COMSOL 6.3 - Fully Coupled

The attribute node (

) uses a damped version of Newton’s method or, for stationary problems, a double dogleg method to handle parameters for a fully coupled solution approach. It can be used with the Stationary Solver and the Time-Dependent Solver.

An alternative to the fully coupled approach is given by the segregated solver, which you control with the Segregated node. Although several and attribute subnodes can be attached to an operation node, only one can be active at any given time.

For more information about the settings below, see:

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The Fully Coupled Attribute and the Double Dogleg Method

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Damped Newton Methods

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Termination Criterion for the Fully Coupled and Segregated Attribute Nodes

General

Select a for linear systems that appear in the corresponding solver configuration. The available solvers are attribute nodes of the types Direct and Iterative.

Method and Termination

Select a to control which damping factor to use in the damped Newton iterations. Select:

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to let the solver automatically determine a damping factor in each iteration of Newton’s method. Go to Automatic (Newton) and Automatic Highly Nonlinear (Newton) for settings.

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to let the solver use a globalization method that uses information about the nonlinear residual as well as the system matrix to estimate the damping that minimizes the residual. Go to Backtracking (Newton) for settings.

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to manually specify a constant damping factor that is used in all iterations of Newton’s method. Go to Constant (Newton) for settings.

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if the solver does not converge with first. It is similar to but this method can make the solver more careful when solving highly nonlinear problems. Go to Automatic (Newton) and Automatic Highly Nonlinear (Newton) for settings.

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For stationary problems, choose to use the double dogleg nonlinear solver. Go to Double Dogleg for settings.

Automatic (Newton) and Automatic Highly Nonlinear (Newton)

For or enter values or expressions for:

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, to specify a damping factor for the first Newton iteration. The default value is 1 for Automatic (Newton) and 1.0·10−4 for Automatic highly nonlinear (Newton).

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, to specify the smallest allowed damping factor. The default value is 1.0·10−4 for Automatic (Newton) and 1.0·10−8 for Automatic highly nonlinear (Newton).

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, to specify a factor that limits how much the damping factor is allowed to change in a Newton iteration. The damping factor can change up or down by at most this factor. The default is 10.

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, to specify the maximum for the allowed absolute increase in the damping factor for a Newton iteration, as opposed to the value, which limits how much the damping factor can change relative to the current damping factor. The default value of 1 means that this setting is not active. A smaller value imposes a restriction (0–1 is the valid range for this value).

The automatic Newton solver can get stuck at the minimum damping factor resulting in no convergence. If you enable the , the Newton solver can try additional steps starting with a damping factor equal to the value of the . This might help to recover from a state where the solver is stuck at the minimum damping factor.

Select an option from the list: (the default), , or .


	The default, Automatic, is equivalent to On for stationary problems and Off for time-dependent problems. For stationary parametric continuation problems, Automatic corresponds to On when solving for the first parameter value and Off when solving for subsequent parameter values.

Choose if a damping factor smaller than the is required. The nonlinear solver then terminates. For , the nonlinear solver takes a Newton step using the constant damping factor, which is defined in the field. The default is 0.75 when or is selected.

From the list, available for stationary and parametric studies, choose (the default), , or . is the behavior in versions of the COMSOL Multiphysics software earlier than version 6.2. If you choose , the automatic scale factors are not taken into account when updating the weights. When the option is selected as the strategy for the following algorithm is used:

For the calculation of the weights U, the current, possibly damped, approximation to the solution is used. The weights are computed as

, where Sj is the average of

for all DOFs i for a fixed j, times a factor equal to 10−5 for highly nonlinear problems or 0.1 otherwise.

The newly computed weights,

, are compared to the weights currently used, Wi,j, using a threshold value Wthr. This value is given by the user as the (default value: 100). If any of the weights satisfies

, the weights are updated to.

When the weights are updated the Newton algorithm is restarted, old step-size norms are cleared, and a new damping factor will be used. The new damping factor is controlled by , where the strategies are (the default) or . For there is an option to select a (default value: 1). For the damping factor will be the .

Then continue with the Termination Technique and Termination Criterion settings that follow.

Backtracking (Newton)

This method, inexact Newton backtracking (INB), uses information about the nonlinear residual as well as the system matrix to estimate the damping that minimizes the residual. This way a faster convergence can be achieved as compared with using a constant damping factor. For more information about INB, see Ref. 15. For more information about the choice of parameters, see Ref. 16. When used for a segregated solver, backtracking is applied for each segregated step individually. Two types of backtracking methods are supported: one using a full estimate and a method using constant damping factors whenever the step taken by the solver does not improve the residual, which makes that method computationally effective. Models that use a fully coupled solver and that only need mild damping are examples of models that are well suited for the constant step algorithm. For :

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First select one of the two methods mentioned above from the Backtracking method list: (the default) or .

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If you chose , select the checkbox to let the algorithm always determine and apply a damping at least once per nonlinear iteration. This is done regardless of whether the general criteria for backtracking are fulfilled, so even if the solver is converging nicely. For certain models this can improve the stability of the solvers.

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If you chose , enter a value between 0 and 1 in the field (default: 0.5). The method evaluates the residual for every backtracking step it takes. Choosing appropriate values for the , , and can help to reduce the number of backtracking steps.

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Enter values for the (default: 1) and the (default: 0.1) to specify the maximally and minimally allowed damping that is applied during one nonlinear iteration. Especially at the beginning of a solving process it can happen that the convergence is rather slow and that the backtracking algorithm would determine an arbitrarily small damping, effectively trapping the solvers there. In order to prevent this behavior, you can set a minimum damping factor, forcing the solvers to advance. This approach can also be used to accelerate the convergence for problems where generally only mild damping is needed.

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All other settings are identical to those for Constant (Newton) below (except that there is no setting).

Then continue with the Termination Technique — Backtracking (Newton) and Constant (Newton) and Termination Criterion settings that follow.

Constant (Newton)

For :

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Enter a value for the to specify a constant damping factor for Newton’s method. The default is 1.

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With a Time-Dependent Solver, select the checkbox to force the nonlinear solver to terminate as soon as the convergence is estimated to be too slow. The default is 0.9. Enter a limit on the convergence rate in the field as required. The nonlinear convergence rate is estimated as γ = (en/e1)(1/(n−1)), where en is the error estimate for iteration n. This can be seen as the average convergence rate after n steps (n>1). If γ \ γlimit (γlimit is the limit on nonlinear convergence rate), then the nonlinear solver will terminate (as if it fails). This means that the current time step will be disqualified, and a new nonlinear solve attempt will be performed with a reduced time step. For problems where the convergence rate can be slow, this option can be used to avoid unnecessary nonlinear iterations (because the solver will in those cases not converge anyways within the allotted iterations specified in the field).

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With a Time-Dependent Solver, choose a : (the default), , or :

reuses the Jacobian for several nonlinear systems whenever deemed possible.

computes a new Jacobian for all iterations of Newton’s method.

computes a new Jacobian on the first iteration of each time step.

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With a Time-Dependent Solver, by default, a stricter tolerance is used at each Newton iteration step when Jacobian is not updated. This approach typically leads to more robust time stepping. Therefore, more nonlinear iterations might be required in each time step, and more Jacobian updates might be needed. To use the minimal Jacobian update used in earlier versions of the COMSOL Multiphysics software, select the checkbox.

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With a Stationary Solver or a parametric solver, choose a : , (the default), or :

reuses the Jacobian for several nonlinear systems whenever deemed possible.

computes a new Jacobian for all iterations of Newton’s method.

computes a new Jacobian on the first iteration of each parameter step.

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The checkbox, available for stationary and parametric studies, is selected by default. When selected it is the behavior in earlier versions of COMSOL Multiphysics; see Termination Criterion for the Fully Coupled and Segregated Attribute Nodes. If the automatic scale factors have changed by two orders of magnitude this will be indicated in the window. If you clear this checkbox, the automatic scale factors are not taken into account when updating the weights.

Then continue with the Termination Technique — Backtracking (Newton) and Constant (Newton) and Termination Criterion settings that follow.

Double Dogleg

If is selected for stationary problems:

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Enter a value for the , to specify a damping factor for the first Newton iteration. The default value is 1.0·10−4.

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Choose a : or . scales the equations based on the field-wise sizes of the initial residual. If is selected, the algorithm terminates on the relative residual based on the initial residual.

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The double dogleg solver can restart, if the solver iterations are considered as stagnated. The stagnation is defined by a given number of consecutive iterations that all have a Newton damping factor less than 0.1 and a step size smaller than 0.1 times the tolerance. By default, the restart is active with 7 iterations before the restart. To turn off the restart, clear the checkbox. The number of iterations before restart must be a positive integer.

Continue with the Termination Technique settings that follow.

Termination Technique

For any , select a to control how the Newton iterations are terminated. Select:

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to terminate the Newton iterations when the estimated relative error is smaller than a specified tolerance. Then enter the to limit the number of Newton iterations. When the maximum number of iterations have been performed, Newton’s method is terminated even if the tolerance is not fulfilled.

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to terminate the Newton iterations when the estimated tolerance is smaller than a specified tolerance or after a specified number of iterations, whichever comes first. Then enter the to specify a fixed number of iterations to perform.

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If or are set as the , then enter a to modify the tolerance used for termination of the Newton iterations. The actual tolerance used is this factor times the value specified in the field in the sections of the Stationary Solver and Time-Dependent Solver.

Termination Technique — Backtracking (Newton) and Constant (Newton)

When or is the nonlinear method, you can also choose to terminate the Newton iterations after a fixed number of iterations. Enter the to specify a fixed number of iterations to perform.

Termination Criterion

For all termination techniques except , the controls how the Newton iterations are terminated. Select:

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to terminate the Newton iterations on a solution-based estimated relative error.

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to terminate the Newton iterations on a residual-based estimated relative error.

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to terminate the Newton iterations on the minimum of the solution-based and residual-based estimated relative errors. Enter a scalar multiplying the residual error estimate. The default is 1000.

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to terminate the Newton iterations on a solution-based estimated relative error and a residual-based estimated relative error (that is, a more stringent termination criterion). Enter a scalar multiplying the residual error estimate. The default is 1000.

For time-dependent solvers and when or is selected as , the convergence check is applied before and after solving for the linear system. Doing so avoids unnecessary resources spent on solving the linear system when the solution is already converged. It can also avoid stagnation issues in iterative linear solvers when started with an initial guess too close to the true solution. For segregated steps the termination criterion of the segregated solver can enforce solving the linear system in certain cases, despite residual termination wanting to terminate (see Segregated).

Adaptive Tolerance in the Linear Solver — Backtracking (Newton) and Constant (Newton)

When or is the nonlinear method, select the checkbox to activate such an adaptive tolerance. Based on an estimate of the global convergence, the algorithm then tries to find an optimal tolerance for the linear solver. Doing so can avoid expensive oversolving of the linear system when still far away from global convergence. When selected, also specify these additional settings:

First, choose a convergence estimator from the list:

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(the default): It uses full vectorial information of the linear and nonlinear residuals (needs to compute the linear residual).

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: It uses only the norms of the linear and nonlinear residuals (needs to compute the linear residual).

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: It uses norms of only the nonlinear residuals (no need of the linear residual).

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: It uses CFL suppression (see the checkbox below) to the initial tolerance without using an additional algorithm for determining the adaptive tolerance. The resulting tolerance will be interpolated between the initial tolerance and the target tolerance based on the current value of the CFL number. The interpolation is done logarithmically in the CFL number and logarithmically in the tolerance. This option is only available in model using a pseudo time-stepping method. The only additional setting is an initial tolerance (see below). For other models using a stationary solver, the is available instead (see below).

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: It uses a suppression mechanism based on the nonlinear error estimate (for stationary solvers not using pseudo time stepping only).

Also specify these additional settings:

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In model using a pseudo time-stepping method, the checkbox is selected by default. If selected, the algorithm will apply a suppression to the adaptive tolerance based on the progression of the CFL number. Doing so insures that the adaptive tolerance reaches the user-requested tolerance when the model converges, at the cost of a slight increase in the local number of linear iterations (that is, the number of linear iterations needed at a constant value of the CFL number). Given the increased stability, many models will converge faster using this option, thereby reducing the global amount of linear and nonlinear iterations.

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For other models, the checkbox is selected by default. If selected, the algorithm will apply a suppression to the adaptive tolerance based on the nonlinear error estimate.

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An initial tolerance in the field (default: 0.1).

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A maximal tolerance in the field (default: 0.9).

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A threshold for safeguard purposes in the field (default: 0.1). This is the value for the tolerance above which the safeguards are applied. The safeguards are there to prevent a too fast decrease of the tolerance. Selecting a too low value can prevent the adaptive tolerance to become smaller fast enough, leading to convergence issues. Selecting a too high value, in turn, can allow for an accidentally small tolerance, leading to resources wasted on unnecessary precision.

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If you chose , also specify a damping exponent between 1 and 2 and a damping factor between 0 and 1 in the (default: 2) field and (default: 1) field, respectively, to define a nonlinear residual.

The full and partial error estimates usually have a similar behavior whereas the tolerance determined by the nonlinear residual tends to be larger, especially when the damping is different from 1.

Stabilization and Acceleration — Backtracking (Newton) and Constant (Newton)

When or is the nonlinear method, you can select one of the following methods for stabilization and acceleration of the nonlinear convergence from the list:

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(the default) to not use any stabilization or acceleration method.

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to use a pseudo time-stepping method to stabilize convergence toward steady state for a stationary solver. Pseudo time stepping is not available for time-dependent solvers. See Pseudo Time Stepping for more information. For the pseudo time-stepping method, specify the following controller parameters:

. Choose (the default) or . The simple controller corresponds to the PID controller in versions of the COMSOL Multiphysics software earlier than version 6.2. The interpolated controller blends different types of controllers when the error reaches the .

. The default is 5.

. The default is 10,000. The solver does not converge unless the target CFL number is reached.

The checkbox is selected by default. This setting controls how the CFL number is changed once the is reached. If you clear this checkbox, the CFL number can continue to increase until the error tolerance is fulfilled.

. The default is 0.65.

. The default is 0.05.

. The default is 0.1.

Select the checkbox to activate Anderson acceleration for the pseudo time-stepping method. See below for more information about the settings for Anderson acceleration.

Select the checkbox to override updates of the Jacobian for the segregated steps. The value (default: 100), which is the value of the CFL number where overriding of the Jacobian update becomes active. That is, the overriding becomes active for larger CFL numbers than the threshold. From the list, choose (the default) or , which updates the Jacobian at least once and then only when the nonlinear solver fails during parameter stepping. It reuses the Jacobian for several nonlinear systems whenever deemed possible.

Select the checkbox to enable that the residual smoothing term is applied to the residual on each iteration in the pseudo time-stepping solver. The value in the field (default: 0.1) controls the magnitude of the residual smoothing term.

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, which is a nonlinear convergence acceleration method that uses information from previous Newton iterations to accelerate convergence. The Anderson acceleration method is primarily intended for acceleration of nonlinear iterations in transport problems involving, for example, crosswind diffusion stabilization. It is useful is for solving linear or almost linear problems using the segregated solver, where convergence can be improved and the performance increased. You can control the number of iteration increments to store using the field (default: 10) and the mixing parameter as a value between 0 and 1 using the field (default: 1.0). The field (default 0) contains the number of iterations between pseudo time stepping becomes inactive and Anderson acceleration becomes active. Enter a threshold value in the field (default: 10). This threshold value controls if the Anderson step or the Newton step is used in the nonlinear step. If the norm of the new step is less than the threshold times the norm of the previous step, the Anderson step is used. Otherwise, the Newton step is used. Lowering the value for the can improve robustness at the price of performance.

When used for pseudo time stepping, you can enter a value (default: 100), which is the value of the CFL number where Anderson acceleration becomes active and pseudo time stepping becomes inactive. You can also choose (the default) or from the list. Select if the solver should set the target CFL number once the threshold is reached, or select if the PID controller should still control the CFL number based on the error estimate.

Results while Solving

See Results While Solving in the Common Study Step Settings section. Also see Getting Results While Solving.


	For an example using a Stationary Solver, see Tubular Reactor with Nonisothermal Cooling Jacket: Application Library path COMSOL_Multiphysics/Chemical_Engineering/tubular_reactor. For an example using a Time-Dependent Solver, see Effective Diffusivity in Porous Materials: Application Library path COMSOL_Multiphysics/Diffusion/effective_diffusivity.