// triangle-winding-number second-order approximation
// t is triangle, p is 'center' of cluster of dipoles, q is evaluation point
static double triwn_order2(ref Triangle3d t, ref Vector3d p, ref Vector3d xn, ref double xA, ref Vector3d q)
{
Vector3d dpq = (p - q);
double len = dpq.Length;
double len3 = len * len * len;
// first-order approximation - integrated_normal_area * \grad(G)
double order1 = (xA / MathUtil.FourPI) * xn.Dot(dpq / len3);
// second-order hessian \grad^2(G)
Matrix3d xqxq = new Matrix3d(ref dpq, ref dpq);
xqxq *= 3.0 / (MathUtil.FourPI * len3 * len * len);
double diag = 1 / (MathUtil.FourPI * len3);
Matrix3d hessian = new Matrix3d(diag, diag, diag) - xqxq;
// second-order LHS - integrated second-order area matrix (formula 26)
Vector3d centroid = new Vector3d(
(t.V0.x + t.V1.x + t.V2.x) / 3.0, (t.V0.y + t.V1.y + t.V2.y) / 3.0, (t.V0.z + t.V1.z + t.V2.z) / 3.0);
Vector3d dcp = centroid - p;
Matrix3d o2_lhs = new Matrix3d(ref dcp, ref xn);
double order2 = xA * o2_lhs.InnerProduct(ref hessian);
return(order1 + order2);
}
// point-winding-number second-order approximation
// x is dipole point, p is 'center' of cluster of dipoles, q is evaluation point
static double pointwn_order2(ref Vector3d x, ref Vector3d p, ref Vector3d xn, ref double xA, ref Vector3d q)
{
Vector3d dpq = (p - q);
Vector3d dxp = (x - p);
double len = dpq.Length;
double len3 = len * len * len;
// first-order approximation - area*normal*\grad(G)
double order1 = (xA / MathUtil.FourPI) * xn.Dot(dpq / len3);
// second-order hessian \grad^2(G)
Matrix3d xqxq = new Matrix3d(ref dpq, ref dpq);
xqxq *= 3.0 / (MathUtil.FourPI * len3 * len * len);
double diag = 1 / (MathUtil.FourPI * len3);
Matrix3d hessian = new Matrix3d(diag, diag, diag) - xqxq;
// second-order LHS area * \outer(x-p, normal)
Matrix3d o2_lhs = new Matrix3d(ref dxp, ref xn);
double order2 = xA * o2_lhs.InnerProduct(ref hessian);
return(order1 + order2);
}