非均质材料的时域自适应—等效力、热分析
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摘要
实际工程问题中常会涉及非均质材料时间相关的力学、传热分析。这类问题的数值模拟具有重要的工程应用背景与理论探讨价值。一种直接的方式是分别考虑非均质材料组分的物理/几何特性,将问题在空间/时域离散后进行计算,这往往会导致计算量过大,甚至不可行。一个变通的策略是将非均质材料考虑成一种宏观均质材料,进行等效求解,从而大幅降低计算量。本文以时域自适应算法为核心,结合不同的空间等效技术,分别以粘弹性节理岩体及非均质线性瞬态热传导问题为研究对象,探讨了非均质材料时间相关的等效数值求解方法,并特别关注时域的求解精度和效率等计算细节,主要工作包括：
分别对单向和双向正交粘弹性节理岩体,给出了相应的简单等效假定,利用时域自适应算法,在代表元上将节理与岩石的应力/应变、本构方程,等效应力/应变展开,得到递推格式的等效本构方程,结合有限元方法,导出了求解等效粘弹性场问题的递推求解格式。在模型的数值验证中,考虑了不同节理密度/倾角、节理-岩石不同本构组合、不同结构几何形状、以及不同时间步长的影响,并将等效解和ANSYS得到的非均质解进行了比较。
以上采用的简单等效假定易于理解和操作,但在数学与物理上不够严格。为此,在具有周期性微结构的非均质材料线性瞬态温度场的等效分析中,引入了更为严格的空间等效技术——渐进展开均匀化方法,推导了相关等效热物性参数的表达式,建立了等效递推控制方程及等效温度场的递推求解模型。通过与ANSYS非均质解的比较,对模型的精度、效率等进行了考察,在计算中考虑了不同单胞形式、不同组分比等的影响。
计算结果表明,所提单向/双向正交粘弹性节理岩体的等效模型,以及具有周期性结构的非均质材料的线性瞬态温度场的等效模型,都可提供快速且计算精度合理的数值结果,也可方便地与无网格、离散元等其它成熟的空间数值求解方法接口,并可通过分段的自适应计算,避免在时段大小变化时可能出现的计算误差。
论文的工作有望为非均质材料时间相关的等效数值求解提供有价值的参考。
The time-dependent solution of mechanical and heat conduction for heterogeneous material often appeared in various fields of engineering. These problems are thus of great importance for engineering and theory. A direct way is to consider the geometrical and mechanical property of component in the heterogeneous material, respectively, and then discretize the problem in space/time domain, but this caused a huge computational cost or would be impossible. Another way is to consider the heterogeneous material as macroscopical homogenous material and equivalent analysis, the computational cost will reduce sharply. Based on an adaptive algorithm in the time domain and different space-homogenization method, this paper presented a numerical method for time-dependent equivalent analysis of heterogeneous material by concerning the viscoelastic jointed rock and linear transient heat conduction problem of heterogeneous material, especially focused on the computing accuracy and cost in time domain. The main works are as follows:
Expanded the stress/strain, constitutive equation and equivalent stress/strain of the jointed rock in base cell, recursive equivalent constitutive models of unidirectional /orthogonal viscoelastic jointed rock and recursive finite element model of the related equivalent fields are proposed based on different equivalent assumptions. Regarding to the different joint density/dip angel, different combinations of constitutive relationships for rock and joint, different geometrical and different time step sizes, numerical examples are presented to verify the proposed model and the results are compared with the FE based heterogeneous solutions via ANSYS.
The simple equivalent assumption is easy to understand and operate, however, it is not very enough strict in the frame of mathematics and physics. An equivalent finite element model for the transient thermal field of heterogeneous material with periodic microstructure is obtained by introducing a more strict equivalent method—asymptotic expansion homogenization (AHE), and the derivation of the related equivalent thermal properties and the formulation of the equivalent governing equation are given. The comparisons between the proposed model and the heterogeneous solutions via ANSYS verified its efficiency and accuracy considering different unit cell and ratio of components.
Numerical examples show that the proposed equivalent models of unidirectional /orthogonal viscoelastic jointed rock and transient thermal field of heterogeneous material with periodic microstructure. This method can interface with different spatial numerical method such as mesh less method and DEM (Discrete Element Method), as well as FEM (Finite Element Method), and avoid the error accumulation caused by the change of the time interval. Finally, we hope this thesis will provide valuable reference for the research on time-dependent equivalent analysis of heterogeneous material.
分别对单向和双向正交粘弹性节理岩体,给出了相应的简单等效假定,利用时域自适应算法,在代表元上将节理与岩石的应力/应变、本构方程,等效应力/应变展开,得到递推格式的等效本构方程,结合有限元方法,导出了求解等效粘弹性场问题的递推求解格式。在模型的数值验证中,考虑了不同节理密度/倾角、节理-岩石不同本构组合、不同结构几何形状、以及不同时间步长的影响,并将等效解和ANSYS得到的非均质解进行了比较。
以上采用的简单等效假定易于理解和操作,但在数学与物理上不够严格。为此,在具有周期性微结构的非均质材料线性瞬态温度场的等效分析中,引入了更为严格的空间等效技术——渐进展开均匀化方法,推导了相关等效热物性参数的表达式,建立了等效递推控制方程及等效温度场的递推求解模型。通过与ANSYS非均质解的比较,对模型的精度、效率等进行了考察,在计算中考虑了不同单胞形式、不同组分比等的影响。
计算结果表明,所提单向/双向正交粘弹性节理岩体的等效模型,以及具有周期性结构的非均质材料的线性瞬态温度场的等效模型,都可提供快速且计算精度合理的数值结果,也可方便地与无网格、离散元等其它成熟的空间数值求解方法接口,并可通过分段的自适应计算,避免在时段大小变化时可能出现的计算误差。
论文的工作有望为非均质材料时间相关的等效数值求解提供有价值的参考。
The time-dependent solution of mechanical and heat conduction for heterogeneous material often appeared in various fields of engineering. These problems are thus of great importance for engineering and theory. A direct way is to consider the geometrical and mechanical property of component in the heterogeneous material, respectively, and then discretize the problem in space/time domain, but this caused a huge computational cost or would be impossible. Another way is to consider the heterogeneous material as macroscopical homogenous material and equivalent analysis, the computational cost will reduce sharply. Based on an adaptive algorithm in the time domain and different space-homogenization method, this paper presented a numerical method for time-dependent equivalent analysis of heterogeneous material by concerning the viscoelastic jointed rock and linear transient heat conduction problem of heterogeneous material, especially focused on the computing accuracy and cost in time domain. The main works are as follows:
Expanded the stress/strain, constitutive equation and equivalent stress/strain of the jointed rock in base cell, recursive equivalent constitutive models of unidirectional /orthogonal viscoelastic jointed rock and recursive finite element model of the related equivalent fields are proposed based on different equivalent assumptions. Regarding to the different joint density/dip angel, different combinations of constitutive relationships for rock and joint, different geometrical and different time step sizes, numerical examples are presented to verify the proposed model and the results are compared with the FE based heterogeneous solutions via ANSYS.
The simple equivalent assumption is easy to understand and operate, however, it is not very enough strict in the frame of mathematics and physics. An equivalent finite element model for the transient thermal field of heterogeneous material with periodic microstructure is obtained by introducing a more strict equivalent method—asymptotic expansion homogenization (AHE), and the derivation of the related equivalent thermal properties and the formulation of the equivalent governing equation are given. The comparisons between the proposed model and the heterogeneous solutions via ANSYS verified its efficiency and accuracy considering different unit cell and ratio of components.
Numerical examples show that the proposed equivalent models of unidirectional /orthogonal viscoelastic jointed rock and transient thermal field of heterogeneous material with periodic microstructure. This method can interface with different spatial numerical method such as mesh less method and DEM (Discrete Element Method), as well as FEM (Finite Element Method), and avoid the error accumulation caused by the change of the time interval. Finally, we hope this thesis will provide valuable reference for the research on time-dependent equivalent analysis of heterogeneous material.
引文
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