| A new type of double-variables second-order explicit time-history integration method(EPI method)is proposed to address the mainstream explicit integration methods,which have problems of insufficient accuracy stability,initial value.Through different definitions of super-convergence factors,SSEPI method and DSEPI method are established,and the stability conditions of these methods are derived.The error theories for SSEPI and DSEPI methods are systematically established in various aspects such as stability,accuracy,numerical damping,algorithm frequency,etc.Then,a fast local nonlinear double-variables second-order explicit integration method(NDSEPI method)was established by using DSEPI method.The linear and nonlinear time-history analysis of explicit integration methods has been widely applied in the field of civil engineering.EPIs avoid the shortcomings of existing explicit integration methods and can be used for time-history analysis of isolated structures to improve the efficiency and accuracy of engineering.The main research works and conclusions of this article are as follows:(1)A new double-variables second-order explicit integration method(EPI method)based on the Hamilton system theory is proposed to address the problem of need-? results at zero time in the central difference method.This method abandons the linear acceleration assumption adopted by existing mainstream integration methods,and starts from Hamiltonian dynamic theory,a time-domain explicit integration format with displacement and momentum terms as variables for motion equations is established by introducing dual variables.Then,Taylor expansion is performed on the aligned secondary exponential matrix and Gaussian integration is performed on nonhomogeneous terms.The method avoids multiple iterative solutions in single step calculations adn large-scale calculations such as aggregating matrices,solving ordinary differential equations,matrix inversion,matrix multiplication.And using double-variables recursive calculation to achieve a zero moment starting calculation scheme,which can simultaneously obtain displacement and velocity results,is an efficient integration method.(2)Combined with stability analysis of EPI method and the need for filtering high-order mode,a super-convergence factor is introduced to establish a single-factor double-variables second-order explicit integration method(SSEPI method).Based on the stability defect analysis of the EPI method,in this method a super-convergence factor is introduced into the quadratic expansion term of the recursive matrix,a new general formula for solving motion equations is established,and then the stability conditions of the SSEPI method is derived based on the spectral radius principle;A predictable numerical damping calculation formula was derived based on the orthogonality of the vibration mode;The algorithm frequency calculation formula was derived based on the comparison with the analytical solution theory formula.Numerical examples show that the SSEPI method can significantly improve the performance of the EPI method and has a similar performance of the classical implicit integration method Wilson-θ.The numerical damping of the method can effectively filter out high-order modes while ensuring the accuracy of low-order modes.(3)To address the accuracy stability issues of the central difference method and the need to preserve mid to high order modes,a double-variables second-order explicit integration method(DSEPI method)was established by introducing super-convergence factors and,which is based on the bivariate principle of the EPI method.In this method super-convergence factors are introduced into the displacement and momentum components of the quadratic expansion term of the recursive matrix,the calculation on the momentum term is simplified,and a new general formula for solving motion equations is established.Then,based on the principle of spectral radius and the orthogonality of vibration modes,the stability conditions of the DSEPI method were derived and it was proved that the DSEPI method does not have numerical damping;The algorithm frequency calculation formula was derived based on the comparison with the analytical solution theory formula.Through example verification,it is shown that the DSEPI method solves stability of EPI and applicability issues of SSEPI,and stability conditions and numerical damping are consistent with the central difference method.However,it is superior to the central difference method in high-order amplitude accuracy,amplitude accuracy stability,initial value processing,and has better computational robustness.(4)The error theory of the time-history integration method was established,including the numerical damping,phase difference,amplitude accuracy,and other issues of the algorithm.The numerical damping calculation formulas for SSEPI method and DSEPI method were derived;the theory of algorithmic frequency calculation is established for interpreting the phase difference with the analytical solution,and based on this theory,it has been proven that both the DSEPI method and the central difference method can obtain accurate amplitudes under sudden applied loadings.The displacement theoretical solution of the DSEPI method under harmonic loadings is derived,it clarified the error magnitude of the DSEPI method solution compared to the analytical solution,and demonstrated the stability of the algorithm at critical time steps.(5)Explored the difficulty of calculating the required results at zero time and the problem of insufficient accuracy stability under harmonic loads using the central difference method.The EPI method utilizes the principle that double-variables can obtain displacement and velocity results in a single step recursion,avoiding the problem of processing initial values at zero time using the central difference method and the problem of solving velocity through linear acceleration assumption.During the calculation,the initial displacement,initial velocity,and initial acceleration can be directly applied at zero time.It is beneficial for establishing a unified program architecture in large-scale computing programs.(6)Considering the nonlinear characteristics of isolated structures,a fast nonlinear double-variables second-order explicit precise integration method(NDSEPI method)was established based on the DSEPI method.Based on the substructure concept,this method shifts the local nonlinear damping force term in the recursive matrix of the DSEPI method to the right hand side,so that the left matrix of the general formula remains unchanged and the right nonlinear change.Examples show that the NDSEPI method,central difference method,Newmark-β method can obtain almost consistent results.(7)Based on the DSEPI method for time history analysis,an entire design method(TAED method)for isolated structures is established,and the average power spectrum modulation iteration method and multi-wave mean method are proposed to address the problem of excessive dispersion of time history analysis results;In response to the problem of low design efficiency caused by multiple sets of results in time history analysis,a pre-combined internal force extraction method based on component stress characteristics and maximum envelope is proposed;A safety assessment method based on incremental dynamic analysis with DSEPI method and probability analysis is proposed to address the issue of multiple failure modes in isolated structures.The stability,rationality,and accuracy have been verified through multiple engineering examples.In this article a class of double-variables second-order explicit integration methods is proposed.Theoretical analysis and numerical examples demonstrate that they have good efficiency,accuracy,and stability.They are superior to the central difference method used in mainstream commercial software in initial value processing,amplitude accuracy,and algorithm stability.They are a new,effective and original breakthrough type of explicit integration method. |
