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annotate +scheme/LaplaceCurvilinear.m @ 554:6e71d4f8b155 feature/grids/laplace_refactor

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Better names for the metric coefficients
author Jonatan Werpers <jonatan@werpers.com>
date Tue, 29 Aug 2017 11:21:02 +0200
parents 63d5c07943dd
children 2a856a589510
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450
8d455e49364f Copy Wave2dCurve to new scheme LaplaceCurvilinear
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1 classdef LaplaceCurvilinear < scheme.Scheme
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2 properties
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3 m % Number of points in each direction, possibly a vector
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4 h % Grid spacing
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5
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6 grid
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7
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8 order % Order accuracy for the approximation
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9
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10 a,b % Parameters of the operator
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11
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12
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13 % Inner products and operators for physical coordinates
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14 D % Laplace operator
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15 H, Hi % Inner product
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16 e_w, e_e, e_s, e_n
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17 d_w, d_e, d_s, d_n % Normal derivatives at the boundary
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18 H_w, H_e, H_s, H_n % Boundary inner products
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19 Dx, Dy % Physical derivatives
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20 M % Gradient inner product
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21
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22 % Metric coefficients
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23 J, Ji
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24 a11, a12, a22
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25 x_u
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26 x_v
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27 y_u
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28 y_v
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30 % Inner product and operators for logical coordinates
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31 H_u, H_v % Norms in the x and y directions
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32 Hi_u, Hi_v
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33 Hu,Hv % Kroneckerd norms. 1'*Hx*v corresponds to integration in the x dir.
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34 Hiu, Hiv
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35 du_w, dv_w
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36 du_e, dv_e
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37 du_s, dv_s
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38 du_n, dv_n
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39 gamm_u, gamm_v
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40 lambda
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41 end
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42
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43 methods
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44 % Implements a*div(b*grad(u)) as a SBP scheme
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45 % TODO: Implement proper H, it should be the real physical quadrature, the logic quadrature may be but in a separate variable (H_logic?)
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46
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47 function obj = LaplaceCurvilinear(g ,order, a, b, opSet)
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48 default_arg('opSet',@sbp.D2Variable);
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49 default_arg('a', 1);
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50 default_arg('b', 1);
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51
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52 if b ~=1
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53 error('Not implemented yet')
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54 end
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56 assert(isa(g, 'grid.Curvilinear'))
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58 m = g.size();
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59 m_u = m(1);
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60 m_v = m(2);
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61 m_tot = g.N();
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62
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63 h = g.scaling();
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64 h_u = h(1);
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65 h_v = h(2);
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66
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67
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68 % 1D operators
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69 ops_u = opSet(m_u, {0, 1}, order);
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70 ops_v = opSet(m_v, {0, 1}, order);
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71
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72 I_u = speye(m_u);
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73 I_v = speye(m_v);
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74
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75 D1_u = ops_u.D1;
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76 D2_u = ops_u.D2;
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77 H_u = ops_u.H;
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78 Hi_u = ops_u.HI;
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79 e_l_u = ops_u.e_l;
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80 e_r_u = ops_u.e_r;
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81 d1_l_u = ops_u.d1_l;
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82 d1_r_u = ops_u.d1_r;
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83
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84 D1_v = ops_v.D1;
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85 D2_v = ops_v.D2;
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86 H_v = ops_v.H;
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87 Hi_v = ops_v.HI;
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88 e_l_v = ops_v.e_l;
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89 e_r_v = ops_v.e_r;
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90 d1_l_v = ops_v.d1_l;
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91 d1_r_v = ops_v.d1_r;
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92
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93
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94 % Logical operators
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95 Du = kr(D1_u,I_v);
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96 Dv = kr(I_u,D1_v);
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97 obj.Hu = kr(H_u,I_v);
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98 obj.Hv = kr(I_u,H_v);
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99 obj.Hiu = kr(Hi_u,I_v);
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100 obj.Hiv = kr(I_u,Hi_v);
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101
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102 e_w = kr(e_l_u,I_v);
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103 e_e = kr(e_r_u,I_v);
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104 e_s = kr(I_u,e_l_v);
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105 e_n = kr(I_u,e_r_v);
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106 obj.du_w = kr(d1_l_u,I_v);
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107 obj.dv_w = (e_w'*Dv)';
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108 obj.du_e = kr(d1_r_u,I_v);
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109 obj.dv_e = (e_e'*Dv)';
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110 obj.du_s = (e_s'*Du)';
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111 obj.dv_s = kr(I_u,d1_l_v);
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112 obj.du_n = (e_n'*Du)';
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113 obj.dv_n = kr(I_u,d1_r_v);
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114
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115
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116 % Metric coefficients
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117 coords = g.points();
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118 x = coords(:,1);
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119 y = coords(:,2);
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120
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121 x_u = Du*x;
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122 x_v = Dv*x;
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123 y_u = Du*y;
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124 y_v = Dv*y;
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125
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126 J = x_u.*y_v - x_v.*y_u;
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127 a11 = 1./J .* (x_v.^2 + y_v.^2);
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128 a12 = -1./J .* (x_u.*x_v + y_u.*y_v);
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129 a22 = 1./J .* (x_u.^2 + y_u.^2);
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130 lambda = 1/2 * (a11 + a22 - sqrt((a11-a22).^2 + 4*a12.^2));
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131
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132 obj.x_u = x_u;
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133 obj.x_v = x_v;
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134 obj.y_u = y_u;
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135 obj.y_v = y_v;
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136
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137
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138 % Assemble full operators
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139 L_12 = spdiag(a12);
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140 Duv = Du*L_12*Dv;
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141 Dvu = Dv*L_12*Du;
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142
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143 Duu = sparse(m_tot);
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144 Dvv = sparse(m_tot);
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145 ind = grid.funcToMatrix(g, 1:m_tot);
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147 for i = 1:m_v
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148 D = D2_u(a11(ind(:,i)));
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149 p = ind(:,i);
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150 Duu(p,p) = D;
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151 end
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152
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153 for i = 1:m_u
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154 D = D2_v(a22(ind(i,:)));
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155 p = ind(i,:);
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156 Dvv(p,p) = D;
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157 end
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158
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159
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160 % Physical operators
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161 obj.J = spdiag(J);
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162 obj.Ji = spdiag(1./J);
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163
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164 obj.D = obj.Ji*a*(Duu + Duv + Dvu + Dvv);
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165 obj.H = obj.J*kr(H_u,H_v);
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166 obj.Hi = obj.Ji*kr(Hi_u,Hi_v);
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167
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168 obj.e_w = e_w;
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169 obj.e_e = e_e;
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170 obj.e_s = e_s;
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171 obj.e_n = e_n;
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172
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173
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174 obj.Dx = spdiag( y_v./J)*Du + spdiag(-y_u./J)*Dv;
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175 obj.Dy = spdiag(-x_v./J)*Du + spdiag( x_u./J)*Dv;
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176
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177
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178 % Misc.
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179 obj.m = m;
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180 obj.h = [h_u h_v];
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181 obj.order = order;
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182 obj.grid = g;
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183
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184 obj.a = a;
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185 obj.b = b;
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186 obj.a11 = a11;
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187 obj.a12 = a12;
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188 obj.a22 = a22;
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189 obj.lambda = lambda;
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190
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191 obj.gamm_u = h_u*ops_u.borrowing.M.d1;
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192 obj.gamm_v = h_v*ops_v.borrowing.M.d1;
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193 end
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194
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195
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196 % Closure functions return the opertors applied to the own doamin to close the boundary
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197 % Penalty functions return the opertors to force the solution. In the case of an interface it returns the operator applied to the other doamin.
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198 % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
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199 % type is a string specifying the type of boundary condition if there are several.
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200 % data is a function returning the data that should be applied at the boundary.
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201 % neighbour_scheme is an instance of Scheme that should be interfaced to.
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202 % neighbour_boundary is a string specifying which boundary to interface to.
347
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203 function [closure, penalty] = boundary_condition(obj, boundary, type, parameter)
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204 default_arg('type','neumann');
347
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205 default_arg('parameter', []);
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206
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207 [e, d_n, d_t, a_n, a_t, s, gamm, halfnorm_inv , ~, ~, ~, scale_factor] = obj.get_boundary_ops(boundary);
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208 switch type
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209 % Dirichlet boundary condition
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210 case {'D','d','dirichlet'}
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211 % v denotes the solution in the neighbour domain
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212 tuning = 1.2;
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213 % tuning = 20.2;
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214 [e, d_n, d_t, a_n, a_t, s, gamm, halfnorm_inv_n, halfnorm_inv_t, halfnorm_t] = obj.get_boundary_ops(boundary);
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215
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216 A_n = spdiag(a_n);
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217 A_t = spdiag(a_t);
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218
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219 F = (s * A_n * d_n' + s * A_t*d_t')';
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220
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221 u = obj;
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222
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223 b1 = gamm*u.lambda./u.a11.^2;
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224 b2 = gamm*u.lambda./u.a22.^2;
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225
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226 tau = -1./b1 - 1./b2;
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227 tau = tuning * spdiag(tau);
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228 sig1 = 1;
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229
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230 penalty_parameter_1 = halfnorm_inv_n*(tau + sig1*halfnorm_inv_t*F*e'*halfnorm_t)*e;
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231
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232 closure = obj.Ji*obj.a * penalty_parameter_1*e';
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233 penalty = -obj.Ji*obj.a * penalty_parameter_1;
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234
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235
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236 % Neumann boundary condition
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237 case {'N','n','neumann'}
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238 A_n = spdiag(a_n);
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239 A_t = spdiag(a_t);
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240 d = (A_n * d_n' + A_t*d_t')';
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241
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242 tau1 = -s;
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243 tau2 = 0;
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244 tau = obj.a * obj.Ji*(tau1*e + tau2*d);
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245
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246 closure = halfnorm_inv*tau*d';
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247 penalty = -halfnorm_inv*tau;
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248
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249 % Characteristic boundary condition
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250 case {'characteristic', 'char', 'c'}
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251 default_arg('parameter', 1);
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252 beta = parameter;
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253
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254 A_n = spdiag(a_n);
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255 A_t = spdiag(a_t);
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256 d = s*(A_n * d_n' + A_t*d_t')'; % outward facing normal derivative
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257
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258 tau = -obj.a * 1/beta*obj.Ji*e;
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259
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260 closure{1} = halfnorm_inv*tau*spdiag(scale_factor)*e';
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261 closure{2} = halfnorm_inv*tau*beta*d';
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262 penalty = -halfnorm_inv*tau;
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263
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264 % Unknown, boundary condition
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265 otherwise
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266 error('No such boundary condition: type = %s',type);
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267 end
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268 end
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269
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270 function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
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271 % u denotes the solution in the own domain
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272 % v denotes the solution in the neighbour domain
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273 tuning = 1.2;
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274 % tuning = 20.2;
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275 [e_u, d_n_u, d_t_u, a_n_u, a_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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276 [e_v, d_n_v, d_t_v, a_n_v, a_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);
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277
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278 A_n_u = spdiag(a_n_u);
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279 A_t_u = spdiag(a_t_u);
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280 A_n_v = spdiag(a_n_v);
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281 A_t_v = spdiag(a_t_v);
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282
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283 F_u = (s_u * A_n_u * d_n_u' + s_u * A_t_u*d_t_u')';
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284 F_v = (s_v * A_n_v * d_n_v' + s_v * A_t_v*d_t_v')';
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285
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286 u = obj;
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287 v = neighbour_scheme;
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288
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289 b1_u = gamm_u*u.lambda(I_u)./u.a11(I_u).^2;
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290 b2_u = gamm_u*u.lambda(I_u)./u.a22(I_u).^2;
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291 b1_v = gamm_v*v.lambda(I_v)./v.a11(I_v).^2;
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292 b2_v = gamm_v*v.lambda(I_v)./v.a22(I_v).^2;
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293
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294 tau = -1./(4*b1_u) -1./(4*b1_v) -1./(4*b2_u) -1./(4*b2_v);
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295 tau = tuning * spdiag(tau);
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296 sig1 = 1/2;
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297 sig2 = -1/2;
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298
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299 penalty_parameter_1 = halfnorm_inv_u_n*(e_u*tau + sig1*halfnorm_inv_u_t*F_u*e_u'*halfnorm_u_t*e_u);
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300 penalty_parameter_2 = halfnorm_inv_u_n * sig2 * e_u;
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301
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302
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303 closure = obj.Ji*obj.a * ( penalty_parameter_1*e_u' + penalty_parameter_2*F_u');
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304 penalty = obj.Ji*obj.a * (-penalty_parameter_1*e_v' + penalty_parameter_2*F_v');
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305 end
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306
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307 % Ruturns the boundary ops and sign for the boundary specified by the string boundary.
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308 % The right boundary is considered the positive boundary
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309 %
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310 % I -- the indecies of the boundary points in the grid matrix
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311 function [e, d_n, d_t, a_n, a_t, s, gamm, halfnorm_inv_n, halfnorm_inv_t, halfnorm_t, I, scale_factor] = get_boundary_ops(obj, boundary)
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312
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313 % gridMatrix = zeros(obj.m(2),obj.m(1));
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314 % gridMatrix(:) = 1:numel(gridMatrix);
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315
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316 ind = grid.funcToMatrix(obj.grid, 1:prod(obj.m));
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317
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318 switch boundary
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319 case 'w'
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320 e = obj.e_w;
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321 d_n = obj.du_w;
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322 d_t = obj.dv_w;
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323 s = -1;
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324
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325 I = ind(1,:);
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326 a_n = obj.a11(I);
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327 a_t = obj.a12(I);
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328 scale_factor = sqrt(obj.x_v(I).^2 + obj.y_v(I).^2);
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329 case 'e'
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330 e = obj.e_e;
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331 d_n = obj.du_e;
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332 d_t = obj.dv_e;
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333 s = 1;
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334
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335 I = ind(end,:);
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336 a_n = obj.a11(I);
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337 a_t = obj.a12(I);
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338 scale_factor = sqrt(obj.x_v(I).^2 + obj.y_v(I).^2);
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339 case 's'
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340 e = obj.e_s;
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341 d_n = obj.dv_s;
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342 d_t = obj.du_s;
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343 s = -1;
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344
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345 I = ind(:,1)';
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346 a_n = obj.a22(I);
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347 a_t = obj.a12(I);
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348 scale_factor = sqrt(obj.x_u(I).^2 + obj.y_u(I).^2);
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349 case 'n'
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350 e = obj.e_n;
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351 d_n = obj.dv_n;
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352 d_t = obj.du_n;
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353 s = 1;
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354
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355 I = ind(:,end)';
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356 a_n = obj.a22(I);
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357 a_t = obj.a12(I);
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358 scale_factor = sqrt(obj.x_u(I).^2 + obj.y_u(I).^2);
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359 otherwise
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360 error('No such boundary: boundary = %s',boundary);
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361 end
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362
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363 switch boundary
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364 case {'w','e'}
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365 halfnorm_inv_n = obj.Hiu;
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366 halfnorm_inv_t = obj.Hiv;
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367 halfnorm_t = obj.Hv;
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368 gamm = obj.gamm_u;
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369 case {'s','n'}
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370 halfnorm_inv_n = obj.Hiv;
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371 halfnorm_inv_t = obj.Hiu;
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372 halfnorm_t = obj.Hu;
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373 gamm = obj.gamm_v;
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374 end
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375 end
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376
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377 function N = size(obj)
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378 N = prod(obj.m);
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379 end
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380
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381
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382 end
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383 end