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comparison +scheme/Beam2d.m @ 0:48b6fb693025

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Initial commit.
author Jonatan Werpers <jonatan@werpers.com>
date Thu, 17 Sep 2015 10:12:50 +0200
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children d52e5cdb6eff
comparison
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1 classdef SchmBeam2d < noname.Scheme
2 properties
3 m % Number of points in each direction, possibly a vector
4 N % Number of points total
5 h % Grid spacing
6 u,v % Grid
7 x,y % Values of x and y for each grid point
8 order % Order accuracy for the approximation
9
10 D % non-stabalized scheme operator
11 M % Derivative norm
12 alpha
13
14 H % Discrete norm
15 Hi
16 H_x, H_y % Norms in the x and y directions
17 Hx,Hy % Kroneckerd norms. 1'*Hx*v corresponds to integration in the x dir.
18 Hi_x, Hi_y
19 Hix, Hiy
20 e_w, e_e, e_s, e_n
21 d1_w, d1_e, d1_s, d1_n
22 d2_w, d2_e, d2_s, d2_n
23 d3_w, d3_e, d3_s, d3_n
24 gamm_x, gamm_y
25 delt_x, delt_y
26 end
27
28 methods
29 function obj = SchmBeam2d(m,lim,order,alpha,opsGen)
30 default_arg('opsGen',@sbp.Higher);
31 default_arg('a',1);
32
33 if length(m) == 1
34 m = [m m];
35 end
36
37 m_x = m(1);
38 m_y = m(2);
39
40 xlim = lim{1};
41 ylim = lim{2};
42
43 [x, h_x] = util.get_grid(xlim{:},m_x);
44 [y, h_y] = util.get_grid(ylim{:},m_y);
45
46 ops_x = opsGen(m_x,h_x,order);
47 ops_y = opsGen(m_y,h_y,order);
48
49 I_x = speye(m_x);
50 I_y = speye(m_y);
51
52
53
54
55 D4_x = sparse(ops_x.derivatives.D4);
56 H_x = sparse(ops_x.norms.H);
57 Hi_x = sparse(ops_x.norms.HI);
58 e_l_x = sparse(ops_x.boundary.e_1);
59 e_r_x = sparse(ops_x.boundary.e_m);
60 d1_l_x = sparse(ops_x.boundary.S_1);
61 d1_r_x = sparse(ops_x.boundary.S_m);
62 d2_l_x = sparse(ops_x.boundary.S2_1);
63 d2_r_x = sparse(ops_x.boundary.S2_m);
64 d3_l_x = sparse(ops_x.boundary.S3_1);
65 d3_r_x = sparse(ops_x.boundary.S3_m);
66
67 D4_y = sparse(ops_y.derivatives.D4);
68 H_y = sparse(ops_y.norms.H);
69 Hi_y = sparse(ops_y.norms.HI);
70 e_l_y = sparse(ops_y.boundary.e_1);
71 e_r_y = sparse(ops_y.boundary.e_m);
72 d1_l_y = sparse(ops_y.boundary.S_1);
73 d1_r_y = sparse(ops_y.boundary.S_m);
74 d2_l_y = sparse(ops_y.boundary.S2_1);
75 d2_r_y = sparse(ops_y.boundary.S2_m);
76 d3_l_y = sparse(ops_y.boundary.S3_1);
77 d3_r_y = sparse(ops_y.boundary.S3_m);
78
79
80 D4 = kr(D4_x, I_y) + kr(I_x, D4_y);
81
82 % Norms
83 obj.H = kr(H_x,H_y);
84 obj.Hx = kr(H_x,I_x);
85 obj.Hy = kr(I_x,H_y);
86 obj.Hix = kr(Hi_x,I_y);
87 obj.Hiy = kr(I_x,Hi_y);
88 obj.Hi = kr(Hi_x,Hi_y);
89
90 % Boundary operators
91 obj.e_w = kr(e_l_x,I_y);
92 obj.e_e = kr(e_r_x,I_y);
93 obj.e_s = kr(I_x,e_l_y);
94 obj.e_n = kr(I_x,e_r_y);
95 obj.d1_w = kr(d1_l_x,I_y);
96 obj.d1_e = kr(d1_r_x,I_y);
97 obj.d1_s = kr(I_x,d1_l_y);
98 obj.d1_n = kr(I_x,d1_r_y);
99 obj.d2_w = kr(d2_l_x,I_y);
100 obj.d2_e = kr(d2_r_x,I_y);
101 obj.d2_s = kr(I_x,d2_l_y);
102 obj.d2_n = kr(I_x,d2_r_y);
103 obj.d3_w = kr(d3_l_x,I_y);
104 obj.d3_e = kr(d3_r_x,I_y);
105 obj.d3_s = kr(I_x,d3_l_y);
106 obj.d3_n = kr(I_x,d3_r_y);
107
108 obj.m = m;
109 obj.h = [h_x h_y];
110 obj.order = order;
111
112 obj.alpha = alpha;
113 obj.D = alpha*D4;
114 obj.u = x;
115 obj.v = y;
116 obj.x = kr(x,ones(m_y,1));
117 obj.y = kr(ones(m_x,1),y);
118
119 obj.gamm_x = h_x*ops_x.borrowing.N.S2/2;
120 obj.delt_x = h_x^3*ops_x.borrowing.N.S3/2;
121
122 obj.gamm_y = h_y*ops_y.borrowing.N.S2/2;
123 obj.delt_y = h_y^3*ops_y.borrowing.N.S3/2;
124 end
125
126
127 % Closure functions return the opertors applied to the own doamin to close the boundary
128 % Penalty functions return the opertors to force the solution. In the case of an interface it returns the operator applied to the other doamin.
129 % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
130 % type is a string specifying the type of boundary condition if there are several.
131 % data is a function returning the data that should be applied at the boundary.
132 % neighbour_scheme is an instance of Scheme that should be interfaced to.
133 % neighbour_boundary is a string specifying which boundary to interface to.
134 function [closure, penalty_e,penalty_d] = boundary_condition(obj,boundary,type,data)
135 default_arg('type','dn');
136 default_arg('data',0);
137
138 [e,d1,d2,d3,s,gamm,delt,halfnorm_inv] = obj.get_boundary_ops(boundary);
139
140 switch type
141 % Dirichlet-neumann boundary condition
142 case {'dn'}
143 alpha = obj.alpha;
144
145 % tau1 < -alpha^2/gamma
146 tuning = 1.1;
147
148 tau1 = tuning * alpha/delt;
149 tau4 = s*alpha;
150
151 sig2 = tuning * alpha/gamm;
152 sig3 = -s*alpha;
153
154 tau = tau1*e+tau4*d3;
155 sig = sig2*d1+sig3*d2;
156
157 closure = halfnorm_inv*(tau*e' + sig*d1');
158
159 pp_e = halfnorm_inv*tau;
160 pp_d = halfnorm_inv*sig;
161 switch class(data)
162 case 'double'
163 penalty_e = pp_e*data;
164 penalty_d = pp_d*data;
165 case 'function_handle'
166 penalty_e = @(t)pp_e*data(t);
167 penalty_d = @(t)pp_d*data(t);
168 otherwise
169 error('Wierd data argument!')
170 end
171
172 % Unknown, boundary condition
173 otherwise
174 error('No such boundary condition: type = %s',type);
175 end
176 end
177
178 function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
179 % u denotes the solution in the own domain
180 % v denotes the solution in the neighbour domain
181 [e_u,d1_u,d2_u,d3_u,s_u,gamm_u,delt_u, halfnorm_inv] = obj.get_boundary_ops(boundary);
182 [e_v,d1_v,d2_v,d3_v,s_v,gamm_v,delt_v] = neighbour_scheme.get_boundary_ops(neighbour_boundary);
183
184 tuning = 2;
185
186 alpha_u = obj.alpha;
187 alpha_v = neighbour_scheme.alpha;
188
189 tau1 = ((alpha_u/2)/delt_u + (alpha_v/2)/delt_v)/2*tuning;
190 % tau1 = (alpha_u/2 + alpha_v/2)/(2*delt_u)*tuning;
191 tau4 = s_u*alpha_u/2;
192
193 sig2 = ((alpha_u/2)/gamm_u + (alpha_v/2)/gamm_v)/2*tuning;
194 sig3 = -s_u*alpha_u/2;
195
196 phi2 = s_u*1/2;
197
198 psi1 = -s_u*1/2;
199
200 tau = tau1*e_u + tau4*d3_u;
201 sig = sig2*d1_u + sig3*d2_u ;
202 phi = phi2*d1_u ;
203 psi = psi1*e_u ;
204
205 closure = halfnorm_inv*(tau*e_u' + sig*d1_u' + phi*alpha_u*d2_u' + psi*alpha_u*d3_u');
206 penalty = -halfnorm_inv*(tau*e_v' + sig*d1_v' + phi*alpha_v*d2_v' + psi*alpha_v*d3_v');
207 end
208
209 % Ruturns the boundary ops and sign for the boundary specified by the string boundary.
210 % The right boundary is considered the positive boundary
211 function [e,d1,d2,d3,s,gamm, delt, halfnorm_inv] = get_boundary_ops(obj,boundary)
212 switch boundary
213 case 'w'
214 e = obj.e_w;
215 d1 = obj.d1_w;
216 d2 = obj.d2_w;
217 d3 = obj.d3_w;
218 s = -1;
219 gamm = obj.gamm_x;
220 delt = obj.delt_x;
221 halfnorm_inv = obj.Hix;
222 case 'e'
223 e = obj.e_e;
224 d1 = obj.d1_e;
225 d2 = obj.d2_e;
226 d3 = obj.d3_e;
227 s = 1;
228 gamm = obj.gamm_x;
229 delt = obj.delt_x;
230 halfnorm_inv = obj.Hix;
231 case 's'
232 e = obj.e_s;
233 d1 = obj.d1_s;
234 d2 = obj.d2_s;
235 d3 = obj.d3_s;
236 s = -1;
237 gamm = obj.gamm_y;
238 delt = obj.delt_y;
239 halfnorm_inv = obj.Hiy;
240 case 'n'
241 e = obj.e_n;
242 d1 = obj.d1_n;
243 d2 = obj.d2_n;
244 d3 = obj.d3_n;
245 s = 1;
246 gamm = obj.gamm_y;
247 delt = obj.delt_y;
248 halfnorm_inv = obj.Hiy;
249 otherwise
250 error('No such boundary: boundary = %s',boundary);
251 end
252 end
253
254 function N = size(obj)
255 N = prod(obj.m);
256 end
257
258 end
259 end