非均质饱和多孔介质准静态行为分析的耦合多尺度及耦合升尺度有限元法
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摘要
在科学研究和实际工程当中,我们遇到的许多问题都具有非均质以及多尺度的性质,如地下水在非均质地层介质中的渗流、复合材料中的热传导、非均质材料在外部载荷作用下的变形等。使用常规数值算法在解析介质非均质性的细网格模型上求解这些问题时,常常会由于自由度数太大导致计算量过大而遇到困难。因此,人们发展了许多种多尺度和升尺度算法,用以在适合进行数值分析的粗网格模型上求解这些问题,如多尺度有限元法(MSFEM)以及各种升尺度算法等。如今对这类算法的研究已经形成了一个广泛的研究领域。与此同时,在常见的实际工程问题当中,多种物理现象常常是耦合在一起的。以地下水、石油、天然气等资源的开采和地面高大建筑物的压力等因素造成的地面沉降为例,岩土介质的变形通常伴随着孔隙流体的流动、热量和物质的传输、污染物的传输以及各种物理与化学反应等现象。此类问题的控制方程往往具有多相多场耦合特征,使用传统数值算法求解时,由于非均质及多尺度特性的存在,为了获得有意义的解,需要在反映非均质性质的细尺度模型上求解,这将导致此类耦合问题求解的计算量和存储需求过大,乃至无法求解。多尺度分析方法是解决此类问题的一种有效方法,到目前为止,国内外已发展了多种求解单一物理场问题十分有效的多尺度或升尺度算法；然而,对于多相与多物理场耦合作用问题,由于问题的复杂性,其多尺度计算方法方面的成果还相对较少,发展求解大规模非均质多相多物理场耦合问题的多尺度或升尺度计算方法仍是需要深入研究的课题,相关研究工作具有重要的理论意义与工程应用价值。
为了实现这一目的,本文发展了耦合多尺度、扩展多尺度以及耦合升尺度有限元法求解框架,在便于进行数值模拟的粗网格模型上求解工程中常见的大规模非均质流固耦合问题。与常规数值算法相比,应用这三种数值算法求解非均质耦合问题时,可以较大程度地降低计算量,减少存储需求。
首先,本文发展了耦合多尺度有限元法(CMSFEM)用于在粗网格模型上求解非均质饱和多孔介质在外载荷作用下的变形与固结问题。在CMSFEM中,基于多尺度有限元法(MSFEM)的基本思想,构建了饱和多孔介质流体相多尺度基函数,与MSFEM不同,CMSFEM还构建了在粗网格单元上用于固相位移场插值的多尺度基函数；在此基础上,采用超样本技术来提高多尺度基函数的计算精度。在所构造的基函数的基础上,非均质多相耦合问题细网格有限元模型就可以转化为粗网格模型进行求解,在这一方法中,非均质饱和多孔介质各非均匀参数场如渗透率、弹性模量、密度、孔隙率等,均得到了考虑。本文给出了多个数值算例,并与传统有限元法在细网格模型上求得的结果进行了比较,结果显示,耦合多尺度有限元法可以被成功地用来求解非均质流固耦合问题,与传统有限元法相比,极大地降低了存储需求和计算量。
其次,本文发展了扩展多尺度有限元法(EMSFEM)作为在粗尺度模型上求解非均质问题的一般性多尺度算法。算法中构建了新型的固相位移场多尺度基函数,使用超样本技术为多尺度基函数建立振荡边界条件,对处于模型边界上的粗网格单元,采用修正的超样本技术形成超样本单元；在多尺度基函数中,通过引入不同方向位移之间的耦合作用项,使得多尺度单元成为混合插值型单元,提高了多尺度基函数对材料非均质特性.的模拟能力,大大提高了传统多尺度有限元法的计算精度。文中给出多个算例,并与传统有限元法进行比较,结果显示,作为求解非均质问题的一般性多尺度算法,扩展多尺度有限元法可以被成功地用来求解单相非均质介质以及非均质饱和多孔介质固结问题,与传统有限元法相比,在保证一定计算精度的同时,降低了计算量和存储需求。
最后,本文发展了耦合升尺度有限元法(CUFEM)用于在粗网格模型上求解非均质饱和多孔介质在外载荷作用下的变形与固结问题。它同样也是一个求解多场耦合问题的的有效算法,本文使用CUFEM求解非均质饱和多孔介质的固结问题以说明其基本求解过程。在CUFEM中,首先求解非均质饱和多孔介质粗网格模型上的每个粗网格单元计算的等效渗透率张量和等效弹性模量张量,并进一步利用超样本技术提升等效参数的计算精度。在粗网格单元宏观矩阵的计算过程中,使用常规有限元形函数在粗网格单元上进行数值积分的方法来考虑非均质标量参数场所带来的非均质性,如密度、压缩系数等。文中给出了具体数值算例以验证算法的有效性。算例分析表明,由耦合升尺度有限元法在粗网格模型上求得的结果与常规有限元法在细网格模型上求解所得结果吻合较好,与耦合多尺度有限元法以及扩展多尺度有限元法一样,耦合升尺度有限元法在粗网格模型上求解非均质问题时,较大程度地减少了计算量和存储需求。
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.
为了实现这一目的,本文发展了耦合多尺度、扩展多尺度以及耦合升尺度有限元法求解框架,在便于进行数值模拟的粗网格模型上求解工程中常见的大规模非均质流固耦合问题。与常规数值算法相比,应用这三种数值算法求解非均质耦合问题时,可以较大程度地降低计算量,减少存储需求。
首先,本文发展了耦合多尺度有限元法(CMSFEM)用于在粗网格模型上求解非均质饱和多孔介质在外载荷作用下的变形与固结问题。在CMSFEM中,基于多尺度有限元法(MSFEM)的基本思想,构建了饱和多孔介质流体相多尺度基函数,与MSFEM不同,CMSFEM还构建了在粗网格单元上用于固相位移场插值的多尺度基函数；在此基础上,采用超样本技术来提高多尺度基函数的计算精度。在所构造的基函数的基础上,非均质多相耦合问题细网格有限元模型就可以转化为粗网格模型进行求解,在这一方法中,非均质饱和多孔介质各非均匀参数场如渗透率、弹性模量、密度、孔隙率等,均得到了考虑。本文给出了多个数值算例,并与传统有限元法在细网格模型上求得的结果进行了比较,结果显示,耦合多尺度有限元法可以被成功地用来求解非均质流固耦合问题,与传统有限元法相比,极大地降低了存储需求和计算量。
其次,本文发展了扩展多尺度有限元法(EMSFEM)作为在粗尺度模型上求解非均质问题的一般性多尺度算法。算法中构建了新型的固相位移场多尺度基函数,使用超样本技术为多尺度基函数建立振荡边界条件,对处于模型边界上的粗网格单元,采用修正的超样本技术形成超样本单元；在多尺度基函数中,通过引入不同方向位移之间的耦合作用项,使得多尺度单元成为混合插值型单元,提高了多尺度基函数对材料非均质特性.的模拟能力,大大提高了传统多尺度有限元法的计算精度。文中给出多个算例,并与传统有限元法进行比较,结果显示,作为求解非均质问题的一般性多尺度算法,扩展多尺度有限元法可以被成功地用来求解单相非均质介质以及非均质饱和多孔介质固结问题,与传统有限元法相比,在保证一定计算精度的同时,降低了计算量和存储需求。
最后,本文发展了耦合升尺度有限元法(CUFEM)用于在粗网格模型上求解非均质饱和多孔介质在外载荷作用下的变形与固结问题。它同样也是一个求解多场耦合问题的的有效算法,本文使用CUFEM求解非均质饱和多孔介质的固结问题以说明其基本求解过程。在CUFEM中,首先求解非均质饱和多孔介质粗网格模型上的每个粗网格单元计算的等效渗透率张量和等效弹性模量张量,并进一步利用超样本技术提升等效参数的计算精度。在粗网格单元宏观矩阵的计算过程中,使用常规有限元形函数在粗网格单元上进行数值积分的方法来考虑非均质标量参数场所带来的非均质性,如密度、压缩系数等。文中给出了具体数值算例以验证算法的有效性。算例分析表明,由耦合升尺度有限元法在粗网格模型上求得的结果与常规有限元法在细网格模型上求解所得结果吻合较好,与耦合多尺度有限元法以及扩展多尺度有限元法一样,耦合升尺度有限元法在粗网格模型上求解非均质问题时,较大程度地减少了计算量和存储需求。
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.
引文
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