*{10}c} 0\\ 0 \\ 1 \end{array} \right ),& {}\\ & f(x) = \left (\begin{array}{*{10}c} 1\\ 0 \end{array} \right ). & {}\\ \end{array}$$

The approximating functions for computing φ ⁽⁰⁾(t) and φ ⁽ⁱ⁾(t) are piecewise cubic Hermite splines on 12 knots; that is, the interval 0 ≤ t ≤ 2 is divided into 12 equal subintervals. Then the characteristic matrix (with entries rounded off to 5 decimal places) is

$$\displaystyle{ \mathcal{C} = \left (\begin{array}{rrr} - 0.01627& - 0.01083& - 0.00370\\ - 0.01083 & - 0.00892 & 0.00542 \\ - 0.00370& 0.00542& 0.00185 \end{array} \right ). }$$

Here,

$$\displaystyle{ \det \mathcal{C} = 1.08273 \times 10^{-6}. }$$

The graphs of the components Φ _{α i} of Φ are shown in Figure 10.2.

Fig. 10.2

Graphs of the Φ _{α i} .

As a second example, consider the ‘expanding’ ellipse ∂ S of parametric equations

$$\displaystyle\begin{array}{rcl} x_{1}(t)& =& 2k\cos (\pi t), {}\\ x_{2}(t)& =& k\sin (\pi t),\quad 0 \leq t \leq 2. {}\\ \end{array}$$

The graph of $\det \mathcal{C}$ as a function of k is shown in Figure 10.3.

Fig. 10.3

Graph of $\det \mathcal{C}$.

Here, $\det \mathcal{C} = 0$ for k = 0. 22546 and k = 0. 26934.