$$ t _ {i n} = \int_ {0} ^ {t} f (t) d t. $$ Typically the amplitude variation is a step function stepping from zero to one at the time the airbag should be deployed. This amplitude variation has the effect of offsetting the inflation time from the analysis time. Input File Usage: Use the following options: \*AMPLITUDE, NAME=amplitude\_name \*FLUID INFLATOR ACTIVATION, INFLATION TIME AMPLITUDE=amplitude\_name # Modifying the mass flow rate If the mass flow rate is prescribed directly in the inflator property definition, you can modify it by specifying an amplitude definition during a step. However, if the mass flow rate is calculated by using tank test data or the dual pressure method, the amplitude definition will be ignored. Input File Usage: Use the following options: \*AMPLITUDE, NAME=amplitude\_name \*FLUID INFLATOR ACTIVATION, MASS FLOW AMPLITUDE=amplitude\_name # Activation in multiple steps By default, when you modify the activation of a fluid inflator definition or activate a new fluid inflator definition, all existing fluid inflator activations in the step remain. When modifying an existing activation, all applicable parameters must be respecified. Activated inflator definitions remain active in subsequent steps unless deactivated. You can choose to deactivate all fluid inflator definitions in the model and optionally reactivate new ones. If you deactivate any fluid inflator definition in a step, all fluid inflator definitions must be respecified. Input File Usage: Use the following option to modify an existing fluid inflator activation or to specify an additional fluid inflator activation (default): \*FLUID INFLATOR ACTIVATION, OP=MOD Use the following option to deactivate all fluid inflator definitions in the model and optionally reactivate new ones: \*FLUID INFLATOR ACTIVATION, OP=NEW # Additional reference • Wang, J. T., and O. J. Nefske, “A New CAL3D Airbag Inflation Model,” SAE paper 880654, 1988. # 11.6 Mass scaling • “Mass scaling,” Section 11.6.1 # 11.6.1 MASS SCALING Products: Abaqus/Explicit Abaqus/CAE # References • “Explicit dynamic analysis,” Section 6.3.3 • “Adjust and/or redistribute mass of an element set,” Section 2.6.1 • “Output,” Section 4.1.1 • \*FIXED MASS SCALING • \*VARIABLE MASS SCALING • “Configuring a dynamic, explicit procedure” in “Configuring general analysis procedures,” Section 14.11.1 of the Abaqus/CAE User’s Guide, in the HTML version of this guide • “Configuring a dynamic fully coupled thermal-stress procedure using explicit integration” in “Configuring general analysis procedures,” Section 14.11.1 of the Abaqus/CAE User’s Guide, in the HTML version of this guide # Overview Mass scaling is often used in Abaqus/Explicit for computational efficiency in quasi-static analyses and in some dynamic analyses that contain a few very small elements that control the stable time increment. Mass scaling can be used to: • scale the mass of the entire model or scale the masses of individual elements and/or element sets; • scale the mass on a per step basis in a multistep analysis; and • scale the mass at the beginning of the step and/or throughout the step. Mass scaling can be performed by: • scaling the masses of all specified elements by a user-supplied constant factor; • scaling the masses of all specified elements by the same value so that the minimum stable time increment for any element in the element set is equal to a user-supplied time increment; • scaling the masses of only the elements in the element set whose element stable time increments are less than a user-supplied time increment so that the element stable time increment for these elements becomes equal to the user-supplied time increment; • scaling the masses of all specified elements so that their element stable time increments each become equal to the user-supplied time increment; and • scaling automatically based on mesh geometry and initial conditions for bulk metal rolling analyses. # 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,” Section 6.3.3), 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,” Section 1.3.6 of the Abaqus Example Problems Guide, 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,” Section 1.3.10 of the Abaqus Benchmarks Guide, 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 Abaqus/CAE Usage: Step module: Create Step: General, Dynamic, Explicit or Dynamic, Temp-disp, Explicit: Mass scaling: Use scaling definitions below: Create: Semi-automatic mass scaling, Scale: At beginning of step # 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. 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. 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 Abaqus/CAE Usage: Step module: Create Step: General, Dynamic, Explicit or Dynamic, Temp-disp, Explicit: Mass scaling: Use scaling definitions below: Create: Semi-automatic mass scaling, Scale: Throughout step # 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 Abaqus/CAE Usage: Step module: Create Step: General, Dynamic, Explicit or Dynamic, Temp-disp, Explicit: Mass scaling: Use scaling definitions below: Create: Semi-automatic mass scaling, Scale: At beginning of step, Scale by factor: scale\_factor # 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-byelement 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-byelement 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,” Section 36.4.5, and “Contact constraint enforcement methods in Abaqus/Explicit,” Section 38.2.3). 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-byelement stability limit or is specified directly, the time increment used will be calculated according to the rules described in “Explicit dynamic analysis,” Section 6.3.3. # 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 Abaqus/CAE Usage: Step module: Create Step: General, Dynamic, Explicit or Dynamic, Temp-disp, Explicit: Mass scaling: Use scaling definitions below: Create: Semi-automatic mass scaling, Scale: At beginning of step or Throughout step, Scale to target time increment of: dt, Scale element mass: Uniformly to satisfy target 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

Abaqus/CAE Usage:

Step module: Create Step: General, Dynamic, Explicit or Dynamic, Temp-disp, Explicit: Mass scaling: Use scaling definitions below: Create: Semi-automatic mass scaling, Scale: At beginning of step or Throughout step, Scale to target time increment of: dt, Scale element mass: If below minimum target

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

Abaqus/CAE Usage:

Step module: Create Step: General, Dynamic, Explicit or Dynamic, Temp-disp, Explicit: Mass scaling: Use scaling definitions below: Create: Semi-automatic mass scaling, Scale: At beginning of step or Throughout step, Scale to target time increment of: dt, Scale element mass: Nonuniformly to equal target

# 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