Optimal Numerical Integration Method for Higher Order Polygonal Finite Elements and its application in microstructure modeling

Abstract

In conventional Finite Element Method (FEM), triangular and quadrilateral elements are commonly used finite elements and their counterparts with tetrahedral and hexahedral elements in three-dimensions to discretize the domain. However, finite element meshes with these elements often lead to difficulty in meshing complicated geometries like polycrystals. To achieve good accuracy in simulation results, the mesh refinement method must be used which leads to heavy computation. These difficulties can be overcome by generalizing the element geometry to have an arbitrary number of sides. The combination of this kind of element with conventional finite elements has been studied recently and popularly known as the Polygonal Finite Element Method (PFEM). In this thesis, first, a numerical scheme is developed to generalize the recently proposed numerical integration scheme called optimal numerical integration method to higher-order polygonal finite elements. It employs a concept of n-point integration for an n-sided polygon. Its application in statistical modeling of the microstructure is demonstrated in subsequent chapters. Wachspress shape functions are used for the interpolation as it possesses linear interpolation order on the boundary whereas higher-order interpolation in the interior of the element. It enables accurate prediction of high-stress gradient due to material discontinuities with lesser computational cost compared to the detailed internal meshing of the grains. The proposed optimal integration scheme is validated by solving 2D cantilever under tensile and shear loadings. The strain energy convergence results are analyzed with respect to the fraction of higher-order polygons introduced in the mesh increasingly, which is similar to the hp-refinement technique used in conventional FEM. The proposed method shows good accuracy and comparable convergence rate with respect to known PFEM at a less computational cost. The second part of the thesis deals with the application of PFEM in statistical modeling of the Titanium alloy microstructure. Using the grain area statistics obtained from the Scanning Electron Microscope (SEM) images, a statistically equivalent mesh is generated. The orientation assignment of each grain is derived from a normal distribution with specified mean and standard deviation values. For different means and standard deviations of grain orientations, a large number of meshes are generated and analyzed using PFEM to study the correlation of effective properties with grain orientation, statistically. Apart from the correlation study of effective properties, the stress localization effect is also captured which is physically caused due to the large misorientation of neighboring grains. The statistical study of effective properties estimation and stress localization is carried out only for HCP grains. In the last part of the thesis, the numerical performance of the PFEM used in the study is demonstrated considering two cases, all HCP grains with the same orientation and all HCP grains with different orientations, respectively. The results showed that the assumed interpolation and mesh size is capable to capture high-stress gradient accurately. The deterministic study of stress localization and the occurrence of local yielding in the presence of two different types of grains, i.e., HCP and lamella and with different orientations in the microstructure is also presented. This deterministic study is done to demonstrate an actual microstructure and can be studied statistically in the future. The thesis concludes with the discussion on three-dimensional modeling of microstructure, homogenization method for basket-weave microstructure and deformation mechanisms in polycrystals.