Mercurial > repos > public > sbplib
annotate +scheme/LaplaceCurvilinear.m @ 545:dfb27a7e801f feature/grids
Add todo to Beam
| author | Jonatan Werpers <jonatan@werpers.com> |
|---|---|
| date | Mon, 21 Aug 2017 11:20:20 +0200 |
| parents | 08b6281ba2a9 |
| children | c5a7a13c03dc |
| rev | line source |
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1 classdef LaplaceCurvilinear < scheme.Scheme |
| 0 | 2 properties |
| 3 m % Number of points in each direction, possibly a vector | |
| 4 h % Grid spacing | |
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5 |
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6 grid |
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7 |
| 0 | 8 order % Order accuracy for the approximation |
| 9 | |
| 10 D % non-stabalized scheme operator | |
| 11 M % Derivative norm | |
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12 a,b |
| 0 | 13 J, Ji |
| 14 a11, a12, a22 | |
| 15 | |
| 16 H % Discrete norm | |
| 17 Hi | |
| 18 H_u, H_v % Norms in the x and y directions | |
| 19 Hu,Hv % Kroneckerd norms. 1'*Hx*v corresponds to integration in the x dir. | |
| 20 Hi_u, Hi_v | |
| 21 Hiu, Hiv | |
| 22 e_w, e_e, e_s, e_n | |
| 23 du_w, dv_w | |
| 24 du_e, dv_e | |
| 25 du_s, dv_s | |
| 26 du_n, dv_n | |
| 27 gamm_u, gamm_v | |
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28 lambda |
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29 |
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30 Dx, Dy % Physical derivatives |
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31 |
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32 x_u |
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33 x_v |
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34 y_u |
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35 y_v |
| 0 | 36 end |
| 37 | |
| 38 methods | |
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39 % Implements a*div(b*grad(u)) as a SBP scheme |
| 539 | 40 % TODO: Implement proper H, it should be the real physical quadrature, the logic quadrature may be but in a separate variable (H_logic?) |
| 41 | |
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42 function obj = LaplaceCurvilinear(g ,order, a, b, opSet) |
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43 default_arg('opSet',@sbp.D2Variable); |
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44 default_arg('a', 1); |
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45 default_arg('b', 1); |
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46 |
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47 |
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48 if b ~=1 |
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49 error('Not implemented yet') |
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50 end |
| 0 | 51 |
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52 assert(isa(g, 'grid.Curvilinear')) |
| 0 | 53 |
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54 m = g.size(); |
| 0 | 55 m_u = m(1); |
| 56 m_v = m(2); | |
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57 m_tot = g.N(); |
| 0 | 58 |
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59 h = g.scaling(); |
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60 h_u = h(1); |
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61 h_v = h(2); |
| 0 | 62 |
| 63 % Operators | |
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64 ops_u = opSet(m_u, {0, 1}, order); |
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65 ops_v = opSet(m_v, {0, 1}, order); |
| 0 | 66 |
| 67 I_u = speye(m_u); | |
| 68 I_v = speye(m_v); | |
| 69 | |
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70 D1_u = ops_u.D1; |
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71 D2_u = ops_u.D2; |
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72 H_u = ops_u.H; |
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73 Hi_u = ops_u.HI; |
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74 e_l_u = ops_u.e_l; |
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75 e_r_u = ops_u.e_r; |
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76 d1_l_u = ops_u.d1_l; |
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77 d1_r_u = ops_u.d1_r; |
| 0 | 78 |
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79 D1_v = ops_v.D1; |
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80 D2_v = ops_v.D2; |
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81 H_v = ops_v.H; |
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82 Hi_v = ops_v.HI; |
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83 e_l_v = ops_v.e_l; |
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84 e_r_v = ops_v.e_r; |
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85 d1_l_v = ops_v.d1_l; |
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86 d1_r_v = ops_v.d1_r; |
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87 |
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88 Du = kr(D1_u,I_v); |
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89 Dv = kr(I_u,D1_v); |
| 0 | 90 |
| 91 % Metric derivatives | |
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92 coords = g.points(); |
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93 x = coords(:,1); |
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94 y = coords(:,2); |
| 0 | 95 |
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96 x_u = Du*x; |
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97 x_v = Dv*x; |
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98 y_u = Du*y; |
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99 y_v = Dv*y; |
| 0 | 100 |
| 101 J = x_u.*y_v - x_v.*y_u; | |
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102 a11 = 1./J .* (x_v.^2 + y_v.^2); |
| 0 | 103 a12 = -1./J .* (x_u.*x_v + y_u.*y_v); |
| 104 a22 = 1./J .* (x_u.^2 + y_u.^2); | |
| 105 lambda = 1/2 * (a11 + a22 - sqrt((a11-a22).^2 + 4*a12.^2)); | |
| 106 | |
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107 % Assemble full operators |
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108 L_12 = spdiags(a12, 0, m_tot, m_tot); |
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109 Duv = Du*L_12*Dv; |
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110 Dvu = Dv*L_12*Du; |
| 0 | 111 |
| 112 Duu = sparse(m_tot); | |
| 113 Dvv = sparse(m_tot); | |
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114 ind = grid.funcToMatrix(g, 1:m_tot); |
| 0 | 115 |
| 116 for i = 1:m_v | |
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117 D = D2_u(a11(ind(:,i))); |
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118 p = ind(:,i); |
| 0 | 119 Duu(p,p) = D; |
| 120 end | |
| 121 | |
| 122 for i = 1:m_u | |
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123 D = D2_v(a22(ind(i,:))); |
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124 p = ind(i,:); |
| 0 | 125 Dvv(p,p) = D; |
| 126 end | |
| 127 | |
| 128 obj.H = kr(H_u,H_v); | |
| 129 obj.Hi = kr(Hi_u,Hi_v); | |
| 130 obj.Hu = kr(H_u,I_v); | |
| 131 obj.Hv = kr(I_u,H_v); | |
| 132 obj.Hiu = kr(Hi_u,I_v); | |
| 133 obj.Hiv = kr(I_u,Hi_v); | |
| 134 | |
| 135 obj.e_w = kr(e_l_u,I_v); | |
| 136 obj.e_e = kr(e_r_u,I_v); | |
| 137 obj.e_s = kr(I_u,e_l_v); | |
| 138 obj.e_n = kr(I_u,e_r_v); | |
| 139 obj.du_w = kr(d1_l_u,I_v); | |
| 140 obj.dv_w = (obj.e_w'*Dv)'; | |
| 141 obj.du_e = kr(d1_r_u,I_v); | |
| 142 obj.dv_e = (obj.e_e'*Dv)'; | |
| 143 obj.du_s = (obj.e_s'*Du)'; | |
| 144 obj.dv_s = kr(I_u,d1_l_v); | |
| 145 obj.du_n = (obj.e_n'*Du)'; | |
| 370 | 146 obj.dv_n = kr(I_u,d1_r_v); |
| 376 | 147 |
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148 obj.x_u = x_u; |
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149 obj.x_v = x_v; |
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150 obj.y_u = y_u; |
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151 obj.y_v = y_v; |
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152 |
| 0 | 153 obj.m = m; |
| 154 obj.h = [h_u h_v]; | |
| 155 obj.order = order; | |
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156 obj.grid = g; |
| 0 | 157 |
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158 obj.a = a; |
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159 obj.b = b; |
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160 obj.J = spdiags(J, 0, m_tot, m_tot); |
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161 obj.Ji = spdiags(1./J, 0, m_tot, m_tot); |
| 0 | 162 obj.a11 = a11; |
| 163 obj.a12 = a12; | |
| 164 obj.a22 = a22; | |
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165 obj.D = obj.Ji*a*(Duu + Duv + Dvu + Dvv); |
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166 obj.lambda = lambda; |
| 0 | 167 |
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168 obj.Dx = spdiag( y_v./J)*Du + spdiag(-y_u./J)*Dv; |
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169 obj.Dy = spdiag(-x_v./J)*Du + spdiag( x_u./J)*Dv; |
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170 |
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171 obj.gamm_u = h_u*ops_u.borrowing.M.d1; |
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172 obj.gamm_v = h_v*ops_v.borrowing.M.d1; |
| 0 | 173 end |
| 174 | |
| 175 | |
| 176 % Closure functions return the opertors applied to the own doamin to close the boundary | |
| 177 % Penalty functions return the opertors to force the solution. In the case of an interface it returns the operator applied to the other doamin. | |
| 178 % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'. | |
| 179 % type is a string specifying the type of boundary condition if there are several. | |
| 180 % data is a function returning the data that should be applied at the boundary. | |
| 181 % neighbour_scheme is an instance of Scheme that should be interfaced to. | |
| 182 % neighbour_boundary is a string specifying which boundary to interface to. | |
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183 function [closure, penalty] = boundary_condition(obj, boundary, type, parameter) |
| 0 | 184 default_arg('type','neumann'); |
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185 default_arg('parameter', []); |
| 0 | 186 |
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187 [e, d_n, d_t, coeff_n, coeff_t, s, gamm, halfnorm_inv , ~, ~, ~, scale_factor] = obj.get_boundary_ops(boundary); |
| 0 | 188 switch type |
| 189 % Dirichlet boundary condition | |
| 190 case {'D','d','dirichlet'} | |
|
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191 % v denotes the solution in the neighbour domain |
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192 tuning = 1.2; |
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193 % tuning = 20.2; |
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194 [e, d_n, d_t, coeff_n, coeff_t, s, gamm, halfnorm_inv_n, halfnorm_inv_t, halfnorm_t] = obj.get_boundary_ops(boundary); |
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195 |
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196 a_n = spdiag(coeff_n); |
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197 a_t = spdiag(coeff_t); |
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198 |
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199 F = (s * a_n * d_n' + s * a_t*d_t')'; |
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200 |
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201 u = obj; |
| 0 | 202 |
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203 b1 = gamm*u.lambda./u.a11.^2; |
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204 b2 = gamm*u.lambda./u.a22.^2; |
| 0 | 205 |
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206 tau = -1./b1 - 1./b2; |
|
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207 tau = tuning * spdiag(tau); |
|
360
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208 sig1 = 1; |
| 0 | 209 |
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210 penalty_parameter_1 = halfnorm_inv_n*(tau + sig1*halfnorm_inv_t*F*e'*halfnorm_t)*e; |
| 0 | 211 |
|
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212 closure = obj.Ji*obj.a * penalty_parameter_1*e'; |
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213 penalty = -obj.Ji*obj.a * penalty_parameter_1; |
| 0 | 214 |
| 215 | |
| 216 % Neumann boundary condition | |
| 217 case {'N','n','neumann'} | |
| 218 a_n = spdiags(coeff_n,0,length(coeff_n),length(coeff_n)); | |
| 219 a_t = spdiags(coeff_t,0,length(coeff_t),length(coeff_t)); | |
| 220 d = (a_n * d_n' + a_t*d_t')'; | |
| 221 | |
| 222 tau1 = -s; | |
| 223 tau2 = 0; | |
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224 tau = obj.a * obj.Ji*(tau1*e + tau2*d); |
| 0 | 225 |
| 226 closure = halfnorm_inv*tau*d'; | |
|
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227 penalty = -halfnorm_inv*tau; |
| 0 | 228 |
|
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229 % Characteristic boundary condition |
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230 case {'characteristic', 'char', 'c'} |
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231 default_arg('parameter', 1); |
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232 beta = parameter; |
| 0 | 233 |
|
347
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234 a_n = spdiags(coeff_n,0,length(coeff_n),length(coeff_n)); |
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235 a_t = spdiags(coeff_t,0,length(coeff_t),length(coeff_t)); |
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236 d = s*(a_n * d_n' + a_t*d_t')'; % outward facing normal derivative |
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237 |
|
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238 tau = -obj.a * 1/beta*obj.Ji*e; |
|
347
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239 |
|
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240 closure{1} = halfnorm_inv*tau*spdiag(scale_factor)*e'; |
|
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241 closure{2} = halfnorm_inv*tau*beta*d'; |
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242 penalty = -halfnorm_inv*tau; |
| 0 | 243 |
| 244 % Unknown, boundary condition | |
| 245 otherwise | |
| 246 error('No such boundary condition: type = %s',type); | |
| 247 end | |
| 248 end | |
| 249 | |
| 250 function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary) | |
| 251 % u denotes the solution in the own domain | |
| 252 % v denotes the solution in the neighbour domain | |
|
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253 tuning = 1.2; |
|
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254 % tuning = 20.2; |
|
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255 [e_u, d_n_u, d_t_u, coeff_n_u, coeff_t_u, s_u, gamm_u, halfnorm_inv_u_n, halfnorm_inv_u_t, halfnorm_u_t, I_u] = obj.get_boundary_ops(boundary); |
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256 [e_v, d_n_v, d_t_v, coeff_n_v, coeff_t_v, s_v, gamm_v, halfnorm_inv_v_n, halfnorm_inv_v_t, halfnorm_v_t, I_v] = neighbour_scheme.get_boundary_ops(neighbour_boundary); |
| 0 | 257 |
|
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258 a_n_u = spdiag(coeff_n_u); |
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259 a_t_u = spdiag(coeff_t_u); |
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260 a_n_v = spdiag(coeff_n_v); |
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261 a_t_v = spdiag(coeff_t_v); |
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262 |
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263 F_u = (s_u * a_n_u * d_n_u' + s_u * a_t_u*d_t_u')'; |
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264 F_v = (s_v * a_n_v * d_n_v' + s_v * a_t_v*d_t_v')'; |
| 0 | 265 |
|
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266 u = obj; |
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267 v = neighbour_scheme; |
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268 |
|
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269 b1_u = gamm_u*u.lambda(I_u)./u.a11(I_u).^2; |
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270 b2_u = gamm_u*u.lambda(I_u)./u.a22(I_u).^2; |
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271 b1_v = gamm_v*v.lambda(I_v)./v.a11(I_v).^2; |
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272 b2_v = gamm_v*v.lambda(I_v)./v.a22(I_v).^2; |
|
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273 |
|
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274 tau = -1./(4*b1_u) -1./(4*b1_v) -1./(4*b2_u) -1./(4*b2_v); |
|
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275 tau = tuning * spdiag(tau); |
| 0 | 276 sig1 = 1/2; |
| 83 | 277 sig2 = -1/2; |
| 0 | 278 |
|
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279 penalty_parameter_1 = halfnorm_inv_u_n*(e_u*tau + sig1*halfnorm_inv_u_t*F_u*e_u'*halfnorm_u_t*e_u); |
| 0 | 280 penalty_parameter_2 = halfnorm_inv_u_n * sig2 * e_u; |
| 281 | |
| 282 | |
|
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283 closure = obj.Ji*obj.a * ( penalty_parameter_1*e_u' + penalty_parameter_2*F_u'); |
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284 penalty = obj.Ji*obj.a * (-penalty_parameter_1*e_v' + penalty_parameter_2*F_v'); |
| 0 | 285 end |
| 286 | |
| 287 % Ruturns the boundary ops and sign for the boundary specified by the string boundary. | |
| 288 % The right boundary is considered the positive boundary | |
|
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289 % |
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290 % I -- the indecies of the boundary points in the grid matrix |
|
384
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291 function [e, d_n, d_t, coeff_n, coeff_t, s, gamm, halfnorm_inv_n, halfnorm_inv_t, halfnorm_t, I, scale_factor] = get_boundary_ops(obj, boundary) |
|
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292 |
|
337
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293 % gridMatrix = zeros(obj.m(2),obj.m(1)); |
|
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294 % gridMatrix(:) = 1:numel(gridMatrix); |
|
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|
295 |
|
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|
296 ind = grid.funcToMatrix(obj.grid, 1:prod(obj.m)); |
|
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297 |
| 0 | 298 switch boundary |
| 299 case 'w' | |
| 300 e = obj.e_w; | |
| 301 d_n = obj.du_w; | |
| 302 d_t = obj.dv_w; | |
| 303 s = -1; | |
| 304 | |
|
337
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|
305 I = ind(1,:); |
|
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306 coeff_n = obj.a11(I); |
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307 coeff_t = obj.a12(I); |
|
384
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|
308 scale_factor = sqrt(obj.x_v(I).^2 + obj.y_v(I).^2); |
| 0 | 309 case 'e' |
| 310 e = obj.e_e; | |
| 311 d_n = obj.du_e; | |
| 312 d_t = obj.dv_e; | |
| 313 s = 1; | |
| 314 | |
|
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|
315 I = ind(end,:); |
|
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316 coeff_n = obj.a11(I); |
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|
317 coeff_t = obj.a12(I); |
|
384
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|
318 scale_factor = sqrt(obj.x_v(I).^2 + obj.y_v(I).^2); |
| 0 | 319 case 's' |
| 320 e = obj.e_s; | |
| 321 d_n = obj.dv_s; | |
| 322 d_t = obj.du_s; | |
| 323 s = -1; | |
| 324 | |
|
337
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|
325 I = ind(:,1)'; |
|
89
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326 coeff_n = obj.a22(I); |
|
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|
327 coeff_t = obj.a12(I); |
|
384
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|
328 scale_factor = sqrt(obj.x_u(I).^2 + obj.y_u(I).^2); |
| 0 | 329 case 'n' |
| 330 e = obj.e_n; | |
| 331 d_n = obj.dv_n; | |
| 332 d_t = obj.du_n; | |
| 333 s = 1; | |
| 334 | |
|
337
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|
335 I = ind(:,end)'; |
|
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|
336 coeff_n = obj.a22(I); |
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|
337 coeff_t = obj.a12(I); |
|
384
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|
338 scale_factor = sqrt(obj.x_u(I).^2 + obj.y_u(I).^2); |
| 0 | 339 otherwise |
| 340 error('No such boundary: boundary = %s',boundary); | |
| 341 end | |
| 342 | |
| 343 switch boundary | |
| 344 case {'w','e'} | |
| 345 halfnorm_inv_n = obj.Hiu; | |
| 346 halfnorm_inv_t = obj.Hiv; | |
| 347 halfnorm_t = obj.Hv; | |
| 348 gamm = obj.gamm_u; | |
| 349 case {'s','n'} | |
| 350 halfnorm_inv_n = obj.Hiv; | |
| 351 halfnorm_inv_t = obj.Hiu; | |
| 352 halfnorm_t = obj.Hu; | |
| 353 gamm = obj.gamm_v; | |
| 354 end | |
| 355 end | |
| 356 | |
| 357 function N = size(obj) | |
| 358 N = prod(obj.m); | |
| 359 end | |
| 360 | |
| 361 | |
| 362 end | |
| 363 end |