The Locally Corrected Nyström Method for Electromagnetics

Abstract

The focus of this chapter is to provide a broad overview of the high-order Locally Corrected Nyström (LCN) method. The LCN method is applied herein to the solution of a general hybrid surface/volume integral equation defining the electromagnetic scattering and/or radiation from arbitrary three-dimensional geometries of arbitrary material composition. The chapter starts out by providing a basic understanding of the LCN algorithm and its analogy to the more popular method of moments. The LCN method is then specialized to the solution of hybrid integral equations by providing nuts-and-bolts details regarding the application of the method. LCN discretizations of surface integral equations are presented for curvilinear quadrilateral or triangular meshes. Furthermore, LCN discretizations of volume integral equations are derived for curvilinear hexahedral, tetrahedral, or prism meshes. Quadrature rules suitable for a Nyström discretization and basis functions suitable for local corrections are defined for each topological type. Performing the local corrections for the different kernels encountered is also discussed. Finally a number of numerical examples are presented that span the different integral operators and topological types. For all cases, data is provided that demonstrates the high-order convergence of a LCN solution method. It is hoped that the power, yet simplicity, of the LCN method is conveyed in this chapter, along with enough details for an interested reader to develop their own LCN software.

Appendix 1: Curvilinear Meshing

5.1.1 Local Coordinate Description

The LCN method is a high-order solution method. Hence, as one increases the order of approximation p, the error should converge as O(h ^p). However, the accuracy in the discrete approximation of the currents and the integral operators must be complemented by the accuracy of the discrete representation of the problem geometry. Therefore, a high-order geometric discretization is necessary which can be accomplished using high-order tessellation schemes based on curvilinear cells. In order to support this, one needs to pose local curvilinear coordinate descriptions of these cells.

Consider a local curvilinear coordinate system $({u^1},{u^2},{u^3})$ as illustrated in Fig. 5.13. Let $\vec{r}=\vec{r}\left( {{u^1},{u^2},{u^3}} \right)$ be the position vector of a point within the curvilinear coordinate frame. Vectors that are tangential to each of the curvilinear coordinate curves can be defined and are referred to as the unitary vectors ${{\vec{a}}_i}$ (i = 1, 2, 3), which are depicted in Fig. 5.13. The unitary vectors are defined as

$$ {{\vec{a}}_i}=\dfrac{{\partial \vec{r}}}{{\partial {u^i}}} $$

(5.119)

and have magnitudes that are equal to the differential lengths along the unitary curves.

For a general curvilinear coordinate frame, the unitary vectors are not orthogonal. In order to complete an orthogonal vector space, a set of reciprocal unitary vectors ${{\vec{a}}^i}$ are introduced [61]:

$$ {{\vec{a}}^1}=\dfrac{{{{\vec{a}}_2}\times {{\vec{a}}_3}}}{{\sqrt{g}}},\;{{\vec{a}}^2}=\dfrac{{{{\vec{a}}_3}\times {{\vec{a}}_1}}}{{\sqrt{g}}},\;{{\vec{a}}^3}=\dfrac{{{{\vec{a}}_1}\times {{\vec{a}}_2}}}{{\sqrt{g}}} $$

(5.120)

where

$$ \sqrt{g}={{\vec{a}}_1}\cdot {{\vec{a}}_2}\times {{\vec{a}}_3} $$

(5.121)

is the Jacobian or local differential volume traced by the unitary vectors. The reciprocal unitary vectors have the units of 1/length. Each ${{\vec{a}}^i}$ is orthogonal to surfaces where the unitary coordinate ${u^i}$ is a constant. Furthermore, the reciprocal vectors can also be computed from the gradient as

$$ {{\vec{a}}^1}=\nabla {u^1},\;{{\vec{a}}^2}=\nabla {u^2},\;{{\vec{a}}^3}=\nabla {u^3}. $$

(5.122)

Fig. 5.13

Curvilinear coordinate system

The unitary and reciprocal unitary spaces form a reciprocal vector space such that

$$ {{\vec{a}}_i}\cdot {{\vec{a}}^j}={\delta_{i,j }} $$

(5.123)

where ${\delta_{i,j }}$ is the Kronecker delta function. With the use of the reciprocal bases, any vector can be represented as a superposition of unitary or reciprocal vectors. For example,

$$ \vec{F}=\sum\limits_{i=1}^3 {{f^i}{{\vec{a}}_i}} =\sum\limits_{i=1}^3 {{f_i}{{\vec{a}}^i}} $$

(5.124)

where

$$ {f^i}=\vec{F}\cdot {{\vec{a}}^i} $$

(5.125)

are the contravariant projections of $\vec{F}$, and

$$ {f_i}=\vec{F}\cdot {{\vec{a}}_i} $$

(5.126)

are the covariant projections of $\vec{F}$. From these, a number of differential and integral operators can be formed in the curvilinear coordinate frame. For example, the gradient of a scalar function is expressed as [61]

$$ \nabla \phi =\sum\limits_{i=1}^3 {{{\vec{a}}^i}\dfrac{{\partial \phi }}{{\partial {u^i}}}}, $$

(5.127)

the divergence of a vector field is expressed as

$$ \nabla \cdot \vec{F}=\dfrac{1}{{\sqrt{g}}}\sum\limits_{i=1}^3 {\dfrac{\partial }{{\partial {u^i}}}\left( {\sqrt{g}{f^i}} \right)}, $$

(5.128)

and the curl of a vector field is expressed as

$$ {\begin{array}{lll}\nabla \times \vec{F}=\dfrac{1}{{\sqrt{g}}}\sum\limits_{i=1}^3 {\left( {\dfrac{{\partial {f_k}}}{{\partial {u^j}}}-\dfrac{{\partial {f_j}}}{{\partial {u^k}}}} \right){{\vec{a}}_i}};\,\,\,\,\,\,j=\bmod (i,3)+1;\,\,k=\bmod (i+1,3)+1.\end{array}} $$

(5.129)

Fig. 5.14

Curvilinear surface S bound by contour C

Next, consider a curvilinear surface S bound by contour C. It is assumed that the sides of the contour C are defined by curvilinear coordinate curves. Thus, ${{\vec{a}}_1}$ is tangential to the ${u^1}$ curves (top and bottom), and ${{\vec{a}}_2}$ is tangential to the ${u^2}$ curves (left and right), as illustrated in Fig. 5.14. One can then compute the line integral of a vector field about the contour as:

$$ \begin{array}{rl} \oint\limits_C {\vec{F}({u^1},{u^2})\cdot d\vec{\ell}} &= \int\limits_0^1 {\vec{F}\left( {{u^1},0} \right)\cdot {{\vec{a}}_1}d{u^1}} +\int\limits_0^1 {\vec{F}\left( {1,{u^2}} \right)\cdot {{\vec{a}}_2}d{u^2}} \hfill \\ & \quad -\int\limits_0^1 {\vec{F}\left( {{u^1},1} \right)\cdot {{\vec{a}}_1}d{u^1}} -\int\limits_0^1 {\vec{F}\left( {0,{u^2}} \right)\cdot {{\vec{a}}_2}d{u^2}}. \end{array} $$

(5.130)

Fig. 5.15

Curvilinear quadrilateral cell defined by uniformly spaced nodes

The surface integral over S can also be computed. To this end, the differential surface vector is defined as

$$ d\vec{s}={{\vec{a}}_1}\times {{\vec{a}}_2}d{u^1}d{u^2}=\hat{n}\sqrt{{{g_s}}}d{u^1}d{u^2} $$

(5.131)

where $\sqrt{{{g_s}}}=|{{\vec{a}}_1}\times {{\vec{a}}_2}|$ is the surface Jacobian and $\hat{n}={{\vec{a}}_1}\times {{\vec{a}}_2}/\sqrt{{{g_s}}}.$ Consequently,

$$ \iint\limits_S {\vec{F}\cdot d\vec{s}}=\iint\limits_S {\vec{F}\cdot \left( {{{\vec{a}}_1}\times {{\vec{a}}_2}} \right)d{u^1}d{u^2}}=\iint\limits_S {\vec{F}\cdot \hat{n}\sqrt{{{g_s}}}d{u^1}d{u^2}}. $$

(5.132)

Finally, the volume integral of a function is simply

$$ \iiint\limits_V {\phi {dv}}=\iiint\limits_V {\phi \sqrt{g}{\it du}^1}{\it du}^2 {\it du}^3 $$

(5.133)

where $\sqrt{g}$ is the Jacobian defined by (5.121).

5.1.2 Curvilinear Quadrilateral

A three dimensional surface can be represented to high order using curvilinear quadrilateral or triangular elements. Given a discrete representation of the curvilinear cell, one can fully construct a local curvilinear coordinate system on which vector and scalar function spaces can be defined. Linear operators (e.g., differentiation or integration) can be performed within the local curvilinear coordinate frame using (5.127, 5.128, 5.129, 5.130, 5.131, 5.132, and 5.133). For sake of example, this is performed here for a high-order quadrilateral mesh defined by ${C^0}$ -continuous Lagrange interpolation polynomials. What is presented here can be applied to other ${C^0}$ -continuous elements such as Serendipity elements and can be extended to${C^n}$ -continuous elements such as NURBS or T-Splines.

Consider the p-th order Legendre quadrilateral element illustrated in Fig. 5.15. The element is defined by a total of ${(p+1)^2}$ nodes. For example, a linear quadrilateral element (p = 1) has four nodes, which correspond to the four vertices of the quadrilateral. The global position vectors of the nodes are defined by the discrete position vectors ${{\vec{r}}_{i,j }}$, where i = 0,…,p, and j = 0,…,p. The points are assumed to be uniformly spaced in the local curvilinear coordinate frame. Thus, the (i, j) coordinate of the local coordinate space is $(i/p,j/p)$. Next, a set of discrete interpolation polynomials are introduced in the local curvilinear coordinate space. A p-th order one-dimensional interpolation polynomial is defined as

$$ P_i^{1D}\left( {{u^1}} \right)={R_{p-i }}\left( {p,1-{u^1}} \right){R_i}\left( {p,{u^1}} \right) $$

(5.134)

where [48]

$$ {R_m}\left( {p,{u^1}} \right)=\left\{ \begin{array}{lll} 1,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,m=0 \hfill \\ \dfrac{1}{m!}\prod\limits_{k=0}^{m-1 } {(p{u^1}-k)},\,\,\,\,\,\,m>0 \hfill \\ \end{array} \right.. $$

(5.135)

The set of functions defined by i = 0,…,p spans a complete polynomial space of order p. The functions are purely interpolatory:

$$ P_i^{1D}\left( {\dfrac{k}{p}} \right)=\left\{ \begin{array}{lll} 1,\,\,\,\,\mathrm{if}\ \,\,\,k=i \hfill \\ 0,\kern0.5em \mathrm{otherwise} \hfill \\ \end{array} \right.. $$

(5.136)

A two-dimensional interpolation function space is then be defined from the product of one-dimensional functions:

$$ P_{i,j}^{2D}\left( {{u^1},{u^2}} \right)=P_i^{1D}\left( {{u^1}} \right)P_j^{1D}\left( {{u^2}} \right). $$

(5.137)

Consequently, the position vector of a point lying on the curvilinear quadrilateral patch is given by

$$ \vec{r}\left( {{u^1},{u^2}} \right)=\sum\limits_{j=0}^p {\sum\limits_{i=0}^p {{{\vec{r}}_{i,j }}P_{i,j}^{2D}\left( {{u^1},{u^2}} \right)} } $$

(5.138)

Given the definition of the position vector in the local curvilinear coordinate frame, the unitary vectors are computed from (5.119) to be

$$ \begin{array}{rl} {{\vec{a}}_1}=\dfrac{\partial }{{\partial {u^1}}}\vec{r}\left( {{u^1},{u^2}} \right)=\sum\limits_{j=0}^p {\sum\limits_{i=0}^p {{{\vec{r}}_{i,j }}\dfrac{\partial }{{\partial {u^1}}}P_{i,j}^{2D}\left( {{u^1},{u^2}} \right)} }, \hfill \\ {{\vec{a}}_2}=\dfrac{\partial }{{\partial {u^2}}}\vec{r}\left( {{u^1},{u^2}} \right)=\sum\limits_{j=0}^p {\sum\limits_{i=0}^p {{{\vec{r}}_{i,j }}\dfrac{\partial }{{\partial {u^2}}}P_{i,j}^{2D}\left( {{u^1},{u^2}} \right)} }. \hfill \\ \end{array} $$

(5.139)

From (5.134, 5.135, 5.136, and 5.137), one derives

$$ {\begin{array}{lll}\dfrac{\partial }{{\partial {u^1}}}P_{i,j}^{2D}\left( {{u^1},{u^2}} \right)=P_j^{1D}\left( {{u^2}} \right)\left[ {{R_{p-i }}\left( {p,1-{u^1}} \right){R_i}^{\prime}\left( {p,{u^1}} \right)-{R_{p-i}}^{\prime}\left( {p,1-{u^1}} \right){R_i}\left( {p,{u^1}} \right)} \right]\end{array}} $$

(5.140)

where ${R_i}^{\prime }$ is the derivative of ${R_i}$ with respect to its argument. An analytical expression for ${R_i}^{\prime }$ can be derived from (5.135). A similar expression is derived for $\partial P_{i,j}^{2D }/\partial {u^2}$.

To complete the three-dimensional space, one can define ${{\vec{a}}_3}=\hat{n}={{\vec{a}}_1}\times {{\vec{a}}_2}/\sqrt{{{g_s}}}$ for the local curvilinear surface. The reciprocal unitary vectors are still defined by (5.120) where ${{\vec{a}}^3}=\hat{n}$.

In summary, given the p-th order Lagrangian quadrilateral element illustrated in Fig. 5.15, the position vector and unitary vectors are defined by (5.138) and (5.139) and the reciprocal unitary vectors by (5.120). Thus, the local curvilinear coordinate space is completely defined. A similar interpolation scheme can be introduced for triangular elements [48]. The same principals can also be extended to three-dimensional hexahedral, tetrahedral, triangular prism, and pyramid elements.