| Almost all problems that usually encountered in science and engineering are multiscale in nature. These problems may include the groundwater flow in heterogeneous subsurface formations, the heat transfer in composite materials, the deformation of non-homogeneous materials under certain loading conditions, etc. Traditional numerical methods will encounter difficulties when applied for solving these problems on the resolved fine-scale numerical models due to its large computational requirements. Then many kinds of multiscale and upscaling methods are developed for solving these problems on manageable coarse-scale models such as the multiscale finite element method (MSFEM), the upscaling methods, etc. It has become a huge research field now. At the same time, many natural phenomena are usually coupled with each other in the problems usually encountered in practical engineering. Take the ground subsidence as an example, the deformation of the soil and rocks is usually accompanied by the fluid flow, the heat and mass transfer, the pollutant diffusion, the chemical and physical reaction, etc. These phenomena cannot be decoupled reasonably and solved independently in most cases. Since the heterogeneities and multiscale properties are widespread, direct solution of these coupling problems on resolved fine-scale models with traditional numerical methods are almost impossible. Multiscale methods are effective algorithms for sovling this kinds of problems. So far many kinds of multiscale and upscaling numerical methods have been developed for solving single physical field problems. However, for the complicate multi-phase and multi-physics coupling problems, corresponding multiscale methods are extremely rare. So, to develop new multiscale or upscaling numerical strategies for solving the heterogeneous multi-phase and multi-physics coupling problems has become a great challenge to the researchers. Related investigations will be of great importance for theoretical progress and engineering applications.For this purpose, the coupling multiscale finite element numerical strategy, the extended multiscale finite element numerical strategy and the coupling upscaling finite element numerical strategy are developed in this dissertation for solving the heterogeneous coupling problems usually encountered in practical engineering on manageable coarse-scale models. The applications of these numerical methods for solving the coupling problems can greatly reduce the computing efforts in both computational cost and memory in comparison with traditional numerical methods.Firstly, the coupling multiscale finite element method (CMSFEM) is developed for solving the coupling problems of deformation and consolidation of the heterogeneous saturated porous media under external loading conditions on coarse-scale models. In CMSFEM, based on the basic idea of MSFEM, the multiscale base functions are constructed for the fluid phase. Furthermore, the multiscale base functions are also constructed for the displacement fields of the solid phase. The oversampling technique is adopted to improve the accuracy. In this way, the heterogeneous problems can be solved on coarse-scale models with a multiscale numerical procedure. The heterogeneities induced by non-homogeneous properties of the saturated porous media, such as permeability, elastic modulus, density, porosity, etc, can be all taken into account. Numerical experiments are carried out for several cases in comparison with the traditional finite element method which is applied on the resolved fine-scale models. The numerical results show that CMSFEM can be successfully used for solving this kind of coupling problems. Most impotantly, it reduces the computing effort in both memory and CPU time.Secondly, the extended multiscale finite element method is developed and treated as a general multiscale numerical procedure for solving the heterogeneous problems. In this method, a set of new multiscale base functions for the displacent field of the solid phase is constructed. The oversampling technique is used to generate the oscillatoty boundary conditions for these multiscale base functions. The modified oversampling technique is used to form the oversampling regions for the coarse grid elements on the boundaries of the model. Moreover, additional coupled terms between the displacement fields of different directions are introduced into the multiscale base functions. The multiscale elements then become mixed-interpolation type. The small scale features within the coarse grid elements induced by non-homogeneous materials can be effectively captured by these multiscale base functions. Then the accuracy of the extended multiscale finite element method is improved. Several numerical examples for EMSFEM are carried out in this dissertation in comparision with the traditional finite element method. The numerical results show that EMSFEM can be successfully used for solving the heterogeneous problems which include both the simple one-phase problems and the complex multi-phase problems. Most importantly, it reduces the computing efforts in comparison with traditional finite element method.Finally, the coupling upscaling finite element method (CUFEM) is developed for solving the coupling problems of deformation and consolidation of the heterogeneous saturated porous media under external loading conditions. It is essentially also a numerical framework for solving many kinds of complicated coupling problems. Its application to the consolidation analysis of heterogeneous saturated porous media is presented here to illustrate its numerical procedure. In this method, equivalent permeability tensors and equivalent elastic modulus tensors are calculated for every coarse grid block in the coarse-scale models of the heterogeneous saturated porous media. The oversampling technique is introduced to improve the calculation accuracy of the equivalent parameters. Instead of simple spatial average, a numerical integration process is performed over the fine-scale mesh within every coarse grid element to capture the small scale information induced by non-uniform scalar field properties such as density, compressibility, etc, with traditional shape functions of the standard finite element method. Numerical experiments are carried out to examine the accuracy of the developed method. It shows that the numerical results obtained by CUFEM on the coarse-scale models fit fairly well with the reference solutions obtained by traditional finite element method on the resolved fine-scale models. It reduces dramatically the computing effort in both CPU time and memory for solving the transient coupling problems in comparison with the traditional FEM just as CMSFEM and EMSFEM.
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