Study Of Numerical Manifold Method And Time Integration Method Based On B-spline Interpolation

Numerical manifold method (NMM), as a generalized numerical method withhigh-order accuracy, can deal with discontinuous and continuous problems in a unifiedway. NMM performs numerical computation with finite cover system which iscomposed of two independent cover grids: mathematical grid and physical cover grid.This finite cover system can simplify mesh generation and avoid mesh distortion inlarge deformation, and thus is very applicable for discontinuous and large gradientproblems. In recent years, the applications of NMM have been broadened to many fields.Present numerical manifold methods possess desirable calculation accuracy, butgenerally cost a significant amount of computation time and easily induce linearindependence (LD) and mass non-conservation problems, which to a great extent limitthe wider applications of NMM. Especially, until now, there is no reported researchesconducted for the application of NMM in structural dynamic analysis. In order to avoidaforementioned problems caused by current numerical manifold method，we havedeveloped a novel numerical manifold method on the basis of polynomial B-splinefunctions and then successfully employed this method for structural dynamic responseanalysis. Additionally, the proposed B-spline functions are utilized to devise three newtime integration methods in terms of their good local support property and orderelevating property. The presented time integration method based on B-spline functionsare proved to be very efficient. The main research work and results of this paper can besummarized as following aspects:①A general manifold calculation scheme for thin plate bending is formulated onthe basis of the generalized minimum potential energy principle of NMM. As for linearelastodynamic response analysis, the generalized instantaneous potential energyprinciple corresponding to the NMM is originally proposed as well as the formulation ofthe elastodynamic equilibrium equation. With due consideration of the peculiarity ofNMM, the initialization of the dynamic equilibrium equation coupled with its timeintegration scheme is exclusively designed to calculate the ultimate numerical results.With above theories, the manifold calculation scheme for the bending vibration of thinplate is first formulated. The validity of the proposed dynamic theories is verified withsome representative numerical examples.②New numerical manifold elements are developed by use of polynomial B-spline functions. For B-spline functions with any degree of polynomials, the correspondingone-dimensional and two-dimensional manifold interpolation functions are analyzedand provided as well as their finite cover systems. Especially, for cubic, quartic andquintic B-spline elements, the mathematical expressions of their weight functions andlocal cover functions are given for practical calculation. Furthermore, the effect ofB-spline multiple knot on compatibility of B-spline manifold elements is analyzed andthe mathematical relationship is accordingly provided. With the presented B-splinemanifold elements and elastodynamic theories of NMM, the bending vibration problemsof simply supported continuous beam and thin plate are analyzed. Thereinto, numericalresults demonstrate the validity of the proposed NMM for structural dynamic responseanalysis, while the numerical results comparison shows the superiority of the NMM incomputation accuracy and efficiency over the finite element method (FEM).③With uniform cubic B-spline functions, a B-spline interpolation model on timeinterval is derived. Subsequently, this model is successfully used for the solving ofdynamic response of a single degree-of-freedom (DOF) system, and later, generalizedfor a multiple DOF system. Stability analysis illustrates that, with an adjustablealgorithmic parameter, the proposed cubic B-spline time integration method can attainboth conditional and unconditional stability. Numerical simulations show cubic B-splinetime integration method is applicable for single DOF systems, multiple DOF systemsand dynamic equilibrium equations by finite element discretization. Accuracy analysiscoupled with numerical simulations demonstrates that the proposed method not onlyrenders desirable displacement results, but also gives more accurate velocity andacceleration results than other conventional methods.④A new time integration method is proposed for structural dynamic analysis byuse of quintic B-spline interpolation functions which are adopted to approximate thedisplacements and their first four derivatives on the time intervals. In this way, theproposed quintic B-spline time integration method is applied to solving the differentialequation of motion governing SDOF systems. Then, the proposed method is generalizedfor MDOF systems. Due to the specific requirements of the proposed method, twoinitialization algorithms are introduced for initial calculation. Moreover, accuracy andconvergence of indirect initialization algorithm are verified with numerical simulations.The proposed method can only attain conditional stability by selecting appropriatealgorithmic parameter. Accuracy analysis coupled with numerical simulations illustratethat the proposed method possesses far higher computation accuracy than other considered methods. Meanwhile, the proposed method is more desirable for wavepropagation analysis than the central difference (CD) method.⑤Uniform septuple B-spline functions are deduced according to the explicitdefinition of B-spline functions. Then, uniform septuple B-spline interpolation model ontime interval is formulated with uniform septuple B-spline functions. Based on theproposed interpolation model and collocation method, a new septuple B-spline timeintegration method is developed. The proposed method can attain both conditional andunconditional stability by selecting different collocation points, that is, algorithmicparameters. Analysis of the effect of parameters on calculation accuracy demonstratesthe proposed method possesses high accuracy. Numerical simulations illustrates theproposed method have good numerical dissipation characteristics. Moreover, septupleB-spline time integration method is high-efficient for the solving of dynamicequilibrium equations based on B-spline manifold elements.