[f94d8a2] | 1 | /* |
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| 2 | Functions for WRC implementation of flexible cylinders |
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| 3 | */ |
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| 4 | static double |
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| 5 | AlphaSquare(double x) |
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| 6 | { |
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[e7678b2] | 7 | // Potentially faster. Needs proper benchmarking. |
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| 8 | // add native_powr to kernel_template |
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| 9 | //double t = native_powr( (1.0 + (x/3.12)*(x/3.12) + |
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| 10 | // (x/8.67)*(x/8.67)*(x/8.67)),(0.176/3.0) ); |
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| 11 | //return t; |
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| 12 | |
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[f94d8a2] | 13 | return pow( (1.0 + (x/3.12)*(x/3.12) + |
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| 14 | (x/8.67)*(x/8.67)*(x/8.67)),(0.176/3.0) ); |
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| 15 | } |
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| 16 | |
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| 17 | // |
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| 18 | static double |
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| 19 | Rgsquarezero(double q, double L, double b) |
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| 20 | { |
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[e7678b2] | 21 | const double r = b/L; |
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| 22 | return (L*b/6.0) * |
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| 23 | (1.0 - r*1.5 + 1.5*r*r - 0.75*r*r*r*(1.0 - exp(-2.0/r))); |
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[f94d8a2] | 24 | } |
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| 25 | |
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| 26 | // |
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| 27 | static double |
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| 28 | Rgsquareshort(double q, double L, double b) |
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| 29 | { |
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| 30 | return AlphaSquare(L/b) * Rgsquarezero(q,L,b); |
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| 31 | } |
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| 32 | |
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| 33 | // |
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| 34 | static double |
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| 35 | Rgsquare(double q, double L, double b) |
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| 36 | { |
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| 37 | return AlphaSquare(L/b)*L*b/6.0; |
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| 38 | } |
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| 39 | |
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[e7678b2] | 40 | static inline double |
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[f94d8a2] | 41 | sech_WR(double x) |
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| 42 | { |
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| 43 | return(1/cosh(x)); |
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| 44 | } |
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| 45 | |
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| 46 | static double |
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| 47 | a1long(double q, double L, double b, double p1, double p2, double q0) |
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| 48 | { |
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[e7678b2] | 49 | double C; |
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| 50 | const double onehalf = 1.0/2.0; |
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[f94d8a2] | 51 | |
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| 52 | if( L/b > 10.0) { |
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| 53 | C = 3.06/pow((L/b),0.44); |
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| 54 | } else { |
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| 55 | C = 1.0; |
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| 56 | } |
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| 57 | |
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[e7678b2] | 58 | const double C1 = 1.22; |
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| 59 | const double C2 = 0.4288; |
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| 60 | const double C3 = -1.651; |
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| 61 | const double C4 = 1.523; |
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| 62 | const double C5 = 0.1477; |
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| 63 | const double miu = 0.585; |
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| 64 | |
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| 65 | const double Rg2 = Rgsquare(q,L,b); |
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| 66 | const double Rg22 = Rg2*Rg2; |
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| 67 | const double Rg = sqrt(Rg2); |
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| 68 | const double Rgb = Rg*q0/b; |
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[f94d8a2] | 69 | |
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[e7678b2] | 70 | const double b2 = b*b; |
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| 71 | const double b3 = b*b*b; |
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| 72 | const double b4 = b3*b; |
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| 73 | const double q02 = q0*q0; |
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| 74 | const double q03 = q0*q0*q0; |
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| 75 | const double q04 = q03*q0; |
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| 76 | const double q05 = q04*q0; |
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[f94d8a2] | 77 | |
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[e7678b2] | 78 | const double Rg02 = Rg2*q02; |
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[f94d8a2] | 79 | |
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[e7678b2] | 80 | const double t1 = (b*C*((4.0/15.0 - pow((double)M_E,(-(Rg02/b2))) * |
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| 81 | ((11.0/15.0 + (7.0*b2)/(15.0*Rg02))) + |
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| 82 | (7.0*b2)/(15.0*Rg02)))); |
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[f94d8a2] | 83 | |
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[e7678b2] | 84 | const double t2 = (2.0*b4*(((-1.0) + pow((double)M_E,(-(Rg02/b2))) + |
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| 85 | Rg02/b2))*((1.0 + onehalf*(((-1.0) - |
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| 86 | tanh((-C4 + Rgb/C5))))))); |
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[f94d8a2] | 87 | |
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[e7678b2] | 88 | const double t3 = ((C3*pow(Rgb,((-3.0)/miu)) + |
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| 89 | C2*pow(Rgb,((-2.0)/miu)) + |
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| 90 | C1*pow(Rgb,((-1.0)/miu)))); |
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[f94d8a2] | 91 | |
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[e7678b2] | 92 | const double t4 = ((1.0 + tanh(((-C4) + Rgb)/C5))); |
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[f94d8a2] | 93 | |
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[e7678b2] | 94 | const double t5 = (1.0/(b*p1*pow(q0,((-1.0) - p1 - p2)) - |
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| 95 | b*p2*pow(q0,((-1.0) - p1 - p2)))); |
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[f94d8a2] | 96 | |
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[e7678b2] | 97 | const double t6 = (b*C*(((-((14.0*b3)/(15.0*q03*Rg2))) + |
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| 98 | (14.0*b3*pow((double)M_E,(-(Rg02/b2))))/(15.0*q03*Rg2) + |
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| 99 | (2.0*pow((double)M_E,(-(Rg02/b2)))*q0*((11.0/15.0 + |
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| 100 | (7.0*b2)/(15.0*Rg02)))*Rg2)/b))); |
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[f94d8a2] | 101 | |
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[e7678b2] | 102 | const double t7 = (Rg*((C3*pow(((Rgb)),((-3.0)/miu)) + |
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| 103 | C2*pow(((Rgb)),((-2.0)/miu)) + |
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| 104 | C1*pow(((Rgb)),((-1.0)/miu))))*pow(sech_WR(((-C4) + |
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| 105 | Rgb)/C5),2)); |
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[f94d8a2] | 106 | |
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[e7678b2] | 107 | const double t8 = (b4*Rg*(((-1.0) + pow((double)M_E,(-(Rg02/b2))) + |
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| 108 | Rg02/b2))*pow(sech_WR(((-C4) + Rgb)/C5),2)); |
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[f94d8a2] | 109 | |
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[e7678b2] | 110 | const double t9 = (2.0*b4*(((2.0*q0*Rg2)/b - |
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| 111 | (2.0*pow((double)M_E,(-(Rg02/b2)))*q0*Rg2)/b))*((1.0 + onehalf*(((-1.0) - |
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| 112 | tanh(((-C4) + Rgb)/C5)))))); |
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[f94d8a2] | 113 | |
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[e7678b2] | 114 | const double t10 = (8.0*b4*b*(((-1.0) + pow((double)M_E,(-(Rg02/b2))) + |
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| 115 | Rg02/b2))*((1.0 + onehalf*(((-1.0) - tanh(((-C4) + |
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| 116 | Rgb)/C5)))))); |
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[f94d8a2] | 117 | |
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[e7678b2] | 118 | const double t11 = (((-((3.0*C3*Rg*pow(((Rgb)),((-1.0) - |
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| 119 | 3.0/miu)))/miu)) - (2.0*C2*Rg*pow(((Rgb)),((-1.0) - |
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| 120 | 2.0/miu)))/miu - (C1*Rg*pow(((Rgb)),((-1.0) - |
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| 121 | 1.0/miu)))/miu)); |
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[f94d8a2] | 122 | |
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[e7678b2] | 123 | const double t12 = ((1.0 + tanh(((-C4) + Rgb)/C5))); |
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[f94d8a2] | 124 | |
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[e7678b2] | 125 | const double t13 = (b*C*((4.0/15.0 - pow((double)M_E,(-(Rg02/b2)))*((11.0/15.0 + |
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| 126 | (7.0*b2)/(15.0*q02* Rg2))) + |
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| 127 | (7.0*b2)/(15.0*Rg02)))); |
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[f94d8a2] | 128 | |
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[e7678b2] | 129 | const double t14 = (2.0*b4*(((-1.0) + pow((double)M_E,(-(Rg02/b2))) + |
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| 130 | Rg02/b2))*((1.0 + onehalf*(((-1.0) - tanh(((-C4) + |
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| 131 | Rgb)/C5)))))); |
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[f94d8a2] | 132 | |
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[e7678b2] | 133 | const double t15 = ((C3*pow(((Rgb)),((-3.0)/miu)) + |
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| 134 | C2*pow(((Rgb)),((-2.0)/miu)) + |
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| 135 | C1*pow(((Rgb)),((-1.0)/miu)))); |
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[f94d8a2] | 136 | |
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[e7678b2] | 137 | |
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| 138 | double yy = (pow(q0,p1)*(((-((b*M_PI)/(L*q0))) +t1/L +t2/(q04*Rg22) + |
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| 139 | onehalf*t3*t4)) + (t5*((pow(q0,(p1 - p2))* |
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| 140 | (((-pow(q0,(-p1)))*(((b2*M_PI)/(L*q02) +t6/L +t7/(2.0*C5) - |
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| 141 | t8/(C5*q04*Rg22) + t9/(q04*Rg22) -t10/(q05*Rg22) + onehalf*t11*t12)) - |
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| 142 | b*p1*pow(q0,((-1.0) - p1))*(((-((b*M_PI)/(L*q0))) + t13/L + |
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| 143 | t14/(q04*Rg22) + onehalf*t15*((1.0 + tanh(((-C4) + |
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| 144 | Rgb)/C5))))))))))); |
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[f94d8a2] | 145 | |
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| 146 | return (yy); |
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| 147 | } |
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| 148 | |
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| 149 | static double |
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| 150 | a2long(double q, double L, double b, double p1, double p2, double q0) |
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| 151 | { |
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[e7678b2] | 152 | double C; |
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| 153 | const double onehalf = 1.0/2.0; |
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[f94d8a2] | 154 | |
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| 155 | if( L/b > 10.0) { |
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| 156 | C = 3.06/pow((L/b),0.44); |
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| 157 | } else { |
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| 158 | C = 1.0; |
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| 159 | } |
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| 160 | |
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[e7678b2] | 161 | const double C1 = 1.22; |
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| 162 | const double C2 = 0.4288; |
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| 163 | const double C3 = -1.651; |
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| 164 | const double C4 = 1.523; |
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| 165 | const double C5 = 0.1477; |
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| 166 | const double miu = 0.585; |
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| 167 | |
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| 168 | const double Rg2 = Rgsquare(q,L,b); |
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| 169 | const double Rg22 = Rg2*Rg2; |
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| 170 | const double b2 = b*b; |
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| 171 | const double b3 = b*b*b; |
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| 172 | const double b4 = b3*b; |
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| 173 | const double q02 = q0*q0; |
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| 174 | const double q03 = q0*q0*q0; |
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| 175 | const double q04 = q03*q0; |
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| 176 | const double q05 = q04*q0; |
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| 177 | const double Rg = sqrt(Rg2); |
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| 178 | const double Rgb = Rg*q0/b; |
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| 179 | const double Rg02 = Rg2*q02; |
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| 180 | |
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| 181 | const double t1 = (1.0/(b* p1*pow(q0,((-1.0) - p1 - p2)) - |
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| 182 | b*p2*pow(q0,((-1.0) - p1 - p2)) )); |
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| 183 | |
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| 184 | const double t2 = (b*C*(((-1.0*((14.0*b3)/(15.0*q03*Rg2))) + |
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| 185 | (14.0*b3*pow((double)M_E,(-(Rg02/b2))))/(15.0*q03*Rg2) + |
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| 186 | (2.0*pow((double)M_E,(-(Rg02/b2)))*q0*((11.0/15.0 + |
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| 187 | (7*b2)/(15.0*Rg02)))*Rg2)/b)))/L; |
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| 188 | |
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| 189 | const double t3 = (Rg*((C3*pow(((Rgb)),((-3.0)/miu)) + |
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| 190 | C2*pow(((Rgb)),((-2.0)/miu)) + |
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| 191 | C1*pow(((Rgb)),((-1.0)/miu))))* |
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| 192 | pow(sech_WR(((-C4) +Rgb)/C5),2.0))/(2.0*C5); |
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| 193 | |
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| 194 | const double t4 = (b4*Rg*(((-1.0) + pow((double)M_E,(-(Rg02/b2))) + |
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| 195 | Rg02/b2))*pow(sech_WR(((-C4) + |
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| 196 | Rgb)/C5),2))/(C5*q04*Rg22); |
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| 197 | |
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| 198 | const double t5 = (2.0*b4*(((2.0*q0*Rg2)/b - |
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| 199 | (2.0*pow((double)M_E,(-(Rg02/b2)))*q0*Rg2)/b))* |
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| 200 | ((1.0 + onehalf*(((-1.0) - tanh(((-C4) + |
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| 201 | Rgb)/C5))))))/(q04*Rg22); |
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| 202 | |
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| 203 | const double t6 = (8.0*b4*b*(((-1.0) + pow((double)M_E,(-(Rg02/b2))) + |
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| 204 | Rg02/b2))*((1.0 + onehalf*(((-1) - tanh(((-C4) + |
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| 205 | Rgb)/C5))))))/(q05*Rg22); |
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| 206 | |
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| 207 | const double t7 = (((-((3.0*C3*Rg*pow(((Rgb)),((-1.0) - |
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| 208 | 3.0/miu)))/miu)) - (2.0*C2*Rg*pow(((Rgb)), |
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| 209 | ((-1.0) - 2.0/miu)))/miu - (C1*Rg*pow(((Rgb)), |
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| 210 | ((-1.0) - 1.0/miu)))/miu)); |
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| 211 | |
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| 212 | const double t8 = ((1.0 + tanh(((-C4) + Rgb)/C5))); |
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| 213 | |
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| 214 | const double t9 = (b*C*((4.0/15.0 - pow((double)M_E,(-(Rg02/b2)))*((11.0/15.0 + |
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| 215 | (7.0*b2)/(15*Rg02))) + (7.0*b2)/(15.0*Rg02))))/L; |
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| 216 | |
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| 217 | const double t10 = (2.0*b4*(((-1) + pow((double)M_E,(-(Rg02/b2))) + |
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| 218 | Rg02/b2))*((1.0 + onehalf*(((-1) - tanh(((-C4) + |
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| 219 | Rgb)/C5))))))/(q04*Rg22); |
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| 220 | |
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| 221 | const double yy = ((-1.0*(t1* ((-pow(q0,-p1)*(((b2*M_PI)/(L*q02) + |
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| 222 | t2 + t3 - t4 + t5 - t6 + onehalf*t7*t8)) - b*p1*pow(q0,((-1.0) - p1))* |
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| 223 | (((-((b*M_PI)/(L*q0))) + t9 + t10 + |
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| 224 | onehalf*((C3*pow(((Rgb)),((-3.0)/miu)) + |
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| 225 | C2*pow(((Rgb)),((-2.0)/miu)) + |
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| 226 | C1*pow(((Rgb)),((-1.0)/miu))))* |
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| 227 | ((1.0 + tanh(((-C4) + Rgb)/C5)))))))))); |
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[f94d8a2] | 228 | |
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| 229 | return (yy); |
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| 230 | } |
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| 231 | |
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| 232 | // |
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| 233 | static double |
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[e7678b2] | 234 | a_short(double q, double L, double b, double p1short, |
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| 235 | double p2short, double factor, double pdiff, double q0) |
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[f94d8a2] | 236 | { |
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[e7678b2] | 237 | const double Rg2_sh = Rgsquareshort(q,L,b); |
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| 238 | const double Rg2_sh2 = Rg2_sh*Rg2_sh; |
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| 239 | const double b3 = b*b*b; |
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| 240 | const double t1 = ((q0*q0*Rg2_sh)/(b*b)); |
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| 241 | const double Et1 = exp(t1); |
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| 242 | const double Emt1 = 1.0/Et1; |
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| 243 | const double q02 = q0*q0; |
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| 244 | const double q0p = pow(q0,(-4.0 + p1short) ); |
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| 245 | |
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| 246 | double yy = ((factor/(L*(pdiff)*Rg2_sh2)* |
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[f94d8a2] | 247 | ((b*Emt1*q0p*((8.0*b3*L - 8.0*b3*Et1*L - 2.0*b3*L*p2short + |
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| 248 | 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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[e7678b2] | 249 | 2.0*b*Et1*L*p2short*q02*Rg2_sh - Et1*M_PI*q02*q0*Rg2_sh2 + |
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| 250 | Et1*p2short*M_PI*q02*q0*Rg2_sh2)))))); |
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[f94d8a2] | 251 | |
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| 252 | return(yy); |
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| 253 | } |
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[e7678b2] | 254 | static double |
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| 255 | a1short(double q, double L, double b, double p1short, double p2short, double q0) |
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| 256 | { |
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| 257 | |
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| 258 | double factor = 1.0; |
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| 259 | return a_short(q, L, b, p1short, p2short, factor, p1short - p2short, q0); |
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| 260 | } |
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[f94d8a2] | 261 | |
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| 262 | static double |
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| 263 | a2short(double q, double L, double b, double p1short, double p2short, double q0) |
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| 264 | { |
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[e7678b2] | 265 | double factor = -1.0; |
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| 266 | return a_short(q, L, b, p2short, p1short, factor, p1short-p2short, q0); |
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[f94d8a2] | 267 | } |
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| 268 | |
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| 269 | //WR named this w (too generic) |
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| 270 | static double |
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| 271 | w_WR(double x) |
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| 272 | { |
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| 273 | return 0.5*(1 + tanh((x - 1.523)/0.1477)); |
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| 274 | } |
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| 275 | |
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| 276 | // |
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| 277 | static double |
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| 278 | u1(double q, double L, double b) |
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| 279 | { |
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| 280 | return Rgsquareshort(q,L,b)*q*q; |
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| 281 | } |
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| 282 | |
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| 283 | static double |
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| 284 | u_WR(double q, double L, double b) |
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| 285 | { |
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| 286 | return Rgsquare(q,L,b)*q*q; |
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| 287 | } |
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| 288 | |
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[504abee] | 289 | static double |
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| 290 | Sdebye_kernel(double arg) |
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| 291 | { |
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| 292 | // ORIGINAL |
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[e7678b2] | 293 | double result = 2.0*(exp(-arg) + arg -1.0)/(arg*arg); |
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[504abee] | 294 | |
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| 295 | // CONVERSION 1 from http://herbie.uwplse.org/ |
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| 296 | // |
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| 297 | // exhibits discontinuity - needs more investigation |
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| 298 | //double a1 = 1.0/6.0; |
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| 299 | //double a2 = 1.0/72.0; |
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| 300 | //double a3 = 1.0/24.0; |
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| 301 | //double result = pow((1.0 - a1*arg - (a2+a3)*arg*arg), 2); |
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| 302 | |
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| 303 | return result; |
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| 304 | } |
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[f94d8a2] | 305 | static double |
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| 306 | Sdebye(double q, double L, double b) |
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| 307 | { |
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[504abee] | 308 | double arg = u_WR(q,L,b); |
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| 309 | return Sdebye_kernel(arg); |
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[f94d8a2] | 310 | } |
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| 311 | |
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| 312 | // |
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| 313 | static double |
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| 314 | Sdebye1(double q, double L, double b) |
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| 315 | { |
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[504abee] | 316 | double arg = u1(q,L,b); |
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| 317 | return Sdebye_kernel(arg); |
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| 318 | |
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[f94d8a2] | 319 | } |
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| 320 | |
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| 321 | // |
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| 322 | static double |
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| 323 | Sexv(double q, double L, double b) |
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| 324 | { |
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| 325 | |
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[e7678b2] | 326 | const double C1=1.22; |
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| 327 | const double C2=0.4288; |
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| 328 | const double C3=-1.651; |
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| 329 | const double miu = 0.585; |
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| 330 | const double qRg = q*sqrt(Rgsquare(q,L,b)); |
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| 331 | const double x = pow(qRg, -1.0/miu); |
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[f94d8a2] | 332 | |
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[e7678b2] | 333 | double yy = (1.0 - w_WR(qRg))*Sdebye(q,L,b) + |
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| 334 | w_WR(qRg)*x*(C1 + x*(C2 + x*C3)); |
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[f94d8a2] | 335 | return (yy); |
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| 336 | } |
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| 337 | |
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| 338 | |
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| 339 | static double |
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| 340 | Sexvnew(double q, double L, double b) |
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| 341 | { |
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[e7678b2] | 342 | double yy; |
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| 343 | |
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| 344 | const double C1 =1.22; |
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| 345 | const double C2 =0.4288; |
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| 346 | const double C3 =-1.651; |
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| 347 | const double miu = 0.585; |
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| 348 | const double del=1.05; |
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| 349 | const double qRg = q*sqrt(Rgsquare(q,L,b)); |
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| 350 | const double x = pow(qRg, -1.0/miu); |
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[f94d8a2] | 351 | |
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| 352 | |
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| 353 | //calculating the derivative to decide on the corection (cutoff) term? |
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| 354 | // I have modified this from WRs original code |
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[e7678b2] | 355 | const double qdel = (Sexv(q*del,L,b)-Sexv(q,L,b))/(q*del - q); |
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| 356 | const double C_star2 =(qdel >= 0.0) ? 0.0 : 1.0; |
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[f94d8a2] | 357 | |
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[e7678b2] | 358 | yy = (1.0 - w_WR(qRg))*Sdebye(q,L,b) + |
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| 359 | C_star2*w_WR(qRg)* |
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| 360 | x*(C1 + x*(C2 + x*C3)); |
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[f94d8a2] | 361 | |
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| 362 | return (yy); |
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| 363 | } |
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| 364 | |
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[13ed84c] | 365 | double Sk_WR(double q, double L, double b); |
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[f94d8a2] | 366 | double Sk_WR(double q, double L, double b) |
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| 367 | { |
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[e7678b2] | 368 | // |
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| 369 | const double p1 = 4.12; |
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| 370 | const double p2 = 4.42; |
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| 371 | const double p1short = 5.36; |
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| 372 | const double p2short = 5.62; |
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| 373 | const double q0 = 3.1; |
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[f94d8a2] | 374 | |
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[e7678b2] | 375 | double q0short = fmax(1.9/sqrt(Rgsquareshort(q,L,b)),3.0); |
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| 376 | double Sexvmodify, ans; |
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[f94d8a2] | 377 | |
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[e7678b2] | 378 | const double C =(L/b > 10.0) ? 3.06/pow((L/b),0.44) : 1.0; |
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[f94d8a2] | 379 | |
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[e7678b2] | 380 | if( L > 4*b ) { // Longer Chains |
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| 381 | if (q*b <= 3.1) { //Modified by Yun on Oct. 15, |
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| 382 | Sexvmodify = Sexvnew(q, L, b); |
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| 383 | ans = Sexvmodify + C * (4.0/15.0 + 7.0/(15.0*u_WR(q,L,b)) - |
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[f94d8a2] | 384 | (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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| 385 | |
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[e7678b2] | 386 | } else { //q(i)*b > 3.1 |
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| 387 | ans = a1long(q, L, b, p1, p2, q0)/(pow((q*b),p1)) + |
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| 388 | a2long(q, L, b, p1, p2, q0)/(pow((q*b),p2)) + M_PI/(q*L); |
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| 389 | } |
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| 390 | } else { //L <= 4*b Shorter Chains |
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| 391 | if (q*b <= fmax(1.9/sqrt(Rgsquareshort(q,L,b)),3.0) ) { |
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| 392 | if (q*b<=0.01) { |
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| 393 | ans = 1.0 - Rgsquareshort(q,L,b)*(q*q)/3.0; |
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| 394 | } else { |
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| 395 | ans = Sdebye1(q,L,b); |
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| 396 | } |
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| 397 | } else { //q*b > max(1.9/sqrt(Rgsquareshort(q(i),L,b)),3) |
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| 398 | ans = a1short(q,L,b,p1short,p2short,q0short)/(pow((q*b),p1short)) + |
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| 399 | a2short(q,L,b,p1short,p2short,q0short)/(pow((q*b),p2short)) + |
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| 400 | M_PI/(q*L); |
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| 401 | } |
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[f94d8a2] | 402 | } |
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| 403 | |
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| 404 | return(ans); |
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| 405 | } |
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