A Wavelet Method For Uniformly Solving Nonlinear Problems And Its Application To Quantitative Research On Flexible Structures With Large Deformation

The nonlinear science, which is initiated and rapidly developed in the20th century, has become an important issue in numerous fundamental researches and engineering applications. Accompanying the development of nonlinear science, many unique phenomena and features of nonlinear systems in different fields have been revealed quantitatively. However, most existing solution methods for the nonlinear problems can only be used to treat those with weak nonlinearity. Only when additional special techniques are combined to these existing solution methods, then few individual strong nonlinear problems can be solved. There exist no any universal methods which can be generally applied to the solutions of both weak and strong nonlinear problems in a unified way. Although, by combining with the numerical tracking technology, some of these existing methods also have a certain degree of universality in treating nonlinearity, yet, the solution obtained by such a technique may cause significant deviation due to the accumulated error, especially for the nonlinear systems which are sensitive to initial values. Thus, developing effective methods to solve problems with nonlinearity especially strong nonlinearity has been a thorny issue in nonlinear science for decades. In order to address this tough issue, based on the basic wavelet algorithms which have been developed by the present research group, this dissertation has proposed an universal method which can effectively and acuractely solve the general weak and strong nonlinear problems in a unified way. And the systematic investigations on the accuracy and convergence rate of such a wavelet method have been conducted by means of theoretical derivation and numerical analysis. By using this method, high-precision quantitative results for typical weak and strong nonlinear problems in applied mechanics and physics have been presented.First, we give a rigorous theoretical analysis on the errors of generalized orthogonal Coiflet approximations for square-integrable functions and related derivatives and integrations defined on a bounded interval, which are verified by a series of numerical examples. When applying the proposed numerical method to solve nonlear boundary value problems (NBVPs), the corresponding differential equations (DEs) will be discretized to systems of algebraic equations, errors during this procedure have been analyzed, and a detailed discussion on the closed feature of this wavelet method is presented. By applying this method to study the one-and two-dimensional Bratu equations which have been taken as benchmark testing problems for numerical methods, we find out that, for both of the weak and strong nonlinear cases, the maximum relative errors of the present wavelet solutions under the resolution level6can be about10-8, and orders of convergence rates of this wavelet method are4and2.5for the one-and two-dimensional Bratu questions, respectively, which are much better than many other existing numerical methods. Moreover, the proposed wavelet algorithm with closed property can effectively solve the strong nonlinear problems with multiple solutions. And by using the same solution procedure, a high-precision approximate solution to the classical two-dimensional Bratu equation is obtained, for which exact solution is not available. Further, we employ the proposed method to the solutions of a series of one-dimensional NBVPs to quantitatively study the dependence of accuracy and convergence rate of the method on orders, quasi-linear and full nonlinear features of DEs. Results show that, for the1st-and2nd-order DEs, the maximum relative errors under resolution level6keep around10-8, and the orde of convergence rate can reach even5. And under the same resolution level, for the third-and fourth-order DEs, the maximum relative errors become around10-4, and the order of convergence rate is3. For the2nd-order full nonlinear problems, the maximum relative errors under resolution level6are about10-7, and the order of convergence rate is about5for both weak and strong nonlinear cases. For applications of the proposed method to the solution of structural mechanics problems, we consider large deflection bending of a cantilevered and a simply supported beam. Results demonstrate that the maximum relative errors of the present wavelet method with15nodal points are about10-6and10-4for the bending problems of cantilever and simply supported beams, respectively, when the maximum deflection is even as large as half of the beam length. However for the same problems and under same number of nodes, the maximum relative errors of the finite element method are only10-2For initial and boundary value problems (IBVPs), the proposed wavelet method can reduce the corresponding nonlinear partial differential equations (NPDEs) into a system of nonlinear ordinary differential equations (NODEs), which can be further solved by using the Runge-Kutta method. To justify this algorithm, the Burgers equations in fluid mechanics are taken as numerical examples, results demonstrate that the maximum relative error of the present wavelet method with16spatial grids is about10-9, and the order of convergence rate is about5under the condition of Reynolds number being200. Moreover, the present wavelet method is always efficient for the solutions of Burgers equations with the Reynolds number even up to107. Further, we consider the solutions of nonlinear vibration problems of simply supported beams. For a forced nonlinear vibration problem with known exact solution, we find that the maximum relative error of the present wavelet solution under15spatial nodes can be about10-4when the amplitude reaches even half of the beam length. For nonlinear free vibrations, we study the dependence of the characteristic frequency on the amplitude. We find that the frequency increases as the amplitude increases, and there exists a critical amplitude which is about one quarter of the beam length. When the amplitude being respectively less or larger than this critical value, the increasing rate of the characteristic frequency become increasing or decreasing as the amplitude increases, respectively.At last, by combining an orthotropic shell model in continuum mechanics and the worm-like chain model in polmer theory, we derive a closed-form solution for the contour-length-dependent persistence length of semiflexible tubular polymers, which agrees with experimental measurements very well. And based on such a deformation-mode-and contour-length-dependent persistence length, a microstructure-based worm-like chain model to study the statistical mechanical behaviors of semiflexible tubular polymers is proposed. As an example, we have applied this model to study the mechanical property of microtubules in animal cells, which has a tubular structure in microns, and can be treated as a typical tubular polymer. By using the proposed wavelet method, we further quantitatively study the mechanics of mitotic spindles which are composite structures consisting of microtubules. Results demonstrate that comparing to a solid rod-like structure, microtubule allows better ability in maintaining the normal shape of the spindle, and capturing and positioning organelles.