attempt at generating BCs automatically, but n=m issue

dev/BC_generator.m

@@ -0,0 +1,145 @@
+syms m a_1 a_2
+N = 4;
+BC_a1 = generate_BC(0,0,1,1,0,m,a_1,N)
+BC_a2 = generate_BC(0,0,1,1,0,m,a_2,N)
+
+function BC = generate_BC(LHS_d_region, LHS_phi_region, RHS_d_region, RHS_phi_region, both_Z_region, m, a, N)
+
+    % pass in five binary bits here
+    % false = 0 = out, true = 1 = in
+    % expected correct answer: 0, 0, 1, 1, 0
+
+    % LHS (left hand side)  is defined as the side where orthogonality happens
+    % RHS (right hand side) is defined as the side where coupling integral happens
+
+    Z_m = get_Z(both_Z_region, m, a);
+
+    syms z h
+    % LHS
+    integrand_LHS = get_dphi_p_dr(LHS_phi_region, a) * Z_m;
+    LHS = int(integrand_LHS, z, -h, LHS_d_region) + ...
+        (h - LHS_d_region) * get_C(LHS_phi_region, m, a) .* get_dR_dr(LHS_phi_region, m, a);
+
+    % RHS
+    syms n % summation index
+    integrand_RHS = get_dphi_p_dr(RHS_phi_region, a) * Z_m;
+    Z_n = get_Z(RHS_phi_region, n, a);
+    coupling_integral(n,m) = int(Z_n(n) * Z_m(m), z, -h, RHS_d_region);
+    sum_RHS(n,m) = get_C(RHS_phi_region,n, a) .* get_dR_dr(RHS_phi_region,n, a) * coupling_integral(n,m);
+    RHS = int(integrand_RHS, z, -h, RHS_d_region) + symsum(sum_RHS, n, 0, N);
+
+    % equation
+    BC = LHS == RHS;
+
+end
+
+function d_phi_p_dr = get_dphi_p_dr(in_out, radius)
+    syms d1 d2 h z r
+    region = get_region_from_in_out_radius(in_out,radius);
+    if strcmp(region,'i1') || strcmp(region,'i2')
+        if strcmp(region,'i1')
+            di = d1;
+        else
+            di = d2;
+        end
+        % eq 5
+        phi_p = 1/(2*(h-di)) * ((z+h)^2 - r^2/2);
+        d_phi_p_dr = subs( diff(phi_p, r), r, radius);
+    elseif strcmp(region,'e')
+        d_phi_p_dr = 0;
+    end
+
+end
+
+function lambda = lambda(n,region)
+    syms d1 d2 h
+    % eq 4
+    if strcmp(region,'i1')
+        di = d1;
+    else
+        di = d2;
+    end
+    lambda(n) = n*pi/(h-di);
+end
+
+function Z = get_Z(in_out, n, radius)
+    syms z h
+    region = get_region_from_in_out_radius(in_out,radius);
+
+    if strcmp(region,'i1') || strcmp(region,'i2')
+        % equation 9
+        Z = piecewise(n==0, 1, ...
+                      n>=1, sqrt(2)*cos(lambda(n,region)*(z+h)) ...
+                      );
+    elseif strcmp(region,'e')
+        syms m0 m_k
+        % eq 2.34 in analytical methods book, also eq 16 in Seah and Yeung 2006 
+        N_k(n) = piecewise(n==0, 1/2*(1+sinh(2*m0*h)/(2*m0*h)), ...
+                                  n>=1, 1/2*(1+sin(2*m_k(n)*h)/(2*m_k(n)*h)) ...
+                                  );
+        % equation 14
+        Z = piecewise(n==0, 1/sqrt(N_k(n)) * cosh(m0 * (z+h)), ...
+                      n>=1, 1/sqrt(N_k(n)) * cos(m_k(n) * (z+h)) ...
+                      );
+    end
+end
+
+function C = get_C(in_out, n, radius)
+    region = get_region_from_in_out_radius(in_out,radius);
+    syms C_1n_1 C_1n_2 C_2n_1 B_k
+    symfun(C_1n_1,n); symfun(C_1n_2,n); symfun(C_2n_1,n); symfun(B_k,n);
+    if strcmp(region,'i1')
+        C = [C_1n_1, C_2n_1];
+    elseif strcmp(region,'i2')
+        C = [C_1n_2, 0];
+    elseif strcmp(region,'e')
+        C = B_k;
+    end
+
+end
+
+function dR_dr = get_dR_dr(in_out, n, radius)
+    region = get_region_from_in_out_radius(in_out,radius);
+
+    syms r
+    R = get_R(region, n);
+    dR_dr = subs( diff(R,r), r, radius);
+
+end
+
+function R = get_R(region, n)
+    syms r a2 m0 R m_k
+    symfun(R,n); symfun(m_k,n);
+    if strcmp(region,'i1') || strcmp(region,'i2')
+        % eq 7
+        R_1n(n) = piecewise(n==0, 1/2, n>=1, ...
+                            besseli(0,lambda(n,region)*r)/besseli(0,lambda(n,region)*a2));
+        if strcmp(region,'i1')
+            % eq 8
+            R_2n(n) = sym(0);
+        else
+            R_2n(n) = piecewise(n==0, 1/2*log(r/a2), ...
+                                n>=1, besselk(0,lambda(n,region)*r)/besselk(0,lambda(n,region)*a2));
+        end
+        R = [R_1n, R_2n];
+
+    elseif strcmp(region,'e')
+        % eq 13
+        Lambda_k(n) = piecewise(n==0, besselh(0,1,m0*r)/besselh(0,1,m0*a2), ...
+                                n>=1, besselk(0,m_k(n)*r)/besselk(0,m_k(n)*a2));
+        R = Lambda_k;
+    end
+end
+
+function region = get_region_from_in_out_radius(in_out,radius)
+    syms a_1 a_2
+    if radius==a_1 && in_out==1
+        region = 'i1';
+    elseif (radius==a_1 && in_out==0) || (radius==a_2 && in_out==1)
+        region = 'i2';
+    elseif radius==a_2 && in_out==0
+        region = 'e';
+    else
+        error('invalid radius or in/out value')
+    end
+end