应用半解析无网格方法求解Helmholtz方程

针对高波数的Helmholtz方程的高精度求解，提出了一种半解析无网格方法。该方法基于单位分解(PU)框架，定义了带分析信息的增强覆盖函数，建立场量函数的近似公式表达.由Galerkin弱形式得到离散模型的代数方程，结合边界条件求解.无需网格地构成了Shepard单位分解函数. 用该方法求解了1D、2D Helmholtz方程，研究了不同增强覆盖函数构成的函数近似对计算精度的影响，并探讨了不同波数的选取对计算精度的影响.结果表明，对1D问题，低波数的精度达10-7量级，高波数的精度达10-5量级；对2D问题，精度达10-4量级,该方法明显地提高了计算精度.

Solution of Helmholtz equations using semi-analyzed meshless method

To accurately solve high wave number Helmholtz equation,a semi-analyzed meshless method was presented.Under the framework of partition of unit(PU) in the method,enrichment cover functions with the known analyzed information were defined,and the formulation of approximate field function was built.Algebraic equations of the discrete model were obtained by using Galerkin weak form,and were solved with appropriate boundary conditions enforced.The Shepard function of PU was constructed without meshing.1D and 2D Helmholtz equations were solved using the method,and the impact of different enrichment cover functions on computational accuracy was investigated.The impact of wave number chosen for computational accuracy was discussed.The results show that for 1D problem,the computational accuracy can reach magnitude of 10~(-7) for low wave number,and reach magnitude of 10~(-5) for high wave number.For 2D problem,the computational accuracy can reach magnitude of 10~(-4).The presented method can greatly improve the numerical accuracy.