Is real-time Computational Welding Mechanics feasible? For the question to have meaning, it is necessary to first specify what we mean by a solution and what errors in the solution are tolerable. We define a solution to be the transient temperature, displacement, strain and stress and microstructure evolution evaluated at each point in space and time in a weld. In the authors’ opinion, numerical errors in stress and strain in the range (5% and 10%) are acceptable.

By numerical errors we mean the difference between the numerical solution and an exact solution to the mathematical problem being solved. By real errors we mean the difference between the numerical solution and experimentally measured values in which there are no experimental errors. It is difficult to reduce real errors below 1% because of errors in available data for material properties such as thermal conductivity and Young’s modulus. Numerical errors in the solution that are much larger than 10% often would not be accepted. The numerical error estimates do not include errors in material properties or in microstructure evolution. We assume that errors in geometry will be held to less than the maximum of 1% or 0.5 mm except for some details.

The uncertainties of the computerized simulation of the cross­sectional geometric parameters of welds are investigated by Sudnik et al [25], using laser beam welds as an example. It has been discussed how to estimate the different errors in the simulations, that are modelling, parametrical (e. g., uncertainty in material and process data) and numerical errors.

The error propagation rule according to Gauss together with error sensitivity coefficients is used as the basis. The uncertainties of simulation are formally dealt with in the same manner as it is usual with the uncertainties of testing results. The simulation error is considered as being composed of modeling errors, parametrical errors and numerical errors. Simulation error and testing error together result in the verification error or prediction confidence. The example comprises a C02 laser beam welds in steel simulated by the computer program DB-LASIM resulting in a modeling error of about 10% and a prediction error of about 13% (standard deviations).

The effect of arbitrarily chosen extreme variations of the material properties on the cross-sectional simulation results is additionally visualized by Figure 7-15.

±0.3X(T>TC)’

Figure 7-15: Effect of arbitrarily chosen extreme uncertainties of material properties on the cross-sectional simulation result, i. e. of ±10% enthalpy (a), of

±30% thermal conductivity at T> Tc = 769°C (b), of ±20% absorption coefficient

(c) and of ±20% surface tension, adopted from Sudnik et al. [25].

There is a varying need for accuracy during different stages of the design process. Isaksson and Runnemalm developed a systematic approach based on simplified simulations used to create a weld response matrix (WRM) used in the preliminary design stage of a welding procedure. The WRM should relate changes in design variables with changes in variables that are important for the desired performance.

We begin by considering thermal-microstructure evolution of an arc weld that is a 100 mm long and made with a welding speed of 1.785 mm/s and is completed in 71.4 s. Of course, many production welds use much faster welding speeds. We were (1993) able to do a high resolution 3D transient thermal analysis in only a few seconds. This analysis involved 40 time steps and 8,718 5-node bricks. The mathematics used is described in details in [9] and also in chapter V.

To a large extent, the time domain for the thermal-microstructure analysis in many welds can be decomposed into three stages: starting transient, steady state and stopping transient. If this were done, it would be feasible to achieve real-time analysis today.

There are many reasons why a thermal stress analysis of welds is a much more challenging problem than a thermal-microstructure analysis. The thermal-microstructure analysis only involves material a short distance from the weld path, usually less than ten weld pool diameters. Only this relatively small region near the weld need be analyzed for temperatures and microstructure evolution. Usually, this width is less than 10 cm. In contrast, in a thermal stress analysis the complete structure being welded is in quasi-static equilibrium. Thus thermal stresses generated by the welding process can travel over the complete structure. This makes it much more difficult to do the analysis in a relatively small region around the weld. In particular, it is difficult to choose realistic boundary conditions for a small region around the weld.

Another factor is that the mesh used for thermal-stress analysis must be finer than the mesh used for thermal analysis. For the examples described above we use an 8-node brick for stress and treat the temperature in the element as piece-wise constant. The reason for this is that strain is the gradient of the displacement. The gradient operator essentially reduces the order of the strain field to one less than the order of the displacement field. If the thermal strain is to be consistent with the strain from the displacement gradient, the thermal element should be one order lower than the displacement element.

Another reason that thermal stress is more challenging than thermal analysis is that thermal analysis is strictly positive definite because of the capacitance matrix. On the other hand, the quasi­static thermal stress analysis is only made positive definite by constraining rigid body modes. In addition, a temperature increment in the order of 100 °K generates a thermal strain equivalent to the yield strain. While larger temperature increments do not cause serious difficulties for the thermal solver, temperature increments that generate stress increments larger than the yield strength make it difficult to do an accurate thermal stress analysis.

If one used a 20-node brick with 60 displacement dofs for thermal stress analysis and an 5-node brick with 5 temperature dofs for thermal analysis, the thermal stress analysis would have 60/8=7,5 times more equations to solve. For a regular mesh topology, each global equation has 243 nonzero terms compared to 27 nonzero terms for the thermal solver. Thus a very rough lower bound estimate is that a thermal stress solver is (60/8) ■ (243/27) = 67.5 times more expensive than a thermal solver. It also requires more than 70 times as much memory because stress and internal variables must be stored at Gauss points. These rough estimates agree with our experience that the thermal stress analysis in CWM is roughly 10 times more expensive than the thermal-microstructure analysis when 5-node bricks are used for both analyses. We would expect the stress analysis to be roughly 100 times more expensive if 20-node bricks were used for stress analysis and 5-node bricks for thermal analysis.

The cost of CWM is roughly linearly proportional to the number of elements in the mesh, the number of time steps, the number of nonlinear iterations per time step and the time required for each nonlinear iteration. There are opportunities to optimize the mesh and reduce the number of elements. In particular, the use of shell elements could reduce the number of DOFs in a problem. A shell element usually has five DOFs at a node compared to a brick that has six.. However, near the weld solutions are truly 3D and shell elements could introduce large errors. We have long favored the use of local 3D transient analysis near the weld pool and shell elements farther from the weld pool where the assumptions of shell theory are valid. There are also opportunities to take longer time steps. A steady state Eulerian analysis of an infinitely long weld in a prismatic geometry would only require one solution step. This would require approximately a few seconds of CPU time for thermal analysis and a few minutes for a thermal stress analysis. This could easily be done in real time for a sufficiently long weld but would not capture the starting and stopping transients.

As of July 2004, Goldak Technologies Inc. is able to do a thermal stress analysis of a weld in a complex structure at a speed of 0.5 mm/s or 2.0 m/hr using a single CPU 3.2 GHz Pentium. By the end of 2004, their goal is to increase the computing speed into the range of 2.5 to 5 mm/s or 10 to 20 m/hr. This would be real-time CWM for many, though not all, industrial arc welding processes.