Scipedia

Marbona, H.

Rodríguez, D.

Martínez-Cava, A.

Valero, E.

Active Control of Separated Flow over a Bump Under Unsteady and Turbulent Conditions

Landi, Tommaso

Hoareau, C.

Deü, J.

Citarella, R.

Ohayon, R.

The 9th European Congress on Computational Methods in Applied Sciences and Engineering
ECCOMAS Congress 2024
3–7 June 2024, Lisboa, Portugal
PROJECTION BASED REDUCED ORDER MODEL OF
ELASTO-ACOUSTIC VIBRATIONS COMPUTED WITH
ISOGEOMETRIC ANALYSES
T. LANDI1,2, C. HOAREAU2, J.F.DEU2, R.CITARELLA1, and
R.OHAYON2
1 Department of Industrial Engineering, University of Salerno, Fisciano, SA, Italy
e-mail: tlandi@unisa.it
e-mail: rcitarella@unisa.it
2 Structural Mechanics and Coupled Systems Laboratory, Conservatoire National Des Arts e
Metiers, Paris, France
email: christophe.hoareau@lecnam.it
email: jean-francois.deu@lecnam.net
email: roger.ohayon@lecnam.net
Key words: Finite Element Method, Isogeometric Analysis, Modal Analysis, Projection of
Modal Basis (PROMs)
Summary. This paper demonstrates the advantages of Isogeometric Analysis (IGA) over the
Finite Element Method (FEM). The study applies IGA to a vibro-acoustic problem, comparing
the results with those obtained using an analytical approach and FEM. Specifically, the problem
involves a cylindrical cavity surrounded by an external structure. The analysis is carried out
in three stages: solving the structural problem, addressing the acoustic problem, and finally
tackling the combined vibro-acoustic problem. The comparison highlights the benefits of IGA
over traditional Finite Element Analysis.
1 INTRODUCTION
The study of vibro-acoustic problems holds fundamental importance in various sectors, such
as automotive [1] and aerospace [2] fields. Structures exhibit varying behaviors under the influ-
ence of surrounding fluids, necessitating coupled analysis [3]. Complex computational techniques
such as the Finite Element Method (FEM), Boundary Element Method (BEM), and Fast Mul-
tipole Method (FMM) are employed for the analysis of these problems.
FEM analysis is widely used in industry due to its reliability and effectiveness. The latter
method involves an initial design phase followed by a meshing phase to divide the structure into
numerous small elements or volumes. The meshing phase can be particularly time-consuming
for complex geometries [4]. Additionally, the size of the mesh influences the solution due to the
geometry approximation it provides.
Challenges encountered in FEM have spurred the development of Isogeometric Analysis (IGA)
[4], which integrates the functions used for describing geometry also for the approximation of
the state variables such as pressure, displacement, and temperature. In Computer-Aided De-
sign (CAD), NURBS (Non-Uniform Rational B-Splines) are commonly used to describe curves.
T. Landi1,2, C. Hoareau2, J.F.Deu2, R.Citarella1 and R.Ohayon2
NURBS have advantageous properties for tackling complex problems compared to the Lagrange
functions, typically used in FEM. These properties include always positive bases and higher
continuity, leading to smoother functions [4, 5].
Despite being relatively recent, IGA has already found application in various engineering
fields such as structural mechanics [6]. Hughes et al. [7] noted that as the interpolation polyno-
mial order increases, NURBS remain stable, whereas Lagrange functions tend to diverge. This
stability makes NURBS well-suited for polynomial degree refinement.
Li et al. [8] demonstrated that using smoother shape functions in coupled fluid-structural
problems leads to more accurate solutions compared to Lagrange functions.
Based on a review of literature, IGA analysis appears promising for studying coupled fluid-
structural problems.
This work aims to investigate the vibro-acoustic problem using the Isogeometric Approach.
The paper will show some examples comparing IGA and FEM with an analytical solution [9],
and The Projection of Modal Basis (PROMs) will be used to reduce the computational time for
calculating the natural frequencies and the modal shapes.
2 Numerical Examples
This chapter presents numerical examples obtained from the study and it is divided into three
sections: the first section addresses the structural problem, the second focuses on the acoustic
problem, and the third covers the vibro-acoustic problem. In each section, the results in terms
of natural frequencies and modal shapes obtained from an analytical solution [9], are compared
with those obtained using FEM and IGA methods.
2.1 Structural problem
The structure under investigation is the one shown in [9]. It is an hollow cylinder (Figure 1),
the dimensions and material characteristics of which are given in Table 1.
2
T. Landi1,2, C. Hoareau2, J.F.Deu2, R.Citarella1 and R.Ohayon2
Table 1: Structural charateristics
ρs
[
kg
m3
]
Young Modulus[Pa] Poisson[-] External Radius[m] Thickness[m] Length[m]
2700 7·E10 0.3 1 0.05 3
Figure 1: Structural geometry
The boundary conditions for the structure specify that only the displacement along the axis
of the cylinder is free at both the top and bottom ends, indicating that it is a simply supported
cylinder.
To ensure consistency of results, the FEM and IGA problems are set to similar conditions by
refining the meshes to achieve approximately the same number of degrees of freedom (DOFs),
while maintaining a conservative approach with DOFsFEM > DOFsIGA. The same polynomial
interpolation degree (p=2) is used for both configurations. The FEM and IGA meshes are shown
in Figure 2.
3
T. Landi1,2, C. Hoareau2, J.F.Deu2, R.Citarella1 and R.Ohayon2
(a) Structural FEM mesh (b) Structural IGA mesh
Figure 2: FEM and IGA Structural Meshes
From Figure 2, it is evident that the two meshes are not identical. However, despite this
difference, the degrees of freedom (DOFs) for FEM are 49920, while those for IGA are 49136,
ensuring that DOFsFEM > DOFsIGA. The discrepancy in the meshes arises because the
refinement strategies for FEM and IGA, although based on the same principles, do not yield
the same number of elements [5]. Importantly, this difference does not lead to an increase
in computational time, as the matrix assembly time is significantly less compared to the time
required to solve the linear system, which computes the eigenvalues and eigenvectors of the
system.
The linear system that needs to be solved to obtain the eigenvalues and eigenvectors of the
system is:
(
K− ω2sM
)
(U) = 0 (1)
Where:
K = Structural stiffness matrix;
M = Structural mass matrix;
U = Structural unknown displacement amplitude;
ωs = Structural circular natural frequency;
In Table 2, the first 10 frequencies obtained through the analytical solution [9], the FEM
analysis, and IGA are shown. Additionally, the wavenumber of each mode is included (m =
axial wavenumber; n = circumferential wavenumber). In Figure 3 the first three modal shapes,
obtained from IGA, are also shown.
4
T. Landi1,2, C. Hoareau2, J.F.Deu2, R.Citarella1 and R.Ohayon2
Table 2: First 10 structural frequencies and related wave-numbers computed with the analytical
solution, FEM and IGA
m n Analytical solution[Hz] FEM[Hz] IGA[Hz]
1 3 1.357×102 1.357×102 1.357×102
1 2 1.584×102 1.584×102 1.584×102
1 4 2.042×102 2.043×102 2.042×102
2 4 2.880×102 2.880×102 2.880×102
2 3 2.889×102 2.889×102 2.889×102
1 5 3.140×102 3.143×102 3.141×102
1 1 3.151×102 3.151×102 3.151×102
2 5 3.683×102 3.685×102 3.684×102
2 2 3.987×102 3.986×102 3.987×102
3 4 4.231×102 4.233×102 4.231×102
(a) Modal shape at 1.357×102 Hz (b) Modal shape at 1.584×102 Hz
(c) Modal shape at 2.042×102 Hz
Figure 3: First three modal shapes computed with IGA
To better figure out the obtained results, it is useful to plot the percentage error of the FEM
5
T. Landi1,2, C. Hoareau2, J.F.Deu2, R.Citarella1 and R.Ohayon2
and IGA frequencies compared to the reference analytical frequencies. The percentage error can
be defined as in the Equation 2 and its trend is shown in Figure 4
Error [%] =
∣∣∣∣( fIGA/FEMfANALYTICAL − 1
)
× 100
∣∣∣∣ (2)
Figure 4: Error between the structural frequencies obtained from FEM/IGA and the analytical
solution
As can be seen from Figure 4, although the FEM method has a low error (the maximum is
around 1.1%), the Isogeometric Approach has a lower error, which proves its higher efficiency
over FEM.
Additionally, a convergence analysis was conducted for the structural modes, evaluating the
frequency obtained by increasing the degrees of freedom of the system. The convergence analysis
for the first and third modes are shown in Figure 5.
(a) Convergence analysis for the structural
mode at 135.719 Hz
(b) Convergence analysis for the struc-
tural mode at 204.171 Hz
Figure 5: Convergence analysis for the structural modes at 135.719 Hz and 204.171 Hz
6
T. Landi1,2, C. Hoareau2, J.F.Deu2, R.Citarella1 and R.Ohayon2
Figure 5 shows that the convergence of the IGA model is higher than that of the FEM one.
2.2 Acoustic problem
The procedure followed for the structural part is repeated for the acoustic side. The conditions
imposed on the fluid cavity are those of rigid walls. The geometry is shown in Figure 6, material
properties and dimensions are listed in Table 3. The FEM and IGA meshes are shown in Figure
7. Also in this case, the condition DOFsFEM > DOFsIGA is satisfied (DOFsFEM = 16663
and DOFsIGA = 15135).
Table 3: Fluid Properties
ρf
[
kg
m3
]
c0
[
m
s
]
Diameter[m] Length[m]
1.2 340 0.95 3
Figure 6: Fluid geometry
7
T. Landi1,2, C. Hoareau2, J.F.Deu2, R.Citarella1 and R.Ohayon2
(a) Fluid FEM mesh (b) Fluid IGA mesh
Figure 7: FEM and IGA Fluid Meshes
Solving the Equation 3, it is possible to get the eigenfrequencies and the eigenvectors for the
acoustic problem.
(
H− ω2fQ
)
(P ) = 0 (3)
Where:
H = Acoustic stiffness matrix;
Q = Acoustic mass matrix;
P = Acoustic unknown pressure amplitude;
ωf = Acoustic circular natural frequency;
The obtained frequencies are shown in Table 4. m,n and k are the wave numbers in the axial,
circumferential and radial direction, respectively.
Table 4: First 10 acoustic frequencies and related wave-numbers computed with analytical
solution, FEM and IGA
m n k Analytical solution[Hz] FEM[Hz] IGA[Hz]
1 0 0 56.667 56.667 56.667
0 1 1 104.875 104.875 104.875
2 0 0 113.333 113.352 113.338
1 1 0 119.205 119.206 119.205
2 1 0 154.413 154.426 154.416
3 0 0 170.00 170.134 170.036
0 2 1 173.972 173.972 173.972
1 2 0 182.968 182.968 182.968
3 1 0 199.747 199.861 199.777
2 2 0 207.631 207.641 207.633
8
T. Landi1,2, C. Hoareau2, J.F.Deu2, R.Citarella1 and R.Ohayon2
The first three acoustic modal shapes, computed with IGA, are shown in Figure 8.
(a) Modal shape at 56.667 Hz (b) Modal shape at 104.875 Hz
(c) Modal shape at 113.338 Hz
Figure 8: IGA acoustic modal shapes of the first three modes
Following the Equation 2, it is possible to plot the percentage error for the acoustic FEM/IGA
mode frequencies respect to the analytical ones.
9
T. Landi1,2, C. Hoareau2, J.F.Deu2, R.Citarella1 and R.Ohayon2
Figure 9: Error between the FEM and IGA acoustic modes respect to the analytical ones
As can be seen from Figure 9, although the FEM method has a low error (the maximum is
around 1.1%), the Isogeometric Approach has a lower error, which proves its higher efficiency
over FEM.
Additionally, a convergence analysis was conducted for the acoustic modes, evaluating the
frequency obtained by increasing the degrees of freedom of the system. The convergence analysis
for one mode is shown in Figure 10.
Figure 10: Convergence analysis for the acoustic mode at 293.540Hz
2.3 Vibro-acoustic problem
In this last section, the vibroacoustic problem is solved using FEM and IGA in combination
with Projection of Modal Basis (PROMs) and compared with analytical results. In Figure 11, the
10
T. Landi1,2, C. Hoareau2, J.F.Deu2, R.Citarella1 and R.Ohayon2
problem under investigation is depicted (cylinder filled with fluid). The material and geometric
properties for both the structure and the fluid are as shown in Tables 1 and 3, respectively.
The boundary conditions are also as stated earlier. Figure 12 shows the FEM and IGA meshes
considered (DOFsFEM = 56281 and DOFsIGA = 55021).
Figure 11: Vibroacoustic Geometry
(a) Vibroacoustic FEM mesh (b) Vibroacoustic IGA mesh
Figure 12: FEM and IGA Vibroacoustic Meshes
To solve the vibroacoustic problem, in order to find out the eigenvectors and the eigenvalues,
it is necessary to solve the Equation 4.
−ω2 [ M 0−C Q
]
+
K − 1ρf ·CT
0 H
(U
P
)
=
(
0
0
)
(4)
11
T. Landi1,2, C. Hoareau2, J.F.Deu2, R.Citarella1 and R.Ohayon2
Where C is the coupling matrix.
The resolution of this problem is computationally very demanding; therefore, the solution is
obtained leveraging on the projection of modal basis.
First, the acoustic and structural problems are solved separately using Equations 1 and 3.
Then, by using the uncoupled modes, it is possible to solve Equation 5.−ω2c [ Mr 0−Cr Qr
]
+
Kr − 1ρf ·CrT
0 Hr
(η
ξ
)
=
(
0
0
)
(5)
Where:
Mr = ϕs
TM ϕs = Reduced structural mass matrix;
Qr = ϕf
T Q ϕf = Reduced acoustic mass matrix;
Cr = ϕf
TC ϕs = Reduced coupling matrix;
Kr = ϕs
TK ϕs = Reduced structure stiffness matrix;
Hr =ϕf
TH ϕf = Reduced acoustic stiffness matrix;
ϕs = Structural modal basis;
ϕf = Acoustic modal basis;
η = Structural modal displacement;
ξ = Acoustic modal pressure;
ωc = Coupled circular natural frequency;
The reduced problem (Equation 5) can be solved more easily than the full problem (Equation
4). In fact, while in the full problem the matrix size is related to the number of degrees of freedom
(DOFs) of the system, in the reduced problem it is related to the number of modes considered,
with DOFsr ≪ DOFs.
In this case, given that the range of interest is [0-200]Hz, to calculate the coupled modes of
the system using Equation 5, all acoustic and structural modes up to 450Hz are considered.
Table 5 shows the first 8 coupled frequencies calculated using the analytical solution [9],
FEM, and IGA and their wave number.
Additionally, Figure 13 shows the modal shapes of the first two vibroacoustic modes computed
with IGA.
12
T. Landi1,2, C. Hoareau2, J.F.Deu2, R.Citarella1 and R.Ohayon2
Table 5: First 8 Vibroacoustic frequencies and related wave-numbers computed with Analytical
solution, FEM and IGA
m n k Analytical solution[Hz] FEM[Hz] IGA[Hz]
1 0 0 56.664 56.667 56.667
0 1 1 105.054 104.867 104.875
2 0 0 113.329 113.339 113.334
1 1 0 119.176 119.205 119.205
1 3 0 135.479 135.891 135.533
2 1 0 154.399 154.406 154.398
1 2 1 157.628 157.347 158.234
0 2 1 174.194 174.754 174.108
(a) Vibroacoustic mode at 56.667 Hz (b) Vibroacoustic mode at 135.533 Hz
Figure 13: Vibroacoustic modes at 56.667 Hz and 135.533 Hz
Figure 14 shows the error (Equation 2), which is useful for demonstrating the efficiency of
IGA over FEM.
13
T. Landi1,2, C. Hoareau2, J.F.Deu2, R.Citarella1 and R.Ohayon2
Figure 14: Error between the FEM and IGA Vibroacoustic Modes respect to the Analytical
ones
As can be seen from Figure 14, although the FEM method has a low error (the maximum
is around 0.85%), also in the vibroacoustic case, the Isogeometric Approach has a lower error,
which proves its high efficiency over FEM.
3 CONCLUSIONS
In conclusion, this paper has demonstrated the advantages of the IGA approach over the
FEM one by comparing the results with those of an analytical solution. As shown in Figures
4, 9, and 14, the frequency error obtained with IGA is more accurate than that obtained with
FEM. The analyses were conducted under the same operating conditions, ensuring that both
the IGA and FEM problems were evaluated under identical hypotheses.
Additionally, convergence analyses for both structural and acoustic parts also indicate supe-
rior performance of the IGA model compared to the FEM one. The IGA approach consistently
demonstrates higher convergence rates, which highlights its effectiveness in solving complex
problems with greater accuracy.
Moreover, further investigation is necessary to explore how the solution evolves with an
increasing polynomial degree. As reported in [7], while the FEM method shows a divergence in
the solution with higher polynomial degrees, the IGA method demonstrates improved behavior.
This suggests that IGA not only provides better initial accuracy but also maintains stability
and accuracy as the polynomial degree increases.
Overall, these findings underscore the potential of the IGA approach in various engineering
applications, particularly where high accuracy and convergence are critical. Future research
should continue to explore the capabilities of IGA, especially in more complex and varied problem
settings, to fully harness its advantages over traditional FEM.
14
T. Landi1,2, C. Hoareau2, J.F.Deu2, R.Citarella1 and R.Ohayon2
REFERENCES
[1] R. Citarella, T. Landi , L. Caivano, G. D’Errico, F. Raffa , M. Romano, E. Armentani,
Structural and Vibro-Acoustics Optimization of a Car Body Rear Part. Appl. Sci. 2023, 13,
3552. https://doi.org/10.3390/app13063552
[2] R. Citarella ; L. Federico, Advances in Vibroacoustics and Aeroacustics of Aerospace and
Automotive Systems. Appl. Sci. 2018, 8, 366. https://doi.org/10.3390/app8030366
[3] R., Ohayon ; C., Soize, “Contents,” in Advanced Computational Vibroacoustics: Reduced-
Order Models and Uncertainty Quantification, Cambridge: Cambridge University Press,
2014, pp. v–viii
[4] J.A. Cottrell,T.J. Hughes, Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD
and FEA; John Wiley and Sons: Hoboken, NJ, USA, 2009, DOI:10.1002/9780470749081
[5] L. Piegl , W. Tiller, The NURBS Book, Springer-Verlag, New York, NY, USA, 1996
[6] T. Elguedj, Y. Bazilevs, V.M. Calo, T.J.R. Hughes, F-bar projection method for finite
deformation elasticity and plasticity using NURBS based isogeometric analysis, Int J Mater
Form 1 (2008) 1091–1094. https://doi.org/10.1007/s12289-008-0209-7.v
[7] T.J.R. Hughes, A. Reali, G. Sangalli, Duality and unified analysis of discrete approximations
in structural dynamics and wave propagation: Comparison of p-method finite elements with
k-method NURBS, Computer Methods in Applied Mechanics and Engineering 197 (2008)
4104–4124. https://doi.org/10.1016/j.cma.2008.04.006.
[8] W. Li, Y. Chai, M. Lei, G.R. Liu, Analysis of coupled structural-acoustic problems based
on the smoothed finite element method (S-FEM), Engineering Analysis with Boundary
Elements 42 (2014) 84–91. https://doi.org/10.1016/j.enganabound.2013.08.009.
[9] W. Larbi, J.-F. Deü, A 3D state-space solution for free-vibration analysis of a radially
polarized laminated piezoelectric cylinder filled with fluid, Journal of Sound and Vibration
330 (2011) 162–181. https://doi.org/10.1016/j.jsv.2010.08.004.
15


Projection Based Reduced Order Model of Elasto-Acoustic Vibrations Computed with Isogeometric Analyses

Kell, D.

Haynes, T.

Phase Field Modelling of CuCrZr Alloy for Nuclear Fusion

Pratap, Vikrant

Gilchrist, Michael

Tripathi, Bharat

Hyperparameter Optimization for Causal Marching Physics Informed Neural Network for Hyperelasticity

Huang, T.

Lidtke, A.

Hendrickson, K.

Terwisga, T.

Weymouth, G.

Characterizing numerical surface tension in the conservative volume-of-fluid method

Nejadseyfi, O.

The 9th European Congress on Computational Methods in Applied Sciences and Engineering
ECCOMAS Congress 2024
3–7 June 2024, Lisboa, Portugal
SUSTAINABLE MANUFACTURING VIA ROBUST
OPTIMIZATION AND TAILORED SCATTER
O. NEJADSEYFI1
1 Computational Design and Mechanics, Precision and Microsystems Engineering Department,
Mechanical Engineering Faculty
Delft University of Technology
Mekelweg 2, 2628 CD, Delft, the Netherlands
e-mail: o.nejadseyfi@tudelft.nl
Key words: Uncertainty Quantification, Surrogate Modeling, Tailoring Variations, Gaussian
Process, Sustainable Manufacturing, Exploration and Exploitation
Summary. Manufacturing industries contribute to 16.7% of global CO2 emissions. Any waste
generated during manufacturing processes such as defective or rejected products due to poor
tolerances or quality issues, results in excessive energy consumption and unnecessary carbon
emission. Most manufacturing waste and quality fluctuations arise from uncertainty and varia-
tion in raw materials and process conditions. Moreover, manufacturing with recycled materials
introduces challenges related to quality control. For this purpose, uncertainty quantification via
surrogate models and robust optimization are among the promising methods to reduce waste
during manufacturing processes. Inverse robust optimization, also known as tailored scatter,
tailored variation, or tailored uncertainty is a recently-introduced method that explores design-
ing uncertainty for a given robust performance. These inverse problems face computational
complexities due to process non-linearities, correlations and problem dimensionality. In this
work, an efficient implementation of robust optimization using Robustimizer software is applied
and its potential for tailored variation is presented. An additive manufacturing process, laser
powder bed fusion, is used to demonstrate the potential of presented methods in increasing effi-
ciency, and reducing environmental footprints in a more accurate and computationally efficient
way. Robust process settings are achieved leading to minimal variation in melt pool size in the
presence of uncertainties of material and process. In addition, the tailored scatter approach is
implemented to provide a methodology for tighter control of noise variables in the laser powder
bed fusion process considering uncertainties.
1 INTRODUCTION
Manufacturing industries are responsible for 16.7% of global CO2 emissions [1]. The 16.7%
figure includes contributions from components which are defective, have poor tolerances, or do
not meet the performance criteria due to quality issues. This excessive energy consumption
and unnecessary carbon emissions, which have its root in variations of raw material and process
conditions can be significantly reduced by robust optimization methods. [2, 3]
Most manufacturing processes are single-stage operation meaning that the transformation of
raw material to the final product occurs in a single step. The output of the process depends on
many variables some of which are easy to control while others are difficult, expensive or energy-
consuming to control. The later are called noise variables in robust optimization and can be
O. Nejadseyfi
described using a probability distribution. Robust optimization aims to find the optimal values
for controllable variables (Design variables) while considering the statistics of noise variables to
achieve the required performance and has been applied in many disciplines [4, 5, 6]
Additive manufacturing is rapidly evolving, enabling the production of highly intricate and
functional parts. It significantly reduces material waste compared to traditional subtractive
methods and reduces the lead times for low-batch production. Various methods of additive
manufacturing encounter specific challenges, particularly when it comes to predicting part per-
formance. Changes in material properties or process conditions can have a significant impact,
making it difficult to ensure consistent outcomes.
In this study, an additive manufacturing process, laser powder bed fusion (LPBF), is used to
demonstrate how robust optimization and tailored scatter can contribute to sustainable process
optimization, minimizing the impacts on unwanted input. Section 2 introduces efficient formu-
lations of robust optimization using Robustimizer and the concept of tailored scatter. Section 3
introduces the additive manufacturing model used in this work. The results are presented and
discussed in Section 4 .
2 Robust optimization and tailored scatter
While numerical simulations and analytical models provide insight about LPBF process, the
major challenge that is still open is how to obtain the model parameters. Statistical methods
provide a reasonable approach by assigning distributions to uncertain variables instead of as-
suming fixed values. This approach is also practical because, in reality, measuring the exact
value of a parameter is tedious and susceptible to measurement and environmental errors.
A surrogate model-based robust optimization procedure consists of several building blocks.
First a design of experiments (DOE) is generated in the combined design and noise space. Then
the responses of the process model are evaluated for the discrete DOE points. In the next step,
a surrogate model is constructed, which is an approximate representation of the process. The
search for the robust optimum design parameter consists of uncertainty quantification and the
repetitive evaluation of objective function value and constraints, which subsequently leads to the
optimal design variables. Since the reliance on a surrogate model might lead to loss of accuracy,
iterative improvement of the surrogate model is usually applied [7].
In the context of surrogate modeling, exploration and exploitation are two key concepts that
relate to how the surrogate model is utilized. Exploration refers to the process of gathering
information about the input output relationship across the entire input space. Exploitation,
on the other hand, involves utilizing the surrogate model to make predictions or decisions that
maximize the desired outcome within the known regions of interest. Striking a balance between
exploration and exploitation is crucial in surrogate modeling to ensure that the model is both
accurate and efficient in finding the robust optimum. New DOE points are added to the initial
DOE to improve the surrogate model of the process. A new infill point is selected in the combined
design and noise space and added to the initial DOE. This procedure can be repeated until the
updated surrogate model does not lead to further improvement of the predicted optimum design
variables. For each building block of robust optimization, various methods exist and a variety
of functions and toolboxes can be combined to perform robust optimization [8, 9]. In this study,
Robustimizer software is used offering various possibilities for all these steps in one easy-to-use
graphical interface.
2
O. Nejadseyfi
Table 1: Model parameters for Ti-6Al-4V
Name Symbol Value
Laser absorptivity (-) A 0.36
Thermal conductivity (W/(mK)) k 7.4
Boiling temperature (K) Tb 3560
Table 2: Ranges of design variables
Name Symbol Upper bound Lower bound
Scanning speed (m/s) v 0.5 1.5
Laser power (w) P 100 200
3 Additive manufacturing via laser powder bed fusion
LPBF is one of the promising additive manufacturing methods in which the powder is selec-
tively melted layer-by-layer using a laser beam. Manufacturing challenges arise while producing
the components. Specifically in LPBF, various types of defects can be introduced such as balling,
keyhole-induced pores, lack of fusion, and cracks if the processing parameters are not optimized.
Many recent studies have focused on modeling and simulation of LPBF using different levels
of complexity [10]. Most studies focus on predicting the melt pool size which is an important
metric for print quality and defect formation[11, 12]. Existing models and simulations include
many parameters which might not be easy to determine. Therefore, robust optimization opens
a new horizon in investigating these processes with a statistical approach.
In this work an analytical model of melt pool depth is chosen to demonstrate how uncertainties
in process modeling can be taken into account during process optimization. The keyhole mode
melting [13, 14] is selected in which instabilities can cause keyhole-induced pore formation among
other issues. In this case, the melt pool depth can be described by:
D =
AP
2πkTb
ln
a+ α/v
a
(1)
where A is laser absorptivity (-), P is laser power (W), k is the thermal conductivity (W/(m.K)),
Tb is the boiling temperature (K), a is laser beam size (m), α is the thermal diffusivity (m2/s),
and v is the scanning speed (m/s).
The model parameters are grouped into three categories. Some model parameters are assumed
to be fixed. The nominal values for these parameters are listed in table 1. Laser power and
scanning speed are chosen to be the design variables which can be adjusted to achieve the desired
performance within their lower and upper bounds (Table 2). Moreover, thermal diffusivity
and laser beam size are assumed to be noise variables meaning that their exact values are
unknown. These can be assumed as probabilistic inputs to the model with given means and
standard deviations as shown in Table 3. Laser beam size can be considered a source of process
variability, influenced by factors such as optical alignment, thermal effects, lens quality, and
environmental conditions. In addition, thermal diffusivity can be considered as a source of
material uncertainty due to inconsistencies in material composition, and complexity of thermal
response of the material under process. The aim is to identify the design variables necessary to
achieve a consistent melt pool depth, considering uncertainties.
3
O. Nejadseyfi
Table 3: Statistics of noise variables
Name Symbol Mean Standard deviation
Thermal diffusivity (m2/s) α 8e-6 0.5e-6
Laser beam size (m) a 40e-6 0.5e-6
4 Results and discussion
To achieve a constant melt pool depth in the presence of noise, the following objective is
implemented in the robust optimization framework:
minimize
x
(
µD(x)−Dt
)2
+ 3
(
σD(x)
)2
(2)
where µD defines the mean depth of the melt pool, σD is the standard deviation, and Dt is the
target depth which is set to 100µm in this work. An initial DOE of 20 points in a 4D space of
design-noise variables is created via combination of full factorial design and Latin hypercube and
the results are evaluated using Equation (1). A surrogate model based on Gaussian processes
which is an approximate of this function is created in Robustimizer and is used to search for
robust optimum. This approach is used to compare the accuracy of the results and it must
be noted that in general the relationship between input and output is not explicitly known. A
modified infill criterion based on [7] is used to search for the robust optimum by adding new
points to the initial DOE. Figure 1 represents the initial DOE with unfilled circles and the added
points in the combined design-noise space with filled circles. The darker and larger the filled
circle, the later it was added to the DOE. 15 points are added sequentially until the expected
improvement value converged to a small number (< 0.001). Figure 2 shows the search for the
robust optimum via exploration and exploitation. The darker and larger the filled circle, the
later the optimum was found in the search process. After adding 15 points adaptively, the
optimum was found at x = [121.81, 0.50].
The ground truth objective function in the design space using Equation (1) is represented
by contour lines in Figure 2. The reference robust optimization solution is at x = [122.77, 0.50],
which means the surrogate-based approach could predict the optimum with only 0.8% deviation.
To achieve a target melt pool depth without considering uncertainties, process optimization in
a deterministic manner leads to x = [198.68, 0.86]. This highlights the importance of considering
noise in the optimization process, as ignoring it can lead to suboptimal settings. In contrast, a
robust approach can lead to significantly different process settings. It is interesting to see that
higher stability against noise is achieved with less laser power and scanning speed.
Inverse robust optimization also known as tailored scatter, is a recently introduced method
[15] that explores the possibilities of designing uncertainty for a given robust performance.
This approach identifies the largest possible noise variations that still ensure a tightened robust
performance. The formulation applied in this work is as follows:
min
x,σz
1∑
iwiσ̄i
s.t.
(
µD(x,σz)−Dt
)2
+ 3
(
σD(x,σz)
)2(
µD(x,σn)−Dt
)2
+ 3
(
σD(x,σn)
)2 − Ctol ≤ 0 (3)
where wi are the weights corresponding to the cost of tightly controlling the noise variable i,
σ̄i =
σzi
σni
, σzi is the optimization variable representing the standard deviation for noise variable
4
O. Nejadseyfi
Figure 1: Initial DOE shown with unfilled circles and the additional points added through exploration
and exploitation shown with filled circles in the (a) design space and (b) noise space. (The darker and
larger the filled circle, the later it was added to the DOE)
Figure 2: Search for the robust optimum via exploration and exploitation. ( The darker and larger the
filled circle, the later the optimum was found in the search process)
i, σni is the nominal standard deviation for noise variable i, and Ctol represents the tightening
ratio of the objective value to the nominal objective value.
Figure 3 shows the tolerable variations for two Ctol values with various control cost weights.
This figure indicates how to achieve a tighter tolerance on the melt-pool depth by tightening
the control on two noise variables. In general, variations in laser beam size have a smaller
5
O. Nejadseyfi
Figure 3: Tolerable scatter for (a) Ctol = 0.25 and (b) Ctol = 0.5 for various control cost weights
influence on overall performance. However, if the cost of controlling the thermal diffusivity is
around three times that of the laser beam size, the variation in both noise variables needs to be
controlled upto 20% of the nominal variation, to achieve a melt pool depth variation of 25% of
the original depth variation. These results provide insight for decision-making in controlling the
LPBF process from a statistical point of view.
5 CONCLUSIONS
This study explored the application of robust optimization to the laser powder bed fusion
process, addressing key challenges related to manufacturing quality. A surrogate model-based
robust optimization procedure was effectively applied to minimize variations and achieve a con-
sistent melt pool depth, which is crucial for enhancing print quality and reducing defects. By
employing the tailored scatter technique, the potential for controlled reduction of uncertainties
and improvement of performance was demonstrated. The approach facilitated the identification
of the largest possible noise variations that ensured required performance. This work contributed
to a deeper understanding of how statistical approaches can be employed to achieve a more ro-
bust process for reducing manufacturing-related variations, paving the way for more efficient,
reliable and sustainable production methods.
REFERENCES
[1] J. Olivier and J. Peters, “Trends in global co2 and total greenhouse gas emissions,” PBL
Netherlands Environmental Assessment Agency, 2020.
[2] O. Nejadseyfi, H. Geijselaers, E. Atzema, M. Abspoel, and A. van den Boogaard, “Ac-
counting for non-normal distribution of input variables and their correlations in robust
optimization,” Optimization and Engineering, pp. 1–27, 2021.
6
O. Nejadseyfi
[3] H. Li, Z. Huang, T. Yang, H. Wang, Z. Fu, J. Chen, and S. Wu, “Robust optimization of
prismatic lithium-ion cells for reducing thermal performance fluctuations and manufacturing
costs,” Journal of Energy Storage, vol. 72, p. 108391, 2023.
[4] A. Yaqoubi, F. Sabouhi, A. Bozorgi-Amiri, and M. S. Amalnick, “A light robust optimiza-
tion model for a hierarchical healthcare network,” Kybernetes, vol. 53, no. 3, pp. 990–1014,
2024.
[5] S. Singh and M. Biswal, “A robust optimization model under uncertain environment: An ap-
plication in production planning,” Computers & Industrial Engineering, vol. 155, p. 107169,
2021.
[6] O. Nejadseyfi, H. J. Geijselaers, and A. H. van den Boogaard, “Evaluation and assess-
ment of non-normal output during robust optimization,” Structural and multidisciplinary
optimization, vol. 59, pp. 2063–2076, 2019.
[7] D. R. Jones, M. Schonlau, and W. J. Welch, “Efficient global optimization of expensive
black-box functions,” Journal of Global optimization, vol. 13, pp. 455–492, 1998.
[8] S. Kitayama and K. Yamazaki, “Sequential approximate robust design optimization using
radial basis function network,” International Journal of Mechanics and Materials in Design,
vol. 10, pp. 313–328, 2014.
[9] O. Nejadseyfi, H. J. M. Geijselaers, and A. H. van den Boogaard, “Robust optimiza-
tion based on analytical evaluation of uncertainty propagation,” Engineering Optimization,
pp. 1–23, 11 2018.
[10] H. Bikas, P. Stavropoulos, and G. Chryssolouris, “Additive manufacturing methods and
modelling approaches: a critical review,” The International Journal of Advanced Manufac-
turing Technology, vol. 83, pp. 389–405, 2016.
[11] J. Yang, J. Han, H. Yu, J. Yin, M. Gao, Z. Wang, and X. Zeng, “Role of molten pool mode
on formability, microstructure and mechanical properties of selective laser melted ti-6al-4v
alloy,” Materials & Design, vol. 110, pp. 558–570, 2016.
[12] B. Liu, G. Fang, L. Lei, and X. Yan, “Predicting the porosity defects in selective laser
melting (slm) by molten pool geometry,” International Journal of Mechanical Sciences,
vol. 228, p. 107478, 2022.
[13] G. G. Gladush and I. Smurov, Physics of laser materials processing: theory and experiment,
vol. 146. Springer Science & Business Media, 2011.
[14] J.-N. Zhu, E. Borisov, X. Liang, E. Farber, M. Hermans, and V. Popovich, “Predictive ana-
lytical modelling and experimental validation of processing maps in additive manufacturing
of nitinol alloys,” Additive Manufacturing, vol. 38, p. 101802, 2021.
[15] O. Nejadseyfi, H. Geijselaers, E. Atzema, M. Abspoel, and A. Van Den Boogaard, “From
specified product tolerance to acceptable material and process scatter: an inverse robust
optimization approach,” International Journal of Material Forming, vol. 13, pp. 467–478,
2020.
7


Sustainable Manufacturing via Robust Optimization and Tailored Scatter

O'Sullivan, A.

Mangani, L.

Ferrer, E.

Study on the Effect of the Upcoming Turbulent Boundary Layer on Automotive Test Case

Bublík, O.

Hybrid Finite Volume-Neural Network Method Applied to Fluid Flow Problems

Delage, M.

Jacot-Descombes, G.

Scantamburlo, D.

Commend, S.

Wood Constitutive Law Implementation in Finite Element Software ZSoil to Model Dowel Embedment Tests on Glulam

BARIK, LOKANATH

Swain, A.

Vibration analysis of FGM sandwich panel with cut-outs using refined HSDT based on isogeometric analysis

Not a member yet