CHAPTER 2. APPLICATION OF FULL-WAVE SCATTERING MODELS TO BENCH-
3.2 High-order Nystr¨ om Method
This section provides an overview of the LC-NM considered in this thesis. The reader may refer to Section 4.3 for a full description of this method. The primary purpose of this section is to describe the sources of error in the LC-NM. This LC-NM is equivalent to a collocation-BEM in which surface fields are expanded using an interpolation technique and where the collocation and field interpolation points coincide with quadrature points on the mesh elements. Traditionally, the NM proceeds (i) by posing the underlying boundary integral equations on an exact geometric representation of the boundary, and (ii) discretizing the integrals using global quadrature rules, by which it implicitly commits to the existence of a global parametric description of the boundary. The assumption of a global parametric description is too restrictive on the types of defects that can be modeled since most CAD representations do not admit such descriptions. Restriction of the method to exact geometries excludes approximate representations of boundaries that are obtained, for example, from coarsening of features and subsequent refinements. We relax both the assumptions so as to allow the application of the NM to approximate geometric representations based on boundary-element meshes. It is in this sense that the term LC-NM is used throughout this thesis.
Consider a defect in an otherwise homogeneous unbounded elastic solid. The scattering problem involves finding the displacement and stress fields scattered from the defect in the presence of an
incident displacement field given by uI(x). This problem can be reformulated in terms of the conventional boundary integral equations (CBIEs), which are given by (see Section4.2.6)
− Z S G+(x, x0)· t(x0) + h ˆ n0· Σ(1)+ (x, x0)iT · u(x0) dS(x0) +1 2u(x) = u I(x), (3.1a) − Z S G−(x, x0)· t(x0) +hnˆ0· Σ(1)− (x, x0)iT · u(x0) dS(x0)−1 2u(x) = 0. (3.1b)
for x∈ S, where u and t are the displacement and traction fields, respectively, S is the surface of the defect, and Σ(1)± (x, x0) and G±(x, x0) are the fundamental solutions of stress and displacement fields, respectively. The “+” and “-” subscripts distinguish fundamental solutions obtained with elastic material constants of the host and the defect, respectively.
Solutions to the CBIE-formulation provide the total displacement and traction fields on the surface of the defect. The scattered fields outside the defect can be computed from the surface fields using representation formulas. All numerical examples presented in this chapter involve the application of the LC-NM to the CBIE-formulation for solving the scattering problem. We assume a surface mesh of the defect that consists of a set of points on S which can be mapped onto triangular patches through a prescribed connectivity between the points. If the surface of the defect is spec- ified, for instance, in terms of a CAD representation, the first step involves constructing a surface mesh of the above-mentioned type from the CAD representation. Then, an approximation of S is de- veloped via patch-wise parametric interpolation using polynomial functions. Because of conformal interpolation, the resulting approximation ˜S consists of continuous but non-overlapping (except at the edges) curved triangular elements. Specifically, there are two interpolation parameters, ξ1 and ξ2, which lie in the reference triangular patch defined by Ω ={(ξ1, ξ2)∈ R2 | ξ1, ξ2> 0, ξ1+ξ2 ≤ 1}. For an interpolation scheme of order M , the interpolation functions are polynomials of degree M in ξ1 and ξ2. We consider mesh interpolations of orders 1 and 2 in this implementation.
Equations (3.1) are rewritten by replacing S with the approximation surface ˜S. On each element of ˜S, the displacement and traction fields are expanded using an interpolation technique wherein the interpolation nodes coincide with quadrature points on the element. All the quadrature points are internal to the element. Therefore, this interpolation technique can be considered as an expansion
of fields using non-conformal basis functions. The components of displacement and traction fields at the interpolation/quadrature points form the DOFs that need to be determined. According to the foregoing field interpolation, the integrals in Equation (3.1) can be written in the following form − Z ˜ S dS(x0) T(1)+ (x, x0)· u(x0) = Np X p=1 Nn X i=1 − Z ∆p dS(x0) T(1)+ (x, x0)·hL(p)i (x0) u(p)iα e(p)α (x0)i, (sum over α = 1, 2, 3) (3.2) where T(1)+ (x, x0) = hˆn0· Σ(1)+ (x, x0)iT, Np is the total number of elements, Nn is the number of interpolation points per element, ∆p is the integration domain corresponding to the element with index p, L(p)i : ∆p → R is an interpolation function such that L(p)i (y(p)j ) = δij for all interpolation nodes y(p)j (j = 1 to Nn) in the pth element; e(p)α (x0) for α = 1, 2, 3 are basis vectors at the point x0 such that u(x0) = u(p)iαe(p)α (x0) (sum over α), and u(p)iα are unknowns which need to be determined. The interpolation functions L(p)i , when written as functions of the interpolation parameters ξ1 and ξ2, are polynomials of degree 0, 1, 2 or 3. See Section 4.3for details.
Rewriting (3.1) by approximating all integrals as shown in (3.2) yields two equations containing the DOFs. These equations are valid for all x ∈ ˜S. A finite-dimensional system of equations is obtained by performing collocation at the interpolation/quadrature points and equating the components along the dual basis vectors. Let x0 = y(q)j be a collocation point representing the jth interpolation point in the element with index q. Let ˜e(q)β (x0), for β = 1, 2 and 3, represent the dual basis vectors of the basis given by e(q)α (x0). For convenience, let e0 represent ˜e(q)β (x0). The equation corresponding to component β at x0 is
Np X p=1 Nn X i=1 t(p)iα Z ∆p ˜ L(p)i (x0) e0· G+(x0, x0)· e(p)α (x0) dS(x0)+ u(p)iα− Z ∆p L(p)i (x0) e0· T(1)+ (x0, x0)· e(p)α (x0) dS(x0) +1 2u (q) jβ = e0· uI(x0). (3.3)
Observe that the coefficients of DOFs are in the form of integrals over the domain ∆p. These coefficients constitute elements of the matrix which needs to be inverted for determining the DOFs.
If x0 is sufficiently away from ∆p, the coefficients can be computed using the same quadrature rule that defines the interpolation nodes, as shown below:
Z ∆p dS(x0) e0· T(1)+ (x0, x0)· h L(p)i (x0) e(p)α (x0)i= Nn X j=1 wje0· T(1)+ (x0, y(p)j )· h L(p)i (y(p)j ) e(p)α (y(p)j )i (3.4a) = wie0· T(1)+ (x0, y(p)i )· e(p)α (x (p) i ), (3.4b)
where wj are the quadrature weights. Therefore, each coefficient can be computed with a single evaluation of the kernel function. More importantly, there is no interpolation error in the evaluated values of coefficients since quadrature points coincide with interpolation nodes. See Section 4.3.3
for the degree of exactness (order) of the quadrature rules used in this implementation. When x0 is one of the quadrature points inside ∆p, the integrand in (3.4a) is singular. Similarly, when x0 is close to ∆p but not inside it, the integrand is nearly-singular. In both cases, singularity subtraction techniques are applied to compute the coefficients [30, Chap. 4]. The resulting values of coefficients have both quadrature and interpolation errors. Most UNDE applications require only the scattering spectra of the defect in the far-field. These far-field quantities can be expressed in the form of surface integrals over the defect, and computed using the same quadrature rule that is applied in field interpolation. The far-field quantities determined in this fashion have quadrature error but are free from field-interpolation error. Summing up, the primary sources of error are:
1. Interpolation error in geometry description.
2. Error in computation of matrix elements, which includes (a) quadrature error in near-field matrix elements, (b) interpolation error in near-field matrix elements, and
(c) quadrature error in far-field matrix elements. 3. Quadrature error in calculating scattering spectra.