Domain of Action and Energy Distribution

5 Discussion of Results

5.1 Extension of Analytical Heat Conduction Models and Evaluation

In this section, the results presented in chapter 4 are discussed. The focus is to emphasise the applicability of analytical temperature field models in connection with a heuristic global optimisation algorithm to solve the inverse heat conduction problem in welding simulation.

x

-coordinate in mm

-5.0 -2.5 0.0 2.5 5.0

100 150 200 250 300

Normal distribution Exponential distribution Parabolic distribution

Unitenergyinm-1

Fig. 5.1 Comparison of different energy distributions for a constant bounding of 2.5 mm

The sensitivity of the energy distribution was investigated as shown in Fig. 4.13 - Fig. 4.17, page 76 - 78. It has already been mentioned that the normal as well as the exponential distribution are governed by a single distribution parameter x_e. In the case of heat source models presented here x_e is the distance to the heat source centre where the value of energy input has fallen to 1 / e of its maximum value. In contrast to that Goldak [93] de-fines a normal distribution by the distance to the heat source centre, where the energy input has decreased to 5 % of the maximum value in the centre.

For the normal as well as the exponential distribution the resulting sensitivity is straightfor-ward. The greater the distribution parameter x_e the smaller is the heat intensity that acts on the solid. This is also evident, if a heat source boundary is taken into account as given in this test case. It can be seen in Fig. 4.13 and Fig. 4.14 that the intensity decreases, if x_e increases and that it becomes more and more uniform. If x_e is set to 25 mm the resulting energy distribution is almost rectangular.

The behaviour of the new parabolic heat source type that has been introduced is different to the previously discussed energy distributions. As mentioned, it is determined by three coef-ficients which define the parabolic distribution. The sensitivity is not as straightforward as it can be seen in Fig. 4.15 - Fig. 4.17. In case of the coefficient a₀ the energy distribution is more sensitive due to a decrease of the coefficient than to an increase. For both cases the magnitude of the unit energy near the heat source boundary at x^¢ = -6mm as well as x^ = 6mm remains unaltered. This is different for the symmetry line at x = 0 where the values for the unit energy change significantly. They increase, if a₀ decreases. This behav-iour holds only up to a value of x » 2mm and is then inverted. Similar to the boundary the magnitude of the unit energy remains also unaltered at x » 2mm . The behaviour for the

remaining parameters a₁ and a₂ is principally the same and characterised by a region where the values remain unaltered that is located at x » 2.5mm .

In general, it can be stated that the newly introduced parabolic heat source allows the con-struction of various parabolic energy distributions. In the current test case a symmetric bounding has been employed. It is reasonable, if the energy distribution in transversal or longitudinal direction to the source movement is of interest. The same holds, if a distribution in thickness direction is considered. The heat source is located at the top surface and is symmetrically distributed in positive and negative z-direction whereas the bounding is governed by the thickness of the specimen.

Nevertheless, the separate adjustment of the coefficients of the parabola may be solvable while a manual simultaneously adjustment is not a trivial task. Hence, an optimisation algo-rithm should be applied. Another fact that has to be taken into account is that a negative energy input has to be avoided. In contrast to the normal or exponential energy distribution this can occur. A specific criterion can be formulated on basis of the discriminant of the parabolic equation. In particular, the discriminant has to be smaller than zero in order to avoid any root of the homogenous parabolic distribution.

Another interesting fact regarding the parabolic energy distribution is that it includes the linear energy distribution as special case if the following substitutions are made

e

a

a k

a

0 1 2

1

0

=

=

-=

(5.1)

with the parameter k_e defined as

e

e

k e 1

ex

= - ^(5.2)

which refers to the solution that has been published by Karkhin [90]. Furthermore, the linear energy distribution allows deriving the widely used double conical heat source. For this purpose the energy distribution in x and h-direction is assumed to be normal. In thickness direction the parameter k_e( )z_e defines a linear distribution.

In addition to the influence of the bounding or the energy distribution the general transient behaviour of the integrand function has been examined. In this context it has to be men-tioned that the energy distribution f( )x is derived from the integrand function Q( , )x t , if the time is set to zero. With regards to the transient behaviour of the integrand function the main conclusion that can be made is that the higher the time of heat diffusion the more the original shape of energy distribution or integrand function, respectively, is damped. This fact is exemplarily illustrated in Fig. 5.2. Here, for the three investigated energy distributions of Fig. 4.18 - Fig. 4.20 the integrand function changes to a normal distribution even if the F OU-RIER number is considerably small. Again, it has to be noted that independently on the dis-tribution and time the integral within the limits of the boundary of the energy disdis-tribution

remains unit. Nevertheless, it can be realised that the effect of heat source bounding is only visible for the integrand function, if the time is set to zero. For greater instances of time the action of bounding diffuses. Especially, the effect that the initial condition of energy distribu-tion diffuses even after short instances of time has also discussed and applied by Rykalin [1]. He argued that the modelling of the temperature field due to a normal distributed source is similar as to consider the temperature due to the action of a point source under assump-tion of a small time shift (time of heat diffusion).

-30 -20 -10 0 10 20 30

0 20 40 60 80 100 120

Normal distribution Exponential distribution Parabolic distribution

x

-coordinate in mm

Integrand functionin mQx(,)t-1

Fig. 5.2 Comparison of integrand function of different energy distributions for a Fourier number of 0.0237

Im Dokument A Contribution to the Solution of the Inverse Heat Conduction Problem in Welding Simulation (Seite 135-138)