Compact Difference Methods And Theoretical Analysis For Fourth Order Vibration Problems

Beam and plate are the basic configurations of structural mechanics,which are widely applied to such engineering practices as bridge construction,civil engineering,mechanical system and others.The mathematical models of elastic beam-plate vibration under a foreign exciting are described by partial differential equations of fourth order through the Hooke law and Hamilton principle,as well as the visco-elastic mathematical models are described by partial differential equation of fourth order coupled with a time-fractional derivative through the fractional Kelvin-Voigt law and the generalized Hamilton principle.Thus,it will be significant to efficiently simulate the beam and plate vibration processes,especially the resonance phenomenon,for making scientific decisions to lessen the rates of catastrophic failure,increase the safety and reduce the overall cost of system maintenance,and furthermore for providing better understanding of the mechanism of the vibrations.The classic numerical methods for elastic beam and plate models are mainly finite difference,finite element,finite volume and mixed finite element methods.These classic numerical methods require to solve a large-scale linear algebraic system at each time level since higher-order differential operators(second-order derivatives in time and fourth-order differential operators in space)should be discretized,which are unfavorable for long time simulations.Furthermore,the mathematical analysis theory and numerical simulation theory have not established systematically for the recently proposed fractional visco-elastic models due to the appearance of the time-fractional operator and its non-locality and weaksingularity.In this doctoral dissertation,we establish the numerical methods for fourth order vibration model and obtain the following main results:Proper orthogonal decomposition-compact difference algorithm and theoretical analysis for elastic plate vibration models.For the elastic plate vibration models,by combining the merits of the compact difference and the proper orthogonal decomposition techniques,we design an efficient proper orthogonal decomposition-compact difference algorithm.That is,the fourth-order partial differential equation is decomposed as a second-order differential system by introducing an intermediate variable,then 3-point compact difference schemes in 1-d case or 9-point compact difference schemes in 2-d case for second-order differential operator are used to achieve fourth-order spacial convergence,and the central difference is used to the time derivative.To lessen the computing time and improve the computing efficiency,the proper orthogonal decomposition technique is applied to reduce the scale of algebraic system generated by the compact difference scheme on each time level.The uniqueness and unconditional stability,the convergence rates of second-order in time and fourth-order in space are rigorously proved for displacement,strain and bending moment of the elastic plate vibration models by the energy analysis.The numerical experiments are conducted to show that,whether or not the resonance phenomenon appears,the proper orthogonal decomposition-compact difference scheme has ideal convergence rates and possess 30 times computing efficiency faster than the compact difference scheme does.This validates the feasibility of the proposed proper orthogonal decomposition-compact difference method by numerical analysis and numerical experiment.Mathematical analysis theory for the fractional viscoelastic vibration models.For the visco-elastic beam model,by using the fractional-order Sobolev space theory and the Galerkin argument,we prove that the stability of the weak solution to the fractional viscoelastic beam vibration model with respect to the external excitations,and prove that the existence and uniqueness,the stability and the regularity of the weak solution to the fractional viscoelastic plate vibration models under appropriate assumptions.This establishes the mathematical analysis theory for the fractional viscoelastic vibration models.Compact difference method and theoretical analysis for visco-elastic beam vibration models.For the visco-elastic beam model,we use the weighted Gr¨unwald difference to discrete the time fractional derivative,and still use the central difference and the compact difference to discrete the temporal and spacial derivative of second order respectively,then propose the weighted Gr¨unwald difference-compact difference scheme.We use the Grenander-Szeg¨o theorem and Stirling formula to give a deeper discussion on the Toeplitz matrix generated by the weighed Gr¨unwald difference operator and prove that the eigenvalues of the Toeplitz matrix have their positive lower and upper bounds.Based on these bounds,we further prove rigorously that the unconditional stability,the convergence rates of second-order in time and fourth-order in space for displacement,strain and bending moment.Numerical experiments verify the reliability of the theoretical results.Proper orthogonal decomposition-compact difference algorithm and theoretical analysis for visco-elastic plate vibration models.For the visco-elastic plate models,borrowing the ideas of the visco-elastic beam case,we propose the weighted Gr¨unwald difference-compact difference scheme,then combining the proper orthogonal decomposition technique,we propose the proper orthogonal decomposition-compact difference scheme for the visco-elastic plate vibration models.The unconditional stability,the convergence rates of secondorder in time and fourth-order in space are rigorously proved for displacement,strain and bending moment by the energy analysis method.The numerical results show that the proper orthogonal decomposition-compact difference scheme possess nearly 40 times computing efficiency faster than the compact difference scheme does.Once again,we demonstrate the feasibility of the propose proper orthogonal decomposition-compact difference scheme by numerical analysis and numerical experiments.Numerical experiments.The numerical experiments conducted in this dissertation are not only for the verification of the established numerical analysis theory,also include the examples with lively engineering backgrounds,as well as for non-resonance case and resonance cases.These imply that the proposed compact difference schemes,proper orthogonal decomposition-compact difference schemes can be good candidates for long time and large-scale simulation of beam and plate vibration models.