Multiscale Finite Element Method For Dynamic And Nonlinear Analyses Of Heterogeneous Materials

As well known, many materials in the nature have multiscale features, such as porous media, animal bones, composite materials in the large machinery. The differences between the maximum and minimum characteristic sizes of these materials vary considerably. In general, one must refine the grids to make the size of the meshes less than the smallest characteristic size of the materials when solving the structures with multiscale features by using the traditional methods such as the finite element method, and finite difference method. This will be bound to spend a lot of computational resources and time. For some complex large-scale problems, these direct methods even can not obtain the solutions as a result of the restrictions of the computer memory, especially for the dynamic problems. During the past years, many researches have developed various kinds of multiscale computational methods for these practical problems, such as the homogeneous method and the representative volume element method. However, these multiscale methods are usually based on the periodic assumption, and they are usually difficult to obtain the fine-scale solution. Therefore, seeking an efficient multiscale computational method almost becomes a research hotspot recently.Firstly, the Extended Multiscale Finite Element Method (EMsFEM) for the linear elastic mechanical analysis of the heterogeneous materials is introduced briefly in this dissertation. The main idea of this method is to construct the multiscale base functions numerically by solving the partial differential equations on the sub domains in different directions separately. These base functions can capture the heterogeneities of the materials in the coarse element effectively and efficiently. Then, the original problems are only solved at the macroscopic level, which can reduce the computational cost significantly. For the solid mechanical problems, the additional coupling terms, which can improve the computational accuracy greatly, are introduced into the numerical base functions of the coarse element to consider the Poisson effect within the coarse element among different directions. In this dissertation, we briefly introduce the basic idea of the EMsFEM, four commonly used methods to construct the numerical base functions, macroscopic analysis, downscaling technique, etc.Inspired by the additional coupling terms in the numerical base functions, a new four-node plane generalized isoparametric element is developed based on the traditional four-node plane quadrilateral element (Q4). The additional coupling terms are introduced into the displacement interpolation functions. Then, poisson effect within the element is taken into consideration in this newly developed element. Compared with the traditional Q4element, this new element can improve the computational accuracy significantly without increasing any additional degree of freedom. There is no gap between the adjacent elements, so this new element is a conforming element. This dissertation verifies that this element can meet the requirements of patch test and investigates the influence on the calculation results by the different forms of additional coupling terms.Geometrically nonlinear problem is one of the most common problems in the solid mechanics. In this dissertation, an efficient multiscale computational formulation is developed for the large displacement-small strain analysis of the heterogeneous materials by combining the EMsFEM and the co-rotational coordinate method. This formulation makes that the EMsFEM can be effectively applied to geometrically nonlinear problems. The main steps are as follows. Firstly, a single heterogeneous unit cell will be equivalent into a coarse element (macroscopic element) by virtue of the numerical base functions. Then, the co-rotational coarse element formulations can be employed on the macroscopic scale. The tangent stiffness matrix of the heterogeneous structure can be also calculated on the macroscopic scale. Thus, the macroscopic nodal displacements of the coarse-scale mesh can be obtained. Finally, the microscopic nodal displacements of the sub grids within the unit cell can be calculated by using the numerical base functions once again. Furthermore, one can also obtain the microscopic stress results naturally.In addition, an efficient multiscale computational method is also developed for the dynamic analysis of the heterogeneous materials. For the static problems, the displacement of the structure is associated with the stiffness of the structure and the external force directly. While for the dynamic problems, the inertial force of the structure also needs to be considered. The multiscale numerical base functions in the original EMsFEM are obtained by solving the static equilibrium equations in the unit cell. However, the displacement of the structure is also related to the inertial force for the dynamic problems. The multiscale numerical base functions in the original EMsFEM do not take the dynamic effect of the unit cell into consideration. This will give rise to a significant calculation error inevitably for the dynamic problems. Therefore in order to reduce the computational errors and improve the accuracy of the original EMsFEM, it seems necessary to take the effect of the inertial force of the unit cell into account during the calculation. Just because of this, the mode base functions are introduced into the original multiscale base functions. Besides, the coarse element used in the EMsFEM for2D problems is usually four-node which can only accurately describe the simple low-order deformation, and this four-node coarse element would be powerless for the high-order complex deformation in the dynamic problems. One will have to refine the coarse-scale meshes when calculating the2D dynamic problems with the four-node coarse element, and this will reduce greatly the computational efficiency and advantages of the EMsFEM. Therefore, in order to ensure the computational accuracy without increasing too much amount of computational cost, the multi-node coarse elements are proposed for the dynamic analysis of the heterogeneous materials.Furthermore, the elasto-plastic dynamic analysis of the heterogeneous materials is investigated based on the proposed multi-node coarse element. A new correction technique of the local displacement is applied to deal with the un-projected microscopic unbalance forces within the coarse element. When plastic deformation occurs during the computation, the plastic unbalanced forces will be produced. Part of these forces will be projected to the macroscopic equations by the multiscale base functions while the rest of these forces will be not. If no further action is taken, these un-projected forces will be lost and cannot be recovered. Therefore, to improve the computational accuracy, a correction technique is implemented to deal with the un-projected forces.Finally, an adaptive algorithm is proposed for the coarse-scale mesh based on the proposed multi-node coarse element for the linear elastic problems. A nearly optimal distribution of macroscopic nodes on the fixed coarse-scale mesh can be found by using the proposed adaptive algorithm, which can achieve that more macroscopic nodes are distributed on the region with larger displacement gradient and fewer macroscopic nodes are distributed on the region with smaller displacement gradient.