| The Cauchy problems of elliptic equations arise in many areas of science, such as wave propagation, vibration, electromagnetic scattering, nondestructive testing, geophysics and cardiology. For the Cauchy problems, the boundary conditions are incomplete, which will lead to some inverse boundary determination problems. The solution of the Cauchy problem is unique, but it is well known that they are severely ill-posed in the sense of Hadamard, i.e. the solutions do not depend continuously on Cauchy data and a small perturbation in the data may result in large change in the solution. Therefore, it is interesting to establish accurate, robust and fast numerical algorithm for the Cauchy problems.This dissertation concerns about the ill-posedness for the Cauchy problem of the Helmholtz equation and Cauchy-Navier equation, and the effective and stable numerical algorithms for the corresponding Cauchy problems. The content includes three parts.Part one is Chapter1. In Chapter1, we first give a brief introduction to the back-ground of the two elliptic equations, then we give a short review of the Cauchy problems.Part two consists of Chapter2and Chapter3. We propose a potential function method for the Cauchy problems about Helmholtz equation in Chapter2, and extend the method to the Cauchy-Navier equation in Chapter3.In Chapter2. We propose a potential function method with regularization for the Cauchy problems connected with the Helmholtz equation. We give the denseness result about the potential function, the ill-posedness of this problem, and the convergence result. At last, some examples are given for the effectiveness of the proposed method.Let D C Rd(d=2,3) be a bounded and simply connected domain with regular boundary, and (?) be a portion of the boundary (?)D. We assume that (?) is connected, Σ=(?)D\Γ. Consider the following Cauchy problem:Given Cauchy data fD and fN on Γ, find u satisfiesWhere n is the unit normal to the boundary (?)D directed into the exterior of D and the wave number k≥0.Theorem1Let u∈H3/2(D) satisfy the elliptic equation. B, D are bounded and con-nected domains with D (?) B such that B\D is connected and (?)D∈C2. Then for every ε>0, there exists a density function ψ∈L2((?)B), such that satisfies andDefine the trace operator N: L2((?)B)â†'L2(Γ)×L2(Γ) byThen, the following property of the operator N holds. Theorem2The operator AT: L2((?)B)â†'L2(Γ)×L2(Γ) is compact and injective.Thus, we get the following integral equationsDue to the ill-posedness, we need to consider the perturbed equations, i.e., the mea-sured noisy data fδ∈L2(Γ)×L2(Γ) satisfying where fδ=‖f‖L2(Γ)×L2(Γ)×noise. Thus, we have‖Nψ-fδ‖L2(Γ)×L2(Γ)≤δ+ε. We solve the integral equations by combined minimum norm algorithm, i.e..The minimum norm solution of (2) is to solve the following equation By introducing the regularization operator we can get the regularized solution (?)αδ=Rαfδ, and choose the regularization parameter via finding the zero of G(α):=‖N(?)—f‖2—(δ+ε)2. The regularization parameter will be obtained numerically by Newton’s method.Theorem3Let e be sufficiently small positive constant and δ+ε<‖fδ‖l2(Γ)×L2(Γ)· Let the regularized solution (?)αδ for all δ∈(0,δ0) satisfy‖N(?)αδ(δ)—fδ‖l2(Γ)×L2(Γ)=δ+ε, and ψ=N*z∈N*(L2(Γ)×L2(Γ)) with‖z‖L2(Γ)×L2(Γ)≤E. ThenHere ψ∈L2((?)B) is the kernel of some single-layer potential function νψ which satisfies the approximation properties in Theorem1.Now, define the single-layer potential function Ï…(?)αδ(δ) of the form:Then we have the following result:Theorem4Let the assumptions in Theorem3hold. Then Moreover, the following estimate on boundary Σ holds The positive constants C1and C2depend only on k, D, B and E.In general, we need to get the numerical solution via the following equation where PN:L2((?)B)â†'VN is the projection operator, VB (?) L2((?)B) is subspace of L2((?)B) satisfying (?)N∞=1VM=L2((?)B), and the dimension of VN is N.The operator N*N is compact, thusThe above equation makes us know that we can get N satisfyingTheorem5The numerical solution (?)N,αδ and (?)αδ satisfy the following estimate:Theorem6Let the assumptions in Theorem3hold. Then Moreover, the following estimate on boundary Σ holdsThe positive constants C1and C2depend only on k, D, B and E.We conclude the numerical alrithrim as following:Given ε,δ, we can get a via finding the zero of G(α):=‖N(?)-(?)‖2-(δ+ε)2, and then we can get N from (3), so we will get density function φM,αδ, finally we will get the approximation of u|Σ and (?)un|Σ.At last, some examples are given for the effectiveness of the proposed method.In Chapter3. We investigate the potential function method with regularization for the Cauchy problems of Navier equation. We give the denseness result about the elastic potential function, the ill-posedness of this problem, and the convergence result. At last, some examples are given for the effectiveness of the proposed method. Let D (?) Rd be a bounded and simply connected domain with regular boundary, and Γ be a portion of the boundary (?)D. We assume that Γ is connected, Σ=(?)D\Γ. Consider the following Cauchy problem: Given Cauchy data f and t on Γ, find u satisfies where Δ*u=μΔu+(λ+μ)(?)(?)?u, n is the unit normal to the boundary (?)D directed into the exterior of D, and Tn=2μ?(?)+λndiv+μn×(?)×. Define the elastic potential function where D (?) B.Theorem7Let u∈H3/2(D) satisfy Navier equation (4). B, D are bounded and con-nected domains with D (?) B such that B\D is connected and (?)D∈C2. Then for every ε>0, there exists a density function (?)∈L2((?)B), such that S(?) satisfies andDefine the operator N: L2((?)B)â†'L2(Γ)×L2(Γ) byThen, the following property of the operator N holds.Theorem8The operator N: L2((?)B)â†'L2(Γ)×L2(Γ) is compact and injectiveThen, we write the problem as follows: to find (?)∈L2((?)B), such that and S(?)(x)|D will be our approximation of u on D. In general, let us consider the perturbed equations, i.e., the measured data with noisy hδ∈L2(Γ)×L2(Γ) satisfiesHere, noise is the noise level. Thus, we get‖N(?)—hδ‖L2(Γ)×L2(Γ)≤δ+ε. We solve the integral equations by combined minimum norm algorithm. The minimum norm solution of (5) is the solution of the following equation By introducing the regularization operatorwe can achieve the regularized solution (?)αδ=Rαhδ, and choose the regularization param-eter via finding the zero of G(α):=‖N(?)—h‖2—(δ+ε)2. The regularization parameter will be obtained numerically by Newton’s method.Theorem9Let ε be sufficiently small positive constant and δ+ε <‖hδ‖L2(Γ)×L2(Γ)· Let the regularized solution (?)αδ(δ) for all δ∈(0,δ0) satisfy and (?)=N*z∈N*(L2(Γ)×L2(Γ)) with‖Z‖L2(Γ)×L2(Γ)≤E.Then Here (?)∈L2((?)B) is the kernel of some single-layer potential function S(?) which satisfies the approximation properties in Theorem7.Define the elastic potential function S(?)αδ(δ) as follows Then we have the following result. Theorem10Let the assumptions in Theorem9hold. Then Moreover, the following estimate on boundary Σ holds The positive constants C1,C2depend only on λ,μ,D,ωB and E.At last, some examples are given for the effectiveness of the proposed method.The third part is Chapter4. The first part of this chapter is the boundary deter-mination problem in2D elasticity by potential function method. In the second part, we propose a method of fundamental solution for the equilibrium equation in two-dimensional linear elasticity, and then investigate the Cauchy problem, boundary determination, and rigid inclusions via the numerical algorithm.1. The boundary determination problem in2D elasticityConsider an isotropic linear elastic material which occupies an open bounded simply connected domain D∈R2with piecewise smooth boundary, and Y is a portion of the boundary3D. In this section, we consider the following problem: given Cauchy data/and t on Y, find u satisfies where n is the unit normal to the boundary3D directed into the exterior of D, and Tn=2μn?(?)+λndiv+μn×(?)×,μ＞0,λ+2μ>0. Assuming that Γ is connected, a smooth curve in half plane {(x1,x2)|x2>0}, and the two endpoints on x2=0. The inverse problem is that given a function f0determine the unspecified smooth curve s(x)≠(?) from the Cauchy data f and t on Γ, and s(x) satisfies s(x) is in half plane {(x1,x2)|x2<0}, and the two endpoints of s(x) are on x2=0. It is not required that Γ∪s(x)=(?)D, which means that Γ∪s(x)=(?)D or s(x)(?)(?)D/Γ. If s(x)(?)(?)D/Γ, we assume that s(x) and Γ were connected by two blackbody segments. Under above assumptions, we will approximate the solution u by some elastic single-layer potential function of the following form where D (?) B.Then, we define the trace operator N: L2((?)B)â†'L2(Γ)×L2(Γ) byThen, we can get the following property of the operator A/". Theorem11The operator N: L2((?)B)â†'L2(Γ)×L2(Γ) is compact and injective.After this, we can write the problem of approximating solution u as follows: to find φ∈L2((?)B), such that and Sφ(x)|D will be our approximation of u on D.We know that if φ is approximated, then we can get the approximation Sφ of u. Our aim here is to get the numerical approximation φαδ*of φ through Tikhonov regularization with Morozov discrepancy principle.Finally, finding an approximation s0(x) to the unknown curve s(x) via the following equation where u0=Sφαδ|s(x)At last, some examples are given for the effectiveness of the proposed method.2. Method of fundamental solutions for the inverse problems in2D elas-ticityConsider an isotropic linear elastic material which occupies an open bounded simply connected domain D∈R2with piecewise smooth boundary. In the absence of body forces, the equilibrium equation with respect to the displacement vector u(x), are also known as Navier equation. Γ is a portion of the boundary dD, and Tnu=2μn·(?)u+λndivu+and v are the shear modulus and Poisson ratio.I. Cauchy Problem: Given Cauchy data f and t on Γ, find u satisfies μΔu+(λ+μ)(?)·u=0, in D, u=f, Tnu=t, on Γ, where n is the unit normal to the boundary (?)D directed into the exterior of D. We will determine the displacement and the traction vectors on a curve s(x).II. Boundary Determination:Given Cauchy data f and t on Γ, μ>0,λ+2μ>0, find u satisfies μΔu+(λ+μ)(?)·u=0, in D, u=f, Tnu=t, on Γ, u=f0(x), on s(x), where n is the unit normal to the boundary3D directed into the exterior of D, f0is the known function on s(x). The inverse problem in this paper is to determine the unspecified boundary or a curve s(x)(?) D/Γ, and s(x)≠(?), in the domain from the Cauchy data f and t, and satisfying whereIII. Rigid Inclusions:Consider an isotropic linear elastic material which occupies an open bounded simply connected domain Ω∈R2with regular boundary (?)Ω. Our goal is to detect an unknown void D(?)Ω from measurements of the traction and displacement taken on the boundary (?)Ω. This situation commonly arises, for instance, in fracture mechanics when some de-fects stem from the manufacturing process, or when the elastic properties of the material deteriorate due to the occurrence of possible damage.We consider the following rigid inclusions determining problem: given Cauchy data f and t on (?)Ω, our goal is to determine a rigid inclusion with a smooth boundary (?)D and compactly contained in Ω, i.e. D (?)Ω, such that Ω\D is an annular domain, μ>0, λ+2μ>0, find u satisfies μΔu+(λ+μ)(?)(?)?u=0, inΩ\D, u=f, Tnu=t, on (?)Ω, u=0, on (?)D, where n is the unit normal to the boundary (?)Ω, directed into the exterior of D. The inverse problem in this paper is to determine the rigid inclusion boundary (?)D from the Cauchy data f and tThe main purpose of this chapter is to provide a numerical method for these problems based on the method of fundamental solutions.The method of fundamental solutions (MFS) assumes an approximation of the form: Here yj(j=1,…, M) are the coordinates of suitably chosen source points.A mathematical description of a physical phenomenon in general must be invariant in some appropriate sense before it can be of any use in physical/engineering applications. Based on this consideration, the invariant MFS assumes an approximation of the following form where c=(c1, C2)T is a constant vector. However, an appended constant in (6) is not the most important. The major difference between the conventionally MFS and the invariant MFS is the constraint about the coefficients aj and bj as follows In the collocation method, they are determined from the interpolation conditions.At last, some examples are given for the three problems to verify the effectiveness of the proposed method. |
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