The recent advances in the Materials Engineering have led to the development of new materials with customized microstructure in which the properties of its constituents and their geometric distribution have a considerable effect on determination of the macroscopic properties of the substance. Direct inclusion of the material microstructure in the analysis on a macro level is challenging since spatial meshes created for the analysis should have enough resolution to be able to accurately capture the geometry of the microstructure. In most cases this leads to a huge finite element model which requires a substantial amount of computational resources. To circumvent this limitation a number of homogenization techniques were developed. By considering a small element of the material, referred to as Representative Volume Element (RVE), homogenization methods make it possible to include the effects of a material’s microstructure on the overall properties at the macro level. However, complexity of the microstructure geometry and the necessity of satisfying periodic boundary conditions introduce additional difficulties into the analysis procedure. In this dissertation we propose a hybrid homogenization method that combines Asymptotic homogenization with MeshFree Solution Structures Method (SSM). Our approach allows realistic inclusion of complex geometry of the microstructure that can be captured from micrographs or micro CT scans. In addition to unprecedented flexibility in handling complex geometries, this method also provides a completely automatic analysis procedure. Using meshfree solution structures simplifies meshing to creating a simple cartesian grid which only needs to contain the domain. This also eliminates manual modifications which usually needs to be performed on meshes created from image data. A computational platform is developed in C++ based on meshfree/asymptotic method. In this platform also a novel meshfree solution structure is designed to provide exact satisfaction of periodic boundary conditions for boundary value problems such as homogenization. Performance of the developed platform is tested over 2D and 3D domains against previously published data and/or conventional finite element methods. After getting satisfactory results, homogenized properties are used to compute localized stress and strain distributions over inhomogeneous structures. Furthermore, effects of geometric features of pores/inclusions on homogenized mechanical properties is investigated and it is demonstrated that the developed platform could provide an automated quantitative analysis tool for studying effects of different design parameters on homogenized properties.

In this work, a pioneering application of the Solution Structure Method (SSM) for structural dynamics problems is presented. Vibration analysis is an important aspect of any design-analysis cycle for which reliable computational methods are required. Unlike many meshfree methods, SSM is capable of {\it exact treatment of all prescribed boundary conditions}. In addition, the method is capable of using basis functions which do not conform to the shape of the geometric model. Together, this defines an unprecedented geometric flexibility of the SSM. This work focused on the development of numerical algorithms for 2D in-plane and 3D natural vibration analysis and 2D in-plane dynamic response. The convergence and numerical properties of the method were evaluated by comparing meshfree results with those obtained using traditional Finite Element Analysis implemented in Solidworks and ANSYS. The numerical experiments presented in this work illustrate that the Solution Structure Method possesses good convergence and in some cases, such as geometries with partially fixed boundaries, this method converges much more rapidly than traditional FEA. Finally, in addition to complex boundary conditions, this method can easily handle complex geometries without losing favorable convergence properties.