# libscuff API: Main Flow Routines

This page documents the main flow portion of the libscuff API -- that is, the routines that implement the key steps in the procedure for solving electromagnetic scattering problems.

The surface-integral-equation/boundary-element-method (SIE/BEM) approach to Maxwell's equations implemented by scuff-em is discussed on the implementation notes page (and in more detail in the technical documents referenced therein); the key equation is this linear system of equations:

in which the RHS vector ${\mathbf{e} \choose \mathbf{h}}$ describes incident fields impinging on one or more scatterers, the matrix $\mathbf M$ describes interactions among those scatterers, and the unknown vector ${\mathbf{k} \choose \mathbf{n}}$ describes surface currents induced on the scatterers by the incident fields. (In what follows, we will refer to the matrix here as M, to the vector on the left-hand side as KN, and to the vector on the right-hand side as RHS). After assembling and solving this system for the surface currents, we can use them to compute various physical quantities of interest (scattered fields, power/force/torque, etc.).

The steps in the main flow of a libscuff scattering problem are the following:

• Create an RWGGeometry object from a .scuffgeo file.

• Assemble the BEM matrix M at a given frequency (and Bloch vector for periodic geometries).

• Assemble the RHS vector RHS for a given incident field.

• Solve the linear system M*KN=RHS for the surface-current vector KN.

• Perform various post-processing operations on the KN vector to compute physical quantities of interest.

The libscuff API routines for each of these steps are discussed below.

# 1. Creating an RWGGeometry

The RWGGeometry class constructor takes a string argument that should be the name of a .scuffgeo file.

C++:

 RWGGeometry *G = new RWGGeometry("MyGeometry.scuffgeo");


Python:

 G =  scuff.RWGGeometry("MyGeometry.scuffgeo");


# 2. Assembling the SIE/BEM matrix

The BEM matrix is assembled by the AssembleBEMMatrix() class method of RWGGeometry. The simplest invocation of this function requires only that you specify the angular frequency:

C++:

 HMatrix *M = G->AssembleBEMMatrix(3.197);


python:

 M =  G.AssembleBEMMatrix(3.197);


The return value of this function is a pointer to a newly-allocated instance of HMatrix, a simple matrix class provided by libscuff. If you later want to recompute the matrix at a different frequency, you can pass M back as the second argument of AssembleBEMMatrix to avoid reallocating new storage:

 // first computation; allocates a new matrix
HMatrix *M = G->AssembleBEMMatrix(3.197);

// subsequent computation; reuses the existing matrix
G->AssembleBEMMatrix(5.134, M);


The angular-frequency parameter to AssembleBEMMatrix has type cdouble, which is libscuff shorthand for a complex number consisting of two double-values:

typedef std::complex<double> cdouble;

 cdouble Omega(2.3, 4.5);

HMatrix *M = G->AssembleBEMMatrix(Omega);


# 3. Assembling the RHS vector

The RHS vector is assembled by the AssembleRHSVector() class method provided by RWGGeometry. Before you can call this routine, you must define the incident field in your scattering geometry. This is done by instantiating a class derived from the IncField base class defined in libscuff.h.

For most users, this will be a one-liner, because libscuff comes with several built-in classes derived from IncField that implement commonly-encountered types of incident field, including plane waves, Gaussian laser beams, and the fields of pointlike dipole radiators. For an instance, you can define a plane wave traveling in the positive z direction with E- field polarized in the x direction like this:

 double pwDir[3]  = { 0.0, 0.0, 1.0 };
cdouble pwPol[3] = { 1.0, 0.0, 0.0 };
pw = new PlaneWave(pwPol, pwDir);

 pw = scuff.PlaneWave([1,0,0], [0,0,1])


For details on how to create plane waves and other types of incident fields, as well as information on how to create your own derived subclass of IncField to represent an arbitrary incident field, see the IncField documentation.

Having defined our incident field, we pass it, and the frequency, to AssembleRHSVector:

 pw = new PlaneWave(pwPol, pwDir);
HVector *RHS = AssembleRHSVector(Omega, pw);

 pw = scuff.PlaneWave([1,0,0], [0,0,1])
RHS  = G.AssembleRHSVector(Omega, pw)


As with AssembleBEMMatrix, these invocations of AssembleRHSVector will cause a new HVector to be allocated. If you later want to reuse the same vector to store a different right-hand side, you can pass it as the third argument:

 // first computation; allocates a new vector
pw1 = new PlaneWave(pwPol1, pwDir1);
HVector *RHS = G->AssembleRHSVector(Omega, pw1);

// subsequent computation; reuses the existing vector
pw2 = new PlaneWave(pwPol2, pwDir2);
G->AssembleRHSVector(Omega, pw, RHS);


# 4. Solving the BEM system

Having assembled the BEM matrix and the RHS vector, the next step is to do some numerical linear algebra to solve the BEM system for the vector of surface-current expansion coefficients.

## Solving the BEM system in C++ programs

From a C++ program, the easiest way to do this is to use the simple interface to LAPACK provided by the libscuff matrix/vector support layer. Specifically, after assembling the BEM matrix M you say M->LUFactorize() to replace M with its LU-factorization; then, after assembling the RHS vector RHS you say M->LUSolve(RHS) to replace RHS with the solution of the system M*KN=RHS.

Note that you only have to call LUFactorize() once on a given BEM matrix, after which you can make any number of calls to LUSolve() to solve the linear system for any number of right-hand side vectors. If you subsequently reassemble the BEM matrix (say, at a different frequency), you must call LUFactorize() again before starting to do LUSolves..

RWGGeometry *G=new RWGGeometry("MyGeometry.scuffgeo");

HMatrix *M=G->AllocateBEMMatrix();

HMatrix *KN=G->AllocateRHSVector();

for( nf=0; nf&lt;NumFreqs; nf++ )
{
G->AssembleBEMMatrix( OmegaValues[nf], M );
M->LUFactorize();

for( ni=0; ni&lt;NumIncidentFields; ni++ )
{
G->AssembleRHSVector( OmegaValues[nf], IncFields[ni], KN );
M->LUSolve(KN);

// now call GetFields() and/or do other things with the solution vector KN

};
};


### Solving the BEM system in Python programs

Probably the easiest thing to do here is to use numpy.linalg.solve:

import numpy;
KN = numpy.linalg.solve(M, RHS)


Alternatively, you can access the LAPACK wrappers provided by libscuff:

 M2 = scuff.HMatrix(M)
M2.LUFactorize()
KN = RHS
M2.LUSolve(LHS2)


# 5. Postprocessing to compute scattered fields and other quantities

The final step in the main flow is to use the solution to the BEM system to compute various physical quantities of interest.

## Field components at arbitrary points in space

Field components are computed by the GetFields class method of RWGGeometry. This method offers a choice of several calling conventions, ranging from simplest to most powerful. But before we can discuss these we need to recall a peculiarity of the SIE/BEM approach to scattering problems.

### Selecting the Incident, Scattered, or Total Fields

To understand the calling convention for GetFields, we must first remember that in an SIE/BEM scattering problem there are three types of fields we can compute:

• the incident fields, which depend on the IncField object passed to AssembleRHSVector,

• the scattered fields, which depend on the HVector object obtained by solving the linear BEM system, and

• the total fields, representing the sum of incident and scattered contributions.

Thus the first two parameters to GetFields are an IncField pointer and an HVector pointer:

RWGGeometry::GetFields( const IncField *IF, const HVector *KN, ... /* other arguments */


Here IF should be the same IncField pointer you passed to AssembleRHSVector, while KN should be the solution to the BEM system, as computed (for instance) by saying something like M->LUSolve(KN).

You can select the type of field computed by GetFields by passing NULL values for one of these parameters to omit the corresponding field contributions. More specifically,

• If you pass NULL for the KN argument, the fields computed will be the incident fields.

• If you pass NULL for the IF argument, the fields computed will be the scattered fields.

• If you pass non-NULL values for both arguments, the fields computed will be the total fields.

Note that the term "scattered fields" is actually somewhat imprecise, because the fields arising from the surface currents are in fact the total fields in regions that do not contain field sources. For example, consider a compact dielectric object irradiated by a plane wave. Outside the object, we have an incident field
(the field of the plane wave) and a scattered field (the field arising from the surface currents on the sphere), and the total field is the sum of these two contributions. However, inside the object we have only the contributions of the surface currents; the field arising from these currents is already the total field inside the sphere with no contribution from the incident field. For evaluation points inside the object, the scattered and total fields as computed by GetFields are identical, while the incident field is zero.

#### Field components at a single point

The simplest way to use GetFields is to compute the Cartesian components of the E and H fields at a single point x. The function prototype in this case is

RWGGeometry::GetFields( const IncField *IF,
const HVector *KN,
const cdouble Omega,
const double X[3],
cdouble EH[6] );


where the additional inputs beyond those discussed above are

• Omega: the angular frequency
• X: the cartesian components of the evaluation point
• KN: the surface-current expansion vector

and, on return, the output vector EH contains the field components:

• EH[0..2] are the cartesian components of the scattered E field,
• EH[3..5] are the cartesian components of the

#### Scattered field components at multiple points

To compute the scattered-field components at N evaluation points, you could just repeat the above procedure N times, but you will get better throughput by first [creating an HMatrix][MatrixVector] of dimension N x 3, whose rows are the Cartesian components of the evaluation points, and then passing this HMatrix in place of the X parameter to GetFields:

HMatrix *RWGGeometry::GetFields( const IncField *IF,
const HVector *KN,
const cdouble Omega,
const HMatrix *XMatrix,
HMatrix *FMatrix = 0);


In this case, the return value is a newly allocated HMatrix of dimension N x 6, whose rows are the cartesian components of the scattered fields at your evaluation points, in the order Ex, Ey, Ez, Hx, Hy, Hz. (If you already have a matrix of the correct size lying around, perhaps from a previous call to GetFields, you can pass it as the FMatrix parameter to suppress allocation of a new HMatrix).

In python, you can create an Nx3 XMatrix either by reading from a text file (containing the three cartesian coordinates of each evaluation point on a separate line, with blank lines and lines beginning with # ignored):

 X=scuff.HMatrix("MyEvalPointFile")

Fields=G.GetFields(PW, KN, Omega, X)

# now we have Fields[0][0..5] = (Ex,Ey,Ez,Hx,Hy,Hz) at point 1
#                   [1][0..5] = (Ex,Ey,Ez,Hx,Hy,Hz) at point 2
# etc.


or by creating an XMatrix of size N x 3 and
setting the entries by hand. For example, if we want the fields at two evaluation points with coordinates (0.1,0.2,0.3) and (0.4,0.5,0.6), we can say

 X=scuff.HMatrix(2, 3); # for 2 eval points

X->SetEntry( 0, 0, 0.1 ); # x coord of first point
X->SetEntry( 0, 1, 0.2 ); # y coord of first point
X->SetEntry( 0, 2, 0.3 ); # z coord of first point

X->SetEntry( 1, 0, 0.4 ); # x coord of second point
X->SetEntry( 1, 1, 0.5 ); # y coord of second point
X->SetEntry( 1, 2, 0.6 ); # z coord of second point

Fields=G.GetFields(PW, KN, Omega, X)

# now we have Fields[0][0..5] = (Ex,Ey,Ez,Hx,Hy,Hz) at point 1
#                   [1][0..5] = (Ex,Ey,Ez,Hx,Hy,Hz) at point 2
# etc.


Alternatively, you can create X as a numpy.array:

 X=numpy.array( [ [0.1, 0.4], [0.2,0.5], [0.3, 0.6]]);

Fields=G.GetFields(PW, KN, Omega, X)


Note that, when constructed as a numpy.array, X must have "shape" (3,N), not (N,3) as you might expect.