Civil Engineering Reference
In-Depth Information
In this report we present application of the geometric-constraints method to the modeling and analysis
of welds which typically result from a range of different types of welding processes, including gas metal
arc (GMA) and shielded metal arc (SMA) or gas tungsten arc (GTA), consisting of either single or multiple
passes. We present a series of prototype case studies of welds typical of both bead-on-plate and bead-in-
groove welding of stainless steel. An interesting feature of our approach is that it represents an extension
of the method of distributed heat sources.
This method has been shown to be well-posed for providing
an approximate representation of GMA welding processes where the combined influence of arc energy
distribution and filler-metal additions can result in relatively complex shapes of the transverse cross section
of the weldment. In particular, these complex shapes include the formation of weld crater/weld finger
shapes, or in general, complex non-semi elliptical shapes, which are observed frequently in GMA weld-
ments. The geometric-constraints method presented in this report represents a relatively more quantitative
approach to simulating the type of weldments which typically result from single and multipass GMA
welding processes than methods where the solution is driven primarily by a source term in the energy
equation. We consider a series of cases, giving examples of analysis where different types of information
concerning a given weld are assumed to be known. We attempt to examine the practical issue of what is
to be considered sufficient information for assuring a reasonable level of accuracy for the solution and its
effective uniqueness when
3
information about a given weld is either incomplete or fragmented.
Our presentation of the mathematical foundation of the geometric constraints method, which is
in terms of constrained optimization theory,
a priori
and of the associated mathematical properties of
pseudo-steady-state temperature fields, is relatively complete. It is not, however, rigorous. Our pre-
sentation is for the most part in terms of general illustrations of the underlying mathematical concepts
rather than a rigorous mathematical development. In particular, our discussion of the mathematical
properties of the steady state temperature field are in terms of either continuous or discrete field
representations, depending upon which type of representation provides a more convenient description
of a given property.
4, 5, 6
6.2
Physical Model of Welding Process
The model system to be specified is that of the temperature field of a dynamic weld in the pseudo steady-
state. That is to say, the model system is characterized by quasi-steady energy transport in a coordinate
system that is fixed in the reference frame of a moving energy source. The outermost boundaries of the
model system are defined by the sides of a finite-sized rectangular region. Although the model system is
formally that of a general time dependent system, only the pseudo steady-state solution is of relevance with
respect to geometric-constraint information. In our development, the time step and volumetric discretiza-
tion of the system serve as a means of constructing a weighted sum for the purpose of optimizing the
discrete temperature field corresponding to a given set of geometric constraints. The input energy source
is effected via specification of an upstream boundary condition on the model system. The equations
governing the model system are
()
T
()
T
C
P
T
------
[
kT
()
T
]
qT
()
q
s
x
()
(6.1)
t
where the quantities
(
T
),
C
(
T
), and
k
(
T
) are the temperature dependent density, heat capacity, and
p
conductivity, respectively, and
t
qT
()
d
H
(6.2)
t
where the quantity
H
is the change in energy per unit volume due to the temperature dependent source
term
q
(
T
). The source term
q
(
x
) is associated with the input of energy into the system resulting
s
