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Changes to the matrix due to large rotations or load stiffness are not computed in a geometrically nonlinear analysis.

# Using matrices in linear perturbation analyses

Matrices can be used in a static perturbation analysis as well as in a natural frequency extraction analysis using the Lanczos or AMS eigensolver. For certain quantities (such as participation factors and global inertia properties) to be computed properly, the coordinates of the nodes associated with the matrices should be defined in the model using matrices. Matrices can also be used in modal analysis procedures using the high-performance SIM architecture; namely, steady-state dynamic, modal dynamic, random response, response spectrum, and complex frequency extraction analyses. Matrices can be used in the substructure generation and matrix generation procedures as well.

Matrices cannot be used in the direct-solution steady-state dynamic analysis procedure and in modal procedures that are not based on the high-performance SIM architecture.

# Constraints and transformations

Kinematic constraints (for example, coupling constraints, linear constraint equations, multi-point constraints, or surface-based tie constraints) can be applied to any nodes in a model containing matrices. Since kinematic constraints in Abaqus/Standard are usually imposed by eliminating degrees of freedom at the dependent nodes, matrix nodes should not be used as dependent nodes.

To apply contact constraints on matrix nodes, a node-based surface must be defined on these nodes and this surface should be used as the slave surface in the contact pair definition.

Nodal transformations defined at nodes that appear in the matrix do not affect the matrix. The matrix entries corresponding to these nodes are assumed to be in the local coordinates defined by the nodal transformations.

# Initial conditions

Initial conditions can be specified as usual; however, only node-based initial conditions can be applied to nodes that appear in matrices. See “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1.

# Boundary conditions

Boundary conditions can be specified as usual. See “Boundary conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.3.1. Matrix nodes can be defined as driven nodes in a submodel analysis (see “Submodeling: overview,” Section 10.2.1); they cannot be defined as driving nodes in a global model. For shell-to-solid submodeling, matrix nodes that are defined as driven nodes are treated as lying within the center zone no matter how far they are from the shell reference surface.

# Loads

Concentrated nodal forces can be applied at displacement degrees of freedom (1–6) of any node as usual. Distributed pressure forces can be applied to surface elements defined over matrix nodes (see “Surface

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elements,” Section 32.7.1). Body forces cannot be applied to parts of the model represented by matrices. User-defined loads can be applied with the same restrictions as above for distributed pressure forces and body forces.

Predefined fields can be applied at any nodes as usual (see “Predefined field variables” in “Predefined fields,” Section 34.6.1, and “Predefined temperature” in “Predefined fields,” Section 34.6.1); however, matrix data are not affected by predefined fields. For example, if temperatures are specified as a predefined field on nodes that appear on a matrix, only the elements that share these nodes with the matrix experience thermal strains if thermal expansion is specified for those elements. The matrix does not experience any thermal strains, but it may experience linear elastic forces due to displacements at shared nodes.

# Elements

All elements that can be used in static stress analysis are available (see “Choosing the appropriate element for an analysis type,” Section 27.1.3).

# Output

All nodal output variables that apply to static analysis are available (see “Abaqus/Standard output variable identifiers,” Section 4.2.1).

# Limitations

The following are known limitations to using matrices:

• Matrices cannot be used in a model containing parts and assemblies.   
• Matrices containing acoustic pressure and mechanical degrees of freedom will disable the coupled acoustic structural eigenvalue extraction.   
• By default, using the matrix data containing internal nodes in text format is not supported. Usage of such matrices in text format can be allowed for some special cases. This feature should be used with caution.   
• In an Abaqus/Standard analysis using matrix input data for the mass matrix, inertia quantities for the global model that are reported in the data (.dat) file, including coordinates of the center of mass and moments of inertia, may be calculated incorrectly.   
• Matrices cannot be used in analyses with inertia relief loads.   
• Matrices cannot be used in direct steady-state dynamic analysis or in mode-based analyses that do not use the SIM-based architecture.

# Input file template

\*HEADING

\*NODE

Data lines to specify nodes

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\*NSET, NSET=NSET1, UNSORTED

Data lines to specify a node set with the nodes in a particular order

\*BOUNDARY

Data lines to specify zero-valued boundary conditions

\*MATRIX INPUT, NAME=MAT1, SCALE FACTOR=sval

Data lines to specify a stiffness matrix

\*MATRIX INPUT, NAME=MAT2, SCALE FACTOR=sval

Data lines to specify a mass matrix

\*MATRIX INPUT, NAME=MAT3, SCALE FACTOR=sval

Data lines to specify a viscous damping matrix

\*MATRIX INPUT, NAME=MAT4, INPUT=input\_file\_name

\*MATRIX INPUT, NAME=MAT5, INPUT=input\_file\_name

\*MATRIX INPUT, NAME=MAT6, INPUT=sim\_file\_name, MATRIX=STIFFNESS

\*MATRIX ASSEMBLE, STIFFNESS=MAT1, MASS=MAT2,

VISCOUS DAMPING=MAT3, STRUCTURAL DAMPING=MAT4

\*MATRIX ASSEMBLE, STIFFNESS=MAT6, MASS=MAT5

\*MATRIX ASSEMBLE, STIFFNESS=MAT6, MASS=MAT5, NSET=NSET1

\*STEP(,NLGEOM)(,PERTURBATION)

Use NLGEOM to include nonlinear geometric effects; it will remain active in all subsequent steps.

\*STATIC

\*BOUNDARY

Data lines to prescribe zero-valued or nonzero boundary conditions

\*CLOAD and/or \*DLOAD

Data lines to specify loads

\*END STEP

\*STEP

\*FREQUENCY

\*BOUNDARY

Data lines to prescribe zero-valued or nonzero boundary conditions

\*END STEP

\*STEP

\*STEADY STATE DYNAMICS

\*CLOAD and/or \*DLOAD

Data lines to specify loads

\*END STEP

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# 2.12 Probability density function

• “Probability density function,” Section 2.12.1

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# 2.12.1 PROBABILITY DENSITY FUNCTION

Products: Abaqus/Explicit Abaqus/Viewer

# References

• “Discrete element method,” Section 15.1.1   
• “Discrete particle elements,” Section 33.1.1   
• \*PARTICLE GENERATOR   
• \*PARTICLE GENERATOR MIXTURE   
• \*PROBABILITY DENSITY FUNCTION   
• \*DISCRETE SECTION

# Overview

A probability density function:

• is used to define the statistical distributions of a continuous random variable; and   
• can be defined for uniform, normal, log-normal, piecewise linear, and discrete distributions.

# Introduction

There are many examples of randomness associated with data. Particle sizes in a granular media such as gravel are an example. Randomness observed in data can be described by statistical distributions. Pseudo-random numbers that are generated based on statistical distributions are used to capture randomness in data in a numerical simulation.

# Applications

The size distribution of particle species generated by a particle generator can be described by statistical distributions.

# Probability density function

A probability density function (PDF) describes the probability of the value of a continuous random variable falling within a range. If the random variable can only have specific values (like throwing dice), a probability mass function (PMF) would be used to describe the probabilities of the outcomes. The plot on the left in Figure 2.12.1–1 shows a PDF for the random variable . The probability that the random variable has a value in the range and is . The probability that the random variable will be in the range $a \leq x \leq b$ is given by:

$$
\operatorname * {P r} [ a \leq x \leq b ] = \int_ {a} ^ {b} f (x) d x.
$$

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The probability that the random variable is in the range and is one; i.e.,

$$
\int_ {- \infty} ^ {\infty} f (x) d x = 1.
$$

The area under the PDF curve is, therefore, always unity.

![](images/page-348_a7b17aef1e378064206d4d8211fa668637e5ac7ece707a09c598e67ec9e0d295.jpg)

<details>
<summary>line</summary>
| x          | f(x)  |
|------------|-------|
| x + dx     | f(x)  |
</details>

![](images/page-348_b94cb02733f378a2aa13281f65e4dfc4b22a877dd37d8a4e4905fc2e91185a56.jpg)

<details>
<summary>bar</summary>

| x | f(x) |
|---|---|
| 2 | 1/36 |
| 3 | 2/36 |
| 4 | 3/36 |
| 5 | 4/36 |
| 6 | 5/36 |
| 7 | 6/36 |
| 8 | 5/36 |
| 9 | 4/36 |
| 10 | 3/36 |
| 11 | 2/36 |
| 12 | 1/36 |
</details>

Figure 2.12.1–1 Probability distributions of continuous and discrete variables.

The plot on the right in Figure 2.12.1–1 shows a PMF where the horizontal axis shows the specific values of the random variable and the vertical axis shows the corresponding probabilities.

Abaqus/Explicit supports uniform, normal (Gaussian), log-normal, piecewise linear, and discrete probability density functions. To define a probability density function, you must assign it a name and specify its type.

Input File Usage: \*PROBABILITY DENSITY FUNCTION, NAME=PDF\_name, $\scriptstyle \mathrm { T Y P E } = P D F \_ t y p e$

# Uniform probability density function

Uniform distributions (shown in Figure 2.12.1–2) have many applications, particularly in the numerical simulation of random processes. The following function describes a uniform probability density function for a random variable between $x _ { \mathrm { m i n } }$ and $x _ { \mathrm { m a x } }$ :

$$
f (x) = \left\{ \begin{array}{c c} \frac {1}{x _ {\max} - x _ {\min}} & x _ {\min} \leq x \leq x _ {\max} \\ 0 & \text {otherwise} \end{array} \right..
$$

The mean is $\begin{array} { r } { \mu = \frac { x _ { \mathrm { m i n } } + x _ { \mathrm { m a x } } } { 2 } } \end{array}$ and the variance is $\begin{array} { r } { \sigma ^ { 2 } = \frac { ( x _ { \mathrm { m a x } } - x _ { \mathrm { m i n } } ) ^ { 2 } } { 1 2 } } \end{array}$ . You specify $x _ { \mathrm { m i n } }$ and $x _ { \mathrm { m a x } }$ for the uniform distribution.

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![](images/page-349_5e94ef1722bd3fd4434bc64b93ddf49cbb39f587344de08c39e61b7455b1b165.jpg)

<details>
<summary>line</summary>

| x    | f(x) |
| ---- | ---- |
| 0    | 1    |
| 0.25 | 1    |
| 0.5  | 1    |
| 0.75 | 1    |
| 1    | 1    |
</details>

Figure 2.12.1–2 Uniform PDF.

Input File Usage: \*PROBABILITY DENSITY FUNCTION, TYPE=UNIFORM

# Normal probability density function

Normal distributions (shown in Figure 2.12.1–3) have many applications in science and engineering; for example, errors in experimental measurements are often assumed to have a normal distribution. The following function describes a normal probability density function:

$$
f (x) = \frac {1}{\sqrt {2 \pi} \sigma} e ^ {\left[ \frac {- (x - \mu) ^ {2}}{2 \sigma^ {2}} \right]}.
$$

The mean is standard de $\begin{array} { r } { \mu = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } x _ { i } } \end{array}$ , and the variance is normal distribution $\begin{array} { r } { \sigma ^ { 2 } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } ( x _ { i } - \mu ) ^ { 2 } } \end{array}$ . You specify the mean, $\mu ,$ and

Input File Usage: \*PROBABILITY DENSITY FUNCTION, TYPE=NORMAL

# Log-normal probability density function

Log-normal distributions (shown in Figure 2.12.1–4) are used in describing many natural phenomena. They are commonly used to describe particle size distributions in soils. The following function describes a log-normal probability density function:

$$
f (x) = \left\{ \begin{array}{c c} \frac {1}{\sqrt {2 \pi} \sigma x} e ^ {\left[ \frac {- [ \ln (x) - \mu ] ^ {2}}{2 \sigma^ {2}} \right]} & x > 0 \\ 0 & \mathrm{otherwise} \end{array} \right..
$$

The mean, $\hat { \mu } ,$ , and standard deviation, ${ \hat { \sigma } } ,$ in the space are related to $\mu$ and $\sigma$ as follows:

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![](images/page-350_6e4a0436cd2d42d2a62598415cfdb4bc4ef0aca1c4985997609b7ac37a991f5c.jpg)

<details>
<summary>line</summary>

| x   | f(x) |
| --- | ---- |
| -3  | 0.00 |
| -2  | 0.05 |
| -1  | 0.25 |
| 0   | 0.40 |
| 1   | 0.25 |
| 2   | 0.05 |
| 3   | 0.00 |
</details>

Normal density function   
Figure 2.12.1–3 Normal PDF.

$$
\hat {\mu} = e ^ {\left[ \mu + \frac {1}{2} \sigma^ {2} \right]}
$$

$$
\hat {\sigma^ {2}} = e ^ {2 \mu} e ^ {\sigma^ {2}} (e ^ {\sigma^ {2}} - 1),
$$

where the parameters $\begin{array} { r } { \mu = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \ln ( x _ { i } ) } \end{array}$ and $\mu$ $\begin{array} { r } { \sigma = \sqrt { \frac { 1 } { N } \sum _ { i = 1 } ^ { N } ( \ln ( x _ { i } ) - \mu ) ^ { 2 } } } \end{array}$ and $\sigma$ are the mean and standard deviation in the space; given by respectively. You specify the mean, $\hat { \mu } ,$ and the standard deviation, , in the space for the log-normal distribution.

![](images/page-350_3bc2ed315eeb41f8a7b9095dbbe6cd6fb539ec844e5bb330e2859af332b36c18.jpg)

<details>
<summary>line</summary>
| x   | f(x) |
| --- | ---- |
| 0   | 0.0  |
| 0.5 | 0.6  |
| 1   | 0.5  |
| 2   | 0.2  |
| 3   | 0.1  |
| 4   | 0.0  |
</details>

Lognormal density function   
Figure 2.12.1–4 Log-normal PDF.

Input File Usage: \*PROBABILITY DENSITY FUNCTION, TYPE=LOGNORMAL
