A technique based on a skeleton-section template for parameterizing finite element (FE) models is reported and applied to shape optimization of thin-walled beam components. The template consists of a skeletal curve and a set of cross-sectional profiles. The skeletal curve can be used to derive global model variations, while the cross section is designed to obtain local deformations of the given shape. A mesh deformation method based on the radial basis functions (RBF) interpolation is employed to derive the shape variations. Specifically, the skeleton-embedding space and an anisotropic distance metric are introduced to improve the RBF deformation method. To validate the applicability of the proposed template-based parameterization technique to general shape optimization frameworks, two proof-of-concept numerical studies pertaining to crashworthiness design of an S-shaped frame were implemented. The first case study focused on global deformations with the skeletal curve, and the second treated the cross-sectional profiles as design parameters to derive local reinforcements on the model. Both studies showed the efficiency of the proposed method in generation of quality shape variants for optimization. From the numerical results, considerable amount of improvements in crashworthiness performances of the S-shaped frame were observed as measured by the peak crushing force (PCF) and the energy absorption. We conclude that the proposed template-based parameterization technique is suitable for shape optimization tasks.

References

References
1.
Donders
,
S.
,
Takahashi
,
Y.
,
Hadjit
,
R.
,
Van Langenhove
,
T.
,
Brughmans
,
M.
,
Van Genechten
,
B.
, and
Desmet
,
W.
,
2009
, “
A Reduced Beam and Joint Concept Modeling Approach to Optimize Global Vehicle Body Dynamics
,”
Finite Elem. Anal. Des.
,
45
(
6–7
), pp.
439
455
.
2.
Mihaylova
,
P.
,
Baldanzini
,
N.
, and
Pierini
,
M.
,
2013
, “
Potential Error Factors in 1D Beam Fe Modeling for the Early Stage Vehicle Design
,”
Finite Elem. Anal. Des.
,
74
, pp.
53
66
.
3.
Tai
,
K.
, and
Chee
,
T.
,
2000
, “
Design of Structures and Compliant Mechanisms by Evolutionary Optimization of Morphological Representations of Topology
,”
ASME J. Mech. Des.
,
122
(
4
), pp.
560
566
.
4.
Cui
,
G. Y.
,
Tai
,
K.
, and
Wang
,
B. P.
,
2002
, “
Topology Optimization for Maximum Natural Frequency Using Simulated Annealing and Morphological Representation
,”
AIAA J.
,
40
(
3
), pp.
586
589
.
5.
Zhang
,
W.
,
Yuan
,
J.
,
Zhang
,
J.
, and
Guo
,
X.
,
2016
, “
A New Topology Optimization Approach Based on Moving Morphable Components (MMC) and the Ersatz Material Model
,”
Struct. Multidiscip. Optim.
,
53
(
6
), pp.
1243
1260
.
6.
Zuo
,
W.
, and
Saitou
,
K.
,
2017
, “
Multi-Material Topology Optimization Using Ordered Simp Interpolation
,”
Struct. Multidiscip. Optim.
,
55
(
2
), pp.
477
491
.
7.
Yang
,
L.
,
Li
,
B.
,
Lv
,
Z.
,
Hou
,
W.
, and
Hu
,
P.
,
2016
, “
Finite Element Mesh Deformation With the Skeleton-Section Template
,”
Comput.-Aided Des.
,
73
, pp.
11
25
.
8.
Duddeck
,
F.
, and
Zimmer
,
H.
,
2013
, “
Modular Car Body Design and Optimization by an Implicit Parameterization Technique Via SFE Concept
,”
FISITA 2012 World Automotive Congress
, pp.
413
424
.
9.
Rayamajhi
,
M.
,
Hunkeler
,
S.
, and
Duddeck
,
F.
,
2014
, “
Geometrical Compatibility in Structural Shape Optimisation for Crashworthiness
,”
Int. J. Crashworthiness
,
19
(
1
), pp.
42
56
.
10.
Zuo
,
W.
,
2015
, “
Bi-Level Optimization for the Cross-Sectional Shape of a Thin-Walled Car Body Frame With Static Stiffness and Dynamic Frequency Stiffness Constraints
,”
Proc. Inst. Mech. Eng., Part D: J. Automob. Eng.
,
229
(
8
), pp.
1046
1059
.
11.
Zuo
,
W.
, and
Bai
,
J.
,
2016
, “
Cross-Sectional Shape Design and Optimization of Automotive Body With Stamping Constraints
,”
Int. J. Automot. Technol.
,
17
(
6
), pp.
1003
1011
.
12.
Zuo
,
W.
,
Lu
,
Y.
,
Zhao
,
X.
, and
Bai
,
J.
,
2018
, “
Cross-Sectional Shape Design of Automobile Structure Considering Rigidity and Driver's Field of View
,”
Adv. Eng. Software
,
115
, pp.
161
167
.
13.
Shimada
,
K.
,
2011
, “
Current Issues and Trends in Meshing and Geometric Processing for Computational Engineering Analyses
,”
ASME J. Comput. Inf. Sci. Eng.
,
11
(
2
), p.
021008
.
14.
Hughes
,
T. J.
,
Cottrell
,
J. A.
, and
Bazilevs
,
Y.
,
2005
, “
Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement
,”
Comput. Methods Appl. Mech. Eng.
,
194
(
39–41
), pp.
4135
4195
.
15.
Kiendl
,
J.
,
Schmidt
,
R.
,
Wüchner
,
R.
, and
Bletzinger
,
K.-U.
,
2014
, “
Isogeometric Shape Optimization of Shells Using Semi-Analytical Sensitivity Analysis and Sensitivity Weighting
,”
Comput. Methods Appl. Mech. Eng.
,
274
, pp.
148
167
.
16.
Zhang
,
X.
,
Xia
,
Y.
,
Hu
,
Q.
, and
Hu
,
P.
,
2017
, “
Efficient Isogeometric Formulation for Vibration Analysis of Complex Spatial Beam Structures
,”
Eur. J. Mech.-A/Solids
,
66
, pp.
212
231
.
17.
Staten
,
M. L.
,
Owen
,
S. J.
,
Shontz
,
S. M.
,
Salinger
,
A. G.
, and
Coffey
,
T. S.
,
2011
, “
A Comparison of Mesh Morphing Methods for 3D Shape Optimization
,”
20th International Meshing Roundtable
, Paris, France, Oct. 23–26, pp.
293
311
.
18.
Sieger
,
D.
,
Menzel
,
S.
, and
Botsch
,
M.
,
2014
, “
RBF Morphing Techniques for Simulation-Based Design Optimization
,”
Eng. Comput.
,
30
(
2
), pp.
161
174
.
19.
Hojjat
,
M.
,
Stavropoulou
,
E.
, and
Bletzinger
,
K.-U.
,
2014
, “
The Vertex Morphing Method for Node-Based Shape Optimization
,”
Comput. Methods Appl. Mech. Eng.
,
268
, pp.
494
513
.
20.
Mihaylova
,
P.
,
Baldanzini
,
N.
,
Pratellesi
,
A.
, and
Pierini
,
M.
,
2012
, “
Beam Bounding Box–A Novel Approach for Beam Concept Modeling and Optimization Handling
,”
Finite Elem. Anal. Des.
,
60
, pp.
13
24
.
21.
Li
,
B.
,
Yang
,
L.
,
Jiang
,
H.
,
Sun
,
W.
, and
Hu
,
P.
,
2017
, “
Concurrent Editing of Automotive Styling and Structure With Wireframe-Pair
,”
Proc. Inst. Mech. Eng., Part D: J. Automob. Eng.
,
231
(
6
), pp.
828
841
.
22.
Rendall
,
T.
, and
Allen
,
C.
,
2008
, “
Unified Fluid–Structure Interpolation and Mesh Motion Using Radial Basis Functions
,”
Int. J. Numer. Methods Eng.
,
74
(
10
), pp.
1519
1559
.
23.
Biancolini
,
M.
,
Viola
,
I.
, and
Riotte
,
M.
,
2014
, “
Sails Trim Optimisation Using CFD and RBF Mesh Morphing
,”
Comput. Fluids
,
93
, pp.
46
60
.
24.
Wang
,
W.
,
Jüttler
,
B.
,
Zheng
,
D.
, and
Liu
,
Y.
,
2008
, “
Computation of Rotation Minimizing Frames
,”
ACM Trans. Graph.
,
27
(
1
), p.
2
.
25.
Wendland
,
H.
,
2006
, “
Computational Aspects of Radial Basis Function Approximation
,”
Studies in Computational Mathematics
, Vol.
12
,
Elsevier
, Amsterdam, The Netherlands, pp.
231
256
.
26.
Zhou
,
Y.
,
Lan
,
F.
, and
Chen
,
J.
,
2011
, “
Crashworthiness Research on S-Shaped Front Rails Made of Steel–Aluminum Hybrid Materials
,”
Thin-Walled Struct.
,
49
(
2
), pp.
291
297
.
27.
Liu
,
S.-T.
,
Tong
,
Z.-Q.
,
Tang
,
Z.-L.
, and
Zhang
,
Z.-H.
,
2014
, “
Design Optimization of the S-Frame to Improve Crashworthiness
,”
Acta Mech. Sin.
,
30
(
4
), pp.
589
599
.
28.
Nguyen
,
V. S.
,
Wen
,
G.
,
Yin
,
H.
, and
Van
,
T. P.
,
2014
, “
Optimisation Design of Reinforced s-Shaped Frame Structure Under Axial Dynamic Loading
,”
Int. J. Crashworthiness
,
19
(
4
), pp.
385
393
.
29.
Couckuyt
,
I.
,
Dhaene
,
T.
, and
Demeester
,
P.
,
2014
, “
ooDACE Toolbox: A Flexible Object-Oriented Kriging Implementation
,”
J. Mach. Learn. Res.
,
15
, pp.
3183
3186
.http://jmlr.csail.mit.edu/papers/volume15/couckuyt14a/couckuyt14a.pdf
30.
Couckuyt
,
I.
,
Forrester
,
A.
,
Gorissen
,
D.
,
De Turck
,
F.
, and
Dhaene
,
T.
,
2012
, “
Blind Kriging: Implementation and Performance Analysis
,”
Adv. Eng. Software
,
49
, pp.
1
13
.
31.
Yang
,
R. J.
,
Wang
,
N.
,
Tho
,
C. H.
,
Bobineau
,
J. P.
, and
Wang
,
B. P.
,
2001
, “
Metamodeling Development for Vehicle Frontal Impact Simulation
,”
ASME J. Mech. Des.
,
127
(
5
), pp.
1014
1020
.
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