The Fully Barycentric Rational Interpolation Collocation Method For The Dynamic Analysis Of The Structure

The dynamic governing equations for general engineering structure, such as beams, plates and shells, are partial differential equations with initial and boundary value conditions. There are only few analytical solutions to the partial differential equations. Consequently, people often solve partial differential equations with initial and boundary value conditions by numerical method, and a lot of numerical examples demonstrate that using the numerical method can obtain the numerical solutions which meets the engineering requirements. As a result, the target of a numerical method is higher computation efficiency, better numerical stability, good node adaptability and high precision. At present, many effective numerical methods have been developed. It is not only to provide accuracy of the numerical solution of structural dynamic analysis, at the same time, a large number of the numerical solution is also able to analyze structural dynamic properties of all kinds of law.Barycentric rational interpolation collocation method for solving differential equations as a numerical method, has advantages of simple computational formulas, easy programming, high precision and good node adaptability. Traditional interpolation method based on Lagrange interpolation polynomial, but it is numerical unstable. The Runge phenomenon illustrates this problem. Barycentric interpolation formula has excellent numerical stability, and barycentric interpolation polynomial can be machine-precision approximation of an arbitrary smooth function. Barycentric Lagrange interpolation can be obtained by rewriting Lagrange interpolation formula into barycentric formula. Barycentric Lagrange interpolation with better numerical stability generally use the interpolation nodes of special distribution which is dense at both ends and sparse at mid-range such as Chebyshev points to get higher interpolation approximate accuracy. Floater proposed a barycentric rational interpolation format, which has better numerical stability and higher interpolation accuracy in terms of equidistant nodes or in the special distribution nodes. To approximate unknown function using barycentric rational interpolation, we can more easily choose the type of computational nodes.Collocation method is mainly used solving boundary value problems. For the initial value and boundary value problems depending on the time, the usual practice is to calculate by using collocation method in the spatial fields and using time difference in the time domain. In this paper, the fully barycentric rational interpolation collocation method is applied to solve the vibration problem in the spatial domain and time domain using the collocation method. A new approach with high-precision is obtained to solve the structural dynamics.To approximate unknown function using barycentric rational interpolation, the differentiation matrices of unknown function are constructed, and the barycentric rational interpolation collocation method are proposed. Unknown function is approximated using fully barycentric rational interpolation on the computational nodes in temporal and spatial fields, and the numerical solution is obtained for all nodes in the whole domain one-time.The dynamic governing equations and initial-value and boundary-value conditions are discreted by using barycentric rational interpolation. The discrete algebraic equations of equation and definite conditions are obtained. The numerical solution of the dynamic governing equations is gotten by solving system of algebraic equations with replacement method for applying boundary and initial conditions.The fully barycentric rational interpolation collocation method is applied to the problem of solving the wave equation, such as horizontal string vibration equations, vibration equations of the pole and so on. Several examples on horizontal string vibration and the pole vibration are given, including the free vibration and forced vibration, and the deflections of the vibration is analyzed and calculated.The fully barycentric rational interpolation collocation method is applied to the problem of solving the beam vibration. Several examples on the free vibration and forced vibration of the cross section beam and the uniform beam are given, and the deflection and moment of the vibration are analyzed and calculated.The numerical examples demonstrated that the fully barycentric rational interpolation collocation method has advantages of simple computational formulas, easy programming and high precision.