def add_instrumentation(self, mgr, observer): Depositor.add_instrumentation(self, mgr, observer) # instrumentation from pytools.log import IntervalTimer, EventCounter self.element_activation_counter = EventCounter( "n_el_activations", "#Advective rec. elements activated this timestep") self.element_kill_counter = EventCounter( "n_el_kills", "#Advective rec. elements retired this timestep") self.advective_rhs_timer = IntervalTimer( "t_advective_rhs", "Time spent evaluating advective RHS") self.active_elements_log = ActiveAdvectiveElements(observer) mgr.add_quantity(self.element_activation_counter) mgr.add_quantity(self.element_kill_counter) mgr.add_quantity(self.advective_rhs_timer) mgr.add_quantity(self.active_elements_log) mgr.set_constant("el_activation_threshold", self.activation_threshold) mgr.set_constant("el_kill_threshold", self.kill_threshold) mgr.set_constant("adv_upwind_alpha", self.upwind_alpha) mgr.set_constant("filter_amp", self.filter_amp) mgr.set_constant("filter_amp", self.filter_order)
def main(): logmgr = LogManager("mylog.dat", "w") # , comm=... # set a run property logmgr.set_constant("myconst", uniform(0, 1)) add_run_info(logmgr) add_general_quantities(logmgr) add_simulation_quantities(logmgr) vis_timer = IntervalTimer("t_vis", "Time spent visualizing") logmgr.add_quantity(vis_timer) logmgr.add_quantity(Fifteen("fifteen")) logmgr.add_watches(["step.max", "t_sim.max", "t_step.max"]) for istep in range(200): logmgr.tick_before() dt = uniform(0.01, 0.1) set_dt(logmgr, dt) sleep(dt) # Illustrate custom timers if istep % 10 == 0: with vis_timer.start_sub_timer(): sleep(0.05) # Illustrate warnings capture if uniform(0, 1) < 0.05: warn("Oof. Something went awry.") logmgr.tick_after() logmgr.close()
def add_instrumentation(self, mgr, observer): mgr.set_constant("depositor", self.__class__.__name__) for key, value in self.log_constants.iteritems(): mgr.set_constant(key, value) from pytools.log import IntervalTimer, EventCounter,\ time_and_count_function self.deposit_timer = IntervalTimer("t_deposit", "Time spent depositing") self.deposit_counter = EventCounter("n_deposit", "Number of depositions") self.deposit_densities = time_and_count_function( self.deposit_densites, self.deposit_timer, self.deposit_counter, 1 + self.method.dimensions_velocity) self.deposit_j = time_and_count_function( self.deposit_j, self.deposit_timer, self.deposit_counter, self.method.dimensions_velocity) self.deposit_rho = time_and_count_function(self.deposit_rho, self.deposit_timer, self.deposit_counter) mgr.add_quantity(self.deposit_timer) mgr.add_quantity(self.deposit_counter)
def __init__(self, method, maxwell_op): self.method = method self.maxwell_op = maxwell_op self.bound_maxwell_op = maxwell_op.bind(self.method.discretization) from pytools.log import IntervalTimer self.field_solve_timer = IntervalTimer("t_field", "Time spent in field solver")
def __init__(self, discr, units, depositor, pusher, dimensions_pos, dimensions_velocity, debug=set()): self.units = units self.discretization = discr self.debug = debug self.depositor = depositor self.pusher = pusher self.dimensions_mesh = discr.dimensions self.dimensions_pos = dimensions_pos self.dimensions_velocity = dimensions_velocity dims = (dimensions_pos, dimensions_velocity) self.mesh_data = _internal.MeshData(discr.dimensions) self.mesh_data.fill_from_hedge(discr) # subsystem init self.depositor.initialize(self) self.pusher.initialize(self) # instrumentation from pytools.log import IntervalTimer, EventCounter self.find_el_timer = IntervalTimer("t_find", "Time spent finding new elements") self.find_same_counter = EventCounter( "n_find_same", "#Particles found in same element") self.find_by_neighbor_counter = EventCounter( "n_find_neighbor", "#Particles found through neighbor") self.find_by_vertex_counter = EventCounter( "n_find_by_vertex", "#Particles found by vertex") self.find_global_counter = EventCounter( "n_find_global", "#Particles found by global search")
def main(write_output=True, allow_features=None): from grudge.timestep import RK4TimeStepper from grudge.mesh import make_ball_mesh, make_cylinder_mesh, make_box_mesh from grudge.visualization import \ VtkVisualizer, \ SiloVisualizer, \ get_rank_partition from math import sqrt, pi from grudge.backends import guess_run_context rcon = guess_run_context(allow_features) epsilon0 = 8.8541878176e-12 # C**2 / (N m**2) mu0 = 4 * pi * 1e-7 # N/A**2. epsilon = 1 * epsilon0 mu = 1 * mu0 dims = 3 if rcon.is_head_rank: if dims == 2: from grudge.mesh import make_rect_mesh mesh = make_rect_mesh(a=(-10.5, -1.5), b=(10.5, 1.5), max_area=0.1) elif dims == 3: from grudge.mesh import make_box_mesh mesh = make_box_mesh(a=(-10.5, -1.5, -1.5), b=(10.5, 1.5, 1.5), max_volume=0.1) if rcon.is_head_rank: mesh_data = rcon.distribute_mesh(mesh) else: mesh_data = rcon.receive_mesh() #for order in [1,2,3,4,5,6]: discr = rcon.make_discretization(mesh_data, order=3) if write_output: vis = VtkVisualizer(discr, rcon, "dipole") from analytic_solutions import DipoleFarField, SphericalFieldAdapter from grudge.data import ITimeDependentGivenFunction sph_dipole = DipoleFarField( q=1, #C d=1 / 39, omega=2 * pi * 1e8, epsilon=epsilon0, mu=mu0, ) cart_dipole = SphericalFieldAdapter(sph_dipole) class PointDipoleSource(ITimeDependentGivenFunction): def __init__(self): from pyrticle.tools import CInfinityShapeFunction sf = CInfinityShapeFunction(0.1 * sph_dipole.wavelength, discr.dimensions) self.num_sf = discr.interpolate_volume_function( lambda x, el: sf(x)) self.vol_0 = discr.volume_zeros() def volume_interpolant(self, t, discr): from grudge.tools import make_obj_array return make_obj_array([ self.vol_0, self.vol_0, sph_dipole.source_modulation(t) * self.num_sf ]) from grudge.mesh import BTAG_ALL, BTAG_NONE if dims == 2: from grudge.models.em import TMMaxwellOperator as MaxwellOperator else: from grudge.models.em import MaxwellOperator op = MaxwellOperator( epsilon, mu, flux_type=1, pec_tag=BTAG_NONE, absorb_tag=BTAG_ALL, current=PointDipoleSource(), ) fields = op.assemble_eh(discr=discr) if rcon.is_head_rank: print("#elements=", len(mesh.elements)) stepper = RK4TimeStepper() # diagnostics setup --------------------------------------------------- from pytools.log import LogManager, add_general_quantities, \ add_simulation_quantities, add_run_info if write_output: log_file_name = "dipole.dat" else: log_file_name = None logmgr = LogManager(log_file_name, "w", rcon.communicator) add_run_info(logmgr) add_general_quantities(logmgr) add_simulation_quantities(logmgr) discr.add_instrumentation(logmgr) stepper.add_instrumentation(logmgr) from pytools.log import IntervalTimer vis_timer = IntervalTimer("t_vis", "Time spent visualizing") logmgr.add_quantity(vis_timer) from grudge.log import EMFieldGetter, add_em_quantities field_getter = EMFieldGetter(discr, op, lambda: fields) add_em_quantities(logmgr, op, field_getter) from pytools.log import PushLogQuantity relerr_e_q = PushLogQuantity("relerr_e", "1", "Relative error in masked E-field") relerr_h_q = PushLogQuantity("relerr_h", "1", "Relative error in masked H-field") logmgr.add_quantity(relerr_e_q) logmgr.add_quantity(relerr_h_q) logmgr.add_watches([ "step.max", "t_sim.max", ("W_field", "W_el+W_mag"), "t_step.max", "relerr_e", "relerr_h" ]) if write_output: point_timeseries = [(open("b-x%d-vs-time.dat" % i, "w"), open("b-x%d-vs-time-true.dat" % i, "w"), discr.get_point_evaluator( numpy.array([i, 0, 0][:dims], dtype=discr.default_scalar_type))) for i in range(1, 5)] # timestep loop ------------------------------------------------------- mask = discr.interpolate_volume_function(sph_dipole.far_field_mask) def apply_mask(field): from grudge.tools import log_shape ls = log_shape(field) result = discr.volume_empty(ls) from pytools import indices_in_shape for i in indices_in_shape(ls): result[i] = mask * field[i] return result rhs = op.bind(discr) t = 0 try: from grudge.timestep import times_and_steps step_it = times_and_steps( final_time=1e-8, logmgr=logmgr, max_dt_getter=lambda t: op.estimate_timestep( discr, stepper=stepper, t=t, fields=fields)) for step, t, dt in step_it: if write_output and step % 10 == 0: sub_timer = vis_timer.start_sub_timer() e, h = op.split_eh(fields) sph_dipole.set_time(t) true_e, true_h = op.split_eh( discr.interpolate_volume_function(cart_dipole)) visf = vis.make_file("dipole-%04d" % step) mask_e = apply_mask(e) mask_h = apply_mask(h) mask_true_e = apply_mask(true_e) mask_true_h = apply_mask(true_h) from pyvisfile.silo import DB_VARTYPE_VECTOR vis.add_data(visf, [("e", e), ("h", h), ("true_e", true_e), ("true_h", true_h), ("mask_e", mask_e), ("mask_h", mask_h), ("mask_true_e", mask_true_e), ("mask_true_h", mask_true_h)], time=t, step=step) visf.close() sub_timer.stop().submit() from grudge.tools import relative_error relerr_e_q.push_value( relative_error(discr.norm(mask_e - mask_true_e), discr.norm(mask_true_e))) relerr_h_q.push_value( relative_error(discr.norm(mask_h - mask_true_h), discr.norm(mask_true_h))) if write_output: for outf_num, outf_true, evaluator in point_timeseries: for outf, ev_h in zip([outf_num, outf_true], [h, true_h]): outf.write("%g\t%g\n" % (t, op.mu * evaluator(ev_h[1]))) outf.flush() fields = stepper(fields, t, dt, rhs) finally: if write_output: vis.close() logmgr.save() discr.close()
def main(write_output=True): from math import sqrt, pi, exp from os.path import join from grudge.backends import guess_run_context rcon = guess_run_context() epsilon0 = 8.8541878176e-12 # C**2 / (N m**2) mu0 = 4 * pi * 1e-7 # N/A**2. epsilon = 1 * epsilon0 mu = 1 * mu0 output_dir = "maxwell-2d" import os if not os.access(output_dir, os.F_OK): os.makedirs(output_dir) from grudge.mesh.generator import make_disk_mesh mesh = make_disk_mesh(r=0.5, max_area=1e-3) if rcon.is_head_rank: mesh_data = rcon.distribute_mesh(mesh) else: mesh_data = rcon.receive_mesh() class CurrentSource: shape = (3, ) def __call__(self, x, el): return [0, 0, exp(-80 * la.norm(x))] order = 3 final_time = 1e-8 discr = rcon.make_discretization(mesh_data, order=order, debug=["cuda_no_plan"]) from grudge.visualization import VtkVisualizer if write_output: vis = VtkVisualizer(discr, rcon, join(output_dir, "em-%d" % order)) if rcon.is_head_rank: print("order %d" % order) print("#elements=", len(mesh.elements)) from grudge.mesh import BTAG_ALL, BTAG_NONE from grudge.models.em import TMMaxwellOperator from grudge.data import make_tdep_given, TimeIntervalGivenFunction op = TMMaxwellOperator(epsilon, mu, flux_type=1, current=TimeIntervalGivenFunction( make_tdep_given(CurrentSource()), off_time=final_time / 10), absorb_tag=BTAG_ALL, pec_tag=BTAG_NONE) fields = op.assemble_eh(discr=discr) from grudge.timestep import LSRK4TimeStepper stepper = LSRK4TimeStepper() from time import time last_tstep = time() t = 0 # diagnostics setup --------------------------------------------------- from pytools.log import LogManager, add_general_quantities, \ add_simulation_quantities, add_run_info if write_output: log_file_name = join(output_dir, "maxwell-%d.dat" % order) else: log_file_name = None logmgr = LogManager(log_file_name, "w", rcon.communicator) add_run_info(logmgr) add_general_quantities(logmgr) add_simulation_quantities(logmgr) discr.add_instrumentation(logmgr) stepper.add_instrumentation(logmgr) from pytools.log import IntervalTimer vis_timer = IntervalTimer("t_vis", "Time spent visualizing") logmgr.add_quantity(vis_timer) from grudge.log import EMFieldGetter, add_em_quantities field_getter = EMFieldGetter(discr, op, lambda: fields) add_em_quantities(logmgr, op, field_getter) logmgr.add_watches( ["step.max", "t_sim.max", ("W_field", "W_el+W_mag"), "t_step.max"]) # timestep loop ------------------------------------------------------- rhs = op.bind(discr) try: from grudge.timestep import times_and_steps step_it = times_and_steps( final_time=final_time, logmgr=logmgr, max_dt_getter=lambda t: op.estimate_timestep( discr, stepper=stepper, t=t, fields=fields)) for step, t, dt in step_it: if step % 10 == 0 and write_output: e, h = op.split_eh(fields) visf = vis.make_file( join(output_dir, "em-%d-%04d" % (order, step))) vis.add_data(visf, [ ("e", discr.convert_volume(e, "numpy")), ("h", discr.convert_volume(h, "numpy")), ], time=t, step=step) visf.close() fields = stepper(fields, t, dt, rhs) assert discr.norm(fields) < 0.03 finally: if write_output: vis.close() logmgr.close() discr.close()
def make_timer(self, name, description=None): from pytools.log import IntervalTimer return IntervalTimer(name, description)
def __init__(self): from pytools.log import IntervalTimer, EventCounter self.force_timer = IntervalTimer( "t_force", "Time spent calculating forces")
def main(write_output=True, dir_tag=TAG_NONE, neu_tag=TAG_NONE, rad_tag=TAG_ALL, flux_type_arg="upwind", dtype=np.float64, debug=[]): from math import sin, cos, pi, exp, sqrt # noqa from hedge.backends import guess_run_context rcon = guess_run_context() dim = 2 if dim == 1: if rcon.is_head_rank: from hedge.mesh.generator import make_uniform_1d_mesh mesh = make_uniform_1d_mesh(-10, 10, 500) elif dim == 2: from hedge.mesh.generator import make_rect_mesh if rcon.is_head_rank: mesh = make_rect_mesh(a=(-0.5, -0.5), b=(0.5, 0.5), max_area=0.008) elif dim == 3: if rcon.is_head_rank: from hedge.mesh.generator import make_ball_mesh mesh = make_ball_mesh(max_volume=0.0005) else: raise RuntimeError("bad number of dimensions") if rcon.is_head_rank: print "%d elements" % len(mesh.elements) mesh_data = rcon.distribute_mesh(mesh) else: mesh_data = rcon.receive_mesh() from hedge.timestep.runge_kutta import LSRK4TimeStepper stepper = LSRK4TimeStepper(dtype=dtype) from hedge.models.wave import StrongWaveOperator from hedge.mesh import TAG_ALL, TAG_NONE # noqa source_center = np.array([0.1, 0.22]) source_width = 0.05 source_omega = 3 import hedge.optemplate as sym sym_x = sym.nodes(2) sym_source_center_dist = sym_x - source_center op = StrongWaveOperator( -1, dim, source_f=sym.CFunction("sin")( source_omega * sym.ScalarParameter("t")) * sym.CFunction("exp")( -np.dot(sym_source_center_dist, sym_source_center_dist) / source_width**2), dirichlet_tag=dir_tag, neumann_tag=neu_tag, radiation_tag=rad_tag, flux_type=flux_type_arg) discr = rcon.make_discretization(mesh_data, order=4, debug=debug, default_scalar_type=dtype, tune_for=op.op_template()) from hedge.visualization import VtkVisualizer if write_output: vis = VtkVisualizer(discr, rcon, "fld") from hedge.tools import join_fields fields = join_fields( discr.volume_zeros(dtype=dtype), [discr.volume_zeros(dtype=dtype) for i in range(discr.dimensions)]) # {{{ diagnostics setup from pytools.log import LogManager, \ add_general_quantities, \ add_simulation_quantities, \ add_run_info if write_output: log_file_name = "wave.dat" else: log_file_name = None logmgr = LogManager(log_file_name, "w", rcon.communicator) add_run_info(logmgr) add_general_quantities(logmgr) add_simulation_quantities(logmgr) discr.add_instrumentation(logmgr) from pytools.log import IntervalTimer vis_timer = IntervalTimer("t_vis", "Time spent visualizing") logmgr.add_quantity(vis_timer) stepper.add_instrumentation(logmgr) from hedge.log import LpNorm u_getter = lambda: fields[0] logmgr.add_quantity(LpNorm(u_getter, discr, 1, name="l1_u")) logmgr.add_quantity(LpNorm(u_getter, discr, name="l2_u")) logmgr.add_watches(["step.max", "t_sim.max", "l2_u", "t_step.max"]) # }}} # {{{ timestep loop rhs = op.bind(discr) try: from hedge.timestep import times_and_steps step_it = times_and_steps( final_time=4, logmgr=logmgr, max_dt_getter=lambda t: op.estimate_timestep( discr, stepper=stepper, t=t, fields=fields)) for step, t, dt in step_it: if step % 10 == 0 and write_output: visf = vis.make_file("fld-%04d" % step) vis.add_data(visf, [ ("u", discr.convert_volume(fields[0], kind="numpy")), ("v", discr.convert_volume(fields[1:], kind="numpy")), ], time=t, step=step) visf.close() fields = stepper(fields, t, dt, rhs) assert discr.norm(fields) < 1 assert fields[0].dtype == dtype finally: if write_output: vis.close() logmgr.close() discr.close()
def main(write_output=True, allow_features=None, flux_type_arg=1, bdry_flux_type_arg=None, extra_discr_args={}): from math import sqrt, pi from hedge.models.em import TEMaxwellOperator from hedge.backends import guess_run_context rcon = guess_run_context(allow_features) epsilon0 = 8.8541878176e-12 # C**2 / (N m**2) mu0 = 4*pi*1e-7 # N/A**2. c = 1/sqrt(mu0*epsilon0) materials = {"vacuum" : (epsilon0, mu0), "dielectric" : (2*epsilon0, mu0)} output_dir = "2d_cavity" import os if not os.access(output_dir, os.F_OK): os.makedirs(output_dir) # should no tag raise an error or default to free space? def eps_val(x, el): for key in materials.keys(): if el in material_elements[key]: return materials[key][0] raise ValueError, "Element does not belong to any material" def mu_val(x, el): for key in materials.keys(): if el in material_elements[key]: return materials[key][1] raise ValueError, "Element does not belong to any material" # geometry of cavity d = 100e-3 a = 150e-3 # analytical frequency and transverse wavenumbers of resonance f0 = 9.0335649907522321e8 h = 2*pi*f0/c l = -h*sqrt(2) # substitute the following and change materials for a homogeneous cavity #h = pi/a #l =-h def initial_val(discr): # the initial solution for the TE_10-like mode def initial_Hz(x, el): from math import cos, sin if el in material_elements["vacuum"]: return h*cos(h*x[0]) else: return -l*sin(h*d)/sin(l*(a-d))*cos(l*(a-x[0])) from hedge.tools import make_obj_array result_zero = discr.volume_zeros(kind="numpy", dtype=numpy.float64) H_z = make_tdep_given(initial_Hz).volume_interpolant(0, discr) return make_obj_array([result_zero, result_zero, H_z]) if rcon.is_head_rank: from hedge.mesh.reader.gmsh import generate_gmsh mesh = generate_gmsh(CAVITY_GEOMETRY, 2, force_dimension=2) mesh_data = rcon.distribute_mesh(mesh) else: mesh_data = rcon.receive_mesh() # Work out which elements belong to each material material_elements = {} for key in materials.keys(): material_elements[key] = set(mesh_data.tag_to_elements[key]) order = 3 #extra_discr_args.setdefault("debug", []).append("cuda_no_plan") #extra_discr_args.setdefault("debug", []).append("dump_optemplate_stages") from hedge.data import make_tdep_given from hedge.mesh import TAG_ALL op = TEMaxwellOperator(epsilon=make_tdep_given(eps_val), mu=make_tdep_given(mu_val), \ flux_type=flux_type_arg, \ bdry_flux_type=bdry_flux_type_arg, dimensions=2, pec_tag=TAG_ALL) # op = TEMaxwellOperator(epsilon=epsilon0, mu=mu0, # flux_type=flux_type_arg, \ # bdry_flux_type=bdry_flux_type_arg, dimensions=2, pec_tag=TAG_ALL) discr = rcon.make_discretization(mesh_data, order=order, tune_for=op.op_template(), **extra_discr_args) # create the initial solution fields = initial_val(discr) from hedge.visualization import VtkVisualizer if write_output: from os.path import join vis = VtkVisualizer(discr, rcon, join(output_dir, "cav-%d" % order)) # monitor the solution at a point to find the resonant frequency try: point_getter = discr.get_point_evaluator(numpy.array([75e-3, 25e-3, 0])) #[0.25, 0.25, 0.25])) except RuntimeError: point_getter = None if rcon.is_head_rank: print "---------------------------------------------" print "order %d" % order print "---------------------------------------------" print "#elements=", len(mesh.elements) from hedge.timestep.runge_kutta import LSRK4TimeStepper stepper = LSRK4TimeStepper(dtype=discr.default_scalar_type, rcon=rcon) #from hedge.timestep.dumka3 import Dumka3TimeStepper #stepper = Dumka3TimeStepper(3, dtype=discr.default_scalar_type, rcon=rcon) # diagnostics setup --------------------------------------------------- from pytools.log import LogManager, add_general_quantities, \ add_simulation_quantities, add_run_info if write_output: from os.path import join log_file_name = join(output_dir, "cavity-%d.dat" % order) else: log_file_name = None logmgr = LogManager(log_file_name, "w", rcon.communicator) add_run_info(logmgr) add_general_quantities(logmgr) add_simulation_quantities(logmgr) discr.add_instrumentation(logmgr) stepper.add_instrumentation(logmgr) from pytools.log import IntervalTimer vis_timer = IntervalTimer("t_vis", "Time spent visualizing") logmgr.add_quantity(vis_timer) #from hedge.log import EMFieldGetter, add_em_quantities #field_getter = EMFieldGetter(discr, op, lambda: fields) #add_em_quantities(logmgr, op, field_getter) logmgr.add_watches( ["step.max", "t_sim.max", #("W_field", "W_el+W_mag"), "t_step.max"] ) # timestep loop ------------------------------------------------------- rhs = op.bind(discr) final_time = 10e-9 if point_getter is not None: from os.path import join pointfile = open(join(output_dir, "point.txt"), "wt") done_dt = False try: from hedge.timestep import times_and_steps from os.path import join step_it = times_and_steps( final_time=final_time, logmgr=logmgr, max_dt_getter=lambda t: op.estimate_timestep(discr, stepper=stepper, t=t, fields=fields)) for step, t, dt in step_it: if step % 10 == 0 and write_output: sub_timer = vis_timer.start_sub_timer() e, h = op.split_eh(fields) visf = vis.make_file(join(output_dir, "cav-%d-%04d") % (order, step)) vis.add_data(visf, [ ("e", discr.convert_volume(e, kind="numpy")), ("h", discr.convert_volume(h, kind="numpy")),], time=t, step=step ) visf.close() sub_timer.stop().submit() fields = stepper(fields, t, dt, rhs) if point_getter is not None: val = point_getter(fields) #print val if not done_dt: pointfile.write("#%g\n" % dt) done_dt = True pointfile.write("%g\n" %val[0]) finally: if write_output: vis.close() logmgr.close() discr.close() if point_getter is not None: pointfile.close()
def simple_wave_entrypoint(dim=2, num_elems=256, order=4, num_steps=30, log_filename="grudge.dat"): cl_ctx = cl.create_some_context() queue = cl.CommandQueue(cl_ctx) from mpi4py import MPI comm = MPI.COMM_WORLD num_parts = comm.Get_size() n = int(num_elems**(1. / dim)) from meshmode.distributed import MPIMeshDistributor mesh_dist = MPIMeshDistributor(comm) if mesh_dist.is_mananger_rank(): from meshmode.mesh.generation import generate_regular_rect_mesh mesh = generate_regular_rect_mesh(a=(-0.5, ) * dim, b=(0.5, ) * dim, n=(n, ) * dim) from pymetis import part_graph _, p = part_graph(num_parts, xadj=mesh.nodal_adjacency.neighbors_starts.tolist(), adjncy=mesh.nodal_adjacency.neighbors.tolist()) part_per_element = np.array(p) local_mesh = mesh_dist.send_mesh_parts(mesh, part_per_element, num_parts) else: local_mesh = mesh_dist.receive_mesh_part() vol_discr = DGDiscretizationWithBoundaries(cl_ctx, local_mesh, order=order, mpi_communicator=comm) source_center = np.array([0.1, 0.22, 0.33])[:local_mesh.dim] source_width = 0.05 source_omega = 3 sym_x = sym.nodes(local_mesh.dim) sym_source_center_dist = sym_x - source_center sym_t = sym.ScalarVariable("t") from grudge.models.wave import StrongWaveOperator from meshmode.mesh import BTAG_ALL, BTAG_NONE op = StrongWaveOperator( -0.1, vol_discr.dim, source_f=( sym.sin(source_omega * sym_t) * sym.exp(-np.dot(sym_source_center_dist, sym_source_center_dist) / source_width**2)), dirichlet_tag=BTAG_NONE, neumann_tag=BTAG_NONE, radiation_tag=BTAG_ALL, flux_type="upwind") from pytools.obj_array import join_fields fields = join_fields( vol_discr.zeros(queue), [vol_discr.zeros(queue) for i in range(vol_discr.dim)]) from pytools.log import LogManager, \ add_general_quantities, \ add_run_info, \ IntervalTimer, EventCounter # NOTE: LogManager hangs when using a file on a shared directory. logmgr = LogManager(log_filename, "w", comm) add_run_info(logmgr) add_general_quantities(logmgr) log_quantities =\ {"rank_data_swap_timer": IntervalTimer("rank_data_swap_timer", "Time spent evaluating RankDataSwapAssign"), "rank_data_swap_counter": EventCounter("rank_data_swap_counter", "Number of RankDataSwapAssign instructions evaluated"), "exec_timer": IntervalTimer("exec_timer", "Total time spent executing instructions"), "insn_eval_timer": IntervalTimer("insn_eval_timer", "Time spend evaluating instructions"), "future_eval_timer": IntervalTimer("future_eval_timer", "Time spent evaluating futures"), "busy_wait_timer": IntervalTimer("busy_wait_timer", "Time wasted doing busy wait")} for quantity in log_quantities.values(): logmgr.add_quantity(quantity) bound_op = bind(vol_discr, op.sym_operator()) def rhs(t, w): val, rhs.profile_data = bound_op(queue, profile_data=rhs.profile_data, log_quantities=log_quantities, t=t, w=w) return val rhs.profile_data = {} dt = 0.04 dt_stepper = set_up_rk4("w", dt, fields, rhs) logmgr.tick_before() for event in dt_stepper.run(t_end=dt * num_steps): if isinstance(event, dt_stepper.StateComputed): logmgr.tick_after() logmgr.tick_before() logmgr.tick_after() def print_profile_data(data): print("""execute() for rank %d: \tInstruction Evaluation: %f%% \tFuture Evaluation: %f%% \tBusy Wait: %f%% \tTotal: %f seconds""" % (comm.Get_rank(), data['insn_eval_time'] / data['total_time'] * 100, data['future_eval_time'] / data['total_time'] * 100, data['busy_wait_time'] / data['total_time'] * 100, data['total_time'])) print_profile_data(rhs.profile_data) logmgr.close()
def add_instrumentation(self, logmgr): from pytools.log import \ add_simulation_quantities, \ add_general_quantities, \ add_run_info, ETA from pyrticle.log import add_particle_quantities, add_field_quantities, \ add_beam_quantities, add_currents setup = self.setup from pyrticle.log import StateObserver self.observer = StateObserver(self.method, self.maxwell_op) self.observer.set_fields_and_state(self.fields, self.state) add_run_info(logmgr) add_general_quantities(logmgr) add_simulation_quantities(logmgr) add_particle_quantities(logmgr, self.observer) add_field_quantities(logmgr, self.observer) if setup.beam_axis is not None and setup.beam_diag_axis is not None: add_beam_quantities(logmgr, self.observer, axis=setup.beam_diag_axis, beam_axis=setup.beam_axis) if setup.tube_length is not None: from hedge.tools import unit_vector add_currents( logmgr, self.observer, unit_vector(self.method.dimensions_velocity, setup.beam_axis), setup.tube_length) self.method.add_instrumentation(logmgr, self.observer) self.f_rhs_calculator.add_instrumentation(logmgr) if hasattr(self.stepper, "add_instrumentation"): self.stepper.add_instrumentation(logmgr) mean_beta = self.method.mean_beta(self.state) gamma = self.method.units.gamma_from_beta(mean_beta) logmgr.set_constant("dt", self.dt) logmgr.set_constant("beta", mean_beta) logmgr.set_constant("gamma", gamma) logmgr.set_constant("v", mean_beta * self.units.VACUUM_LIGHT_SPEED()) logmgr.set_constant("Q0", self.total_charge) logmgr.set_constant("n_part_0", setup.nparticles) logmgr.set_constant("pmass", setup.distribution.mean()[3][0]) logmgr.set_constant("chi", setup.chi) logmgr.set_constant("phi_decay", setup.phi_decay) logmgr.set_constant("shape_radius_setup", setup.shape_bandwidth) logmgr.set_constant("shape_radius", self.method.depositor.shape_function.radius) logmgr.set_constant("shape_exponent", self.method.depositor.shape_function.exponent) from pytools.log import IntervalTimer self.vis_timer = IntervalTimer("t_vis", "Time spent visualizing") logmgr.add_quantity(self.vis_timer) logmgr.add_quantity(ETA(self.nsteps)) logmgr.add_watches(setup.watch_vars)
def mpi_communication_entrypoint(): cl_ctx = cl.create_some_context() queue = cl.CommandQueue(cl_ctx) from mpi4py import MPI comm = MPI.COMM_WORLD i_local_rank = comm.Get_rank() num_parts = comm.Get_size() from meshmode.distributed import MPIMeshDistributor, get_partition_by_pymetis mesh_dist = MPIMeshDistributor(comm) dim = 2 dt = 0.04 order = 4 if mesh_dist.is_mananger_rank(): from meshmode.mesh.generation import generate_regular_rect_mesh mesh = generate_regular_rect_mesh(a=(-0.5, ) * dim, b=(0.5, ) * dim, n=(16, ) * dim) part_per_element = get_partition_by_pymetis(mesh, num_parts) local_mesh = mesh_dist.send_mesh_parts(mesh, part_per_element, num_parts) else: local_mesh = mesh_dist.receive_mesh_part() vol_discr = DGDiscretizationWithBoundaries(cl_ctx, local_mesh, order=order, mpi_communicator=comm) source_center = np.array([0.1, 0.22, 0.33])[:local_mesh.dim] source_width = 0.05 source_omega = 3 sym_x = sym.nodes(local_mesh.dim) sym_source_center_dist = sym_x - source_center sym_t = sym.ScalarVariable("t") from grudge.models.wave import StrongWaveOperator from meshmode.mesh import BTAG_ALL, BTAG_NONE op = StrongWaveOperator( -0.1, vol_discr.dim, source_f=( sym.sin(source_omega * sym_t) * sym.exp(-np.dot(sym_source_center_dist, sym_source_center_dist) / source_width**2)), dirichlet_tag=BTAG_NONE, neumann_tag=BTAG_NONE, radiation_tag=BTAG_ALL, flux_type="upwind") from pytools.obj_array import join_fields fields = join_fields( vol_discr.zeros(queue), [vol_discr.zeros(queue) for i in range(vol_discr.dim)]) # FIXME # dt = op.estimate_rk4_timestep(vol_discr, fields=fields) # FIXME: Should meshmode consider BTAG_PARTITION to be a boundary? # Fails because: "found faces without boundary conditions" # op.check_bc_coverage(local_mesh) from pytools.log import LogManager, \ add_general_quantities, \ add_run_info, \ IntervalTimer, EventCounter log_filename = None # NOTE: LogManager hangs when using a file on a shared directory. # log_filename = 'grudge_log.dat' logmgr = LogManager(log_filename, "w", comm) add_run_info(logmgr) add_general_quantities(logmgr) log_quantities =\ {"rank_data_swap_timer": IntervalTimer("rank_data_swap_timer", "Time spent evaluating RankDataSwapAssign"), "rank_data_swap_counter": EventCounter("rank_data_swap_counter", "Number of RankDataSwapAssign instructions evaluated"), "exec_timer": IntervalTimer("exec_timer", "Total time spent executing instructions"), "insn_eval_timer": IntervalTimer("insn_eval_timer", "Time spend evaluating instructions"), "future_eval_timer": IntervalTimer("future_eval_timer", "Time spent evaluating futures"), "busy_wait_timer": IntervalTimer("busy_wait_timer", "Time wasted doing busy wait")} for quantity in log_quantities.values(): logmgr.add_quantity(quantity) # print(sym.pretty(op.sym_operator())) bound_op = bind(vol_discr, op.sym_operator()) # print(bound_op) # 1/0 def rhs(t, w): val, rhs.profile_data = bound_op(queue, profile_data=rhs.profile_data, log_quantities=log_quantities, t=t, w=w) return val rhs.profile_data = {} dt_stepper = set_up_rk4("w", dt, fields, rhs) final_t = 4 nsteps = int(final_t / dt) print("rank=%d dt=%g nsteps=%d" % (i_local_rank, dt, nsteps)) # from grudge.shortcuts import make_visualizer # vis = make_visualizer(vol_discr, vis_order=order) step = 0 norm = bind(vol_discr, sym.norm(2, sym.var("u"))) from time import time t_last_step = time() logmgr.tick_before() for event in dt_stepper.run(t_end=final_t): if isinstance(event, dt_stepper.StateComputed): assert event.component_id == "w" step += 1 print(step, event.t, norm(queue, u=event.state_component[0]), time() - t_last_step) # if step % 10 == 0: # vis.write_vtk_file("rank%d-fld-%04d.vtu" % (i_local_rank, step), # [("u", event.state_component[0]), # ("v", event.state_component[1:])]) t_last_step = time() logmgr.tick_after() logmgr.tick_before() logmgr.tick_after() def print_profile_data(data): print("""execute() for rank %d: \tInstruction Evaluation: %f%% \tFuture Evaluation: %f%% \tBusy Wait: %f%% \tTotal: %f seconds""" % (i_local_rank, data['insn_eval_time'] / data['total_time'] * 100, data['future_eval_time'] / data['total_time'] * 100, data['busy_wait_time'] / data['total_time'] * 100, data['total_time'])) print_profile_data(rhs.profile_data) logmgr.close() logger.debug("Rank %d exiting", i_local_rank)
def main(write_output=True, dir_tag=TAG_NONE, neu_tag=TAG_NONE, rad_tag=TAG_ALL, flux_type_arg="upwind"): from math import sin, cos, pi, exp, sqrt # noqa from hedge.backends import guess_run_context rcon = guess_run_context() dim = 2 if dim == 1: if rcon.is_head_rank: from hedge.mesh.generator import make_uniform_1d_mesh mesh = make_uniform_1d_mesh(-10, 10, 500) elif dim == 2: from hedge.mesh.generator import make_rect_mesh if rcon.is_head_rank: mesh = make_rect_mesh(a=(-1, -1), b=(1, 1), max_area=0.003) elif dim == 3: if rcon.is_head_rank: from hedge.mesh.generator import make_ball_mesh mesh = make_ball_mesh(max_volume=0.0005) else: raise RuntimeError("bad number of dimensions") if rcon.is_head_rank: print "%d elements" % len(mesh.elements) mesh_data = rcon.distribute_mesh(mesh) else: mesh_data = rcon.receive_mesh() discr = rcon.make_discretization(mesh_data, order=4) from hedge.timestep.runge_kutta import LSRK4TimeStepper stepper = LSRK4TimeStepper() from hedge.visualization import VtkVisualizer if write_output: vis = VtkVisualizer(discr, rcon, "fld") source_center = np.array([0.7, 0.4]) source_width = 1 / 16 source_omega = 3 import hedge.optemplate as sym sym_x = sym.nodes(2) sym_source_center_dist = sym_x - source_center from hedge.models.wave import VariableVelocityStrongWaveOperator op = VariableVelocityStrongWaveOperator( c=sym.If(sym.Comparison(np.dot(sym_x, sym_x), "<", 0.4**2), 1, 0.5), dimensions=discr.dimensions, source=sym.CFunction("sin")(source_omega * sym.ScalarParameter("t")) * sym.CFunction("exp")( -np.dot(sym_source_center_dist, sym_source_center_dist) / source_width**2), dirichlet_tag=dir_tag, neumann_tag=neu_tag, radiation_tag=rad_tag, flux_type=flux_type_arg) from hedge.tools import join_fields fields = join_fields( discr.volume_zeros(), [discr.volume_zeros() for i in range(discr.dimensions)]) # {{{ diagnostics setup from pytools.log import LogManager, \ add_general_quantities, \ add_simulation_quantities, \ add_run_info if write_output: log_file_name = "wave.dat" else: log_file_name = None logmgr = LogManager(log_file_name, "w", rcon.communicator) add_run_info(logmgr) add_general_quantities(logmgr) add_simulation_quantities(logmgr) discr.add_instrumentation(logmgr) from pytools.log import IntervalTimer vis_timer = IntervalTimer("t_vis", "Time spent visualizing") logmgr.add_quantity(vis_timer) stepper.add_instrumentation(logmgr) from hedge.log import LpNorm u_getter = lambda: fields[0] logmgr.add_quantity(LpNorm(u_getter, discr, 1, name="l1_u")) logmgr.add_quantity(LpNorm(u_getter, discr, name="l2_u")) logmgr.add_watches(["step.max", "t_sim.max", "l2_u", "t_step.max"]) # }}} # {{{ timestep loop rhs = op.bind(discr) try: from hedge.timestep.stability import \ approximate_rk4_relative_imag_stability_region max_dt = (1 / discr.compile(op.max_eigenvalue_expr())() * discr.dt_non_geometric_factor() * discr.dt_geometric_factor() * approximate_rk4_relative_imag_stability_region(stepper)) if flux_type_arg == "central": max_dt *= 0.25 from hedge.timestep import times_and_steps step_it = times_and_steps(final_time=3, logmgr=logmgr, max_dt_getter=lambda t: max_dt) for step, t, dt in step_it: if step % 10 == 0 and write_output: visf = vis.make_file("fld-%04d" % step) vis.add_data(visf, [ ("u", fields[0]), ("v", fields[1:]), ], time=t, step=step) visf.close() fields = stepper(fields, t, dt, rhs) assert discr.norm(fields) < 1 finally: if write_output: vis.close() logmgr.close() discr.close()
def main(write_output=True): from hedge.timestep.runge_kutta import LSRK4TimeStepper from math import sqrt, pi, exp from hedge.backends import guess_run_context rcon = guess_run_context() epsilon0 = 8.8541878176e-12 # C**2 / (N m**2) mu0 = 4 * pi * 1e-7 # N/A**2. epsilon = 1 * epsilon0 mu = 1 * mu0 c = 1 / sqrt(mu * epsilon) pml_width = 0.5 #mesh = make_mesh(a=np.array((-1,-1,-1)), b=np.array((1,1,1)), #mesh = make_mesh(a=np.array((-3,-3)), b=np.array((3,3)), mesh = make_mesh( a=np.array((-1, -1)), b=np.array((1, 1)), #mesh = make_mesh(a=np.array((-2,-2)), b=np.array((2,2)), pml_width=pml_width, max_volume=0.01) if rcon.is_head_rank: mesh_data = rcon.distribute_mesh(mesh) else: mesh_data = rcon.receive_mesh() class Current: def volume_interpolant(self, t, discr): from hedge.tools import make_obj_array result = discr.volume_zeros(kind="numpy", dtype=np.float64) omega = 6 * c if omega * t > 2 * pi: return make_obj_array([result, result, result]) x = make_obj_array(discr.nodes.T) r = np.sqrt(np.dot(x, x)) idx = r < 0.3 result[idx] = (1+np.cos(pi*r/0.3))[idx] \ *np.sin(omega*t)**3 result = discr.convert_volume(result, kind=discr.compute_kind, dtype=discr.default_scalar_type) return make_obj_array([-result, result, result]) order = 3 discr = rcon.make_discretization(mesh_data, order=order, debug=["cuda_no_plan"]) from hedge.visualization import VtkVisualizer if write_output: vis = VtkVisualizer(discr, rcon, "em-%d" % order) from hedge.mesh import TAG_ALL, TAG_NONE from hedge.data import GivenFunction, TimeHarmonicGivenFunction, TimeIntervalGivenFunction from hedge.models.em import MaxwellOperator from hedge.models.pml import \ AbarbanelGottliebPMLMaxwellOperator, \ AbarbanelGottliebPMLTMMaxwellOperator, \ AbarbanelGottliebPMLTEMaxwellOperator op = AbarbanelGottliebPMLTEMaxwellOperator(epsilon, mu, flux_type=1, current=Current(), pec_tag=TAG_ALL, absorb_tag=TAG_NONE, add_decay=True) fields = op.assemble_ehpq(discr=discr) stepper = LSRK4TimeStepper() if rcon.is_head_rank: print "order %d" % order print "#elements=", len(mesh.elements) # diagnostics setup --------------------------------------------------- from pytools.log import LogManager, add_general_quantities, \ add_simulation_quantities, add_run_info if write_output: log_file_name = "maxwell-%d.dat" % order else: log_file_name = None logmgr = LogManager(log_file_name, "w", rcon.communicator) add_run_info(logmgr) add_general_quantities(logmgr) add_simulation_quantities(logmgr) discr.add_instrumentation(logmgr) stepper.add_instrumentation(logmgr) from pytools.log import IntervalTimer vis_timer = IntervalTimer("t_vis", "Time spent visualizing") logmgr.add_quantity(vis_timer) from hedge.log import EMFieldGetter, add_em_quantities field_getter = EMFieldGetter(discr, op, lambda: fields) add_em_quantities(logmgr, op, field_getter) logmgr.add_watches( ["step.max", "t_sim.max", ("W_field", "W_el+W_mag"), "t_step.max"]) from hedge.log import LpNorm class FieldIdxGetter: def __init__(self, whole_getter, idx): self.whole_getter = whole_getter self.idx = idx def __call__(self): return self.whole_getter()[self.idx] # timestep loop ------------------------------------------------------- t = 0 pml_coeff = op.coefficients_from_width(discr, width=pml_width) rhs = op.bind(discr, pml_coeff) try: from hedge.timestep import times_and_steps step_it = times_and_steps( final_time=4 / c, logmgr=logmgr, max_dt_getter=lambda t: op.estimate_timestep( discr, stepper=stepper, t=t, fields=fields)) for step, t, dt in step_it: if step % 10 == 0 and write_output: e, h, p, q = op.split_ehpq(fields) visf = vis.make_file("em-%d-%04d" % (order, step)) #pml_rhs_e, pml_rhs_h, pml_rhs_p, pml_rhs_q = \ #op.split_ehpq(rhs(t, fields)) j = Current().volume_interpolant(t, discr) vis.add_data( visf, [ ("e", discr.convert_volume(e, "numpy")), ("h", discr.convert_volume(h, "numpy")), ("p", discr.convert_volume(p, "numpy")), ("q", discr.convert_volume(q, "numpy")), ("j", discr.convert_volume(j, "numpy")), #("pml_rhs_e", pml_rhs_e), #("pml_rhs_h", pml_rhs_h), #("pml_rhs_p", pml_rhs_p), #("pml_rhs_q", pml_rhs_q), #("max_rhs_e", max_rhs_e), #("max_rhs_h", max_rhs_h), #("max_rhs_p", max_rhs_p), #("max_rhs_q", max_rhs_q), ], time=t, step=step) visf.close() fields = stepper(fields, t, dt, rhs) _, _, energies_data = logmgr.get_expr_dataset("W_el+W_mag") energies = [value for tick_nbr, value in energies_data] assert energies[-1] < max(energies) * 1e-2 finally: logmgr.close() if write_output: vis.close()
def main(write_output=True, allow_features=None, flux_type_arg=1, bdry_flux_type_arg=None, extra_discr_args={}): from hedge.mesh.generator import make_cylinder_mesh, make_box_mesh from hedge.tools import EOCRecorder, to_obj_array from math import sqrt, pi # noqa from analytic_solutions import ( # noqa RealPartAdapter, SplitComplexAdapter, CylindricalFieldAdapter, CylindricalCavityMode, RectangularWaveguideMode, RectangularCavityMode) from hedge.models.em import MaxwellOperator logging.basicConfig(level=logging.DEBUG) from hedge.backends import guess_run_context rcon = guess_run_context(allow_features) epsilon0 = 8.8541878176e-12 # C**2 / (N m**2) mu0 = 4 * pi * 1e-7 # N/A**2. epsilon = 1 * epsilon0 mu = 1 * mu0 eoc_rec = EOCRecorder() cylindrical = False periodic = False if cylindrical: R = 1 d = 2 mode = CylindricalCavityMode(m=1, n=1, p=1, radius=R, height=d, epsilon=epsilon, mu=mu) # r_sol = CylindricalFieldAdapter(RealPartAdapter(mode)) # c_sol = SplitComplexAdapter(CylindricalFieldAdapter(mode)) if rcon.is_head_rank: mesh = make_cylinder_mesh(radius=R, height=d, max_volume=0.01) else: if periodic: mode = RectangularWaveguideMode(epsilon, mu, (3, 2, 1)) periodicity = (False, False, True) else: periodicity = None mode = RectangularCavityMode(epsilon, mu, (1, 2, 2)) if rcon.is_head_rank: mesh = make_box_mesh(max_volume=0.001, periodicity=periodicity) if rcon.is_head_rank: mesh_data = rcon.distribute_mesh(mesh) else: mesh_data = rcon.receive_mesh() for order in [4, 5, 6]: #for order in [1,2,3,4,5,6]: extra_discr_args.setdefault("debug", []).extend( ["cuda_no_plan", "cuda_dump_kernels"]) op = MaxwellOperator(epsilon, mu, flux_type=flux_type_arg, bdry_flux_type=bdry_flux_type_arg) discr = rcon.make_discretization(mesh_data, order=order, tune_for=op.op_template(), **extra_discr_args) from hedge.visualization import VtkVisualizer if write_output: vis = VtkVisualizer(discr, rcon, "em-%d" % order) mode.set_time(0) def get_true_field(): return discr.convert_volume(to_obj_array( mode(discr).real.astype(discr.default_scalar_type).copy()), kind=discr.compute_kind) fields = get_true_field() if rcon.is_head_rank: print "---------------------------------------------" print "order %d" % order print "---------------------------------------------" print "#elements=", len(mesh.elements) from hedge.timestep.runge_kutta import LSRK4TimeStepper stepper = LSRK4TimeStepper(dtype=discr.default_scalar_type, rcon=rcon) #from hedge.timestep.dumka3 import Dumka3TimeStepper #stepper = Dumka3TimeStepper(3, dtype=discr.default_scalar_type, rcon=rcon) # {{{ diagnostics setup from pytools.log import LogManager, add_general_quantities, \ add_simulation_quantities, add_run_info if write_output: log_file_name = "maxwell-%d.dat" % order else: log_file_name = None logmgr = LogManager(log_file_name, "w", rcon.communicator) add_run_info(logmgr) add_general_quantities(logmgr) add_simulation_quantities(logmgr) discr.add_instrumentation(logmgr) stepper.add_instrumentation(logmgr) from pytools.log import IntervalTimer vis_timer = IntervalTimer("t_vis", "Time spent visualizing") logmgr.add_quantity(vis_timer) from hedge.log import EMFieldGetter, add_em_quantities field_getter = EMFieldGetter(discr, op, lambda: fields) add_em_quantities(logmgr, op, field_getter) logmgr.add_watches( ["step.max", "t_sim.max", ("W_field", "W_el+W_mag"), "t_step.max"]) # }}} # {{{ timestep loop rhs = op.bind(discr) final_time = 0.5e-9 try: from hedge.timestep import times_and_steps step_it = times_and_steps( final_time=final_time, logmgr=logmgr, max_dt_getter=lambda t: op.estimate_timestep( discr, stepper=stepper, t=t, fields=fields)) for step, t, dt in step_it: if step % 50 == 0 and write_output: sub_timer = vis_timer.start_sub_timer() e, h = op.split_eh(fields) visf = vis.make_file("em-%d-%04d" % (order, step)) vis.add_data(visf, [ ("e", discr.convert_volume(e, kind="numpy")), ("h", discr.convert_volume(h, kind="numpy")), ], time=t, step=step) visf.close() sub_timer.stop().submit() fields = stepper(fields, t, dt, rhs) mode.set_time(final_time) eoc_rec.add_data_point(order, discr.norm(fields - get_true_field())) finally: if write_output: vis.close() logmgr.close() discr.close() if rcon.is_head_rank: print print eoc_rec.pretty_print("P.Deg.", "L2 Error") # }}} assert eoc_rec.estimate_order_of_convergence()[0, 1] > 6