Design And Analysis Of Self-Starting Step-by-Step Time Integration Algorithms For Structural Dynamics

Partial differential equations(PDEs)governing dynamics of structures are reduced to a set of time-continuous ordinary differential equations(ODEs),also known as semidiscrete equations of motion,after the spatial discretizations,such as finite element,isogeometric analysis,and virtual element methods,are performed.This thesis focuses mainly on developing,analyzing,and using efficient direct time integration methods to solve the semi-discrete equations of motion.The main contents of this thesis are stated as follows.(1)Proposing the conditions to achieve the identical order of accuracy for time integration algorithms and refining the general technique to analytically derive amplitude and phase errors for time integration algorithms.Designers can use the accuracy analysis given in this thesis to derive the orders of accuracy for any solution variables for time integration algorithms,including naturally displacement,velocity,and acceleration variables in dynamics.On the other hand,designers can analytically calculate amplitude and phase errors for time integration algorithms so that the performance differences among various integration algorithms can be compared accurately not numerically.(2)Proving that the traditional implicit algorithms cannot simultaneously achieve selfstarting feature,identical second-order accuracy,unconditional stability,single-solve,controllable numerical high-frequency dissipation,and zero-order overshoots.This thesis designs,analyzes,and develops two implicit integration algorithms by using auxiliary variables to achieve the six requirements above.The two novel implicit algorithms are significantly superior to the published implicit algorithms,such as the TPO/G-α method,for overshoot,amplitude,and phase errors.(3)Proposing optimal two-and three-sub-step implicit integration algorithms with second-order accuracy.This thesis designs,analyzes,and develops second-order implicit integration algorithms via employing the composite sub-step technique.The novel sub-step implicit algorithms achieve self-starting feature,identical second-order accuracy,unconditional stability,identical effective stiffness matrix(optimal spectral characteristics),controllable numerical high-frequency dissipation,and zero-order overshoots.The optimal two-and three-sub-step implicit algorithms still have great advantages over the single-sub-step implicit methods when considering the same computational costs.Besides,this thesis also designs directly self-starting versions corresponding to the optimal two-and three-sub-step implicit algorithms.(4)Proposing an optimal single-sub-step explicit integration method(GSSE)with second-order accuracy.GSSE achieves self-starting,identical second-order accuracy,maximized conditional stability,single-solve,the explicit treatment of velocity,and controllable numerical dissipation.There are no time integration methods to simultaneously achieve the characteristics above,so GSSE fills this gap.On the other hand,this thesis also extends GSSE to the implicit treatment of velocity and thus proposes an alternative to the central difference methods(GSSI).GSSI not only shares the central difference methods’ advantages but also makes up for their disadvantages.For example,GSSI provides larger conditional stability bounds(exceeding 2)than the central difference methods in the damped case.(5)Generalizing the two-sub-step Noh–Bathe explicit algorithm to the s-sub-step case and optimizing a family of generalized s-sub-step explicit algorithms.When considering the same computational costs,although each extended explicit algorithm’ conditional stability bound increases with the increase of sub-steps,the increasement decreases gradually.Therefore,developing explicit integration algorithms with more than six sub-steps does not attain significant performance improvement.In addition,this thesis also constructs,analyzes,and optimizes a family of generalized ssub-step explicit algorithms,whose members can not only treat the bifurcation point as a user-specified parameter but also optimize numerical dissipation imposed in the low-frequency range.(6)Proposing two efficient families of high-order implicit integration algorithms(DSUCIn and SUCIn).DSUCIn and SUCIn achieve self-starting,identical highorder accuracy,unconditional stability,controlled numerical dissipation,and zeroorder overshoots.There are no time integration methods with real-valued parameters to achieve the characteristics above,so DSUCIn and SUCIn fill this gap.The order of accuracy increases with the increase of sub-steps,but they are developed to achieve at most sixth-order accuracy.The two implicit methods with exceeding sixth-order accuracy need more sub-steps to achieve unconditional stability,so this thesis only analyzes and develops implicit algorithms up to six sub-steps and sixth-order accuracy.(7)Proposing two efficient and high-order explicit integration algorithms.Both explicit algorithms achieve self-starting,identical third-order accuracy,maximized conditional stability,the explicit treatment of velocity,and controllable numerical dissipation.There are no time integration methods to achieve the characteristics above,so the two novel high-order explicit algorithms fill the gap.When considering the same computational costs,the published explicit algorithms cannot achieve identical high-order accuracy,maximized conditional stability,or controllable numerical dissipation.On the other hand,when considering the same order of accuracy,the published explicit algorithms may require more computational costs and cannot achieve maximized conditional stability bounds or controllable numerical dissipation.(8)Proposing a general method to analyze dispersion characteristics of wave propagations using time integration algorithms.When solving wave propagations,numerical solutions with high-order accuracy requires analyzing coupled errors imposed by spatial and temporal discretizations.The analysis technique proposed in this thesis is general and applicable to all time integration methods.Typically,the optimal CFL numbers of some second-order implicit dissipative algorithms are derived to accurately solve wave propagation problems.Numerical results are in agreement well with the theoretical analysis.