Technology Database  Metallurgy THE METHOLOGY OF PARAMETRIC OPTIMIZASHION OF DETAILS AND DESIGNS OF COMPLEX CONFIGURATIONS
THE
METHOLOGY OF PARAMETRIC OPTIMIZASHION OF DETAILS AND DESIGNS OF COMPLEX
CONFIGURATIONS
©
B.N.Poliakov
Prof., Technical
sciences Doctor
Contact to the
author:
bpoliakov@hotmail.com
Summary
The opportunity is substantiated and the algorithm of stage-by-stage
consecutive parametric optimization of carrier details and designs of the
complex configurations, based on correct application of strict mathematical
methods is offered, serviceability and efficiency which is proved of many years
practice of designing of the equipment of heavy mechanical engineering
_________________________________________________________________________
Keywords: parametric optimization,
carrier details, designs, method
of finite elements, stress-strain state, thermoelastic
states
At designing heavily loaded carrier, foundation and case
details or designs of the complex configurations, used in the equipment of
rolling shops and subject to influence of technological and thermal loadings,
inevitably there is a problem of a choice of the best, optimum from a position
of this or that criterion of quality, parameters of details and regimes of
cooling.
The statements and decisions of problems of optimum designing
for real details enough complex, and frequently unique forms possess a series of
features. The mathematical model making an information basis of statement anyone
optimization of a problem, because of variety of designs of details should be
enough universal, with opportunities of a wide variation boundary conditions,
variables designing, area of change of managing parameters and restrictions as
equalities or inequalities. To these qualities statement of the boundary-value
problem, described by system of the differential equations in private
derivatives with corresponding boundary and initial conditions to the greatest
degree satisfies, one of which versions is the model of method of finite
elements ( MFE). Therefore the main difficulty of optimization of details and
designs of complex configurations with a position of such widespread criterions
of quality as the minimal (set) metal consumption, the maximal (set) rigidity or
the minimal (set) intensity and so on, consists in high dimension and bulkiness
of the mathematical models, describing their strained, deformed or thermoelastic
states (TES), which can be in most cases submitted in the matrix form, as
software of calculations on the computer of the stress-strain state ( SSS) and
TES. One more feature of the decision of such problems: exactness and
reliability of the decision depend on a level information process of designing,
i.e. completeness and accuracy of knowledge of conditions of load capacity,
character of communications and degrees of freedom of a detail, thermophysics
parameters, service properties of a material and so on, and this knowledge are
rather approached and are limited. Therefore from the engineering point of view
achievement mathematical a strict and exact optimum has no practical sense more
often, as boundary and initial conditions of a boundary-value problem are never
known with the big accuracy and always are available various, even mathematical
not formalization, technological restrictions, “promoting” reception of only
rational decision. Besides cost of calculation of one variant of a design,
especially three-dimensional SSS or TES, can be very high, and, hence, the big
practical importance complexity, so gets, and speed of convergence of algorithm
of optimization. However nevertheless it is necessary to notice, that the
opportunity of reception of the correct optimum decision is extremely important,
as, first, this natural and primeval aspiration of the person to perfection and
true and, secondly, at all not realizing it in a design, allows the engineer to
define directions of reduction of weight of a detail at designing or at
preservation of metal consumption to lower load capacity and to raise its
uniformity so, to increase durability.
The analysis of the requirements showed to optimization of
typical carrier details and constructions of the rolling equipment, allows to
formulate the following kinds of mathematical statement of problems of
optimization.
Let the elastic body, occupying area 
in with border 
is considered. As criterion of an optimality 
shall choose a maximum of equivalent stress 
in researched area
Y( X ) =
σequ(X,Z),
where Z - the vector of coordinates;
X - the vector of varied parameters of designing (the
constructive sizes of elements),
= VUS.
Problem 1. It is required to find a vector X*,
which informs a minimum to criterion of an optimality
Y(X*) = 
σequ (X,Z) ( 1
)
at restrictions such as equalities
 (z,u,q)
= 0,
Z 
V; ( 2 )
v(z,u,q) = 0,
Z
S ( 3 )
and inequalities
σequ(X*,
Z)[ σ ],  ZV;
( 4 )
/ui(X*,Z)/ [u], ZV,
i = 1,2,3; ( 5 )
X*
X , ( 6 )
where
- the differential operator,
determining a regional problem of the theory of elasticity;
v - the operator, assigning boundary
conditions;
u - a vector of displacements;
q - a vector of loadings;
X - the permissible set of varied parameters, determined by
geometrical, constructive and technological conditions;
[ σ
]- the condition of strength, dependent on a material of a design, a kind of a
strained state, character of a loading of the accepted settlement scheme and
other factors;
[ u ] - the top border of restrictions of the displacements,
dependent on permissible values of the clearances between the details, required
rigidity, etc.
However for some details, for example such as the housing of
sheet hot or cold rolling mill, the determining parameter is rigidity. Therefore
as goal function it is necessary accept a level of displacements of ui. Then a
problem of optimization can to be put as follows.
Problem 2. To find a vector X*, which informs a minimum to
criterion of an optimality
Y ( X*) = 
/ui (X, Z )/
at presence of restrictions such as equality ( 2 ), ( 3 ) and
inequalities ( 4 ), ( 5 ), and also i accepts value 1, either 2 or 3.
Except for problems 1 and 2, for series of details it is
necessary to allocate also a problem of minimization of mass, which can be
formulated as follows:
Problem 3. To find vector X*, which informs a minimum to
criterion of an optimality
Y ( X*) =
Ω (X) =
where γ - density of a material, at presence of
restrictions ( 2 ) - ( 4 ).
The given mathematical statements demonstrate the approach to
problems of optimization by a principle of “a black box “, when the
condition of object is described by two groups of parameters: entrance (independent)
variables of designing (varied parameters of a design is a vector X) and exit (dependent)
parameters of quality of functioning of object of Y. Such the approach is simple
and convenient for the majority of real monolithic and welded designs, when
optimization is preceded, as a rule, with alternative calculations and the
careful analysis of the SSS or TES. Application of this principle for
parametrical optimization of designs, in which calculation of the SSS is not so
expensive also quantity of parameters of designing is limited , and it is
required to model set of variants of restrictions and conditions loading (for
example, for search of critical conditions of destruction) is especially
practical.
The choice of an effective method of the decision of a
problem of optimization depends on features of change of the SSS of a detail at
a variation of its parameters. By optimization of the form for models of
behaviour of designs based on MFE, there is big freedom of a choice of the
varied parameters, determining statement and efficiency of a problem. But even
at discrete nodular representation of flat model MFE of a stress is necessary to
define in several hundreds points. Usually at the analysis of the SSS are
limited to consideration of the points, lying on a contour of a detail, but also
in this case their number reaches several tens. It troubles the analysis of
results of calculation. Therefore is expedient to build criterion function for
the characteristic points, taking into account features of the SSS or TES all
detail, for example, for points of a maximum of intensity or a maximum of
equivalent stresses.
Thus, for the majority of problems of optimization at
designing details of a complex configuration the following features are
characteristic:
1. As a rule, communication between criterion of an
optimality Y and a vector X has not obvious character, and is carried out
through system of the differential equations of the second order.
2. The mathematical model of optimized object is
expressed not in an obvious analytical kind, and in the form of the operator,
realized as software of calculations of the SSS or TES on a computer.
3. The big dimension of space of optimized
parameters.
4. Significant expenses of machine time for calculation of
one value of criterion function.
5. The mixed composition of restrictions on
characteristics of object, i.e. at mathematical statement there are both
equality and inequalities.
At the decision of such complex problems of optimization
strict mathematical methods arise significant difficulties. However to their
decision is possible to apply a combination of known methods: planning of
computing experiments and the decision of problems of mathematical programming.
The method of planning of experiments allows to execute approximation of the
numerical decision : to find obvious dependence for criterion function and
functions of restrictions, to estimate substantiality of influence of factors on
criterion of optimization in various ranges of change of managing parameters and
to reduce quantity of calculation variants. Then the problem of optimization is
easily to reduce to a problem of linear or nonlinear programming, applying for
the decision, for example, a simplex - method of the Nelder - Mid or a method of
the quickest descent.
The decision of a problem of parametrical optimization at
designing details of complex configurations from a position of this or that
criterion of quality is offered to be carried out stage by stage in the
following sequence (see Fig.):
1. To develop or accept mathematical model of an optimized
detail, which is represented as software of calculations of the SSS or TES on
the computer. Further this model to consider as “ a black box “. On its
entry varied parameters move, and on an output parameters of the SSS or TES
object of designing are removed.
2. With use of the accepted model after a series of trial
calculations to reveal variables of designing (design parameters ) most
essentially influencing on a pressure and a deformations. If in result the
configuration of the detail meeting the set requirements (for example, to a
level of pressure, metal consumption, etc.) is determined, calculations can be
stopped.
3. If the variant of a design does not meet the set
requirements , that on the revealed variables of designing, most essentially
influencing on exit parameters of a detail, to plan computing experiment.
4. To execute a series of calculations under the plan of full
factorial experiment, thus the maximal number of variants equally 2n,
where n-quantity of parameters varied at two levels; it is possible to apply to
decrease of variability and fractional factorial experiment.
5. In an environs of an initial point (a projected detail
corresponding to the prototype) on the basis of statistical methods : the
software of correlation and plural regression the analysis of a matrix of
planning to execute local approximation of characteristics of system (functions
of the response) the linear or nonlinear equations of regress, representing to
dependence of the maximal values of stresses and deformations in characteristic
points of a detail from its design parameters and their pair interactions.
6. To execute analysis the received obvious dependence (the
equations of regress) for goal functions of restrictions with revealing most
statistically significant design parameters. If the received information has not
enough for decision-making and designing of a rational design to proceed to the
following stage.
7. Using strict methods of the decision of problems of
mathematical programming, at presence of system of restrictions on variables of
designing and a level of stresses and deformations in a series of points, and
also accepting goal function (a level of stresses, metal consumption, etc.) to
receive a rational (optimum) variant of a design, that is carried out
effectively enough at small expenses of computer times.
For realization of the submitted algorithm ( see Fig. )
optimization of designs had been developed the corresponding software.
Fig. The block - scheme of algorithm of the stage-by-stage
decision of a problem of multi variant designing of optimum details of a complex
configuration
The formed algorithm of the stage-by-stage, consecutive
decision of a problem of parametric optimization of complex designs possesses a
number of the advantages confirmed in practice [1, 2]:
1) association and close interaction of programs of
optimization with the software for researches of the SSS or TES designs, that
provides fast convergence and economically of algorithm;
2) the account of a significant share of experience,
intuition and common sense of the qualified design engineer and his active
participation in decision-making process;
3) Formed on a stage No 5 equations of regress, approximating
communication of the SSS in characteristic points with variables of designing,
can be used directly in calculation practice at designing for the analysis of
the SSS or TES and a choice of rational parameters without attraction of a
computer, and also by development CAD carrier and thermo-loaded details,
construction of their parametric series and for unification; besides, the
equations of regress allow to estimate sensitivity of criterion of an optimality
to parameters (to define gradients at movement to an optimum) and to choose the
best design decision;
4) An opportunity of addition with the known software,
realizing in real time a dialogue regime of correction of all components of
boundary conditions, including and configurations of a detail, that to a
considerable degree approaches process of optimization to the organization of
the closed system of the automated synthesis of optimum constructions.
Performance capability and efficiency of the created
algorithm of consecutive parametric optimization and its the software are proved
convincingly on numerous examples of improvement of designs of housings of
blooming mills, of continuous - billet and sheet hot and cold rolling mills, of
universal joints and rods of various designs, of typical carrier details and of
designs mining, drilling and other equipment of heavy mechanical engineering [1,
2].
For further information contact Dr.B.Poliakov
e-mail bpoliakov@hotmail.com
REFERENCES
1. B.N.Poliakov and et.al.
The load capacity, carrier
capacity and durability of the rolling equipment.-M.:Metallurgy, 1990.- 320pp. (
The Library of the Congress USA ).
2. B.N.Poliakov. Increase quality of the technologies and
longevity of the equipment the rolling mills. Part 1.Ekaterinburg: Publishers
Sverdlovsk engineering- pedagogical institute, 1993. - 208pp.( The Library of
the Congress USA ).
Source: SciTecLibrary.ru
Publishing date: November 9, 2011
To make an inquiry on the technology you are interested in
Back
| 
