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Plastic integrated circuit (IC) packages are usually composite components made of multiple bonded materials with different mechanical and thermal properties. High stresses often occur at corners between different components within the packages where discontinuities of geometry or material properties are present. Delaminations usually initiate from these corners when packages undergo adverse thermal or moisture environments. Hence, in order to prevent delaminations from occurring and to improve package reliability performance, it is crucial to accurately and efficiently evaluate the stresses at corners within the package. However, with conventional finite element methods, it is always a challenge to give an accurate description of the stresses at the corners since these corners represent stress singularity points. In this paper, an effective method is developed to precisely evaluate the stresses in the vicinity of internal corners within the packages. A new variable-order singular boundary element is constructed with a built-in accurate description of the stresses at the corner. This method is versatile for solving general corner problems involving wedges, two-material and three-material corners and interfacial cracks that are common in the IC packages. This method is verified by solving a bimaterial interface crack problem with known solution. Comparisons are made with other conventional methods such as displacement-based quarter-point singular elements and normal quadratic elements, on the calculation of interfacial stress intensity factors. Results show that the new method has significant advantages in giving more accurate results with much less computational resources needed. The new method is applied to a typical plastic IC package with multiple internal corners and interface cracks. The possible failure sites and failure modes within the package are predicted and the results agree well with package evaluations done in the industry.