Curve-fitting uses a nonlinear regression process or optionally a generic algorithm method. Your error norms can be either normalized or non-normalized. Normalized error norms (the default regression option) generally give better results than the non-normalized error norms, since normalized error gives equal weight to all of your data points.
The solution-control parameters of a nonlinear regression include:
Number of iterations
Residual tolerance
Coefficient change tolerance
The solution stops when both the residual tolerance and the coefficient change tolerance of your error norm are met, or if the number of iterations criteria is met. When using nonlinear regression, you must initialize your coefficients to appropriate values.
For the generic algorithm methods, also specify the lower and upper bounds of each coefficient.
The following topics are available for specifying solution-control parameters:
TBFT,SOLVE,MATID,Option1,Option2,Option3,Option4, ...,Option7! set control parameters and solve
where:
MATID = Material reference ID number |
Option1 = Category
– AML |
Option2 = Model name. See Table 7.4: Valid Material Models for Hyperelastic Parameter-Fitting for the models available. |
Option3 = Order or number of your coefficients.
See Table 7.4: Valid Material Models for Hyperelastic Parameter-Fitting for possible values. |
Option4 = Curve-fitting procedure: 0 =
non-normalized least squares, 1 = normalized least squares |
Option5 = Maximum number of iterations |
Option6 = Tolerance of residual changes |
Option7 = Tolerance of coefficient
changes |
By default, the program uses the Levenberg-Marquardt algorithm, a nonlinear regression process, to perform the optimization process. The algorithm is a nonlinear optimization procedure that uses a combination of Gauss-Newton and gradient-descent algorithms to step through the optimization process. The generic algorithm is also available for performing the optimization.
To change solver parameters:
TBFT,SET,MATID,AML,GENR,Option3,Parname,Parval
|
where:
MATID = Material reference ID number |
Option3 = The specified name for your
curve-fitting model. |
Parname = Solver parameter name (ALGO or
IPOP). |
Parval = Value of the solver parameter. |
When Parname = ALGO, the solver algorithm changes from
the default nonlinear regression process (keyword DEFA in the
Parval position) to generic algorithms (keyword GA in the
Parval position). When using generic algorithms, define
initial population size and bounds for each parameter; bounds must have a nonzero
range of values (where the maximum and minimum values cannot be equal).
To define parameter bounds:
TBFT,BNDS,MATID,AML,GENR,Option3,PARINDEX,LBNDS,UBNDS
|
where:
MATID = Material reference ID number |
Option3 = The specified name for your
curve-fitting model. |
PARINDEX = Parameter index. |
LBNDS = Lower value of the parameter
bounds. |
UBNDS = Upper value of the parameter
bounds. |
Define initial population via Parname = IPOP and
Parval = an integer value (typically 100 to 300).
To obtain a more accurate solution, use a generic algorithm (GA) followed by the default nonlinear regression option solution.
Other solution parameters are also available (TBFT).
Complex plasticity models can involve 10 to 30 or even more parameters, depending on the models selected and their order. It is not possible to capture all behaviors in a single solve operation. You can, however, use a multistep solution approach for such models.
You can input multiple sets of experimental data to solve for various plasticity behaviors. Example 3 shows how to assign attributes to various experimental data and perform the multistep solve.
Initiate up to three internal solve operations (TBFT,PSOLVE) as needed: one for simple time-independent plasticity, one for time-dependent plasticity, and one for static recovery:
TBFT,PSOLVE,MATID,AML,GENR,Option3,Option4,
..., Option7 |
where:
MATID = Material reference ID number |
Option3 = The specified name for your
curve-fitting model. |
Option4 = Error norm: 0 (non-normalized) or 1
(normalized) (default). |
Option5 = Maximum number of iterations. |
Option6 = Tolerance of residual
changes. |
Option7 = Tolerance of coefficient
changes. |
Other solution parameters are also available (TBFT).
ViscoElasticity fitting can be challenging when it includes data for Shear Modulus Decay, Bulk Modulus Decay, and temperature dependency. It is not possible to capture all these behaviors in a single solve operation. However, you can use a multistep solution approach in this situation as described here.
You can input multiple sets of experimental data at different temperatures to solve for both Shear and Bulk Modulus decay.
Initiate up to three solve operations (TBFT,SOLVE) as needed:
a solve for the shear terms
a solve for the bulk terms
a solve that includes all experimental data.
For viscoelastic curve-fitting using the shift function, follow these steps to perform the regression:
Solve the shear coefficients:
Optional: Solve for any bulk coefficients:
Set the partial solve option (TBFT,SET,,,,,COMP,PBULK). Set the reference temperature at which your partial solution is performed (TBFT,SET,,,,,TREF,TX). The reference temperature must be the same for both shear and bulk. Only data at temperature TX is used to estimate bulk coefficients. Perform the solve operation (TBFT,SOLVE). Optional: Solve the shift function coefficients (or solve for all three coefficients):
If you do not use the shift function, discrete sets of coefficients must be generated at each temperature to account for temperature dependency. Follow these steps to perform the regression:
Set the temperature-dependency flag (TBFT, SET,,,,,TDEP,1).
Solve the shear coefficients:
Set the partial solve option (TBFT,SET,,,,,COMP,PSHEAR). Set the reference temperature at which your partial solution is performed (TBFT,SET,,,,,TREF,TX). Only data at temperature TX is used to estimate shear coefficients. Initialize the coefficients. Perform the solve operation (TBFT,SOLVE). Repeat these steps for all temperatures. Optional: Solve for any bulk coefficients:
Set the partial solve option (TBFT,SET,,,,,COMP,PBULK). Set the reference temperature at which your partial solution is performed (TBFT,SET,,,,,TREF,TX). The reference temperature must be the same for both shear and bulk. Only data at temperature TX is used to estimate bulk coefficients. Initialize the coefficients. Perform the solve operation (TBFT,SOLVE). Repeat these steps for all temperatures.
Alternatively, you can solve for both shear and bulk data at the same time.
Set the temperature dependency flag (TBFT,SET,,,,,TDEP,1).
Solve for all coefficients:
Set the partial solve option (TBFT,SET,,,,,COMP,DEFA). Set the reference temperature at which your partial solution is performed (TBFT,SET,,,,,TREF,TX). Only data at temperature TX is used to estimate shear and bulk coefficients. Initialize the coefficients. Perform the solve operation (TBFT,SOLVE). Repeat these steps for all temperatures.
Set solution options (
Oper= SET on TBFT):TBFT,SET, MATID,AML,GENR,Option3,Option4,Option5where:
MATID= Material reference ID numberOption3= User supplied name.Option4= COMP.Option5= PSHEA (for shear only) or PBULK (for bulk only) or DEFA (for default or all parameters).Specify procedure types, tolerances, and number of iterations (
Oper= SOLVE on TBFT):TBFT,SOLVE, MATID,AML,GENR,Option3,Option4, ...,Option7where:
MATID= Material reference ID numberOption3= User supplied name.Option4= Curve-fitting procedure: 0 = non-normalized least squares, 1 = normalized least squares (default)Option5= Maximum number of iterations.Option6= Tolerance of residual changes.Option7= Tolerance of coefficient changes.Other solving parameters are available (TBFT).