Configuration-dependent loads (`structsolve.callbacks`)#

Calls to the callables of Analysis used by the non-linear solvers

The callables may depend on the load factor and on the configuration through their optional keyword arguments, which are passed only when the callable declares them explicitly in its signature (a **kwargs does not count), such that the callables written for the previous versions are called exactly as before:

inc: the current load factor, passed to calc_fint, calc_kC and calc_kG. It allows configuration-dependent loads, e.g. follower pressures, whose configuration-dependent part enters calc_fint and whose load stiffness enters calc_kC or calc_kG, both scaled by the load factor.
c: the current configuration, passed to calc_fext, which must then return the load vector of that configuration for a unit load factor. The arc-length solvers use it in the predictor and in the constraint instead of the constant reference vector calc_fext(inc=1.).
lbd: the current load factor, passed to calc_fext together with c, for a load vector that depends on the load factor as well, e.g. prescribed displacements imposed exactly as linear constraints. The keyword inc keeps its meaning, the factor that multiplies the returned vector, and the solvers keep calling calc_fext(inc=1., ...). When calc_fext declares lbd:
- the arc-length solvers pass lbd with c in the predictor and in the constraint;
- the arc-length solvers scale the displacements with the linear solution of the load vector at c = 0, lbd = 0, instead of the linear solution of the reference vector calc_fext(inc=1.), and the Newton-Raphson solver uses this linear solution as the predictor of its first step;
- the convergence criterion also considers the norm of the load factor times the load vector, see newton_raphson._check_convergence();
- calc_fext(c=c, lbd=lbd) is called at the configurations where the tangent stiffness matrix is updated, right after calc_kC and calc_kG at the same c and load factor, such that the callables may share their computations, and the last load vector evaluated is the reference of the convergence checks until the next update.
lbd is never passed by the linear static analysis, Analysis.static(NLgeom=False).

Notes#

With inc, the residual remains

\[\boldsymbol{R}(\boldsymbol{c}, \lambda) = \boldsymbol{F}_{int}(\boldsymbol{c}, \lambda) - \lambda \boldsymbol{F}_{ext}\]

with calc_kC + calc_kG its exact Jacobian \(\partial \boldsymbol{R}/\partial \boldsymbol{c}\).

With c, calc_fext(inc=1., c=c) is \(-\partial \boldsymbol{R}/\partial \lambda\) at \(c\).

With lbd, prescribed displacements imposed exactly as linear constraints \(\boldsymbol{c} = \boldsymbol{Z} \boldsymbol{c}_r + \boldsymbol{c}_{p0} + \lambda \boldsymbol{c}_{p1}\) give the reduced residual

\[\boldsymbol{R}_r(\boldsymbol{c}_r, \lambda) = \boldsymbol{Z}^\top \left( \boldsymbol{F}_{int}(\boldsymbol{Z} \boldsymbol{c}_r + \boldsymbol{c}_{p0} + \lambda \boldsymbol{c}_{p1}) - \lambda \boldsymbol{F}_{ext} \right)\]

which has \(-\partial \boldsymbol{R}_r/\partial \lambda = \boldsymbol{Z}^\top(\boldsymbol{F}_{ext} - \boldsymbol{K}_T \boldsymbol{c}_{p1})\), with \(\boldsymbol{K}_T\) evaluated at the current configuration and load factor, the vector that calc_fext(inc=1., c=c, lbd=lbd) must return. The convergence criterion also considers the norm of \(-\lambda \, \partial \boldsymbol{R}/\partial \lambda\), since under a prescribed displacement alone the reference vector and the reduced internal force vector both vanish at equilibrium, and the displacements of the arc-length solvers are scaled with the linear solution of \(-\partial \boldsymbol{R}/\partial \lambda\) at \(c = 0, \lambda = 0\), since the reference vector calc_fext(inc=1.) may then be null.

structsolve.callbacks.accepts_kwarg(func, name)#

Tell whether func declares the keyword argument name

The argument is declared when it is a named parameter of the signature, positional-or-keyword or keyword-only; a **kwargs does not count. A callable whose signature cannot be read by inspect.signature() does not declare it.

Parameters:

funccallable: Function or method to inspect.
namestr: Name of the keyword argument.

Returns:

acceptsbool: True when func has a parameter name that can be passed by keyword.

Configuration-dependent loads (structsolve.callbacks)#

Notes#

Configuration-dependent loads (`structsolve.callbacks`)#