Application of the finite element P1 / P1 to the simulation of a MAG welding
operation
Eric Feulvarch1 , Jean-Christophe Roux1 , a , Robin Chatelin1 , and Jean-Michel Bergheau1
1
Univ Lyon, ENISE, LTDS, UMR 5513 CNRS, 58 rue Jean Parot, F-42023 SAINT-ÉTIENNE cedex 2, France
Abstract. This work shows the ability of the finite element P1/P1 to simulate a welding operation. In the
first part, while the P1/P1 finite element is known to be not LBB-stable, the properties of this element and the
conditions of its correct use are studied. A new simple and robust stabilized formulation that prevents parasites
pressure modes is proposed in the second part. Finally, a simulation example of a circular MAG welding
operation is presented.
1 Introduction
2.2 Finite element discretization
In the general context of large transformations, the theoretical analysis of problems involving elasto-viscoplasticity
in solid mechanics is very difficult. Indeed, these problems
induce geometrical configuration changes and the integration in time of tensorial quantities such as stresses while
satisfying the material objectivity [1]. Moreover, they are
very complex to analyze mathematically. To avoid this
difficulty, we propose to base our discussion on a configuration which corresponds to hot forming processes [2] by
considering an elasto-viscoplastic behavior without hardening associated to the von Mises criterion generally used
for solid metals.
Following the usual finite element procedure, the finite element P1 / P1 consists in approximating the velocity u and
the pressure p as follows:
2 Properties and use of the P1 / P1 element
2.1 Weak formulation
Assuming that the elastic part of the strain rate is spherical,
and within a Langrangian framework, the weak formulation of the problem governing the momentum in a domain
Ω of boundary ∂Ω is ([3]):
Find functions u, p such that for all functions v, q,
Z
 Z
2

s
s


µ ∇ · u ∇ · v dV
2
µ
∇
u
:
∇
v
dV
−




Ω
Ω 3


Z
Z
(1a)




(p)
−
p
∇
·
v
dV
−
T
v
dS
=
0




Ω
ΓT


Z
Z



1



q ∇ · u dV +
q p0 dV = 0
(1b)

Ω
Ω α
where u is the material velocity, p is the hydrostatic pressure, ∇ s u is the strain rate tensor, µ is the dynamic viscosity, α is the bulk modulus, T(p) is a prescribed “stress
vector” and p0 = ∂p
∂t .
a e-mail: [email protected]
uh (x) =
N
X
ui Ni (x)
i=1
ph (x) =
N
X
pi Ni (x) ⇒ p0 h (x) =
N
X
p0 i Ni (x)
i=1
i=1
In these expressions N denotes the number of nodes, ui ,
pi and p0 i the values of the functions u, p and p0 at node i
and Ni (x) the linear shape function associated to this node.
Following Galerkin’s standard approach, functions v and
q are taken of the same form.
2.3 Volumetric locking
A very interesting feature of the finite element P1 / P1 is
that it is not sensitive to volumetric locking phenomena
well known in computational mechanics for incompressible material flows [4]. Concerning the Stokes problem
(α → +∞ in eq.(1b)), the locking phenomenon comes
from the fact that all solutions {uh } must not be reduced
to {0} for achieving a physically acceptable solution. In
other words, the incompressibility equation (1b) must not
impose the uniqueness of the velocity solution:
Z
D E
h
∀q ,
qh ∇ · uh dV = qh [B] {uh } = 0
Ω
=⇒
[B] {uh } = {0}
=⇒
6
{uh } = {0}
where {qh } ≡ (qi )16i6N and {uh } ≡ (uik )16i6N; 16k63 denote
the vectors of nodal values of the functions qh and uh respectively; [B] ≡ (Bi, jk )16i, j6N; 16k63 is a matrix defined by
Z
∂N j
Bi, jk ≡
Ni
dV
∂xk
Ω
One can note that the finite element P1 / P1 has three times
more unknowns for the velocity than for the pressure in
3D. Thus, the matrix [B] has always more columns than
lines and we get rank (B) 6 N < 3 N. This ensures that all
solutions {uh } are never reduced to {0}.
Locking phenomena are therefore avoided.
i; ∆xh is the displacement increment; αi is the value of the
bulk modulus α at node i.
While placing ourself in a compressible case, we propose to add to the equation (2) a term of the following
form:
Z
∆t
Gi uh .∇∆ph dV
(3)
Ωt+∆t
2.4 Uniqueness of the pressure solution
In the literature, the finite element P1 / P1 is known for
being unable to model incompressible material flows (α →
+∞) [5, 6]. Indeed, the uniqueness of the pressure solution
in equation (1a) is ensured by verifying:
Z
h
∀v ,
ph ∇ · vh dV = 0 ⇒ [B]T {ph } = {0} ⇒ {ph } = {0}
Ω
Unfortunately, this is not verified by the element P1 / P1
h
because there
R can exist at least one field p , 0 such that
h
h
h
∀v we get Ω p ∇ · v dV = 0 (see [7]). For this reason,
the finite element P1 / P1 is not LBB-stable [8]. One way
to satisfy the LBB condition is to employ, for the velocity,
an interpolation of a higher order than the one used for the
pressure. The well known tetrahedral elements P2 / P1 or
P1+ / P1 can be cited for example [4].
For compressible solid metals, the bulk modulus α
does not tend toward +∞ and has a finite value. From the
numerical point of view, the recent work of Al Akhrass
et al. [9] shows that there is no problem when using the
same interpolation for the kinematics and the pressure in
von Mises elasto-plasticity. Obviously, the value of α must
not be too high in order to guarantee a compressible material flow as mentioned by Duvaut and Lions [10].
Unfortunately, numerical experiments exhibit pressure
oscillations [3] which need to be well-addressed to compute satisfactory solutions. The handling procedure for the
stabilization is described the next section.
3 Stabilized formulation for the pressure
solution
The idea which consists in adding a term of diffusion to
eliminate spurious modes of pressure was initially introduced by Brezzi and Pitkaranta [11] for the Stokes problem in the incompressible case. The
R authors have proposed
to add a term of the form βh2 Ω ∇qh .∇ph dV in the discretized weak form relative to the incompressibility constraint. This approach presents an important difficulty who
lies in the choice of the positive constant β.
We propose to add an anisotropic term of diffusion so
as to improve the conditioning of the mixed problem finite elements in large transformations [8]. By applying
the standard P1/P1 discretization, the pressure increment
∆pi is obtained at each node i by solving the following
equation at each time t + ∆t:
Z
Z
∆pi
Ni dV = 0
(2)
Ni ∇ · ∆xh dV +
αi Ωt+∆t
Ωt+∆t
where Ωt∆t represents the geometrical configuration deformed at time t + ∆t; Ni is the shape fonction of node
where ∆t is the time step; uh is the speed; ∆ph is the pressure increment.
The advantage of this approach is to add a term that
is homogenous with the finite element equations for equilibrium constraints, and then improves the conditioning of
the underlying linear systems.
The nodal function Gi is built following the SUPG
method. Contrary to the standards methods SU or SUPG,
the terms in eq. (2) are not homogeneous with the other
terms of the eq. (3). Moreover,Rthis term is directly added
i
N dV and ensures the
to the compressibility term ∆p
αi
Ωt+∆t i
uniqueness of the pressure independently of the LBB condition. Finally, this approach does not require to solve additional equations as it is the case for the sub-grid scale
methods ASGS and OSGS [12, 13].
4 Example – Circular MAG welding
operation [3]
The welding operation is made using the MAG (Metal Active Gas) process and consist in a single-pass technique
with a welding speed of 660mm/min. The welding stage
starts and stops at the same point corresponding to the
cross-section located at the angle value of 0o . The welded
parts are a tube with a 20mm internal radius and a thickness of 12.5mm and a 8mm thickness holed plate linked
with a T-Joint, and all components, including the filler
metal, are made of ferritic steel of type S 355 (see [14]
for the data).
Figures 1 and 2 show the distributions of the von Mises
residual stress and the residual hydrostatic pressure for
the 2 simulations comparing Q1 and stabilized P1 / P1 elements. A slight difference can be observed especially in
the starting and stopping areas. This could be reduced using finer tetrahedral meshes. The results are in good agreement and show that the element P1 / P1-stabilized can be
very convenient for the multiphysical simulation of manufacturing processes.
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NUMIFORM 2016
Figure 1. Distribution of von Mises equivalent residual stress (MPa) – Q1 and P1/P1-stabilized elements ([3]).
Figure 2. Distribution of residual hydrostatic pressure (MPa) – Q1 and P1/P1-stabilized elements ([3]).
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