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3D diagram of a rectangular block with a downward arrow indicating force or displacement (no text or symbols)

Figure 12.3.2–1 Proportional-loading, linear-response example: small deflection of a cantilever. 

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3D diagram of a curved structural component with directional arrows indicating force or movement (no text or symbols)

Figure 12.3.2–2 Monotonic response example: large deflection of a cantilever. Therefore, it is appropriate to base the remeshing on the value of the error indicator in the last increment for any choice of error indicator variable. # General response example Figure 12.3.2–3 illustrates a case where the loading characteristics change dramatically during the analysis. Your choice of error indicator in this case will depend on the material model. The element energy density error indicator, ENDENERI, will account for the complexity of load history (and lead to an adapted mesh that provides an accurate solution through the analysis) regardless of the material type. If plastic deformation occurs, you also have the option to use the equivalent plastic strain, PEEQERI, or plastic strain, PEERI, error indicators. Plastic strain and the plastic strain error indicator generally do not capture history effects; for example, they do not account for peak straining in models undergoing symmetric strain reversals. This example, however, involves no strain reversals; therefore, PEERI would be a valid error indicator choice. 

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3D diagram showing three spheres on a rectangular block with directional arrows indicating motion (no text or symbols)

Figure 12.3.2–3 General response example: block subjected to a rigid indenter. # General multistep response example: die forming and springback Figure 12.3.2–4 illustrates a further generalization of a general response. Here, a forming operation is simulated, and different steps are used for different stages of the operation. 

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Step 1: Clamp the workpiece Step 2: Forming Step 3: Springback Punch Die Workpiece Die

Figure 12.3.2–4 General multistep response example. In this case the response of the model varies from step to step. You will typically want the error indicator to capture the extreme of the model’s response to the load history adequately. However, you do not know if any particular increment captures this extreme. Therefore, you should select an error indicator variable that records the solution history. # 12.3.3 SOLUTION-BASED MESH SIZING Products: Abaqus/Standard Abaqus/CAE # References • “Adaptive remeshing: overview,” Section 12.3.1 • “Selection of error indicators influencing adaptive remeshing,” Section 12.3.2 • “Understanding ALE adaptive meshing,” Section 14.6 of the Abaqus/CAE User’s Guide • “Advanced meshing techniques,” Section 17.14 of the Abaqus/CAE User’s Guide # Overview Solution-based mesh sizing: • is performed in Abaqus/CAE; and • operates on error indicator output variables and your remeshing rule parameters (see “Creating a remeshing rule,” Section 17.21.1 of the Abaqus/CAE User’s Guide) to determine an improved element size distribution for your mesh. # Basic operation of the sizing method 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 12.3.3–1 illustrates the sizing operation. Figure 12.3.3–1 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 12.3.3–2 provides an illustration of an error indicator and corresponding base solution distribution on a generalized slice through a model. 

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```mermaid graph TD A["Remesh iteration i"] --> B["Base solution distribution"] B --> C["Lowest base solution"] B --> D["Lowest error indicator"] B --> E["Greatest error indicator"] B --> F["Error indicator distribution"] F --> G["Sizing method"] G --> H["Target element size in the next remesh iteration"] H --> I["Mesh creation"] I --> J["Remesh iteration i+1"] ```

Figure 12.3.3–1 Sizing method operation and interaction with meshing. Figure 12.3.3–2 illustrates the following error indicator characteristics: • In regions where the value of the base solution is high, such as for element “i” in Figure 12.3.3–2, 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. base solution variable error indicator 

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| position | solution | | -------------- | -------- | | element i | C_b_i | | element j | C_b_j |

Figure 12.3.3–2 Error indicator and base solution distribution. • In regions where the base solution is low, such as for element $^ { \ast \mathrm { , \ast \mathrm { , } } }$ in Figure 12.3.3–2, 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 $$ \left(\frac {\mathbf {c _ {e}}}{\mathbf {c _ {b}}}\right) \times 1 0 0 \Rightarrow \eta , $$ where $\mathbf { c _ { e } }$ is a measure of the error indicator and $\mathbf { c _ { b } }$ 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. Abaqus/CAE Usage: Mesh module: Create Remeshing Rule: Sizing Method # 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. $\eta _ { \mathrm { m a x } }$ controls the sizing in regions where the base solution (such as stress) is highest, and $\eta _ { \mathrm { m i n } }$ 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. Abaqus/CAE Usage: Mesh module: Create Remeshing Rule: Sizing Method: Method: choose Minimum/Maximum control 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, $\eta _ { \mathrm { m a x } }$ and $\eta _ { \mathrm { m i n } }$ . 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. Abaqus/CAE Usage: Mesh module: Create Remeshing Rule: Sizing Method: Error Targets; choose Automatic error target reduction Specifying the error targets As an alternative to automatic error target reduction, you can specify the two error targets, $\eta _ { \mathrm { m a x } }$ and $\eta _ { \mathrm { m i n } }$ . Figure 12.3.3–2 illustrates these two locations. $\eta _ { \mathrm { m a x } }$ is applied to element $i ,$ and $\eta _ { \mathrm { m i n } }$ is applied to element $j .$ . Using the value of the two error targets, Abaqus/CAE applies a sizing method that attempts to meet both $\eta _ { \mathrm { m i n } }$ and $\eta _ { \mathrm { m a x } }$ at their respective locations. Abaqus/CAE Usage: Mesh module: Create Remeshing Rule: Sizing Method: Error Targets; choose Fixed error targets; enter the maximum base solution error indicator target, $\eta _ { \mathrm { m a x } }$ , and the minimum base solution error indicator target, . # 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 12.3.3–3. 

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Weak setting Maximum Base Solution Strong setting Minimum Base Solution

Figure 12.3.3–3 The impact of the bias factor on the element size distribution. 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. Abaqus/CAE Usage: Mesh module: Create Remeshing Rule: Sizing Method: Mesh Bias; drag the slider to a setting between Weak and Strong # 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. Abaqus/CAE Usage: Mesh module: Create Remeshing Rule: Sizing Method: Method: choose Uniform error distribution 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. Abaqus/CAE Usage: Mesh module: Create Remeshing Rule: Sizing Method: Error Targets; choose Automatic error target reduction # 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. Abaqus/CAE Usage: Mesh module: Create Remeshing Rule: Sizing Method: Error Targets: choose Fixed error target; enter the error indicator target, # 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 12.3.3–1. Table 12.3.3–1 Default sizing method for each error indicator.

Solution Quantity	Error indicator variable	Default sizing method
Element energy density	ENDENERI	Uniform error distribution
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. Abaqus/CAE Usage: Mesh module: Create Remeshing Rule: Sizing Method: Method: choose Default methods and parameters # 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 12.3.3–4 shows the stress result for a simple loading of a plate with a hole.