The implementation of periodic boundary conditions (PBCs) is one of the most important and difficult steps in the computational analysis of structures and materials. This is especially true in cases such as mechanical metamaterials which typically possess intricate geometries and designs which makes finding and implementing the correct PBCs a difficult challenge. In this work, we analyze one of the most common PBCs implementation technique, as well as implement and validate an alternative generic method which is suitable to simulate any possible 2D microstructural geometry with a quadrilateral unit cell regardless of symmetry and mode of deformation. A detailed schematic of how both these methods can be employed to study 3D systems is also presented.
相似文献Advanced 3D MEMS packaging technologies involving the encapsulation of devices at wafer-level are being developed in order to achieve further minimization and cost reduction of consumer electronic devices. Compression molding using epoxy molding compounds is one technique being considered for wafer-level encapsulation. Excessive out-of-plane deformation has been reported in wafer-level compression molding trials using blank wafers which would negatively impact device reliability and the implementation of successive processes to the molded wafer. This paper presents finite element models of the molded wafer, with and without embedded dies which simulate the observed multi-state warpage characteristics. Molded wafer warpage measurements were also carried out in order to verify the applicability of the small and large deformation theories for layered plates and to verify the finite element model of the molded blank wafer. Possible factors (non-planar mold layer thickness and anisotropic wafer elastic properties) leading to asymmetric warpage in molded blank wafers were also investigated. From the molded wafer model with embedded dies the effects of flip-chip die dimensions and wafer thickness on the out-of-plane deformation together with possible reliability issues were analyzed.
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