Mass Scaling

Introduction

The explicit dynamics procedure is typically used to solve two classes of problems: transient dynamic response calculations and quasi-static simulations involving complex nonlinear effects (most commonly problems involving complex contact conditions). Because the explicit central difference method is used to integrate the equations in time (see Explicit Dynamic Analysis), the discrete mass matrix used in the equilibrium equations plays a crucial role in both computational efficiency and accuracy for both classes of problems. When used appropriately, mass scaling can often improve the computational efficiency while retaining the necessary degree of accuracy required for a particular problem class. However, the mass scaling techniques most appropriate for quasi-static simulations may be very different from those that should be used for dynamic analyses.

Quasi-Static Analysis

For quasi-static simulations incorporating rate-independent material behavior, the natural time scale is generally not important. To achieve an economical solution, it is often useful to reduce the time period of the analysis or to increase the mass of the model artificially (“mass scaling”). Both alternatives yield similar results for rate-independent materials, although mass scaling is the preferred means of reducing the solution time if rate dependencies are included in the model because the natural time scale is preserved.

Mass scaling for quasi-static analysis is usually performed on the entire model. However, when different parts of a model have different stiffness and mass properties, it may be useful to scale only selected parts of the model or to scale each of the parts independently. In any case, it is never necessary to reduce the mass of the model from its physical value, and it is generally not possible to increase the mass arbitrarily without degrading accuracy. A limited amount of mass scaling is usually possible for most quasi-static cases and will result in a corresponding increase in the time increment used by Abaqus/Explicit and a corresponding reduction in computational time. However, you must ensure that changes in the mass and consequent increases in the inertial forces do not alter the solution significantly.

Although mass scaling can be achieved by modifying the densities of the materials in the model, the methods described in this section offer much more flexibility, especially in multistep analyses.

See Rolling of thick plates for a discussion of using mass scaling in a quasi-static analysis.

Dynamic Analysis

The natural time scale is always important in dynamic analysis, and an accurate representation of the physical mass and inertia in the model is required to capture the transient response. However, many complex dynamic models contain a few very small elements, which will force Abaqus/Explicit to use a small time increment to integrate the entire model in time. These small elements are often the result of a difficult mesh generation task. By scaling the masses of these controlling elements at the beginning of the step, the stable time increment can be increased significantly, yet the effect on the overall dynamic behavior of the model may be negligible.

During an impact analysis, elements near the impact zone typically experience large amounts of deformation. The reduced characteristic lengths of these elements result in a smaller global time increment. Scaling the mass of these elements as required throughout the simulation can significantly decrease the computation time. For cases in which the compressed elements are impacting a stationary rigid body, increases in mass for these small elements during the simulation will have very little effect on the overall dynamic response.

Mass scaling for truly dynamic events should almost always occur only for a limited number of elements and should never significantly increase the overall mass properties of the model, which would degrade the accuracy of the dynamic solution.

See Impact of a copper rod for a discussion of using mass scaling in a dynamic analysis.

Stable Time Increments

Throughout this section the term “element stable time increment” refers to the stable time increment of a single element. The term “element-by-element stable time increment” refers to the minimum element stable time increment within a specific element set. The term “stable time increment” refers to the stable time increment of the entire model, regardless of whether the global estimator or the element-by-element estimator is used.

Introducing Mass Scaling into a Model

Two types of mass scaling are available in Abaqus/Explicit: fixed mass scaling and variable mass scaling. These two types of mass scaling can be applied separately, or they can be applied together to define an overall mass scaling strategy. The mass scaling can also apply globally to the entire model or, alternatively, on an element set by element set basis.

Fixed Mass Scaling

Fixed mass scaling is performed once at the beginning of the step for which it is specified. Two basic approaches are available for fixed mass scaling: you can define a mass scaling factor directly, or you can define a desired minimum stable time increment for which the mass scaling factors are determined by Abaqus/Explicit.

If both variable mass scaling and fixed mass scaling are specified in a step, the element original mass is scaled once at beginning of that step based on the specified fixed mass scaling. It is then further scaled at the beginning and periodically during that step based on the specified variable mass scaling.

Fixed mass scaling provides a simple means to modify the mass properties of a quasi-static model at the beginning of an analysis or to modify the masses of a few small elements in a dynamic model so that they do not control the stable time increment size. Since the scaling operation is performed only once at the beginning of the step for which the mass scaling is defined, fixed mass scaling is computationally efficient.

Input File Usage

FIXED MASS SCALING

Variable Mass Scaling

Variable mass scaling is used to scale the mass of elements at the beginning of a step and periodically during that step. When using this type of mass scaling, you define a desired minimum stable time increment: mass scaling factors will be calculated automatically and applied, as required, throughout the step.

Variable mass scaling is most useful when the stiffness properties that control the stable time increment change drastically during a step. This situation can occur in both quasi-static bulk forming and dynamic simulations in which elements are highly compressed or crushed.

Input File Usage

VARIABLE MASS SCALING

Defining a Scale Factor Directly

Defining a scale factor directly is useful for quasi-static analyses in which the kinetic energy in the model should remain small. You can define a fixed mass scaling factor that is applied to the original mass of all elements in a specified element set. The masses of the elements will be scaled at the beginning of the step and held fixed throughout the step unless further modified by variable mass scaling.

Input File Usage

FIXED MASS SCALING, FACTOR=scale_factor

For example, the following option scales the masses of elements contained in element set elset by a factor of 10:

FIXED MASS SCALING, FACTOR=10., ELSET=elset

Defining a Desired Element-by-Element Stable Time Increment

You can define a desired element-by-element stable time increment for an element set for fixed or variable mass scaling. Abaqus/Explicit will then determine the necessary mass scaling factors. There are three mutually exclusive methods available to scale the mass of the model when a desired element-by-element stable time increment is defined. Each method is described in detail later in this section.

To determine the stable time increment used during an increment, Abaqus/Explicit first determines the smallest stable time increment on an element-by-element basis. Then, a global estimation algorithm determines a stable time increment based on the highest frequency of the model. The larger of the two estimates determines the stable time increment used. In general, the stable time increment determined by the global estimator will be greater than the stable time increment determined by the element-by-element estimator. When fixed or variable mass scaling is used with a specified element-by-element stable time increment to scale the mass of a set of elements, the element-by-element stable time increment estimate is being affected directly. If all of the elements in the model are being scaled by a single mass scaling definition, the element-by-element estimate will equal the value assigned to the element-by-element stable time increment unless the penalty method is being used to enforce contact constraints. Penalty contact can cause the element-by-element estimate to be slightly below the value assigned to the element-by-element stable time increment (see Contact Controls for General Contact in Abaqus/Explicit and Contact Constraint Enforcement Methods in Abaqus/Explicit). The actual stable time increment used may be greater than the value assigned to the element-by-element stable time increment because of the use of the global estimator. If mass scaling is performed on only a portion of the model, the elements that are not scaled may have element stable time increments that are less than the value assigned to the element-by-element stable time increment and in that case will control the element-by-element stable time increment estimate. As a result, if only portions of the model are being scaled, the time increment used will generally not equal the value assigned to the element-by-element stable time increment.

If the fixed time increment size for the explicit dynamic step is based on the initial element-by-element stability limit or is specified directly, the time increment used will be calculated according to the rules described in Explicit Dynamic Analysis.

Scaling the Mass Uniformly

Scaling the mass uniformly is useful for quasi-static analyses in which the kinetic energy in the model should remain small. This approach is similar to defining a scale factor directly. In both cases the masses of all the elements specified are scaled uniformly by a single factor. However, with this method the mass scaling factor is determined by Abaqus/Explicit instead of being user specified. A single mass scaling factor is applied uniformly to all the elements so that the minimum stable time increment within these elements is equal to the value assigned to the element-by-element stable time increment, dt.

Input File Usage

Use either of the following options:

FIXED MASS SCALING, TYPE=UNIFORM, DT=dt
VARIABLE MASS SCALING, TYPE=UNIFORM, DT=dt

Scaling Only Elements with Element Stable Time Increments below the Specified Element-by-Element Stable Time Increment

Scaling elements with element stable time increments below a user-specified value is appropriate for both quasi-static and dynamic analyses. It is useful for increasing the element stable time increment of the most critical elements.

When the mesh at the beginning of an analysis or a step contains a few very small elements that control the stable time increment size, use fixed mass scaling to scale the masses of those elements and start the step with a desired time increment value. Increasing the mass of only these controlling elements means that the stable time increment can be increased significantly, yet the effect on the overall behavior of the model may be negligible.

For analyses in which evolving deformation creates a limited number of small elements, use variable mass scaling to scale the masses of those elements, thereby limiting the reduction in the stable time increment.

Input File Usage

Use either of the following options:

FIXED MASS SCALING, TYPE=BELOW MIN, DT=dt
VARIABLE MASS SCALING, TYPE=BELOW MIN, DT=dt

Scaling All Elements to Have Equal Element Stable Time Increments

Scaling all elements such that they have the same stable time increment effectively contracts the eigenspectrum of the model; that is, it reduces the range between the lowest and highest natural frequency of the model. Because of the drastic change in mass properties, this approach is appropriate only for quasi-static analyses. It implies that some elements may have mass scaling factors that are less than one.

Input File Usage

Use either of the following options:

FIXED MASS SCALING, TYPE=SET EQUAL DT, DT=dt
VARIABLE MASS SCALING, TYPE=SET EQUAL DT, DT=dt

Global and Local Mass Scaling

Specifying an element set for either fixed or variable mass scaling scales the mass of a localized region of the model. Omitting an element set implies that mass scaling will be performed for all elements. A global definition can be overwritten by a local definition for a given element set by repeating the mass scaling definition with an element set specified.

Input File Usage

Use either of the following options:

FIXED MASS SCALING, ELSET=elset
VARIABLE MASS SCALING, ELSET=elset

Example 1

Different mass scaling factors may be useful when materials with vastly different wave speeds or mesh refinements are present in an analysis. In this example a scale factor of 50 may be desirable for the masses of all elements in a quasi-static analysis, except for a few elements for which a mass scaling factor of 500 is used.

FIXED MASS SCALING, FACTOR=50.0
FIXED MASS SCALING, FACTOR=500.0, ELSET=elset1

The first fixed mass scaling definition scales the masses of all elements in the model by a factor of 50. The second fixed mass scaling definition overrides the first definition for the elements contained in element set elset1 by scaling their masses by a factor of 500.

Example 2

An alternative method of scaling the masses of elements in elset1 is to assign a stable time increment to them and allow Abaqus/Explicit to determine the mass scaling factors.

FIXED MASS SCALING, FACTOR=50.0
FIXED MASS SCALING, DT=.5E-6, TYPE=BELOW MIN, ELSET=elset1

The first fixed mass scaling definition scales the masses in the entire model by a factor of 50. The second fixed mass scaling definition overrides the first definition by scaling the masses of any elements in elset1 whose stable time increments are less than .5 × 10⁻⁶.

Mass Scaling at the Beginning of the Step

Fixed mass scaling is used to prescribe mass scaling only at the beginning of a step and always scales the original element masses. When the scale factor is defined directly, the mass is scaled by the value assigned to the scale factor. If the element-by-element stable time increment, dt, is specified, the mass scaling is based on this value. If both the scale factor and the element-by-element stable time increment are specified, the mass is first scaled by the value assigned to the scale factor and then possibly scaled again, depending on the value assigned to the element-by-element stable time increment and the type of fixed mass scaling chosen.

Local mass scaling can be defined for a specific element set. If no element set is specified, the fixed mass scaling definition will apply to all elements in the model. Only one fixed mass scaling definition is permitted per element set. Multiple fixed mass scaling definitions cannot contain overlapping element sets. Local mass scaling definitions will overwrite global definitions for the specified element sets.

Input File Usage

FIXED MASS SCALING, FACTOR=factor, DT=dt, TYPE=type, ELSET=elset

Example

Assume that for a quasi-static analysis a mass scaling factor of 50 is applied to all the elements in the model. Furthermore, assume that even after being scaled by a factor of 50, a few extremely small or poorly shaped elements are causing the stable time increment to be less than a desired minimum. To increase the stable time increment, the following option is used:

FIXED MASS SCALING, FACTOR=50., TYPE=BELOW MIN, DT=.5E-6

The specified scale factor causes the masses of all the elements in the model to be scaled by a factor of 50. If any element's stable time increment is still below 0.5 × 10⁻⁶ after being scaled by a factor of 50.0, its mass will be scaled such that its stable time increment is equal to 0.5 × 10⁻⁶.

Mass Scaling throughout the Step

Variable mass scaling with a specified element-by-element stable time increment is used to define mass scaling that is to be performed at the beginning and throughout the step. Either the frequency in increments or the number of intervals must be specified to define how frequently mass scaling is to be performed. In increments other than those in which mass scaling is performed, the time increment used will generally be different from the value assigned to the element-by-element stable time increment.

Local mass scaling can be defined for a specific element set. If no element set is specified, the variable mass scaling definition will apply to all elements in the model. Only one variable mass scaling definition is permitted per element set. Multiple variable mass scaling definitions cannot contain overlapping element sets. Local mass scaling definitions will overwrite global definitions for the specified element sets.

Input File Usage

VARIABLE MASS SCALING, DT=dt, TYPE=type, ELSET=elset

Calculating the Mass Scaling at Equally Spaced Increments

You can specify the number of increments between mass scaling calculations. For example, specifying a frequency of 5 will cause mass scaling to be performed at the beginning of the step and at increments 5, 10, 15, etc.

Care should be taken when choosing the value of the frequency, since performing mass scaling every few increments during an analysis may result in noticeable additional computational cost per increment.

Input File Usage

VARIABLE MASS SCALING, TYPE=type, DT=dt, FREQUENCY=n

Calculating the Mass Scaling at Equally Spaced Time Intervals

Alternatively, you can specify the number of equally spaced time intervals at which the mass scaling calculations are to be performed. For example, specifying 5 intervals in a step with a duration of one second will cause mass scaling to be performed at the beginning of the step and at times of .2 , .4, .6, .8, and 1.0 seconds.

Input File Usage

VARIABLE MASS SCALING, TYPE=type, DT=dt, NUMBER INTERVAL=n

Different Mass Scaling at the Beginning and during the Step

There are cases where it is desirable to include mass scaling at the beginning of a step that may be modified further throughout the step.

Input File Usage

Use both of the following options:

FIXED MASS SCALING, FACTOR=factor, TYPE=type, DT=dt_init
VARIABLE MASS SCALING, TYPE=type, DT=dt_min, FREQUENCY=n
or NUMBER INTERVAL=n

Example

Assume that in a dynamic impact analysis, a few extremely small or poorly shaped elements exist in the mesh and consequently control the stable time increment. To prevent these elements from controlling the stable time increment, it is desirable to scale their masses at the beginning of the step. In addition, elements in a region of the mesh will develop severe distortions as a result of impact with a fixed rigid surface. Consequently, elements in the impact zone may eventually control the stable time increment.

Since the elements in the impact zone are essentially stationary against the rigid surface, selectively scaling their masses will guarantee that the overall dynamic response is not adversely affected. Mass scaling these elements by prescribing a time increment to limit the reduction in the element-by-element stable time increment may decrease run time substantially.

For example, specify fixed mass scaling for all elements in the model with stable time increments below a value of 1.0 × 10⁻⁶. In addition, specify variable mass scaling for the elements in the impact zone (elset1) with stable time increments below a value of 0.5 × 10⁻⁶. In this case all the elements in the model are checked at the beginning of the step. If any have stable time increments less than 1.0 × 10⁻⁶, their masses are scaled (independently) such that the element-by-element stable time increment equals 1.0 × 10⁻⁶. This scaling remains in effect throughout the step and is not further modified, except for those elements in elset1. The variable mass scaling definition causes the elements contained in elset1 to be scaled throughout the step so that their stable time increments do not become less than 0.5 × 10⁻⁶. Because only elements in elset1 are scaled during the step, it is possible that a stable time increment less than 0.5 × 10⁻⁶ may result.

Mass Scaling in a Multiple Step Analysis or an Abaqus/Explicit to Abaqus/Explicit Import Analysis

When mass scaling is applied, Abaqus/Explicit automatically carries forward the following to the subsequent step or import analysis:

the scaled element masses at the end of one step or a previous analysis
any variable mass scaling methods specified in that step or analysis

This approach ensures that continuity is carried forward automatically to the subsequent step or the import analysis. This ensures continuity in the mass matrix at the subsequent step/analysis and continued application of the variable mass scaling methods. However, you can reset the element masses to their original values or recompute the element masses by using a new fixed mass scaling method at the beginning of the subsequent step or the import analysis. You can also remove the variable mass scaling methods inherited from the prior step/analysis or replace an inherited method with a new variable mass scaling method.

To reset the initial mass matrix, specify a fixed mass scaling method in the subsequent step or in the import analysis. Similarly, specify a variable mass scaling method in the subsequent step or in the import analysis to discontinue all of the variable mass scaling methods of the prior step/analysis. The examples below illustrate the following special cases: continuous mass matrix with no further mass scaling, and reverting the mass matrix to the original state with no further mass scaling.

Very large changes in element mass across the steps due to mass scaling may lead to precision problems in the mass calculations. These precision problems may give rise to erroneous or misleading results. When large changes in element masses are desired in such situations, it is recommended that fixed mass scaling be used in the new step or in the import analysis to reset the element masses to their original values before using additional mass scaling definitions, as required, to scale the element masses to their desired values.

Continuous Mass Matrix with No Further Scaling

To define a continuous mass matrix with no further scaling, remove any variable mass scaling definitions inherited from the prior step/analysis by redefining a new variable mass scaling definition.

Input File Usage

Use the following option without any parameters in a new step or an import analysis:

VARIABLE MASS SCALING

Example

Assume that during the first step of a quasi-static analysis, elements experience distortions that cause the stable time increment to decrease dramatically. Furthermore, assume that the deformation during the second step is not large enough to have any further effect on the stable time increment. Similarly, assume the same behavior for an analysis followed by an import analysis.

HEADING
…
STEP
…
FIXED MASS SCALING, FACTOR=1.1
VARIABLE MASS SCALING, TYPE=BELOW MIN, DT=1.E-5, FREQUENCY=10
…
END STEP
STEP
…
VARIABLE MASS SCALING
…
END STEP

During the first step/analysis the fixed mass scaling increases the element mass by the factor 1.1. The variable mass scaling definition scales the mass of the entire model at the beginning of the step and every tenth increment such that the element-by-element stable time increment equals at least 1 × 10⁻⁵. The variable mass scaling definition in the second step or the import analysis replaces the one continued from the first step/analysis. This particular definition of variable mass scaling without any parameters in the second step/analysis also prevents any further mass scaling during the second step/analysis. The scaled mass matrix from the first step/analysis is carried over to be used during the entire second step/analysis.

Reverting the Mass Matrix to the Original State

You can introduce a fixed mass scaling method in the subsequent step or the import analysis to discontinue all of the mass scaling methods of the prior step/analysis. Further, if the default specification of fixed mass scaling is used, element masses revert to their original values at the beginning of the subsequent step or the import analysis. Therefore, specify just the default fixed mass scaling method to prevent the scaled mass of the previous step/analysis from being used in a new step or an import analysis. This approach is useful when you are going from a quasi-static simulation step/analysis where mass scaling is appropriate to a dynamic step/analysis in which no scaling is desired.

Input File Usage

Use both of the following options without any parameters:

FIXED MASS SCALING
VARIABLE MASS SCALING

Example

Assume that an analysis contains a quasi-static step followed by a dynamic step, or assume a quasi-static analysis is followed by a dynamic analysis. Mass scaling can be performed during the quasi-static step/analysis but turned off during the dynamic step/analysis.

HEADING
STEP
…
FIXED MASS SCALING, FACTOR=1.1
VARIABLE MASS SCALING, TYPE=BELOW MIN, DT=1.E-5, FREQUENCY=10
END STEP
STEP
FIXED MASS SCALING
VARIABLE MASS SCALING
END STEP

During the first step/analysis the fixed mass scaling increases the element mass by the factor 1.1. The variable mass scaling definition scales the mass of the entire model at the beginning of the step and every tenth increment such that the element-by-element stable time increment equals at least 1 × 10⁻⁵. The new fixed mass scaling definition without any parameters in the second step or in the import analysis then reverts the mass matrix back to the original state. The new variable mass scaling definition replaces all the variable mass scaling definitions inherited from the first step/analysis. Further, since the new variable mass scaling definition has no parameters, no mass scaling is applied during the second step/analysis. Therefore, the mass matrix for the second step or the import analysis reverts to that of the original state.

Mass Contribution from External Programs Connected to Abaqus via Co-Simulation

Co-simulation can lead to mass and/or rotary inertia from external programs being added to the Abaqus model during a step. However, that contribution along with other quantities imported from the external program must be removed once the co-simulation step is completed. If co-simulation is expected to add mass and/or rotary inertia to the Abaqus model, Abaqus automatically reverts the mass matrix back to the original state once such a co-simulation step is completed. You need to respecify any mass scaling that must be continued beyond the co-simulation step.

When Mass Scaling Is or Is Not Used

The following entities are not affected by mass scaling:

Thermal solution response in a fully coupled thermal-stress analysis
Gravity loads, viscous pressure loads
Adiabatic heat calculations
Equation of state materials
Fluid and fluid link elements
Surface-based fluid cavities
Spring and dashpot elements

Densities associated with any of the relevant items in this list will remain unscaled. Mass, rotary inertia, infinite, and rigid elements can be scaled. However, because none of the elements has an associated stable time increment, they can be scaled only using either a user-specified scale factor or an element-by-element stable time increment applied uniformly. If the element-by-element stable time increment is specified, at least one element with a stable time increment must be included in the mass scaling definition. Rotary inertia in shell, beam, and pipe elements is based on the scaled mass.

The mass of infinite elements can be scaled; however, the infinite elements will not act as quiet boundaries unless the densities of each adjacent deformable element are scaled by the same factor. The mass of both elements will be scaled by the same factor if they are both included in the same fixed or variable mass scaling definition.

Automatic Mass Scaling for Analysis of Bulk Metal Rolling

Bulk metal rolling is generally considered a quasi-static process, but the process is often modeled with Abaqus/Explicit because of its ability to handle the contact problem well. To achieve an economical solution with Abaqus/Explicit, it is often useful to increase the mass of the product artificially. However, the mass scaling factor must be chosen such that the changes in the mass and the corresponding changes in the inertial forces do not alter the solutions significantly. Choosing too high a scaling factor will not produce quasi-static results. Choosing too low a scaling factor, while conservative, will result in long run times. Rolling variable mass scaling can be used to make the choice of the optimal scaling factor automatic for this process.

The automatic strategy is based on the semi-automatic method of scaling all elements to have equal element stable time increments. The method is made automatic by determining the appropriate value for the target stable time increment from several parameters of the rolling process. The value used for the target stable time increment, $Δ t$ , is based on the average element length in the rolling direction, $L_{e}$ ; the feed rate, V; and the number of nodes in the cross-section of the product, n. The feed rate is defined as the average velocity of the product in the rolling direction during steady-state conditions. The value of $Δ t$ is adjusted during the analysis to account for the actual value of the feed rate. You must specify estimated values for the average velocity, the average element length in the rolling direction, and the number of nodes in the cross-section of the product.

The mass of any element will never drop below its original mass. This is different from the method of scaling all elements to have equal element stable time increments. Imposing this restriction means that rolling problems that have significant inertial effects will not have their mass adjusted automatically when they are analyzed as quasi-static.

To achieve a good result, it is recommended that:

the product be meshed by extruding a two-dimensional cross-section of the product;
the average element length in the rolling direction not vary significantly along the length of the product;
the product have an initial velocity in the rolling direction approximately equal to the steady-state feed rate;
the element size in the cross-section be equal to or less than the size in the rolling direction; and
no other mass scaling be used on elements scaled with rolling automatic variable mass scaling.

Input File Usage

VARIABLE MASS SCALING, ELSET=elset1, FREQUENCY=n, 
TYPE=ROLLING, FEED RATE=V, EXTRUDED LENGTH= $L_{e}$ ,  
CROSS SECTION NODES=n

Output

Output variable EMSF provides the element mass scaling factor. Output variable DMASS provides the total percent change in mass of the model as a result of mass scaling. Output variable DMASS is not available on an element set basis.

Output variable EDT provides the element stable time increment. The element stable time increment includes the effect of mass scaling.