1 | /* |
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2 | Functions for WRC implementation of flexible cylinders |
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3 | */ |
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4 | double Sk_WR(double q, double L, double b); |
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5 | |
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6 | |
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7 | static double |
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8 | AlphaSquare(double x) |
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9 | { |
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10 | // Potentially faster. Needs proper benchmarking. |
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11 | // add native_powr to kernel_template |
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12 | //double t = native_powr( (1.0 + (x/3.12)*(x/3.12) + |
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13 | // (x/8.67)*(x/8.67)*(x/8.67)),(0.176/3.0) ); |
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14 | //return t; |
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15 | |
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16 | return pow( (1.0 + (x/3.12)*(x/3.12) + |
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17 | (x/8.67)*(x/8.67)*(x/8.67)),(0.176/3.0) ); |
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18 | } |
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19 | |
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20 | // |
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21 | static double |
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22 | Rgsquarezero(double q, double L, double b) |
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23 | { |
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24 | const double r = b/L; |
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25 | return (L*b/6.0) * |
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26 | (1.0 - r*1.5 + 1.5*r*r - 0.75*r*r*r*(1.0 - exp(-2.0/r))); |
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27 | } |
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28 | |
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29 | // |
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30 | static double |
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31 | Rgsquareshort(double q, double L, double b) |
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32 | { |
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33 | return AlphaSquare(L/b) * Rgsquarezero(q,L,b); |
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34 | } |
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35 | |
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36 | // |
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37 | static double |
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38 | Rgsquare(double q, double L, double b) |
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39 | { |
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40 | return AlphaSquare(L/b)*L*b/6.0; |
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41 | } |
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42 | |
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43 | static inline double |
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44 | sech_WR(double x) |
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45 | { |
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46 | return(1/cosh(x)); |
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47 | } |
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48 | |
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49 | static double |
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50 | a1long(double q, double L, double b, double p1, double p2, double q0) |
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51 | { |
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52 | double C; |
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53 | const double onehalf = 1.0/2.0; |
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54 | |
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55 | if( L/b > 10.0) { |
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56 | C = 3.06/pow((L/b),0.44); |
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57 | } else { |
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58 | C = 1.0; |
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59 | } |
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60 | |
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61 | const double C1 = 1.22; |
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62 | const double C2 = 0.4288; |
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63 | const double C3 = -1.651; |
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64 | const double C4 = 1.523; |
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65 | const double C5 = 0.1477; |
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66 | const double miu = 0.585; |
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67 | |
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68 | const double Rg2 = Rgsquare(q,L,b); |
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69 | const double Rg22 = Rg2*Rg2; |
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70 | const double Rg = sqrt(Rg2); |
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71 | const double Rgb = Rg*q0/b; |
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72 | |
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73 | const double b2 = b*b; |
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74 | const double b3 = b*b*b; |
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75 | const double b4 = b3*b; |
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76 | const double q02 = q0*q0; |
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77 | const double q03 = q0*q0*q0; |
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78 | const double q04 = q03*q0; |
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79 | const double q05 = q04*q0; |
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80 | |
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81 | const double Rg02 = Rg2*q02; |
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82 | |
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83 | const double t1 = (b*C*((4.0/15.0 - pow((double)M_E,(-(Rg02/b2))) * |
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84 | ((11.0/15.0 + (7.0*b2)/(15.0*Rg02))) + |
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85 | (7.0*b2)/(15.0*Rg02)))); |
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86 | |
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87 | const double t2 = (2.0*b4*(((-1.0) + pow((double)M_E,(-(Rg02/b2))) + |
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88 | Rg02/b2))*((1.0 + onehalf*(((-1.0) - |
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89 | tanh((-C4 + Rgb/C5))))))); |
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90 | |
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91 | const double t3 = ((C3*pow(Rgb,((-3.0)/miu)) + |
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92 | C2*pow(Rgb,((-2.0)/miu)) + |
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93 | C1*pow(Rgb,((-1.0)/miu)))); |
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94 | |
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95 | const double t4 = ((1.0 + tanh(((-C4) + Rgb)/C5))); |
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96 | |
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97 | const double t5 = (1.0/(b*p1*pow(q0,((-1.0) - p1 - p2)) - |
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98 | b*p2*pow(q0,((-1.0) - p1 - p2)))); |
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99 | |
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100 | const double t6 = (b*C*(((-((14.0*b3)/(15.0*q03*Rg2))) + |
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101 | (14.0*b3*pow((double)M_E,(-(Rg02/b2))))/(15.0*q03*Rg2) + |
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102 | (2.0*pow((double)M_E,(-(Rg02/b2)))*q0*((11.0/15.0 + |
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103 | (7.0*b2)/(15.0*Rg02)))*Rg2)/b))); |
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104 | |
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105 | const double t7 = (Rg*((C3*pow(((Rgb)),((-3.0)/miu)) + |
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106 | C2*pow(((Rgb)),((-2.0)/miu)) + |
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107 | C1*pow(((Rgb)),((-1.0)/miu))))*pow(sech_WR(((-C4) + |
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108 | Rgb)/C5),2)); |
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109 | |
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110 | const double t8 = (b4*Rg*(((-1.0) + pow((double)M_E,(-(Rg02/b2))) + |
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111 | Rg02/b2))*pow(sech_WR(((-C4) + Rgb)/C5),2)); |
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112 | |
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113 | const double t9 = (2.0*b4*(((2.0*q0*Rg2)/b - |
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114 | (2.0*pow((double)M_E,(-(Rg02/b2)))*q0*Rg2)/b))*((1.0 + onehalf*(((-1.0) - |
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115 | tanh(((-C4) + Rgb)/C5)))))); |
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116 | |
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117 | const double t10 = (8.0*b4*b*(((-1.0) + pow((double)M_E,(-(Rg02/b2))) + |
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118 | Rg02/b2))*((1.0 + onehalf*(((-1.0) - tanh(((-C4) + |
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119 | Rgb)/C5)))))); |
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120 | |
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121 | const double t11 = (((-((3.0*C3*Rg*pow(((Rgb)),((-1.0) - |
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122 | 3.0/miu)))/miu)) - (2.0*C2*Rg*pow(((Rgb)),((-1.0) - |
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123 | 2.0/miu)))/miu - (C1*Rg*pow(((Rgb)),((-1.0) - |
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124 | 1.0/miu)))/miu)); |
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125 | |
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126 | const double t12 = ((1.0 + tanh(((-C4) + Rgb)/C5))); |
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127 | |
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128 | const double t13 = (b*C*((4.0/15.0 - pow((double)M_E,(-(Rg02/b2)))*((11.0/15.0 + |
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129 | (7.0*b2)/(15.0*q02* Rg2))) + |
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130 | (7.0*b2)/(15.0*Rg02)))); |
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131 | |
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132 | const double t14 = (2.0*b4*(((-1.0) + pow((double)M_E,(-(Rg02/b2))) + |
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133 | Rg02/b2))*((1.0 + onehalf*(((-1.0) - tanh(((-C4) + |
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134 | Rgb)/C5)))))); |
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135 | |
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136 | const double t15 = ((C3*pow(((Rgb)),((-3.0)/miu)) + |
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137 | C2*pow(((Rgb)),((-2.0)/miu)) + |
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138 | C1*pow(((Rgb)),((-1.0)/miu)))); |
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139 | |
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140 | |
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141 | double yy = (pow(q0,p1)*(((-((b*M_PI)/(L*q0))) +t1/L +t2/(q04*Rg22) + |
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142 | onehalf*t3*t4)) + (t5*((pow(q0,(p1 - p2))* |
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143 | (((-pow(q0,(-p1)))*(((b2*M_PI)/(L*q02) +t6/L +t7/(2.0*C5) - |
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144 | t8/(C5*q04*Rg22) + t9/(q04*Rg22) -t10/(q05*Rg22) + onehalf*t11*t12)) - |
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145 | b*p1*pow(q0,((-1.0) - p1))*(((-((b*M_PI)/(L*q0))) + t13/L + |
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146 | t14/(q04*Rg22) + onehalf*t15*((1.0 + tanh(((-C4) + |
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147 | Rgb)/C5))))))))))); |
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148 | |
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149 | return (yy); |
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150 | } |
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151 | |
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152 | static double |
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153 | a2long(double q, double L, double b, double p1, double p2, double q0) |
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154 | { |
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155 | double C; |
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156 | const double onehalf = 1.0/2.0; |
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157 | |
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158 | if( L/b > 10.0) { |
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159 | C = 3.06/pow((L/b),0.44); |
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160 | } else { |
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161 | C = 1.0; |
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162 | } |
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163 | |
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164 | const double C1 = 1.22; |
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165 | const double C2 = 0.4288; |
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166 | const double C3 = -1.651; |
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167 | const double C4 = 1.523; |
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168 | const double C5 = 0.1477; |
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169 | const double miu = 0.585; |
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170 | |
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171 | const double Rg2 = Rgsquare(q,L,b); |
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172 | const double Rg22 = Rg2*Rg2; |
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173 | const double b2 = b*b; |
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174 | const double b3 = b*b*b; |
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175 | const double b4 = b3*b; |
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176 | const double q02 = q0*q0; |
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177 | const double q03 = q0*q0*q0; |
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178 | const double q04 = q03*q0; |
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179 | const double q05 = q04*q0; |
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180 | const double Rg = sqrt(Rg2); |
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181 | const double Rgb = Rg*q0/b; |
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182 | const double Rg02 = Rg2*q02; |
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183 | |
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184 | const double t1 = (1.0/(b* p1*pow(q0,((-1.0) - p1 - p2)) - |
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185 | b*p2*pow(q0,((-1.0) - p1 - p2)) )); |
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186 | |
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187 | const double t2 = (b*C*(((-1.0*((14.0*b3)/(15.0*q03*Rg2))) + |
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188 | (14.0*b3*pow((double)M_E,(-(Rg02/b2))))/(15.0*q03*Rg2) + |
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189 | (2.0*pow((double)M_E,(-(Rg02/b2)))*q0*((11.0/15.0 + |
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190 | (7*b2)/(15.0*Rg02)))*Rg2)/b)))/L; |
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191 | |
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192 | const double t3 = (Rg*((C3*pow(((Rgb)),((-3.0)/miu)) + |
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193 | C2*pow(((Rgb)),((-2.0)/miu)) + |
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194 | C1*pow(((Rgb)),((-1.0)/miu))))* |
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195 | pow(sech_WR(((-C4) +Rgb)/C5),2.0))/(2.0*C5); |
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196 | |
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197 | const double t4 = (b4*Rg*(((-1.0) + pow((double)M_E,(-(Rg02/b2))) + |
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198 | Rg02/b2))*pow(sech_WR(((-C4) + |
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199 | Rgb)/C5),2))/(C5*q04*Rg22); |
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200 | |
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201 | const double t5 = (2.0*b4*(((2.0*q0*Rg2)/b - |
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202 | (2.0*pow((double)M_E,(-(Rg02/b2)))*q0*Rg2)/b))* |
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203 | ((1.0 + onehalf*(((-1.0) - tanh(((-C4) + |
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204 | Rgb)/C5))))))/(q04*Rg22); |
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205 | |
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206 | const double t6 = (8.0*b4*b*(((-1.0) + pow((double)M_E,(-(Rg02/b2))) + |
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207 | Rg02/b2))*((1.0 + onehalf*(((-1) - tanh(((-C4) + |
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208 | Rgb)/C5))))))/(q05*Rg22); |
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209 | |
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210 | const double t7 = (((-((3.0*C3*Rg*pow(((Rgb)),((-1.0) - |
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211 | 3.0/miu)))/miu)) - (2.0*C2*Rg*pow(((Rgb)), |
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212 | ((-1.0) - 2.0/miu)))/miu - (C1*Rg*pow(((Rgb)), |
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213 | ((-1.0) - 1.0/miu)))/miu)); |
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214 | |
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215 | const double t8 = ((1.0 + tanh(((-C4) + Rgb)/C5))); |
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216 | |
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217 | const double t9 = (b*C*((4.0/15.0 - pow((double)M_E,(-(Rg02/b2)))*((11.0/15.0 + |
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218 | (7.0*b2)/(15*Rg02))) + (7.0*b2)/(15.0*Rg02))))/L; |
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219 | |
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220 | const double t10 = (2.0*b4*(((-1) + pow((double)M_E,(-(Rg02/b2))) + |
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221 | Rg02/b2))*((1.0 + onehalf*(((-1) - tanh(((-C4) + |
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222 | Rgb)/C5))))))/(q04*Rg22); |
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223 | |
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224 | const double yy = ((-1.0*(t1* ((-pow(q0,-p1)*(((b2*M_PI)/(L*q02) + |
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225 | t2 + t3 - t4 + t5 - t6 + onehalf*t7*t8)) - b*p1*pow(q0,((-1.0) - p1))* |
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226 | (((-((b*M_PI)/(L*q0))) + t9 + t10 + |
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227 | onehalf*((C3*pow(((Rgb)),((-3.0)/miu)) + |
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228 | C2*pow(((Rgb)),((-2.0)/miu)) + |
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229 | C1*pow(((Rgb)),((-1.0)/miu))))* |
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230 | ((1.0 + tanh(((-C4) + Rgb)/C5)))))))))); |
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231 | |
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232 | return (yy); |
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233 | } |
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234 | |
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235 | // |
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236 | static double |
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237 | a_short(double q, double L, double b, double p1short, |
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238 | double p2short, double factor, double pdiff, double q0) |
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239 | { |
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240 | const double Rg2_sh = Rgsquareshort(q,L,b); |
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241 | const double Rg2_sh2 = Rg2_sh*Rg2_sh; |
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242 | const double b3 = b*b*b; |
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243 | const double t1 = ((q0*q0*Rg2_sh)/(b*b)); |
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244 | const double Et1 = exp(t1); |
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245 | const double Emt1 = 1.0/Et1; |
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246 | const double q02 = q0*q0; |
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247 | const double q0p = pow(q0,(-4.0 + p1short) ); |
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248 | |
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249 | double yy = ((factor/(L*(pdiff)*Rg2_sh2)* |
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250 | ((b*Emt1*q0p*((8.0*b3*L - 8.0*b3*Et1*L - 2.0*b3*L*p2short + |
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251 | 2.0*b3*Et1*L*p2short + 4.0*b*L*q02*Rg2_sh + 4.0*b*Et1*L*q02*Rg2_sh - |
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252 | 2.0*b*Et1*L*p2short*q02*Rg2_sh - Et1*M_PI*q02*q0*Rg2_sh2 + |
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253 | Et1*p2short*M_PI*q02*q0*Rg2_sh2)))))); |
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254 | |
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255 | return(yy); |
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256 | } |
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257 | static double |
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258 | a1short(double q, double L, double b, double p1short, double p2short, double q0) |
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259 | { |
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260 | |
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261 | double factor = 1.0; |
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262 | return a_short(q, L, b, p1short, p2short, factor, p1short - p2short, q0); |
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263 | } |
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264 | |
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265 | static double |
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266 | a2short(double q, double L, double b, double p1short, double p2short, double q0) |
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267 | { |
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268 | double factor = -1.0; |
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269 | return a_short(q, L, b, p2short, p1short, factor, p1short-p2short, q0); |
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270 | } |
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271 | |
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272 | //WR named this w (too generic) |
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273 | static double |
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274 | w_WR(double x) |
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275 | { |
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276 | return 0.5*(1 + tanh((x - 1.523)/0.1477)); |
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277 | } |
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278 | |
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279 | // |
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280 | static double |
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281 | u1(double q, double L, double b) |
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282 | { |
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283 | return Rgsquareshort(q,L,b)*q*q; |
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284 | } |
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285 | |
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286 | static double |
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287 | u_WR(double q, double L, double b) |
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288 | { |
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289 | return Rgsquare(q,L,b)*q*q; |
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290 | } |
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291 | |
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292 | static double |
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293 | Sdebye_kernel(double arg) |
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294 | { |
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295 | // ORIGINAL |
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296 | double result = 2.0*(exp(-arg) + arg -1.0)/(arg*arg); |
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297 | |
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298 | // CONVERSION 1 from http://herbie.uwplse.org/ |
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299 | // |
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300 | // exhibits discontinuity - needs more investigation |
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301 | //double a1 = 1.0/6.0; |
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302 | //double a2 = 1.0/72.0; |
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303 | //double a3 = 1.0/24.0; |
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304 | //double result = pow((1.0 - a1*arg - (a2+a3)*arg*arg), 2); |
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305 | |
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306 | return result; |
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307 | } |
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308 | static double |
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309 | Sdebye(double q, double L, double b) |
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310 | { |
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311 | double arg = u_WR(q,L,b); |
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312 | return Sdebye_kernel(arg); |
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313 | } |
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314 | |
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315 | // |
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316 | static double |
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317 | Sdebye1(double q, double L, double b) |
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318 | { |
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319 | double arg = u1(q,L,b); |
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320 | return Sdebye_kernel(arg); |
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321 | |
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322 | } |
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323 | |
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324 | // |
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325 | static double |
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326 | Sexv(double q, double L, double b) |
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327 | { |
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328 | |
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329 | const double C1=1.22; |
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330 | const double C2=0.4288; |
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331 | const double C3=-1.651; |
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332 | const double miu = 0.585; |
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333 | const double qRg = q*sqrt(Rgsquare(q,L,b)); |
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334 | const double x = pow(qRg, -1.0/miu); |
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335 | |
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336 | double yy = (1.0 - w_WR(qRg))*Sdebye(q,L,b) + |
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337 | w_WR(qRg)*x*(C1 + x*(C2 + x*C3)); |
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338 | return (yy); |
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339 | } |
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340 | |
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341 | |
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342 | static double |
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343 | Sexvnew(double q, double L, double b) |
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344 | { |
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345 | double yy; |
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346 | |
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347 | const double C1 =1.22; |
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348 | const double C2 =0.4288; |
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349 | const double C3 =-1.651; |
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350 | const double miu = 0.585; |
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351 | const double del=1.05; |
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352 | const double qRg = q*sqrt(Rgsquare(q,L,b)); |
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353 | const double x = pow(qRg, -1.0/miu); |
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354 | |
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355 | |
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356 | //calculating the derivative to decide on the corection (cutoff) term? |
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357 | // I have modified this from WRs original code |
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358 | const double qdel = (Sexv(q*del,L,b)-Sexv(q,L,b))/(q*del - q); |
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359 | const double C_star2 =(qdel >= 0.0) ? 0.0 : 1.0; |
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360 | |
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361 | yy = (1.0 - w_WR(qRg))*Sdebye(q,L,b) + |
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362 | C_star2*w_WR(qRg)* |
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363 | x*(C1 + x*(C2 + x*C3)); |
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364 | |
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365 | return (yy); |
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366 | } |
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367 | |
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368 | double Sk_WR(double q, double L, double b) |
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369 | { |
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370 | // |
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371 | const double p1 = 4.12; |
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372 | const double p2 = 4.42; |
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373 | const double p1short = 5.36; |
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374 | const double p2short = 5.62; |
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375 | const double q0 = 3.1; |
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376 | |
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377 | double q0short = fmax(1.9/sqrt(Rgsquareshort(q,L,b)),3.0); |
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378 | double Sexvmodify, ans; |
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379 | |
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380 | const double C =(L/b > 10.0) ? 3.06/pow((L/b),0.44) : 1.0; |
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381 | |
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382 | if( L > 4*b ) { // Longer Chains |
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383 | if (q*b <= 3.1) { //Modified by Yun on Oct. 15, |
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384 | Sexvmodify = Sexvnew(q, L, b); |
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385 | ans = Sexvmodify + C * (4.0/15.0 + 7.0/(15.0*u_WR(q,L,b)) - |
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386 | (11.0/15.0 + 7.0/(15.0*u_WR(q,L,b)))*exp(-u_WR(q,L,b)))*(b/L); |
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387 | |
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388 | } else { //q(i)*b > 3.1 |
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389 | ans = a1long(q, L, b, p1, p2, q0)/(pow((q*b),p1)) + |
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390 | a2long(q, L, b, p1, p2, q0)/(pow((q*b),p2)) + M_PI/(q*L); |
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391 | } |
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392 | } else { //L <= 4*b Shorter Chains |
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393 | if (q*b <= fmax(1.9/sqrt(Rgsquareshort(q,L,b)),3.0) ) { |
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394 | if (q*b<=0.01) { |
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395 | ans = 1.0 - Rgsquareshort(q,L,b)*(q*q)/3.0; |
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396 | } else { |
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397 | ans = Sdebye1(q,L,b); |
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398 | } |
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399 | } else { //q*b > max(1.9/sqrt(Rgsquareshort(q(i),L,b)),3) |
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400 | ans = a1short(q,L,b,p1short,p2short,q0short)/(pow((q*b),p1short)) + |
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401 | a2short(q,L,b,p1short,p2short,q0short)/(pow((q*b),p2short)) + |
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402 | M_PI/(q*L); |
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403 | } |
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404 | } |
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405 | |
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406 | return(ans); |
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407 | } |
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