Reaching Equilibrium

A model may need to reach mechanical equilibrium after gravity initialization, loading, excavation, material-point generation, conversion from zones, or another change in the model configuration. For a quasi-static calculation, equilibrium represents a state in which the applied loads, internal forces, and boundary reactions are sufficiently balanced and the remaining material motion is negligible.

MPoint advances the mechanical solution explicitly through a sequence of calculation cycles. The commands model cycle and model step may be used to perform a specified number of cycles. In most quasi-static analyses, however, it is preferable to use model solve so that cycling continues until a selected equilibrium criterion is satisfied.

For example:

model solve ratio-average 1e-4

The appropriate solve limit depends on the model, loading sequence, material behavior, required accuracy, and quantities of interest. A single force-ratio target should therefore not be treated as a universal measure of equilibrium.

Convergence Measures

At each calculation cycle, material-point mass, momentum, and forces are transferred to the background-grid nodes. The resulting nodal forces are used to calculate acceleration and update the motion of the material points. A node is locally balanced when the sum of the forces acting on it approaches zero.

The maximum out-of-balance force is the largest remaining force component after all force contributions have been assembled. Although this quantity is useful for observing whether the residual force is decreasing, it has units of force and depends on the model dimensions, grid spacing, material properties, and loading magnitude. It should not generally be used alone to compare convergence between different models.

Force ratios provide normalized measures of equilibrium. Using the norm

\[\left\langle \mathbf{v} \right\rangle = |v_x|+|v_y|+|v_z|,\]

the local force ratio at background node \(j\) may be expressed generally as

\[r_j = \frac{ \left\langle\sum_k\mathbf{f}_{jk}\right\rangle }{ \sum_k\left\langle\mathbf{f}_{jk}\right\rangle },\]

where \(\mathbf{f}_{jk}\) represents an individual force contribution acting at the node. The numerator represents the remaining out-of-balance force, while the denominator represents the magnitude of the forces being applied.

The commonly used solve measures include:

  • \(ratio-average\) — measures the overall force imbalance throughout the model. This is the default criterion that will be used if a model solve command is given. The default ratio is 1e-5;

  • \(ratio-local\) — represents the largest local force ratio;

  • \(ratio-maximum\) — compares the maximum unbalanced force with a representative total force; and

  • \(unbalanced-maximum\) — uses the maximum residual force directly.

The average force ratio is generally useful for monitoring the overall model response, but it can conceal a localized region that is still moving or adjusting. This may occur near boundaries, interfaces, contact regions, highly deformable material, or regions with large differences in stiffness. The maximum or local ratio should therefore be examined when local behavior is important.

Monitoring the Solution

Equilibrium should be evaluated using both convergence measures and the physical response of the model. Useful model histories include:

history interval 100

model history name 'Average Force Ratio' ...
mechanical ratio-average

model history name 'Maximum Force Ratio' ...
mechanical ratio-maximum

model history name 'Maximum Unbalanced Force' ...
mechanical unbalanced-maximum

Material-point displacement and velocity should also be monitored at locations important to the analysis. For example:

[monitor_position = vector(5.0,10.0)]

mpoint history name 'Monitor Displacement' ...
displacement-y position [monitor_position]

mpoint history name 'Monitor Velocity' ...
velocity-y position [monitor_position]

A model may be considered sufficiently equilibrated when:

  • the selected force ratio has decreased below an acceptable target;

  • material-point velocities have become small and are not increasing;

  • displacements, stresses, and other quantities of interest have stabilized;

  • the model no longer exhibits significant oscillation or progressive movement; and

  • applied loads are consistent with the calculated reactions and internal response.

The histories should be examined over a sufficient number of cycles. A temporarily small force ratio does not necessarily indicate equilibrium if displacement or velocity continues to change.

Because the MPoint background grid does not permanently follow the material, a particular background node may interact with different material points as the calculation proceeds. Histories of representative material points are therefore especially useful for evaluating whether the physical material has stopped moving.

Using Damping

Numerical damping may be used to remove kinetic energy and help a quasi-static model approach equilibrium. By default, local damping is active with a default coefficient of 0.8 (see Mechanical Damping). A different local damping coefficient may be assigned to the MPoint background nodes using:

mpoint node damping local 0.6

Note that for a dynamic model the default damping is 0 (see Solution Control).

Combined damping may also be used:

mpoint node damping combined 0.8

The damping coefficient should be selected according to the response of the model. Too little damping may result in persistent oscillation and slow convergence, whereas excessive damping may cause the model to approach equilibrium unnecessarily slowly or obscure important transient behavior.

Loads should also be applied gradually when an instantaneous change would introduce an unwanted dynamic disturbance. Gravity, footing loads, excavation, or changes in boundary conditions may be divided into stages, with the model partially or fully solved after each increment.

Note

Numerical damping is appropriate for obtaining a quasi-static state. This should not be used to suppress motion that is part of the physical dynamic problem being simulated. Usually a small amount of Rayleigh Damping is used in dynamic problems.

Models That Do Not Reach Equilibrium

Not every MPoint calculation is expected to reach static equilibrium. Problems involving landslides, granular collapse, penetration, impact, post-failure motion, material discharge, or continuing flow may retain significant velocity and deformation throughout the analysis.

For these problems, the calculation should normally be advanced to a specified physical time or number of cycles rather than stopped using a force-ratio criterion. For example:

model solve time 5.0

The model response should then be evaluated using displacement, velocity, energy, stress, strain, and material-point trajectories. A force ratio that does not decrease toward a small value may indicate continuing physical motion rather than failure of the numerical solution.

Practical Equilibrium Procedure

A general procedure for establishing and checking equilibrium in MPoint is:

  1. Apply appropriate boundary conditions and verify that they cover the required background nodes.

  2. Establish the initial stress state where applicable.

  3. Apply gravity and other loads gradually when sudden loading is not part of the intended physical problem.

  4. Use suitable background-node damping for a quasi-static calculation.

  5. Record the average and maximum force ratios.

  6. Record displacement and velocity histories at representative material points.

  7. Solve to a selected force-ratio limit.

  8. Confirm that the histories have stabilized and inspect regions of localized motion, plastic deformation, contact, or stiffness contrast.

  9. Repeat the solve using a stricter criterion when the results of interest continue to change significantly.

A model should be considered equilibrated only when the numerical convergence measures and the physical response both indicate that additional cycling would not materially change the results of interest.