1 | /* |
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2 | Functions for WRC implementation of flexible cylinders |
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3 | */ |
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4 | |
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5 | static double |
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6 | Rgsquare(double L, double b) |
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7 | { |
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8 | const double x = L/b; |
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9 | // Use Horner's method to evaluate Pedersen eq 15: |
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10 | // alpha^2 = [1.0 + (x/3.12)^2 + (x/8.67)^3] ^ (0.176/3) |
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11 | const double alphasq = |
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12 | pow(1.0 + x*x*(1.027284681130835e-01 + 1.534414548417740e-03*x), |
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13 | 5.866666666666667e-02); |
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14 | return alphasq*L*b/6.0; |
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15 | } |
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16 | |
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17 | static double |
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18 | Rgsquareshort(double L, double b) |
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19 | { |
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20 | const double r = b/L; // = 1/n_b in Pedersen ref. |
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21 | return Rgsquare(L, b) * (1.0 + r*(-1.5 + r*(1.5 + r*0.75*expm1(-2.0/r)))); |
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22 | } |
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23 | |
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24 | static double |
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25 | w_WR(double x) |
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26 | { |
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27 | // Pedersen eq. 16: |
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28 | // w = [1 + tanh((x-C4)/C5)]/2 |
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29 | const double C4 = 1.523; |
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30 | const double C5 = 0.1477; |
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31 | return 0.5 + 0.5*tanh((x - C4)/C5); |
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32 | } |
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33 | |
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34 | static double |
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35 | Sdebye(double qsq) |
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36 | { |
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37 | #if FLOAT_SIZE>4 |
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38 | # define DEBYE_CUTOFF 0.25 // 1e-15 error |
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39 | #else |
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40 | # define DEBYE_CUTOFF 1.0 // 4e-7 error |
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41 | #endif |
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42 | |
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43 | if (qsq < DEBYE_CUTOFF) { |
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44 | const double x = qsq; |
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45 | // mathematica: PadeApproximant[2*Exp[-x^2] + x^2-1)/x^4, {x, 0, 8}] |
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46 | const double A1=1./15., A2=1./60, A3=0., A4=1./75600.; |
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47 | const double B1=2./5., B2=1./15., B3=1./180., B4=1./5040.; |
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48 | return ((((A4*x + A3)*x + A2)*x + A1)*x + 1.) |
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49 | / ((((B4*x + B3)*x + B2)*x + B1)*x + 1.); |
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50 | } else { |
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51 | return 2.*(expm1(-qsq) + qsq)/(qsq*qsq); |
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52 | } |
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53 | } |
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54 | |
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55 | static double |
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56 | a_long(double q, double L, double b/*, double p1, double p2, double q0*/) |
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57 | { |
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58 | const double p1 = 4.12; |
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59 | const double p2 = 4.42; |
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60 | const double q0 = 3.1; |
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61 | |
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62 | // Constants C1, ..., C5 derived from least squares fit (Pedersen, eq 13,16) |
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63 | const double C1 = 1.22; |
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64 | const double C2 = 0.4288; |
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65 | const double C3 = -1.651; |
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66 | const double C4 = 1.523; |
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67 | const double C5 = 0.1477; |
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68 | const double miu = 0.585; |
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69 | |
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70 | const double C = (L/b>10.0 ? 3.06*pow(L/b, -0.44) : 1.0); |
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71 | //printf("branch B-%d q=%g L=%g b=%g\n", C==1.0, q, L, b); |
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72 | const double r2 = Rgsquare(L,b); |
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73 | const double r = sqrt(r2); |
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74 | const double qr_b = q0*r/b; |
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75 | const double qr_b_sq = qr_b*qr_b; |
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76 | const double qr_b_4 = qr_b_sq*qr_b_sq; |
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77 | const double qr_b_miu = pow(qr_b, -1.0/miu); |
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78 | const double em1_qr_b_sq = expm1(-qr_b_sq); |
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79 | const double sech2 = 1.0/square(cosh((qr_b-C4)/C5)); |
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80 | const double w = w_WR(qr_b); |
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81 | |
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82 | const double t1 = pow(q0, 1.0 + p1 + p2)/(b*(p1-p2)); |
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83 | const double t2 = C/(15.0*L) * ( |
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84 | + 14.0*b*b*em1_qr_b_sq/(q0*qr_b_sq) |
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85 | + 2.0*q0*r2*exp(-qr_b_sq)*(11.0 + 7.0/qr_b_sq)); |
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86 | const double t11 = ((C3*qr_b_miu + C2)*qr_b_miu + C1)*qr_b_miu; |
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87 | const double t3 = r*sech2/(2.*C5)*t11; |
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88 | const double t4 = r*(em1_qr_b_sq + qr_b_sq)*sech2 / (C5*qr_b_4); |
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89 | const double t5 = -4.0*r*qr_b*em1_qr_b_sq/qr_b_4 * (1.0 - w); |
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90 | const double t10 = 2.0*(em1_qr_b_sq + qr_b_sq)/qr_b_4 * (1.0 - w); //=Sdebye*(1-w) |
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91 | const double t6 = 4.0*b/q0 * t10; |
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92 | const double t7 = r*((-3.0*C3*qr_b_miu -2.0*C2)*qr_b_miu -1.0*C1)*qr_b_miu/(miu*qr_b); |
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93 | const double t9 = C*b/L * (4.0 - exp(-qr_b_sq) * (11.0 + 7.0/qr_b_sq) + 7.0/qr_b_sq)/15.0; |
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94 | const double t12 = b*b*M_PI/(L*q0*q0) + t2 + t3 - t4 + t5 - t6 + t7*w; |
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95 | const double t13 = -b*M_PI/(L*q0) + t9 + t10 + t11*w; |
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96 | |
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97 | const double a1 = pow(q0,p1)*t13 - t1*pow(q0,-p2)*(t12 + b*p1/q0*t13); |
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98 | const double a2 = t1*pow(q0,-p1)*(t12 + b*p1/q0*t13); |
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99 | |
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100 | const double ans = a1*pow(q*b, -p1) + a2*pow(q*b, -p2) + M_PI/(q*L); |
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101 | return ans; |
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102 | } |
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103 | |
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104 | static double |
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105 | _short(double r2, double exp_qr_b, double L, double b, double p1short, |
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106 | double p2short, double q0) |
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107 | { |
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108 | const double qr2 = q0*q0 * r2; |
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109 | const double b3 = b*b*b; |
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110 | const double q0p = pow(q0, -4.0 + p1short); |
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111 | |
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112 | double yy = 1.0/(L*r2*r2) * b/exp_qr_b*q0p |
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113 | * (8.0*b3*L |
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114 | - 8.0*b3*exp_qr_b*L |
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115 | + 2.0*b3*exp_qr_b*L*p2short |
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116 | - 2.0*b*exp_qr_b*L*p2short*qr2 |
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117 | + 4.0*b*exp_qr_b*L*qr2 |
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118 | - 2.0*b3*L*p2short |
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119 | + 4.0*b*L*qr2 |
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120 | - M_PI*exp_qr_b*qr2*q0*r2 |
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121 | + M_PI*exp_qr_b*p2short*qr2*q0*r2); |
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122 | |
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123 | return yy; |
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124 | } |
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125 | static double |
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126 | a_short(double qp, double L, double b |
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127 | /*double p1short, double p2short*/, double q0) |
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128 | { |
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129 | const double p1short = 5.36; |
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130 | const double p2short = 5.62; |
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131 | |
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132 | const double r2 = Rgsquareshort(L,b); |
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133 | const double exp_qr_b = exp(r2*square(q0/b)); |
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134 | const double pdiff = p1short - p2short; |
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135 | const double a1 = _short(r2,exp_qr_b,L,b,p1short,p2short,q0)/pdiff; |
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136 | const double a2= -_short(r2,exp_qr_b,L,b,p2short,p1short,q0)/pdiff; |
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137 | const double ans = a1*pow(qp*b, -p1short) + a2*pow(qp*b, -p2short) + M_PI/(qp*L); |
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138 | return ans; |
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139 | } |
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140 | |
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141 | |
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142 | static double |
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143 | Sexv(double q, double L, double b) |
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144 | { |
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145 | // Pedersen eq 13, corrected by Chen eq A.5, swapping w and 1-w |
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146 | const double C1=1.22; |
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147 | const double C2=0.4288; |
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148 | const double C3=-1.651; |
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149 | const double miu = 0.585; |
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150 | const double qr = q*sqrt(Rgsquare(L,b)); |
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151 | const double qr_miu = pow(qr, -1.0/miu); |
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152 | const double w = w_WR(qr); |
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153 | const double t10 = Sdebye(qr*qr)*(1.0 - w); |
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154 | const double t11 = ((C3*qr_miu + C2)*qr_miu + C1)*qr_miu; |
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155 | |
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156 | return t10 + w*t11; |
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157 | } |
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158 | |
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159 | // Modified by Yun on Oct. 15, |
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160 | static double |
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161 | Sexv_new(double q, double L, double b) |
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162 | { |
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163 | const double qr = q*sqrt(Rgsquare(L,b)); |
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164 | const double qr2 = qr*qr; |
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165 | const double C = (L/b > 10.0) ? 3.06*pow(L/b, -0.44) : 1.0; |
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166 | const double t9 = C*b/L * (4.0 - exp(-qr2) * (11.0 + 7.0/qr2) + 7.0/qr2)/15.0; |
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167 | |
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168 | const double Sexv_orig = Sexv(q, L, b); |
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169 | |
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170 | // calculating the derivative to decide on the correction (cutoff) term? |
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171 | // Note: this is modified from WRs original code |
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172 | const double del=1.05; |
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173 | const double qdel = (Sexv(q*del,L,b) - Sexv_orig)/(q*(del - 1.0)); |
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174 | |
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175 | if (qdel < 0) { |
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176 | //printf("branch A1-%d q=%g L=%g b=%g\n", C==1.0, q, L, b); |
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177 | return t9 + Sexv_orig; |
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178 | } else { |
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179 | //printf("branch A2-%d q=%g L=%g b=%g\n", C==1.0, q, L, b); |
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180 | const double w = w_WR(qr); |
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181 | const double t10 = Sdebye(qr*qr)*(1.0 - w); |
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182 | return t9 + t10; |
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183 | } |
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184 | } |
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185 | |
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186 | |
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187 | static double |
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188 | Sk_WR(double q, double L, double b) |
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189 | { |
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190 | const double Rg_short = sqrt(Rgsquareshort(L, b)); |
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191 | double q0short = fmax(1.9/Rg_short, 3.0); |
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192 | double ans; |
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193 | |
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194 | |
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195 | if( L > 4*b ) { // L > 4*b : Longer Chains |
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196 | if (q*b <= 3.1) { |
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197 | ans = Sexv_new(q, L, b); |
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198 | } else { //q(i)*b > 3.1 |
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199 | ans = a_long(q, L, b /*, p1, p2, q0*/); |
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200 | } |
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201 | } else { // L <= 4*b : Shorter Chains |
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202 | if (q*b <= q0short) { // q*b <= fmax(1.9/Rg_short, 3) |
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203 | //printf("branch C-%d q=%g L=%g b=%g\n", square(q*Rg_short)<DEBYE_CUTOFF, q, L, b); |
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204 | // Note that q0short is usually 3, but it will be greater than 3 |
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205 | // small enough b, depending on the L/b ratio: |
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206 | // L/b == 1 => b < 2.37 |
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207 | // L/b == 2 => b < 1.36 |
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208 | // L/b == 3 => b < 1.00 |
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209 | // L/b == 4 => b < 0.816 |
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210 | // 2017-10-01 pkienzle: moved low q approximation into Sdebye() |
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211 | ans = Sdebye(square(q*Rg_short)); |
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212 | } else { // q*b > max(1.9/Rg_short, 3) |
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213 | //printf("branch D q=%g L=%g b=%g\n", q, L, b); |
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214 | ans = a_short(q, L, b /*, p1short, p2short*/, q0short); |
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215 | } |
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216 | } |
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217 | |
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218 | return ans; |
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219 | } |
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