# <StaticODESolver />¶

Implementation of a Newton-Raphson static ODE solver.

The solver does a serie of Newton-Raphson iterations where at each iteration $$k$$, the following linear system is solved:

$\begin{split}\boldsymbol{K}(\boldsymbol{u}^k) \cdot \delta \boldsymbol{u}^{k+1} &= - \boldsymbol{R}(\boldsymbol{u}^k) \\ \boldsymbol{u}^{k+1} & = \boldsymbol{u}^k + \delta \boldsymbol{u}^{k}\end{split}$

where the stiffness matrix $$\boldsymbol{K}$$ is the derivative of the residual with respect to the displacement, i.e. $$\boldsymbol{K} = \frac{\partial \boldsymbol{R}}{\partial \boldsymbol{u}}$$ and is typically accumulated by the addKtoMatrix method of forcefields. If an iterative linear solver is used, it is possible that the stiffness matrix is never accumulated, instead the operation $$\boldsymbol{K}(\boldsymbol{u}^k) \cdot \delta \boldsymbol{u}^{k+1}$$ is done through the addDForce method of forcefields. The residual vector $$\boldsymbol{R}(\boldsymbol{u}^k)$$ is accumulated by the addForce method of forcefields.

Attribute

Format

Default

Description

printLog

bool

false

Output informative messages at the initialization and during the simulation.

newton_iterations

int

1

Number of newton iterations between each load increments (normally, one load increment per simulation time-step).

correction_tolerance_threshold

float

1e-5

Relative convergence criterion: The newton iterations will stop when the norm of correction $$\frac{|\delta \boldsymbol{u}^{k}|}{\sum_{i=0}^k|\delta \boldsymbol{u}^{i}|}$$ reaches this threshold.

residual_tolerance_threshold

float

1e-5

Relative convergence criterion: The newton iterations will stop when the relative norm of the residual $$\frac{|\boldsymbol{R}_k|}{|\boldsymbol{R}_0|}$$ at iteration k is lower than this threshold. Use a negative value to disable this criterion.

absolute_residual_tolerance_threshold

float

1e-15

Absolute convergence criterion: The newton iterations will stop when the absolute norm of the residual $$|\boldsymbol{R}_k|$$ at iteration k is lower than this threshold. This criterion is also used to detect the absence of external forces and skip useless Newton iterations. Use a negative value to disable this criterion.

pattern_analysis_strategy

option

BEGINNING_OF_THE_TIME_STEP

Define when the pattern of the system matrix should be analyzed to extract a permutation matrix. If the sparsity and location of the coefficients of the system matrix doesn’t change much during the simulation, then this analysis can be avoided altogether, or computed only one time at the beginning of the simulation. Else, it can be done at the beginning of the time step, or even at each reformation of the system matrix if necessary. The default is to analyze the pattern at each time step.

Options:
• NEVER

• BEGINNING_OF_THE_SIMULATION

• BEGINNING_OF_THE_TIME_STEP (default)

• ALWAYS

linear_solver

LinearSolver

None

Linear solver used for the resolution of the system. Will be automatically found in the current context node if none is supplied.

converged

bool

N/A

Whether or not the last call to solve converged.

## Quick example¶

XML

<Node>
<StaticODESolver newton_iterations="10" correction_tolerance_threshold="1e-8" residual_tolerance_threshold="1e-8" printLog="1" />
<LLTSolver backend="Pardiso" />
</Node>


Python

node.addObject('StaticODESolver', newton_iterations=10, correction_tolerance_threshold=1e-8, residual_tolerance_threshold=1e-8, printLog=True)

• squared_residuals (list [numpy.double]) – The list of squared residual norms ($$|r|^2$$) of every newton iterations of the last solve call.
• squared_initial_residual (numpy.double) – The initial squared residual ($$|r_0|^2$$) of the last solve call.