B-spline And Subinterval Method For Unconditional Stability

Subinterval method includes two basic theoretical issues: computation stability and accuracy; while the application of uniformly divided piecewise nth B-spline also includes two basic theoretical issues: how to expand B-spline in the case n is even and how to construct generalized nth B-spline basis function.By research of previous subjects, this paper achieves following six consequences:1. The expansion way in nth B-spline expansion theorem when n is even is supplemented; in general sense, a new constructing approach of uniformly divided piecewise nth B-spline basis function is given.The dynamic analysis of subinterval method shows that the higher is the times of n, the higher is the accuracy. Also it shows that the basis function of nth B-spline has better approximation property and adaptability. So it can provide a new calculational tool for calculational methods, such as the static and dynamic analysis of structures etc.2. A new computation approach of subinterval method—nestification method is given. This method is based on the basic principle of subinterval method whose physical concept is well defined. It is convenient to deal with time domain initial condition and termination condition.Correspondent with nestification method of subinterval method, in general sense, a new constructing approach of nestification method of uniformly divided piecewise nth B-spline basis function is given.The accuracy analysis shows that the accuracy of nestification method of subinterval method is higher than one of subinterval method.3 The construction form of uniformly divided piecewise B-spline functionφ_n(X) is extended and thus the primary property such as density, piecewise smoothness and expansion theorem are also extended. As a consequence, the cognitive range of uniformly divided piecewise B-spline functionφ_n(X) is extended. 4. Through the construction ofφ_n(X), a time domain subdivision reconstruction method is given and this is the development of time finite element concept, which can be applied in research of time finite element.By the basis and dimension of time domain expansion, the construction of uniformly divided piecewise nth B-spline basis function and nth B-spline basis function of nestification method is extended.5. According to the basic principle of Wilson-θmethod, the method of integral forθin recurrence format and sum-form of generalized displacement coordinate qi (t)about integral forθis proposed. This realizes the unconditional stability of subinterval method recurrence format, and the detailed mode of cubic B-spline (M =4) unconditional stable recurrence format is obtained.6. Comparison of recent frequently used algorithm.The accuracy requirement of Wilson-θalgorithm etc. is limited in the range ofΔt/T_r≤1/10, in which T_r is the least auto-oscillation periodicity of mode contributing most to constructional reaction. As the construction method in this paper adopts the J mode, the interval of high accuracy degree [ I ,II] of recurrence format can be enlarged which leads to the advancement compared with Wilson-θalgorithm.