operates on error indicator output variables and your remeshing rule
parameters (see
Creating a remeshing rule)
to determine an improved element size distribution for your mesh.
The sizing method calculates new element sizes during the adaptive remeshing
process.
Abaqus/CAE
applies the sizing method to a field of error indicator variables and their
corresponding base solution variables over the region defined by the remeshing
rule. The output of a sizing method is a set of scalar sizes located at the
nodes in the region defined by the remeshing rule.
Figure 1
illustrates the sizing operation.
Figure 1
shows the base solution and error indicator distributions after the first
remesh iteration. The sizing method determines that the element size should be
reduced in the region of greatest error indicator and increased in the region
of the lowest error indicator. The mesh that is generated from these target
element sizes is illustrated.
Characteristics of Error Indicators
The sizing method and parameter settings that you select have a significant
impact on how adaptive remeshing changes the error indicator distribution in
your model. You may, for example, choose a sizing method that aggressively
reduces error indicators only near a stress riser. In other cases, where the
global response of your structure is more important than local effects, you may
choose a sizing method that attempts to reduce the error indicators to a
uniform level throughout the region. To understand how the sizing methods
affect the error indicators, you should first understand typical
characteristics of the error indicator variables.
Figure 2
provides an illustration of an error indicator and corresponding base solution
distribution on a generalized slice through a model.
Figure 2
illustrates the following error indicator characteristics:
In regions where the value of the base solution is high, such as for
element “i” in
Figure 2,
error indicator values can be low relative to local values of the base
solution. In many cases you may want to use mesh refinement to drive these
error indicators even lower.
In regions where the base solution is low, such as for element “j” in
Figure 2,
error indicator values can be high relative to the local values of the base
solution. In many cases you may not be interested in obtaining an accurate
solution in these regions.
These characteristics can affect your decision on which sizing method to use
and what parameters to set in the sizing method.
Sizing Methods
Sizing methods employ the concept of an error target,
,
which is expressed in a normalized percentage form and which defines a general
goal
where is a
measure of the error indicator and is a
measure of the base solution. Based on your definition of the error targets
when you created the remeshing rule,
Abaqus/CAE
creates a size distribution that attempts to meet your error target in the
subsequent analysis job using the remeshed model. The specific meaning of an
error target depends on your choice of the sizing method.
Abaqus/CAE
provides two fundamental sizing methods: Minimum/maximum
control and Uniform error distribution. You can
also choose a third method, Default method and parameters,
which results in
Abaqus/CAE
choosing one of the fundamental sizing methods for you, based on your choice of
error indicator variable.
Minimum/Maximum Control
The minimum/maximum control method provides the most flexibility in the
remeshing of your model. This method has the following characteristics:
Two error indicator targets for controlling the sizing.
controls the sizing in regions where the base solution (such as stress) is
highest, and
controls the sizing in regions where the base solution is lowest.
A continuous variation in error targets between regions of high and low
base solution values, with a bias factor parameter provided to control the
variation.
To avoid excessive refinement at elements with a small base solution, a
global averaged element base is chosen when the element base solution is
smaller than the global averaged element base.
If singularities are present in the remeshing rule region, this method
will fail to satisfy the error target because the maximum base solution, which
occurs at the location of the singularity, is unbounded.
You can either allow
Abaqus
to choose the targets automatically, or you can specify the error targets.
Similarly, you can accept the default bias factor displayed by
Abaqus/CAE,
or you can specify a qualitative bias factor.
Allowing Abaqus/CAE to Choose the Error Targets
If you specify the minimum/maximum error control method without setting
error targets,
Abaqus/CAE
automatically chooses the error targets,
and .
Both targets are computed as a fraction of the error indicator result in the
previous remesh iteration analysis. Automatic error target reduction is a good
choice for mesh refinement studies, where you have no specific error target
goal but want to see the impact of mesh refinement on your results.
Specifying the Error Targets
As an alternative to automatic error target reduction, you can specify the
two error targets,
and .
Figure 2
illustrates these two locations.
is applied to element ,
and
is applied to element .
Using the value of the two error targets,
Abaqus/CAE
applies a sizing method that attempts to meet both
and
at their respective locations.
Bias Factor
You can use the bias factor definition in the remeshing rule to further
tune the distribution of sizing between maximum and minimum base solution
locations. The bias factor defines the gradient of the size distribution
between these two extremes in your remesh region, as shown in
Figure 3.
You can set this factor between two qualitative extremes, “weak” and
“strong.” At the weak extreme, element sizes will increase most quickly at
locations moving away from the maximum base solution. At the strong extreme,
element sizes will increase most slowly. The default setting is a bias toward
the strong extreme.
Uniform Error Distribution
The uniform error distribution method provides a single error indicator
target, ,
for controlling the sizing.
Abaqus/CAE
applies a sizing method such that the total error in the remeshing rule region
is distributed uniformly across all the elements and satisfies the given error
indicator target. This method attempts to satisfy the error indicator target
collectively for the whole remeshing rule region but not at every element.
Therefore, the presence of singularities will not prevent the adaptivity
process from achieving the error target.
Allowing Abaqus/CAE to Choose the Error Target
If you specify the uniform error distribution method without setting an
error target,
Abaqus/CAE
automatically chooses the error target, .
The target is computed as a fraction of the error indicator result in the
previous remesh iteration analysis. Automatic error target reduction is a good
choice for mesh refinement studies, where you have no specific error target
goal but want to see the impact of mesh refinement on your results.
Specifying the Error Target
As an alternative to the automatic error target reduction, you can specify
the single error target, .
When you use the uniform error distribution method,
Abaqus/CAE
compares the error target to a global norm of a normalized form of the error
indicator. Such an approach ensures a globally converging mesh within the
region.
Default Sizing Methods and Parameters
This method results in application of the Automatic error target
reduction form of either the Minimum/maximum
control or Uniform error distribution method,
with the method applied based on the error indicator variable according to
Table 1.
Table 1. Default sizing method for each error indicator.
Solution Quantity
Error indicator variable
Default sizing method
Element energy density
ENDENERI
Uniform error distribution
von Mises stress
MISESERI
Minimum/maximum control
Equivalent plastic strain
PEEQERI
Minimum/maximum control
Plastic strain
PEERI
Minimum/maximum control
Creep strain
CEERI
Minimum/maximum control
Heat flux
HFLERI
Uniform error distribution
Electric flux
EFLERI
Minimum/maximum control
Electric potential gradient
EPGERI
Minimum/maximum control
When your remeshing rule refers to multiple error indicators, sizing methods
will be applied independently to each error indicator variable with the
resulting local size based on the smallest size calculated from each sizing
method.
Example: Plate with a Circular Stress Riser
The difference between the basic behavior of the minimum/maximum control and
the uniform error distribution methods is illustrated by a simple example.
Figure 4
shows the stress result for a simple loading of a plate with a hole.
Minimum/Maximum Control
Figure 5
illustrates the adaptive mesh that was generated by
Abaqus/CAE
when the user selected the minimum/maximum control method and specified the two
error targets (
and ).
In this example =5%
and =1%,
and the mesh bias is set to the default setting. These settings result in a
mesh that focuses tightly around the hole, the stress riser, while
transitioning smoothly to a relatively coarse mesh away from the hole.
Uniform Error Distribution
Figure 6
illustrates the adaptive mesh that was generated by
Abaqus/CAE
when the user selected the uniform error distribution method and specified the
single uniform error indicator target ().
In this example =1%.
This setting results in a mesh that focuses around the hole, the stress riser,
while also refining the mesh in less stressed regions.
Impact of Additional Remeshing Rule Settings
You specify the sizing method when you create a remeshing rule, and the
sizing method calculates new element sizes during the adaptive remeshing
process. However, the following additional settings in the remeshing rule can
affect the mesh generated by
Abaqus/CAE,
regardless of the sizing method that you selected:
region selection,
step and frame selection,
size constraints,
approximate maximum number of elements, and
refinement and coarsening rate factors.
Region Selection
Sizing methods are defined across sets of elements, corresponding to the
regions over which the remeshing rules were applied in
Abaqus/CAE.
Within each set of elements,
Abaqus/CAE
applies the sizing operation to the error indicator variables specified in the
remeshing rule. The results of the sizing operation are based on the
extrapolation of whole element calculations to the nearest nodes, and the
results are node based.
Step and Frame Selection
Abaqus
applies sizing operations to error indicator variables from only the last
available frame in a specified step. See
Error Indicator Characteristics
for a discussion of how your selection of the step, frame, and error indicator
can affect your ability to capture the response in transient analyses.
Size Constraints
When you create the remeshing rule, you can constrain the sizing operation
from specifying elements greater than or less than size constraints that you
define for the remesh rule region.
Abaqus/CAE
provides default settings for these constraints.
The default minimum element size constraint is 1% of the default
boundary seed size for the part instance to which the remeshing rule is
applied.
The default maximum element size constraint is 10 times the default
boundary seed size for the part instance to which the remeshing rule is
applied.
Approximate Maximum Number of Elements
For a remeshing rule you can specify an approximate limit for the maximum
number of elements. By using this constraint, you can control the cost of your
analysis and ensure that unreasonably large meshes are not created. If the
target error requires more elements than the specified limit when this
constraint is defined,
Abaqus/CAE
will reduce the target error internally so that the generated elements
approximately satisfy the specified element count. The use of this constraint
may prevent an adaptivity process from achieving the error targets. By default,
this constraint is not active.
Refinement and Coarsening Rate Factors
The refinement and coarsening factors that you specify define a constraint
on the mesh size in terms of iteration to iteration changes to the mesh. These
factors modulate the aggressivity of the sizing methods. The refinement factor
controls the refinement of the mesh or the introduction of smaller elements.
The coarsening factor controls the coarsening of the mesh or the introduction
of larger elements.
Abaqus/CAE
provides default settings for these rate factors, which are designed to prevent
excessive coarsening or prohibitively expensive refinement in a single remesh
iteration.
The refinement factor can have a significant effect on the convergence of
the adaptive meshing procedure. Once you have settled on sizing method
parameters that work well for your application, you may be able to achieve
faster and more efficient mesh convergence by increasing the refinement factor.
In cases where your adaptivity process is not converging well, however, an
increased refinement factor could result in an excessive increase in elements
in a remesh iteration.
Reconciling Overlapping Remeshing Rules
Abaqus/CAE
imposes no restrictions on the region or the steps associated with your
remeshing rules. You can apply multiple remeshing rules and, hence, sizing
functions to the same region at the same time. Similarly, you can specify
remeshing rules that overlap one another. When
Abaqus/CAE
generates the new mesh, it considers all of the remeshing rules at all of the
locations and uses the smallest calculated element size to drive the meshing
algorithm.