不确定性结构的无网格伽辽金法研究
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摘要
随着计算机技术和数值计算方法的发展,科学计算在科学与工程中的作用逐渐提升,已由辅助性工具转变为与理论、实验并重的科研手段之一。在工程结构分析和设计模型中,通常建立在确定性物理意义上,即把分析过程中各种因素作为确定的物理量来进行处理。而在工程结构的实际分析中存在着与初始条件、几何参数、材料性质、边界条件、加工装配以及外力载荷相关的误差或不确定性,如果在计算分析时将这些不确定参数简单地视为确定性参数来处理,则可能会得出不尽合理甚至错误的计算结果。随机有限元、区间有限元和模糊有限元等数值模拟算法已经广泛应用在工程结构不确定性分析中,但它们都是由有限元演变而来,受自身缺点的限制,在处理撞击、加工成型、裂纹扩展、材料相变,以及特大变形等问题时遇到了越来越大的困难。无网格法是一种很有发展前途的数值方法,它克服了对网格的依赖,彻底或部分消除了网格的划分,在处理以上问题时具有明显优势,因此受到了越来越多科学工作者的关注。
无网格法在最近十几年发展颇为迅速,在基础理论和工程应用方面均取得了相当大的进展,主要包括：算法的数学论证、计算效率、边界条件处理、快速稳定的域积分技术、有限元和无网格法的广义化,以及针对有限元求解困难甚至无法求解的难题的算法改进与应用等。这些研究成果都是基于确定性模型的,没有考虑工程结构中普遍存在的不确定性问题。如何将无网格法拓展到不确定性领域,是一个极具理论与现实意义的研究方向。
基于移动最小二乘近似的无网格伽辽金法容易形成病态方程组,计算复杂,从而限制了其使用。本文基于局部加权正交基函数,对其进行改进,提出了改进无网格伽辽金法。摄动理论、区间数学和模糊集合论等,为不确定性问题分析提供了强大的数学工具,无网格伽辽金法在处理某些问题时较有限元方法更具优势,将无网格伽辽金法与摄动理论、区间数学、模糊集合论相结合,对不确定性问题进行研究是一个有益的探索。基于此,本文提出了摄动随机局部正交无网格伽辽金法、区间局部正交无网格伽辽金法和模糊局部正交无网格伽辽金法,并成功应用到含不确定因素力学问题之中,进一步丰富和发展了无网格法处理不确定性结构力学问题的理论和方法。主要工作有：
1、建立了基于局部加权正交基函数的改进无网格伽辽金法,使得基函数及其导数的表达式简洁明了,并保持正交基函数的性质,求逆简单,避免矩阵产生奇异,易于编程实现,提高了计算效率。通过数值算例将改进无网格伽辽金法与传统无网格伽辽金法进行比较,验证了该方法的高效性。
2、通过对改进无网格伽辽金法和摄动法的研究,提出了摄动随机局部正交无网格伽辽金法,并对无网格伽辽金法的随机变分原理和离散方案进行了详细推导,采用罚函数法施加本质边界条件,对含随机参数的杆、梁及带圆孔方板进行了分析,结果表明该方法正确、可行、高效。
3、通过对区间数学的研究,结合改进无网格伽辽金法提出了区间局部正交无网格伽辽金法,并把裂纹尖端应力场的奇异解析函数引入到局部加权正交基函数中,利用区间数分解方法求解区间平衡方程,详细推导出区间J积分公式,对含裂纹结构的不确定性问题进行了分析,算例结果表明该方法对于含不确定参数问题的求解是正确有效的。
4、提出了模糊局部正交无网格伽辽金法,给出了模糊最小势能原理,推导出模糊局部正交无网格伽辽金法平衡方程,并可根据输入模糊数的隶属函数,给出结构响应量的可能性分布,利用模糊局部正交无网格伽辽金法求得节点模糊位移值后采用直接位移法计算模糊应力强度因子,提供了一种计算具有不确定参数含裂纹结构的裂纹尖端应力强度因子的新途径。
5、研究了不确定因素下的含界面裂纹结构的界面端奇异应力场问题,建立了含界面裂纹结构的区间局部正交无网格伽辽金法分析模型,使用增强基函数模拟裂纹尖端应力场的奇异性,采用衍射准则来处理裂纹的不连续性,提供了含界面裂纹结构的应力强度因子的无网格计算方法,为无网格伽辽金法开辟一个新的应用范畴,在变量的概率统计特性未知而其区间范围已知的条件下,有效地给出结构响应的区间范围,这些数据能够为不确定性结构分析和设计提供有益的参考。
With the developments of the computer technique and the numerical computing methods, the effects of numerical simulation in the science and engineering areas are enhanced gradually, the utility of which has been shifted from an assistant tool to an essential means of the scientific research, and plays an important role both in the theoretical and the experimental areas. Generally, we establish a engineering structural analysis model and the design model based on deterministic physical significance. That is, we deal with the factors as deterministic ones during analysis process. However, errors and uncertainties that caused by loads, the initial conditions, the boundary constraints, manufacture and assembling widely exist in the real engineering world. If these uncertain factors are roughly concerned as deterministic parameters, we may receive conflict or unreasonable results on occasion. Some numerical solution, such as the stochastic Finite Element (FE), the interval FE and the fuzzy FE, have been widely used in the uncertain analysis of the engineering structures. However, they are all evolved from the FEM, and they encounter difficulties in dealing with some problems caused by their own drawbacks, such as crashes, shaping, crack propagation, phase change in materials, and super-large deformation. Element-Free Method (EFM) is a promising method, which can overcome the dependency on meshes, and eliminate the mesh partition thoroughly or partly. It possesses an apparent advantage rather than FEM over the above problems, and attracts more and more scientists'interests.
The EFM has developed so fast in the past decades, many contributions have been made both in basic theoretical research and engineering application, which focused on algorithm proof, efficiency of calculation, boundary constraints application, quick and stable domain integration technique, generalization of FEM and EFM, and improvement and application that overcome the challenge of FEM. But all these results were based on deterministic models; they did not consider the uncertainties which exited comprehensively in engineering structures. Hereby, how to extend the EFM to uncertain problems becomes a significant task.
The traditional Element Free Galerkin method (EFGM) based on moving least square approximate may lead to ill conditioned system of equations, and will limit its application as a result. Based on the improved local weighted orthogonal basis function, an improved element-free Galerkin method was proposed in this paper, and named as the Local Orthogonal Element Free Galerkin method (LOEFGM). Perturbation theorem, interval mathematics and fuzzy set theorem provide powerful mathematical tools for analysis of uncertain problems. EFGM takes apparent advantage than FEM when dealing with some special problems. It will be a valuable orientation if combine the EFGM with perturbation theorem, interval mathematical, and fuzzy set theorem to solve uncertain problems. We hereby proposed the Perturbation Stochastic Local Orthogonal Element Free Galerkin method (PSLOEFGM), the Interval Local Orthogonal Element Free Galerkin method (ILOEFGM), and the Fuzzy Local Orthogonal Element Free Galerkin method (FLOEFGM). And we also applied the methods mentioned above to solve uncertain problems successfully. This will enrich and develop the EFM theory and method in dealing with structral mechanic problems with uncertainties.
The main achievements of this dissertation are:
Based on local weighted orthogonal basis functions, the improved Element Free Galerkin method so called the Local Orthogonal Element Free Galerkin method (LOEFGM) was established. LOEFGM can also preserve the character of the orthogonal basis function, but it makes the formulation simplify, and what's more, it is easy to calculate the inverse matrix and can avoid the singularity of matrix, thus, it takes advantage of programming and promotes the computation efficiency. The numeral example was taken to compare LOEFGM with the traditional EFGM, the results showed that LOEFGM was more efficient than the traditional EFGM.
By studying the improved EFGM and the perturbation method, the perturbation stochastic local orthogonal EFGM was proposed, and the random variational principle and the discrete scheme of the EFM were deduced in detail. The natural boundary conditions were added by using the penalty function. The numerical examples of bar, the beam and the square plate with circular hole, which include random parameters were studied, respectively. The results showed that the proposed method is correct, feasible and efficient.
By studying the interval mathematics, combined with the inner product space and the improved EFGM, the interval local orthogonal EFGM was proposed. The singular function of the linear elastic fracture mechanics was added as an enhance function into the local weighted orthogonal basis function of the moving least square approximation. The interval number decomposition method was used to solve the interval equilibrium equations. The interval J-integral formula was deduced in detail. The uncertain problem of the cracked structure was analyzed. From the results of the example, it can be seen that the interval local orthogonal EFGM is correct and effective in solving the uncertain problems.
The fuzzy local orthogonal EFGM was proposed and the fuzzy minimum potential energy principle was given. Then, the fuzzy local orthogonal element-free equilibrium equations were deduced. According to the membership function of the given fuzzy numbers, the possibility distributions of the structural response quantities can be obtained; the fuzzy local orthogonal EFGM was used to obtain the fuzzy displacements of the nodes, and then the direct displacement method was used to compute the fuzzy stress intensity factor. The method discussed here provides a new pathway of computing the stress intensity factor in the crack tip of the cracked structures which including uncertain parameters.
The research was carried out which focused on the interface edge singular stress field of the structure with interface crack under uncertainties. The Interval Local Orthogonal Element Free Galerkin analysis model was established on the structure with interface crack. The singularity of stress field of the crack tip was simulated with enriched basis functions, and the discontinuousness of crack was dealed with diffraction method. This provided an EFM for the calculation of stress intensity factor of structure with interface crack, and also provided a new application category of EFGM. The proposed method can present the response interval of structures effectively, even if the probability statistics is unknown whereas the range of the interval is given. This datum provides a valuable reference for uncertain structure analysis and design.
无网格法在最近十几年发展颇为迅速,在基础理论和工程应用方面均取得了相当大的进展,主要包括：算法的数学论证、计算效率、边界条件处理、快速稳定的域积分技术、有限元和无网格法的广义化,以及针对有限元求解困难甚至无法求解的难题的算法改进与应用等。这些研究成果都是基于确定性模型的,没有考虑工程结构中普遍存在的不确定性问题。如何将无网格法拓展到不确定性领域,是一个极具理论与现实意义的研究方向。
基于移动最小二乘近似的无网格伽辽金法容易形成病态方程组,计算复杂,从而限制了其使用。本文基于局部加权正交基函数,对其进行改进,提出了改进无网格伽辽金法。摄动理论、区间数学和模糊集合论等,为不确定性问题分析提供了强大的数学工具,无网格伽辽金法在处理某些问题时较有限元方法更具优势,将无网格伽辽金法与摄动理论、区间数学、模糊集合论相结合,对不确定性问题进行研究是一个有益的探索。基于此,本文提出了摄动随机局部正交无网格伽辽金法、区间局部正交无网格伽辽金法和模糊局部正交无网格伽辽金法,并成功应用到含不确定因素力学问题之中,进一步丰富和发展了无网格法处理不确定性结构力学问题的理论和方法。主要工作有：
1、建立了基于局部加权正交基函数的改进无网格伽辽金法,使得基函数及其导数的表达式简洁明了,并保持正交基函数的性质,求逆简单,避免矩阵产生奇异,易于编程实现,提高了计算效率。通过数值算例将改进无网格伽辽金法与传统无网格伽辽金法进行比较,验证了该方法的高效性。
2、通过对改进无网格伽辽金法和摄动法的研究,提出了摄动随机局部正交无网格伽辽金法,并对无网格伽辽金法的随机变分原理和离散方案进行了详细推导,采用罚函数法施加本质边界条件,对含随机参数的杆、梁及带圆孔方板进行了分析,结果表明该方法正确、可行、高效。
3、通过对区间数学的研究,结合改进无网格伽辽金法提出了区间局部正交无网格伽辽金法,并把裂纹尖端应力场的奇异解析函数引入到局部加权正交基函数中,利用区间数分解方法求解区间平衡方程,详细推导出区间J积分公式,对含裂纹结构的不确定性问题进行了分析,算例结果表明该方法对于含不确定参数问题的求解是正确有效的。
4、提出了模糊局部正交无网格伽辽金法,给出了模糊最小势能原理,推导出模糊局部正交无网格伽辽金法平衡方程,并可根据输入模糊数的隶属函数,给出结构响应量的可能性分布,利用模糊局部正交无网格伽辽金法求得节点模糊位移值后采用直接位移法计算模糊应力强度因子,提供了一种计算具有不确定参数含裂纹结构的裂纹尖端应力强度因子的新途径。
5、研究了不确定因素下的含界面裂纹结构的界面端奇异应力场问题,建立了含界面裂纹结构的区间局部正交无网格伽辽金法分析模型,使用增强基函数模拟裂纹尖端应力场的奇异性,采用衍射准则来处理裂纹的不连续性,提供了含界面裂纹结构的应力强度因子的无网格计算方法,为无网格伽辽金法开辟一个新的应用范畴,在变量的概率统计特性未知而其区间范围已知的条件下,有效地给出结构响应的区间范围,这些数据能够为不确定性结构分析和设计提供有益的参考。
With the developments of the computer technique and the numerical computing methods, the effects of numerical simulation in the science and engineering areas are enhanced gradually, the utility of which has been shifted from an assistant tool to an essential means of the scientific research, and plays an important role both in the theoretical and the experimental areas. Generally, we establish a engineering structural analysis model and the design model based on deterministic physical significance. That is, we deal with the factors as deterministic ones during analysis process. However, errors and uncertainties that caused by loads, the initial conditions, the boundary constraints, manufacture and assembling widely exist in the real engineering world. If these uncertain factors are roughly concerned as deterministic parameters, we may receive conflict or unreasonable results on occasion. Some numerical solution, such as the stochastic Finite Element (FE), the interval FE and the fuzzy FE, have been widely used in the uncertain analysis of the engineering structures. However, they are all evolved from the FEM, and they encounter difficulties in dealing with some problems caused by their own drawbacks, such as crashes, shaping, crack propagation, phase change in materials, and super-large deformation. Element-Free Method (EFM) is a promising method, which can overcome the dependency on meshes, and eliminate the mesh partition thoroughly or partly. It possesses an apparent advantage rather than FEM over the above problems, and attracts more and more scientists'interests.
The EFM has developed so fast in the past decades, many contributions have been made both in basic theoretical research and engineering application, which focused on algorithm proof, efficiency of calculation, boundary constraints application, quick and stable domain integration technique, generalization of FEM and EFM, and improvement and application that overcome the challenge of FEM. But all these results were based on deterministic models; they did not consider the uncertainties which exited comprehensively in engineering structures. Hereby, how to extend the EFM to uncertain problems becomes a significant task.
The traditional Element Free Galerkin method (EFGM) based on moving least square approximate may lead to ill conditioned system of equations, and will limit its application as a result. Based on the improved local weighted orthogonal basis function, an improved element-free Galerkin method was proposed in this paper, and named as the Local Orthogonal Element Free Galerkin method (LOEFGM). Perturbation theorem, interval mathematics and fuzzy set theorem provide powerful mathematical tools for analysis of uncertain problems. EFGM takes apparent advantage than FEM when dealing with some special problems. It will be a valuable orientation if combine the EFGM with perturbation theorem, interval mathematical, and fuzzy set theorem to solve uncertain problems. We hereby proposed the Perturbation Stochastic Local Orthogonal Element Free Galerkin method (PSLOEFGM), the Interval Local Orthogonal Element Free Galerkin method (ILOEFGM), and the Fuzzy Local Orthogonal Element Free Galerkin method (FLOEFGM). And we also applied the methods mentioned above to solve uncertain problems successfully. This will enrich and develop the EFM theory and method in dealing with structral mechanic problems with uncertainties.
The main achievements of this dissertation are:
Based on local weighted orthogonal basis functions, the improved Element Free Galerkin method so called the Local Orthogonal Element Free Galerkin method (LOEFGM) was established. LOEFGM can also preserve the character of the orthogonal basis function, but it makes the formulation simplify, and what's more, it is easy to calculate the inverse matrix and can avoid the singularity of matrix, thus, it takes advantage of programming and promotes the computation efficiency. The numeral example was taken to compare LOEFGM with the traditional EFGM, the results showed that LOEFGM was more efficient than the traditional EFGM.
By studying the improved EFGM and the perturbation method, the perturbation stochastic local orthogonal EFGM was proposed, and the random variational principle and the discrete scheme of the EFM were deduced in detail. The natural boundary conditions were added by using the penalty function. The numerical examples of bar, the beam and the square plate with circular hole, which include random parameters were studied, respectively. The results showed that the proposed method is correct, feasible and efficient.
By studying the interval mathematics, combined with the inner product space and the improved EFGM, the interval local orthogonal EFGM was proposed. The singular function of the linear elastic fracture mechanics was added as an enhance function into the local weighted orthogonal basis function of the moving least square approximation. The interval number decomposition method was used to solve the interval equilibrium equations. The interval J-integral formula was deduced in detail. The uncertain problem of the cracked structure was analyzed. From the results of the example, it can be seen that the interval local orthogonal EFGM is correct and effective in solving the uncertain problems.
The fuzzy local orthogonal EFGM was proposed and the fuzzy minimum potential energy principle was given. Then, the fuzzy local orthogonal element-free equilibrium equations were deduced. According to the membership function of the given fuzzy numbers, the possibility distributions of the structural response quantities can be obtained; the fuzzy local orthogonal EFGM was used to obtain the fuzzy displacements of the nodes, and then the direct displacement method was used to compute the fuzzy stress intensity factor. The method discussed here provides a new pathway of computing the stress intensity factor in the crack tip of the cracked structures which including uncertain parameters.
The research was carried out which focused on the interface edge singular stress field of the structure with interface crack under uncertainties. The Interval Local Orthogonal Element Free Galerkin analysis model was established on the structure with interface crack. The singularity of stress field of the crack tip was simulated with enriched basis functions, and the discontinuousness of crack was dealed with diffraction method. This provided an EFM for the calculation of stress intensity factor of structure with interface crack, and also provided a new application category of EFGM. The proposed method can present the response interval of structures effectively, even if the probability statistics is unknown whereas the range of the interval is given. This datum provides a valuable reference for uncertain structure analysis and design.
引文
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