Source code for sfepy.solvers.ts_solvers

"""
Time stepping solvers.
"""
from inspect import signature
from functools import partial
import numpy as nm

from sfepy.base.base import (get_default, output, assert_,
                             Struct, IndexedStruct)
from sfepy.base.timing import Timer
from sfepy.linalg.utils import output_array_stats
from sfepy.solvers.solvers import TimeSteppingSolver, NonlinearSolver
from sfepy.solvers.ls import RMMSolver
from sfepy.solvers.ts_controllers import FixedTSC
from sfepy.solvers.ts import TimeStepper, VariableTimeStepper

[docs] def standard_ts_call(call): """ Decorator handling argument preparation and timing for time-stepping solvers. """ def _standard_ts_call(self, vec0=None, nls=None, init_fun=None, prestep_fun=None, poststep_fun=None, status=None, **kwargs): timer = Timer(start=True) nls = get_default(nls, self.nls, 'nonlinear solver has to be specified!') init_fun = get_default(init_fun, lambda ts, vec0: vec0) prestep_fun = get_default(prestep_fun, lambda ts, vec: None) poststep_fun = get_default(poststep_fun, lambda ts, vec: None) result = call(self, vec0=vec0, nls=nls, init_fun=init_fun, prestep_fun=prestep_fun, poststep_fun=poststep_fun, status=status, **kwargs) elapsed = timer.stop() if status is not None: status['time'] = elapsed status['n_step'] = self.ts.n_step return result return _standard_ts_call
# # General solvers. #
[docs] class StationarySolver(TimeSteppingSolver): """ Solver for stationary problems without time stepping. This class is provided to have a unified interface of the time stepping solvers also for stationary problems. """ name = 'ts.stationary' def __init__(self, conf, nls=None, context=None, **kwargs): TimeSteppingSolver.__init__(self, conf, nls=nls, context=context, **kwargs) self.ts = TimeStepper(0.0, 1.0, n_step=1, is_quasistatic=True) @standard_ts_call def __call__(self, vec0=None, nls=None, init_fun=None, prestep_fun=None, poststep_fun=None, status=None, **kwargs): ts = self.ts nls = get_default(nls, self.nls) vec0 = init_fun(ts, vec0) vec0 = prestep_fun(ts, vec0) vec = nls(vec0) vec = poststep_fun(ts, vec) return vec
[docs] class SimpleTimeSteppingSolver(TimeSteppingSolver): """ Implicit time stepping solver with a fixed time step. """ name = 'ts.simple' _parameters = [ ('t0', 'float', 0.0, False, 'The initial time.'), ('t1', 'float', 1.0, False, 'The final time.'), ('dt', 'float', None, False, 'The time step. Used if `n_step` is not given.'), ('n_step', 'int', 10, False, 'The number of time steps. Has precedence over `dt`.'), ('quasistatic', 'bool', False, False, """If True, assume a quasistatic time-stepping. Then the non-linear solver is invoked also for the initial time."""), ] def __init__(self, conf, nls=None, context=None, **kwargs): TimeSteppingSolver.__init__(self, conf, nls=nls, context=context, **kwargs) self.ts = TimeStepper.from_conf(self.conf) nd = self.ts.n_digit format = '====== time %%e (step %%%dd of %%%dd) =====' % (nd, nd) self.format = format self.verbose = self.conf.verbose
[docs] def solve_step0(self, nls, vec0): if self.conf.quasistatic: vec = nls(vec0) else: res = nls.fun(vec0) err = nm.linalg.norm(res) output('initial residual: %e' % err, verbose=self.verbose) vec = vec0.copy() return vec
[docs] def solve_step(self, ts, nls, vec, prestep_fun=None): return nls(vec)
[docs] def output_step_info(self, ts): output(self.format % (ts.time, ts.step + 1, ts.n_step), verbose=self.verbose)
@standard_ts_call def __call__(self, vec0=None, nls=None, init_fun=None, prestep_fun=None, poststep_fun=None, status=None, **kwargs): """ Solve the time-dependent problem. """ ts = self.ts nls = get_default(nls, self.nls) vec0 = init_fun(ts, vec0) self.output_step_info(ts) if ts.step == 0: vec0 = prestep_fun(ts, vec0) vec = self.solve_step0(nls, vec0) vec = poststep_fun(ts, vec) ts.advance() else: vec = vec0 for step, time in ts.iter_from(ts.step): self.output_step_info(ts) vec = prestep_fun(ts, vec) vect = self.solve_step(ts, nls, vec, prestep_fun) vect = poststep_fun(ts, vect) vec = vect return vec
[docs] def get_min_dt(adt): red = adt.red while red >= adt.red_max: red *= adt.red_factor dt = adt.dt0 * red return dt
[docs] def adapt_time_step(ts, status, adt, context=None, verbose=False): """ Adapt the time step of `ts` according to the exit status of the nonlinear solver. The time step dt is reduced, if the nonlinear solver did not converge. If it converged in less then a specified number of iterations for several time steps, the time step is increased. This is governed by the following parameters: - `red_factor` : time step reduction factor - `red_max` : maximum time step reduction factor - `inc_factor` : time step increase factor - `inc_on_iter` : increase time step if the nonlinear solver converged in less than this amount of iterations... - `inc_wait` : ...for this number of consecutive time steps Parameters ---------- ts : VariableTimeStepper instance The time stepper. status : IndexedStruct instance The nonlinear solver exit status. adt : Struct instance The object with the adaptivity parameters of the time-stepping solver such as `red_factor` (see above) as attributes. context : object, optional The context can be used in user-defined adaptivity functions. Not used here. Returns ------- is_break : bool If True, the adaptivity loop should stop. """ is_break = False if status.condition == 0: if status.n_iter <= adt.inc_on_iter: adt.wait += 1 if adt.wait > adt.inc_wait: if adt.red < 1.0: adt.red = adt.red * adt.inc_factor ts.set_time_step(adt.dt0 * adt.red) output('+++++ new time step: %e +++++' % ts.dt, verbose=verbose) adt.wait = 0 else: adt.wait = 0 is_break = True else: adt.red = adt.red * adt.red_factor if adt.red < adt.red_max: is_break = True else: ts.set_time_step(adt.dt0 * adt.red, update_time=True) output('----- new time step: %e -----' % ts.dt, verbose=verbose) adt.wait = 0 return is_break
[docs] class AdaptiveTimeSteppingSolver(SimpleTimeSteppingSolver): """ Implicit time stepping solver with an adaptive time step. Either the built-in or user supplied function can be used to adapt the time step. """ name = 'ts.adaptive' _parameters = SimpleTimeSteppingSolver._parameters + [ ('adapt_fun', 'callable(ts, status, adt, context, verbose)', None, False, """If given, use this function to set the time step in `ts`. The function return value is a bool - if True, the adaptivity loop should stop. The other parameters below are collected in `adt`, `status` is the nonlinear solver status, `context` is a user-defined context and `verbose` is a verbosity flag. Solvers created by :class:`Problem <sfepy.discrete.problem.Problem>` use the Problem instance as the context."""), ('dt_red_factor', 'float', 0.2, False, 'The time step reduction factor.'), ('dt_red_max', 'float', 1e-3, False, 'The maximum time step reduction factor.'), ('dt_inc_factor', 'float', 1.25, False, 'The time step increase factor.'), ('dt_inc_on_iter', 'int', 4, False, """Increase the time step if the nonlinear solver converged in less than this amount of iterations for `dt_inc_wait` consecutive time steps."""), ('dt_inc_wait', 'int', 5, False, 'The number of consecutive time steps, see `dt_inc_on_iter`.'), ] def __init__(self, conf, nls=None, context=None, **kwargs): TimeSteppingSolver.__init__(self, conf, nls=nls, context=context, **kwargs) self.ts = VariableTimeStepper.from_conf(self.conf) get = self.conf.get adt = Struct(red_factor=get('dt_red_factor', 0.2), red_max=get('dt_red_max', 1e-3), inc_factor=get('dt_inc_factor', 1.25), inc_on_iter=get('dt_inc_on_iter', 4), inc_wait=get('dt_inc_wait', 5), red=1.0, wait=0, dt0=0.0) self.adt = adt adt.dt0 = self.ts.get_default_time_step() self.ts.set_n_digit_from_min_dt(get_min_dt(adt)) self.format = '====== time %e (dt %e, wait %d, step %d of %d) =====' self.verbose = self.conf.verbose self.adapt_time_step = self.conf.adapt_fun if self.adapt_time_step is None: self.adapt_time_step = adapt_time_step
[docs] def solve_step(self, ts, nls, vec, prestep_fun): """ Solve a single time step. """ status = IndexedStruct(n_iter=0, condition=0) while 1: vect = nls(vec, status=status) is_break = self.adapt_time_step(ts, status, self.adt, self.context, verbose=self.verbose) if is_break: break vec = prestep_fun(ts, vec) return vect
[docs] def output_step_info(self, ts): output(self.format % (ts.time, ts.dt, self.adt.wait, ts.step + 1, ts.n_step), verbose=self.verbose)
# # Elastodynamics solvers. #
[docs] def transform_equations_ed(equations, materials): """ Transform equations and variables for :class:`ElastodynamicsBaseTS`-based time stepping solvers. The displacement variable name is automatically detected by seeking the second time derivative, i.e. the 'dd' prefix in variable names. """ from sfepy.terms.terms import Terms, Term from sfepy.discrete.variables import FieldVariable from sfepy.discrete.equations import Equations, Equation eq_mass = [] eq_damping = [] eq_other = [] new_var_primary = set() variables = equations.variables for eq in equations: for term in eq.terms: for name in term.names.state: der = term.arg_derivatives[name] if ((der is not None) and isinstance(der, int) and (der > 0)): new_var_primary.add(name) if der == 2: eq_mass.append(term) else: eq_damping.append(term) else: eq_other.append(term) continue assert len(new_var_primary) == 1 uname = new_var_primary.pop() vname = variables[uname].dual_var_name duname = 'd' + uname dduname = 'dd' + uname dvname = 'd' + vname ddvname = 'dd' + vname new_variables = variables.copy() if new_variables[uname]._order is None: raise ValueError('state variable orders have to be specified when using' ' auto_transform_equations!') num = len(new_variables.state) new_variables.extend([ FieldVariable(duname, 'unknown', variables[uname].field, order=num), FieldVariable(dduname, 'unknown', variables[uname].field, order=num + 1), FieldVariable(dvname, 'test', variables[uname].field, primary_var_name=duname), FieldVariable(ddvname, 'test', variables[uname].field, primary_var_name=dduname), ]) for it, term in enumerate(eq_mass.copy()): aux = ','.join([ii.strip() for ii in term.arg_str.split(',')]) arg_str = aux.replace(f',{vname},', f',{ddvname},') new_term = term.__class__(term.name, arg_str, term.integral, term.region) new_term.setup(allow_derivatives=False) new_term.assign_args(new_variables, materials, user=None) eq_mass[it] = new_term for it, term in enumerate(eq_damping.copy()): aux = ','.join([ii.strip() for ii in term.arg_str.split(',')]) arg_str = aux.replace(f',{vname},', f',{dvname},') new_term = term.__class__(term.name, arg_str, term.integral, term.region) new_term.setup(allow_derivatives=False) new_term.assign_args(new_variables, materials, user=None) eq_damping[it] = new_term if not len(eq_damping): mterm = eq_mass[0] # Dummy damping to introduce du, could use a single cell region? dterm = Term.new( f'dw_zero({dvname}, {duname})', mterm.integral, mterm.region, **{dvname : new_variables[dvname], duname : new_variables[duname]}, ) dterm.setup(allow_derivatives=False) dterm.assign_args(new_variables, materials, user=None) eq_damping = [dterm] new_equations = Equations([Equation('M', Terms(eq_mass), setup=False), Equation('C', Terms(eq_damping), setup=False), Equation('K', Terms(eq_other), setup=False),]) var_names = { 'u' : uname, 'du' : duname, 'ddu' : dduname, 'extra' : set(equations.variables.di.var_names) - set([uname, duname, dduname]) } return new_equations, var_names
[docs] def gen_multi_vec_packing(di, names, extra_variables=False): """ Return DOF vector (un)packing functions for nlst. For multiphysical problems (non-empty `ie` slice for extra variables) the `unpack()` function accepts an additional argument `mode` that can be set to 'full' or 'nls'. The following DOF ordering must be obeyed: - The full DOF vector: | ``---iue---|-iv-|-ia-`` | ``-iu-|-ie-|`` """ iu = di.indx[names['u']] iv = di.indx[names['du']] ia = di.indx[names['ddu']] ie = slice(iu.stop, iv.start) iue = slice(0, iv.start) assert_(iu.stop == di.n_dof[names['u']]) assert_(iv.start == ie.stop) assert_(ia.start == iv.stop) if not extra_variables: assert_(ie.stop == ie.start) n_arg = 3 def unpack(vec, mode=None): return vec[iu], vec[iv], vec[ia] else: n_arg = 4 def unpack(vec, mode='full'): if mode == 'nls': return vec[iu], vec[ie] else: return vec[iu], vec[ie], vec[iv], vec[ia] def pack(*args): return nm.concatenate(args) indices = dict(indices=(iue, iu, ie, iv, ia), n_dof=di.n_dof_total, n_uedof=ie.stop, n_arg=n_arg) unpack.__dict__.update(indices) pack.__dict__.update(indices) return unpack, pack
def _cache(obj, attr, dep): def decorate(fun): def new_fun(*args, **kwargs): if dep: val = getattr(obj, attr) if val is None: val = fun(*args, **kwargs) setattr(obj, attr, val) else: val = fun(*args, **kwargs) return val return new_fun return decorate
[docs] class ElastodynamicsBaseTS(TimeSteppingSolver): """ Base class for elastodynamics solvers. Assumes block-diagonal matrix in `u`, `v`, `a`. """ _common_parameters = [ ('t0', 'float', 0.0, False, 'The initial time.'), ('t1', 'float', 1.0, False, 'The final time.'), ('dt', 'float', None, False, 'The time step. Used if `n_step` is not given.'), ('n_step', 'int', 10, False, 'The number of time steps. Has precedence over `dt`.'), ('is_linear', 'bool', False, False, 'If True, the problem is considered to be linear.'), ('var_names', 'dict', None, False, """The mapping of variables with keys 'u', 'du', 'ddu' and 'extra', and values corresponding to the names of the actual variables. See `var_names` returned from :func:`transform_equations_ed()`"""), ] def __init__(self, conf, nls=None, tsc=None, context=None, **kwargs): TimeSteppingSolver.__init__(self, conf, nls=nls, tsc=tsc, context=context, **kwargs) self.conf.quasistatic = False if self.tsc is None: self.tsc = FixedTSC({}) if isinstance(self.tsc, FixedTSC): # Using TimeStepper instead of VariableTimeStepper ensures the # final time is reached "exactly". self.ts = TimeStepper.from_conf(self.conf) else: self.ts = VariableTimeStepper.from_conf(self.conf) nd = self.ts.n_digit format = '====== time %%e (step %%%dd of %%%dd) =====' % (nd, nd) self.format = format self.verbose = self.conf.verbose self.constant_matrices = None self.matrix = None
[docs] def get_matrices(self, nls, vec, unpack=None): if self.conf.is_linear and self.constant_matrices is not None: out = self.constant_matrices else: aux = nls.fun_grad(vec) if unpack is not None: iue, iu, ie, iv, ia = unpack.indices aux = nls.fun_grad(vec) M = aux[ia, ia] C = aux[iv, iv] K = aux[iue, iue] else: assert_((len(vec) % 3) == 0) i3 = len(vec) // 3 K = aux[:i3, :i3] C = aux[i3:2*i3, i3:2*i3] M = aux[2*i3:, 2*i3:] out = (M, C, K) if self.conf.is_linear: M.eliminate_zeros() C.eliminate_zeros() K.eliminate_zeros() self.constant_matrices = (M, C, K) return out
[docs] def get_a0(self, nls, u0, e0, v0, unpack): iue, iu, ie, iv, ia = unpack.indices vec = nm.r_[u0, e0, v0, nm.zeros_like(v0)] aux = nls.fun(vec) r = aux[iu] + aux[iv] + aux[ia] M = self.get_matrices(nls, vec, unpack)[0][iu, iu] a0 = nls.lin_solver(-r, mtx=M) nls.lin_solver.clear() output_array_stats(a0, 'initial acceleration', verbose=self.verbose) return a0
[docs] def get_initial_vec(self, nls, vec0, init_fun, prestep_fun, poststep_fun): if not set(self.conf.var_names.keys()).issuperset(['ddu', 'du', 'u']): raise ValueError( "var_names have to contain 'u', 'du', 'ddu' keys!" ) unpack, pack = gen_multi_vec_packing( self.di, self.conf.var_names, extra_variables=self.get('extra_variables', False), ) self.unpack = unpack self.pack = pack ts = self.ts vec0 = init_fun(ts, vec0) output(self.format % (ts.time, ts.step + 1, ts.n_step), verbose=self.verbose) if ts.step == 0: vec0 = prestep_fun(ts, vec0) if unpack.n_arg == 4: u0, e0, v0, _ = unpack(vec0) else: u0, v0, _ = unpack(vec0) e0 = nm.empty(0, dtype=u0.dtype) a0 = self.get_a0(nls, u0, e0, v0, unpack) if unpack.n_arg == 4: vec = pack(u0, e0, v0, a0) else: vec = pack(u0, v0, a0) vec = poststep_fun(ts, vec) ts.advance() else: vec = vec0 return vec, unpack, pack
[docs] def clear_lin_solver(self, clear_constant_matrices=True): self.nls.lin_solver.clear() self.matrix = None if clear_constant_matrices: self.constant_matrices = None
@standard_ts_call def __call__(self, vec0=None, nls=None, init_fun=None, prestep_fun=None, poststep_fun=None, status=None, **kwargs): sig = signature(init_fun) if len(sig.parameters) == 3: init_fun = partial(init_fun, self) sig = signature(poststep_fun) if len(sig.parameters) == 3: poststep_fun = partial(poststep_fun, self) vec, unpack, pack = self.get_initial_vec( nls, vec0, init_fun, prestep_fun, poststep_fun) ts = self.ts dt0 = self.tsc.get_initial_dt(ts, vec, unpack=unpack) if not isinstance(self.tsc, FixedTSC): ts.set_time_step(dt0, update_time=True) while 1: output(self.format % (ts.time, ts.step + 1, ts.n_step), verbose=self.verbose) # step, time = time step to compute = n+1 # step-1, time-ts.dt = previous known step data = n # adaptivity modifies dt and time. while 1: # Previous step state q(t_n). # Both prestep_fun() and poststep_fun() apply EBCs to vec/vect # in case active_only is False. # TODO: Generalized alpha: EBCs for current time t_{n+1}. but # loads should be applied in the mid-step time t_{n+1-a}. vec = prestep_fun(ts, vec) vect = self.step(ts, vec, nls, pack, unpack, prestep_fun=prestep_fun) if isinstance(self.tsc, FixedTSC): new_dt = ts.dt break new_dt, status = self.tsc(ts, vec, vect, unpack=unpack) output('dt:', ts.dt, 'new dt:', new_dt, 'status:', status, verbose=self.verbose) if new_dt != ts.dt: self.clear_lin_solver(clear_constant_matrices=False) if status.result == 'accept': break ts.set_time_step(new_dt, update_time=True) # Current step state q(t_{n+1}). vect = poststep_fun(ts, vect) if ts.nt >= 1: break if new_dt != ts.dt: ts.set_time_step(new_dt, update_time=False) ts.advance() vec = vect return vec
[docs] class VelocityVerletTS(ElastodynamicsBaseTS): """ Solve elastodynamics problems by the explicit velocity-Verlet method. The algorithm can be found in [1]_. It is mathematically equivalent to the :class:`CentralDifferenceTS` method. The current implementation code is essentially the same, as the mid-time velocities are not used for anything other than computing the new time velocities. .. [1] Swope, William C.; H. C. Andersen; P. H. Berens; K. R. Wilson (1 January 1982). "A computer simulation method for the calculation of equilibrium constants for the formation of physical clusters of molecules: Application to small water clusters". The Journal of Chemical Physics. 76 (1): 648 (Appendix). doi:10.1063/1.442716 """ name = 'ts.velocity_verlet' _parameters = [ ] + ElastodynamicsBaseTS._common_parameters
[docs] def create_nlst(self, nls, dt, u0, v0, a0): vm = v0 + 0.5 * dt * a0 u1 = u0 + dt * vm def v1(a): return vm + 0.5 * dt * a nlst = nls.copy() def fun(at): vec = nm.r_[u1, vm, at] aux = nls.fun(vec) i3 = len(at) rt = aux[:i3] + aux[i3:2*i3] + aux[2*i3:] return rt @_cache(self, 'matrix', self.conf.is_linear) def fun_grad(at): vec = None if self.conf.is_linear else nm.r_[u1, vm, at] M = self.get_matrices(nls, vec)[0] return M nlst.fun = fun nlst.fun_grad = fun_grad nlst.v1 = v1 nlst.u1 = u1 return nlst
[docs] def step(self, ts, vec, nls, pack, unpack, **kwargs): """ Solve a single time step. """ dt = ts.dt ut, vt, at = unpack(vec) nlst = self.create_nlst(nls, dt, ut, vt, at) atp = nlst(at) vtp = nlst.v1(atp) utp = nlst.u1 vect = pack(utp, vtp, atp) return vect
[docs] class CentralDifferenceTS(ElastodynamicsBaseTS): r""" Solve elastodynamics problems by the explicit central difference method. It is the same method as obtained by using :class:`NewmarkTS` with :math:`\beta = 0`, :math:`\gamma = 1/2`, but uses a simpler code. It is also mathematically equivalent to the :class:`VelocityVerletTS` method. The current implementation code is essentially the same. This solver supports, when used with :class:`RMMSolver <sfepy.solvers.ls.RMMSolver>`, the reciprocal mass matrix algorithm, see :class:`MassTerm <sfepy.terms.terms_mass.MassTerm>`. """ name = 'ts.central_difference' _parameters = [ ] + ElastodynamicsBaseTS._common_parameters def _create_nlst_a(self, nls, dt, ufun, vfun, cc, cache_name, is_rmm=False): nlst = nls.copy() if is_rmm: def fun(at): ut = ufun() zz = nm.zeros_like(ut) vec = nm.r_[ut, zz, zz] aux = nls.fun(vec) i3 = len(at) rt = aux[:i3] return rt @_cache(self, cache_name, self.conf.is_linear) def fun_grad(at): M = self.get_matrices(nls, None)[0] return M else: def fun(at): vec = nm.r_[ufun(), vfun(at), at] aux = nls.fun(vec) i3 = len(at) rt = aux[:i3] + aux[i3:2*i3] + aux[2*i3:] return rt @_cache(self, cache_name, self.conf.is_linear) def fun_grad(at): vec = (None if self.conf.is_linear else nm.r_[ufun(), vfun(at), at]) M, C = self.get_matrices(nls, vec)[:2] Kt = M + cc * C return Kt nlst.fun = fun nlst.fun_grad = fun_grad nlst.u = ufun nlst.v = vfun return nlst
[docs] def create_nlst(self, nls, dt, u0, v0, a0): dt2 = dt**2 def v(a): return v0 + dt * 0.5 * (a0 + a) def u(): return u0 + dt * v0 + dt2 * 0.5 * a0 if isinstance(nls.lin_solver, RMMSolver): import scipy.sparse as sps class NoNLS(NonlinearSolver): def __call__(self, vec_x0, conf=None, fun=None, fun_grad=None, lin_solver=None, iter_hook=None, status=None): vec_r = self.fun(vec_x0) # Dummy all-zero matrix to make standard_call() happy. mtx_a = sps.csr_matrix((vec_r.shape[0], vec_r.shape[0])) return self.lin_solver(-vec_r, mtx=mtx_a) nlst = self._create_nlst_a(nls, dt, u, v, 0.5 * dt, 'matrix', is_rmm=True) nlst = NoNLS(Struct(name='nonls', kind='nls.nonls'), fun=nlst.fun, fun_grad=nlst.fun_grad, lin_solver=nlst.lin_solver, iter_hook=nlst.iter_hook, status=nlst.status, context=nlst.context, u=nlst.u, v=nlst.v) # nlst.lin_solver.a0 = a0 else: nlst = self._create_nlst_a(nls, dt, u, v, 0.5 * dt, 'matrix') return nlst
[docs] def step(self, ts, vec, nls, pack, unpack, **kwargs): """ Solve a single time step. """ dt = ts.dt ut, vt, at = unpack(vec) nlst = self.create_nlst(nls, dt, ut, vt, at) atp = nlst(at) vtp = nlst.v(atp) utp = nlst.u() vect = pack(utp, vtp, atp) return vect
[docs] class NewmarkTS(ElastodynamicsBaseTS): r""" Solve elastodynamics problems by the Newmark method. The method was introduced in [1]. Common settings [2]: ==================== ======== ==== ===== ========== name kind beta gamma Omega_crit ==================== ======== ==== ===== ========== trapezoidal rule: implicit 1/4 1/2 unconditional linear acceleration: implicit 1/6 1/2 :math:`2\sqrt{3}` Fox-Goodwin: implicit 1/12 1/2 :math:`\sqrt{6}` central difference: explicit 0 1/2 2 ==================== ======== ==== ===== ========== All of these methods are 2-order of accuracy. [1] Newmark, N. M. (1959) A method of computation for structural dynamics. Journal of Engineering Mechanics, ASCE, 85 (EM3) 67-94. [2] Arnaud Delaplace, David Ryckelynck: Solvers for Computational Mechanics """ name = 'ts.newmark' extra_variables = True _parameters = [ ('beta', 'float', 0.25, False, 'The Newmark method parameter beta.'), ('gamma', 'float', 0.5, False, 'The Newmark method parameter gamma.'), ] + ElastodynamicsBaseTS._common_parameters
[docs] def create_nlst(self, nls, dt, gamma, beta, u0, e0, v0, a0, pack, unpack): dt2 = dt**2 iue, iu, ie, iv, ia = pack.indices cc0 = (1.0 - gamma) * dt cc = gamma * dt ck0 = (0.5 - beta) * dt2 ck = beta * dt2 def v(a): return v0 + cc0 * a0 + cc * a def u(a): return u0 + dt * v0 + ck0 * a0 + ck * a if iue == iu: def fun(at): vec = nm.r_[u(at), v(at), at] aux = nls.fun(vec) rt = aux[iu] + aux[iv] + aux[ia] return rt @_cache(self, 'matrix', self.conf.is_linear) def fun_grad(at): vec = None if self.conf.is_linear else nm.r_[u(at), v(at), at] M, C, K = self.get_matrices(nls, vec, unpack) Kt = M + cc * C + ck * K return Kt else: # Extra variables present. def fun(aet): at = aet[iu] vec = nm.r_[u(at), aet[ie], v(at), at] aux = nls.fun(vec) rt = nm.empty(pack.n_uedof, aux.dtype) rt[iu] = aux[iu] + aux[iv] + aux[ia] rt[ie] = aux[ie] return rt @_cache(self, 'matrix', self.conf.is_linear) def fun_grad(aet): if self.conf.is_linear: vec = None else: at = aet[iu] vec = nm.r_[u(at), aet[ie], v(at), at] M, C, K = self.get_matrices(nls, vec, unpack) Kt = K.copy() Kt[:, iu] *= ck Kt[iu, iu] += M + cc * C return Kt nlst = nls.copy() nlst.fun = fun nlst.fun_grad = fun_grad nlst.u = u nlst.v = v return nlst
[docs] def step(self, ts, vec, nls, pack, unpack, **kwargs): """ Solve a single time step. """ dt = ts.dt conf = self.conf ut, et, vt, at = unpack(vec) nlst = self.create_nlst(nls, dt, conf.gamma, conf.beta, ut, et, vt, at, pack, unpack) aetp = nlst(pack(at, et)) atp, etp = unpack(aetp, mode='nls') vtp = nlst.v(atp) utp = nlst.u(atp) vect = pack(utp, etp, vtp, atp) return vect
[docs] class GeneralizedAlphaTS(ElastodynamicsBaseTS): r""" Solve elastodynamics problems by the generalized :math:`\alpha` method. - The method was introduced in [1]. - The method is unconditionally stable provided :math:`\alpha_m \leq \alpha_f \leq \frac{1}{2}`, :math:`\beta >= \frac{1}{4} + \frac{1}{2}(\alpha_f - \alpha_m)`. - The method is second-order accurate provided :math:`\gamma = \frac{1}{2} - \alpha_m + \alpha_f`. This is used when `gamma` is ``None``. - High frequency dissipation is maximized for :math:`\beta = \frac{1}{4}(1 - \alpha_m + \alpha_f)^2`. This is used when `beta` is ``None``. - The default values of :math:`\alpha_m`, :math:`\alpha_f` (if `alpha_m` or `alpha_f` are ``None``) are based on the user specified high-frequency dissipation parameter `rho_inf`. Special settings: - :math:`\alpha_m = 0` corresponds to the HHT-:math:`\alpha` method. - :math:`\alpha_f = 0` corresponds to the WBZ-:math:`\alpha` method. - :math:`\alpha_m = 0`, :math:`\alpha_f = 0` produces the Newmark method. [1] J. Chung, G.M.Hubert. "A Time Integration Algorithm for Structural Dynamics with Improved Numerical Dissipation: The Generalized-:math:`\alpha` Method" ASME Journal of Applied Mechanics, 60, 371:375, 1993. """ name = 'ts.generalized_alpha' _parameters = [ ('rho_inf', 'float', 0.5, False, """The spectral radius in the high frequency limit (user specified high-frequency dissipation) in [0, 1]: 1 = no dissipation, 0 = asymptotic annihilation."""), ('alpha_m', 'float', None, False, r'The parameter :math:`\alpha_m`.'), ('alpha_f', 'float', None, False, r'The parameter :math:`\alpha_f`.'), ('beta', 'float', None, False, r'The Newmark-like parameter :math:`\beta`.'), ('gamma', 'float', None, False, r'The Newmark-like parameter :math:`\gamma`.'), ] + ElastodynamicsBaseTS._common_parameters def __init__(self, conf, nls=None, tsc=None, context=None, **kwargs): ElastodynamicsBaseTS.__init__(self, conf, nls=nls, tsc=tsc, context=context, **kwargs) conf = self.conf rho_inf = conf.rho_inf alpha_m = get_default(conf.alpha_m, (2.0 * rho_inf - 1.0) / (rho_inf + 1.0)) alpha_f = get_default(conf.alpha_f, rho_inf / (rho_inf + 1.0)) beta = get_default(conf.beta, 0.25 * (1.0 - alpha_m + alpha_f)**2) gamma = get_default(conf.gamma, 0.5 - alpha_m + alpha_f) self.pars = (alpha_m, alpha_f, gamma, beta) output('parameters rho_inf, alpha_m, alpha_f, beta, gamma:', verbose=self.verbose) output(rho_inf, alpha_m, alpha_f, beta, gamma, verbose=self.verbose) def _create_nlst_a(self, nls, dt, ufun, vfun, afun, cm, cc, ck, cache_name): nlst = nls.copy() def fun(at): vec = nm.r_[ufun(at), vfun(at), afun(at)] aux = nls.fun(vec) i3 = len(at) rt = aux[:i3] + aux[i3:2*i3] + aux[2*i3:] return rt @_cache(self, cache_name, self.conf.is_linear) def fun_grad(at): vec = (None if self.conf.is_linear else nm.r_[ufun(at), vfun(at), afun(at)]) M, C, K = self.get_matrices(nls, vec) Kt = cm * M + cc * C + ck * K return Kt nlst.fun = fun nlst.fun_grad = fun_grad return nlst
[docs] def create_nlst(self, nls, dt, alpha_m, alpha_f, gamma, beta, u0, v0, a0): dt2 = dt**2 def u1(a): return u0 + dt * v0 + dt2 * ((0.5 - beta) * a0 + beta * a) def v1(a): return v0 + dt * ((1.0 - gamma) * a0 + gamma * a) def um(a): return (1.0 - alpha_f) * u1(a) + alpha_f * u0 def vm(a): return (1.0 - alpha_f) * v1(a) + alpha_f * v0 def am(a): return (1.0 - alpha_m) * a + alpha_m * a0 nlst = self._create_nlst_a(nls, dt, um, vm, am, (1.0 - alpha_m), (1.0 - alpha_f) * gamma * dt, (1.0 - alpha_f) * beta * dt2, 'matrix') nlst.u = u1 nlst.v = v1 return nlst
[docs] def step(self, ts, vec, nls, pack, unpack, **kwargs): """ Solve a single time step. """ dt = ts.dt alpha_m, alpha_f, gamma, beta = self.pars # Previous step state q(t_n). ut, vt, at = unpack(vec) nlst = self.create_nlst(nls, dt, alpha_m, alpha_f, gamma, beta, ut, vt, at) # Notation: a = \alpha_f, t = t_{n+1}, t - dt = t_n. # Set time to t_{n+1-a} = (1 - a) t + a (t - dt) = t - a dt ts.set_substep_time(- alpha_f * dt) atp = nlst(at) # Restore t_{n+1}. ts.restore_step_time() vtp = nlst.v(atp) utp = nlst.u(atp) # Current step state q(t_{n+1}). vect = pack(utp, vtp, atp) return vect
[docs] class BatheTS(ElastodynamicsBaseTS): """ Solve elastodynamics problems by the Bathe method. The method was introduced in [1]. [1] Klaus-Juergen Bathe, Conserving energy and momentum in nonlinear dynamics: A simple implicit time integration scheme, Computers & Structures, Volume 85, Issues 7-8, 2007, Pages 437-445, ISSN 0045-7949, https://doi.org/10.1016/j.compstruc.2006.09.004. """ name = 'ts.bathe' _parameters = [ ] + ElastodynamicsBaseTS._common_parameters def __init__(self, conf, nls=None, context=None, **kwargs): ElastodynamicsBaseTS.__init__(self, conf, nls=nls, context=context, **kwargs) self.matrix1 = None self.ls1 = None self.ls2 = None def _create_nlst_u(self, nls, dt, vfun, afun, cm, cc, cache_name): nlst = nls.copy() def fun(ut): vt = vfun(ut) at = afun(vt) vec = nm.r_[ut, vt, at] aux = nls.fun(vec) i3 = len(at) rt = aux[:i3] + aux[i3:2*i3] + aux[2*i3:] return rt @_cache(self, cache_name, self.conf.is_linear) def fun_grad(ut): if self.conf.is_linear: vec = None else: vt = vfun(ut) at = afun(vt) vec = nm.r_[ut, vt, at] M, C, K = self.get_matrices(nls, vec) Kt = cm * M + cc * C + K return Kt nlst.fun = fun nlst.fun_grad = fun_grad nlst.v = vfun nlst.a = afun return nlst
[docs] def create_nlst1(self, nls, dt, u0, v0, a0): """ The first sub-step: the trapezoidal rule. """ dt4 = 4.0 / dt def v(u): return dt4 * (u - u0) - v0 def a(v): return dt4 * (v - v0) - a0 nlst = self._create_nlst_u(nls, dt, v, a, dt4 * dt4, dt4, 'matrix1') if self.ls1 is None: self.ls1 = nls.lin_solver.copy() nlst.lin_solver = self.ls1 return nlst
[docs] def create_nlst2(self, nls, dt, u0, u1, v0, v1): """ The second sub-step: the three-point Euler backward method. """ dt1 = 1.0 / dt dt4 = 4.0 * dt1 dt3 = 3.0 * dt1 def v(u): return dt1 * u0 - dt4 * u1 + dt3 * u def a(v): return dt1 * v0 - dt4 * v1 + dt3 * v nlst = self._create_nlst_u(nls, dt, v, a, dt3 * dt3, dt3, 'matrix') if self.ls2 is None: self.ls2 = nls.lin_solver.copy() nlst.lin_solver = self.ls2 return nlst
[docs] def clear_lin_solver(self, clear_constant_matrices=True): ElastodynamicsBaseTS.clear_lin_solver( self, clear_constant_matrices=clear_constant_matrices, ) self.ls1 = self.ls2 = None self.matrix1 = None
[docs] def step(self, ts, vec, nls, pack, unpack, prestep_fun): """ Solve a single time step. """ dt = ts.dt ut, vt, at = unpack(vec) nlst1 = self.create_nlst1(nls, dt, ut, vt, at) ut1 = nlst1(ut) vt1 = nlst1.v(ut1) at1 = nlst1.a(vt1) ts.set_substep_time(0.5 * dt) vec1 = pack(ut1, vt1, at1) vec1 = prestep_fun(ts, vec1) nlst2 = self.create_nlst2(nls, dt, ut, ut1, vt, vt1) ut2 = nlst2(ut1) vt2 = nlst2.v(ut2) at2 = nlst2.a(vt2) ts.restore_step_time() vec2 = pack(ut2, vt2, at2) return vec2