APPLICATION OF THE MOMENT METHOD FOR
OPTIMIZATION OF WELDING PROCESSES
V. Melyukov
Vyatka State University, Kirov, Russia
The technology of various kinds of materials treatment by concentrated sources of
energy (mechanical, heat treatment, welding, build-up welding, thermal cutting etc.)
is characterized by existence of the general problem of determination of the mode
ensuring the necessary or the closest to the desired mechanical and operating
properties of the material being processed.
In developing technological processes of materials treatment by concentrated energy
sources the problem of the mode determination is often solved by empirical methods
depending on the experience and knowledge of the technologist with further
experimental check of the selected modes. The modes in this case are chosen from the
reference book, available data, recommendations or are prescribed proceeding from
the accumulated technological experience and then the mode is adjusted on full-scale
specimens.
The mode is also often determined by computational methods developed in the theory
of thermal welding processes (1). These methods allow to determine the mode of
materials treatment by concentrated energy sources on the basis of a computational
experiment on solution of a direct problem of thermal conductivity T  Aq , where A
is the integral operator establishing the correspondence between the input value q
(cause of the thermal process) and the output T (consequence of the thermal process).
A PROBLEM OF THE CONTROL OF WELDING PROCESS
The direct problem of thermal conductivity does not break the cause-and-effect
relation typical of a real thermal process: “source-temperature field”. However,
analysis of approaches and ways of determining the mode show that the problem of
mode determination by its setting is an inverse problem characterized by a breakdown
of the cause-and-effect relation of a real thermal process. In formulation of the
problem of the mode determination it is necessary to restore the cause of the thermal
process – the source by the consequences (required properties of the material or the
prescribed temperature field T ' built with account of these properties). Mathematical
formalization of the mode determination problem as an inverse problem reduces to the
dependence of
q  A 1T
which is incorrect in the real process environment.
Mathematical model of welding process
Let us consider the mathematical model of heat welding process of butt joint for two
thin plates (Fig.1b).
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Fig. 1. Distribution of temperatures in the zone of butt welding of the plates: a actual; b - prescribed
The weld source is moved along but with constant speed  and mathematical model
of heat conductivity is valid only in those regions of the plates which have attained
the so-called quasi-stationary thermal state, which is characterized by the fact that to
an observer moving with the weld source, the distribution of temperature around the
source does not change with time. Let us assume that the temperature is constant
across the thickness of the plate and for above we have two –dimensional heat
conductivity process. The differential equation of heat may thus be written as follows
(2):
T a   2T  2T  q

[1]
 

x v  x 2 y 2  vcv
Hear T ( x, y ) is the temperature of every point in moving system of coordination
x, y attached to the weld source; q ( x, y ) is the function of the volume power desity;
a - the coefficient of thermal diffusivity; сv -the volume specific heat of plates. The
surfaces of the plates are assumed to be heat-insulated, but at the side and ends of the
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plate the heat exchange is considered in accordance with the boundary conditions of
the first kind
T ( x,0)  1 ( x),
T ( x, l )   2 ( x),
T (0, y )   ( y ),
[2]
or of the second kind
 T 
 T 
  q 2 ( x)
[3]
    q1 ( x),
  
 y  y 0
 y  y l
 T 
0


 x  x 
where  - is the coefficient of the thermal conductivity of the plates, l- the summary
width of the plates.
Solution of heat conductivity problem
The solution of the heat conductivity equation [1] under the boundary conditions [2,
3] is obtained with the help of the Fourie finit transformation and inverse
transformation. In the case of boundary conditions of the first kind we use the sine
transformation
l
Tn ( x)   T ( x, y ) sin  n ydy
0
where  n  n / l
The transformation from the image Tn (x) to the original T ( x, y ) is performed with the
help of inverse transformation
2 
T  x, y    Tn ( x) sin  n y .
l n 1
It is easy to verify that the solution of this case can be put in following way (3)
xl
Q( x, y )    q ( , ) K ( x,  , y, )dd
[4]
00
where Q ( x, y ) is the function of the parameters T ( x, y ),  ( y ), 1 ( x),  2 ( x). The
kernel K of the integral equation is determined by the expression
2 
K ( x,  , y, )   n ( x,  , ) sin  n y
l n 1
k
 n ( x,  , )  (ak n ) 1 e 2 n x (e  k2 n  e  k1n ) sin  n
where
[5]
kn  2
v2
  n2
2
4a
k1, 2 n 
v
v2

  n2
2
2a
4a
FORMULATION AND SOLUTION OF THE OPTIMIZATION PROBLEM
In technology of welding production and other kinds of materials treatment by
concentrated energy sources there constantly arise problems related to improved
efficiency of the process, its operation speed and high precision in a whole range of
parameters. Many technological processes of materials treatment by concentrated
energy sources are performed at the modes that are far from being optimal, do not
take full advantage of the energy and functional possibilities of concentrated sources
and do not allow to reach treatment quality indicators that could be reached. An
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especially acute problem of effective determination of concentrated sources and
optimal modes of their effect arises in development of technological processes for
new little-studied materials.
The most successful solution of the problem can be reached on the basis of simulation
of inverse problems for systems with distributed parameters and application of the
optimal control theory. Numerical simulation of optimal systems and development of
computational algorithms allow to find original technical solutions in development of
high technologies and new ways of materials treatment by concentrated energy
sources.
Setting of the problem of the optimum control
The problem of optimal control of a thermal cycle will be defined on an example of
carbon and low-alloy construction steel most widely used in industry and
constructions. This definition of the problem can be extended to other materials
sensitive to overheating in the high-temperature interval and to the speed of cooling in
the interval of low temperatures. For example, these properties are typical of
austenitic steel, zirconium and molybdenum alloys. Long-term exposure to high
temperatures results in considerable growth of the grain and reduced corrosion
resistance of these alloys in the heat-affected zone.
To reduce overheating the thermal cycle must be characterized by a sharp peak of
heating-cooling in the interval of the temperatures ( T АС3 ; Tm ). The value Tm
determines the melting temperature here. To eliminate hardening structures the time
t р of the metal stay in the interval of the temperatures ( T Ar 1 , Tmin ) must be increased.
During the welding process not all points of the metal have the thermal cycle that
goes through the interval ( T АС3 , Tm ) at maximum temperatures. If we consider the
temperature distribution during heating perpendicular to the plates joint (Fig. 1 a).
The weld sources have limited sizes of heat spot and a restriction of power.
Let h be a length of weld heat spot (Fig.1b), accordingly the function q ( x, y ) of
power density must satisfy the condition of finiteness (4)
x  (0, h)
q( x, y ),
q ( x, y )  
x  (,0)  (h, )
 0,
In the case the problem of optimization length of weld pool may be set.
The power density q of weld source must satisfy the condition
[6]
0  q( x, y)  qmax
In building the prescribed temperature distribution T  it is necessary to take account of
the conditions ensuring the necessary properties and quality of the weld joint. These
conditions can be determined on the basis of the dependence of mechanical and
operating properties of the weld joint on the temperature of heating in the welding
process, temporary parameters of the thermal cycle, chemical composition of the
metal being welded etc.
To formulate the problem of determination of welding mode as the inverse problem it
is necessary to built the temperature T ( y ) in the cross-section x  y (Fig.1,b) taking
account of the minimum temperatures and the least time of heating during the welding
process. To reduce metal overheating it is necessary to limit the maximum heating
temperatures. This condition can be taken into account in prescribing the temperature
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value Tm and its distribution over the weld seam width or area of the local heat
treatment from y1 to y 2 .
To account for the establishment of the quasi-stationary thermal state in equation [1],
the time   of heating is non-open parameter and it being found by the following
expression:    h / v. That is why we must formulate the problem of minimization of
the spot length h . The problem of minimization of parameter h may be solved with
the help of the moment method of the optimum control theory for the systems with
distributed parameters (4).
The temperature distribution T  is the prescribed parameter in this problem of
optimum control and accordingly T  ( y ) must be a continuous and smooth function
(5). Figure 1 b shows continuous and smooth distribution of the temperature T ' (y)
with account of the conditions
T ' ( y1  0)  T ' ( y1  0)  0 ,
T ' ( y 2  0)  T ' ( y 2  0)  0 .
[7]
Solution of the optimizing problem
Let us transform the control so that the constraint [6], imposed on the control function
q ( x, y ) appeared to symmetrical to the origin of the coordinates of the space of
controls. With this end we introduce the functions
u( x, y)  q( x, y) / cv  c
Condition [6] becomes
 c  u ( x, y )  c
[8]
Where c  q max / 2cv
The function of weld source with condition [7], [8] is piecewise-constant function.
The equation [4] is more easily handled, if q ( x, y ) be replaced by the following
expression:
q( x, y)  (u( x, y)  c)cv
[9]
Putting the expression [9] in [4] and performing the calculation with x  h we find
the integral equation
h l
Q(h, y)    u ( , ) K (h,  , y, )dd
0 0
According to the moment method, the optimum control u ( x, y ) is determined by the
expression

u( x, y)  csign    n (h, x, y)
n 0
n
The coordinate h and the system of numbers  1, 2 ,... are determined from the solution
of the following problem: to find
h l

1
min     n n (h, x, y) dxdy 

c
n 1
0 0
under the condition

  Q  ( h)  1
n 1
n
n
where
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
Qn (h)  Tn  Fn  c(ak n  n ) 1 (1  (1) n ) k1n1 (e k2 n h  e  kn h )  k 2n1 (e k2 n h  1)
Here T , Fn are the coefficients of Fourier series for T ( y ) and  ( y )
According to the inverse transformation we get from [9] the expression of the
optimum control of the weld source in the heating period

q
q
q( x, y )  max  max sign   n n (h, x, y )
[10]
2
2
n 1
PRACTICAL RESULT
Dollezhal Research and Development Institute of Power Engineering uses the method
of mathematical modeling and optimum control when developing technological
processes of the original designs made of zirconium-niobium alloys for operating
nuclear power plants and research reactors. Welded joints of zirconium alloys work in
corrosive media, under pressure and at high temperatures (7, 8).
Fig. 2. Microstructure of seam’s metal: a – non-optimum mode, b – optimum mode.
The welded joint is the zone of structural heterogeneity and non-uniformity of the
elastoplastic state. The problem of increasing the corrosion resistance and plasticity of
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
welded structures of zirconium alloys is solved by modern scientific and
technological methods using numeral modeling of the optimum regime for electron
beam welding and following local thermo cyclic treatment of the welded joint with
the same electron beam.
The micrographs of structure of welding joint of the zirconium-niobium alloy are
shown in Fig.2. The microstructure of seams metal after electron beam welding by
usual (non-optimum) mode is shown in Fig.2a. This microstructure indicates the
grains of welded metal and the rough marten site needles. Fig. 2b shows the
microstructures of seam’s metal after electron beam welding by optimum mode [10].
The grains of the metal are smaller, the boundaries of marten site needles are
dispersed and  - phase of niobium is picked out.
REFERENCES
1. Rykalin N: Calculation of thermal processes in welding. – Moscow. Machgiz,
1951.
2. Carslaw H., Jaeger.J.: Conduction of heat in solids. Moscow, Nauka, 1964.
3. Melyukov V.: Optimizing the thermal regime of the welding process. Welding
International.1996.
4. Butkovsky A.: Theory of optimal control of systems with distribute parameters.M.: Nauka, 1969.
5. Melyukov V.: Optimizing of the welding mode: Textbook.-Kirov: VyatSu
publishing house, 2006.
6. Zaimovsky A.: Zirconium alloys in nuclear power engineering – М.:
Energoizdat. 1982.
7. Melyukov V.: Method of electron-beam welding, Patent ru, Pat № 225 9264,
Augest, 2005
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