### 6.1.2 General and linear perturbation procedures

Products: ABAQUS/Standard  ABAQUS/Explicit  ABAQUS/CAE

### Overview

An analysis step during which the response can be either linear or nonlinear is called a general analysis step. An analysis step during which the response can be linear only is called a linear perturbation analysis step. General analysis steps can be included in an ABAQUS/Standard or ABAQUS/Explicit analysis; linear perturbation analysis steps are available only in ABAQUS/Standard.

A clear distinction is made in ABAQUS/Standard between general analysis and linear perturbation analysis procedures. Loading conditions are defined differently for the two cases, time measures are different, and the results should be interpreted differently. These distinctions are defined in this section.

ABAQUS/Standard treats a linear perturbation analysis as a linear perturbation about a preloaded, predeformed state. ABAQUS/Foundation, a subset of ABAQUS/Standard, is limited entirely to linear perturbation analysis but does not allow preloading or predeformed states.

### General analysis steps

A general analysis step is one in which the effects of any nonlinearities present in the model can be included. The starting condition for each general step is the ending condition from the last general step, with the state of the model evolving throughout the history of general analysis steps as it responds to the history of loading. If the first step of the analysis is a general step, the initial conditions for the step can be specified directly (Initial conditions, Section 27.2.1).

ABAQUS always considers total time to increase throughout a general analysis. Each step also has its own step time, which begins at zero in each step. If the analysis procedure for the step has a physical time scale, as in a dynamic analysis, step time must correspond to that physical time. Otherwise, step time is any convenient time scale—for example, 0.0 to 1.0—for the step. The step times of all general analysis steps accumulate into total time. Therefore, if an option such as creep (available only in ABAQUS/Standard) whose formulation depends on total time is used in a multistep analysis, any steps that do not have a physical time scale should have a negligibly small step time compared to the steps in which a physical time scale does exist.

#### Sources of nonlinearity

Nonlinear stress analysis problems can contain up to three sources of nonlinearity: material nonlinearity, geometric nonlinearity, and boundary nonlinearity.

##### Material nonlinearity

ABAQUS offers models for a wide range of nonlinear material behaviors (see Combining material behaviors, Section 16.1.3). Many of the materials are history dependent: the material's response at any time depends on what has happened to it at previous times. Thus, the solution must be obtained by following the actual loading sequence. The general analysis procedures are designed with this in view.

##### Geometric nonlinearity

It is possible in ABAQUS to define a problem as a “small-displacement” analysis, which means that geometric nonlinearity is ignored in the element calculations—the kinematic relationships are linearized. By default, large displacements and rotations are accounted for in contact constraints even if the small-displacement element formulations are used for the analysis; i.e., a large-sliding contact tracking algorithm is used (see Contact formulation for ABAQUS/Standard contact pairs, Section 29.2.2, and Contact formulation for ABAQUS/Explicit contact pairs, Section 29.4.4). The elements in a small-displacement analysis are formulated in the reference (original) configuration, using original nodal coordinates. The errors in such an approximation are of the order of the strains and rotations compared to unity. The approximation also eliminates any possibility of capturing bifurcation buckling, which is sometimes a critical aspect of a structure's response (see Unstable collapse and postbuckling analysis, Section 6.2.4). You must consider these issues when interpreting the results of such an analysis.

The alternative to a “small-displacement” analysis in ABAQUS is to include large-displacement effects. In this case most elements are formulated in the current configuration using current nodal positions. Elements therefore distort from their original shapes as the deformation increases. With sufficiently large deformations, the elements may become so distorted that they are no longer suitable for use; for example, the volume of the element at an integration point may become negative. In this situation ABAQUS will issue a warning message indicating the problem. In addition, ABAQUS/Standard will cut back the time increment before making further attempts to continue the solution. ABAQUS/Explicit also offers element failure models to allow elements that reach high strains to be removed from a model; see Dynamic failure models, Section 18.2.8, for details.

For each step of an analysis you specify whether a small- or large-displacement formulation should be used (i.e., whether geometric nonlinearity should be ignored or included). By default, ABAQUS/Standard uses a small-displacement formulation and ABAQUS/Explicit uses a large-displacement formulation. The default value for the formulation in an import analysis is the same as the value at the time of import. If a large-displacement formulation is used during any step of an analysis, it will be used in all following steps in the analysis; there is no way to turn it off.

Almost all of the elements in ABAQUS use a fully nonlinear formulation. The exceptions are the cubic beam elements in ABAQUS/Standard and the small-strain shell elements (those shell elements other than S3/S3R, S4, S4R, and the axisymmetric shells) in which the cross-sectional thickness change is ignored so that these elements are appropriate only for large rotations and small strains. Except for these elements, the strains and rotations can be arbitrarily large.

The calculated stress is the “true” (Cauchy) stress. For beam and shell elements the stress components are given in local directions that rotate with the material. For all other elements the stress components are given in the global directions unless a local orientation (Orientations, Section 2.2.5) is used at a point. For small-displacement analysis the infinitesimal strain measure is used, which is output with the strain output variable E; strain output specified with output variables LE and NE is the same as with E.

 Input File Usage: Use the following option to specify that a large-displacement formulation should be used for the step: `*STEP, NLGEOM=YES (default in ABAQUS/Explicit)`Use the following option to specify that a small-displacement formulation should be used for the step:`*STEP, NLGEOM=NO (default in ABAQUS/Standard)`Omitting the NLGEOM parameter is equivalent to using the default value.

 ABAQUS/CAE Usage: Step module: Create Step: select any step type: Basic: Nlgeom: Off (for a small-displacement formulation) or On (for a large-displacement formulation)

##### Boundary nonlinearity

Contact problems are a common source of nonlinearity in stress analysis—see Contact interaction analysis: overview, Section 29.1.1. Other sources of boundary nonlinearity are nonlinear elastic springs, films, radiation, multi-point constraints, etc.

In a general analysis step the loads must be defined as total values. The rules for applying loads in a general, multistep analysis are defined in Applying loads: overview, Section 27.4.1.

#### Incrementation

The general analysis procedures in ABAQUS offer two approaches for controlling incrementation. Automatic control is one choice: you define the step and, in some procedures, specify certain tolerances or error measures. ABAQUS then automatically selects the increment size as it develops the response in the step. Direct user control of increment size is the alternative approach, whereby you specify the incrementation scheme. The direct approach is sometimes useful in repetitive analyses with ABAQUS/Standard, where you have a good “feel” for the convergence behavior of the problem. The methods for selecting automatic or direct incrementation are discussed in the individual procedure sections.

In nonlinear problems in ABAQUS/Standard the challenge is always to obtain a convergent solution in the least possible computational time. In these cases automatic control of the time increment is usually more efficient because ABAQUS/Standard can react to nonlinear response that you cannot predict ahead of time. Automatic control is particularly valuable in cases where the response or load varies widely through the step, as is often the case in diffusion-type problems such as creep, heat transfer, and consolidation. Ultimately, automatic control allows nonlinear problems to be run with confidence in ABAQUS/Standard without extensive experience with the problem.

Strong nonlinearities typically do not present difficulties in ABAQUS/Explicit because of the small time increments that are characteristic of an explicit dynamic analysis product.

#### Stabilization of unstable problems in ABAQUS/Standard

Some static problems can be naturally unstable, for a variety of reasons.

##### Unconstrained rigid body motions

Instability may occur because unconstrained rigid body motions exist. ABAQUS/Standard may be able to handle this type of problem with automatic viscous damping (see Adjusting contact controls in ABAQUS/Standard, Section 29.2.12) when rigid body motions exist during the approach of two bodies that will eventually come into contact.

 Input File Usage: Use one of the following options: ```*CONTACT CONTROLS, APPROACH *CONTACT CONTROLS, STABILIZE```

 ABAQUS/CAE Usage: Automatic viscous damping is not supported in ABAQUS/CAE.

##### Localized buckling behavior or material instability

Instability may also be caused by localized buckling behavior or by material instability; such instabilities are especially significant when no time-dependent behavior exists in the material modeling. The static, general analysis procedures in ABAQUS/Standard can stabilize this type of problem if you request it (see Static stress analysis, Section 6.2.2; Quasi-static analysis, Section 6.2.5; Steady-state transport analysis, Section 6.4.1; Fully coupled thermal-stress analysis, Section 6.5.4; or Coupled pore fluid diffusion and stress analysis, Section 6.7.1).

 Input File Usage: Use one of the following options: ```*STATIC, STABILIZE *VISCO, STABILIZE *STEADY STATE TRANSPORT, STABILIZE *COUPLED TEMPERATURE-DISPLACEMENT, STABILIZE *SOILS, CONSOLIDATION, STABILIZE```

 ABAQUS/CAE Usage: Step module: Create Step: General: any valid step type: Basic: Use stabilization with dissipated energy fraction

### Linear perturbation analysis steps

Linear perturbation analysis steps are available only in ABAQUS/Standard (ABAQUS/Foundation is essentially the linear perturbation functionality in ABAQUS/Standard). The response in a linear analysis step is the linear perturbation response about the base state. The base state is the current state of the model at the end of the last general analysis step prior to the linear perturbation step. If the first step of an analysis is a perturbation step, the base state is determined from the initial conditions (Initial conditions, Section 27.2.1). In ABAQUS/Foundation the base state is always determined from the initial state of the model.

Linear perturbation analyses can be performed from time to time during a fully nonlinear analysis by including the linear perturbation steps between the general response steps. The linear perturbation response has no effect as the general analysis is continued. The step time of linear perturbation steps, which is taken arbitrarily to be a very small number, is never accumulated into the total time. A simple example of this method is the determination of the natural frequencies of a violin string under increasing tension (see Vibration of a cable under tension, Section 1.4.3 of the ABAQUS Benchmarks Manual). The tension of the string is increased in several geometrically nonlinear analysis steps. After each of these steps, the frequencies can be extracted in a linear perturbation analysis step.

If geometric nonlinearity is included in the general analysis upon which a linear perturbation study is based, stress stiffening or softening effects and load stiffness effects (from pressure and other follower forces) are included in the linear perturbation analysis.

Load stiffness contributions are also generated for centrifugal and Coriolis loading. In direct steady-state dynamic analysis Coriolis loading generates an imaginary antisymmetric matrix. This contribution is accounted for currently in solid and truss elements only and is activated by using the unsymmetric matrix storage and solution scheme in the step.

#### Linear perturbation procedures

The following purely linear perturbation procedures are available in ABAQUS/Standard:

Except for these procedures and the static procedure (explained below), all other procedures can be used only in general analysis steps (in other words, they are not available with ABAQUS/Foundation). All linear perturbation procedures except for the complex eigenvalue extraction procedure are available with ABAQUS/Foundation.

#### Linear static perturbation analysis

A linear static stress analysis (Static stress analysis, Section 6.2.2) can be conducted in ABAQUS/Standard.

 Input File Usage: Use both of the following options to conduct a linear static perturbation analysis: ```*STEP, PERTURBATION *STATIC```Omitting the PERTURBATION parameter on the *STEP option implies that a general static analysis is required.

 ABAQUS/CAE Usage: Step module: Create Step: Linear perturbation: Static, Linear perturbation

Load magnitudes (including the magnitudes of prescribed boundary conditions) during a linear perturbation analysis step are defined as the magnitudes of the load perturbations only. Likewise, the value of any solution variable is output as the perturbation value only—the value of the variable in the base state is not included.

Multiple load cases can be analyzed simultaneously for static and direct-solution steady-state dynamic linear perturbation steps. See Multiple load case analysis, Section 6.1.3, for a description of this capability.

#### Restrictions

A linear perturbation analysis is subject to the following restrictions:

• Since a linear perturbation analysis has no time period, amplitude references (Amplitude curves, Section 27.1.2) can be used only to specify loads or boundary conditions as functions of frequency (in a steady-state dynamics analysis) or to define base motion (in mode-based dynamics procedures).

• A general implicit dynamic analysis (Implicit dynamic analysis using direct integration, Section 6.3.2) cannot be interrupted to perform perturbation analyses: before performing the perturbation analysis, ABAQUS/Standard requires that the structure be brought into static equilibrium.

• During a linear perturbation analysis step, the model's response is defined by its linear elastic (or viscoelastic) stiffness at the base state. Plasticity and other inelastic effects are ignored. For hyperelasticity (Hyperelastic behavior of rubberlike materials, Section 17.5.1) or hypoelasticity (Hypoelastic behavior, Section 17.4.1), the tangent elastic moduli in the base state are used. If cracking has occurred—for example, in the concrete model (Concrete smeared cracking, Section 18.5.1)—the damaged elastic (secant) moduli are used.

• Contact conditions cannot change during a linear perturbation analysis; they remain as they were defined in the base state. Frictional slipping is not allowed during linear perturbation analyses; all points in contact are assumed to be sticking if friction is present. The exception is real or complex eigenvalue extraction analysis where, regardless of the friction coefficient, slipping conditions are assumed at contact nodes for which a velocity differential is imposed by the motion of the reference frame or the transport velocity.

• The effects of temperature and field variable perturbations are ignored for materials that are dependent on temperature and field variables. However, temperature perturbations will produce perturbations of thermal strain.