边界元法中高阶单元奇异积分的一个新正则化算法及其应用研究
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摘要
本文综述了边界元法中几乎奇异积分问题的国内外研究现状,目前边界元法关于线性单元几乎奇异积分问题算法较为成熟,但对高阶单元尤其是三维高阶单元几乎奇异积分计算缺乏一种通行、高效的解决方法,这妨碍了边界元法的工程应用。
文中首先对边界元法线性单元几乎奇异积分正则化算法思想作了简要回顾和总结,并将其应用于三维声场边界元分析。在此基础上对边界元法高阶单元几乎奇异积分进行系统研究,以二次单元为例,创立了一种计算高阶单元各类几乎奇异积分的半解析算法,包括弱、强和超几乎奇异积分。并且将本文建立的半解析算法应用于二维和三维位势及其薄体问题、二维弹性力学及其层合结构边界元法分析。全文主要创新点概括如下：
1.拓展了线性单元几乎奇异积分正则化算法在三维声场边界元分析中的应用。将三维声场基本解函数进行Taylor级数展开,分离出奇异积分和非奇异积分两个部分。采用线性单元正则化算法计算其中的奇异积分部分,从而解决了三维声场边界元法分析中的几乎奇异积分难题。声场问题算例表明,本文算法计算精度较常规边界元法显著提高,可以为基于近边界点声学参量准确计算为基础的各类声学分析提供重要的参考依据。本文基于基本解函数Taylor级数展开的正则化算法思想,也为基本解为非多项式形式的边界元几乎奇异积分正则化提供了解决思路,拓宽了线性单元正则化算法的应用领域。
2.创立了二维位势问题边界元法高阶单元几乎奇异积分的一个新的正则化算法。本文分析了边界积分方程高阶单元中几乎奇异积分的原因,不失一般性,以二维问题3节点二次单元为例剖析了二次单元的几何特征,定义了源点到高阶曲线单元的接近度概念,分离出二维位势积分方程积分式中近似核函数。对积分核应用扣除法(Subtraction)技巧,通过扣除与积分核具有同样奇异性的近似积分核来消除几乎奇异性,建立了一个新的半解析算法,成功地计算出接近度为10-14的几乎强奇异积分和接近度为10-7的几乎超奇异积分。该算法应用于二维位势和薄体问题分析,结果表明本文算法可以计算距离边界非常近的内点位势和位势导数,并具有很高的计算精度。
3.针对3节点二次曲线单元,将二维位势问题的半解析正则化算法思想应用于二维弹性力学边界元分析,通过局部坐标变换,分离出二维弹性力学积分方程积分项中的近似奇异核函数,再采用扣除法消除了几乎强奇异和几乎超奇异性并推导出几乎奇异积分部分的解析计算公式,建立了弹性力学边界元法几乎强奇异和几乎超奇异积分的半解析算法。本文将半解析算法同多域边界元法联合应用,成功地求解了弹性力学薄体和层合结构的近边界内点位移和应力,算例结果表明边界元法高阶单元比线性单元以及有限元法更具有优势。
4.创立了三维位势问题边界元法高阶单元几乎奇异积分的半解析正则化算法。以8节点四边形二次曲面单元为例,分别在整体坐标系、局部直角坐标系和局部极坐标系下剖析单元的几何特征,提出了源点到高阶曲面单元的接近度概念。分离出三维位势问题基本解中几乎奇异积分核函数的近似函数,然后通过坐标变换使其近似函数面积分中的两个积分变量分离,从而可以依次单独计算积分。以此为基础,对积分核应用扣除法技巧,把几乎奇异面积分转化为非奇异积分和奇异积分两项之和,其中非奇异积分项用常规Gauss数值积分计算,而奇异积分项在局部极坐标系下推导出对极变量积分的解析计算公式,对角变量积分用常规Gauss数值积分计算。本文半解析算法应用于三维位势问题及其薄体问题边界元分析,成功地计算出其中的几乎强奇异和几乎超奇异积分。
本文半解析算法技术同样适用于其他高阶边界单元几乎奇异积分的计算,从而一般性地解决了二维和三维边界元法中高阶单元几乎强奇异和几乎超奇异积分的计算难题。该方法被成功应用于边界元法位势问题和弹性力学问题分析,使得边界元法在二维和三维薄域(薄体)问题计算方面比有限元法更具有优势。
The present status of the researches for evaluation of the nearly singular integrals in boundary element methods (BEM) is investigated and reviewed. Up to now, some regularization algorithms about the nearly singular integrals on the linear elements of BEM have been established successfully. However, the calculation of the nearly singular integrals on the high-order elements, especially in three-dimensional BEM (3-D BEM), is still a very difficult problem, which has been handicapping the applications of BEM in engineering.
The regularization algorithm of the nearly singular integrals on the linear elements in BEM is introduced firstly. In this thesis, the algorithm is developed to analyzing three-dimensional (3-D) acoustics problem by BEM. Then, By means of the idea of the regularization algorithm for the linear element, a kind of novel semi-analytical algorithms for the high-order elements are proposed to calculate the nearly strongly singular and hyper-singular integrals in2-D and3-D BEM, where the quadratic element is taken as a sample. And then, the present semi-analytical algorithms are performed to deal with the nearly singular integrals in the BE analysis with the high-order elements for two-dimensional (2-D) and3-D potentials and their thin domain problems,2-D elasticity and its thin-walled structures. The main contributions in the thesis can be shown in the following.
1. The regularization algorithm of the nearly singular integrals on the linear element is developed to the BE analysis of3-D acoustics. In the present thesis, the fundamental solutions of3-D acoustics are expanded as the Taylor series so that the boundary integral expressions are separated as both the non-singular integral parts and the singular integral parts where the later is equivalent to the lead singular term of the fundamental solutions. Consequently, the regularization algorithm is applied to calculating the nearly strongly singular and hyper-singular integrals in the BE analysis of3-D acoustics. Some examples are shown that the present computed results are more accurate than ones of the conventional BEM. It is noted that the accurate physical quantities at the inner pointes which are very close to the boundary are valuable for the applications of acoustic analysis in engineering. As an idea, the present technique of the Taylor series expansion with respect to some functions in the fundamental solution can be generalized to solving the nearly singular integrals in other boundary integral equations where explicit forms of their fundamental solutions are not rational functions.
2. A novel semi-analytical algorithm is proposed to deal with the nearly singular integrals on high-order elements in2-D BEM. In the present thesis, by the geometrical analysis for the high-order element, the relative distance from a source point to the integral element is defined as approach degree (denoted as e1) which can measure possible influence of the singularity of the integrals, where3-node quadratic curve element is taken as a sample without lost generality. Then the equivalent singular functions are separated from the integral kernels about the high-order elements in2-D potential boundary integral equation by means of an asymptotic analysis for the nearly singular integrals with respect to the local coordinate on the element. And the subtract ion strategy is applied to removing the singularities of the integral expressions. Therefore, the new semi-analytical algorithm is established, which can accurately evaluate the nearly strongly singular integrals within the range of e1>10-14and the nearly hyper-singular integrals within the range of e1>10-7. The semi-analytical algorithm has been successfully applied to the BE analysis of2-D potential and its thin domain problems, which can also obtain the inner potentials and fluxes close to the boundary.
3. The idea of the above semi-analytical algorithm is expanded to the BE analysis of2-D elasticity. The same manipulation as the above is done by taking3-node quadratic curve element as a sample. In the discrete boundary integral equation with the high-order elements, the equivalent singular functions are separated from the integral kernel functions by the local coordinate system transformation. Then the nearly strongly singularity and hyper-singularity on the integral elements are removed by the use of the subtraction technique. The formulations of the semi-analytical algorithm for computing the nearly strongly singular and hyper-singular integrals are obtained in terms of tedious manipulation. The semi-analytical algorithm has been successfully employed in the multi-domain BE analysis of2-D elasticity to solving the inner displacements and stresses close to the boundary. Several benchmark numerical examples demonstrate that the present regularized algorithm about the high-order elements is more accurate and efficient than one about the linear elements in BEM as well as finite element method for analyzing the thin-walled structures and laminate structures.
4. Another new semi-analytical algorithm is proposed to deal with the nearly singular integrals on high-order elements in3-D BEM. Through the geometrical analysis for the high-order curve surface element under the local Cartesian coordinate and polar coordinate systems, the relative distance from a source point to the integral element is defined as approach degree which is a main factor led to the nearly singular surface integrals, where8-node quadrilateral curve surface element is taken as a sample without lost generality. The approximate singular parts of the fundamental solutions in3-D potential boundary integral equations are separated from the integral kernel functions by means of the transformations of two local coordinate systems and the asymptotic analysis for the nearly singular integrals in the local polar coordinate. Consequently the nearly strongly singularity and hyper-singularity on the integral surface elements are eliminated by subtracting the approximate singular parts from the integral kernels instead of the direct numerical quadrature. The singular surface integrals related to the equivalent singular functions can be separately calculated about two integral variables in turn. It follows that the integral with respect to polar variable p is firstly represented with the analytical formulations and then the leading integral with respect to variable θ is numerically calculated with the conventional Gauss method, which is named as the semi-analytical algorithm by the thesis for evaluating the nearly singular integrals in3-D BEM. The present semi-analytical approach has been applied to the BE analysis of3-D potential and its thin domain problems. Both the nearly strongly and hyper-singular integrals are computed accurately.
It is noted that the proposed semi-analytical algorithms can be developed, without any difficulty, to calculate the nearly strongly and hyper-singular integrals on other high-order elements in2-D and3-D BEM, which indicates that the puzzle of the evaluations of the nearly strongly and hyper-singular integrals in BEM has been solved well in the present thesis. The present algorithms have been successfully employed to the BE analysis of potential and elasticity with the high-order elements, which makes the BEM have more advantage than FEM in analyzing2-D and3-D thin body problems.
文中首先对边界元法线性单元几乎奇异积分正则化算法思想作了简要回顾和总结,并将其应用于三维声场边界元分析。在此基础上对边界元法高阶单元几乎奇异积分进行系统研究,以二次单元为例,创立了一种计算高阶单元各类几乎奇异积分的半解析算法,包括弱、强和超几乎奇异积分。并且将本文建立的半解析算法应用于二维和三维位势及其薄体问题、二维弹性力学及其层合结构边界元法分析。全文主要创新点概括如下：
1.拓展了线性单元几乎奇异积分正则化算法在三维声场边界元分析中的应用。将三维声场基本解函数进行Taylor级数展开,分离出奇异积分和非奇异积分两个部分。采用线性单元正则化算法计算其中的奇异积分部分,从而解决了三维声场边界元法分析中的几乎奇异积分难题。声场问题算例表明,本文算法计算精度较常规边界元法显著提高,可以为基于近边界点声学参量准确计算为基础的各类声学分析提供重要的参考依据。本文基于基本解函数Taylor级数展开的正则化算法思想,也为基本解为非多项式形式的边界元几乎奇异积分正则化提供了解决思路,拓宽了线性单元正则化算法的应用领域。
2.创立了二维位势问题边界元法高阶单元几乎奇异积分的一个新的正则化算法。本文分析了边界积分方程高阶单元中几乎奇异积分的原因,不失一般性,以二维问题3节点二次单元为例剖析了二次单元的几何特征,定义了源点到高阶曲线单元的接近度概念,分离出二维位势积分方程积分式中近似核函数。对积分核应用扣除法(Subtraction)技巧,通过扣除与积分核具有同样奇异性的近似积分核来消除几乎奇异性,建立了一个新的半解析算法,成功地计算出接近度为10-14的几乎强奇异积分和接近度为10-7的几乎超奇异积分。该算法应用于二维位势和薄体问题分析,结果表明本文算法可以计算距离边界非常近的内点位势和位势导数,并具有很高的计算精度。
3.针对3节点二次曲线单元,将二维位势问题的半解析正则化算法思想应用于二维弹性力学边界元分析,通过局部坐标变换,分离出二维弹性力学积分方程积分项中的近似奇异核函数,再采用扣除法消除了几乎强奇异和几乎超奇异性并推导出几乎奇异积分部分的解析计算公式,建立了弹性力学边界元法几乎强奇异和几乎超奇异积分的半解析算法。本文将半解析算法同多域边界元法联合应用,成功地求解了弹性力学薄体和层合结构的近边界内点位移和应力,算例结果表明边界元法高阶单元比线性单元以及有限元法更具有优势。
4.创立了三维位势问题边界元法高阶单元几乎奇异积分的半解析正则化算法。以8节点四边形二次曲面单元为例,分别在整体坐标系、局部直角坐标系和局部极坐标系下剖析单元的几何特征,提出了源点到高阶曲面单元的接近度概念。分离出三维位势问题基本解中几乎奇异积分核函数的近似函数,然后通过坐标变换使其近似函数面积分中的两个积分变量分离,从而可以依次单独计算积分。以此为基础,对积分核应用扣除法技巧,把几乎奇异面积分转化为非奇异积分和奇异积分两项之和,其中非奇异积分项用常规Gauss数值积分计算,而奇异积分项在局部极坐标系下推导出对极变量积分的解析计算公式,对角变量积分用常规Gauss数值积分计算。本文半解析算法应用于三维位势问题及其薄体问题边界元分析,成功地计算出其中的几乎强奇异和几乎超奇异积分。
本文半解析算法技术同样适用于其他高阶边界单元几乎奇异积分的计算,从而一般性地解决了二维和三维边界元法中高阶单元几乎强奇异和几乎超奇异积分的计算难题。该方法被成功应用于边界元法位势问题和弹性力学问题分析,使得边界元法在二维和三维薄域(薄体)问题计算方面比有限元法更具有优势。
The present status of the researches for evaluation of the nearly singular integrals in boundary element methods (BEM) is investigated and reviewed. Up to now, some regularization algorithms about the nearly singular integrals on the linear elements of BEM have been established successfully. However, the calculation of the nearly singular integrals on the high-order elements, especially in three-dimensional BEM (3-D BEM), is still a very difficult problem, which has been handicapping the applications of BEM in engineering.
The regularization algorithm of the nearly singular integrals on the linear elements in BEM is introduced firstly. In this thesis, the algorithm is developed to analyzing three-dimensional (3-D) acoustics problem by BEM. Then, By means of the idea of the regularization algorithm for the linear element, a kind of novel semi-analytical algorithms for the high-order elements are proposed to calculate the nearly strongly singular and hyper-singular integrals in2-D and3-D BEM, where the quadratic element is taken as a sample. And then, the present semi-analytical algorithms are performed to deal with the nearly singular integrals in the BE analysis with the high-order elements for two-dimensional (2-D) and3-D potentials and their thin domain problems,2-D elasticity and its thin-walled structures. The main contributions in the thesis can be shown in the following.
1. The regularization algorithm of the nearly singular integrals on the linear element is developed to the BE analysis of3-D acoustics. In the present thesis, the fundamental solutions of3-D acoustics are expanded as the Taylor series so that the boundary integral expressions are separated as both the non-singular integral parts and the singular integral parts where the later is equivalent to the lead singular term of the fundamental solutions. Consequently, the regularization algorithm is applied to calculating the nearly strongly singular and hyper-singular integrals in the BE analysis of3-D acoustics. Some examples are shown that the present computed results are more accurate than ones of the conventional BEM. It is noted that the accurate physical quantities at the inner pointes which are very close to the boundary are valuable for the applications of acoustic analysis in engineering. As an idea, the present technique of the Taylor series expansion with respect to some functions in the fundamental solution can be generalized to solving the nearly singular integrals in other boundary integral equations where explicit forms of their fundamental solutions are not rational functions.
2. A novel semi-analytical algorithm is proposed to deal with the nearly singular integrals on high-order elements in2-D BEM. In the present thesis, by the geometrical analysis for the high-order element, the relative distance from a source point to the integral element is defined as approach degree (denoted as e1) which can measure possible influence of the singularity of the integrals, where3-node quadratic curve element is taken as a sample without lost generality. Then the equivalent singular functions are separated from the integral kernels about the high-order elements in2-D potential boundary integral equation by means of an asymptotic analysis for the nearly singular integrals with respect to the local coordinate on the element. And the subtract ion strategy is applied to removing the singularities of the integral expressions. Therefore, the new semi-analytical algorithm is established, which can accurately evaluate the nearly strongly singular integrals within the range of e1>10-14and the nearly hyper-singular integrals within the range of e1>10-7. The semi-analytical algorithm has been successfully applied to the BE analysis of2-D potential and its thin domain problems, which can also obtain the inner potentials and fluxes close to the boundary.
3. The idea of the above semi-analytical algorithm is expanded to the BE analysis of2-D elasticity. The same manipulation as the above is done by taking3-node quadratic curve element as a sample. In the discrete boundary integral equation with the high-order elements, the equivalent singular functions are separated from the integral kernel functions by the local coordinate system transformation. Then the nearly strongly singularity and hyper-singularity on the integral elements are removed by the use of the subtraction technique. The formulations of the semi-analytical algorithm for computing the nearly strongly singular and hyper-singular integrals are obtained in terms of tedious manipulation. The semi-analytical algorithm has been successfully employed in the multi-domain BE analysis of2-D elasticity to solving the inner displacements and stresses close to the boundary. Several benchmark numerical examples demonstrate that the present regularized algorithm about the high-order elements is more accurate and efficient than one about the linear elements in BEM as well as finite element method for analyzing the thin-walled structures and laminate structures.
4. Another new semi-analytical algorithm is proposed to deal with the nearly singular integrals on high-order elements in3-D BEM. Through the geometrical analysis for the high-order curve surface element under the local Cartesian coordinate and polar coordinate systems, the relative distance from a source point to the integral element is defined as approach degree which is a main factor led to the nearly singular surface integrals, where8-node quadrilateral curve surface element is taken as a sample without lost generality. The approximate singular parts of the fundamental solutions in3-D potential boundary integral equations are separated from the integral kernel functions by means of the transformations of two local coordinate systems and the asymptotic analysis for the nearly singular integrals in the local polar coordinate. Consequently the nearly strongly singularity and hyper-singularity on the integral surface elements are eliminated by subtracting the approximate singular parts from the integral kernels instead of the direct numerical quadrature. The singular surface integrals related to the equivalent singular functions can be separately calculated about two integral variables in turn. It follows that the integral with respect to polar variable p is firstly represented with the analytical formulations and then the leading integral with respect to variable θ is numerically calculated with the conventional Gauss method, which is named as the semi-analytical algorithm by the thesis for evaluating the nearly singular integrals in3-D BEM. The present semi-analytical approach has been applied to the BE analysis of3-D potential and its thin domain problems. Both the nearly strongly and hyper-singular integrals are computed accurately.
It is noted that the proposed semi-analytical algorithms can be developed, without any difficulty, to calculate the nearly strongly and hyper-singular integrals on other high-order elements in2-D and3-D BEM, which indicates that the puzzle of the evaluations of the nearly strongly and hyper-singular integrals in BEM has been solved well in the present thesis. The present algorithms have been successfully employed to the BE analysis of potential and elasticity with the high-order elements, which makes the BEM have more advantage than FEM in analyzing2-D and3-D thin body problems.
引文
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