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# Piecewise linear probability density function

A piecewise linear probability density function can be used to approximate general distributions that are not well represented by the other PDF forms discussed above. With a piecewise linear probability density function, you specify PDF values at discrete points. Abaqus/Explicit considers linear variations in the PDF between these points, as shown in Figure 2.12.1–5. The PDF is zero below the first data point and above the last data point.

![](images/page-351_1a5357c6e7308df4e978739e7e132d6d8a1d706fd825aa732a396d02522cbbe6.jpg)

<details>
<summary>line</summary>

| x       | f(x) |
| ------- | ---- |
| X₁      | Low  |
| X₂      | Medium |
| Xₖ      | High |
| Xₖ₊₁    | High |
| Xₙ      | Medium |
</details>

Figure 2.12.1–5 Piecewise linear PDF.

As mentioned earlier, the area under a PDF is unity. Abaqus/Explicit will renormalize the specified PDF data to achieve this requirement. This renormalization of data values allows you to specify relative PDF values that may be obtained from a histogram. A histogram contains the data in the form of a table of random variable ranges and the percentage or number that fall within those ranges. As shown in Figure 2.12.1–6, you specify a table of the midpoint value of each range in the histogram and the corresponding count:

$$
\begin{array}{l} x _ {1} \quad c (x _ {1}) \\ \left\{ \begin{array}{c c} x _ {2} & c (x _ {2}) \\ x _ {3} & c (x _ {3}) \\ \ldots & \ldots \end{array} \right. \\ x _ {n} \quad c (x _ {n}) \\ \end{array}
$$

As mentioned above, Abaqus/Explicit will renomalize these data to create the piecewise linear probability density function.

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![](images/page-352_20de3a46616f8b20801209c281d8a38ba714219071aa3bd553f31b84eff529a6.jpg)

<details>
<summary>line</summary>

| x       | Counts |
| ------- | ------ |
| x₁      | 0      |
| x₂      | 0.5    |
| xₖ      | 1.0    |
| xₖ₊₁    | 1.0    |
| xₙ      | 0.5    |
</details>

Figure 2.12.1–6 Histogram.

There may be situations where the random variable has continuous values over certain ranges and discrete values elsewhere. Figure 2.12.1–7 shows the use of a piecewise linear probability density function to approximate such distributions where the discrete values are approximated by continuous random variables spanning a very narrow range of values (for example, the discrete value $x _ { 7 }$ is approximated by the continuous range from to ).

![](images/page-352_a583d1fb1c46ccc47465c93a65af3eb3b669f1209564749ba2785fa4644f311a.jpg)

<details>
<summary>line</summary>

| x    | f(x) |
| ---- | ---- |
| X₁   | 0    |
| X₂   | 0    |
| X₃   | 1    |
| X₄   | 0    |
| X₅   | 0    |
| X₆   | 0    |
| X₇   | 1    |
| X₈   | 0    |
| X₉   | 0    |
| X₁₀  | 0    |
| X₁₁  | 0    |
| X₁₂  | 1    |
| X₁₃  | 0    |
| X₁₄  | 0    |
| X₁₅  | 1    |
| X₁₆  | 1    |
| X₁₇  | 0    |
| X₁₈  | 0    |
</details>

Figure 2.12.1–7 Approximating a discrete probability distribution using a piecewise linear PDF.

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

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# Discrete probability density function

Some applications have only certain specific outcomes. These applications can be represented by a discrete probability density function, as shown in Figure 2.12.1–8. A simple example is throwing of a pair of dice. Only the outcomes of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 are possible, with the probabilities of 1/36, 2/36, 3/36, 4/36, 5/36, 6/36, 5/36, 4/36, 3/36, 2/36, and 1/36, respectively. A very specific case of a discrete probability density function is the case when only one value occurs with the probability of 1. To specify a discrete probability density function, you provide a table of the specific values of the random variable along with the corresponding probability:

$$
\left\{ \begin{array}{c c} x _ {1} & p _ {1} \\ x _ {2} & p _ {2} \\ x _ {3} & p _ {3} \\ \ldots & \ldots \\ x _ {n} & p _ {n} \end{array} \right.
$$

Abaqus/Explicit will renormalize the specified probabilities to ensure that they sum up to 1.

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

![](images/page-353_145a7e30e8b51872c49882aab81f974565179e9f1acde2446492d64b42669739.jpg)

<details>
<summary>bar</summary>
| x | f(x_k) |
|---|---|
| x₁ | f₁ |
| x₂ | f₄ |
| x₃ | fₖ |
| x₄ | f₄ |
| x_k | f₃ |
| x_n | f₂ |
</details>

Figure 2.12.1–8 Discrete PDF.

# Truncated probability density function

The normal and log-normal probability density functions have open-ended characteristics. These PDFs can be truncated to enforce upper and lower bounds on the value of the random variable. Figure 2.12.1–9

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shows a truncated normal distribution ${ \hat { f } } ( x )$ where all values of the random variable $\textit { x } < \textit { x } _ { m i n }$ and $x > x _ { m a x }$ from the untruncated normal distribution $f ( x )$ have been rejected.

$$
\hat {f} (x) = \left\{ \begin{array}{c c} 0 & x <   x _ {m i n} \\ \frac {f (x)}{k} & x _ {m i n} \leq x \leq x _ {m a x} \\ 0 & x > x _ {m i n} \end{array} \right.,
$$

where

$$
k = \int_ {x _ {m i n}} ^ {x _ {m a x}} f (x) d x.
$$

The factor represents the probability that the random variable is in the range from $x _ { m i n }$ to $x = x _ { m a x }$ for the untruncated PDF.

![](images/page-354_a5bf2143a2ce433d9f4749d910cdc50f165f5e31df196b04a20bb0347c7e757e.jpg)

<details>
<summary>line</summary>
| x       | f(x) (original PDF) | f(x) (truncated PDF) |
| ------- | ------------------- | -------------------- |
| 0       | 0                   | 0                    |
| X_min   | ~0.8                | ~0.9                 |
| X_max   | ~0.2                | ~0.3                 |
</details>

Figure 2.12.1–9 Truncated PDF.

You specify the lower and upper limits of the random variable along with the mean and standard deviation for these types of PDFs. The uniform and the piecewise linear distributions have lower and upper limits for the random variable built into the definition of the PDF and, therefore, do not require renormalization because of truncation.

# Output

No output is available for probability density functions.

# Limitations

Probability density functions are supported only for the size distributions of PD3D elements created using a particle generator.

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The following example illustrates the use of a probability density function for particle size distribution:

```c
*HEADING
...
*PARTICLE GENERATOR, NAME=generator_name, TYPE=PD3D,
MAXIMUM NUMBER OF PARTICLES=number
**
*PARTICLE GENERATOR INLET, SURFACE=inlet_surf
**
*PARTICLE GENERATOR MIXTURE
gen_SET1, gen_SET2
**
*PROBABILITY DENSITY FUNCTION, NAME=PDF_gen_SET1, TYPE=NORMAL
Data line to define PDF
*PROBABILITY DENSITY FUNCTION, NAME=PDF_gen_SET2, TYPE=LOGNORMAL
Data line to define PDF
**
*DISCRETE SECTION, ELSET=gen_SET1
PDF_gen_SET1
*DISCRETE SECTION, ELSET=gen_SET2
PDF_gen_SET2
...
*END STEP 
```

# Additional references

• Benjamin, J. R., and C. A. Cornell, “Probability, Statistics, and Decision for Civil Engineers,” McGraw-Hill, 1970.   
• Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical Recipes in Fortran 77, The Art of Scientific Computing,” University of Cambridge, 1992.   
• Saucier, R., “Computer Generation of Statistical Distributions,” Army Research Laboratory, 2000.

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# 3. Job Execution

Execution procedures: overview 3.1

Execution procedures 3.2

Environment file settings 3.3

Managing memory and disk resources 3.4

Parallel execution 3.5

File extension definitions 3.6

Fortran unit numbers 3.7

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# 3.1 Execution procedures: overview

• “Execution procedure for Abaqus: overview,” Section 3.1.1

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