Distribution Definition

A distribution:

is a spatially varying field defined over elements, nodes, or element faces in an Abaqus model;
can be used to define shell thicknesses for shell elements with displacement degrees of freedom;
can be used to define beam radii for beam elements with a solid circular section when numerical integration across the section is required;
can be used to define shell stiffness;
can be used to define local coordinate systems on solid continuum and shell elements that have displacement degrees of freedom;
can be used to define local material directions or fiber directions on solid continuum and shell elements with anisotropic materials (such as anisotropic hyperelastic materials) or, in Abaqus/Explicit, fabric materials;
can be used to define orientation angles on the layers of composite shell elements that have displacement degrees of freedom;
can be used to define orientation angles for connector elements;
can be used to define thicknesses on the layers of conventional composite shell elements;
can be used to specify initial contact clearances;
can be used to specify the volume fraction, aspect ratio, and second-order orientation tensor of a constituent in a multiscale material; and
can be used in an adjoint design sensitivity analysis to specify scale factors and a quantity related to the scale factor derivatives for scaling the mass, stiffness, or design stress material attribute on an element-by-element basis; to specify scale factors for material thermal conductivity on an element-by-element basis; or to specify nodal adjustments (displacements) on a node-by-node basis; and
in an Abaqus/Standard analysis can be used to define mass density, linear elastic material behavior, and thermal expansion for solid continuum elements; shell offsets; orientation angles and thicknesses on the layers of composite solid continuum elements; local coordinate systems on membrane elements; and membrane thickness on an element-by-element basis.

This page discusses:

Distributions
Specifying the Location of a Distribution
Defining a Distribution Table
Defining Distributions by Importing Field Data from an Output Database File
Applying Distributions
Examples

Distributions

A distribution is a spatial analogy of an amplitude definition (see Amplitude Curves). Amplitude definitions are used to provide arbitrary time variations of loads, displacements, and other prescribed variables. Distributions are used to specify arbitrary spatial variations of selected element properties, material properties, local coordinate systems, boundary conditions, and spatial variations of initial contact clearances.

The two main components of a distribution are its location and field data. The location identifies where the distribution is defined, either on elements, nodes, or element faces. Field data are a specified number of floating point values defined for each element, node, or element face in the distribution.

To define a distribution, you must assign it a unique name. You must also specify the number and physical dimension of each data value in the distribution by referring to a distribution table.

Specifying the Location of a Distribution

You can define a distribution on elements or nodes. Distributions on nodes are supported only for defining initial contact clearances as described in Contact Initialization for General Contact in Abaqus/Explicit. All other applications of distributions require distributions defined on elements.

There is no limit on the number of distributions to which a given element or node may belong. Elements and nodes cannot be combined within the same distribution definition.

Defining a Distribution on Elements

Defining a distribution on elements requires you to specify field data for each element or element set included in the distribution definition. All distributions on elements require that default data be defined. Default data are used for all elements that are not specifically assigned a value in the distribution.

Defining a Distribution on Nodes

Defining a distribution on nodes requires you to specify field data for each node or node set included in the distribution definition.

Defining a Distribution Table

Every distribution definition must refer to a distribution table. A distribution table defines the number of field data items needed for each element or node in a distribution. The distribution table also defines the physical dimension of each data value in a distribution. A distribution table can be referred to as many times as needed by different distributions. The distribution table consists of a list of predefined labels shown in Table 1. The combination of labels needed for a given distribution is determined by how the distribution is applied.

Table 1. Distribution table labels—Abaqus/Standard and Abaqus/Explicit.
Label	Physical dimension	Number of data items per label
ANGLE	angle in degrees	1
COORD3D	(L, L, L)	3
DENSITY	ML⁻³	1
DIR3D	dimensionless	3
DISP3D	(L, L, L)	3
EXPANSION	$θ$ ⁻¹	1
LENGTH	L	1
MODULUS	FL⁻²	1
ORIENTS	dimensionless	6
ORITENS	dimensionless	6
RATIO	dimensionless	1
SHELLSTIFF1	FL^-1	1
SHELLSTIFF2	F	1
SHELLSTIFF3	FL	1

Defining Distributions by Importing Field Data from an Output Database File

For two- and three-dimensional continuum elements and three-dimensional conventional shell elements, you can define distribution data using output variables from a particular step and increment or a user-specified time in the output database (.sim) file from a previous analysis. For more information, see General Capability for Importing External Fields.

If the previous analysis is performed with third-party software, the results file must be converted to the .sim file format.
You can use any appropriate result from the output database by specifying an output variable identifier (see Abaqus/Standard Output Variable Identifiers and Abaqus/Explicit Output Variable Identifiers, for available output variable identifiers).
There are some variables that can be read from the output database (.sim) file for which you cannot request output. For example, you cannot request output for the variable ORIENT; it is written automatically to the output database (.sim) file by Abaqus/Standardand Abaqus/Explicit if an orientation is associated with the results data in the analysis.
The location of the results can be either at nodes or elements. Results data requested at integration points are mapped to the centroid of the target element, and only data at the first section point in the shell element are imported. No interpolation or averaging is performed through the thickness.
When importing tensor or vector fields, the field data are transformed to the global system before mapping; mapped data are then transformed to the local system if specified in the target element.

You can specify a source region (node or element set in the previous model) if data are imported only from a subset of the previous model. Sometimes a source region is also specified to eliminate ambiguity during mapping. You can specify a target region if data are specified only on a subset of the current model.

You must specify the full name of the output database file including the file extension .sim.

You can import only results data requested on two- and three-dimensional continuum elements and three-dimensional conventional shell elements.

You can specify mapping tolerances and special tensor averaging methods if mapping is performed. If the model in the previous analysis is repositioned in the current analysis, you must specify the translation and rotation so that the source region can be repositioned before data are imported, except in the following cases:

Scalar data are imported from a matching mesh.
Tensor data are imported from a matching mesh, and there is no rotation between the source region and the target region.

Applying Distributions

The data defined in a distribution are not used in an Abaqus analysis unless the distribution is referred to by name by a feature that supports distributions, and the distribution is applied only to the elements, nodes, or element faces that are associated with the referenced feature. In addition, a distribution definition can be referenced more than one time in a given model. These points are illustrated in the examples below.

If an element in an Abaqus/Standard or Abaqus/Explicit analysis is declared rigid (see Rigid Body Definition) any distributions used to define element properties, material properties (with the exception of density), or local coordinate systems are ignored.

Examples

The simple examples below illustrate how distributions are defined. A large number of illustrative example problems using distributions can be found in Spatially varying element properties.

Example 1

A distribution for shell thickness is defined and applied to two different shell section definitions through the SHELL THICKNESS parameter—as noted above the distribution dist0 would not be used if it is not referred to by a feature that supports distributions. See Using a Shell Section Integrated during the Analysis to Define the Section Behavior for more details. The distribution table defines both the number of data values (one) and the physical dimension (LENGTH) of the thickness data. The thicknesses defined in distribution dist0 are assigned only to shell elements that belong to the element set elset1 or elset2. The default thickness (t₀) defined in the first data line of dist0 will be assigned to all elements in elset1 and elset2 that are not explicitly assigned a thickness in dist0.

DISTRIBUTION TABLE, NAME=tab0
	LENGTH
DISTRIBUTION, NAME=dist0, LOCATION=element, TABLE=tab0
	                     , t₀
	element set or number, t₁
	element set or number, t₂
 …
SHELL SECTION, ELSET=elset1, SHELL THICKNESS=dist0
SHELL SECTION, ELSET=elset2, SHELL THICKNESS=dist0

Example 2

A distribution for spatially varying isotropic elastic material behavior is defined and applied to a material definition (Linear Elastic Behavior). This material is then referred to by a solid section definition. This is important because like any material definition, a material defined by a distribution is not used unless it is referred to by a section definition, and then it is applied only to the elements associated with the section definition. The distribution table defines both the number of data values (two) and the physical dimensions (MODULUS and RATIO) of the isotropic elastic data. Other material behaviors (in this case plasticity) can also be included in the material definition. The default elastic constants (E₀, $ν$ ₀) in distribution dist1 will be assigned to all elements in elset3 that are not explicitly assigned elastic constants in dist1.

DISTRIBUTION TABLE, NAME=tab1
	MODULUS, RATIO
DISTRIBUTION, NAME=dist1, LOCATION=element, TABLE=tab1
	                     , E₀,  $ν$ ₀
	element set or number, E₁,  $ν$ ₁
	element set or number, E₂,  $ν$ ₂
 …
MATERIAL, NAME=MAT
ELASTIC
	dist1
PLASTIC
 …
SOLID SECTION, ELSET=elset3, MATERIAL=MAT

Example 3

A spatially varying local coordinate system ( Orientations) is defined by specifying both spatially varying coordinates for points a and b as well as a spatially varying additional rotation angle. This orientation is then referred to by a general shell section definition. This is important because like any orientation definition, an orientation defined by a distribution is not used unless it is referred to by a section definition, and then it is applied only to the elements associated with the section definition. The distribution table for the coordinates specifies COORD3D twice to indicate that data for two three-dimensional coordinates points must be specified for each element in the distribution.

DISTRIBUTION TABLE, NAME=tab2
	COORD3D, COORD3D
DISTRIBUTION, NAME=dist2, LOCATION=element, TABLE=tab2
	                     , aX₀,aY₀,aZ₀,bX₀,bY₀,bZ₀ 
	element set or number, aX₁,aY₁,aZ₁,bX₁,bY₁,bZ₁
	element set or number, aX₂,aY₂,aZ₂,bX₂,bY₂,bZ₂
 …
DISTRIBUTION TABLE, NAME=tab3
	ANGLE
DISTRIBUTION, NAME=dist3, LOCATION=element, TABLE=tab3
                      ,  $θ$ ₀ 
 element set or number,  $θ$ ₁
 element set or number,  $θ$ ₂
 …
ORIENTATION, NAME=ORI, DEFINITION=COORDINATES
	dist2
 3, dist3
SHELL GENERAL SECTION, ELSET=elset4, ORIENTATION=ORI

Example 4

Spatially varying thicknesses and orientation angles are defined on the layers of a composite shell element. The distribution table for the thicknesses specifies LENGTH, and the distribution table for the orientation angles specifies ANGLE. A distribution of thicknesses is used on layers 1 and 3, while a distribution of angles is used on layers 2 and 3.

DISTRIBUTION TABLE, NAME=tableThick
	LENGTH
DISTRIBUTION, NAME=thickPly1, LOCATION=element, TABLE=tableThick
	                     , t₀
	element set or number, t₁
	element set or number, t₂
 …
DISTRIBUTION, NAME=thickPly3, LOCATION=element, TABLE=tableThick
	                     , t₀
	element set or number, t₁
	element set or number, t₂
 …
DISTRIBUTION TABLE, NAME=tableOriAngle
	ANGLE
DISTRIBUTION, NAME=oriAnglePly2, LOCATION=element, 
TABLE=tableOriAngle
	                     ,  $ϕ$ ₀
	element set or number,  $ϕ$ ₁
	element set or number,  $ϕ$ ₂
 …
DISTRIBUTION, NAME=oriAnglePly3, LOCATION=element, 
TABLE=tableOriAngle
	                     ,  $ϕ$ ₀
	element set or number,  $ϕ$ ₁
	element set or number,  $ϕ$ ₂
 …
SHELL SECTION, ELSET=elset1, COMPOSITE
 thickPly1, 3, mat1, 0.
        1., 3, mat2, oriAnglePly2
 thickPly3, 3, mat3, oriAnglePly3

Example 5

A distribution for a spatially varying volume fraction is defined in distribution distVF. A distribution for a spatially varying aspect ratio is defined in distribution distAR, and a distribution for the second-order orientation tensor is defined in distribution distOriTens. These distributions are then applied to the definition of a constituent, which is then used in the material definition of a multiscale material with mean-field homogenization (Mean-Field Homogenization). The distribution table for the volume fraction and aspect ratio specifies RATIO, and the distribution table for the orientation tensor specifies ORITENS.

DISTRIBUTION TABLE, NAME=vfTable
	RATIO
DISTRIBUTION TABLE, NAME=arTable
	RATIO
DISTRIBUTION TABLE, NAME=oriTable
	ORITENS
DISTRIBUTION, NAME=distVF, LOCATION=element, TABLE=vfTable
	element set or number, v_f
DISTRIBUTION, NAME=distAR, LOCATION=element, TABLE=arTable
	element set or number, ar
DISTRIBUTION, NAME=distOriTens, LOCATION=element, TABLE=oriTable
	element set or number, a₁₁,a₂₂,a₃₃,a₁₂,a₁₃,a₂₃
 …
MATERIAL, NAME=MAT
MEAN FIELD HOMOGENIZATION
CONSTITUENT
 distVF, distAR, distOriTens