【作者】 王凯；

【导师】 唐春安；

【作者基本信息】 大连理工大学 ， 岩土工程， 2024， 博士

【摘要】 随着我国对地热能、核能等低碳清洁能源需求的日益增长,深地热开采和核废料地质储存等岩石热破裂相关问题引起了研究者们的广泛关注。数值模拟方法凭借成本低廉、高效快捷、便于进行参数分析等优势,已逐渐成为分析上述问题的主要手段。实际岩石工程问题普遍具有空间尺度大、时间跨度长及荷载高度时间相关等特点,这对数值模拟方法的求解精度和计算效率提出了严峻挑战。为此,本文在总结、分析前人研究成果的基础上,以传统数值流形方法为基本理论和技术框架,从空间和时间两个维度入手,对岩石热破裂分析中涉及的瞬态热弹性微分方程求解和准静态裂纹扩展实现等环节进行改进和优化,建立了一种全新的时空高精度数值流形方法,并将其应用于以深地热开采和核废料地质储存为背景的数值模拟研究。本文的主要研究内容如下:(1)详细梳理了数值流形法框架下岩石热破裂分析各环节的实施方法;通过引入Delaunay三角剖分算法,实现了在复杂形状流形元上的准确积分,使得数值流形法在双重覆盖的基础上进一步摆脱了对数学网格的依赖;基于瞬态热弹性问题微分方程、边界条件、数值流形法中的逼近函数,采用变分原理和加权残值法导出了矩阵形式的瞬态热弹性问题控制方程,为后续高精度求解方案的创建提供了理论基础。(2)创建了一种能够准确捕捉裂纹尖端奇异性的高精度空域离散化方案。在传统数值流形方法构造的结构化网格的基础上,提供了确定待细分单元和多级细分方法的实施流程;为处理悬挂节点问题,引入了适用于可变边中节点单元的一致形状函数来构建数学片上的权函数,设计了一种特定的剖分算法来实现过渡单元内流形元上的数值积分;为减少内存占用和时间消耗,提出了裂尖随动细分策略来抑制裂纹扩展过程中系统自由度数目的快速增长。将裂尖增强函数与上述方法构成的网格局部多级细分技术相结合,共同应用于热-力荷载下的岩石断裂分析和裂纹扩展轨迹模拟。数值结果表明,当前空域离散化方案可在保证求解精度的前提下有效节省计算机内存占用和时间消耗。(3)创建了两种能够高精度求解非齐次瞬态热传导微分方程的时域积分方案,即解析求解方案和增维求解方案。两种方案均引入了精细时间步长积分法来处理控制方程的齐次项。区别在于,解析求解方案直接对由等效热荷载向量F_T进行泰勒级数展开或傅里叶级数展开后得到的非齐次项NHT进行高精度求解,而增维求解方案则是利用增维技术将非齐次瞬态热传导微分方程等价地转化为齐次瞬态热传导微分方程,避免了对后续非齐次项的处理。两种方案在执行瞬态传热分析后的结果显示,它们都完整保留了精细时间步长积分法高精度、稳定和收敛的优点。从理论角度证明了解析求解方案在处理瞬态传热问题上的无条件稳定性和收敛性,并对它进行了全面的误差分析。进一步,将该方案应用于岩石瞬态热断裂分析。数值结果表明,对于具有不同裂纹构型和高度时间相关荷载形式的模型,当前方案均能获得比广泛使用的向后差分方案更加精确的模拟结果,且计算效率更高。(4)提出了一种瞬态传热模式下准静态裂纹扩展过程中的热自由度继承策略。通过将物理片上的局部逼近映射到形成热传导矩阵时的高斯积分点和温度加载点上的方式,实现了相邻时间步内发生裂纹扩展时,本文所提时域积分方案对下一时步温度场直接且准确的计算。融合新创建的高精度空域离散化方案和高精度时域积分方案,建立了一种全新的时空高精度数值流形方法。相关数值算例的模拟结果表明,本文方法在分析岩石热破裂问题时能够实现求解精度和计算效率的双赢,可为深地热开采和核废料地质储存等实际岩石工程问题研究提供更加可靠、高效的数值计算辅助。更多还原

【Abstract】 With the continuous growth of China’s demand for low-carbon clean energy such as geothermal energy and nuclear energy,issues related to rock thermal fracturing,deep geothermal extraction,and high-level radioactive waste geological storage have attracted widespread attention among researchers.Numerical simulation methods,due to their low cost,high efficiency,and ease of parameter analysis,have gradually become the primary means of analyzing the aforementioned issues.Real-world rock engineering problems generally exhibit characteristics such as large spatial scales,long time spans,and time-dependent loading,posing significant challenges to the accuracy and computational efficiency of numerical simulation methods.Therefore,based on the summary and analysis of previous research results,this paper improves and optimizes the implementation methods of transient heat conduction differential equation solving and quasi-static crack propagation involved in rock thermal fracturing simulation,taking the traditional numerical manifold method as the basic theoretical and technical framework,from the perspectives of both space and time.A high-precision numerical manifold method is established and applied to numerical simulation research with deep geothermal extraction and nuclear waste geological storage as application backgrounds.The main work and conclusions of this paper are as follows:(1)Detailed scrutiny and analysis of the implementation methods of various aspects of rock thermal fracturing simulation under the numerical manifold framework are provided.By introducing the Delaunay triangulation algorithm and updated local polar coordinate system representation for crack tips,accurate integration on complex-shaped manifold elements is achieved,enabling the numerical manifold method to further alleviate its dependence on mathematical grids based on double coverage.Based on the transient thermoelastic problem’s differential equations,boundary conditions,and approximation functions in the numerical manifold method,matrix-form transient thermoelastic problem control equations are derived using variational principles and weighted residual methods,providing a foundation for subsequent high-precision integration schemes.(2)A high-precision spatial discretization scheme that accurately captures the singularity of crack tips is created.Based on the structured grid constructed by the traditional numerical manifold method,the implementation process for determining the subdivision elements and multi-level subdivision methods is provided.To address the hanging node problem,uniform shape functions applicable to variable-midside-node elements are introduced to construct weight functions on mathematical patches,and a specific partitioning algorithm is designed to achieve numerical integration on manifold elements within transition units.To reduce memory consumption and time consumption,a crack tip adaptive subdivision strategy is proposed to suppress the rapid growth of degrees of freedom during crack propagation.The crack tip enhancement function is combined with the local multi-level subdivision technique formed by the above methods,applied together in fracture analysis under thermal-mechanical loading and simulation of crack propagation trajectories.Numerical results indicate that the current spatial discretization scheme can effectively save computer memory and time consumption while ensuring solution accuracy.(3)Two high-precision time-domain integration schemes for accurately solving nonhomogeneous transient heat conduction differential equations are developed: analytical solution scheme and dimension-increasing solution scheme.Both schemes incorporate precise time step integration method to integrate the matrix-form control equations obtained from the traditional numerical manifold method over time.The difference lies in the approach: the analytical solution strategy directly achieves high-precision solution of nonhomogeneous term NHT obtained by polynomial expansion or Fourier expansion of the equivalent heat load vector FT,while the dimension-increasing solution strategy utilizes dimension-increasing techniques to equivalently transform nonhomogeneous transient heat conduction equations into homogeneous ones,avoiding the need for subsequent solution processes for nonhomogeneous terms.Results after performing transient heat transfer analysis with both schemes demonstrate that they retain the advantages of high precision,stability,and convergence of the precise time step integration method.The unconditional stability and convergence of the analytical solution scheme in handling transient heat transfer problems are theoretically proved,and a comprehensive error analysis is conducted.Furthermore,this scheme is applied to transient thermal fracture analysis of rocks.Numerical results show that for models with different crack configurations and highly time-dependent loading forms,the current scheme can obtain more accurate simulation results than traditional finite difference schemes and with higher computational efficiency.(4)A thermal degree of freedom inheritance strategy during quasi-static crack propagation under transient heat transfer conditions is proposed.By mapping the local approximation on the physical patch to the Gaussian integration points and temperature loading points when forming the thermal conduction matrix,the time-domain integration method achieves direct and accurate calculation of the temperature field for the next time step.Integrating the newly created highprecision spatial discretization scheme and high-precision time-domain integration scheme,a new spatiotemporal high-precision numerical manifold method is established.Results of relevant numerical examples demonstrate that the proposed method can achieve a win-win situation of solution accuracy and computational efficiency when analyzing rock thermal fracturing problems,providing more reliable and efficient numerical computational assistance for practical rock engineering problems such as deep geothermal exploitation and geological storage of high-level radioactive waste.更多还原