This paper is to discuss an approach which combines B-spline patches and transfinite interpolation to establish a linear algebraic system for solving partial differential equations and modify the WEB-spline method dev...This paper is to discuss an approach which combines B-spline patches and transfinite interpolation to establish a linear algebraic system for solving partial differential equations and modify the WEB-spline method developed by Klaus Hollig to derive this new idea.First of all,the authors replace the R-function method with transfinite interpolation to build a function which vanishes on boundaries. Secondly,the authors simulate the partial differential equation by directly applying differential operators to basis functions,which is similar to the RBF method rather than Hollig’s method.These new strategies then make the constructing of bases and the linear system much more straightforward.And as the interpolation is brought in,the design of schemes for solving practical PDEs can be more flexible. This new method is easy to carry out and suitable for simulations in the fields such as graphics to achieve rapid rendering.Especially when the specified precision is not very high,this method performs much faster than WEB-spline method.展开更多
A Fourier-Chebyshev spectral method is proposed in this paper for solving the cavitation problem in nonlinear elasticity. The interpolation error for the cavitation solution is analyzed, the elastic energy error estim...A Fourier-Chebyshev spectral method is proposed in this paper for solving the cavitation problem in nonlinear elasticity. The interpolation error for the cavitation solution is analyzed, the elastic energy error estimate for the discrete cavitation solution is obtained, and the convergence of the method is proved. An algorithm combined a gradient type method with a damped quasi-Newton method is applied to solve the discretized nonlinear equilibrium equations. Numerical experiments show that the Fourier-Chebyshev spectral method is efficient and capable of producing accurate numerical cavitation solutions.展开更多
基金partially supported by NKBRSF under Grant No.2011CB302404NSFC under Grant Nos. 10871195,10925105,60821002,and 50875027
文摘This paper is to discuss an approach which combines B-spline patches and transfinite interpolation to establish a linear algebraic system for solving partial differential equations and modify the WEB-spline method developed by Klaus Hollig to derive this new idea.First of all,the authors replace the R-function method with transfinite interpolation to build a function which vanishes on boundaries. Secondly,the authors simulate the partial differential equation by directly applying differential operators to basis functions,which is similar to the RBF method rather than Hollig’s method.These new strategies then make the constructing of bases and the linear system much more straightforward.And as the interpolation is brought in,the design of schemes for solving practical PDEs can be more flexible. This new method is easy to carry out and suitable for simulations in the fields such as graphics to achieve rapid rendering.Especially when the specified precision is not very high,this method performs much faster than WEB-spline method.
文摘A Fourier-Chebyshev spectral method is proposed in this paper for solving the cavitation problem in nonlinear elasticity. The interpolation error for the cavitation solution is analyzed, the elastic energy error estimate for the discrete cavitation solution is obtained, and the convergence of the method is proved. An algorithm combined a gradient type method with a damped quasi-Newton method is applied to solve the discretized nonlinear equilibrium equations. Numerical experiments show that the Fourier-Chebyshev spectral method is efficient and capable of producing accurate numerical cavitation solutions.
