Bayesian optimization for inverse calibration of expensive computer models: A case study for Johnson-Cook model in machining

Abstract

Inverse model calibration for identifying the constitutive model parameters can be computationally demanding for expensive-to-evaluate simulation models. This paper presents a modified Bayesian optimization (BO) method, denoted as BO-bound, that incorporates theoretical bounds on the quantity of interest. A case study for the inverse calibration of the Johnson Cook (J-C) flow stress model parameters is presented using machining (cutting) force data. The results show fast calibration of the five J-C parameters within 25 simulations. In general, the BO-bound method is applicable for inverse calibration of any expensive simulation models as well as optimization problems with known bounds.

Full text

Bayesian optimization for inverse calibration of expensive computer models: a
case study for Johnson-Cook model in machining
Jaydeep Karandikara,1, Anirban Chaudhurib, Timothy Noc, Scott Smitha, Tony Schmitza,c
aOak Ridge National Laboratory, Oak Ridge, TN
bUniversity of Texas at Austin, Austin, TX, 78712, USA
cUniversity of Tennessee at Knoxville, Knoxville, TN, 37996, USA
Abstract
Inverse model calibration for identifying the constitutive model parameters can be computationally demanding for
expensive-to-evaluate simulation models. This paper presents a modified Bayesian optimization (BO) method, de-
noted as BO-bound, that incorporates theoretical bounds on the quantity of interest. A case study for the inverse
calibration of the Johnson Cook (J-C) flow stress model parameters is presented using machining (cutting) force data.
The results show fast calibration of the five J-C parameters within 25 simulations. In general, the BO-bound method
is applicable for inverse calibration of any expensive simulation models as well as optimization problems with known
bounds.
Keywords: Inverse model calibration, Bayesian optimization, machining, finite element model calibration, bounded
objective function
1. Introduction
Finite element (FE) methods are widely used for modeling complex manufacturing processes. However, FE meth-
ods require a new constitutive model for each material system to describe its behavior (e.g., flow stress behavior
during the shearing action in metal cutting). The constitutive model parameters can be determined through direct
methods that employ high strain rate/temperature testing to measure the behavior or by inverse methods that use man-
ufacturing process data to infer the parameters. For complex constitutive models with many parameters, the inverse
method is preferred [1,2,3]. The inverse method finds the optimal parameter set that minimizes the difference be-
tween the predicted process variables, such as force and temperature, from the FE model and the experimental values
from the manufacturing tests; this is done by iteratively modifying the model parameters for each FE simulation [2].
The methods described in the literature for inverse calibration of the constitutive model parameters in manufacturing
use gradient-based methods, evolutionary algorithms, or hybrid approaches [2]. A major limitation of the existing
methods is the relatively large number of simulations required for inverse calibration, especially when the gradient
information is not readily available. This can make the methods computationally prohibitive for expensive-to-evaluate
FE models, which can take many hours to complete. To address this challenge, this paper presents a Bayesian opti-
mization (BO) method [4,5,6,7] that accounts for the bounded nature of the error functions used in deterministic
inverse calibration of the constitutive model parameters.
In this work, the inverse calibration of the Johnson-Cook (J-C) model [1] is used as a case-study. The J-C model
is widely-used in the machining community for modeling material flow stress behavior in machining operations. Note
1Corresponding author. Tel.: +1 8655744641 E-mail address: karandikarjm@ornl.gov (Jaydeep Karandikar).
2This manuscript has been authored in part by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy
(DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a
nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for
US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public
Access Plan (http://energy.gov/downloads/doe-public-access-plan).
Preprint submitted to Manufacturing Letters March 7, 2022

that, in general, the described BO approach can be used for calibrating any other model and manufacturing process.
The J-C model empirically describes the material flow stress as a function of the strain, strain rate, and temperature as
flow stress =A+Bn"1+Cln ˙
˙0!#"1− Tmat −Tref
Tmelt −Tref !m#,(1)
where is the plastic strain, ˙is the plastic strain rate, ˙0is the reference strain rate, Tmat is the material temperature,
Tref is the reference temperature at or below which there is no temperature dependence of the yield stress, and Tmelt
is the material melting temperature. The J-C model parameters are the yield strength of the material under refer-
ence conditions A, the strain hardening constant B, the strain rate strengthening coefficient C, the strain hardening
coefficient n, and the thermal softening coefficient m. Direct calibration of the J-C model parameters for machin-
ing with tension/compression split-Hopkinson pressure bar tests is difficult due to the high strains, strain rates, and
temperatures [8,9,10,11].
There have been various methods presented in the literature for inverse calibration of J-C model parameters.
These include iterative gradient-based search methods [12,13], response surface methodology [14], evolutionary
algorithms such as particle swarm and genetic algorithms [15,16,17], and Bayesian calibration [8]. This work focuses
on deterministic inverse model calibration using BO, which is a derivative-free global optimizer [4,5]. Gradient-
based methods can also be used for efficient local optimization when adjoints are available. In absence of adjoints
(e.g., when using commercial/proprietary FE software), gradients estimated through finite difference can result in
a computationally challenging number of expensive simulations. The global BO is computationally comparable to
local gradient-based solvers when adjoints are available and can be more efficient in the absence of adjoints for
low to medium dimensional problems. The objective function in BO is approximated using a Gaussian process
(GP) regression surrogate that provides the prediction mean and the prediction uncertainty. The GP surrogate is
refined sequentially using an acquisition function to make the optimal sampling decisions in every iteration that
contribute towards converging to the global optimum. The expected improvement (EI) is the most popular acquisition
function [4,6]. In this work, the normalized mean absolute error (NMAE) between the experimental and the simulated
process variables is used as the objective for model calibration, which has a theoretical lower bound of zero. However,
the generic EI function does not account for the bounded nature of objective functions encountered in the inverse
calibration problems and this limitation is addressed here.
The two primary contributions of the paper are: (1) the BO-bound method, which is a BO method that incorporates
the theoretical bounds in the underlying quantity of interest; and (2) the application of the BO-bound method to the
inverse calibration of expensive computer models using NMAE as the objective function. To address the limitation of
the EI acquisition function in BO not accounting for bounds, a closed-form expression for EI was derived for the BO-
bound method that can incorporate the theoretical bounds of the objective function. Note that the described BO-bound
method can be used for any optimization problem where there exists a known bound (upper or lower) on the objective
function. An alternate application could be the optimization of process parameters in laser additive manufacturing to
minimize part porosity, where the part porosity has a lower bound of zero.
The remainder of the paper is organized as follows. Section 2gives an overview of the standard BO method
and describes the proposed BO-bound method that incorporates the NMAE lower bound of zero. Section 3presents
results for inverse calibration of the J-C flow stress model parameters for machining and a comparison of the BO-
bound method with the standard BO method. Section 4presents conclusions and future work.
2. Bayesian optimization for inverse calibration of J-C model parameters
The inverse calibration problem for identifying the J-C model parameters by minimizing the NMAE between the
simulated and experimental observations is detailed in Section 2.1. The standard BO method for inverse calibration is
then presented in Section 2.2. Finally, the proposed BO-bound method that incorporates the non-negative property of
the NMAE is described in Section 2.3.
2.1. Inverse calibration problem setup
For a given set of parameters specific to a workpiece material and tool geometry (such as rake angle and rake face
shape), FE simulation can be used to predict the tangential component of the cutting force, Fsim
T, which is aligned
2

with the cutting speed direction, and the normal component of the cutting force, Fsim
N, which is perpendicular to the
machined surface. In this case, the inputs to the system are the five J-C parameters x:={A,B,C,n,m} ∈ X ⊆ R5,
where Xdenotes the parameter search space. The system output is denoted by y=f(x), where f:X 7→ Ω⊆R, which
is the NMAE between the experimental and simulated tangential and normal force components obtained through the
expensive FE model simulation at xas given by
f(x) :=NMAE(A,B,C,n,m)=1
2






|Fsim
T(x)−Fexpt
T|
Fexpt
T
+|Fsim
N(x)−Fexpt
N|
Fexpt
N





.(2)
The superscript sim denotes results from the FE model and the superscript expt denotes the experimental results. The
optimization problem is defined as x∗=arg min
x∈X
f(x). Although minimizing the NMAE results in non-unique solutions
for the J-C model parameters, each solution is considered valid for the goal of accurately predicting the cutting force
components through FE simulations [1,8,9].
2.2. BO for inverse calibration of expensive functions
BO is a GP-based method for sequentially converging to the global optimum. Given ksamples of the NMAE,
the next sampling location xk+1to simulate and update the GP surrogate is selected by maximizing an acquisition
function, J(x), as xk+1=arg max
x∈X
J(x).
EI is the most popular acquisition function and uses the expected value of improvement to balance trade-offs be-
tween exploration (global search) and exploitation (local search) [4,6]. For a minimization problem, the improvement
function I(x) at any xis defined based on improving beyond the current observed best solution yk
min after ksample
evaluations as
I(x) :=(yk
min − Y(x),Y(x)≤yk
min
0,Y(x)>yk
min,(3)
where Y(x)∼ N(µ(x), σ(x)) is the GP prediction at any xwith µ(x) denoting the GP prediction mean and σ(x))
denoting the GP prediction standard deviation that gives a measure of uncertainty in the prediction. The EI acquisition
function at any given xis [4]
E[I(x)] =yk
min −µ(x)Φ(β(x)) +σ(x)φ(β(x)),(4)
where Edenotes expectation, β(x)=(yk
min −µ(x))/σ(x), Φ(.) is the standard normal cumulative distribution function,
and φ(.) is the standard normal probability density function (see Appendix A.1 for derivation and a demonstration in
Example 1).
2.3. BO-bound for inverse calibration of expensive functions with theoretical bounds
For the inverse calibration problem, the lower bound for NMAE is known to be zero. The GP surrogate is data-
driven and cannot directly incorporate the underlying theoretical bounds on the quantity of interest. As a result, the
GP can predict negative values for NMAE, which are unattainable. Within the standard BO framework, the incorrect
negative NMAE predictions from the GP surrogate are propagated to the EI acquisition function, which could lead to a
misinformed choice of the sampling location. In this work, the objective function theoretical bounds are incorporated
within the BO framework by redefining the improvement function. The modified improvement function IB(x) restricts
the maximum predicted improvement to be yk
min as
IB(x) :=








yk
min,Y(x)≤0
yk
min − Y(x),0<Y(x)≤yk
min
0,Y(x)>yk
min,
(5)
When Y(x)≤0, the maximum improvement is restricted to yk
min by suppressing the GP prediction to zero when
predicted to be negative. Note that IB(x) can be similarly defined for any given bounds on the objective function
making the BO-bound method applicable to other cases beyond the inverse calibration problem.
The modified EI at any given x can be derived in closed-form (see Appendix A.2 for derivation) as
E[IB(x)] = Φ(α(x))yk
min +yk
min −µ(x)(Φ(β(x)) −Φ(α(x)))+σ(x)(φ(β(x)) −φ(α(x))),(6)
3

where α(x)=(0 −µ(x))/σ(x). The BO method with the modified EI function for objective functions with theoretical
bounds is denoted as BO-bound. Appendix A provides a comparison between the EI values in standard BO and
BO-bound through Example 1.
3. Results
The BO-bound method was evaluated using an experimental result for orthogonal (or two-dimensional) turning
reported in [8]. The material was Aluminum 6061-T6. The cutting tool had a rake angle of 15◦and a relief angle of
8◦. The cutting speed was 60 m/min, the feed was 0.3 mm/rev, and the depth of cut was 1 mm. The tangential cutting
force component Fexpt
Twas 224.1 N and normal cutting force component Fexpt
Nwas 95.1 N. The proposed BO-bound
method was first compared with the standard BO method by fixing three of the five J-C parameters to illustrate the
efficiency and robustness of the method. Subsequently, BO-bound method was applied for the inverse calibration of
all five J-C parameters using expensive FE simulations. The procedure for the BO/BO-bound method is described in
Algorithm 1.
Algorithm 1 BO/BO-bound pseudo-code
Input: Number of initial samples kinit, total number of simulations ktotal , NMAE simulation model f(.)
Output: Optimal J-C parameters x∗, optimal NMAE f(x∗)
1: Initial Latin hypercube sampling (LHS) of kinit points
2: Determine the force values, Fsim
Tand Fsim
Nby FE simulation and the corresponding NMAE values {f(xi)}k
i=1using
Eq. (2)
3: k=kinit
4: while k≤ktotal do
5: Fit GP surrogate to the available training data {xi,f(xi)}k
i=1
6: Select next sample xk+1that maximizes the acquisition function in Eq. (4) for BO or Eq. (6) for BO-bound
7: Run FE simulation at the selected sample xk+1and determine NMAE value f(xk+1) using Eq. (2)
8: k=k+1
9: end while
10: Find optimal parameters x∗=arg min {f(xi)}k
i=1Optimal NMAE is given by f(x∗)
11: return x∗,f(x∗)
To compare the standard BO and BO-bound methods, the mode 1 values reported in [8] were used to fix three
of five parameter values as C=0.0142,n=0.035,and m=1.47 along with varying A∈[50,350] MPa and
B∈[40,300] MPa. The objective of the BO and BO-bound method is to find the {A,B}combination that leads
to the same forces as measured in the experiment, using the minimum number of objective function simulations.
The orthogonal cutting FE simulations were completed using Third Wave Systems’ AdvantEdgeTM. The friction
coefficient (for relative sliding between the chip and rake face) was selected as 0.8 [8]. The computation time for each
simulation was one hour. For the J-C model, ˙0was taken as 1 and Tref was 20◦C.
Two initial LHS samples were used to train the GP model for NMAE followed by 10 iterations. The Matern kernel
was used for the GP model with length scale bounds as 0.001 and 100. The efficiency of any BO method depends
on the initial design of experiment (in this case, LHS). To evaluate the robustness of the BO and BO-bound methods
Algorithm 1was repeated 100 times each with a different initial LHS of two samples. To computationally enable
completing 100 repeats of BO, each with 12 objective function simulations, a second-order polynomial model was
first fitted to the Fsim
Tand Fsim
Nas a function of Aand Bto act as a surrogate for the FE simulations (see Appendix
Bfor details). This enabled rigorously testing the two BO methods without requiring the expensive FE simulation at
each iteration.
Fig. 1shows the results for the 100 repeats of BO and BO-bound completed using the polynomial surrogate model
for Fsim
Tand Fsim
N. In Fig. 1(a), each dot represents the best {A,B}combination at the end of 12 simulations from
the 100 repeats and the contour plot show NMAE from the polynomial surrogate model. BO-bound shows better
performance compared to standard BO as seen from the NMAE contour plots in Fig. 1(a) and the NMAE convergence
plot in Fig. 1(c). From the 100 repeats, 80 cases for the BO-bound method have NMAE less than 1% as compared
4

to 66 for BO as seen from the histogram of NMAE after 12 simulations in Fig. 1(b). In this case, BO-bound takes
nine simulations (including the two initial samples) as compared to 11 simulations required by standard BO to reach
a median NMAE of less than 1% leading to computational savings of 18% as seen from Fig. 1(c). The confidence
bands representing 25 and 75 percentile range from 100 repeats shown in Fig. 1(c) show that BO-bound is more
robust than BO. The results show that incorporating the theoretical bounds through the BO-bound method results in
improved convergence as compared to the standard BO method.
50 100 150 200 250 300 350
A (MPa)
50
100
150
200
250
300
B (MPa)
BO (100 repeats)
BO-bound (100 repeats)
0
9
18
27
36
45
54
63
72
NMAE (%)
(a)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
NMAE (%)
0
10
20
30
40
50
60
# of occurences
BO
BO-bound
(b)
0 2 4 6 8 10 12
# of simulations
0
2
4
6
8
10
12
14
NMAE (%)
BO
BO-bound
(c)
Figure 1: Comparison of standard BO with BO-bound for the two-dimensional test showing (a) optimal parameters obtained and (b) NMAE
histogram after 12 simulations, and (c) convergence of median NMAE w.r.t. number of simulations along with confidence bands representing 25
and 75 percentiles for 100 repeats.
For the second study, the BO-bound method was tested for calibrating all the five J-C model parameters using the
FE simulation in each iteration. The J-C model parameter ranges were selected based on a literature review [8,18,
19,20,21,22,23,24,25] to be: A∈[50,350] MPa, B∈[40,220] MPa, C∈[0.001,0.05], n∈[0.05,0.5], and
m∈[1,1.5]. Five initial LHS samples were used to train the GP model for NMAE followed by 20 iterations using
BO-bound. The optimal J-C model parameter values were A =202.38 MPa, B =77.46 MPa, C =0.015, n =0.30, and
m=1.21, with NMAE equal to 0.82%. The sequence of tests and results are shown in Appendix C. Note that the
inverse model calibration is done for experimental forces at a single cutting parameter set. The J-C model parameters
would not be optimal for cutting forces measured at different cutting parameters. To illustrate, Appendix C shows
the simulated and experimental forces at 5◦and 25◦rake angles using calibrated J-C model parameters at 15◦rake
angle. For experimental cutting force data at different cutting parameters, an average value of NMAE can be used for
the inverse calibration. The process will require running the FE simulations at different cutting parameters in parallel,
calculating the NMAE at each (using Eq. 2), and taking the average NMAE value as the output yfor the BO-bound
method.
To evaluate the robustness of the result, the BO-bound was repeated five times with different initial LHS. For
comparison, the standard BO method was also completed for the five repeats. Table 1shows the best values for the
J-C model parameter for each repeat for the BO-bound and BO method. As seen from Table 1, the best NMAE is less
than 1.7% for each repeat of BO-bound with a total computational cost of 25 FE simulations in each case. Table 1
also shows that the NMAE is less for BO-bound compared to the standard BO in four out of five repetitions. As noted,
the standard BO does not consider the zero bound for NMAE. Table 1confirms that the inverse calibration of the J-C
model parameter from machining forces does not have a unique solution; multiple feasible solutions give force values
close to the experimental values [8,18]. Note that even though the flow stress-strain curve for each parameter set is
different, they are considered equally valid for the goal of accurately predicting the machining forces.
4. Conclusions
A BO method for calibration of expensive FE simulation models that accounts for known bounds of error func-
tions used as the objective function in inverse model calibration was presented. A case study for inverse calibration
of the J-C model parameters was demonstrated. The objective function was the NMAE between the experimental and
5

Table 1: Optimal values for the J-C parameters for five repeats of the BO-bound and BO method
# Method A(MPa) B(MPa) C n m Fsim
T(N) Fsim
N(N) NMAE (%)
1BO-bound 202.38 94.97 0.015 0.30 1.21 225.75 94.25 0.82
BO 270.22 77.45 0.043 0.33 1.49 230.46 96.29 2.04
2BO-bound 272.92 90.74 0.027 0.20 1.13 225.28 94.6 0.52
BO 324.91 183.04 0.037 0.38 1.40 226.60 95.38 0.70
3BO-bound 220.66 136.85 0.037 0.09 1.21 231.4 95.01 1.67
BO 105.94 204.34 0.036 0.05 1.48 225.08 94.31 0.63
4BO-bound 279.48 53.48 0.034 0.25 1.27 223.77 93.13 1.10
BO 350.00 40.00 0.005 0.29 1.50 223.05 92.33 1.69
5BO-bound 292.70 150.63 0.011 0.11 1.09 227.97 93.66 1.62
BO 209.67 201.23 0.012 0.13 1.00 224.56 99.62 2.47
simulated force components in the tangential and normal directions obtained from an expensive-to-evaluate FE simu-
lation. The proposed BO-bound method considers the non-negativity of the NMAE by suppressing the GP predictions
beyond zero through restricting the improvement function values used in the acquisition function of EI. Furthermore,
the BO-bound method can generally be applied to any optimization problem with known theoretical bounds on a
computationally expensive objective function.
Results on calibrating Aand Bshowed that the BO-bound method converges faster than the standard BO approach
and is more robust; BO-bound method led to ∼18% computational savings as compared to the standard BO approach
for median NMAE from 100 repetitions to reach below 1%. The calibration of all five J-C model parameters showed
that the BO-bound method achieves fast convergence with NMAE from each of the five repetitions reaching within
1.7% using only 25 FE simulation and outperforms the standard BO method in four out of five repeats.
Future work will include modifying the BO-bound method to include additional variables (such as cutting tem-
perature and chip thickness), experimental results from different process parameters, and noise in the experimental
results. Parallel BO methods [7] will be explored to take advantage of parallelizing the FE simulations in each BO
iteration. A multi-objective optimization routine will also be evaluated where the error in each force prediction is
modeled separately to determine the optimal FE model parameters.
Acknowledgements
This work has been supported in part by the DOE Office of Energy Efficiency and Renewable Energy (EERE),
Manufacturing Science Division, and used resources at the Manufacturing Demonstration Facility, a DOE-EERE User
Facility at Oak Ridge National Laboratory. The second author acknowledges support from Department of Energy
award number DE-SC0021239.
The authors would also like to thank Dr. Patxi Fernandez-Zelaia, Oak Ridge National Laboratory, for sharing the
experimental data used in the paper.
Appendix A. Expected improvement acquisition function
In this section, the EI acquisition function used in the standard BO is described in Appendix A.1 followed by the
derivation of EI for BO-bound in Appendix A.2. An illustrative example is provided in Example 1to further explain
BO and BO-bound acquisition functions.
Appendix A.1. Derivation of EI for standard BO
Let yk
min =min{y(1),...,y(k)}be the current best function value after ksimulations, where {y(1) ,...,y(k)}are ob-
servations at {x1,...,xk}, respectively. The GP prediction at any xis a normal distribution with the mean µ(x) and
the standard deviation σ(x). The improvement function I(x) for a minimization problem is given by Eq. (3). The
6

probability of improvement when Y(x)≤yk
min is PI(x)=P[Y(x)≤yk
min]= Φ(β(x)),where β(x)=(yk
min −µ(x))/σ(x).
The EI is given by
E[I(x)] =PI(x)E[yk
min − Y(x)|Y(x)≤yk
min]+(1 −PI(x)) ×0=PI(x)(yk
min −E[Y(x)|Y(x)≤yk
min])
= Φ(β(x)) yk
min − µ(x)−σ(x)φ(β(x))
Φ(β(x)) !!=yk
min −µ(x)Φ(β(x)) +σ(x)φ(β(x)),(A.1)
Appendix A.2. Derivation of EI for BO-bound with bounded objective functions
The modified improvement function IB(x) is given by Eq. (5), which restricts the maximum improvement to yk
min
when Y(x)≤0 since NMAE has a lower bound of 0. The probability of improvement when Y(x)≤0 is
PI1(x)=P[Y(x)≤0] = Φ(α(x)),(A.2)
where α(x)=(0 −µ(x))/σ(x). The probability of lying within 0 <Y(x)≤yk
min is
PI2(x)=P[0 <Y(x)≤yk
min]= Φ(β(x)) −Φ(α(x)).(A.3)
The EI for bounded improvement function in BO-bound is given by
E[IB(x)] =PI1(x)E[yk
min|Y(x)≤0] +PI2(x)E[yk
min − Y(x)|0<Y(x)≤yk
min]+(1 −PI1(x)−PI2(x)) ×0
=PI1(x)yk
min +PI2(x)(yk
min −E[Y(x)|0<Y(x)≤yk
min])
= Φ(α(x))yk
min +(Φ(β(x)) −Φ(α(x))) yk
min − µ(x)−σ(x) φ(β(x)) −φ(α(x))
Φ(β(x)) −Φ(α(x)) !!!
= Φ(α(x))yk
min +yk
min −µ(x)(Φ(β(x)) −Φ(α(x)))+σ(x)(φ(β(x)) −φ(α(x))).
(A.4)
Example 1 (Illustrative example showing BO and BO-bound EI).Let the true function be y(x)=(6x−2)2sin(12x−
4) −6.02 with a known lower bound of zero (global minimum is zero at x =0.76). Fig. A.2(a) shows the GP fit using
four training data at {xi}4
i=1={0,0.45,0.65,1}with corresponding observations {y(i)}4
i=1={9.04,6.50,3.81,21.85}.
Based on the current y observations, the best function value is ymin =3.81 at x =0.65. Fig. A.2(b) shows the EI in
ymin as a function of x for BO and BO-bound and can be seen to be different. The maximum value of EI is marked
with a star for BO and BO-bound in Fig. A.2(b) to show the difference in the next sample selection. The maximum
EI is equal to 0.98 and occurs at x =0.22 for standard BO while the maximum EI is equal to 0.7 and occurs at
x=0.58 for BO-bound. The BO-bound method penalizes the negative prediction values at x =0.22 and restricts
the maximum improvement for IB(x)to ymin =3.81; this reduces the EI for BO-bound at x =0.22 and switches the
maximum location to x =0.58.
Appendix B. Second-order polynomial fit for algorithm testing
In this work, 25 FE simulations were completed using an equally-spaced 5 ×5 grid and the force values for
each sample were recorded. A second-order polynomial fit to this data was used as the FE surrogate in BO to
eliminate the need to complete expensive FE simulations at each BO iteration in the first study to rigorously analyze
the BO approaches. Fig. B.3 shows the leave-one-out cross-validation results for the surrogate model. The polynomial
surrogate model for Fsim
Tand Fsim
Nare
b
Fsim
T=2.98339 +0.698A+0.811B−0.000374A2−0.000470B2−0.001083AB
b
Fsim
N=−3.2744 +0.354A+0.384B−0.000356A2−0.000253B2−0.000715AB
7

0.0 0.2 0.4 0.6 0.8 1.0
x
−5
0
5
10
15
20
25
y
ytrue
µ(GP mean)
data
σ(GP std. dev.)
(a)
0.0 0.2 0.4 0.6 0.8 1.0
x
0.0
0.2
0.4
0.6
0.8
1.0
EI
BO
BO-bound
(b)
Figure A.2: Illustrative example showing (a) GP surrogate fit with true objective function ytrue and (b) EI for BO and BO-bound with the maximum
locations marked with a star.
50 100 150 200 250 300
FE simulation Fsim
T
50
100
150
200
250
300
Polynomial fit prediction b
Fsim
T
(a)
50 60 70 80 90 100 110 120
FE simulation Fsim
N
50
60
70
80
90
100
110
120
Polynomial fit prediction b
Fsim
N
(b)
Figure B.3: Leave one out cross-validation plot for the second-order polynomial model for predicting (a) Fsim
Tand (b) Fsim
N.
Appendix C. Sequence of simulations for the BO-bound method
Table C.2 shows the results with the first five rows showing the initial LHS samples followed by 20 subsequent
simulations determined by the BO-bound method. The optimal J-C model parameters are highlighted in bold. The
NMAE for the optimal J-C model parameters after 25 simulation was 0.82%.
Table C.3 shows the simulated and experimental forces at 5◦and 25◦rake angles using calibrated J-C model
parameters at 15◦rake angle.
8

Table C.2: NMAE values for J-C model parameter simulations using BO-bound for one repetition
#A(MPa) B(MPa) C n m Fsim
T(N) Fsim
N(N) NMAE (%)
1 270.23 77.46 0.043 0.33 1.49 271.91 110.90 18.97
2 195.15 101.69 0.014 0.31 1.20 230.46 96.29 2.04
3 75.02 179.63 0.036 0.08 1.01 181.00 75.70 19.82
4 115.54 298.35 0.023 0.47 1.32 596.84 242.34 160.58
5 338.04 217.70 0.001 0.18 1.15 281.61 120.64 26.26
6 192.36 104.21 0.014 0.31 1.20 235.42 96.47 3.25
7 202.38 94.97 0.015 0.30 1.21 225.75 94.25 0.82
8 194.82 94.12 0.016 0.29 1.21 221.10 92.30 2.14
9 209.22 96.31 0.014 0.31 1.22 232.60 96.80 2.79
10 194.34 89.04 0.015 0.31 1.22 217.66 90.79 3.70
11 205.60 99.29 0.015 0.30 1.21 229.74 95.46 1.45
12 201.00 93.58 0.014 0.29 1.20 219.75 91.43 2.90
13 201.18 100.40 0.016 0.30 1.22 230.55 95.49 1.64
14 350.00 40.00 0.050 0.05 1.50 253.39 88.71 9.89
15 190.18 40.00 0.050 0.05 1.50 179.40 71.70 22.28
16 350.00 40.00 0.024 0.05 1.50 228.91 83.81 7.01
17 350.00 40.00 0.032 0.050 1.320 226.96 80.89 8.11
18 350.00 79.27 0.031 0.050 1.465 249.40 88.50 9.11
19 350.00 45.69 0.005 0.050 1.385 205.98 78.34 12.85
20 350.00 40.12 0.032 0.127 1.482 241.89 92.00 5.60
21 350.00 65.74 0.020 0.152 1.500 246.82 98.93 7.08
22 350.00 40.00 0.032 0.316 1.500 254.59 103.47 11.20
23 306.36 40.00 0.024 0.109 1.500 217.33 85.88 6.36
24 350.00 40.00 0.020 0.203 1.500 235.00 95.42 2.60
25 342.64 40.00 0.010 0.251 1.500 224.66 93.74 0.84
Table C.3: Using the calibrated J-C model parameters at different rake angles.
Rake angle (◦)Fexpt
T(N) Fexpt
N(N) Fsim
T(N) Fsim
N(N) NMAE (%)
Calibration 15 224.1 95.1 225.75 94.25 0.82
Test 5 261.2 171.3 289.7 178.3 7.49
Test 25 178.5 43.3 170.1 43.1 2.69
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