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comparison +scheme/LaplaceCurvilinear.m @ 592:4422c4476650 feature/utux2D

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Merge with feature/grids
author Martin Almquist <martin.almquist@it.uu.se>
date Mon, 11 Sep 2017 14:17:15 +0200
parents 33b962620e24
children 07f8311374c6
comparison
equal deleted inserted replaced
591:39554f2de783 592:4422c4476650
5 5
6 grid 6 grid
7 7
8 order % Order accuracy for the approximation 8 order % Order accuracy for the approximation
9 9
10 D % non-stabalized scheme operator 10 a,b % Parameters of the operator
11 M % Derivative norm 11
12 a,b 12
13 % Inner products and operators for physical coordinates
14 D % Laplace operator
15 H, Hi % Inner product
16 e_w, e_e, e_s, e_n
17 d_w, d_e, d_s, d_n % Normal derivatives at the boundary
18 H_w, H_e, H_s, H_n % Boundary inner products
19 Dx, Dy % Physical derivatives
20 M % Gradient inner product
21
22 % Metric coefficients
13 J, Ji 23 J, Ji
14 a11, a12, a22 24 a11, a12, a22
15 25 x_u
16 H % Discrete norm 26 x_v
17 Hi 27 y_u
28 y_v
29
30 % Inner product and operators for logical coordinates
18 H_u, H_v % Norms in the x and y directions 31 H_u, H_v % Norms in the x and y directions
32 Hi_u, Hi_v
19 Hu,Hv % Kroneckerd norms. 1'*Hx*v corresponds to integration in the x dir. 33 Hu,Hv % Kroneckerd norms. 1'*Hx*v corresponds to integration in the x dir.
20 Hi_u, Hi_v
21 Hiu, Hiv 34 Hiu, Hiv
22 e_w, e_e, e_s, e_n
23 du_w, dv_w 35 du_w, dv_w
24 du_e, dv_e 36 du_e, dv_e
25 du_s, dv_s 37 du_s, dv_s
26 du_n, dv_n 38 du_n, dv_n
27 gamm_u, gamm_v 39 gamm_u, gamm_v
28 lambda 40 lambda
29
30 Dx, Dy % Physical derivatives
31
32 x_u
33 x_v
34 y_u
35 y_v
36 end 41 end
37 42
38 methods 43 methods
39 % Implements a*div(b*grad(u)) as a SBP scheme 44 % Implements a*div(b*grad(u)) as a SBP scheme
45 % TODO: Implement proper H, it should be the real physical quadrature, the logic quadrature may be but in a separate variable (H_logic?)
46
40 function obj = LaplaceCurvilinear(g ,order, a, b, opSet) 47 function obj = LaplaceCurvilinear(g ,order, a, b, opSet)
41 default_arg('opSet',@sbp.D2Variable); 48 default_arg('opSet',@sbp.D2Variable);
42 default_arg('a', 1); 49 default_arg('a', 1);
43 default_arg('b', 1); 50 default_arg('b', 1);
44 51
45
46 if b ~=1 52 if b ~=1
47 error('Not implemented yet') 53 error('Not implemented yet')
48 end 54 end
49 55
50 assert(isa(g, 'grid.Curvilinear')) 56 assert(isa(g, 'grid.Curvilinear'))
56 62
57 h = g.scaling(); 63 h = g.scaling();
58 h_u = h(1); 64 h_u = h(1);
59 h_v = h(2); 65 h_v = h(2);
60 66
61 % Operators 67
68 % 1D operators
62 ops_u = opSet(m_u, {0, 1}, order); 69 ops_u = opSet(m_u, {0, 1}, order);
63 ops_v = opSet(m_v, {0, 1}, order); 70 ops_v = opSet(m_v, {0, 1}, order);
64 71
65 I_u = speye(m_u); 72 I_u = speye(m_u);
66 I_v = speye(m_v); 73 I_v = speye(m_v);
81 e_l_v = ops_v.e_l; 88 e_l_v = ops_v.e_l;
82 e_r_v = ops_v.e_r; 89 e_r_v = ops_v.e_r;
83 d1_l_v = ops_v.d1_l; 90 d1_l_v = ops_v.d1_l;
84 d1_r_v = ops_v.d1_r; 91 d1_r_v = ops_v.d1_r;
85 92
93
94 % Logical operators
86 Du = kr(D1_u,I_v); 95 Du = kr(D1_u,I_v);
87 Dv = kr(I_u,D1_v); 96 Dv = kr(I_u,D1_v);
88 97 obj.Hu = kr(H_u,I_v);
89 % Metric derivatives 98 obj.Hv = kr(I_u,H_v);
99 obj.Hiu = kr(Hi_u,I_v);
100 obj.Hiv = kr(I_u,Hi_v);
101
102 e_w = kr(e_l_u,I_v);
103 e_e = kr(e_r_u,I_v);
104 e_s = kr(I_u,e_l_v);
105 e_n = kr(I_u,e_r_v);
106 obj.du_w = kr(d1_l_u,I_v);
107 obj.dv_w = (e_w'*Dv)';
108 obj.du_e = kr(d1_r_u,I_v);
109 obj.dv_e = (e_e'*Dv)';
110 obj.du_s = (e_s'*Du)';
111 obj.dv_s = kr(I_u,d1_l_v);
112 obj.du_n = (e_n'*Du)';
113 obj.dv_n = kr(I_u,d1_r_v);
114
115
116 % Metric coefficients
90 coords = g.points(); 117 coords = g.points();
91 x = coords(:,1); 118 x = coords(:,1);
92 y = coords(:,2); 119 y = coords(:,2);
93 120
94 x_u = Du*x; 121 x_u = Du*x;
100 a11 = 1./J .* (x_v.^2 + y_v.^2); 127 a11 = 1./J .* (x_v.^2 + y_v.^2);
101 a12 = -1./J .* (x_u.*x_v + y_u.*y_v); 128 a12 = -1./J .* (x_u.*x_v + y_u.*y_v);
102 a22 = 1./J .* (x_u.^2 + y_u.^2); 129 a22 = 1./J .* (x_u.^2 + y_u.^2);
103 lambda = 1/2 * (a11 + a22 - sqrt((a11-a22).^2 + 4*a12.^2)); 130 lambda = 1/2 * (a11 + a22 - sqrt((a11-a22).^2 + 4*a12.^2));
104 131
132 obj.x_u = x_u;
133 obj.x_v = x_v;
134 obj.y_u = y_u;
135 obj.y_v = y_v;
136
137
105 % Assemble full operators 138 % Assemble full operators
106 L_12 = spdiags(a12, 0, m_tot, m_tot); 139 L_12 = spdiag(a12);
107 Duv = Du*L_12*Dv; 140 Duv = Du*L_12*Dv;
108 Dvu = Dv*L_12*Du; 141 Dvu = Dv*L_12*Du;
109 142
110 Duu = sparse(m_tot); 143 Duu = sparse(m_tot);
111 Dvv = sparse(m_tot); 144 Dvv = sparse(m_tot);
121 D = D2_v(a22(ind(i,:))); 154 D = D2_v(a22(ind(i,:)));
122 p = ind(i,:); 155 p = ind(i,:);
123 Dvv(p,p) = D; 156 Dvv(p,p) = D;
124 end 157 end
125 158
126 obj.H = kr(H_u,H_v); 159
127 obj.Hi = kr(Hi_u,Hi_v); 160 % Physical operators
128 obj.Hu = kr(H_u,I_v); 161 obj.J = spdiag(J);
129 obj.Hv = kr(I_u,H_v); 162 obj.Ji = spdiag(1./J);
130 obj.Hiu = kr(Hi_u,I_v); 163
131 obj.Hiv = kr(I_u,Hi_v); 164 obj.D = obj.Ji*a*(Duu + Duv + Dvu + Dvv);
132 165 obj.H = obj.J*kr(H_u,H_v);
133 obj.e_w = kr(e_l_u,I_v); 166 obj.Hi = obj.Ji*kr(Hi_u,Hi_v);
134 obj.e_e = kr(e_r_u,I_v); 167
135 obj.e_s = kr(I_u,e_l_v); 168 obj.e_w = e_w;
136 obj.e_n = kr(I_u,e_r_v); 169 obj.e_e = e_e;
137 obj.du_w = kr(d1_l_u,I_v); 170 obj.e_s = e_s;
138 obj.dv_w = (obj.e_w'*Dv)'; 171 obj.e_n = e_n;
139 obj.du_e = kr(d1_r_u,I_v); 172
140 obj.dv_e = (obj.e_e'*Dv)'; 173 %% normal derivatives
141 obj.du_s = (obj.e_s'*Du)'; 174 I_w = ind(1,:);
142 obj.dv_s = kr(I_u,d1_l_v); 175 I_e = ind(end,:);
143 obj.du_n = (obj.e_n'*Du)'; 176 I_s = ind(:,1);
144 obj.dv_n = kr(I_u,d1_r_v); 177 I_n = ind(:,end);
145 178
146 obj.x_u = x_u; 179 a11_w = spdiag(a11(I_w));
147 obj.x_v = x_v; 180 a12_w = spdiag(a12(I_w));
148 obj.y_u = y_u; 181 a11_e = spdiag(a11(I_e));
149 obj.y_v = y_v; 182 a12_e = spdiag(a12(I_e));
150 183 a22_s = spdiag(a22(I_s));
184 a12_s = spdiag(a12(I_s));
185 a22_n = spdiag(a22(I_n));
186 a12_n = spdiag(a12(I_n));
187
188 s_w = sqrt((e_w'*x_v).^2 + (e_w'*y_v).^2);
189 s_e = sqrt((e_e'*x_v).^2 + (e_e'*y_v).^2);
190 s_s = sqrt((e_s'*x_u).^2 + (e_s'*y_u).^2);
191 s_n = sqrt((e_n'*x_u).^2 + (e_n'*y_u).^2);
192
193 obj.d_w = -1*(spdiag(1./s_w)*(a11_w*obj.du_w' + a12_w*obj.dv_w'))';
194 obj.d_e = (spdiag(1./s_e)*(a11_e*obj.du_e' + a12_e*obj.dv_e'))';
195 obj.d_s = -1*(spdiag(1./s_s)*(a22_s*obj.dv_s' + a12_s*obj.du_s'))';
196 obj.d_n = (spdiag(1./s_n)*(a22_n*obj.dv_n' + a12_n*obj.du_n'))';
197
198 obj.Dx = spdiag( y_v./J)*Du + spdiag(-y_u./J)*Dv;
199 obj.Dy = spdiag(-x_v./J)*Du + spdiag( x_u./J)*Dv;
200
201 %% Boundary inner products
202 obj.H_w = H_v*spdiag(s_w);
203 obj.H_e = H_v*spdiag(s_e);
204 obj.H_s = H_u*spdiag(s_s);
205 obj.H_n = H_u*spdiag(s_n);
206
207 % Misc.
151 obj.m = m; 208 obj.m = m;
152 obj.h = [h_u h_v]; 209 obj.h = [h_u h_v];
153 obj.order = order; 210 obj.order = order;
154 obj.grid = g; 211 obj.grid = g;
155 212
156 obj.a = a; 213 obj.a = a;
157 obj.b = b; 214 obj.b = b;
158 obj.J = spdiags(J, 0, m_tot, m_tot);
159 obj.Ji = spdiags(1./J, 0, m_tot, m_tot);
160 obj.a11 = a11; 215 obj.a11 = a11;
161 obj.a12 = a12; 216 obj.a12 = a12;
162 obj.a22 = a22; 217 obj.a22 = a22;
163 obj.D = obj.Ji*a*(Duu + Duv + Dvu + Dvv);
164 obj.lambda = lambda; 218 obj.lambda = lambda;
165
166 obj.Dx = spdiag( y_v./J)*Du + spdiag(-y_u./J)*Dv;
167 obj.Dy = spdiag(-x_v./J)*Du + spdiag( x_u./J)*Dv;
168 219
169 obj.gamm_u = h_u*ops_u.borrowing.M.d1; 220 obj.gamm_u = h_u*ops_u.borrowing.M.d1;
170 obj.gamm_v = h_v*ops_v.borrowing.M.d1; 221 obj.gamm_v = h_v*ops_v.borrowing.M.d1;
171 end 222 end
172 223
180 % neighbour_boundary is a string specifying which boundary to interface to. 231 % neighbour_boundary is a string specifying which boundary to interface to.
181 function [closure, penalty] = boundary_condition(obj, boundary, type, parameter) 232 function [closure, penalty] = boundary_condition(obj, boundary, type, parameter)
182 default_arg('type','neumann'); 233 default_arg('type','neumann');
183 default_arg('parameter', []); 234 default_arg('parameter', []);
184 235
185 [e, d_n, d_t, coeff_n, coeff_t, s, gamm, halfnorm_inv , ~, ~, ~, scale_factor] = obj.get_boundary_ops(boundary); 236 [e, d, gamm, H_b, ~] = obj.get_boundary_ops(boundary);
186 switch type 237 switch type
187 % Dirichlet boundary condition 238 % Dirichlet boundary condition
188 case {'D','d','dirichlet'} 239 case {'D','d','dirichlet'}
189 % v denotes the solution in the neighbour domain
190 tuning = 1.2; 240 tuning = 1.2;
191 % tuning = 20.2; 241 % tuning = 20.2;
192 [e, d_n, d_t, coeff_n, coeff_t, s, gamm, halfnorm_inv_n, halfnorm_inv_t, halfnorm_t] = obj.get_boundary_ops(boundary); 242
193 243 b1 = gamm*obj.lambda./obj.a11.^2;
194 a_n = spdiag(coeff_n); 244 b2 = gamm*obj.lambda./obj.a22.^2;
195 a_t = spdiag(coeff_t); 245
196 246 tau1 = tuning * spdiag(-1./b1 - 1./b2);
197 F = (s * a_n * d_n' + s * a_t*d_t')'; 247 tau2 = 1;
198 248
199 u = obj; 249 tau = (tau1*e + tau2*d)*H_b;
200 250
201 b1 = gamm*u.lambda./u.a11.^2; 251 closure = obj.a*obj.Hi*tau*e';
202 b2 = gamm*u.lambda./u.a22.^2; 252 penalty = -obj.a*obj.Hi*tau;
203
204 tau = -1./b1 - 1./b2;
205 tau = tuning * spdiag(tau);
206 sig1 = 1;
207
208 penalty_parameter_1 = halfnorm_inv_n*(tau + sig1*halfnorm_inv_t*F*e'*halfnorm_t)*e;
209
210 closure = obj.Ji*obj.a * penalty_parameter_1*e';
211 penalty = -obj.Ji*obj.a * penalty_parameter_1;
212 253
213 254
214 % Neumann boundary condition 255 % Neumann boundary condition
215 case {'N','n','neumann'} 256 case {'N','n','neumann'}
216 a_n = spdiags(coeff_n,0,length(coeff_n),length(coeff_n)); 257 tau1 = -1;
217 a_t = spdiags(coeff_t,0,length(coeff_t),length(coeff_t));
218 d = (a_n * d_n' + a_t*d_t')';
219
220 tau1 = -s;
221 tau2 = 0; 258 tau2 = 0;
222 tau = obj.a * obj.Ji*(tau1*e + tau2*d); 259 tau = (tau1*e + tau2*d)*H_b;
223 260
224 closure = halfnorm_inv*tau*d'; 261 closure = obj.a*obj.Hi*tau*d';
225 penalty = -halfnorm_inv*tau; 262 penalty = -obj.a*obj.Hi*tau;
226 263
227 % Characteristic boundary condition
228 case {'characteristic', 'char', 'c'}
229 default_arg('parameter', 1);
230 beta = parameter;
231
232 a_n = spdiags(coeff_n,0,length(coeff_n),length(coeff_n));
233 a_t = spdiags(coeff_t,0,length(coeff_t),length(coeff_t));
234 d = s*(a_n * d_n' + a_t*d_t')'; % outward facing normal derivative
235
236 tau = -obj.a * 1/beta*obj.Ji*e;
237
238 closure{1} = halfnorm_inv*tau*spdiag(scale_factor)*e';
239 closure{2} = halfnorm_inv*tau*beta*d';
240 penalty = -halfnorm_inv*tau;
241 264
242 % Unknown, boundary condition 265 % Unknown, boundary condition
243 otherwise 266 otherwise
244 error('No such boundary condition: type = %s',type); 267 error('No such boundary condition: type = %s',type);
245 end 268 end
248 function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary) 271 function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
249 % u denotes the solution in the own domain 272 % u denotes the solution in the own domain
250 % v denotes the solution in the neighbour domain 273 % v denotes the solution in the neighbour domain
251 tuning = 1.2; 274 tuning = 1.2;
252 % tuning = 20.2; 275 % tuning = 20.2;
253 [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); 276 [e_u, d_u, gamm_u, H_b_u, I_u] = obj.get_boundary_ops(boundary);
254 [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); 277 [e_v, d_v, gamm_v, H_b_v, I_v] = neighbour_scheme.get_boundary_ops(neighbour_boundary);
255
256 a_n_u = spdiag(coeff_n_u);
257 a_t_u = spdiag(coeff_t_u);
258 a_n_v = spdiag(coeff_n_v);
259 a_t_v = spdiag(coeff_t_v);
260
261 F_u = (s_u * a_n_u * d_n_u' + s_u * a_t_u*d_t_u')';
262 F_v = (s_v * a_n_v * d_n_v' + s_v * a_t_v*d_t_v')';
263 278
264 u = obj; 279 u = obj;
265 v = neighbour_scheme; 280 v = neighbour_scheme;
266 281
267 b1_u = gamm_u*u.lambda(I_u)./u.a11(I_u).^2; 282 b1_u = gamm_u*u.lambda(I_u)./u.a11(I_u).^2;
268 b2_u = gamm_u*u.lambda(I_u)./u.a22(I_u).^2; 283 b2_u = gamm_u*u.lambda(I_u)./u.a22(I_u).^2;
269 b1_v = gamm_v*v.lambda(I_v)./v.a11(I_v).^2; 284 b1_v = gamm_v*v.lambda(I_v)./v.a11(I_v).^2;
270 b2_v = gamm_v*v.lambda(I_v)./v.a22(I_v).^2; 285 b2_v = gamm_v*v.lambda(I_v)./v.a22(I_v).^2;
271 286
272 tau = -1./(4*b1_u) -1./(4*b1_v) -1./(4*b2_u) -1./(4*b2_v); 287 tau1 = -1./(4*b1_u) -1./(4*b1_v) -1./(4*b2_u) -1./(4*b2_v);
273 tau = tuning * spdiag(tau); 288 tau1 = tuning * spdiag(tau1);
274 sig1 = 1/2; 289 tau2 = 1/2;
275 sig2 = -1/2; 290
276 291 sig1 = -1/2;
277 penalty_parameter_1 = halfnorm_inv_u_n*(e_u*tau + sig1*halfnorm_inv_u_t*F_u*e_u'*halfnorm_u_t*e_u); 292 sig2 = 0;
278 penalty_parameter_2 = halfnorm_inv_u_n * sig2 * e_u; 293
279 294 tau = (e_u*tau1 + tau2*d_u)*H_b_u;
280 295 sig = (sig1*e_u + sig2*d_u)*H_b_u;
281 closure = obj.Ji*obj.a * ( penalty_parameter_1*e_u' + penalty_parameter_2*F_u'); 296
282 penalty = obj.Ji*obj.a * (-penalty_parameter_1*e_v' + penalty_parameter_2*F_v'); 297 closure = obj.a*obj.Hi*( tau*e_u' + sig*d_u');
298 penalty = obj.a*obj.Hi*(-tau*e_v' + sig*d_v');
283 end 299 end
284 300
285 % Ruturns the boundary ops and sign for the boundary specified by the string boundary. 301 % Ruturns the boundary ops and sign for the boundary specified by the string boundary.
286 % The right boundary is considered the positive boundary 302 % The right boundary is considered the positive boundary
287 % 303 %
288 % I -- the indecies of the boundary points in the grid matrix 304 % I -- the indecies of the boundary points in the grid matrix
289 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) 305 function [e, d, gamm, H_b, I] = get_boundary_ops(obj, boundary)
290 306
291 % gridMatrix = zeros(obj.m(2),obj.m(1)); 307 % gridMatrix = zeros(obj.m(2),obj.m(1));
292 % gridMatrix(:) = 1:numel(gridMatrix); 308 % gridMatrix(:) = 1:numel(gridMatrix);
293 309
294 ind = grid.funcToMatrix(obj.grid, 1:prod(obj.m)); 310 ind = grid.funcToMatrix(obj.grid, 1:prod(obj.m));
295 311
296 switch boundary 312 switch boundary
297 case 'w' 313 case 'w'
298 e = obj.e_w; 314 e = obj.e_w;
299 d_n = obj.du_w; 315 d = obj.d_w;
300 d_t = obj.dv_w; 316 H_b = obj.H_w;
301 s = -1;
302
303 I = ind(1,:); 317 I = ind(1,:);
304 coeff_n = obj.a11(I);
305 coeff_t = obj.a12(I);
306 scale_factor = sqrt(obj.x_v(I).^2 + obj.y_v(I).^2);
307 case 'e' 318 case 'e'
308 e = obj.e_e; 319 e = obj.e_e;
309 d_n = obj.du_e; 320 d = obj.d_e;
310 d_t = obj.dv_e; 321 H_b = obj.H_e;
311 s = 1;
312
313 I = ind(end,:); 322 I = ind(end,:);
314 coeff_n = obj.a11(I);
315 coeff_t = obj.a12(I);
316 scale_factor = sqrt(obj.x_v(I).^2 + obj.y_v(I).^2);
317 case 's' 323 case 's'
318 e = obj.e_s; 324 e = obj.e_s;
319 d_n = obj.dv_s; 325 d = obj.d_s;
320 d_t = obj.du_s; 326 H_b = obj.H_s;
321 s = -1;
322
323 I = ind(:,1)'; 327 I = ind(:,1)';
324 coeff_n = obj.a22(I);
325 coeff_t = obj.a12(I);
326 scale_factor = sqrt(obj.x_u(I).^2 + obj.y_u(I).^2);
327 case 'n' 328 case 'n'
328 e = obj.e_n; 329 e = obj.e_n;
329 d_n = obj.dv_n; 330 d = obj.d_n;
330 d_t = obj.du_n; 331 H_b = obj.H_n;
331 s = 1;
332
333 I = ind(:,end)'; 332 I = ind(:,end)';
334 coeff_n = obj.a22(I);
335 coeff_t = obj.a12(I);
336 scale_factor = sqrt(obj.x_u(I).^2 + obj.y_u(I).^2);
337 otherwise 333 otherwise
338 error('No such boundary: boundary = %s',boundary); 334 error('No such boundary: boundary = %s',boundary);
339 end 335 end
340 336
341 switch boundary 337 switch boundary
342 case {'w','e'} 338 case {'w','e'}
343 halfnorm_inv_n = obj.Hiu;
344 halfnorm_inv_t = obj.Hiv;
345 halfnorm_t = obj.Hv;
346 gamm = obj.gamm_u; 339 gamm = obj.gamm_u;
347 case {'s','n'} 340 case {'s','n'}
348 halfnorm_inv_n = obj.Hiv;
349 halfnorm_inv_t = obj.Hiu;
350 halfnorm_t = obj.Hu;
351 gamm = obj.gamm_v; 341 gamm = obj.gamm_v;
352 end 342 end
353 end 343 end
354 344
355 function N = size(obj) 345 function N = size(obj)
356 N = prod(obj.m); 346 N = prod(obj.m);
357 end 347 end
358
359
360 end 348 end
361 end 349 end