<Coding of sheet, sphere- and cylinder-shaped specimen in the emission mode>
// Total amount of diffusing gas which has entered the sheet at time t
// Ref: Crank et.al. The Mathematics of Diffusion (1975) p. 47, Equation (4.18) [24]
// D: Diffusion coefficient, L: Sheet thickness
// M: Emitted total amount by diffusion at infinite time.
public double Ftn_Sheet(double t) {
	double pi2 = Math.PI × Math.PI;
	double zSum = 0;
	for (int n = 0; n < 50; ++n) {
		int n21 = (2 × n + 1) × (2 × n + 1);
		double add = Math.Exp(−D × n21 × pi2 × t/(L × L))/n21;
		double zSumAdded = zSum + add;
		if (zSum >= zSumAdded) break; // When convergence achieved
		zSum = zSumAdded;
	}
	return M − 8/pi2 × (M × zSum);
}
// Total amount of diffusing gas which has entered the sphere at time t.
// Ref: Crank et.al. The Mathematics of Diffusion (1975) p. 91, Equation (6.20) [24]
// D: Diffusion Coefficient, a: Radius of sphere.
// M: Emitted total amount by diffusion at infinite time.
public double Ftn_Sphere(double t) {
	double pi2 = Math.PI × Math.PI;
	double zSum = 0;
	for (int n = 1; n < 100; ++n) {
		int n2 = n × n;
		double add = Math.Exp(−D × n2 × pi2 × t/(a × a))/n2;
		double zSumAdded = zSum + add;
		if (zSum >= zSumAdded) break; // When convergence achieved
		zSum = zSumAdded;
	}
	return M − 6/pi2 × (M × zSum);
}
// Total amount of diffusing gas which has entered the cylinder at time t.
// Ref: Al Demarez, et. al., Acta Metallurgica (1954), Equation (5) [28]
// D: Diffusion Coefficient, a: Radius of cylinder, L: Length of cylinder
// M: Emitted total amount by diffusion at infinite time.
public double Ftn_Cylinder(double t) {
	double pi2 = Math.PI × Math.PI;
	double zSum = 0;
	for (int n = 0; n < 100; ++n) {
		int n2 = (2 × n + 1) × (2 × n + 1);
		double add = Math.Exp(−n2 × pi2 × D × t/(L × L))/n2;
		double zSumAdded = zSum + add;
		if (zSum >= zSumAdded) break; // When convergence achieved
		zSum = zSumAdded;
	}
	double rSum = 0;
	for (int n = 1; n <= 50; ++n) {
		double b = Bessel0root[n − 1];
		double add = Math.Exp(−D × b × b × t/(a × a))/(b × b);
		double rSumAdded = rSum + add;
		if (rSum >= rSumAdded) break; // When convergence achieved
		rSum = rSumAdded;
	}
	return M − 32/pi2 × (M × zSum × rSum);
}
// Zeros of first kind Bessel function J_0(x)
public static double[] bessel0root =
	new double[] {2.40482556, 5.52007811, 8.65372791, 11.79153444, 14.93091771,
		18.07106397, 21.21163663, 24.35247153, 27.49347913, 30.63460647,
		33.77582021, 36.91709835, 40.05842576, 43.19979171, 46.34118837,
		49.4826099, 52.62405184, 55.76551076, 58.90698393, 62.04846919,
		65.1899648, 68.33146933, 71.4729816, 74.61450064, 77.75602563,
		80.89755587, 84.03909078, 87.18062984, 90.32217264, 93.46371878,
		96.605267951, 99.74681986, 102.88837425, 106.02993092, 109.17148965,
		112.3130503, 115.45461265, 118.59617663, 121.73774209, 124.87930891,
		128.020877, 131.1624463, 134.3040166, 137.445588, 140.5871604,
		143.7287336, 146.8703076, 150.0118825, 153.153458, 156.2950343};
// Nelder-Mead Simplex Optimization.
// Return value is fit error
// Optima(class): Manage all optimization procedure and calc. error and FOM etc.
// Simplex(Class): Polytope of N + 1 vertices in N dim. undetermined param. space.
// Vertex(Class): One of N + 1 vertices of Simplex.
// Coord(Class): Having coord. of vertex and methods, reflect(), expand(), …
// simplexProtocol: ExpFactor = 2.0, ConFactor = 0.5, PrcntInitCoef = 5.0, …
public double Iterate(double[] unknowns) {
	int b, w;
	int loop = 0; // # of loops
	int dim = unknowns.Length;
	Simplex simplex = new Simplex(unknowns, simplexProtocol.PrcntInitCoef);
	// Calculate errors for each vertexes.
	for (int i = 0; i < dim + 1; ++i)
	simplex.s[i].error = optima.ErrorF(simplex.s[i].coord);
	do { // start iteration
		b = simplex.bestVertex(); // find best and worst vertices
		w = simplex.worstVertex();
		// calculate the midPoint of the non-worst values for later calc’s
		double[] mp = simplex.midPoint(w);
		Vertex refl_vrtx = new Vertex(simplex.s[w]); // save the old values
		Coord.reflect(mp, refl_vrtx.coord); // first try a reflection
		refl_vrtx.error = optima.ErrorF(refl_vrtx.coord);
		// is the new vertex better or equal to the old best vertex
		if (refl_vrtx.error < simplex.s[b].error) {
			// try to expand the reflected vertex next
			Vertex expd_vrtx = new Vertex(refl_vrtx);
			Coord.expand(mp, expd_vrtx.coord, simplexProtocol.ExpFactor);
			expd_vrtx.error = optima.ErrorF(expd_vrtx.coord);
			// if the expanded vertex is not as good or better than the best
			// revert to the reflected vertex
			if (expd_vrtx.error <= refl_vrtx.error) simplex.s[w] = expd_vrtx;
			else simplex.s[w] = refl_vrtx;
		} else {
			if (refl_vrtx.error <= simplex.s[w].error) simplex.s[w] = refl_vrtx;
			else {
				Vertex cont_vrtx = new Vertex(simplex.s[w]);
				Coord.contract(mp, cont_vrtx.coord, simplexProtocol.ConFactor);
				cont_vrtx.error = optima.ErrorF(cont_vrtx.coord);
				if (cont_vrtx.error <= simplex.s[w].error) simplex.s[w] = cont_vrtx;
				else {
					simplex.contract_all(b, simplexProtocol.ConFactor);
					for (int i = 0; i < dim + 1; ++i)
						simplex.s[i].error = otima.ErrorF(simplex.s[i].coord);
				}; // else
			}; // else
		};
		++loop;
	} while (!(simplex.s[b].error <= simplexProtocol.Tolerance ||
	loop >= simplexProtocol.InternalLoop));
	// return best result
	for (int i = 0; i < dim; ++i) unknowns[i] = simplex.s[b].coord[i];
	return simplex.s[b].error;
}