!> @brief Module that contains subroutines and functions to sample various energy distributions. MODULE korc_energy_pdfs USE korc_types USE korc_constants USE korc_HDF5 USE korc_hpc IMPLICIT NONE !> @brief KORC derived type that contains information about a given Gamma distribution function @f$f_\Gamma(x,\kappa,\theta)@f$. !! @details We write a given Gamma distribution function in terms of its shape factor @f$\kappa@f$ and scale factor !! @f$\theta@f$, so that: !! !! !! @f$f_\Gamma(x,\kappa,\theta) = \frac{1}{\Gamma(\kappa) \theta^\kappa}x^{\kappa-1}\exp{\left(-x/\theta\right)}@f$. TYPE, PRIVATE :: GAMMA_PARAMS REAL(rp) :: min_energy !< Minimum energy of sampled @f$f_\Gamma(x,\kappa,\theta)@f$ in MeV. REAL(rp) :: max_energy !< Maximum energy of sampled @f$f_\Gamma(x,\kappa,\theta)@f$ in MeV. REAL(rp) :: min_p !< Minimum momentum of sampled @f$f_\Gamma(x,\kappa,\theta)@f$. REAL(rp) :: max_p !< Maximum momentum of sampled @f$f_\Gamma(x,\kappa,\theta)@f$. REAL(rp) :: k !< Shape factor @f$\kappa@f$. REAL(rp) :: t !< Scale factor @f$\theta@f$. END TYPE GAMMA_PARAMS TYPE(GAMMA_PARAMS), PRIVATE :: gamma_pdf_params !< An instance of the KORC derived type GAMMA_PARAMS REAL(rp), PRIVATE, PARAMETER :: co = (C_E*1.0E6)/(C_ME*C_C**2) !< Constant with units to transform @f$\mathcal{E}@f$ in !! @f$f_\Gamma(\mathcal{E},\kappa,\theta)@f$ from MeV to Joules. REAL(rp), PRIVATE, PARAMETER :: minmax_buffer_size = 10.0_rp !< This is the size of the buffer zone in each direction when !!using a Metropolis-Hastings method to sample a distribution. PUBLIC :: get_gamma_distribution PRIVATE :: initialize_gamma_params,& save_gamma_params,& sample_gamma_distribution,& deg2rad,& fRE,& random_norm,& fGamma CONTAINS !> @brief Subroutine that contains calls to subroutine to generate a gamma distribution for the energy distribution of a given !! species in the simulation. !! !! @param[in] params Core KORC simulation parameters. !! @param[in,out] g Relativistic gamma factor @f$\gamma@f$ of the particles in a given species in the simulation. These are so that, they follow !! a Gamma distribution in energy. The parameters of the Gamma distributions are given by the user. !! @param[out] go Mean value of @f$\gamma@f$ of the particles in a given species. used to calculate the minimum required time step to !! resolve in detail the full-orbit dynamics of the particles. SUBROUTINE get_gamma_distribution(params,g,go) TYPE(KORC_PARAMS), INTENT(IN) :: params REAL(rp), DIMENSION(:), ALLOCATABLE, INTENT(INOUT) :: g REAL(rp), INTENT(OUT) :: go call initialize_gamma_params(params) call save_gamma_params(params) call sample_gamma_distribution(params,g,go) END SUBROUTINE get_gamma_distribution !> @brief Subroutine that reads from the input file the parameters of the Gamma distribution !! @f$f_\Gamma(x,\kappa,\theta) = \frac{1}{\Gamma(\kappa) \theta^\kappa}x^{\kappa-1}\exp{\left(-x/\theta\right)}@f$. !! !! @param[in] params Core KORC simulation parameters. !! @param max_energy Maximum energy of sampled @f$f_\Gamma(x,\kappa,\theta)@f$ in MeV. !! @param min_energy Minimum energy of sampled @f$f_\Gamma(x,\kappa,\theta)@f$ in MeV. !! @param k Shape factor @f$\kappa@f$. !! @param t Scale factor @f$\theta@f$. SUBROUTINE initialize_gamma_params(params) TYPE(KORC_PARAMS), INTENT(IN) :: params REAL(rp) :: max_energy REAL(rp) :: min_energy REAL(rp) :: k REAL(rp) :: t NAMELIST /EnergyGammaPDF/ max_energy,min_energy,k,t open(unit=default_unit_open,file=TRIM(params%path_to_inputs),status='OLD',form='formatted') read(default_unit_open,nml=EnergyGammaPDF) close(default_unit_open) gamma_pdf_params%min_energy = min_energy*C_E ! In Joules gamma_pdf_params%max_energy = max_energy*C_E ! In Joules gamma_pdf_params%k = k gamma_pdf_params%t = t gamma_pdf_params%max_p = SQRT((gamma_pdf_params%max_energy/(C_ME*C_C**2))**2 - 1.0_rp) ! In units of mc gamma_pdf_params%min_p = SQRT((gamma_pdf_params%min_energy/(C_ME*C_C**2))**2 - 1.0_rp) ! In units of mc END SUBROUTINE initialize_gamma_params !> @brief Function that converts @f$x@f$ from degrees to radians. !! !! @param x Angle @f$x@f$ in degrees. !! @param deg2rad Angle @f$x@f$ converted to radians. FUNCTION deg2rad(x) REAL(rp), INTENT(IN) :: x REAL(rp) :: deg2rad deg2rad = C_PI*x/180.0_rp END FUNCTION !> @brief Function that calculates the value of the Gamma distribution @f$f_\Gamma(x,\kappa,\theta) = !! \frac{1}{\Gamma(\kappa) \theta^\kappa}x^{\kappa-1}\exp{\left(-x/\theta\right)}@f$. !! !! @param x Variable @f$x@f$ of @f$f_\Gamma(x,\kappa,\theta)@f$. !! @param k Shape factor @f$\kappa@f$ of @f$f_\Gamma(x,\kappa,\theta)@f$. !! @param t Scale factor @f$\theta@f$ of @f$f_\Gamma(x,\kappa,\theta)@f$. !! @param fGamma Computed value of @f$f_\Gamma(x,\kappa,\theta)@f$. FUNCTION fGamma(x,k,t) REAL(rp), INTENT(IN) :: x REAL(rp), INTENT(IN) :: k REAL(rp), INTENT(IN) :: t REAL(rp) :: fGamma fGamma = x**(k - 1.0_rp)*EXP(-x/t)/(GAMMA(k)*t**k) END FUNCTION fGamma !> Evaluation of the energy distribution function @f$f_{RE}(\mathcal{E})@f$ of runaway electrons as function of the normalized momentum !! @f$p' = p/m_ec@f$. Here, @f$p'@f$ is the normalized momentum and @f$m_e@f$ and @f$c@f$ are the electron mass and the speed of light. !! !! @param p Normalized momentum @f$p' = p/m_ec@f$ of a given electron in the simulation. !! @param fRE Computed value of the energy distribution function of runaway electrons. !! @param Eo Normalized energy @f$\mathcal{E}' = \sqrt{1 + p'}@f$ of the the electron with normalized momentum @f$p'@f$. FUNCTION fRE(p) REAL(rp), INTENT(IN) :: p ! momentum in units of mc REAL(rp) :: fRE REAL(rp) :: Eo ! In units of mc^2 Eo = SQRT(p**2.0_rp + 1.0_rp) fRE = fGamma(Eo,gamma_pdf_params%k,gamma_pdf_params%t*co) END FUNCTION fRE !> @brief Gaussian random number generator. !! @details This function returns a deviate of a Gaussian distribution @f$f_G(x;\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}} \exp{\left( -(x-\mu)^2/2\sigma^2 \right)}@f$, !! with mean @f$\mu@f$, and standard deviation @f$\sigma@f$. !! !! We use the Inverse Transform Sampling Method for sampling @f$x@f$. With this method we get @f$x = \sqrt{-2\log{(1-y)}}\cos(2\pi z)@f$, !! where @f$y@f$ and @f$z@f$ are uniform random numbers in the interval @f$[0,1]@f$. !! !! @param[in] mu Mean value @f$\mu@f$ of the Gaussian distribution. !! @param[in] mu Standard deviation @f$\sigma@f$ of the Gaussian distribution. !! @param random_norm Sampled number @f$x@f$ from the Gaussian distribution @f$f_G(x;\mu,\sigma)@f$. !! @param rand1 Uniform random number in the interval @f$[0,1]@f$. !! @param rand2 Uniform random number in the interval @f$[0,1]@f$. FUNCTION random_norm(mean,sigma) REAL(rp), INTENT(IN) :: mean REAL(rp), INTENT(IN) :: sigma REAL(rp) :: random_norm REAL(rp) :: rand1 REAL(rp) :: rand2 call RANDOM_NUMBER(rand1) call RANDOM_NUMBER(rand2) random_norm = SQRT(-2.0_rp*LOG(1.0_rp-rand1))*COS(2.0_rp*C_PI*rand2); END FUNCTION random_norm !> @brief Subroutine that samples a Gamma distribution representing the runaways' (marginal) energy distribution function. !! @details This subroutine uses the Metropolis-Hastings method for sampling the Gamma distribution representing the runaways' !! (marginal) energy distribution function. Unlike the typical Metropolis-Hasting method, after setting the boundaries of the region !! we want to sample, we perform a sampling in a larger region that contains the original sampling area plus a buffer region. !! After finishing the first sampling, we only keep the particles in the original sampling region, the particles in the p_buffer !! are sampled again until all of them lie within the original sampling region. This method ensures that the boundaries are !! well sampled. !! !! @param[in] params Core KORC simulation parameters. !! @param[in,out] g Relativistic gamma factor @f$\gamma@f$ of the particles in a given species in the simulation. These are so that, !! they follow a Gamma distribution in energy. The parameters of the Gamma distributions are given by the user. !! @param[out] go Mean value of @f$\gamma@f$ of the particles in a given species. used to calculate the minimum required time step to !! resolve in detail the full-orbit dynamics of the particles. !! @param p Sampled normalized momentum @f$p' = p/m_ec@f$ of particles in a given particle species. !! @param p_buffer Size along the momentum axis of the buffer used in the Metropolis-Hastings method. !! @param p_test The test value of the normalized momentum used in the Metropolis-Hastings method. !! @param ratio Ratio of probabilities used to determine when a move in during the sampling is kept as part of the sampled chain. !! @param rand_unif A deviate of a uniform random distribution in the interval @f$[0,1]@f$. !! @param p_samples Temporary array to keep the sampled normalized momentum. !! @param deta Step size along the pitch-angle direction of the random walk used in the Metropolis-Hastings sampling. !! @param dp Step size along the momentum direction of the random walk used in the Metropolis-Hastings sampling. !! @param ii Iterator. !! @param ppp Number of particles per MPI processes. !! @param nsamples Number of total samples in the initial condition of a given simulation. !! @param mpierr MPI error status. SUBROUTINE sample_gamma_distribution(params,g,go) TYPE(KORC_PARAMS), INTENT(IN) :: params REAL(rp), DIMENSION(:), ALLOCATABLE, INTENT(INOUT) :: g REAL(rp), INTENT(OUT) :: go REAL(rp), DIMENSION(:), ALLOCATABLE :: p REAL(rp) :: p_buffer REAL(rp) :: p_test REAL(rp) :: ratio REAL(rp) :: rand_unif REAL(rp), DIMENSION(:), ALLOCATABLE :: p_samples REAL(rp) :: deta REAL(rp) :: dp INTEGER :: ii INTEGER :: ppp INTEGER :: nsamples INTEGER :: mpierr ppp = SIZE(g) nsamples = ppp*params%mpi_params%nmpi ALLOCATE(p(ppp)) dp = 1.0_rp if (params%mpi_params%rank.EQ.0_idef) then ALLOCATE(p_samples(nsamples))! Number of samples to distribute among all MPI processes call RANDOM_SEED() call RANDOM_NUMBER(rand_unif) p_buffer = gamma_pdf_params%min_p + (gamma_pdf_params%max_p - gamma_pdf_params%min_p)*rand_unif ii=2_idef do while (ii .LE. 1000000_idef) p_test = p_buffer + random_norm(0.0_rp,dp) do while ((p_test.LT.gamma_pdf_params%min_p).OR.(p_test.GT.gamma_pdf_params%max_p)) p_test = p_buffer + random_norm(0.0_rp,dp) end do ratio = fRE(p_test)/fRE(p_buffer) if (ratio .GE. 1.0_rp) then p_buffer = p_test ii = ii + 1_idef else call RANDOM_NUMBER(rand_unif) if (rand_unif .LT. ratio) then p_buffer = p_test ii = ii + 1_idef end if end if end do call RANDOM_SEED() call RANDOM_NUMBER(rand_unif) p_samples(1) = p_buffer ii=2_idef do while (ii .LE. nsamples) p_test = p_samples(ii-1) + random_norm(0.0_rp,dp) do while ((p_test.LT.gamma_pdf_params%min_p).OR.(p_test.GT.gamma_pdf_params%max_p)) p_test = p_samples(ii-1) + random_norm(0.0_rp,dp) end do ratio = fRE(p_test)/fRE(p_samples(ii-1)) if (ratio .GE. 1.0_rp) then p_samples(ii) = p_test ii = ii + 1_idef else call RANDOM_NUMBER(rand_unif) if (rand_unif .LT. ratio) then p_samples(ii) = p_test ii = ii + 1_idef end if end if end do go = SUM(SQRT(1.0_rp + p_samples**2))/nsamples end if CALL MPI_SCATTER(p_samples,ppp,MPI_REAL8,p,ppp,MPI_REAL8,0,MPI_COMM_WORLD,mpierr) CALL MPI_BCAST(go,1,MPI_REAL8,0,MPI_COMM_WORLD,mpierr) call MPI_BARRIER(MPI_COMM_WORLD,mpierr) g = SQRT(1.0_rp + p**2) DEALLOCATE(p) if (params%mpi_params%rank.EQ.0_idef) then DEALLOCATE(p_samples) end if END SUBROUTINE sample_gamma_distribution !> @brief Surboutine that saves the Gamma distribution parameters to the HDF5 file <i>gamma_distribution_parameters.h5</i>. !! !! @param[in] params Core KORC simulation parameters. !! @param filename String containing the name of the HDF5 file. !! @param gname String containing the group name of a set of KORC parameters. !! @param attr_array An 1-D array with attributes of 1-D real or integer arrays that are passed to KORC interfaces of HDF5 I/O subroutines. !! @param dset Name of data set to be saved to file. !! @param attr A single attributes of real or integer data that is passed to KORC interfaces of HDF5 I/O subroutines. !! @param h5file_id HDF5 file identifier. !! @param group_id HDF5 group identifier. !! @param h5error HDF5 error status. !! @param units Temporary variable used to add physical units to KORC parameters. SUBROUTINE save_gamma_params(params) TYPE(KORC_PARAMS), INTENT(IN) :: params CHARACTER(MAX_STRING_LENGTH) :: filename CHARACTER(MAX_STRING_LENGTH) :: gname CHARACTER(MAX_STRING_LENGTH), DIMENSION(:), ALLOCATABLE :: attr_array CHARACTER(MAX_STRING_LENGTH) :: dset CHARACTER(MAX_STRING_LENGTH) :: attr INTEGER(HID_T) :: h5file_id INTEGER(HID_T) :: group_id INTEGER :: h5error REAL(rp) :: units if (params%mpi_params%rank .EQ. 0) then filename = TRIM(params%path_to_outputs) // "gamma_distribution_parameters.h5" call h5fcreate_f(TRIM(filename), H5F_ACC_TRUNC_F, h5file_id, h5error) gname = "params" call h5gcreate_f(h5file_id, TRIM(gname), group_id, h5error) dset = TRIM(gname) // "/min_energy" attr = "Minimum energy in avalanche PDF (eV)" units = 1.0_rp/C_E call save_to_hdf5(h5file_id,dset,units*gamma_pdf_params%min_energy,attr) dset = TRIM(gname) // "/max_energy" attr = "Maximum energy in avalanche PDF (eV)" units = 1.0_rp/C_E call save_to_hdf5(h5file_id,dset,units*gamma_pdf_params%max_energy,attr) dset = TRIM(gname) // "/max_p" attr = "Maximum momentum in avalanche PDF (me*c^2)" call save_to_hdf5(h5file_id,dset,gamma_pdf_params%max_p,attr) dset = TRIM(gname) // "/min_p" attr = "Maximum momentum in avalanche PDF (me*c^2)" call save_to_hdf5(h5file_id,dset,gamma_pdf_params%min_p,attr) dset = TRIM(gname) // "/k" attr = "Shape factor" call save_to_hdf5(h5file_id,dset,gamma_pdf_params%k,attr) dset = TRIM(gname) // "/t" attr = "Scale factor" call save_to_hdf5(h5file_id,dset,gamma_pdf_params%t,attr) call h5gclose_f(group_id, h5error) call h5fclose_f(h5file_id, h5error) end if END SUBROUTINE save_gamma_params END MODULE korc_energy_pdfs