Research And Application Of Scaled Boundary FEM And Fast Multipole BEM

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.