Comparison with Finite Element Model

The introduced functional-analytical framework is compared with finite element models that obey temperature independent thermo-physical material properties. This allows evaluating the influence of the spatial and temporal discretisation of finite element models. On the other hand, the implementation of the movement of distributed heat sources, e.g. double ellipsoidal ones, on curved trajectories on basis of functional-analytical methods can be validated because no analytical references are available in literature.

As explained in chapter 3.2.2.5 (page 50) the comparison involves the movement on a trajectory with sub-paths that have an arbitrary orientation with respect to the global

coordi-nate system. The analytically calculated temperature field agrees well with the finite ele-ment counterpart with respect to the temperature profile along a reference path as shown in Fig. 5.6 during welding and Fig. 5.7 during the cooling down phase. It is worth noticing that the suggested functional-analytical framework is capable of modelling the transient tem-perature field for volumetric heat sources that move on arbitrary shaped welding trajectories under consideration of plate of finite dimension. This is an extension to the proposed mod-els of Cao [123] and Winczek [124] since they only consider an infinite plate and concen-trated or symmetrically distributed heat sources. Furthermore, the physically necessary bounding of the domain of action of the heat source has been taken into account. In addi-tion it can be demonstrated, that analytical soluaddi-tions can account for double ellipsoidal energy distributions which are in good agreement to the numerical equivalent which was discussed by Goldak [93] but also has been realised by Fachinotti [120].

0 100 200 300 400 500 600 700 800 900

0 5 10 15 20 25 30 35 40 45 50

x-coordinate in mm Analytic Linear FEA

Temperature in °C

Fig. 5.6 Temperature profile along the measuring path at time t = 16 s

0 100 200 300 400 500 600 700 800 900

Analytic Linear FEA

0 5 10 15 20 25 30 35 40 45 50

x-coordinate in mm

Temperature in °C

Fig. 5.7 Temperature profile along the measuring path at time t = 21 s

Supplementary to the extracted temperature profiles the transient temperatures of both models are compared by calculated thermal cycles. Again, as shown in Fig. 5.8 the analyti-cal model and the finite element model produce thermal cycles that are in best agreement.

0 5 10 15 20 25 30 35 40 45 50

0 200 400 600 800 1000 1200

Time in s

Analytic Linear FEA

Temperature in °C

A

B

C

F E

D

Fig. 5.8 Comparison of thermal cycles that are calculated analytically and with a linear finite ele-ment model

Besides the agreement between analytical and linear finite element models the question is why to apply analytical methods. The answer is at hand by help of Table 5.1. Here the computational time for the calculation of the 3D transient temperature field, the temperature profile as well as the thermal cycles is listed for both the finite element as well as the func-tional analytical model. As mentioned in chapter 3.2.2.5 an adaptive meshing was applied in order to ensure a converged FEA solution. Beginning with 3468 nodes that corresponds to an element edge length of 1.5 mm near the heat source, the mesh was locally refined down to an edge length of 0.75 mm that generates approximately 19725 nodes for the entire FEA model. The calculations were performed on an ordinary office computer.

It can be noted that with respect to the three dimensional transient temperature field the difference in calculation time between the finite element and analytical model is in the same range. The situation differs in case of the calculation of temperature profiles or thermal cycles. Here, the analytical approach is between one and two orders of magnitude faster than the numerical counterpart which enhances the inverse problem solution concerning the evaluation of the objective as defined by equation (3.6).

Table 5.1 Comparison of computational time between finite element and analytical approach

Linear finite element

model

Functional-analytical model

Factor Time for

calcula-tion of 3D tempera-ture field

≈ 1620 s 1303 s 1.24

Time for calcula-tion of temperature

profile

≈ 1620 s 3.5 s 462

Time for calcula-tion of thermal

cycles

≈ 1620 s 60 s 27

As introduced in chapter 4.2.2.3 (page 85), the consideration of the heat impermeable boundary requires the introduction of various virtual heat sources acting in a virtual calcula-tion domain. Furthermore, the decomposicalcula-tion of the welding trajectory into sub-paths yields further dummy heat sources for each sub-path. These facts clarify the main property of analytical models in comparison to their numerical counterparts namely that the computa-tional cost is dependent on the complexity of the welding trajectory. However, as outlined previously the solution of the inverse heat conduction problem requires several direct simu-lations. The objective function is defined with respect to experimental reference data as the fusion line in the cross section or further weld characteristics like the thermal cycle. That means that only the required simulation data in dependence on the experimental available reference data has to be calculated. In case of the thermal cycle the problem to be solved is quite clear since only the temperature at a specific point and instance of time has to be calculated. Here the main advantage of the functional-analytical approach should be em-phasised. This is because the temperature at an arbitrary point within the computational

domain and specific time can be calculated independently on neighbouring points/locations and previous instances of time. In case of a numerical method, as finite element analysis, the situation is different since the influence of neighbouring points/nodes is defined by the composition of the global stiffness matrix of the system under consideration and the ele-ment connectivity [189]. In other words, the complete system of equations needs to be solved for every node and every time step in order to get information of the dependent variable at a specific location. If this location is not coinciding with a node, or more precisely GAUSS -point of the discretised system, then the shape function of the element has to be used to evaluate the (interpolated) value of temperature at a specific point. It can be easily seen that the evaluation of temperature is orders of magnitudes faster, if a functional-analytical method is used than a numerical approach. However, the increase in speed re-sults in a reduction of model complexity like the neglect of the temperature dependence of the material properties. The presentation of the algorithm in Fig. 4.43 showed that only a few evaluations of temperatures at specific points are needed to reconstruct the fusion line in the cross section. Of course, the same algorithm could be applied for a numerical model to extract the geometry of the fusion line out of the calculated 3D temperature field. In this case, for every search point the corresponding temperature values have to be interpolated from the nearest GAUSS-points by means of the element shape functions. This requires having the 3D temperature field at a certain instance of time available which, as already discussed, includes the solution of the system of equations for all nodes and previous time steps. Again, the advantage of functional-analytical methods is clarified by fulfilling the same task orders of magnitudes faster.