ClosedOrbitFinder.h 32.1 KB
Newer Older
1 2 3 4
/**
 * @file ClosedOrbitFinder.h
 * The algorithm is based on the paper of M. M. Gordon: "Computation of closed orbits and basic focusing properties for
 * sector-focused cyclotrons and the design of 'cyclops'" (1983)
5 6
 * As template arguments one chooses the type of the variables and the integrator for the ODEs. The supported steppers can
 * be found on
7 8 9 10 11 12
 * http://www.boost.org/ where it is part of the library Odeint.
 *
 * @author Matthias Frey
 * @version 1.0
 */

13 14 15
#ifndef CLOSEDORBITFINDER_H
#define CLOSEDORBITFINDER_H

16
#include <algorithm>
17 18 19
#include <array>
#include <cmath>
#include <functional>
adelmann's avatar
adelmann committed
20
#include <limits>
21
#include <numeric>
adelmann's avatar
adelmann committed
22
#include <string>
23
#include <utility>
24 25
#include <vector>

26
#include "Utilities/Options.h"
27 28 29
#include "Utilities/Options.h"
#include "Utilities/OpalException.h"

frey_m's avatar
frey_m committed
30 31
#include "AbstractObjects/OpalData.h"

32
#include "AbsBeamline/Cyclotron.h"
frey_m's avatar
frey_m committed
33

34 35
// include headers for integration
#include <boost/numeric/odeint/integrate/integrate_n_steps.hpp>
frey_m's avatar
frey_m committed
36 37 38
#include <boost/filesystem.hpp>

extern Inform *gmsg;
39

40
/// Finds a closed orbit of a cyclotron for a given energy
41 42 43
template<typename Value_type, typename Size_type, class Stepper>
class ClosedOrbitFinder
{
44 45 46 47 48 49 50 51 52
    public:
        /// Type of variables
        typedef Value_type value_type;
        /// Type for specifying sizes
        typedef Size_type size_type;
        /// Type of container for storing quantities (path length, orbit, etc.)
        typedef std::vector<value_type> container_type;
        /// Type for holding state of ODE values
        typedef std::vector<value_type> state_type;
frey_m's avatar
frey_m committed
53 54
        
        typedef std::function<void(const state_type&, state_type&, const double)> function_t;
55 56 57

        /// Sets the initial values for the integration and calls findOrbit().
        /*!
58
         * @param E0 is the potential energy (particle energy at rest) [MeV].
adelmann's avatar
adelmann committed
59
         * @param N specifies the number of splits (2pi/N), i.e number of integration steps
60
         * @param cycl is the cyclotron element
61 62
         * @param domain is a boolean (default: true). If "true" the closed orbit is computed over a single sector,
         * otherwise over 2*pi.
63
         */
frey_m's avatar
frey_m committed
64
        ClosedOrbitFinder(value_type E0, size_type N,
65
                          Cyclotron* cycl, bool domain = true);
66 67

        /// Returns the inverse bending radius (size of container N+1)
68
        container_type getInverseBendingRadius(const value_type& angle = 0);
69 70

        /// Returns the step lengths of the path (size of container N+1)
71
        container_type getPathLength(const value_type& angle = 0);
72 73

        /// Returns the field index (size of container N+1)
74
        container_type getFieldIndex(const value_type& angle = 0);
75 76 77 78

        /// Returns the radial and vertical tunes (in that order)
        std::pair<value_type,value_type> getTunes();

79 80 81 82 83 84 85
        /// Returns the closed orbit (size of container N+1) starting at specific angle (only makes sense when computing
        /// the closed orbit for a whole turn) (default value: 0°).
        /// Attention: It computes the starting index of the array. If it's not an integer it just cuts the floating point
        /// part, i.e. it takes the next starting index below. There's no interpolation of the radius.
        /*!
         * @param angle is the start angle for the output. Has to be within [0°,360°[ (default: 0°).
         */
86 87
        container_type getOrbit(value_type angle = 0);

88 89 90 91 92 93
        /// Returns the momentum of the orbit (size of container N+1)starting at specific angle (only makes sense when
        /// computing the closed orbit for a whole turn) (default value: 0°), \f$ \left[ p_{r} \right] = \si{m}\f$.
        /// Attention: It computes the starting index of the array. If it's not an integer it just cuts the floating point
        /// part, i.e. it takes the next starting index below. There's no interpolation of the momentum.
        /*!
         * @param angle is the start angle for the output. Has to be within [0°,360°[ (default: 0°).
94
         * @returns the momentum in \f$ \beta * \gamma \f$ units
95
         */
96
        container_type getMomentum(value_type angle = 0);
97 98 99 100

        /// Returns the average orbit radius
        value_type getAverageRadius();

adelmann's avatar
adelmann committed
101 102
        /// Returns the frequency error
        value_type getFrequencyError();
103 104 105 106 107 108 109 110

        /// Returns true if a closed orbit could be found
        bool isConverged();

        /// Computes the closed orbit
        /*!
         * @param accuracy specifies the accuracy of the closed orbit
         * @param maxit is the maximal number of iterations done for finding the closed orbit
frey_m's avatar
frey_m committed
111
         * @param ekin energy for which to find closed orbit (in tune mode: upper limit of range)
frey_m's avatar
frey_m committed
112
         * @param dE step increase [MeV]
113
         * @param rguess initial radius guess in [mm]
frey_m's avatar
frey_m committed
114
         * @param isTuneMode comptute tunes of all energies in one sweep
115
         */
116 117 118 119
        bool findOrbit(value_type accuracy, size_type maxit,
                       value_type ekin,
                       value_type dE = 1.0, value_type rguess = -1.0,
                       bool isTuneMode = false);
120

121
        /// Fills in the values of h_m, ds_m, fidx_m.
122
        void computeOrbitProperties(const value_type& E);
123

124
    private:
125 126 127 128 129 130 131 132
        /// This function is called by the function getTunes().
        /*! Transfer matrix Y = [y11, y12; y21, y22] (see Gordon paper for more details).
         * @param y are the positions (elements y11 and y12 of Y)
         * @param py2 is the momentum of the second solution (element y22 of Y)
         * @param ncross is the number of sign changes (\#crossings of zero-line)
         */
        value_type computeTune(const std::array<value_type,2>&, value_type, size_type);

frey_m's avatar
frey_m committed
133 134 135 136
        // Compute closed orbit for given energy
        bool findOrbitOfEnergy_m(const value_type&, container_type&, value_type&,
                                 const value_type&, size_type);
        
adelmann's avatar
adelmann committed
137
        /// This function computes nzcross_ which is used to compute the tune in z-direction and the frequency error
138
//         void computeVerticalOscillations();
139 140 141
        
        /// This function rotates the calculated closed orbit finder properties to the initial angle
        container_type rotate(value_type angle, container_type& orbitProperty);
142 143 144 145 146

        /// Stores current position in horizontal direction for the solutions of the ODE with different initial values
        std::array<value_type,2> x_m; // x_m = [x1, x2]
        /// Stores current momenta in horizontal direction for the solutions of the ODE with different initial values
        std::array<value_type,2> px_m; // px_m = [px1, px2]
frey_m's avatar
frey_m committed
147
        /// Stores current position in vertical direction for the solutions of the ODE with different initial values
148
        std::array<value_type,2> z_m; // z_m = [z1, z2]
frey_m's avatar
frey_m committed
149
        /// Stores current momenta in vertical direction for the solutions of the ODE with different initial values
150 151 152 153 154 155 156 157
        std::array<value_type,2> pz_m; // pz_m = [pz1, pz2]

        /// Stores the inverse bending radius
        container_type h_m;
        /// Stores the step length
        container_type ds_m;
        /// Stores the radial orbit (size: N_m+1)
        container_type r_m;
158 159
        /// Stores the vertical oribt (size: N_m+1)
        container_type vz_m;
160 161
        /// Stores the radial momentum
        container_type pr_m;
162 163
        /// Stores the vertical momentum
        container_type vpz_m;
164 165 166 167 168 169 170 171
        /// Stores the field index
        container_type fidx_m;

        /// Counts the number of zero-line crossings in horizontal direction (used for computing horizontal tune)
        size_type nxcross_m;
        /// Counts the number of zero-line crossings in vertical direction (used for computing vertical tune)
        size_type nzcross_m; //#crossings of zero-line in x- and z-direction

172
        /// Is the rest mass [MeV / c**2]
173 174
        value_type E0_m;
        
175
        /// Is the nominal orbital frequency
176 177 178
        /* (see paper of Dr. C. Baumgarten: "Transverse-Longitudinal
         * Coupling by Space Charge in Cyclotrons" (2012), formula (1))
         */
179
        value_type wo_m;
adelmann's avatar
adelmann committed
180
        /// Number of integration steps
181 182 183 184 185 186 187 188 189 190
        size_type N_m;
        /// Is the angle step size
        value_type dtheta_m;

        /// Is the average radius
        value_type ravg_m;

        /// Is the phase
        value_type phase_m;

191
        /**
192 193 194 195
         * Stores the last orbit value (since we have to return to the beginning to check the convergence in the
         * findOrbit() function. This last value is then deleted from the array but is stored in lastOrbitVal_m to
         * compute the vertical oscillations)
         */
snuverink_j's avatar
snuverink_j committed
196
        /* value_type lastOrbitVal_m; */
197

198 199 200 201 202
        /**
         * Stores the last momentum value (since we have to return to the beginning to check the convergence in the
         * findOrbit() function. This last value is then deleted from the array but is stored in lastMomentumVal_m to
         * compute the vertical oscillations)
         */
snuverink_j's avatar
snuverink_j committed
203
        /* value_type lastMomentumVal_m; */
204 205

        /**
206 207 208
         * Boolean which is true by default. "true": orbit integration over one sector only, "false": integration
         * over 2*pi
         */
adelmann's avatar
adelmann committed
209
        bool domain_m;
210

211 212
        /// Defines the stepper for integration of the ODE's
        Stepper stepper_m;
213 214 215 216 217 218 219 220 221 222 223 224 225 226 227
        
        /*!
         * This quantity is defined in the paper "Transverse-Longitudinal Coupling by Space Charge in Cyclotrons" 
         * of Dr. Christian Baumgarten (2012)
         * The lambda function takes the orbital frequency \f$ \omega_{o} \f$ (also defined in paper) as input argument.
         */
        std::function<double(double)> acon_m = [](double wo) { return Physics::c / wo; };
        
        /// Cyclotron unit \f$ \left[T\right] \f$ (Tesla)
        /*!
         * The lambda function takes the orbital frequency \f$ \omega_{o} \f$ as input argument.
         */
        std::function<double(double, double)> bcon_m = [](double e0, double wo) {
            return e0 * 1.0e7 / (/* physics::q0 */ 1.0 * Physics::c * Physics::c / wo);
        };
228
        
229
        Cyclotron* cycl_m;
230 231 232 233 234 235
};

// -----------------------------------------------------------------------------------------------------------------------
// PUBLIC MEMBER FUNCTIONS
// -----------------------------------------------------------------------------------------------------------------------

236 237 238
template<typename Value_type, typename Size_type, class Stepper>
ClosedOrbitFinder<Value_type,
                  Size_type,
frey_m's avatar
frey_m committed
239
                  Stepper>::ClosedOrbitFinder(value_type E0,
240
                                              size_type N, Cyclotron* cycl,
241
                                              bool domain)
242 243 244
    : nxcross_m(0)
    , nzcross_m(0)
    , E0_m(E0)
245
    , wo_m(cycl->getRfFrequ()*1E6/cycl->getCyclHarm()*2.0*Physics::pi)
246 247 248 249
    , N_m(N)
    , dtheta_m(Physics::two_pi/value_type(N))
    , ravg_m(0)
    , phase_m(0)
snuverink_j's avatar
snuverink_j committed
250 251
    /* , lastOrbitVal_m(0.0) */
    /* , lastMomentumVal_m(0.0) */
252 253
    , domain_m(domain)
    , stepper_m()
254
    , cycl_m(cycl)
255
{
frey_m's avatar
frey_m committed
256
    
257
    if ( cycl_m->getFMLowE() > cycl_m->getFMHighE() )
frey_m's avatar
frey_m committed
258 259 260
        throw OpalException("ClosedOrbitFinder::ClosedOrbitFinder()",
                            "Incorrect cyclotron energy (MeV) bounds: Maximum cyclotron energy smaller than minimum cyclotron energy.");
    
adelmann's avatar
adelmann committed
261 262
    // if domain_m = true --> integrate over a single sector
    if (domain_m) {
263
        N_m /=  cycl_m->getSymmetry();
adelmann's avatar
adelmann committed
264
    }
frey_m's avatar
frey_m committed
265 266 267
    
    cycl_m->read(cycl_m->getFieldFlag(cycl_m->getCyclotronType()),
                 cycl_m->getBScale());
268

269 270 271 272 273
    // reserve storage for the orbit and momentum (--> size = 0, capacity = N_m+1)
    /*
     * we need N+1 storage, since dtheta = 2pi/N (and not 2pi/(N-1)) that's why we need N+1 integration steps
     * to return to the origin (but the return size is N_m)
     */
adelmann's avatar
adelmann committed
274 275
    r_m.reserve(N_m + 1);
    pr_m.reserve(N_m + 1);
276 277
    vz_m.reserve(N_m + 1);
    vpz_m.reserve(N_m + 1);
278

279
    // reserve memory of N_m for the properties (--> size = 0, capacity = N_m)
adelmann's avatar
adelmann committed
280 281 282
    h_m.reserve(N_m);
    ds_m.reserve(N_m);
    fidx_m.reserve(N_m);
283 284 285
}

template<typename Value_type, typename Size_type, class Stepper>
286
inline typename ClosedOrbitFinder<Value_type, Size_type, Stepper>::container_type
287
    ClosedOrbitFinder<Value_type, Size_type, Stepper>::getInverseBendingRadius(const value_type& angle)
288
{
289 290 291 292
    if (angle != 0.0)
        return rotate(angle, h_m);
    else
        return h_m;
293 294 295
}

template<typename Value_type, typename Size_type, class Stepper>
296
inline typename ClosedOrbitFinder<Value_type, Size_type, Stepper>::container_type
297
    ClosedOrbitFinder<Value_type, Size_type, Stepper>::getPathLength(const value_type& angle)
298
{
299 300 301 302
    if (angle != 0.0)
        return rotate(angle, ds_m);
    else
        return ds_m;
303 304 305
}

template<typename Value_type, typename Size_type, class Stepper>
306
inline typename ClosedOrbitFinder<Value_type, Size_type, Stepper>::container_type
307
    ClosedOrbitFinder<Value_type, Size_type, Stepper>::getFieldIndex(const value_type& angle)
308
{
309 310
    if (angle != 0.0)
        return rotate(angle, fidx_m);
frey_m's avatar
frey_m committed
311
    return fidx_m;
312 313 314
}

template<typename Value_type, typename Size_type, class Stepper>
315 316 317
std::pair<Value_type,Value_type> ClosedOrbitFinder<Value_type, Size_type, Stepper>::getTunes() {
    // compute radial tune
    value_type nur = computeTune(x_m,px_m[1],nxcross_m);
318

319 320 321 322
    // compute vertical tune
    value_type nuz = computeTune(z_m,pz_m[1],nzcross_m);

    return std::make_pair(nur,nuz);
323 324 325
}

template<typename Value_type, typename Size_type, class Stepper>
326
inline typename ClosedOrbitFinder<Value_type, Size_type, Stepper>::container_type
327
    ClosedOrbitFinder<Value_type, Size_type, Stepper>::getOrbit(value_type angle)
328
{
329 330 331 332
    if (angle != 0.0)
        return rotate(angle, r_m);
    else
        return r_m;
333 334 335 336
}

template<typename Value_type, typename Size_type, class Stepper>
inline typename ClosedOrbitFinder<Value_type, Size_type, Stepper>::container_type
337
    ClosedOrbitFinder<Value_type, Size_type, Stepper>::getMomentum(value_type angle)
338 339
{
    container_type pr = pr_m;
340 341 342
    
    if (angle != 0.0)
        pr = rotate(angle, pr);
343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358
    
    // change units from meters to \beta * \gamma
    /* in Gordon paper:
     * 
     * p = \gamma * \beta * a
     * 
     * where a = c / \omega_{0} with \omega_{0} = 2 * \pi * \nu_{0} = 2 * \pi * \nu_{rf} / h
     * 
     * c: speed of light
     * h: harmonic number
     * v_{rf}: nomial rf frequency
     * 
     * Units:
     * 
     * [a] = m --> [p] = m
     * 
359
     * The momentum in \beta * \gamma is obtained by dividing by "a"
360
     */
361
    value_type factor =  1.0 / acon_m(wo_m);
362
    std::for_each(pr.begin(), pr.end(), [factor](value_type& p) { p *= factor; });
363
    
364
    return pr;
365 366 367
}

template<typename Value_type, typename Size_type, class Stepper>
368 369 370
inline typename ClosedOrbitFinder<Value_type, Size_type, Stepper>::value_type
    ClosedOrbitFinder<Value_type, Size_type, Stepper>::getAverageRadius()
{
371
    return ravg_m;
372 373 374
}

template<typename Value_type, typename Size_type, class Stepper>
375 376
typename ClosedOrbitFinder<Value_type, Size_type, Stepper>::value_type
    ClosedOrbitFinder<Value_type, Size_type, Stepper>::getFrequencyError()
377
{
378
    return phase_m;
379 380 381 382 383 384
}

// -----------------------------------------------------------------------------------------------------------------------
// PRIVATE MEMBER FUNCTIONS
// -----------------------------------------------------------------------------------------------------------------------

frey_m's avatar
frey_m committed
385 386


387
template<typename Value_type, typename Size_type, class Stepper>
388 389
bool ClosedOrbitFinder<Value_type, Size_type, Stepper>::findOrbit(value_type accuracy,
                                                                  size_type maxit,
frey_m's avatar
frey_m committed
390
                                                                  value_type ekin,
frey_m's avatar
frey_m committed
391
                                                                  value_type dE,
frey_m's avatar
frey_m committed
392 393
                                                                  value_type rguess,
                                                                  bool isTuneMode)
394
{
395 396 397 398
    /* REMARK TO GORDON
     * q' = 1/b = 1/bcon
     * a' = a = acon
     */
399

400 401 402
    // resize vectors (--> size = N_m+1, capacity = N_m+1), note: we do N_m+1 integration steps
    r_m.resize(N_m+1);
    pr_m.resize(N_m+1);
403 404
    vz_m.resize(N_m+1);
    vpz_m.resize(N_m+1);
405

406 407
    // store acon locally
    value_type acon = acon_m(wo_m);            // [acon] = m
408 409
    // amplitude of error; Gordon, formula (18) (a = a')
    value_type error = std::numeric_limits<value_type>::max();
410 411 412 413

    /*
     * Christian:
     * N = 1440 ---> N = 720 ---> dtheta = 2PI/720 --> nsteps = 721
414
     *
415
     * 0, 2, 4, ... ---> jeden zweiten berechnene: 1, 3, 5, ... interpolieren --> 1440 Werte
416
     *
417 418
     * Matthias:
     * N = 1440 --> dtheta = 2PI/1440 --> nsteps = 1441
419
     *
420
     * 0, 1, 2, 3, 4, 5, ... --> 1440 Werte
421
     *
422
     */
423

424 425 426
    value_type E;     // starting energy
    value_type E_fin; // final    energy

frey_m's avatar
frey_m committed
427
    if ( isTuneMode ) {
428 429 430 431 432
        E     = cycl_m->getFMLowE();
        E_fin = cycl_m->getFMHighE();
    } else {
        E     = ekin;
        E_fin = ekin;
frey_m's avatar
frey_m committed
433
    }
Andreas Adelmann's avatar
Andreas Adelmann committed
434

frey_m's avatar
frey_m committed
435 436 437 438
    namespace fs = boost::filesystem;
    fs::path dir = OpalData::getInstance()->getInputBasename();
    dir = dir.parent_path() / "data";
    std::string tunefile = (dir / "tunes.dat").string();
439

frey_m's avatar
frey_m committed
440 441
    if ( isTuneMode ) {
        std::ofstream out(tunefile, std::ios::out);
442

frey_m's avatar
frey_m committed
443
        out << std::left
444 445 446
            << std::setw(15) << "energy[MeV]"
            << std::setw(15) << "radius_ini[m]"
            << std::setw(15) << "radius_avg[m]"
frey_m's avatar
frey_m committed
447 448 449 450
            << std::setw(15) << "nu_r"
            << std::setw(15) << "nu_z"
            << std::endl;
    }
451

452 453 454
    // iterate until suggested energy (start with minimum energy)
    // increase energy by dE
    for (; E <= E_fin ; E+=dE) {
frey_m's avatar
frey_m committed
455
        error = std::numeric_limits<value_type>::max();
456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479

        // energy dependent values
        value_type en     = E / E0_m;        // en = E/E0 = E/(mc^2) (E0 is potential energy)
        value_type gamma  = en + 1.0;
        value_type gamma2 = gamma * gamma;   // = gamma^2
        value_type beta   = std::sqrt(1.0 - 1.0 / gamma2);

        // set initial values for radius and radial momentum for lowest energy Emin
        // orbit, [r] = m;  Gordon, formula (20)
        // radial momentum; Gordon, formula (20)

        container_type init;
        //      r            pr   z    pz
        init = {beta * acon, 0.0, 0.0, 1.0};

        if (rguess >= 0.0) {
            init[0] = rguess * 0.001;
        }

        std::fill(  r_m.begin(),   r_m.end(), 0);
        std::fill( pr_m.begin(),  pr_m.end(), 0);
        std::fill( vz_m.begin(),  vz_m.end(), 0);
        std::fill(vpz_m.begin(), vpz_m.end(), 0);

480
        // (re-)set inital values for r and pr
481 482 483
        r_m[0]   = init[0];
        pr_m[0]  = init[1];
        vz_m[0]  = init[2];
484
        vpz_m[0] = init[3];
485

frey_m's avatar
frey_m committed
486
        if ( !this->findOrbitOfEnergy_m(E, init, error, accuracy, maxit) ) {
487
            *gmsg << "ClosedOrbitFinder didn't converge for energy " + std::to_string(E) + " MeV." << endl;
488
            continue;
frey_m's avatar
frey_m committed
489
        }
490

frey_m's avatar
frey_m committed
491
        if ( isTuneMode ) {
492 493
            this->computeOrbitProperties(E);

frey_m's avatar
frey_m committed
494
            std::pair<value_type , value_type > tunes = this->getTunes();
495 496
            value_type reo = this->getOrbit(   cycl_m->getPHIinit())[0];
            value_type peo = this->getMomentum(cycl_m->getPHIinit())[0];
frey_m's avatar
frey_m committed
497

498 499
            *gmsg << std::left
                  << "* ----------------------------" << endl
frey_m's avatar
frey_m committed
500 501
                  << "* Closed orbit info (Gordon units):" << endl
                  << "*" << endl
502 503 504 505
                  << "* kinetic energy:   " << std::setw(12) << E      << " [MeV]" << endl
                  << "* average radius:   " << std::setw(12) << ravg_m << " [m]" << endl
                  << "* initial radius:   " << std::setw(12) << reo    << " [m]" << endl
                  << "* initial momentum: " << std::setw(12) << peo    << " [Beta Gamma]" << endl
frey_m's avatar
frey_m committed
506 507 508 509
                  << "* frequency error:  " << phase_m        << endl
                  << "* horizontal tune:  " << tunes.first    << endl
                  << "* vertical tune:    " << tunes.second   << endl
                  << "* ----------------------------" << endl << endl;
510

frey_m's avatar
frey_m committed
511 512 513 514
            std::ofstream out(tunefile, std::ios::app);
            out << std::left
                << std::setw(15) << E
                << std::setw(15) << reo
515
                << std::setw(15) << ravg_m
frey_m's avatar
frey_m committed
516 517 518 519
                << std::setw(15) << tunes.first
                << std::setw(15) << tunes.second << std::endl;
            out.close();
        }
520
    }
521

522 523 524
    /* store last entry, since it is needed in computeVerticalOscillations(), because we have to do the same
     * number of integrations steps there.
     */
snuverink_j's avatar
snuverink_j committed
525 526
    /* lastOrbitVal_m    = r_m[N_m];        // needed in computeVerticalOscillations() */
    /* lastMomentumVal_m = pr_m[N_m];       // needed in computeVerticalOscillations() */
527

528 529 530
    // remove last entry (since we don't have to store [0,2pi], but [0,2pi[)  --> size = N_m, capacity = N_m+1
    r_m.pop_back();
    pr_m.pop_back();
531

532 533 534 535 536
    /* domain_m = true --> only integrated over a single sector
     * --> multiply by cycl_m->getSymmetry() to get correct phase_m
     */
    if (domain_m)
        phase_m *= cycl_m->getSymmetry();
537

538 539 540
    // returns true if converged, otherwise false
    return error < accuracy;
}
541

frey_m's avatar
frey_m committed
542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560
template<typename Value_type, typename Size_type, class Stepper>
bool ClosedOrbitFinder<Value_type, Size_type, Stepper>::findOrbitOfEnergy_m(
    const value_type& E,
    container_type& init,
    value_type& error,
    const value_type& accuracy,
    size_type maxit)
{
    value_type bint, brint, btint;
    value_type invbcon = 1.0 / bcon_m(E0_m, wo_m);      // [bcon] = MeV*s/(C*m^2) = 10^6 T = 10^7 kG (kilo Gauss)
    
    value_type xold = 0.0;                              // for counting nxcross
    value_type zold = 0.0;                              // for counting nzcross

    // index for reaching next element of the arrays r and pr (no nicer way found yet)
    size_type idx = 0;
    // observer for storing the current value after each ODE step (e.g. Runge-Kutta step) into the containers of r and pr
    auto store = [&](state_type& y, const value_type t)
    {
561 562 563
        r_m[idx]   = y[0];
        pr_m[idx]  = y[1];
        vz_m[idx]  = y[6];
frey_m's avatar
frey_m committed
564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589
        vpz_m[idx] = y[7];

        // count number of crossings (excluding starting point --> idx>0)
        nxcross_m += (idx > 0) * (y[4] * xold < 0);
        xold = y[4];
        
        // number of times z2 changes sign
        nzcross_m += (idx > 0) * (y[10] * zold < 0);
        zold = y[10];
        
        ++idx;
    };
    
    // define initial state container for integration: y = {r, pr, x1, px1, x2, px2,
    //                                                      z, pz, z1, pz1, z2, pz2,
    //                                                      phase}
    state_type y(11);
    
    // difference of last and first value of r (1. element) and pr (2. element)
    container_type err(2);
    // correction term for initial values: r = r + dr, pr = pr + dpr; Gordon, formula (17)
    container_type delta = {0.0, 0.0};
    // if niterations > maxit --> stop iteration
    size_type niterations = 0;
    
    // energy dependent values
590 591
    value_type en     = E / E0_m;                      // en = E/E0 = E/(mc^2) (E0 is potential energy)
    value_type gamma  = en + 1.0;
frey_m's avatar
frey_m committed
592
    value_type p = acon_m(wo_m) * std::sqrt(en * (2.0 + en));     // momentum [p] = m; Gordon, formula (3)
593 594
    value_type gamma2    = gamma * gamma;           // = gamma^2
    value_type invgamma4 = 1.0 / (gamma2 * gamma2); // = 1/gamma^4
595 596
    value_type p2 = p * p;                              // p^2 = p*p

frey_m's avatar
frey_m committed
597 598
    // helper constants
    value_type pr2;                                     // squared radial momentum (pr^2 = pr*pr)
599
    value_type ptheta, invptheta;                       // azimuthal momentum
frey_m's avatar
frey_m committed
600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616

    // define the six ODEs (using lambda function)
    function_t orbit_integration = [&](const state_type &y,
                                       state_type &dydt,
                                       const double theta)
    {
        pr2 = y[1] * y[1];
        if (p2 < pr2)
            throw OpalException("ClosedOrbitFinder::findOrbit()",
                                "p_{r}^2 > p^{2} (defined in Gordon paper) --> Square root of negative number.");

        // Gordon, formula (5c)
        ptheta = std::sqrt(p2 - pr2);
        invptheta = 1.0 / ptheta;
        // interpolate values of magnetic field
        cycl_m->apply(y[0], y[6], theta, brint, btint, bint);
        
snuverink_j's avatar
snuverink_j committed
617
        bint  *= invbcon;
frey_m's avatar
frey_m committed
618 619 620 621 622
        brint *= invbcon;
        btint *= invbcon;

        // Gordon, formula (5a)
        dydt[0] = y[0] * y[1] * invptheta;
623
        // Gordon, formula (5b) (typo in paper! second equal sign is a minus)
frey_m's avatar
frey_m committed
624 625 626
        dydt[1] = ptheta - y[0] * bint;
        // Gordon, formulas (9a) and (9b)
        for (size_type i = 2; i < 5; i += 2) {
627
            dydt[i]   = (y[1] * y[i] + y[0] * p2 * y[i+1] * invptheta * invptheta) * invptheta;
frey_m's avatar
frey_m committed
628 629
            dydt[i+1] = - y[1] * y[i+1] * invptheta - (bint + y[0] * brint) * y[i];
        }
630

frey_m's avatar
frey_m committed
631 632
        // Gordon, formulas (22a) and (22b)
        for (size_type i = 6; i < 12; i += 2) {
633
            dydt[i]   = y[0] * y[i+1] * invptheta;
frey_m's avatar
frey_m committed
634 635 636 637
            dydt[i+1] = (y[0] * brint - y[1] * invptheta * btint) * y[i];
        }

        // integrate phase
638
        dydt[12] = y[0] * invptheta * gamma - 1;
639

frey_m's avatar
frey_m committed
640 641
    };

642
    // integrate until error smaller than user-defined accuracy
frey_m's avatar
frey_m committed
643
    do {
644
        // (re-)set initial values
645
        x_m[0]  = 1.0;               // x1;  Gordon, formula (10)
frey_m's avatar
frey_m committed
646
        px_m[0] = 0.0;               // px1; Gordon, formula (10)
647
        x_m[1]  = 0.0;               // x2;  Gordon, formula (10)
frey_m's avatar
frey_m committed
648 649 650 651 652 653 654 655 656
        px_m[1] = 1.0;               // px2; Gordon, formula (10)
        z_m[0]  = 1.0;
        pz_m[0] = 0.0;
        z_m[1]  = 0.0;
        pz_m[1] = 1.0;
        phase_m = 0.0;
        nxcross_m = 0;               // counts the number of crossings of x-axis (excluding first step)
        nzcross_m = 0;
        idx = 0;                     // index for looping over r and pr arrays
657

frey_m's avatar
frey_m committed
658 659 660 661 662 663 664 665
        // fill container with initial states
        y = {init[0],init[1],
             x_m[0], px_m[0], x_m[1], px_m[1],
             init[2], init[3],
             z_m[0], pz_m[0], z_m[1], pz_m[1],
             phase_m
        };

666 667 668
        try {
            // integrate from 0 to 2*pi (one has to get back to the "origin")
            boost::numeric::odeint::integrate_n_steps(stepper_m, orbit_integration,y,0.0,dtheta_m,N_m,store);
669 670
        } catch(OpalException & ex) {
            *gmsg << ex.where() << " " << ex.what() << endl;
671 672 673
            break;
        }

frey_m's avatar
frey_m committed
674
        // write new state
675
        x_m[0]  = y[2];
frey_m's avatar
frey_m committed
676
        px_m[0] = y[3];
677
        x_m[1]  = y[4];
frey_m's avatar
frey_m committed
678
        px_m[1] = y[5];
679 680

        z_m[0]  = y[8];
frey_m's avatar
frey_m committed
681
        pz_m[0] = y[9];
682
        z_m[1]  = y[10];
frey_m's avatar
frey_m committed
683 684
        pz_m[1] = y[11];
        phase_m = y[12] * Physics::u_two_pi; // / (2.0 * Physics::pi);
685

frey_m's avatar
frey_m committed
686 687
        // compute error (compare values of orbit and momentum for theta = 0 and theta = 2*pi)
        // (Note: size = N_m+1 --> last entry is N_m)
688
        err[0] =  r_m[N_m] -  r_m[0];    // Gordon, formula (14)
frey_m's avatar
frey_m committed
689 690
        err[1] = pr_m[N_m] - pr_m[0];    // Gordon, formula (14)

691
        // correct initial values of r and pr
frey_m's avatar
frey_m committed
692
        value_type invdenom = 1.0 / (x_m[0] + px_m[1] - 2.0);
693 694
        delta[0] = ((px_m[1] - 1.0) * err[0] -  x_m[1] * err[1]) * invdenom; // dr;  Gordon, formula (16a)
        delta[1] = (( x_m[0] - 1.0) * err[1] - px_m[0] * err[0]) * invdenom; // dpr; Gordon, formula (16b)
frey_m's avatar
frey_m committed
695 696 697 698

        // improved initial values; Gordon, formula (17) (here it's used for higher energies)
        init[0] += delta[0];
        init[1] += delta[1];
699 700

        // compute amplitude of the error (Gordon, formula (18)
frey_m's avatar
frey_m committed
701
        error = std::sqrt(delta[0] * delta[0] + delta[1] * delta[1] * invgamma4) / r_m[0];
702 703 704 705 706

        // *gmsg << "iteration " << niterations << " error: " << error << endl;

    } while ((error > accuracy) && (niterations++ < maxit));

707
    if (error > accuracy)
708
        *gmsg << "findOrbit not converged after " << niterations << " iterations with error: " << error << ". Needed accuracy " << accuracy << endl;
709

frey_m's avatar
frey_m committed
710 711 712
    return (error < accuracy);
}

713
template<typename Value_type, typename Size_type, class Stepper>
714 715 716
Value_type ClosedOrbitFinder<Value_type, Size_type, Stepper>::computeTune(const std::array<value_type,2>& y,
                                                                          value_type py2, size_type ncross)
{
717
    // Y = [y1, y2; py1, py2]
718

719 720
    // cos(mu)
    value_type cos = 0.5 * (y[0] + py2);
snuverink_j's avatar
snuverink_j committed
721

722
    value_type mu;
723

724 725
    // sign of sin(mu) has to be equal to y2
    bool negative = std::signbit(y[1]);
726

727
    bool uneven = (ncross % 2);
728

729 730 731
    value_type abscos = std::fabs(cos);
    value_type muPrime;
    if (abscos > 1.0) {
732 733
        // store the number of crossings
        if (uneven)
734
            ncross = ncross + 1;
735

736
        // Gordon, formula (36b)
737
        muPrime = -std::acosh(abscos);    // mu'
738

739
    } else {
740
        muPrime = (uneven) ? std::acos(-cos) : std::acos(cos);    // mu'
741
        /* It has to be fulfilled: 0<= mu' <= pi
742 743 744 745 746
         * But since |cos(mu)| <= 1, we have
         * -1 <= cos(mu) <= 1 --> 0 <= mu <= pi (using above programmed line), such
         * that condition is already fulfilled.
         */
    }
747

748 749
    // Gordon, formula (36)
    mu = ncross * Physics::pi + muPrime;
750

751
    if (abscos < 1.0) {
752 753
        // if sign(y[1]) > 0 && sign(sin(mu)) < 0
        if (!negative && std::signbit(std::sin(mu))) {
754
            mu = ncross * Physics::pi - muPrime;
755
        } else if (negative && !std::signbit(std::sin(mu))) {
756
            mu = ncross * Physics::pi - muPrime + Physics::two_pi;
757 758
        }
    }
759

760
    // nu = mu/theta, where theta = integration domain
761

762
    /* domain_m = true --> only integrated over a single sector --> multiply by cycl_m->getSymmetry() to
adelmann's avatar
adelmann committed
763 764 765
     * get correct tune.
     */
    if (domain_m)
766
        mu *= cycl_m->getSymmetry();
767

768
    return mu * Physics::u_two_pi;
769 770 771
}

template<typename Value_type, typename Size_type, class Stepper>
772
void ClosedOrbitFinder<Value_type, Size_type, Stepper>::computeOrbitProperties(const value_type& E) {
773
    /*
774 775 776 777 778
     * The formulas for h, fidx and ds are from the paper:
     * "Tranverse-Longitudinal Coupling by Space Charge in Cyclotrons"
     * written by Dr. Christian Baumgarten (2012, PSI)
     * p. 6
     */
779

adelmann's avatar
adelmann committed
780
    // READ IN MAGNETIC FIELD: ONLY FOR STAND-ALONE PROGRAM
781
    value_type bint, brint, btint; // B, dB/dr, dB/dtheta
782

783
    value_type invbcon = 1.0 / bcon_m(E0_m, wo_m);
784
    value_type en = E / E0_m;                                  // en = E/E0 = E/(mc^2) (E0 is potential energy)
785
    value_type p = acon_m(wo_m) * std::sqrt(en * (2.0 + en));    // momentum [p] = m; Gordon, formula (3)
786 787 788
    value_type p2 = p * p;
    value_type theta = 0.0;                                             // angle for interpolating
    value_type ptheta;
789

790 791 792 793 794 795 796
    // resize of container (--> size = N_m, capacity = N_m)
    h_m.resize(N_m);
    fidx_m.resize(N_m);
    ds_m.resize(N_m);

    for (size_type i = 0; i < N_m; ++i) {
        // interpolate magnetic field
797
        cycl_m->apply(r_m[i], vz_m[i], theta, brint, btint, bint);
snuverink_j's avatar
snuverink_j committed
798
        bint  *= invbcon;
799 800
        brint *= invbcon;
        btint *= invbcon;
frey_m's avatar
frey_m committed
801
        
802 803 804 805
        // inverse bending radius
        h_m[i] = bint / p;

        // local field index
806
        if (p < pr_m[i])
snuverink_j's avatar
snuverink_j committed
807
            throw OpalException("ClosedOrbitFinder::computeOrbitProperties()",
808 809
                                "p_{r}^2 > p^{2} " + std::to_string(p) + " " + std::to_string(pr_m[i]) + " (defined in Gordon paper) --> Square root of negative number.");

810
        ptheta = std::sqrt(p2 - pr_m[i] * pr_m[i]);
811

812 813 814 815 816 817 818
        fidx_m[i] = (brint * ptheta - btint * pr_m[i] / r_m[i]) / p2; //(bint*bint);

        // path length element
        ds_m[i] = std::hypot(r_m[i] * pr_m[i] / ptheta,r_m[i]) * dtheta_m; // C++11 function

        // increase angle
        theta += dtheta_m;
819
    }
820 821 822

    // compute average radius
    ravg_m = std::accumulate(r_m.begin(),r_m.end(),0.0) / value_type(r_m.size());
823 824
}

825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846
template<typename Value_type, typename Size_type, class Stepper> 
inline typename ClosedOrbitFinder<Value_type, Size_type, Stepper>::container_type
ClosedOrbitFinder<Value_type, Size_type, Stepper>::rotate(value_type angle, container_type &orbitProperty) {

    container_type orbitPropertyCopy = orbitProperty;
    
    // compute the number of steps per degree
    value_type deg_step = N_m / 360.0;

    // compute starting point
    size_type start = deg_step * angle;

    // copy end to start
    std::copy(orbitProperty.begin() + start, orbitProperty.end(), orbitPropertyCopy.begin());
    
    // copy start to end
    std::copy_n(orbitProperty.begin(), start, orbitPropertyCopy.end() - start);

    return orbitPropertyCopy;

}

Andreas Adelmann's avatar
Andreas Adelmann committed
847
#endif