无网格介点法求解Helmholtz方程

Solve Helmholtz equations by the meshless intervention point (MIP) method

作为一种配点型无网格法,无网格介点MIP法具有数值实施简单、计算精度高、运算高效和适用范围广等优点。Helmholtz方程是科学与工程问题中广泛应用的一类特殊方程,因此对MIP法求解此类方程的适用性进行了验证。利用MIP法的d适应性,给出了MIP法求解该方程的两种计算格式。在数值算例中,分别对平面规则域和不规则域上的一般Helmholtz方程,以及轴对称Helmholtz方程进行了数值分析。结果表明,MIP法完全适用于求解Helmholtz方程。而且,MIP法的计算精度和收敛性都优于普通配点法。此外,MIP法的两种计算格式中,L2C0型通常具有更好的计算效果,故建议将该计算格式作为MIP法求解该类方程的标准形式。

As a collocation-type meshless method,the meshless intervention point(MIP)method is simple,accurate and efficient.Furthermore,the method may be suitable for wide applications.The Helmholtz equation is widely used in science and engineering problems.Therefore,the feasibility of solving the Helmholtz equation by the MIP method is verified.Using the d-adaptability of the MIP method,two algorithms for solving the equations are given.In the numerical tests,the Helmholtz equation on a regular domain,on an irregular domain,and on an axisymmetric domain are studied.The results show that the MIP method is fully suitable for solving the Helmholtz equation,and has a higher accuracy and a better convergence than the common collocation method.Besides,for better performance,L2C0 scheme is suggested as the standard form for the MIP method to solve this kind of equations.