Boundary Conditions

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

Integrand functionin mQx(,)t-1

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

0 25 50 75 100 125 150 175 200 0

250 500 750 1000 1250 1500

x-coordinate in mm

0 10 20 30 40 50 60

Temperature Harmonics Image Sources

Number of image sources / harmonics

Temperature in °C

time = 10 s

Fig. 5.3 Temperature profile during welding along the weld centre line and number of image sources needed to maintain the adiabatic boundary condition

0 25 50 75 100 125 150 175 200

200 225 250 275 300 325 350

Temperaturein°C

x-coordinate in mm

0 2 4 6 8 10 12 14 16 18

Numberofimagesources/harmonics

Temperature Harmonics Image Sources

time = 50 s

Fig. 5.4 Temperature profile during cooling down along the weld centre line and number of image sources needed to maintain the adiabatic boundary condition

This circumstance refers to the result that has been shown in Fig. 4.27. If an energy distri-bution has to be approximated a significantly high number of harmonics is needed. Back to the fact emphasised in Fig. 5.3 it can clearly be seen that the number of harmonics de-creases, if the distance to the heat source increases. This contrary behaviour with regards to the image source method is also shown in Fig. 5.4 for the cooling down phase. Here the image source method requires about 14 reflections to achieve a converged temperature field while in case of the FOURIER’S method a constant number of only 3 harmonics is suffi-cient. The reason for that characteristic can be explained by the temperature profile in thickness direction. In direct vicinity to the heat source a large gradient between top and bottom surface occurs. Thus, less image sources but many harmonics are needed to pro-vide the correct temperature field. This situation differs, if the distance to the heat source is increased. Consequently, the temperature gradient along the thickness decreases and more image sources but fewer harmonics are needed. This fact occurs even more distinct during the cooling down mode. The advantage of the FOURIER’S series expansion is that the computational time remains constant during the cooling down while the number of image sources increases continuously.

In general, it can be stated that there is a critical instance of time when the image source method should be applied and when it has to be replaced by the FOURIER’S method. This issue is focussed on in Fig. 5.5. Here a thermal cycle was calculated for the point

P(110,100, 0) on basis of the reference modelling conditions. Again, it can be seen that the number of harmonics increases abruptly, if the heat sources reaches the point under study.

0 10 20 30 40 50 60 70

0 250 500 750 1000 1250 1500

Time in s

0 10 20 30 40 50 60

0.0 3.8 7.6 11.4 15.2 19.0 22.8

Fourier number

Number of image sources / harmonics

Temperature in °C

Temperature Harmonics Image Sources

Fig. 5.5 Thermal cycle recorded at a point on the weld centre line and comparison of method of images and Fourier’s series expansion

After that it decreases continuously. The corresponding contrary behaviour is obtained for the method of image sources. To summarise it can be said that for high FOURIER numbers, i.e. for this test case greater than 4.0, FOURIER’S method can be applied and for small F OU-RIER numbers, here smaller than 4.0, the method of image sources is better suited.

The investigations regarding the performance of the algorithms to consider adiabatic boundaries of the specimen mainly focus on the consideration of the thickness of the plate-like geometry. As discussed previously, the incorporation of side faces requires the same approach but this would increase the computational costs significantly. Because of that the concept of an activation domain was introduced. This means that the computational domain has to be extended by a virtual domain where all the virtually reflected and dummy sources are located. That is the main difference to finite discretisation schemes where the system of equations is only solved for nodes which are located within or at the boundary of the speci-men. The issue also governs the concerns regarding the computational costs of functional-analytical and numerical discretisation approaches. As sketched in Fig. 4.33 the more sub-paths the welding trajectory contains the more dummy sources have to be introduced which are reflected at all bounding surfaces. Therefore, the computational costs are dependent on the complexity of the welding trajectory in the case of functional-analytical methods. This is obviously not the case, if numerical methods are taken.

The concept of an activation domain allows reducing the computational cost significantly.

This method is especially suitable for the reflections with respect to the side faces of the specimen since the dimension in width and length direction is higher than in thickness direc-tion. Because of that fact many reflections or harmonics are needed in thickness direction while mostly only one reflection has to be used for the side faces which can then be even neglected by application of the activation range. Of course, these assumptions can not be made for the cooling down phase since the heat diffusion of heat sources which do not affect the temperature field in the computational domain during welding can now influence that domain during the cooling down phase. Thus, a further requirement of the concept of activation range is that if the real heat source stops acting all reflected heat sources are activated.

To summarize the concept of a virtual computation domain is a consequence of the funda-mental functional-analytical solutions which are defined for the infinite solid. The bounding faces can be incorporated by usage of the image or FOURIER’S method.The series are trun-cated, if a temperature convergence criterion of 1 x 10^-5 is reached. Nevertheless, at least one reflection is needed in order to evaluate this convergence criterion. While in the case of the series expansion in thickness direction this fact has no important meaning since many reflections/harmonics have to be applied this situation changes for the side faces as heat impermeable boundaries. Here, even a single reflection would increase the computational costs significantly. To overcome this problem the concept of an activation range is intro-duced to neglect those heat sources having no influence on the temperature field in the computational domain yet.