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![](images/page-111_f38ebe42ab2e1add2cf63a57abab694cc5167f5990f7f3c477ac7b56ac412b24.jpg)

<details>
<summary>text_image</summary>

a
New Set
Old set
</details>

a is the point through which the nodes are reflected

Figure 2.1.1–6 Reflection of coordinates through a point.   
![](images/page-111_a664d04a2d59c24a488317f35c435799fa843aa74d146031f608a04bc5a7a87f.jpg)

<details>
<summary>text_image</summary>

L
L
pole
node
a
old set
new set
</details>

Figure 2.1.1–7 Projection of existing nodes from a pole node.

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You can create nodes by filling in nodes between two bounds. In this case you specify the two node sets whose members form the bounds, the number of intervals along each line between the bounding nodes, and the increment in node numbers from the node number at the first bound set end.

Let l equal the number of lines of nodes to be created between the two bounding node sets; the number of intervals along each line between the bounding nodes is then given by $l + 1$ .

Let n equal the increment in node numbers from the node number at the first bound set end; for each node $( n _ { A _ { i } } )$ in the first bounding node set, the corresponding node in the other bounding node set $( n _ { B _ { i } } )$ must be numbered such that $( n _ { B _ { i } } - n _ { A _ { i } } ) / n$ is a whole number.

The node sets that define the bounds of the region are used as they exist at the time the node fill definition appears in the input file: only those nodes that have been added to the sets prior to the node fill definition are used. Both sorted and unsorted node sets can be used. Nodes that have not yet been given coordinates are assumed to be at the origin, (0.,0.,0.).

The nodes created by this method lie on straight lines between corresponding nodes in the two sets. If the sets do not have the same number of nodes, the extra nodes in the longer set are ignored. By default, the spacing between nodes along the lines is uniform.

Input File Usage: \*NFILL

# Example

For example, Figure 2.1.1–8 shows a simple quarter-cylinder model.

![](images/page-112_e7bd0640c7f554edc555f27e54cf486d33ad7cf0e5b6899b4e631de71b255836.jpg)

<details>
<summary>text_image</summary>

OUTSIDE A
6501
OUTSIDE B
6101
INSIDE B
1501
6105
1101
6505
INSIDE A
1105
1505
</details>

Figure 2.1.1–8 Filling a three-dimensional region.

The quarter circles INSIDEA (nodes 1101–1105), OUTSIDEA (nodes 1501–1505), INSIDEB (nodes 6101–6105), and OUTSIDEB (6501–6505) have already been defined by specifying their coordinates

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directly or generating them incrementally. The region is filled by first filling the end planes and placing the nodes on those planes into sets A and B and then filling between those sets with the following options:

```txt
*NFILL, NSET=A
INSIDEA, OUTSIDEA, 4, 100
*NFILL, NSET=B
INSIDEB, OUTSIDEB, 4, 100
*NFILL
A, B, 5, 1000 
```

# Concentrating the nodes toward one bound or the other

You can concentrate the nodes toward one bound or the other by specifying b, the ratio of adjacent distances between nodes along each line of nodes generated as the nodes go from the first bounding node set to the second.

Thus, if b is less than one, the nodes are concentrated toward the first bounding node set; if b is greater than one, the nodes are concentrated toward the second bounding set. The value of b must be positive.

The bias intervals along the line from the first bounding node are $L , L / b , L / b ^ { 2 } , L / b ^ { 3 } , L / b ^ { 4 } , L / b ^ { 5 }$ , … (where L is the length of the first interval). In Abaqus/Standard the bias value can be applied at every interval along the line or at every second interval along the line as described later.

Input File Usage: \*NFILL, BIAS=b

# Example

For example, suppose the lines of nodes shown in Figure 2.1.1–9 have already been generated by other methods and placed into node sets INSIDE and OUTSIDE. The following option will fill the region as shown in Figure 2.1.1–10:

```txt
*NFILL, BIAS=0.6
INSIDE, OUTSIDE, 5, 100 
```

Applying the bias value at every second interval along the line

In Abaqus/Standard you can apply the bias value at every second interval along the line. In this case the nodes will be positioned along the line correctly for use with second-order elements, so that the midside nodes are at the middle of the interval between the corner nodes of the elements.

The bias intervals along the line from the first bounding node are L, L, , , , , … (where L is the length of the first interval).

Input File Usage: \*NFILL, BIAS=b, TWO STEP

# Creating quarter-point spacing

In Abaqus/Standard you can create quarter-point spacing for fracture mechanics calculations with second-order isoparametric elements (“Fracture mechanics: overview,” Section 11.4.1). This spacing

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![](images/page-114_0ae7411436a1c63c230188a3c0c57d1247d4c17867a4975749b73c9d40338910.jpg)

<details>
<summary>line</summary>

| Position | Value |
| -------- | ----- |
| Inside   | 105   |
| Inside   | 104   |
| Inside   | 103   |
| Inside   | 102   |
| Inside   | 101   |
| Outside  | 605   |
| Outside  | 604   |
| Outside  | 603   |
| Outside  | 602   |
| Outside  | 601   |
</details>

Figure 2.1.1–9 Node sets defining bias example.   
![](images/page-114_34ead980e124b589073b1494d9f4fcdc1a6f28feb759742b6fb30abaf8af0678.jpg)

<details>
<summary>line</summary>

| Series | Value |
|---|---|
| 1 | 105 |
| 2 | 205 |
| 3 | 305 |
| 4 | 405 |
| 5 | 505 |
| 6 | 605 |
| 7 | 604 |
| 8 | 603 |
| 9 | 602 |
| 10 | 601 |
| 11 | 501 |
| 12 | 401 |
| 13 | 301 |
| 14 | 201 |
| 15 | 101 |
| 16 | 102 |
| 17 | 103 |
| 18 | 203 |
| 19 | 303 |
| 20 | 403 |
| 21 | 503 |
| 22 | 603 |
| 23 | 604 |
| 24 | 605 |
| 25 | 104 |
| 26 | 204 |
| 27 | 304 |
| 28 | 404 |
| 29 | 504 |
| 30 | 604 |
| 31 | 605 |
| 32 | 105 |
| 33 | 205 |
| 34 | 305 |
| 35 | 405 |
| 36 | 505 |
| 37 | 605 |
| 38 | 605 |
| 39 | 605 |
| 40 | 605 |
| 41 | 605 |
| 42 | 605 |
| 43 | 605 |
| 44 | 605 |
| 45 | 605 |
| 46 | 605 |
| 47 | 605 |
| 48 | 605 |
| 49 | 605 |
| 50 | 605 |
| 51 | 605 |
| 52 | 605 |
| 53 | 605 |
| 54 | 605 |
| 55 | 605 |
| 56 | 605 |
| 57 | 605 |
| 58 | 605 |
| 59 | 605 |
| 60 | 605 |
| 61 | 605 |
| 62 | 605 |
| 63 | 605 |
| 64 | 605 |
| 65 | 605 |
| 66 | 605 |
| 67 | 605 |
| 68 | 605 |
| 69 | 605 |
| 70 | 605 |
| 71 | 605 |
| 72 | 605 |
| 73 | 605 |
| 74 | 605 |
| 75 | 605 |
| 76 | 605 |
| 77 | 605 |
| 78 | 605 |
| 79 | 605 |
| 80 | 605 |
| 81 | 605 |
| 82 | 605 |
| 83 | 605 |
| 84 | 605 |
| 85 | 605 |
| 86 | 605 |
| 87 | 605 |
| 88 | 605 |
| 89 | 605 |
| 90 | 605 |
| 91 | 605 |
| 92 | 605 |
| 93 | 605 |
| 94 | 605 |
| 95 | 605 |
| 96 | 605 |
| 97 | 605 |
| 98 | 605 |
| 99 | 605 |
| 100 | 605 |
</details>

Figure 2.1.1–10 Result of bias example.

gives a square root singularity in the strain field at the crack tip by placing the first node away from

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that point at one-quarter of the distance to the second point. The remaining nodes on each line are spaced so that the size of the elements will grow as the square of the distance from the singularity, with the midside nodes exactly at the midsides of the elements. This spacing produces a reasonable mesh gradation for this type of problem; however, better results can be obtained for crude meshes by making the size of the crack element smaller than the quarter-point spacing technique does.

Input File Usage: \*NFILL, SINGULAR

# Example

Figure 2.1.1–11 shows a simple fracture mechanics example.   
![](images/page-115_e8ddfbdc02323904508014cda3750aa91cb59fd95bdd5fe2887f641a264ad927.jpg)

<details>
<summary>network</summary>

| Node Set | Node Count |
| :--- | :--- |
| Node set TOP | 507 |
| Node set TOP | 506 |
| Node set TOP | 505 |
| Node set TOP | 504 |
| Node set TOP | 503 |
| Node set MID | 107 |
| Node set MID | 106 |
| Node set MID | 105 |
| Node set MID | 104 |
| Node set MID | 103 |
| Nodes 101-109 in node set OUTER | 108 |
| Nodes 101-109 in node set OUTER | 109 |
| Nodes 101-109 in node set OUTER | 102 |
| Nodes 101-109 in node set OUTER | 101 |
Nodes 1-9 at crack tip (node set TIP)
</details>

Figure 2.1.1–11 Node fill used in a singular problem.

(The mesh shown is very coarse, and a finer mesh would probably be used in an actual case.) The nodes on the top edge have been placed in node set TOP, those on the horizontal line at the upper end of the focused region are in node set MID, all of the nodes around the focused region are in node set OUTER, and there are multiple nodes at the crack tip in node set TIP. The following options are used to fill in the region as shown in Figure 2.1.1–12 (note the quarter-point nodes adjacent to the crack tip):

```csv
*NFILL, BIAS=0.8
MID, TOP, 4, 100
*NFILL, SINGULAR=1
TIP, OUTER, 5, 20 
```

# Mapping a set of nodes from one coordinate system to another

You can map a set of nodes from one coordinate system to another. You can also rotate, translate, or scale the nodes in a set by using a more direct method instead of coordinate system mapping. These capabilities

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![](images/page-116_07cfd176fcf186cc219a5f427dcc65f1e94d94129bfa9846130f7390cec40a96.jpg)

<details>
<summary>radar</summary>

| Point | Value |
|---|---|
| 1 | 101 |
| 2 | 22 |
| 3 | 42 |
| 4 | 62 |
| 5 | 81 |
| 6 | 101 |
| 7 | 102 |
| 8 | 103 |
| 9 | 203 |
| 10 | 303 |
| 11 | 403 |
| 12 | 503 |
</details>

Figure 2.1.1–12 Node fill used in a singular problem.

are useful for many geometric situations: a mesh can be generated quite easily in a local coordinate system (for example, on the surface of a cylinder) using other methods and then can be mapped into the global (X, Y, Z) system. In other cases some parts of your model need to be translated or rotated along a given axis or scaled with respect to one point.

The mapping capability cannot be used in a model defined in terms of an assembly of part instances.

The following different mappings are provided: a simple scaling; a simple shift and/or rotation; skewed Cartesian; cylindrical; spherical; toroidal; and, in Abaqus/Standard only, blended quadratic. The first five of these mappings are shown in Figure 2.1.1–13. Blended quadratic mapping is shown in Figure 2.1.1–14.

In all cases the coordinates of the nodes in the set are assumed to be defined in the local system: these local coordinates at each node are replaced with the global Cartesian (X, Y, Z) coordinates defined by the mapping. All angular coordinates should be given in degrees.

You can use either coordinates or node numbers to define the new coordinate system, the axis of rotation and translation, or the reference point used for scaling.

The mapping capability can be used several times in succession on the same nodes, if required.

# Scaling the local coordinates before they are mapped

For all mappings except the blended quadratic mapping, you can specify a scaling factor to be applied to the local coordinates before they are mapped.

This facility is useful for “stretching” some of the coordinates that are given. For example, in cases where the local system uses some angular coordinates and some distance coordinates (cylindrical, spherical, etc.), it may be preferable to generate the mesh in a system that uses distance measures in the angular directions and then scale onto the angular coordinate system for the mapping.

Two different scaling methods are available.

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![](images/page-117_f41d5ced530e7ef724198efcbf3aacdbb95a6b300a0ac0570e2241776ca28c84.jpg)

<details>
<summary>text_image</summary>

Z
Y
a
c
b
X
y
z
x
</details>

rectangular

![](images/page-117_c985d6302619c60d4f0845dfdd435da22e30d8a77db6dcdc8a8543517b7b957f.jpg)

<details>
<summary>text_image</summary>

a
b
c
d
X
Y
Z
z
y
x
</details>

skewed Cartesian

![](images/page-117_e825116b2d1e64e7a87629105e77a15cdb49ea4ce25a06f3f62d9c5a49b2d842.jpg)

<details>
<summary>text_image</summary>

Z
b
(R, θ, φ)
φ
R
θ
a
z
y
x
c
(θ = 0)
(φ = 0)
</details>

spherical

![](images/page-117_a60aa30cee9319c5be17f48eca4a52c22353caba5fd4f1ded35675ae3ce569e0.jpg)

<details>
<summary>text_image</summary>

Z
b
R
R, θ, Z
a
θ
c
(θ = 0)
z
y
x
</details>

cylindrical

![](images/page-117_52565175451dcf2afd2a87213596fc59b3b251fe7d390b780b717885d2d9cd9b.jpg)

<details>
<summary>text_image</summary>

z
y
x
a
φ
R
c
(r, θ, φ)
θ
r
b (φ = 0)
</details>

toroidal   
Figure 2.1.1–13 Coordinate systems; angles are in degrees.

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![](images/page-118_aafc7310187c32ad4a89a191ac296fbb77ac1c1e69a0d857340616059a75fb14.jpg)

<details>
<summary>other</summary>

| Point ID | Value  |
| -------- | ------ |
| 1        | 10134  |
| 2        | 10136  |
| 3        | 10138  |
| 4        | 10130  |
| 5        | 5138   |
| 6        | 10126  |
| 7        | 10124  |
| 8        | 10122  |
| 9        | 5122   |
| 10       | 10126  |
| 11       | 10124  |
| 12       | 122    |
| 13       | 126    |
| 14       | 130    |
| 15       | 136    |
| 16       | 138    |
| 17       | 134    |
| 18       | 124    |
| 19       | 122    |
| 20       | 126    |
| 21       | 124    |
| 22       | 122    |
| 23       | 126    |
| 24       | 124    |
| 25       | 122    |
| 26       | 126    |
| 27       | 124    |
| 28       | 122    |
| 29       | 126    |
| 30       | 124    |
| 31       | 122    |
| 32       | 126    |
| 33       | 124    |
| 34       | 122    |
| 35       | 126    |
| 36       | 124    |
| 37       | 122    |
| 38       | 126    |
| 39       | 124    |
| 40       | 122    |
| 41       | 126    |
| 42       | 124    |
| 43       | 122    |
| 44       | 126    |
| 45       | 124    |
| 46       | 122    |
| 47       | 126    |
| 48       | 124    |
| 49       | 122    |
| 50       | 126    |
| 51       | 124    |
| 52       | 122    |
| 53       | 126    |
| 54       | 124    |
| 55       | 122    |
| 56       | 126    |
| 57       | 124    |
| 58       | 122    |
| 59       | 126    |
| 60       | 124    |
| 61       | 122    |
| 62       | 126    |
| 63       | 124    |
| 64       | 122    |
| 65       | 126    |
| 66       | 124    |
| 67       | 122    |
| 68       | 126    |
| 69       | 124    |
| 70       | 122    |
| 71       | 126    |
| 72       | 124    |
| 73       | 122    |
| 74       | 126    |
| 75       | 124    |
| 76       | 122    |
| 77       | 126    |
| 78       | 124    |
| 79       | 122    |
| 80       | 126    |
| 81       | 124    |
| 82       | 122    |
| 83       | 126    |
| 84       | 124    |
| 85       | 122    |
| 86       | 126    |
| 87       | 124    |
| 88       | 122    |
| 89       | 126    |
| 90       | 124    |
| 91       | 122    |
| 92       | 126    |
| 93       | 124    |
| 94       | 122    |
| 95       | 126    |
| 96       | 124    |
| 97       | 122    |
| 98       | 126    |
| 99       | 124    |
| 100      | 10001  |
</details>

![](images/page-118_101bcc4234af7119a5e178e55e699f9b72e9481cb3cdefa90e2b04c3c2da96b6.jpg)

<details>
<summary>natural_image</summary>

Isometric line drawing of a rectangular block with internal diagonal lines and a curved cutout (no text or symbols)
</details>

ORIGINAL CONFIGURATION   
![](images/page-118_c3843279cbf26797a80dab5b3e77ae5fc94c0e230702d1ec940a462fba880254.jpg)

<details>
<summary>radar</summary>

| Label  | Value |
|--------|-------|
| 10134  | 10134 |
| 10136  | 10136 |
| 10138  | 10138 |
| 5134   | 5134  |
| 10130  | 10130 |
| 5138   | 5138  |
| 10001  | 10001 |
| 10126  | 10126 |
| 10124  | 10124 |
| 10122  | 10122 |
| 5122   | 5122  |
| 126    | 126   |
| 124    | 124   |
| 122    | 122   |
| 130    | 130   |
| 136    | 136   |
| 138    | 138   |
</details>

![](images/page-118_8b6ad5d3c864792a295b1a35d6f35f49895bc782e9cb00b4c98e853ea3228cc2.jpg)

<details>
<summary>natural_image</summary>

Isometric line drawing of a 3D geometric object with layered surfaces and a central circular hole (no text or symbols)
</details>

![](images/page-118_d7f6f64d30b83b5de38229a769b3b2a4867c2fcc686988a22a6b53ad9e01d297.jpg)  
MAPPED CONFIGURATION   
Figure 2.1.1–14 Use of blended quadratic mapping to develop a solid mesh onto a curved block.

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# Specifying the scaling factors directly

A first method of scaling the nodes with respect to the origin of the local system is to specify the scale factors directly. In this case the scaling is done at the same time as the mapping from one coordinate system to another.

# Input File Usage:

\*NMAP, NSET=name

first data line

second data line

scale factor for first local coord, scale factor for second local coord,

scale factor for third local coord

# Specifying the scaling with respect to a reference point

Alternatively, you can scale with respect to a point other than the origin. The reference point with respect to which the scaling is done can be defined by using either its coordinates or the user node number.

# Input File Usage:

Use the following option to define the scaling reference point by using its coordinates (default):

\*NMAP, TYPE=SCALE, DEFINITION=COORDINATES

X-coordinate of reference point, Y-coordinate of reference point,

Z-coordinate of reference point

scale factor for first local coord, scale factor for second local coord,

scale factor for third local coord

Use the following option to define the scaling reference point by using its node number:

\*NMAP, TYPE=SCALE, DEFINITION=NODES

Local node number of the reference point

scale factor for first local coord, scale factor for second local coord,

scale factor for third local coord

# Introducing a simple shift and/or rotation by mapping from one coordinate system to another

In the case of a simple shift and/or rotation, point a in Figure 2.1.1–13 defines the origin of the local rectangular coordinate system defining the map. The local -axis is defined by the line joining points a and b. The local – plane is defined by the plane passing through points a, b, and c.

# Input File Usage:

\*NMAP, NSET=name, TYPE=RECTANGULAR

# Introducing a pure shift by specifying the axis and magnitude of the translation

You can define a pure translation (or shift) to move a set of nodes by a prescribed value along a desired axis. You must specify the axis of translation by providing either the coordinates or the two node numbers defining this axis, and you must prescribe the magnitude of the translation.

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<table><tr><td>Input File Usage:</td><td>Use the following option to specify the axis of translation using coordinates (default):*NMAP, NSET=name, TYPE=TRANSLATION, DEFINITION=COORDINATESUse the following option to specify the axis of translation using node numbers:*NMAP, NSET=name, TYPE=TRANSLATION, DEFINITION=NODES</td></tr></table>

# Introducing a pure rotation by specifying the axis, origin, and angle of the rotation

You can define a rotation of a set of nodes by providing the axis of rotation, the origin of rotation, and the magnitude of the rotation. You must specify the axis of rotation by providing either the coordinates or the two node numbers defining this axis. You must specify the origin of the rotation by providing either the coordinates or the node number at the origin of rotation. Finally, you must specify the angle of the rotation in degrees.

Input File Usage: Use the following option to specify the axis of rotation using coordinates (default):

\*NMAP, NSET=name, TYPE=ROTATION, DEFINITION=COORDINATES

Use the following option to specify the axis of rotation using node numbers: \*NMAP, NSET=name, TYPE=ROTATION, DEFINITION=NODES

# Mapping from cylindrical coordinates

For mapping from cylindrical coordinates, point a in Figure 2.1.1–13 defines the origin of the local cylindrical coordinate system defining the map. The line going through point a and point b defines the -axis of the local cylindrical coordinate system. The local – plane for is defined by the plane passing through points a, b, and c.

Input File Usage: \*NMAP, NSET=name, TYPE=CYLINDRICAL

# Mapping from skewed Cartesian coordinates

For mapping from skewed Cartesian coordinates, point a in Figure 2.1.1–13 defines the origin of the local diamond coordinate system defining the map. The line going through point a and point b defines the -axis of the local coordinate system. The line going through point a and point c defines the -axis of the local coordinate system. The line going through point a and point d defines the -axis of the local coordinate system.

Input File Usage: \*NMAP, NSET=name, TYPE=DIAMOND

# Mapping from spherical coordinates

For mapping from spherical coordinates, point a in Figure 2.1.1–13 defines the origin of the local spherical coordinate system defining the map. The line going through point a and point b defines the polar axis of the local spherical coordinate system. The plane passing through point a and perpendicular
