等几何分析方法和比例边界等几何分析方法的研究及其工程应用
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摘要
本文基于等几何分析和比例边界有限元方法,研究开发了高精度、高效率的工程数值计算方法,将其应用于大坝-库水-地基系统的动力分析,并拓展至结构分析、电磁场分析和薄板弯曲和振动分析等工程问题的求解。
等几何分析是近年来发展的一种新型的数值方法,旨在实现CAD和CAE的无缝统一。该方法将CAD中对几何形状进行精确描述的样条函数作为结构分析的形函数,可大大提高函数梯度场的计算精度,并在保持几何形状不变的前提下,方便地实现自适应的非通讯性细分操作。比例边界有限元方法是一种求解偏微分方程的半数值半解析解法,只需对求解区域的边界进行有限元离散,降低求解规模,但又无需基本解,特别适合于求解含有无限域、线弹性裂尖奇异应力场等工程问题。
本文将等几何分析方法和比例边界有限元方法相结合,提出了新型的比例边界等几何分析方法,相比于等几何分析或者比例边界有限元,比例边界等几何分析方法的计算精度和计算效率进一步提高。因此将这种方法推广应用于实际工程中,将大大提高求解精度和效率。另外,此方法能和等几何分析方法进行无缝连接,利用等几何分析和比例边界等几何分析的耦合方法,建立了大坝-库水-无限地基时域动力分析的计算模型,全面考虑了大坝-库水、大坝-无限地基动力相互作用对大坝地震响应的影响。本文还将等几何分析、比例边界有限元方法、比例边界等几何分析的应用领域进行拓展,并对这些方法在实际应用中所遇到的若干理论和数值问题进行了深入研究,并提出了有效的解决方法。主要内容如下：
(1)基于等几何分析和比例边界有限方法,提出了比例边界等几何分析方法。对弹性静、动力学以及电磁场问题分别推导了其离散方程和求解格式,并进行总结和比较。h-细分和p-细分过程仅需针对结构环向表面边界,在该方向上变量在相邻单元交界面处可达到较高的连续阶,而在径向解具有解析特性。相比于其他的数值方法,可达到更高的收敛速度。针对该方法,还研究了各类边界条件的施加策略。
(2)提出了大坝-库水-地基系统动力分析的等几何分析方法-比例边界等几何分析方法时域耦合计算模型,全面考虑了大坝-库水、大坝-无限地基动力相互作用的影响,并将其应用到高拱坝的地震动力分析中,为大坝的抗震安全评价提供重要参考依据。其中在大坝-库水动力相互作用分析中,考虑了库水可压缩性、库底淤沙对压力波的吸收、无限水域的辐射阻尼效应,并构造了压力-力转换矩阵,提高了流固耦合计算效率。在大坝-无限地基动力相互作用分析中,采用求解加速度脉冲响应函数的高效稳定算法,并构造了高效的加速度脉冲响应函数的离散策略和自适应策略,大大提高了相互作用力的计算精度和效率。利用该模型,分析了大坝-库水-地基动力相互作用对重力坝、拱坝系统地震响应的影响。
(3)针对比例边界有限元、等几何分析、比例边界等几何分析在新的应用领域拓展过程中遇到的若干理论和数值问题,提出了有效的处理方法。构造了新型的NURBS曲面的裁剪交点搜索算法及单元局部重构方式。相对于传统的以全域单元为搜索对象的搜索策略,新型的搜索策略则沿着裁剪曲线的切线方向逐一搜索,提高了搜索效率。提出了Lagrange乘子法解决重控制点问题和非齐次边值问题。发展了等几何分析在电磁场分析中的应用,对静电场问题和波导本征问题进行求解,显著地提高了计算效率和计算精度,对电容和波导等电子元件的设计具有重要参考意义。发展了等几何分析在薄板弯曲与振动分析中的应用,无需引入转角自由度便可满足C’连续的要求,显著地减少了计算自由度,并对稳定参数的取值进行了讨论。提出了相似中心由固定型扩展为移动型的处理方法,为比例边界有限元的进一步拓展,提供了有利条件。
通过本论文的研究可看出,等几何分析方法、比例边界等几何分析方法在实际工程中具有广阔的应用前景,相关问题还需做更为广泛和深入的研究。
Based on the Isogeometric Analysis (IGA) and Scaled Boundary Finite Element Method (SBFEM), a new numerical method termed as the Scaled Boundary Isogeometric Analysis Method (SBIGA) is proposed. SBIGA features high precision and high efficiency. Based on in-depth investigation into various related aspects, this method has been applied to the dynamic analysis of dam-reservoir-foundation system, structure mechanics, electromagnetic analysis and bending and vibration analysis of thin plate.
IGA is a numerical method aiming to seamlessly integral CAD and CAE systems. Spline function employed in CAD geometry is inherited as function of computational analysis, which greatly improves accuracy of the gradient field of variables. And spline function can be very convenient for geometry refinement. SBFEM is a semi-analytical method for solving partial differential equations, in which the problem is reduced by one with only boundary discredited and the fundamental solution is unnecessary. It is superior to solve infinite domain and stress singularity problems containing crack tip.
The newly proposed method SBIGA combines the advantages of IGA and SBFEM, which further improve the accuracy and efficiency beyond IGA and SBFEM. Thus, it is suitable to be applied to the practical engineering structure, where the accuracy and efficiency will be greatly improved. SBIGA can be seamlessly connected with IGA. Thus, an efficient simulation method is established for dam-reservoir-foundation system by coupling IGA and SBIGA, in which both structure-soil interaction and fluid-structure interaction are taken into consideration. IGA, SBFEM and SBIGA are extended to solve problems in new areas, including structure mechanics, electromagnetism, and elastic thin plate problems.
In the process of research, the innovative achievements are as follows:
(1) SBIGA was proposed by combining advantages of IGA and SBFEM. The discretized equation and solution procedure were derived for both elastic and dynamic problem for mechanics and electromagnetism, respectively. Both h-refinement and p-refinement can be easy implemented only one boundary and more flexible continuity can be reach in the circumferential direction while analytical property was hold in the radial direction. Faster ratio of convergence can be reached by SBIGA rather than other methods. The applying strategies for various types of boundary conditions in SBIGA were studied in detail.
(2) An efficient simulation method was established for dam-reservoir-foundation system by coupling IGA and SBIGA, in which both structure-soil interaction and fluid-structure interaction are taken into consideration. In fluid-structure interaction, water compressibility, wave reflection at bottom of reservoir and radiation damping effect of infinite reservoir were efficiently portrayed, and pressure-force transformation matrix was constructed to leave out redundant operations. In soil-structure interaction, a new discretization strategy for acceleration unit-impulse response function was established, and the adaptive strategy was proposed to determine the series of acceleration unit-impulse response function. The coupled method was applied to seismic response analysis of high arch dam aiming to provide an important reference for seismic safety evaluation of dam.
(3) Effective schemes were put forward for several issues encountered in the application of IGA, SBFEM and SBIGA in new areas. New search scheme of intersection point and local reconstruction for trimmed surface was proposed in IGA. Instead of element-based search scheme, the new scheme was oriented to trimmed curve and the interaction point was determined step by step along the tangent direction of the trimmed curve. Lagrange multiplier schemes were proposed for repeated control points and nonhomogeneous boundary value problems. IGA was expanded in electromagnetic field analysis, including problem of electrostatic field and waveguide eigenvalue. It significantly improved the computational efficiency and accuracy, which was important for design of electronic components, such as capacitors and waveguide. IGA was applied in the analysis of bending and vibration of thin plates by employing Nitsche functional. Degrees of freedom significantly were reduced due to eliminating the rotational degree of freedom, while C1continuity was also satisfied. The optimum stability parameters were discussed for bending and vibration, respectively. The concept of moving scaling center was proposed for eccentric domain, which further extended the scope of the method.
It's observed from the research work that IGA and SBIGA have good prospect and potential in engineering applications, and more extensive and in-depth study should be done.
等几何分析是近年来发展的一种新型的数值方法,旨在实现CAD和CAE的无缝统一。该方法将CAD中对几何形状进行精确描述的样条函数作为结构分析的形函数,可大大提高函数梯度场的计算精度,并在保持几何形状不变的前提下,方便地实现自适应的非通讯性细分操作。比例边界有限元方法是一种求解偏微分方程的半数值半解析解法,只需对求解区域的边界进行有限元离散,降低求解规模,但又无需基本解,特别适合于求解含有无限域、线弹性裂尖奇异应力场等工程问题。
本文将等几何分析方法和比例边界有限元方法相结合,提出了新型的比例边界等几何分析方法,相比于等几何分析或者比例边界有限元,比例边界等几何分析方法的计算精度和计算效率进一步提高。因此将这种方法推广应用于实际工程中,将大大提高求解精度和效率。另外,此方法能和等几何分析方法进行无缝连接,利用等几何分析和比例边界等几何分析的耦合方法,建立了大坝-库水-无限地基时域动力分析的计算模型,全面考虑了大坝-库水、大坝-无限地基动力相互作用对大坝地震响应的影响。本文还将等几何分析、比例边界有限元方法、比例边界等几何分析的应用领域进行拓展,并对这些方法在实际应用中所遇到的若干理论和数值问题进行了深入研究,并提出了有效的解决方法。主要内容如下：
(1)基于等几何分析和比例边界有限方法,提出了比例边界等几何分析方法。对弹性静、动力学以及电磁场问题分别推导了其离散方程和求解格式,并进行总结和比较。h-细分和p-细分过程仅需针对结构环向表面边界,在该方向上变量在相邻单元交界面处可达到较高的连续阶,而在径向解具有解析特性。相比于其他的数值方法,可达到更高的收敛速度。针对该方法,还研究了各类边界条件的施加策略。
(2)提出了大坝-库水-地基系统动力分析的等几何分析方法-比例边界等几何分析方法时域耦合计算模型,全面考虑了大坝-库水、大坝-无限地基动力相互作用的影响,并将其应用到高拱坝的地震动力分析中,为大坝的抗震安全评价提供重要参考依据。其中在大坝-库水动力相互作用分析中,考虑了库水可压缩性、库底淤沙对压力波的吸收、无限水域的辐射阻尼效应,并构造了压力-力转换矩阵,提高了流固耦合计算效率。在大坝-无限地基动力相互作用分析中,采用求解加速度脉冲响应函数的高效稳定算法,并构造了高效的加速度脉冲响应函数的离散策略和自适应策略,大大提高了相互作用力的计算精度和效率。利用该模型,分析了大坝-库水-地基动力相互作用对重力坝、拱坝系统地震响应的影响。
(3)针对比例边界有限元、等几何分析、比例边界等几何分析在新的应用领域拓展过程中遇到的若干理论和数值问题,提出了有效的处理方法。构造了新型的NURBS曲面的裁剪交点搜索算法及单元局部重构方式。相对于传统的以全域单元为搜索对象的搜索策略,新型的搜索策略则沿着裁剪曲线的切线方向逐一搜索,提高了搜索效率。提出了Lagrange乘子法解决重控制点问题和非齐次边值问题。发展了等几何分析在电磁场分析中的应用,对静电场问题和波导本征问题进行求解,显著地提高了计算效率和计算精度,对电容和波导等电子元件的设计具有重要参考意义。发展了等几何分析在薄板弯曲与振动分析中的应用,无需引入转角自由度便可满足C’连续的要求,显著地减少了计算自由度,并对稳定参数的取值进行了讨论。提出了相似中心由固定型扩展为移动型的处理方法,为比例边界有限元的进一步拓展,提供了有利条件。
通过本论文的研究可看出,等几何分析方法、比例边界等几何分析方法在实际工程中具有广阔的应用前景,相关问题还需做更为广泛和深入的研究。
Based on the Isogeometric Analysis (IGA) and Scaled Boundary Finite Element Method (SBFEM), a new numerical method termed as the Scaled Boundary Isogeometric Analysis Method (SBIGA) is proposed. SBIGA features high precision and high efficiency. Based on in-depth investigation into various related aspects, this method has been applied to the dynamic analysis of dam-reservoir-foundation system, structure mechanics, electromagnetic analysis and bending and vibration analysis of thin plate.
IGA is a numerical method aiming to seamlessly integral CAD and CAE systems. Spline function employed in CAD geometry is inherited as function of computational analysis, which greatly improves accuracy of the gradient field of variables. And spline function can be very convenient for geometry refinement. SBFEM is a semi-analytical method for solving partial differential equations, in which the problem is reduced by one with only boundary discredited and the fundamental solution is unnecessary. It is superior to solve infinite domain and stress singularity problems containing crack tip.
The newly proposed method SBIGA combines the advantages of IGA and SBFEM, which further improve the accuracy and efficiency beyond IGA and SBFEM. Thus, it is suitable to be applied to the practical engineering structure, where the accuracy and efficiency will be greatly improved. SBIGA can be seamlessly connected with IGA. Thus, an efficient simulation method is established for dam-reservoir-foundation system by coupling IGA and SBIGA, in which both structure-soil interaction and fluid-structure interaction are taken into consideration. IGA, SBFEM and SBIGA are extended to solve problems in new areas, including structure mechanics, electromagnetism, and elastic thin plate problems.
In the process of research, the innovative achievements are as follows:
(1) SBIGA was proposed by combining advantages of IGA and SBFEM. The discretized equation and solution procedure were derived for both elastic and dynamic problem for mechanics and electromagnetism, respectively. Both h-refinement and p-refinement can be easy implemented only one boundary and more flexible continuity can be reach in the circumferential direction while analytical property was hold in the radial direction. Faster ratio of convergence can be reached by SBIGA rather than other methods. The applying strategies for various types of boundary conditions in SBIGA were studied in detail.
(2) An efficient simulation method was established for dam-reservoir-foundation system by coupling IGA and SBIGA, in which both structure-soil interaction and fluid-structure interaction are taken into consideration. In fluid-structure interaction, water compressibility, wave reflection at bottom of reservoir and radiation damping effect of infinite reservoir were efficiently portrayed, and pressure-force transformation matrix was constructed to leave out redundant operations. In soil-structure interaction, a new discretization strategy for acceleration unit-impulse response function was established, and the adaptive strategy was proposed to determine the series of acceleration unit-impulse response function. The coupled method was applied to seismic response analysis of high arch dam aiming to provide an important reference for seismic safety evaluation of dam.
(3) Effective schemes were put forward for several issues encountered in the application of IGA, SBFEM and SBIGA in new areas. New search scheme of intersection point and local reconstruction for trimmed surface was proposed in IGA. Instead of element-based search scheme, the new scheme was oriented to trimmed curve and the interaction point was determined step by step along the tangent direction of the trimmed curve. Lagrange multiplier schemes were proposed for repeated control points and nonhomogeneous boundary value problems. IGA was expanded in electromagnetic field analysis, including problem of electrostatic field and waveguide eigenvalue. It significantly improved the computational efficiency and accuracy, which was important for design of electronic components, such as capacitors and waveguide. IGA was applied in the analysis of bending and vibration of thin plates by employing Nitsche functional. Degrees of freedom significantly were reduced due to eliminating the rotational degree of freedom, while C1continuity was also satisfied. The optimum stability parameters were discussed for bending and vibration, respectively. The concept of moving scaling center was proposed for eccentric domain, which further extended the scope of the method.
It's observed from the research work that IGA and SBIGA have good prospect and potential in engineering applications, and more extensive and in-depth study should be done.
引文
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