Determination of Boundary Values for the Numerical Simulation of

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					Determination of Boundary Values for the Numerical Simulation of Metal
                Forming Processes using Fuzzy Control

                                    Bernd Leisten*, Ralf Luce, Reiner Kopp,
                              Institut für Bildsame Formgebung, RWTH Aachen
                                        Intzestraße 10, D-52056 Aachen
                               Phone: +49-241-805930, Fax: +49-241-8888234
                                email: (leisten,luce,kopp)@ibf.rwth-aachen.de


ABSTRACT: Models of workpieces and tools containing numerous physical parameters (material - and boundary-
values) which are valid for the process are required for the numerical simulation of forming processes. This article
presents the possible use of fuzzy logic to determine boundary values, where the friction-coefficient and the heat
transfer-coefficient receive special attention. Basic dependencies between process parameters and physical sizes are
used. The various process parameters (e.g. temperature of workpiece) are linked to boundary sizes using fuzzy rules.
Fuzzy control methods are used to describe these dependencies since the input and output variables have to be
translated from numbers to fuzzy terms. A comparison with existing measurements shows the usability of fuzzy-
control-methods to specify boundary values. The described system will be part of larger program package for
determining material data and boundary values combining fuzzy-based knowledge processing with formulae and with
facts from a data base.



KEYWORDS: metal forming, FEM-simulation, material data, boundary values, friction, heat transfer, fuzzy-control,
control equipment


INTRODUCTION

Metal forming calls for a number of skills from the user. The plasticity theory is employed to describe the forming
processes, leading to mathematical models of these processes. Thanks to the progress made in computer technology,
the numerical simulation of forming processes is becoming increasingly important in metal forming. The Finite
Element Method (FEM) has proven particularly suitable for the simulation of forming processes and has established
itself as an aid for process analysis and optimisation. Workpiece and tool models have to be made for the simulation
which contain not only the geometry but also numerous physical parameters (material and boundary sizes) valid for
the process. The geometry of the workpiece is represented by finite elements (often squares) (Figure 1).
The actual simulation itself is based on the solution of a system of partial differential equations. The velocities at the
individual nodes in the network are approximated in discrete time steps. The nodes are shifted in the direction of the
velocity after each calculation, producing a sequence of networks for which the target parameters, for example
temperature distribution, then have to be calculated. These calculations are primarily based on the material and
boundary values. The temperature field is thus calculated according to the Fourier thermal conductivity equation using
the heat conductivity, density and specific thermal capacity (l,r,cp). Mechanisms such as friction and heat transfer
between the workpiece and tool are responsible for the generation of heat. Since the numerical approximation of the
node shift itself contains an error propagation, the correct specification of the material and boundary values is
important so as to keep the errors in the calculation as low as possible.
Those material and boundary sizes important for the simulation (see Figure 2) can initially be split into two groups,
the material-specific and the process-dependent parameters. The thermal material values are the heat conductivity l,
the density r and the specific thermal capacity cp. These are independent of the forming process under consideration
and depend on only the material and the temperature. An important mechanical material parameter is the yield stress
kf, which affects the state of stress required for plastic deformation. Apart from the material (analysis and structure) it
also depends on the temperature, true strain and the strain rate. Parameters which depend primarily on the respective
                       Figure 1: Representation of a rolling process for FEM simulation

process are called (process) boundary parameters. The most important of these are the friction coefficient m, the heat
transfer coefficient a and the radiation emission coefficient e. These essentially depend on specific process parameters
such as strain rate, temperature, surface finish and the effect of lubricants. These parameters can only be derived from
the process under consideration. The influence of the boundary parameters on the process target parameters
determined by simulation can be explained by a so-called sensitivity analysis, where a process is calculated using
different values for the boundary parameters. Philipp (1993) shows that a and m can greatly affect the temperature
distribution, required power and geometry of a formed workpiece. Thus, a required power in the workpiece of between
530 kN and 780 kN was calculated by varying a between 0.08 W/mm2 *K and 0.2 W/mm2 *K in a die forging
process.




                  Figure 2: Effect of material and boundary parameters during metal forming

The material and boundary parameters for the process under consideration must be specified when making a model for
FEM simulation. The material values of a variety of metals have already been determined and can be used
accordingly. Nevertheless, new alloys are constantly being developed whose parameters are unknown and have to be
measured. The radiation emission coefficient, heat transfer coefficient and friction coefficient have to be determined
separately for each forming process. Simulation experts can fall back on a wealth of information and are constantly
gathering new experiences. There are no fixed methods of determining these parameters. This is why they are used for
knowledge-based calculations.
METHODS OF DETERMINING BOUNDARY VALUES

Whereas numerous measurements are available for the thermal material parameters and yield stress, experience is
often needed to determine the boundary values for a simulation. Simulation experts refer to existing processes with
similar materials. Table 1 shows some common values for heat transfer coefficients in rolling processes. Important
factors such as the lubricant layer between tool and workpiece cannot be taken into account in these experimental
values. In the case of friction coefficients, values have been assumed for almost friction-less processes between 10-2 -
10-3 and values for static friction (workpiece and tool stick together during forming), between 0.5 and 1 (theoretically
higher too). If there is a lubricant film between the workpiece and tool and if the surface oxidises in hot forming
processes, this results in mixed-film friction which is frequently assumed to be 0.3. Of course, these experimental
values are only very rough. More exact values are needed if processes react sensitively to changes in boundary values.
Alloy                   Low-alloy steels         High-alloy steels        Aluminium               Nickel basis
a [W/(mm2*K)]           0.013                    0.013                    0.015                   0.02

                                 Table 1: Heat transfer coefficients of rolling processes

Storage in databases is ideal for computer-assisted simulations. Databases already exist for numerous material
parameters [INSITU (1998)]. However, it is difficult to find an universal scheme since various dependencies play a
role in this case. As soon as a parameter is no longer simply material and temperature-dependent, the pre-treatment
stages also have to be taken into account. These are predominantly pre-forming and heat treatment. This is why one
will find almost no flow curves in electronic data archives despite numerous measurement results [Doege (1986)].
Since kf is a decisive parameter for the process it is usually measured anew for each application case. The measuring
technology has been perfected to a large extent [Bernrath (1998)].
On the other hand, there are only very few and complicated methods to measure boundary parameters. One way of
determining m and a is the Multi-Level concept developed at the IBF [Kopp (1992)], which employs inverse
modelling. Experiments are hereby compared to FEM variation calculations until a target parameter is identical in the
test and in the simulation. One condition is that the target parameter is the only unknown in the problem to be solved.
This explains the step-by-step approach to determining boundary values.
The heat transfer coefficient is determined with the aid of a cylindrical test piece which is slowly cooled between two
upsetting plates (Figure 3). The temperature in the core of the test piece is hereby measured as a comparative value.
This test is then simulated by FEM whilst varying a until the measured and simulated core temperature curves are
identical. The material values and radiation are required for this. A conical, bent hollow cylinder is upset to determine
m (Figure 3). The comparative value for the subsequent simulation, where m is now varied, is the geometry, i.e. the
outer and inner radius of the conical tube.
These tests provide results for processes with the same temperature curve, the same lubricant, the same surface finish,
etc. They are thus difficult to transfer to new processes. Moreover, they are also time-consuming and costly. The
variety of dependencies between boundary parameters and their importance for the accuracy of a simulation finally led
to the development of a set of rules which should reflect these relationships. The rules which are used here relate to
exactly measurable process parameters (e.g. the workpiece temperature) as well as fuzzy data (e.g. the quality of the
lubrication). However, exact figures are needed as results, which is why fuzzy-control-methods were used.
Fuzzification allows the equal combination of numbers and fuzzy terms. These can be linked by rules, present as
expert knowledge, and converted back into numbers through defuzzification. The fuzzy-based set of rules described in
the following demonstrates the applicability of Fuzzy technology to determine boundary values. However, it is still on
the way to becoming a universal tool since the relationships it has to show are very complex.
                                         Figure 3: Test set-up to determine a (left) and m (right)



FUZZY BASED MODELLING TO DETERMINE BOUNDARY VALUES

The control equipment has been developed for the material and boundary parameters described above. Special
attention was initially paid to the heat transfer coefficient and the friction coefficient, so that the following
explanations are limited to these two parameters. The value and a number, which specifies the importance of the
parameter with respect to a special process, are determined for every size. Figure 4 describes the functionality for
determining values, where process parameters and knowledge in the form of rules produce a result using fuzzy-
control-methods.




                                         Figure 4:Program System as Blackbox
DEPENDENCIES OF MATERIAL DATA UND BOUNDARY VALUES ON PROCESS PARAMETERS

Heat-transfer and friction take place in the boundary layer between workpiece and tool. This layer is generated by the
use of lubricants, the high temperature oxidation (scale) and the roughness of the workpiece. The temperature and
pressure of the tool on the workpiece are also important. Friction is also influenced by the relative velocity between the
two participating partners. A rough set of dependencies ,which are briefly summarized in the following, was initially
developed using numerous existing publications [Heußen (1997)] Refinements can be made in any required depth so
that only the main factors will be considered at first. Indirect influences of process parameters are considered only in
cases of direct necessity.
The friction coefficient increases with a deterioration in the lubricant. Higher friction also occurs with a rough surface
of the workpiece. Temperature has an important indirect influence on friction since high temperatures lead to a
liquefaction of the lubricant and to more scale. In general the friction first increases with a rising temperature and then
decreases. Increasing pressure of the workpiece on the tool leads to a higher friction because of the larger contact area.
The importance of friction decreases with a growing friction coefficient since the frictional heat is lower and the strain
condition in the workpiece stays nearly the same with an increasing adhesion between workpiece and tool. The
importance increases with a large contact area because more friction occurs. If the workpiece is much hotter than the
tool the friction coefficient becomes less important because the frictional heat dissipates very quickly and has less
influence on the process. The effect of frictional heat and thus the importance of the friction coefficient also decreases
if the heat can dissipate slowly.
The heat transfer coefficient increases proportionate to the pressure between tool and workpiece. It also depends on the
thermal conduction of the lubricant and on the thickness of the lubrication layer. If the surface of the workpiece is very
rough, the heat transfer becomes worse. a also increases with an increasing temperature. The importance of the heat
transfer coefficient mainly depends on the difference in temperature between workpiece and tool which means more or
less heat flow. The importance of a also increases with a larger contact area, increasing pressure and increasing strain
rate.
The dependencies presented here affect the target parameters in different ways. The resulting rules thus have to be
weighted differently.


MODELLING OF LINGUISTIC VARIABLES

The process parameters as input variables and the boundary values respective their importance as output values are
represented by linguistic variables in order to express the rules with help of fuzzy logic. Three terms are normally used
for every (linguistic) variable meaning “low”, “middle” and “high”. In the case of lubrication for example they can
also mean “good”, “middle” and “bad”. The base variable is modelled in the range [0;10] ÎÂ . The borders are only
ambiguous letters for the real minima and maxima of the variables. The rules can thus be applied for any forming
process. Rolling processes for example run much faster than die forging processes. The value of the input variable
strain rate can be high for die forging and low for rolling. The influence of strain rate on the boundary values is equal
in both processes. The ambiguous letters are replaced by concrete values during the application. These values can be
entered interactively by the user and parallel to this the data for some processes are saved in a database.
The basic modelling of the linguistic variables is shown in Figure 5. The terms are modelled as triangles. This
specification can in principle be adapted, but this was only used once. This simple expression reduces the complexity
and is based on the rough description of the dependencies for the first construction of the control equipment. Two
boundary conditions determine the modelling:
· A variable cannot be low and high at the same time
· A variable must be low at the beginning of a scope and high at its end
                                                                                                          1
                                                                                        mniedrig( x) = - x + 1, falls x Î[0.0;5.0]
                                                                                                          5
                                                                                                       2
                                                                                        mmittel ( x) = x - 1, falls x Î[2.5;5.0]
                                                                                                       5
                                                                                                         2
                                                                                        mmittel ( x) = - x + 3, falls x Î[5.0;7.5]
                                                                                                         5
                                                                                                      1
                                                                                        mhoch( x) = x - 1, falls x Î[5.0;10.0]
                                                                                                      5




                                          Figure 5: Modelling linguistic variables



USE OF A META-LANGUAGE

The dependencies described above are expressed in a meta-language, which is translated into fuzzy rule blocks by a
special procedure. The dependencies can be described as if-then-rules. The process parameters are used as anticidents
connected with “AND”, the consequents are the boundary values and their importance. In this way the following
simple meta-language is constructed.
IF I <motion> [(AND I <motion>)*]
THEN O <motion>
motion:           increases | decreases
Example:
IF lubrication <decreases> THEN m <increases>
IF contact area <increases> THEN importance(m) <increases>
Only simple dependencies lead to an extension of this language with limiting factors. The friction coefficient for
example decreases with less lubrication, but the influence of lubrication is lower at high workpiece temperatures since
the lubricant becomes liquid. The new keyword “WHERE” is used for the limitation.
WHERE I<limit(a, b, c)>
a, b, c Î [-1.0;1.0]
The input variable I has the influence “a”, if it is low, the influence “b”, if it is middle and the influence “c”, if it is
high. A positive influence strengthens the rule, a negative influence weakens it. The influence is taken into account
when translating meta-rules into fuzzy-rule-blocks.
The variables are constructed by three terms. A fuzzy-rule-block with 3n possible combinations of input variables has
to be constructed using n input and one output variable(s) to express a meta-rule. The following example explains the
translation:
IF lubrication <increases> WHERE temperature <limit(0.0,0.0,0.2)> THEN friction coefficient <decreases>
We thus have two input variables (temperature and lubrication) and one output variable (friction coefficient) which
leads to nine possible rules. Every term is now assigned a value. For output variables this is always: value(low) = 1,
value(middle) = 2, value(high) = 3. If the motion of an input variable is inverse to that of the output variable, as in the
example, the values are defined as: : value(low) = 3, value(middle) = 2, value(high) = 1. The value for the output term
is now calculated for every possible combination of input variables. The values of the input variables that are not
limiting factors are added together and divided by the number of input variables. The corresponding limiting factor is
also added. The resulting values for the example are shown in Table 2.
         Temperature                         low                          middle                           high
lubrication
low                                          3.0                            3.0                            3.2
middle                                       2.0                            2.0                            2.2
high                                         1.0                            1.0                            1.2

                       Table 2: Example of the term values for the output variable friction coefficient

The resulting term of a rule is defined by the derived value. If the value is uneven and lies between zero and three, two
rules have to be developed. These rules are given a Degree of Support (DoS) derived from the decimal place. The 11
rules (7 direct + 2*2 splitted) for the example shown in Table 3 explain this derivation.
                               IF                                                                         THEN
         lubrication                     temperature                        DoS                     friction coeficient
              low                            low                            1.0                            high
              low                           middle                          1.0                            high
              low                            high                           1.0                            high
          middle                             low                            1.0                           middle
          middle                            middle                          1.0                           middle
          middle                             high                           0.8                           middle
          middle                             high                           0.2                            high
              high                           low                            1.0                            low
              high                          middle                          1.0                            low
              high                           high                           0.8                            low
              high                           high                           0.2                           middle

                                          Table 3: Rules in the example rule-block



USE OF FUZZY-CONTROL METHODS

A set of fuzzy-rule-blocks is developed with a set of meta-rules for each output variable. The fuzzy implementation for
each output is carried out with the tool FuzzyTech® from the firm of Inform®. If the input values and process specific
minima and maxima are known they can be converted into the standard range of values [0;10] and fuzzyfication takes
place. The input variables are connected with a logical AND. This corresponds to an aggregation with the minimum
operator. The Mamdani operator [Mamdani (1976)] is used for the implication so that the minimum input
membership is copied to the output term. This simple strategy is sufficient for the existing rules. It could be changed
during the development of the system. The result of every rule-block is derived by the bounded-sum-method. Different
results for one output-term derive from different input variables because of the given development of fuzzy-rules. This
justifies the addition of term-memberships that have to be cut at the value one.
The results of every single rule-block now have to be weighted. This is the same as weighting a meta-rule since every
rule-block is developed from a meta-rule. One example of weighting is the effect of the lubrication. Its influence on the
friction coefficient increases enormously with the “goodness” of lubrication.. A perfect lubrication leads to the
disappearance of friction almost independent of the other factors. The result for the example rule-block must be
weighted much higher than the other rule-blocks. Every rule-block has its own weighting function depending on the
input variables. In many cases the weighting has a constant value of one, which means that there is no special
weighting for the rule. A result for one output variable is derived by using the arithmetic mean of the weighted term-
results. This finally leads to a fuzzy-result for the three terms of the output variable. This is defuzzified using the
Centre of Maximum (CoM) method. The derived value has to be converted into the correct range for the output
variable. The values for the importance are given as a number between one and ten. The fuzzy-result could be used
directly, but the scale is more exact and no result-term is lost. Figure 6 describes the structure to determine the friction
coefficient. This produces one module for every boundary value or importance which can be integrated into other
program systems.




                       Figure 6: Structure of a module to determine the friction coefficient



RESULTS

The results of the control equipment are compared to secured data for boundary values. It was mentioned above that
such values are quite rare (this fact initiated the control equipment) since empirical determination is very costly. The
comparisons of the friction coefficient refer to the test shown in Figure 3. The rules for the heat transfer coefficient
were compared with values from a research project. The determined values are steady in respect to locality and time
throughout the process. This simplification is carried out in the FEM simulation and leads to a difference between
control equipment and test. The control equipment uses the data at the beginning of the process. Parameters such as
temperature and pressure differ in reality for locality and time. The values determined in the tests have to be seen as
average values for the whole process. If this difference is very large, the input variables for the control equipment
should be average values.
The heat transfer coefficient has been thoroughly analysed within a scientifical project for axial-radial-forming
[Heußen (1996)]. This test is carried out by upsetting a tube in a radial direction and then forming the resulting bulge
with four hammers in an axial direction. A value of 0.015 W/mm2*K for a was determined in the test. The control
equipment produced nearly the same value, exactly 0.014 W/mm2*K. The pressure test for a (Figure 3) could not be
reproduced with the control equipment. The results of the test were different to data found in literature. Further
comparisons produced good results, for example for the simulation of a compression test with a cylindrical test piece.
The upsetting test with a conical hollow cylinder (Figure 3) to determine the friction coefficient was performed using
the high alloy steel X5CrNi 18 8 with a hot test piece and a cold tool (20°C) [Philipp (1993)]. The friction coefficients
determined were over 0.2 in tests without lubrication. The values were very low with lubricated test pieces. For the
lubricant-free tests the values matched very well. The values for the control equipment remained within the confidence
interval of the test. The effect of the lubricant did not decrease, though the temperature was very high. This special
case of a very good lubricant was not integrated into the control equipment. Here the effect of the lubricant depends, as
described above, on the temperature. A different limitation for different lubricants can be used to correct this mistake.
However, the inputs and the benefits generally have to be carefully weighed against one another. The results are
shown in Table 4.

         Workpiece        Lubrication      Velocity of Test    Pressure (as            m                 m
        temperature                            [mm/s]            Stress)              Test            Control
           [°C]                                                 [N/mm2]            (Figure 3)        equipment


            900               bad                10               72.24              0.34               0.38


            900               bad                100              72.24              0.55               0.62


            900               bad                100              72.24              0.01               0.22


           1000               good               10               62.76              0.44               0.44


           1000               good               100              62.76               0.9               0.76


           1000               good               100              62.76              0.04               0.24


                           Table 4: Results of the control equipment compared to test results



PROSEPCTS FOR A KNOWLEDGE-BASED DETERMINATION OF MATERIAL DATA AND
BOUNDARY VALUES

The results generally came up to expectations. A fuzzy-based set of rules is able to represent dependencies to
determine boundary values for the simulation of metal forming processes. The system can be further adapted to real
conditions using many possible influences. For example, new meta-rules have to be adapted to new information. The
modelling of linguistic variables can be refined by adding further terms. The fuzzy-control-methods can also be
changed. What is important is an integration into an overall concept which contains a database with facts about
different processes. This allows the standardized use of minima and maxima for the input variables. The control
equipment is integrated into the pre- and postprocessor PEP [Franzke (1997)] for FEM simulation. Minima and
maxima can be entered in a special mask within PEP. The results of the control equipment are directly adapted to the
FEM model. The FEM user needs a compact system to use material and boundary data. A software system called
“material and boundary data knowledge” (in German: StoRa) is currently being developed which integrates the fuzzy-
based determination of boundary values described here. It will be connected to a process- and material database to
allow the combination of facts and rules. The knowledge-based determination of boundary values expands the FEM
simulation of metal forming processes and makes this more accurate. The use of fuzzy-control-methods plays an
important role in this respect.



REFERENCES

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Wien, Austria
Franzke ,Martin, 1997, „PEP-Programmers’s Environment for Pre/Postprocessing“, 15thCAD-FEM User’s Meeting,
Fulda, Germany
Heußen ,Michael, 1996, "Determination of thermal and tribologic boundary conditions" (german), Final Report of
DFG project Schn 463/46
Heußen, Michael, 1997, „Examination of material behaviour of metals in range of the solidus-temperature“ (german),
pp. 193-215, Umformtechnische Schriften 73, Shaker Verlag, Aachen, Germany
INSITU ,1998, „steel“ (Database),INSITU,Aachen,Germany
Kopp ,Reiner; Philipp, Franz-Dieter, 1992, „Physical parameters and boundary conditions for the numerical
simulation of hot forming processes“, steel research 63, pp.392-398
Mamdani , 1976, „Advances in the linguistic synthesis of fuzzy controllers“, Int.J.ofMan-Machine-Studies7, pp.1-13
Philipp , Franz-Dieter, 1993, „Physical Process Data for Numerical Simulation of hot metal-forming processes“
(german), VerlagStahleisen, Düsseldorf, Germany