Smoothed Multiscale Finite Element Method For Heterogeneous Materials

There are various heterogeneous materials in the natural world and the human dailylife, such as bones, porous media, composite materials, alloy and so on. The minimumcharacteristic sizes of these materials are usually very small, in the order of micrometer.On the other hand, their maximum characteristic sizes are usually as big as meters andeven bigger, crossing several orders of magnitude. When analyzing the mechanicalbehaviors of this kind of material in single scale by using the traditional numericalmethods such as the finite element method or finite difference method, a precisesolution could be obtained only when the grid scale of the discrete structure is smallerthan the minimum characteristic size. This can cause large numbers of elements anddegrees of freedom, low computational efficiency, and even no solution., Severalmultiscale algorithms were proposed to solve this problem, such as homogenizationmethod, heterogeneous multiscale method, multiscale finite element method, andvariational multiscale method. Among them the multiscale finite element method, due tothe advantages of no requirement of periodic hypothesis, high efficiency, easy fordownscaling and parallel computation, is becoming a hot spot in the area ofcomputational mechanics.This dissertation proposed a smoothed multiscale finite element method based onthe idea of multiscale finite element method introducing the gradient smoothingtechnology into the assembly of the macro matrix. More over, the smoothed multiscalestochastic collocation method, smoothed extended multiscale finite element method,and mixed smoothed extended multiscale finite element method were proposed basedon the above theory. These methods were used to analyze groundwater flow, fracturedrock seepage, stochastic seepage of the fractured rock, elasticity problems offunctionally graded material plate, and electromechanical coupled problems offunctionally gradient piezoelectric plate. The results showed that, comparing withtraditional numerical methods, the proposed methods could save the computing time,enhance the computing efficiency, and ensure accuracy when solving the mechanicalproperties of heterogeneous materials.The main works of this dissertation include:1. A smoothed multiscale finite element method was proposed to solve the ellipticproblems with oscillating coefficients. The basic idea of this method was to constructnumerical multiscale basis functions to satisfy the degraded elliptical equation and capture parametric spatial variability. The method introduced the gradient smoothingtechnology to assemble the macro matrix of coarse element, reflected the basisfunctions’ fine scale parameter information in coarse scale, and improved computationalaccuracy on the coarse element. The introduction of the gradient smoothing technologyhelped the smoothed multiscale finite element method to avoid problems related totraditional multiscale finite element method, for example to achieve the continuous formof the basis functions through interpolation and compute the gradients of the basisfunctions. In the mean time, the transformation from surface or volume integral to theline or surface integral softened the stiffness matrix of the structure, the structure of thesmoothed subdomains, intensified the analysis of the heterogeneity, and enhancedaccuracy and efficiency. A simulation of the groundwater flow verified the correctnessand effectiveness of the algorithm.2. A smoothed multiscale finite element method was proposed to solve thefractured rock seepage problem. Two multiscale basis functions of fractured media andporous media were constructed on the coarse model to satisfy the correspondingelliptical differential equations and reflect the heterogeneity of the fractured and porousmedia. The finite global information was used to determine the boundary conditions ofthe basis functions, which allowed to capture the global coupled information of thewater flow in the fractured media and the porous media. At last, the gradient smoothingtechnology was used to construct the smoothed hydraulic head gradient matrix andassembled the macro matrix of coarse element. A simulation of the fractured rockseepage problem verified the effectiveness and efficiency of the algorithm.3. A smoothed multiscale stochastic collocation method was proposed to solve thestochastic problems of fluid flow in heterogeneous fractured rocks. Considering therandomness and heterogeneity of the skeleton arrangement and fracture mediadistribution in the fractured rock, the analysis of the fracture rock seepage was definedas a random problem. First, the Karhunen-Loeve expansion method was used todiscretize the random field of the hydraulic conductivity. Then, based on the sparse gridstochastic collocation method, suitable collocation points were chosen and grouped toreduce the random seepage problem to a series of deterministic problem at thecollocation points. Finally, the smoothed multiscale finite element method was used tosolve the problem at the collocation points and the results were statistically analyzed.The numerical simulation results showed that the method, while keeps high precision,has much higher computing efficiency than the traditional stochastic method.4. A smoothed extended multiscale finite element method was proposed to solvethe three-dimensional elastromechanical problem for the functionally graded material.The multiscale basis function were constructed in different coordinate directions tosatisfy the respective simplified governing equations and capture the gradient variationsof the materials. Additional coupling terms were introduced into the base function to consider the bulk expansion and contraction at different directions. A smooth strainmatrix with coupling terms was derived from three-dimensional gradient smoothingtechnology to assemble the macro matrix of the coarse elements. This allowed thegradient smoothing technology to be introduced into the extended multiscale finiteelement method. Numerical simulation examples showed higher precision andefficiency of the proposed algorithms comparing to traditional extended multiscalefinite element method.5. A mixed smoothed extended multiscale finite element method was proposed tosolve the electromechanical coupling problem for functionally graded piezoelectricmaterial. Displacement and electrical field multiscale basis functions that conform to thelaws of physics were constructed to represent the heterogeneity of a material’s differentphysical parameters. A displacement multiscale basis function was constructed bysmoothed extended multiscale finite element method and the electric basis function wasconstructed by smoothed multiscale finite element method. Through the assembly of thecoarse grid macro matrix, the micro information and coupling relationship between thetwo fields were expressed in macro scale. This approach facilitated the problem beingsolved in the coarse grid with high efficiency and precision. Numerical simulationexamples showed the correctness and feasibility of the proposed algorithm.