Figure 6-5, chapter VI, shows a composite mesh for a weld on a pressure vessel. These meshes do not always conform; i. e., nodes and element faces on either side of the interface need not match.

Continuity of temperature or displacement fields across the interface can be maintained by constraints. These constraints are imposed automatically by the code. Composite meshes make it much easier to mesh complex structures because each part of the structure can be meshed independently and the mesh is not required to maintain continuity of the mesh. In some cases, this can greatly reduce the work needed to mesh a structure for CWM. However, evaluating the cost of constraints increases the cost of analyses.

Figure 6-6, chapter VI, shows the moving mesh near the weld pool region with the weld. Also filler metal is being added as the weld moves.

Figure 7-3 shows a Tee-joint mesh including filler metal for the three passes.

Figure 7-3: The weld Tee-joint mesh including filler metal for the three passes

Figure 7-4 shows the temperature near the weld pool with three temperature isosurfaces.

Figure 7-4:.The temperature near the weld pool with three temperature isosurfaces

Somewhat similar efficiencies could be achieved by adaptive meshing, which automatically refines and coarsens a finite element mesh [13].

Adaptive meshing has been used in some studies, for Lagrangian meshes, so as to utilize the degrees of freedom in the computational model better by concentrating them to regions where large gradients occur. Siva Prasad and Sankaranarayanan [19] used triangular constant strain elements. McDill et al. [20, 21, 22 and 23] implemented a graded element that alleviates the refming/coarsening of a finite element mesh consisting of quad elements, in two dimensions, or brick elements, in three dimensions. It has been used successfully for three-dimensional simulations [17] where the computer time was reduced by 60% and with retained accuracy, Figure 7-5, for the electron beam welding of a copper canister.

Figure 7-5: Axial stresses (a) with remeshing at 10, 50, 100 and 200 sec. and (b) without remeshing at 50 and 100 sec., from Lindgren et al. [17 and 24].

Runnemalm and Hyun [18], Figure 7-6, combined this with error measures to create an automatic, adaptive mesh. They showed that it is necessary to account both for thermal and mechanical gradients in

the error measure. The mesh created by the latter error measure is not so easy to foresee even for an experienced user. Therefore, the Figure 7-6: Adaptive meshing based on gradient in (a) thermal field and (b) gradients in thermal and mechanical fields, from [18 and 24]

The first three dimensional residual stress predictions of full welds appear to be by Lindgren and Karlsson [28], who used shell elements when modeling a thin-walled pipe. Karlsson and Josefson [27] modeled the same pipe using solid elements.

с Figure 7-7: Finite element meshes used a) combined solid and shell element model b) solid element model c) shell element model, form [15].

The combination of shell and solid elements in the same model is of special interest in welding simulations. A thin-walled structure can be modeled with shell elements, but a more detailed resolution of the region near the weld will require solid elements. This was done by Gu and Goldak [14] for a thermal simulation of a weld and by Naesstroem et al. [15] for a thermomechanical model of a weld, Figure 7-7.

Shell elements can be successfully used in finite element calculations of thin walled structures. However, in the weld and the heat affected zone (HAZ) shell elements may not be sufficient, since the through thickness stress gradient is high in these regions. Naesstrom et al. [15] presented a combination of eight-nodes solid elements near the weld and four-nodes shell elements elsewhere. This combination of solid elements and shell elements reduces the number of degrees of freedom in the problem in comparison with the use of solid elements only. The residual stress results show that the combined model can be successfully used in structures where high stress and temperature gradients are localized to a narrow region i. e. a fine mesh in the heat affected zone and a coarse mesh of shell elements elsewhere.

McDill et al [16] developed a promising element formulation where a three-dimensional element with eight nodes and only displacements as model unknowns can be a solid or a shell element. It was demonstrated on a small-weld case in their paper. The element is based on the same element as in Lindgren et al [17] and Runnemalm and Hyun [18], making it possible to perform adaptive meshing with a combination of solid and shell elements. It will also be possible to determine adaptively whether an element should be treated as a solid or a shell.

Lindgren L. E.’s [24] recommendations say “the adaptive meshing and parallel computations are currently necessary to solve three­dimensional problems with the same accuracy as in existing two­dimensional models. The Eulerian approach is effective but less general”.

Composite meshes are similar to adaptive meshes in that both are based on constraints, i. e., some nodes are declared to be linearly dependent on other nodes. Composite meshes differ from adaptive meshes in that fine elements need not be children of coarse elements, i. e., fine elements need not be formed by refining coarse elements.

Figure 7-4 shows an example of the weld pool being meshed with element boundaries on the liquid/metal interface. This weld pool mesh is parameterized so that weld pool shape can be defined dynamically during the analysis. We call this a parametric conforming weld pool mesh.

The weld pool dimensions for a double ellipsoid weld pool were:

- Front ellipsoid width, depth and length in meters are (0.006, 0.000914 and 0.006).

- Rear ellipsoid width, depth and length are (0.006, 0.000914 and 0.012).

In Figure 7-4 note that except very close to the weld pool, the temperature varies linearly through the thickness of the wall. This implies that almost everywhere only 2 nodes through the wall are needed to accurately capture the temperature variation. The five nodes that we used were not necessary and a significant reduction in computing time could have been achieved by using elements that were linear through the thickness of the wall and quadratic or cubic on the wall surfaces.

Figure 7-8: The temperature distribution in a composite mesh with fewer mesh parts

Figure 7-8 shows the temperature distribution in a composite mesh with fewer mesh parts. In this case there is no mesh part WP1. Thus this mesh makes no attempt to capture the weld pool geometry by placing element boundaries on the weld pool boundary. We call this a nonconforming weld pool mesh. In this case, elements near the weld pool are less deformed and there are fewer constraints between non-conforming meshes to evaluate.

Figure 7-9 shows displacement vectors near weld pool. Note that displacement vectors behind the weld pool are pulled backwards towards the start of the weld while displacement vectors ahead of the weld pool are pushed forward by the thermal wave advancing with the weld pool. It is the hysteresis of this cycle that leads to residual stress and distortion in welds.

Figure 7-9: Displacement vectors are shown near the weld pool Discussion of progress in CWM

The structure is represented by a FEM mesh that ignores the details of the welded joints. The FEM mesh of the structure can be a separate mesh for each part or it can be single FEM mesh for the complete structure. If an automatic meshing capability is available, then it is convenient to define the geometry of the structure or parts by CAD files such as stereolithographic files.

In addition to the welded structure, a sequence of welds and weld passes for each weld must be specified. Such a sequence is a list of weld joints and for each weld joint a list of weld passes. Both lists are ordered in time. With each weld pass in each weld joint, a start time is specified.

Analyzing multipass welds as a series of single-pass welds is certainly the most rigorous, albeit costly, process. Multi pass welds have been analyzed by Ueda [33], Rybicki [30] and Leung [34]. To reduce the cost of separate analyses for each pass, several passes have often been lumped together in different ways, Figure 7-10. In some, only the last pass in a specific layer is analyzed [33]. In others, it would appear that the volume of the weld deposit of several passes or layers of passes are lumped together and the thermal history of a single pass located in the middle of the deposit
is imposed during the stress analysis [30]. Another technique, lumps the thermal histories from several passes together, the temperature at any point, at any instant in time being the greatest value from any pass. All passes in a single layer, except the last, are lumped together. The last pass in any layer is treated separately [34]. Lumping layers together is inadvisable with the possible exception that the lumped layers remain a small proportion of the total thickness. For extremely large numbers of passes even these techniques may not be adequate.

15 mm

R=48.59 mm

Figure 7-10: Finite element model used by Rybicki and Stonesifer [30], where seven weld passes were lumped into four passes, adopted from [24]

Passes 6 & 7 Passes 4 & 5 Passes 2 & 3 Root pass

A somewhat more speculative approach was used by the Goldak et al [35] to analyze the resurfacing of a thin plate (hydroelectric turbine blade) that involved several hundred passes, figure 7-11. Each pass was 10 cm long; groups of 20 passes were done sequentially to cover 10 cm x 10 cm rectangular surface patches. The orientation of each patch, and the patch sequence were varied in order to minimize the deformation. The residual stress pattern of the multipass case was created by superimposing the residual stress pattern of each individual weld and taking the largest value from any individual pass. By building up the stress pattern for each patch and sequentially applying this pattern to the plate it was possible to

8.56 mm

obtain good qualitative agreement in both the deformed shape and optimum patch sequence. The danger in all these lumping techniques comes from the accuracy with which they model the sequencing effect.

Figure 7-11: Deformed shapes of multipass, resurfaced, hydroelectric turbine blades; superimposed residual stress patterns. A: longitudinal patch sequence, B: transverse path sequence. The distortion is magnified 100 times.

Chakravarti and Goldak et al [35 and 36] developed a finite element model to predict the vertical distortions due to block welds in welding overlays on a flat plate. The finite element predictions are compared to experimental measurements. The FEM analysis uses a three stage model to do the transient thermal analysis, transient thermo-elasto-plastic, three dimensional analysis and elasto-plastic analysis of a plate containing six blocks using total strain from the blocks as initial strain in the plate. The results show reasonable agreement between the finite element predictions and experimental measurements. This gives confidence in the finite element model, and its potential to simulate more complex geometries where simple analytical expressions may be inadequate.

Lindgren’s review [32] recommends that simplifying the multipass welding procedure by some kind of lumping technique must be exercised with care. All lumping and envelope techniques change the temperature history and will affect the transient and residual strains near the weld. Lumping by merging several weld passes that conserve the total heat input is preferred. Thus, the simulation will correspond to a multipass weld but with fewer weld passes than the original.

Figure 7-12 shows a structure to be welded as another example with several weld joints.

Figure 7-12: The transient temperature field is shown on one joint of the saddle in the upper picture. The mesh for this weld joint including filler metal is shown in the lower picture. For analyses this mesh is refined.

The FEM mesh for the structure is usually too coarse to be used for the analysis of a weld. In any case, it does not capture the geometry of the weld joint and the filler metal added in each weld pass. Therefore, a new FEM mesh is made for each weld joint. To do this, the specification of the weld procedure includes a 2D FEM mesh. This 2D mesh is parameterized by thickness of plates or other parameters associated with the weld joint. The parameterization of this 2D mesh is such that it can be made to conform to the geometry of a cross-section of each weld joint. This 2D mesh also includes the

metal added for each pass in the weld joint as a separate part type for each weld pass.

For each weld joint this new weld joint mesh is composed with a subset of the FEM mesh of the structure to make the FEM mesh for the structure that will be the domain to be used for the CWM analysis of each weld pass in that joint. As each weld pass is made, the FEM elements in the filler metal added in the pass are added to the domain. This process is shown in Figure 7-13.

Figure 7-13: The filler metal added for the first and third pass of the Tee-joint weld joint are shown. The insets show the transient temperature field.

The next weld starts with the FEM mesh of the domain of the previous weld and adds the mesh of the next weld joint to a subset of the previous domain. Hence as the structure is fabricated, the mesh changes dynamically during each weld pass to capture the geometry and state of the structure including the geometry and state of welds that have been completed.

Figures 7-12 and 7-14 show transient thermal analyses of welds.

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Figure 7-14: The 1100°K temperature isotherm of the weld in Figure 7-12

We learned two important lessons from our experience with parametric weld pool meshes. First and perhaps most important, the distorted elements in conforming weld pool meshes appear to have far larger discretization errors than the nonconforming weld pool mesh. This should not have been surprising because the effect of element distortion on accuracy has been studied and is reasonably well understood [11 and 12]. The second important lesson that we learned was the need to balance the overhead of working with composite meshes with the cost of solving. Although the cost of solving the nonconforming mesh is slightly higher, the cost of overhead is less and the total cost can be lower.

In pursuing our ultimate objective of CWM analyses of welded structures that have many, multipass welds; we have developed a software environment that separates the structural design, weld joint design and production welding stages. Structural design specifies the parts to be welded; in particular their geometry, their material types and any relevant internal variables and boundary conditions needed to constrain rigid body motion. Weld joint design specifies a curvilinear coordinate system for each weld joint. Each weld pass on each weld joint has a start point, end point and a start time. Production welding specifies the welding procedure for each weld
joint. The welding procedure defines the welding parameters for each pass, e. g., current, voltage, speed, weld pool size, shape and position in a cross-section of the weld joint. These three data sets fully define the process of welding a structure and contain the data needed to perform a computational welding mechanics analysis for fabrication of a complete structure.