摘要
比例边界有限元法(SBFEM)是近年来提出和发展的一种半解析数值方法,它结合了有限元法和边界元法的优点,只需对计算域边界进行数值离散,减少了一个空间维数;在没有离散的径向方向利用解析的方法求解,具有较高的计算精度。运用SBFEM进行计算时,不需要基本解,不存在奇异积分问题;而对于无限域问题,不需要人为引入截断边界,能够自动满足无穷远处的边界条件。SBFEM已成功地应用于固体力学领域,最近又求解了一些流体力学问题,有着很大的应用空间。
SBFEM求解Poisson方程边值问题时,不存在体积分,而且可以精确处理无限域问题,与有限元法和边界元法相比,具有很大的优势。但是,目前该方法对Poisson方程右端项有特殊的要求,限制了其在Poisson方程边值问题上的应用范围。此外,SBFEM目前只能求解线性问题,依然处于发展之中,需要进一步研究。本论文先后引进了切比雪夫多项式逼近和同伦分析方法,对SBFEM进行了改进和发展。
首先,引进切比雪夫多项式逼近,对SBFEM进行了改进,同时还提出了一种更为高效的求解技术,并运用改进后的方法求解若干Poisson方程边值问题。计算结果表明改进后的方法同样保持了高精度的优点,而且大大拓宽了SBFEM所能求解的Poisson方程边值问题的范围,具有广泛的应用前景。
其次,引进一种求解强非线性问题的解析方法――同伦分析方法,并与改进后的SBFEM相结合,提出了一种新的求解非线性边值问题的半解析数值方法――基于同伦的SBFEM。该方法既保持了半解析数值方法的优点,又可以将SBFEM应用于求解非线性问题。本论文运用该新方法求解了一个二维Poisson型非线性边值问题,验证了该方法求解非线性边值问题的可行性和有效性。
快速多极子边界元法(FMBEM)是最近发展起来的一种能够快速计算的数值方法,它克服了传统边界元法计算效率低下、存储量大的缺点,适合于求解大规模问题。本论文还运用该方法求解了海洋工程中圆柱体波浪绕射问题和水下三维复杂结构物附加质量计算问题,充分展现了该方法高效、低存储以及高精度的特性,证明了FMBEM在求解海洋工程中超大规模数值问题中具有巨大的潜力。
本论文的主要创新点为:
(1)首次运用切比雪夫多项式逼近,对SBFEM进行了改进,使其能够求解具有复杂右端项的Poisson方程边值问题,扩大了SBFEM的解题范围。
(2)首次将同伦分析方法与SBFEM结合起来,提出了一种新的求解非线性问题的半解析数值方法――基于同伦的比例边界有限元法。该新方法克服了传统比例边界有限元法仅能求解线性问题的局限性,大大地拓宽了比例边界有限元法的应用领域,为求解非线性工程问题提供了一条有效途径。
(3)应用FMBEM成功求解了海洋工程中大规模势流问题,边界离散单元数目已高达十万,而所需的计算时间仅在一个小时以内,证明该方法在求解海洋工程大规模势流问题中具有广泛的应用前景。
The scaled boundary finite-element method (SBFEM) is a novel semi-analyticnumerical method, combining the advantages of the finite element and the boundaryelement methods. In SBFEM, only the boundary is spatially discretised, leading toa reduction of the spatial dimension by one. In the radial direction, the solutionis analytical, so the simulation precision of this method is high. The SBFEM doesnot need the fundamental solution, thus avoids the problem of singular integral. Foran unbounded domain problem, this method can meet the infinity of the boundarycondition automatically without introducing artificial boundary. In addition, whensolving the boundary-valued problems of Poisson equation, there is no volume integralinvolved. The analytical solution in the radial direction also permits the boundarycondition at infinity to be satisfied rigorously. The SBFEM has been successfullyapplied to solid mechanics, and recently extended to ?uid dynamics.
Although this method has many advantages over the finite element and boundaryelement methods, the SBFEM has special requirements to the right-hand side terms ofthe Poisson equation, limiting its applications to some certain types of Poisson equa-tions. In addition, the SBFEM is only valid for solving linear problems. By introduc-ing the Chebyshev polynomial approximation and homotopy analysis method(HAM),the SBFEM has been improved and developed in this dissertation.
Firstly, the Chebyshev polynomial approximation is introduced to improve theSBFEM. A more e?cient solution technique is proposed, and is applied to someboundary-valued problems of Poisson equation. The results show that the improvedmethod maintains high precision and high e?ciency. This work greatly expands thescope of the SBFEM in solving the Poisson equations.
Secondly, an analytical method for solving strongly nonlinear problems, namedhomotopy analysis method, is introduced and combined with the improved SBFEMsuccessfully. This new method is then applied to solve some 2D Poisson-type nonlinearboundary-valued problems. The feasibility and e?ectiveness of this new method isverified by numerical results. The new method not only maintains the advantages ofthe semi-analytical method, but also extends the SBFEM to nonlinear problems.
The fast multipole boundary element method (FMBEM) is a new fast algorithm,which overcomes the low-e?ciency, high-storage of the traditional boundary element method. It is suitable for solving large-scale problems. In this paper, the FMBEMhas been successfully applied to solve the ocean engineering problems, including linearwaves di?raction around a vertical circular cylinder and the calculation of addedmass coe?cients of 3D underwater bodies. High e?ciency, low storage and highaccuracy of this method are demonstrated by the numerical results, which indicatesthe great potential of the FMBEM in solving large-scale numerical problems in oceanengineering.
The innovation of this dissertation are as follows:
(1)Using the Chebyshev polynomial approximation, we proposed an improvedalgorithm of SBFEM for solving boundary-valued problems of Poisson equation.
(2)Combining the homotopy analysis method with the SBFEM, we proposeda semi-analytic numerical method, namely homotopy-based scaled boundary finite-element method, for solving nonlinear boundary-valued problems.
(3)A fast algorithm, namely Fast multipole boundary-element method was ap-plied to solve the large-scale potential problems in the ocean engineering, and thenumber of the discrete boundary element of the problems can be up to one hundredthousand, but the CPU time is less than one hour.
In conclusion, this dissertation improved the SBFEM, and proposed a new semi-analytic numerical method for nonlinear boundary-valued problems. This dissertationalso indicates that the FMBEM is suitable to solve complicated large-scale potentialproblems in ocean engineering.
引文
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