6.2 Methodology for Inverse Estimation of Material Properties
6.2.1 Heuristic algorithm for optimization
Heuristic optimization used for the inverse determination of thermal diffusivity (Dt) absorptivity (), specific heat (Cp), density (ρ) and beam radius (r) was similar to that used by Eideh and Dixit (2013). Table 6.2 shows the initial ranges of parameters. For each parameter the range was divided into three linguistic zones, viz., low (L), medium (M) and high (H) as shown in Fig. 6.2 for five parameters. Thus, the entire domain was divided into 35 = 243 cells. As an initial guess, the centre values of each parameter were chosen and simulation was carried out. Experimental temperatures at two locations were compared with the analytical results.
Table 6.2 Ranges of the initial parameters
Parameters Symbol Unit Range
Thermal diffusivity Dt m2/s 31059105
Thermal absorptivity ƞ 0.10.9
Specific heat cp J/kg.°C 6001500
Density ρ kg/m3 16002800
Beam radius r m 0.0010.01
The parameters Dt, , cp, ρ and r that minimize the error between experiment measured temperatures and temperatures estimated from simulation are considered as true parameters. The temperatures at two locations are measured experimentally by using two pyrometers. These temperatures are compared with the temperatures predicted from simulation. The root mean squared (RMS) errors of temperatures up to 15 s at two locations are indicated by E1 (at pyrometer one) and E2 (at pyrometer two).
2 1
1 Experimental temperature Simulation temperature Experimental temperature
n i
E ,
n
(6.1)
where n is the number of experimental temperature data at each parameter. The objective was to minimize the combined error given by
12 22
2E E E / , (6.2) The numerical values of thermal conductivities indicate how fast heat flows in a given workpiece. During laser pass, the rate of laser heat flow through a workpiece is expressed as the amount of heat flows per unit time through a unit area, per unit temperature gradient. Thermal conductivity (k) of the sheet metal was calculated by
t p
,
k D c
(6.3) where Dt is thermal diffusivity, ρ is density and cp is specific heatMinimization of the objective function is carried out by a heuristic method.
Inverse modelling is carried out by using ABAQUS® package for determining the
material properties. Prediction errors in temperature are minimizing by using ABAQUS®. The methodology for finding out the material parameter, viz., thermal diffusivity (Dt) absorptivity (), specific heat (Cp), density (ρ) and beam radius (r) during laser bending is as follows:
Step 1: Choose suitable ranges for material parameters Dt, η, Cp, ρ and r.
Step 2: For each parameter, the range is divided into three zones, viz., Low (L), Medium (M) and High (H) zones. Thus, the entire domain is divided into 35
= 243 cells.
Step 3: Choose the middle (M) values of all parameters as initial guess parameters of Dt, η, Cp, ρ and r and calculate RMS error, E using Eq. 6.1 and Eq. 6.2. The RMS errors at two experimental points, E1 (at point P1) and E2 (at point P2), are calculated by using 15 experimental data at each laser parameters.
Step 4: Fixing four parameters as constant, viz., η, Cp, ρ and r, carry out one- dimensional search for the optimum Dt. An effective way to do this is as follows:
If the estimated temperatures at the current two locations (P1 and P2) are greater than the measured temperature, then increase the value of Dt by jumping to the centre of adjacent cell. If the new root mean error (RMS)new
is greater than the old root mean square (RMS)old, then keep the old point as the current point. If the new root mean error (RMS)new is less than the old root mean square (RMS)old, then the current point is replaced by the centre of the new cell.
If the estimated temperatures at the current two locations (P1 and P2) are less than the measured temperature, then decrease the value of Dt by jumping to the one cell behind. If the new root mean error (RMS)new is greater than the old root mean square (RMS)old, then keep the old point as the current point.
If the new root mean error (RMS)new is less than the old root mean square (RMS)old, then the current point is accepted as the centre of current cell.
Step 5: Execute Step 4 for all variables. i.e., one parameter at a time is changed keeping other four parameters constant.
L M H L
M H
Variable 1
Variable 2
Begin First level refinement
Second level refinement
Minimum error cell after 3rd
iteration
Third level refinement
L = Low M = Medium H = High L
M H
L M H
L M H
L M H
Figure 6.2 Two-dimensional graphical representation of search procedure
Step 6: For the further refinement, the optimum cell is further divided as in Step 2.
Repeat the procedure of Step 3 to Step 5. After carrying out this procedure, if the E could not be reduced significantly, then the ranges of estimated and measured temperature need to be reduced. A graphical representation for reducing sizes of the search domain towards optimum is shown in Fig. 6.2.
The basic steps of implementing heuristic optimization technique are summarized in Fig. 6.3.
In order to obtain temperature-dependent properties, this procedure can be followed for different laser powers at fixed scan speed. Different laser powers will produce different temperature distributions in the sheet. Hence, the inverse procedure can find out the thermal properties for different temperature distributions. As a simplification, these properties can be considered as the properties at some average temperature. In this work, the average is taken as the mean of bottom and top surface temperatures. Bottom surface temperature is directly measured by the pyrometer, whereas the top surface temperature is the estimated by FEM package. The key assumption is that it is the average temperature in the vicinity of laser irradiation zone
that is deciding the overall behaviour. Subsequent analysis shows that this assumption works reasonably well, but one should develop a more robust procedure in future.
Start
Choose the suitable ranges of input parameters Divide each range into 3 equal parts.
Put the guess parameters into ABAQUS® software.
Select the middle (center) values for each parameter as initial guess.
After simulation, note down the temperature at two locations.
Compare the experimental and simulated temperatures.
Calculate RMS error.
Do not accept the change in the value of
variable.
Accept new value of the variable.
Increase or decrease one variable depending on the sign of (Texp Tsimu) and simulate again.
Carry out this procedure for all variables.
Is
(RMS)new > (RMS)old ?
Is
||Texp – Tsimu||
small ?
Stop
No Yes
Yes
No
, r and cp.
Refine the cell in which the current optimum value lies and repeat the
procedure.
No
Figure 6.3 Flow chart of optimization techniques