Numerical Solutions

One of the major advantages of numerical solutions for the representation of the heat source in welding is that complex geometries and temperature-dependent material characteristics may be taken into account in principle without restrictions, [12]. Furthermore, there is no need for simplifications with respect to travel speed of the moving heat source, i.e. time dependence. This makes the heat source truly moving across the structure in a preferably arbitrary path. In practice, CPU and memory expenses limit the size and complexity of the numerical model.

Perfectly, a fully coupled temperature and fluid flow model should be implemented for the heat source and weld pool in the analysis. As mentioned earlier this calls for full solution of Navier-Stokes equations together with the energy equation. Strictly speaking, this also involves solution of the electromagnetic fields as well. This, however, is extremely time consuming and for the purpose of the present work this kind of analysis is not relevant.

By excluding the fluid flow analysis in the melted weld pool, in reality a liquid, molten material is treated with the theory and modelling techniques of solid mechanics. The characteristics of the melt are hence approximated through appropriate modelling of the material properties as discussed in the previous chapter. On this basis, the moving heat source, the formation of and temperature distribution in the weld pool can be modelled in various ways. One approach is to prescribe the temperature distribution in the weld pool, another to prescribe the heat flux input on the surface or directly in the weld metal. Together with considerations on filler material, these models are treated below.

The disadvantage of the numerical approaches compared to analytical solutions is that no general statement is obtained with the numerical models. They are dedicated for the specific application though the principle can be generally used for similar welded structures. In any case, the temperature fields calculated should be checked by means of temperature measurements, [12]. To what extent this is necessary depends on the purpose of the analysis. At any rate, the questions of the efficiency and distribution of the heat input, the heat losses by convection and radiation, material properties at elevated temperatures etc. call for uncertainties of the analysis result.

5.3.1 Prescribed Temperatures

The approach of prescribing the temperature distribution in the weld pool requires knowledge of the actual temperature as well as the shape and position of the weld pool. This can be either as an estimate or from examinations that can be accomplished in different ways, e.g. thermo-camera during welding, geometrical

determination from micro-samples etc. If the position, i.e. shape of weld pool, and the temperature at the interface between liquid and solid is known and held constant, only the solid part of the structure needs attention. Considerations on temperatures in the weld pool can be ignored. This known field can then be moved across the solid with the appropriate travel speed.

This requires preferably some adaptive meshing technique in order to resolve the weld pool geometry in detail. In e.g. Prasad et al. [65,66], the effect of external constraints and sequence of welding on residual stresses in weldments are studied using adaptive meshes generated based on the temperature calculation for the estimation of residual stresses. This work is carried out with a two-dimensional thermo-elastic-plastic model and uses a Gaussian distributed prescribed heat flux method as described in the following, but the adaptive meshing technique shows an example of how this can be accomplished. Today, several numerical modelling software packages (both finite element, finite volume etc.) offer adaptive meshing techniques, and transient updating of the mesh is used with advantage in many materials processing applications, e.g. that of mechanical cutting, Kalhori [67].

FIGURE 5.3 The local/global approach applied to a hat shape structure, Mourgue et al. [25]. Left: The global 3D shell model with zoom inserted. Right: The local model meshed with 3D solid elements.

The shape of weld pool can also be modelled by predefining an adequately fine mesh in the total length of the weld. It is desirable to have at least four linear elements in each direction of the weld pool, [63]. This is by far the most common applied method. Another approach is that of predefining different meshes across the weld, one that is fine and one that is rather coarse and maybe some in between.

These meshes then replaces each other section-wise as the heat source moves over the structure requiring a certain dense of the mesh in order to capture the temperature field as a result of the varying gradients. Like for the adaptive meshing technique some kind of mapping procedure for the fields between meshes is needed.

CH A P T E R 5 . HE A T SO U R C E MO D E L L I N G

The method is adopted in e.g. [68,69]. A related method referred to as a local/global approach is adopted in [25,26]. This procedure involves a combination of 3D shell elements for the coarse global mesh and truly 3D solids locally around the weld, Figure 5.3.

5.3.2 Prescribed Heat Flux Input on Surface

For the welding of thin plates or other processes with limited penetration, a Gaussian source for the heat flux on the surface of the metal narrowed down to the size of the weld pool is usually a good solution as suggested by Pavelic et al. [70].

This model implies knowledge of the heat source characteristics power from the welding torch as well as the efficiency of the specific process as mentioned earlier.

Also in the case of prescribed heat flux input on the surface, some kind of adaptive meshing technique can be used with advantage. The same condition for the modelling of mesh as for the prescribed temperature models holds in general for all descriptions of the heat source independent of the method applied.

FIGURE 5.4 Physical phenomena driving the heat and fluid flow in the weld pool, Pavlyk and Dilthey [71].

With the heat flux prescribed at the surface, the weld pool shape will approximately be spherical since all the heat in the model is transported by diffusion, [63]. This is not the case for real welds where most heat is transported by advection. The difference is that with diffusion, the heat flows in all directions whereas advection carries the heat downstream only. The physical phenomena driving the weld pool formation and the directions of their actions on the heat and mass transport are schematically shown in Figure 5.4. Large surface flows exist due to the spatial variation of surface tension. This can be either an outward or an inward flow

depending on surface-active elements such as sulphur, Zacharia et al. [15].

Furthermore, the buoyancy force and the electromagnetic body force contribute to the convection in the weld pool together with the arc pressure and volume expansion causing the surface to deform. All these effects together with the complex heat exchange mechanisms are included in the comprehensive thermo-fluid model presented by Pavlyk and Dilthey [71]. The impact and kinetic energy of droplets when they touch the pool surface and the forces exerted on them when they enter into the weld has been analysed by Sun et al. [72]. This was used to determine the volume in which the droplet-heat-content is distributed thereby improving the heat input to the weld compared to the traditional Gaussian distribution. As results of all these physical phenomena acting on the weld pool, the convective flow in the common welding configuration as usual leads to more narrow and deeper weld pools than the spherical approximation obtained with a pure diffusion model.

The kind of models including the effects in the weld pool as indicated in Figure 5.4 are very computational demanding and are in most cases two-dimensional assumptions concentrated on stationary spot gas tungsten arc welding, [15]. In the recent years, more three-dimensional models are seen and in few cases, the transient weld pool is presented as results of a moving arc, e.g. Ohring and Lugt [73].

Applications of deep-penetration laser welding are also seen involving some of the effects affecting the keyhole formation, e.g. Berger et al. [74].

5.3.3 Prescribed Volumetric Heat Flux Input

In the case of deep penetration fusion welds, a semi-ovaloid (double ellipsoid) model can be used, Goldak et al. [75]. Also in this model, the power from the welding torch as well as the efficiency of the process must be estimated a priori.

As this is one of the most respected and hence used heat source models, it deserves notice. It is more convenient than the prescribed temperature method where detailed knowledge of the weld pool geometry is required and in most cases it gives a better description of the heat source distribution than the surface flux models. If the global effects are of interest, the double ellipsoid model can give a good estimate of the weld pool without any experimental measurements. This requires some experience, though, with both the numerical model and the actual welding process under investigation.

Figure 5.5 shows a schematic representation of the semi-ovaloid model. In front and rear of the weld, two ellipsoids define the heat source distribution independently.

Expressions can be found in e.g. [14]. A good estimate is letting the dimension of the ovaloid be approximately 10% smaller than the dimension of the weld pool [12].

Nevertheless, this requires the experimental knowledge from welding the actual weld.

CH A P T E R 5 . HE A T SO U R C E MO D E L L I N G

FIGURE 5.5 Moving semi-ovaloid or double ellipsoid heat source model.

Normal distribution of volumetric heat source density in front and rear of weld, respectively. After Radaj [12], presented by Goldak et al. [75].

5.3.4 Addition of Filler Material

None of the above-mentioned methods of modelling the heat flux discusses the effect of the weld filler, which in relation to the subsequent mechanical calculation naturally plays a role when welding thick plates with single-V-groove welds.

At least two principal methods exist which both require the mesh in the whole structure to be predefined. One way is to change the material properties of the parts not yet active in the model to reflect air-like conditions corresponding to thermal insulation (i.e. adiabatic thermal boundary condition to the active parts or infinite thermal resistance between active and inactive regions) and mechanical very low stiffness. When material is activated, the material properties are changed to match that of the filler metal. This approach often referred to as “quiet elements/cells/control volumes etc.” is rather straightforward to implement in the numerical code but suffers from often quite ill conditioned matrix systems due to the flexibility of the inactive parts of the geometry.

Another way to add filler material is the kill/rebirth method (removing/adding elements). In this method, the inactive parts are defined in the initial configuration with the right properties for the filler metal. They are then removed from the equation system and then added as required. This makes the removed parts of the geometry truly inactive and hence there is no ill conditioning in the matrix system from this.