# NLPCI

Bulk Data Entry Define the parameters for arc-length method to solve post-buckling problems in nonlinear static analysis.

## Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
NLPCI ID TYPE MINALR MAXALR SCALE   DESITER MAXINC
The following continuation lines are used to define additional control parameters for load factor control (LFCTRL), displacement control (DISPCTRL) and the arc-length control (ALCTRL). Each continuation line is optional and can be specified as needed.
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
LFCTRL MAXLF MAXDLF
DISPCTRL MAXDISP G C
ALCTRL OPTION

## Example 1

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
NLPCI 5

## Example 2

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
NLPCI 5 CRIS 0.80 1.20 1.0   8 100
ALCTRL AUTO
LFCTRL 1.50 0.20
DISPCTRL 3.6 256 3

## Definitions

Field Contents SI Unit Example
ID Set identification number.

Must have the same ID as the NLPARM Bulk Data Entry.

No default (Integer > 0)

TYPE Constraint function type.
CRIS (Default)
Crisfield
RIKS
Riks
MRIKS
Modified Riks

MINALR Minimum arc-length ratio.

Default = 0.50 (Real ≥ 0.0)

MAXALR Maximum arc-length ratio.

Default = 1.50 (Real ≥ 0.0)

SCALE Scale parameter. Can be used to consider the different dimensions of displacement and loading in the arc-length constraint equation.

Default = 1.0 (Real ≥0.0)

DESITER Desired number of iterations.

Default = 5 (Integer > 0)

MAXINC Maximum number of increments to terminate the analysis.

Default = 100 (Real > 0.0)

MAXLF Maximum (absolute) value of load factor ( $\lambda$ ) to terminate the analysis.

Default = 1.0 (Real > 0.0)

MAXDLF Maximum of desired incremental load factor ( $\text{Δ}\lambda$ ).

No default (Real > 0.0)

MAXDISP Maximum displacement to terminate the analysis.

No default (Real > 0.0)

G Grid point Identification Number for which the MAXDISP is checked.

No default (Integer > 0)

C Degree of freedom of grid GID.

No default (Integer > 0)

OPTION Arc-length control (ALCTRL) option.
ON (Default)
Always turned on arc-length method.
AUTO
Automatically turns on and off arc-length method.

## Comments

1. The NLPCI Bulk Data Entry is needed to activate the arc-length method. The ID must be the same as the corresponding NLPARM Bulk Data Entry. NLPCI is not a Subcase Information Entry.
2. The initial load factor is determined according to the corresponding NLPARM card using the TTERM field.
3. A LOAD Subcase Information Entry must be specified to define the scalable load. DLOAD is not allowed because there is no definition of time (pseudo time) in arc-length method.
4. The TYPE field defines the type of constraint function for arc-length method. The SCALE field is used to consider the different dimensions of displacement and loading. The default value of SCALE is 1.0, which means the weighting factor of displacement and loading are the same. With SCALE=1.0 the Crisfield method uses a spherical constraint function, and by setting SCALE=0.0 a cylindrical function can be achieved. If SCALE is approaching infinity, the arc-length method behaves like the standard load-controlled method; on the opposite, if SCALE is approaching zero, the arc-length method is similar as the displacement-controlled method. In most cases it is recommended to use the default value 1.0.
5. The DESITER field defines the desired iteration number for the scaling of arc-length in the next increment n+1, by considering the number of iterations in the previous increment $n$ , denoted as ${I}_{n}$ . The relation is:(1)
$\text{Δ}{s}_{n+1}={\left(\frac{DESITER}{{I}_{n}}\right)}^{0.5}\text{Δ}{s}_{n}$
The MINALR and MAXALR fields are used in addition to DESITER, to limit the arc-length ratio.(2)
$MINALR\le \frac{\text{Δ}{s}_{n+1}}{\text{Δ}{s}_{n}}\le MAXALR$

If MINALR and MAXALR are both specified explicitly as 1.0, the arc-length $\text{Δ}s$ remains constant. If the analysis is easy to converge; therefore, ${I}_{n}$ is small (smaller than DESITER) and the arc-length will be expanded to much further than the last increment. Otherwise, the arc-length will be shrunk to much less. If low or rough accuracy is acceptable, DESITER can be increased such that a bigger arc-length than default can be achieved. On the opposite, if the model is highly nonlinear, DESITER can be reduced to use generally smaller arc-lengths.

6. The fields MAXINC, MAXLF, and MAXDISP can all be used to terminate the analysis, as long as their values are reached. MAXINC has a default value of 100. MAXLF means the maximum absolute value of load factor, which defines both the lower and the upper limits. MAXLF has a default (absolute) value of 1.0. The analysis will be terminated if the load factor is either smaller than -1.0 or larger than 1.0. The value MAXLF can be smaller than 1.0; if the value MAXLF is larger than 1.0, it will not stop at load level 1.0 in arc-length method, TTERM in the NLPARM card is ignored for the termination of the analysis.
Note: MAXINC should not be smaller than the number of incremental results defined in NLOUT.
7. The MAXDLF field means maximum desired incremental load factor $\text{Δ}\lambda$ , which has a similar function of DESITER to scale the arc-length. A minimum load increment is not needed.
8. The OPTION field in the ALCTRL continuation line is used for the activation of arc-length control method.
• With the option ON, the arc-length method is always turned on (default).
• With the option AUTO, the arc-length method can be turned on and off automatically for multiple times, before and after the limit points.
9. The theoretical background to the different constraint functions is briefly described as follows. The equilibrium equation in the nonlinear static analysis can be expressed as:(3)
${R}_{n+1}={F}_{\mathrm{int},n+1}\left({u}_{n+1}\right)-{\lambda }_{n+1}P=0$
Where,
$n$
Number of the last converged increment
$n+1$
Next number
${R}_{n+1}$
Unbalanced force vector
${F}_{\mathrm{int},n+1}$
Internal force vector
${u}_{n+1}$
Unknown or displacement vector
${\lambda }_{n+1}$
Load factor
$P$
External force vector
To solve the post-buckling problem, an additional constraint equation is added with the load factor as an additional unknown, which is generally expressed as:(4)
${f}_{n+1}={f}_{n+1}\left({u}_{n+1},{\lambda }_{n+1}\right)$

Where, ${f}_{n+1}$ is a function of both ${u}_{n+1}$ and ${\lambda }_{n+1}$ .

The constraint function of Crisfield method can be expressed as:(5)
${f}_{n+1}={\left({u}_{n+1}-{u}_{n}\right)}^{T}\left({u}_{n+1}-{u}_{n}\right)+{\psi }^{2}{\left({\lambda }_{n+1}-{\lambda }_{n}\right)}^{2}{P}^{T}P-{\left(\text{Δ}{s}_{n+1}\right)}^{2}$
Where,
$\psi$
Scaling factor, due to the different dimensions of displacement and load.
$\text{Δ}{s}_{n+1}$
Desired arc-length in the current increment.
Note: The scaling factor is not the same as the value of SCALE field.
The Riks method is expressed as:(6)
${f}_{n+1}={\left({u}_{n+1}^{1}-{u}_{n}\right)}^{T}\text{Δ}u+\left({\lambda }_{n+1}^{1}-{\lambda }_{n}\right)\text{Δ}\lambda$
Where,
${u}_{n+1}^{1}$
Displacement in the first iteration.
$\text{Δ}u$
Correction of displacement.
$\text{Δ}\lambda$
Correction of load factor in each iteration.
The modified Riks method is expressed as:(7)
${f}_{n+1}={\left({u}_{n+1}-{u}_{n}\right)}^{T}\text{Δ}u+\left({\lambda }_{n+1}-{\lambda }_{n}\right)\text{Δ}\lambda$

Basically, the modified Riks method performs better than the original Riks method because the direction of correction is updated after every iteration, thus the convergence behavior is faster. Generally, the default setting of CRIS method is suggested for the most cases.

10. Subcase continuation with arc-length method is supported. Both the previous and the continuing subcases can be specified in a NLPCI card. In the continuing subcase, the equilibrium equation is expressed as:(8)
$R={F}_{\mathrm{int}}-{F}_{\text{ext}}={F}_{\mathrm{int}}-{P}_{1}-\lambda \left({P}_{2}-{P}_{1}\right)$
Where,
${P}_{1}$
Load vector at the end of the previous subcase.
The previous subcase can be specified with or without a NLPCI card.
${P}_{2}$
Load vector in the continuing subcase.
For example, if you want to apply an additional load based on a constant load. The constant load should be defined in the first subcase as ${P}_{1}$ . Then both the additional load and the constant load will be defined in the continuing subcase as ${P}_{2}$ . The arc-length method will scale the additional load which is ${P}_{2}-{P}_{1}$
11. The MONITOR card can be used together with NLPCI to monitor the load factor in real time.
12. Some parameters of the NLADAPT card have different meanings when used with arc-length method (NLPCI card).
• NLADAPT,DIRECT,YES will turn off arc-length method.
• DTMIN and DTMAX will be the limits of the arc-length factor. The arc-length factor is defined as the ratio of the current arc-length divided by the initial arc-length. If DTMIN is specified, the analysis will be terminated once the arc-length factor is smaller than this minimum value. If DTMAX is specified, the arc-length factor will be limited to ensure that it will not exceed this maximum value.
13. NLOUT can be used to request the output of the increment results. The increment size of load factor near the limit point can be very small, thus the solver may have a lot of increments. Therefore, it is recommended to use a larger NINT in the NLOUT card if detailed history of the results is needed.
14. The arc-length method can also be used along with imperfection. For more details, refer to Imperfection in the User Guide.
15. Restart analysis is currently not supported.