In electronic devices solder joints form a mechanical as well as an electrical connection between the circuit board and the component (e.g. a chip or a resistor). Temperature variations occurring during field use cause crack initiation and crack growth inside the joints. Accurate prediction of the lifetime requires a method to simulate the damage process based on microstructural properties.Numerical simulation of developing cracks and microstructural entities such as grain boundaries and grain junctions gives rise to several problems. The solution contains strong and weak discontinuities as well as weak singularities. To obtain reasonable solutions with the finite element method (FEM) the element edges have to align with the cracks and the grain boundaries, which imposes geometrical restrictions on the mesh choice. Additionally, a large number of elements has to be used in the vicinity of the singularities which increases the computational effort. Both problems can be circumvented with the extended finite element method (X-FEM) by using appropriate enrichment functions.In this thesis the X-FEM will be developed for the simulation of complex microstructural geometries. Due to the anisotropy of the different grains forming a joint and the variety of different microstructural configurations it is not always possible to write the enrichment functions in a closed form. A procedure to determine enrichment functions numerically is explained and tested. As a result, a very simple meshing scheme, which will be introduced here, can be used to simulate developing cracks in solder joint microstructures. Due to the simplicity of the meshing algorithm the simulation can be automated completely. A large number of enrichment functions must be used to realize this. Well-conditioned equation systems, however, cannot be guaranteed for such an approach. To improve the condition number of the X-FEM stiffness matrix and thus the robustness of the solution process a preconditioning technique is derived and applied.This approach makes it possible to develop a new and fully automated procedure for addressing the reliability of solder joints numerically. The procedure relies on the random generation of microstructures. Performing crack growth calculations for a series of these structures makes it possible to address the influence of varying microstructures on the damage process. Material parameters describing the microstructure are determined in an inverse procedure. It will be shown that the numerical results correspond well with experimental observations.

SUMMARYStrain singularities appear in many linear elasticity problems. A very fine mesh has to be used in the vicinity of the singularity in order to obtain acceptable numerical solutions with the finite element method (FEM). Special enrichment functions describing this singular behavior can be used in the extended finite element method (X-FEM) to circumvent this problem. These functions have to be known in advance, but their analytical form is unknown in many cases. Li et al. described a method to calculate singular strain fields at the tip of a notch numerically. A slight modification of this approach makes it possible to calculate singular fields also in the interior of the structural domain. We will show in numerical experiments that convergence rates can be significantly enhanced by using these approximations in the X-FEM. The convergence rates have been compared with the ones obtained by the FEM. This was done for a series of problems including a polycrystalline structure.

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