EVALUATE THE STRAIN ENERGY ERROR FOR THE LASER WELD BY THE H-REFINEMENT OF THE FINITE ELEMENT METHOD

International Journal of Innovative Research in Advanced Engineering (IJIRAE) ISSN: 2349-2163
Issue 09, Volume 6 (September 2019) www.ijirae.com
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EVALUATE THE STRAIN ENERGY ERROR FOR THE
LASER WELD BY THE H-REFINEMENT OF THE FINITE
ELEMENT METHOD
Long Nguyen-Nhut-Phi *
Faculty of Mechanical Engineerin,g
HCMC University of Technology and Education, Viet Nam
longnnp@hcmute.edu.vn
Son Nguyen-Hoai*
Faculty of Civil Engineering,
HCMC University of Technology and Education, Viet Nam
sonnh@hcmute.edu.vn
Quan Nguyen
Faculty of Engineering - Technology,
PDU, Quang Ngai, Viet Nam
nquan@pdu.edu.vn
Manuscript History
Number: IJIRAE/RS/Vol.06/Issue09/SPAE10081
Received: 07, September2019
Final Correction: 17, September 2019
Final Accepted: 20, September 2019
Published: September 2019
Citation: Nhut-Phi, N., Nguyen-Hoai & Nguyen (2019). EVALUATE THE STRAIN ENERGY ERROR FOR THE LASER
WELD BY THE H-REFINEMENT OF THE FINITE ELEMENT METHOD. IJIRAE::International Journal of Innovative
Research in Advanced Engineering, Volume VI, 586-591. doi: 10.26562/IJIRAE.2019.SPAE10081
Editor: Dr.A.Arul L.S, Chief Editor, IJIRAE, AM Publications, India
Copyright: ©2019 This is an open access article distributed under the terms of the Creative Commons Attribution
License, Which Permits unrestricted use, distribution, and reproduction in any medium, provided the original author
and source are credited
Abstract— Currently, the finite element method (FEM) is still one of the useful tools in numerical simulation for
technical problems. With this method, a continuum model presented by a certain number of elements with a
simple approximation field causes the presence of discretization error in solutions. This paper considers the butt
weld by laser which subjected the tension for AISI 1018 steel highness 8 mm. The aim of the study is to use the h-
refinement of the FEM in estimation the strain energy error for the laser weld mentioned. The results show that
the stability of the h-refinement shown by the value of the relative error of the strain energy is quite small,
specifically; FEM is less than 5.7% and extra is no more than 3.7%.
Keywords— Finite element method; error; laser weld; AISI 1018 steel; h-refinement; strain energy;
I. INTRODUCTION
Nowadays, with the increasing use of finite element analysis programs, developing a reliable and fast process for
engineers is necessary. A mesh refinement finite element method (in short, called a refinement method) has been
the subject of the extensive investigation to obtain solutions with pre-specified accuracy with the minimum cost of
model preparation and computation. In a refinement procedure, a finite element mesh is sequentially upgraded in
such a way that the discretization error in the final solution is reduced and reaches the desired level. In h-
refinement, a mesh with a fixed type of low order element is repeatedly refined by reducing the element size h.
The h-refinement procedure can be directly incorporated into the existing finite element codes. For this reason,
the h-refinement method has been more widespread in commercial programs.

__________________________________________________________________________________
The study [1] presented local error evaluation for two 2D models (the Poisson problem in the plane and the plane
elasticity problem) using the adaptive h-type FEM based on interpolation estimates and the 'extraction formulas'
of Babu Ska and Miller. The results compared with those obtained by the ‘ad-hoc’ methods. However, these
formulas made more widespread for certain nonlinear cases, general elliptic boundary value problems. The
results of the document [2] showed the analysis of mathematical models in detail, highlighting the similarities and
differences between one-dimensional and two-dimensional problems through emphasizing the aspects of p and h-
p type of FEM. The ability to evaluate the reliability of any technical data is one of the basic comparison criteria. It
seems that with an appropriately designed grid, the adaptive p-version is suitable for the calculation of structural
mechanical problems in practice but in complex three-dimensional problems based on STRIPE, this study refers
([3], [4]). In [5], the results of h-anisotropic version mesh were compared to h-isotropic version mesh. The
solutions of the anisotropic method are approximate to one-dimensional behaviour in some areas, typically, the
boundary layer problem. By using the estimations of interpolation error for quadrilateral elements without the
hypothesis of the finiteness of the proportion of two sizes measured in two directions to assess the effectiveness
of this method. For more compound problems, this improvement is seen as the stimulation for a few heuristic
plans. However, the performance of this interpolation for curvilinear elements and various problems needs more
study. The research [6] presented the use of the local h-refinement with T-splines by proposing an adaptive
algorithm that creates locally refined meshes through combined with a set of posteriori error estimate to
overcome the disadvantage of iso geometric analysis based on non-uniform rational B-splines (NURBS). This has
resulted in a significant increase in the quality of being efficient. However, in the three-dimensional problem, it is
worrying that the finery of T-junctions is not as local as expected. Based on the aforementioned studies and the
effectiveness of h-version, this paper proposes using the h-refinement of the FEM to evaluate the energy norm
error for the butt weld for AISI 1018 steel by laser. At the same time, there have not been many studies applying h-
version in determining this error for welding in general and laser welding in particular.
II. THE ERROR ESTIMATION FOR THE H-REFINEMENT OF FINITE ELEMENT METHOD
The equilibrium equation (the strong form):
0; uLu f Lu f     (1)
The principle of virtual work (weak form):
( ) 0; u,v Lu f d v

    (2)
The homogenous Dirichlet’s boundary condition:
0 Du on  (3)
Find the energy u V such that it satisfies equation (3):
( , ) ( )B u v L v v V   (4)
Where
  1
: 0
( , ) ( ) : ( )
L( ) . .
sd
N
n
D the infinite dimensional space
the virtual work
V v H v on
B u v v u d
v v f d v t
of internal stresses
the virtual work of external ford ces
 

 

       

 


   



 
The strain energy:
1
( ) ( , )
2
U u B u u (5)
The energy norm: ( )
( , )E
u B u u
 (6)
The error e and the energy norm error  E
e 
in the finite-dimensional space:
he u u  (7)
   hE E
e u u 
  (8)
where u is the exact energy and hu is the approximation element finite energy
The error of the energy norm can be theoretically estimated by equation (9):
( ) ( )h E h E
k
e u u
N    (9)
where k is a positive problem's dependent constants, N the number of degrees of freedom, and  the asymptotic
rate of convergence. Using the Richardson extrapolation technique, the error can be written in term of energy:

__________________________________________________________________________________
2
2
( ) ( ) 2
2h E h E
k
e U U
N     (10)
where 
2
Ω
1
2
U u is the exact strain energy and 
2
Ω
1
2h h
U u is the finite element strain energy.
The relative error of the strain energy:
(%) 100%h
FEM
U U
U


  (11)
1
(%) 100%h h
extra
h
U U
U
 
  (12)
The effectivity index: extra
FEM



 (13)
III.THE LASER WELD WITH THE TENSILE STRESS PROBLEM
Refer to Figure 2 of [7], this study proposes the laser weld model which is subjected to tensile stress as shown in
Figure 1: The base material is AISI 1018 steel with the modulus of elasticity E = 205 GPa, and the Poissons ratio 
= 0.29 ([8], [9]) The dimensions in Figure 1 include the length L = 100 mm, the high H = 8 mm, and the thick t = 1
mm.
(1) The base material (2) The heat-affected zone (3) The zone of fusion
Fig. 1 The scheme of the laser weld with the tensile stress
The exact strain energy U by given ([10], [11]):
2
1
2
V
U
E

 (14)
Replacing the tensile strength of the laser weld after the test  = 562 MPa [4], the volume V = L × H × t = 800 mm3,
and the modulus of elasticity E = 205 GPa into the equation (14), the value of the exact strain energy
U = 0.616281 kJ. The finite element analysis is performed in case of the plane strain.
Figure 2 presents the Matlab code program structure.
Fig. 2 The program structure

__________________________________________________________________________________
IV. NUMERICAL RESULTS
Fig. 3 The finite element model Fig. 4 The deformed model
Fig. 5 The displacement field in the x-direction Fig. 6 The normal stress field in the x-direction
TABLE I- THE H-REFINEMENT ERROR ESTIMATION RESULTS
Mesh
Element
numbers
Dofs
The FEM
strain energy
The extra
strain energy
CPU time
(s)
FEM
(%)
extra
(%)

10x10 500 1122 0.614324438 0.6151587149 4.311 5.634525995 3.682658519 0.653588
11x11 605 1344 0.614415158 0.6151357316 6.466 5.502347809 3.422580574 0.622022
12x12 720 1586 0.614489235 0.6151203606 10.007 5.392015867 3.203154843 0.594055
13x13 845 1848 0.614550606 0.6151100296 15.431 5.298867385 3.015739461 0.569129
14x14 980 2130 0.614602102 0.6151028451 23.057 5.219426836 2.853210237 0.546652
15x15 1125 2432 0.614645795 0.6150978253 33.682 5.151060698 2.71088754 0.526278
16x16 1280 2754 0.614683239 0.6150942646 49.702 5.091743887 2.585019044 0.507688
17x17 1445 3096 0.614715611 0.6150917687 66.657 5.039898373 2.472948899 0.490674
18x18 1620 3458 0.614743821 0.6150899289 92.058 4.994279531 2.372119294 0.474967
19x19 1805 3840 0.614768580 0.6150885422 122.959 4.953894817 2.280763997 0.460398
20x20 2000 4242 0.614790452 0.6150875415 166.091 4.917944624 2.197735303 0.446881

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Fig. 7 The error estimation by the Richardson extrapolation technique
Fig. 8 The convergence curve
V. CONCLUSIONS
This paper implemented an assessment of the strain energy error for the tensile laser weld by the h-refinement of
the FEM in two-dimensional mesh with the quadrilateral element. The base material of the butt weld is AISI 1018
steel highness 8mm. Corresponding to the mesh size, number of elements, and degrees of freedom, Table 1 has
shown the relative error results of the strain energy 4.917944624 5.63452599(%) 5FEM  ,
2.197735303 3.68265851(%) 9extra  and the effective index  in the (0.446881 – 0.653588) range. This result is
reliable, satisfying: 1 ≤  (%) ≤ 10,  ≤ 1.2 ([12])
REFERENCES
1. L. Demkowicz, Ph. Devloo, J.T. Oden, On an h-type mesh-refinement strategy based on minimization of
interpolation errors, Computer Methods in Applied Mechanics and Engineering, Volume 53, Issue 1, Pages 67-
89, ISSN 0045-7825, 1985. https://doi.org/10.1016/0045-7825(85)90076-3
2. I. Babuška, B.Q. Guo, The h, p and h-p version of the finite element method; basis theory and applications,
Advances in Engineering Software, Volume 15, Issues 3–4, Pages 159-174, ISSN 0965-9978, 1992.
https://doi.org/10.1016/0965-9978(92)90097-Y
3. Andersson, B. , Falk, U. , Babus̆ka, I. and Von Petersdorff, T., Reliable stress and fracture mechanics analysis of
complex components using a h–p version of FEM. Int. J. Numer. Meth. Engng., 38: 2135-2163, 1995
https://doi.org/10.1002/nme.1620381302
4. Babuška, I., Petersdorff, v.T. & Andersson, B., Computation of the vertex singularity factors for laplace equation
in 3 dimensions, Tech. Note BN-1140, Institute for Physical Science and Technology, University of Maryland,
USA, 1992.
5. W Rachowicz, An anisotropic h-type mesh-refinement strategy, Computer Methods in Applied Mechanics and
Engineering, Volume 109, Issues 1–2, Pages 169-181, ISSN 0045-7825, 1993 https://doi.org/10.1016/0045-
7825(93)90231-L

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6. Michael R. Dörfel, Bert Jüttler, Bernd Simeon, Adaptive isogeometric analysis by local h-refinement with T-
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ISSN 0045-7825, 2010 https://doi.org/10.1016/j.cma.2008.07.012
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4031-4039, 2017 https://doi.org/10.1557/adv.2017.599
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1018ColdDrawn.pdf
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http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_I/BookSM_Part_I/08_Energy/08_Ene
rgy_02_Elastic_Strain_Energy.pdf
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EVALUATE THE STRAIN ENERGY ERROR FOR THE LASER WELD BY THE H-REFINEMENT OF THE FINITE ELEMENT METHOD

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