There are many numerical methods used for simulation of engineering problems, among which the finite difference method (FDM) and the finite element method (FEM) are the most commonly used. As illustrated in the figure below, the finite difference model gives a point-wise approximation to a problem with an array of grid points subdividing the geometry along each coordinate axis, while the finite element model gives a piece-wise approximation to a problem with an assemblage of elements subdividing the geometry along the boundaries. The FDM solves the governing equations by direct differentiation along each coordinate axis thus it can run very fast. The FEM solves the governing equations by discretization of the domain with elements of a selected shape and assembling them into the entire system, thus it runs usually slower than the FDM. The FDM is mostly used for solving fluid mechanics and heat transfer problems often with stationary boundaries, but it is impossible to use for solving problems with large strain/deformation. The FEM is more advantageous to solve problems with large deformation and can be used for nearly all kinds of engineering problems with complex geometry and material combinations.

SORPAS® is developed based on the finite element method. All numerical models and procedures have been developed and integrated with the welding engineering expertise, which has been fully automated inside the software. The graphical user interface of SORPAS® is also tailor-made for resistance welding. Users are not required to have prior knowledge of the FEM. However, having some basic knowledge of the FEM can help to better understand the simulations and obtain more reliable simulation results. More details will be introduced at the training course (usually 1-2 days). We hereby emphasize only the two basic concepts that have essential influence in FEM simulations.

**Mesh density**

The mesh density or the size of elements has essential influence on the accuracy of FEM calculations regarding distribution of variables in the geometry. As a basic procedure of FEM, the problem domain (geometry and materials) is divided into a number of elements (mesh). This procedure is called the mesh generation. The FEM calculations are mainly based on the values of variables on the nodal points. The more nodal points or the more elements are divided, the more accurate results can be obtained for the distribution of variables. Increasing the total number of elements will increase the nodal points, but also increase the number of calculations and the computation time. It is more efficient to only increase the number of elements in the areas with large changes (or gradients) of variables, leaving fewer elements in the areas with small changes. The total number of elements can thus still be kept in a reasonable number. This is why the mesh density control is introduced to allow users to determine where they need more elements (or higher mesh density). In SORPAS® the mesh generation is fully automated according to the user defined total number of elements and density control points. The figure below shows the influence of the mesh density in a spot welding simulation. It is noticed that the computation time increases drastically with increasing number of elements but the obtained nugget volume changes very little with more than 1000 elements. We have thus recommended that 1000 elements are normally enough for an ordinary spot welding simulation.

**Time step**

In order to calculate the dynamic development of variables, the process time is divided into small time steps during simulations. The FEM calculations will be carried out incrementally at every time step through the entire welding process. The time step has significant influence on the accuracy of FEM simulations especially for the dynamic variables rapidly changing with time. The smaller the time step is, the more accurate results can be obtained regarding the dynamically changing variables, but it will also increase the number of calculations and thus the computation time. The figure below shows the influence of the time step in a spot welding simulation. It is noticed that the computation time decreases drastically with increasing step size. There is no big influence to the nugget volume when the time step size is below 0.5ms. To ensure the reliability of results, we have recommended that a time step of 0.05-0.1ms should be sufficient for most of the resistance welding applications.

**Procedures for making simulations**

Simulation with SORPAS® is a virtual resistance welding process on a computer, where the whole process, from design to welding, is done on the computer without using actual materials and welding equipment. Users will see the results of welding graphically displayed on the computer. In this way, the engineers can evaluate their design of electrodes and weldability of new materials, and optimize process parameter settings before actual welding tests. The procedure for making simulations with SORPAS® includes three steps: 1) data preparation, 2) running simulation and 3) evaluation of results.

The input data needed for simulation with SORPAS® are summarized below:

- Geometry and materials
- Define forms and select materials for electrodes
- Define geometry and select materials for workpieces
- Define thickness and select materials for coatings

- Machine settings
- Define connection of electrodes to the welding machine
- Set welding force
- Set welding current and time

- Simulation controls
- Define time step sizes and select numerical models
- Define automated optimization procedures

After all input data are prepared, the simulation can be started simply by pressing a button; it will then run automatically. The results will be saved through the progress of the simulation, which can be graphically displayed for analysis after the simulation is finished. With the automated optimization procedures, the optimal weld current for achieving a desired size of weld nugget can be obtained.