Efficient Seismic Analysis Method For Solid Structures Based On The Inelasticity-Separated Theory

With the emergency of a large number of large-scale complex strurtures of various forms in recent years,it will inevitably lead to serious consequences if they are damaged under extreme environmental loads.It has been getting more and more attention to clarify the disaster mechanism of such structures.As the most popular numerical analysis method at present,the finite element method is a powerful tool for studing the mechanical characters and disaster mechanism of complex structures under environmental loads.Structural nonlinear analysis generally adopts the variable stiffness solution format by using finite element method that the global stiffness matrix needs to be updated and decomposed once have element enter nonlinear state.With the expansion of structural scale and the improvement of refinement of the finite element model,the calculation cost will increase sharply and the calculation efficiency will become lower.The inelasticity-separated finite element method is a new proposed structural nonlinear analysis method in recent years,the core idea of this method is that decomposing the element material strain into the linear strain and inelastic strain two components,and taking the advantage of the characteristic of local material nonlinearity in which the inelastic strains at the collocation points that exhibit nonlinearity are regarded as additional inelastic degrees of freedom such that the global stiffness matrix can be expressed as a low-rank modification to the global initial elastic stiffness matrix,and the Woodbury formula can be used to improve the computational efficiency.Although the basic theoretical framework of the inelasticity-separated method has been established,there are fewer elememt models that can be used for structure analysis,therefore,it is necessary to carry out research to enrich the inelasticity-separated element model libaray.In this paper,an efficient and refined analysis method for two-dimensional plane and three-dimensional solid structures with material nonlinearity is proposed in element and algorithm level.In element level,the various inelasticity-separated element models are established.In algorithm,an improved Woodbury approximation approach（IWAA）is proposed that the solution of nonlocal material nonlinearity of large-scale refined model can be relized.Finally,combing with viscous-spring artificial boundary and inelasticity-separated method,an efficient analysis method for soil-strucures dynamic interaction problems under seismic excitations is proposed.The main research contents are shown as follows:（1）The inelasticity-separated refined element models for material nonlinearity analysis of two-dimensional plane and three-dimensional solid structures are constructed within the framework of inelasticity-separated method.Based on the material strain decomposition and inelastic strain interpolation format,the elemental governing equation is eatablished by setting the numerial integral points of a finite element to be the inelastic strain interpolation points and using the principle of virtual work.By the static condensation to the inelastic strain vector of an element,the inelasticity-separated elemental governing equation is equal to the elemental governing equation of variable stiffnes format.For high-order solid elements and its corresponding degenerated elements,as long as they can use gaussian or hammer numerical integral scheme for elemet numerical integration,there is no need to construct complex inelastic strain interpolation function such that the element construct difficulty can be greatly reduced.However,due to the introduction of many inelastic strain interpolation points in above elements,the inelastic degrees of freedom are general higher and the exact Woodbury formula or existing improved algorithms are difficult to solve these problems even though the structures are in local nonlinea state.Therefore,it is suggested to directly using the improved Woodbury approximation approach that proposed in chapter 4 to solve those problems and the numerical examples verify the correctness of the constructed element model.（2）The inelasticity-separated sacled boundary finite element method is proposed based on the principle of inelasticity-separated method.Taking the polygon scaled boundary element as an example,the basic format of scaled boundary finite element method is introduced in first and the elemental shape function and strain matrix are obtained through the element characteristics analysis.Then,splitting the polygon scaled boundary element into several sector subelements,in which the strain decomposition and inelastic strain interpolation format among those subelements are independent such that the inelastic strain interpolation function for the entire S-element is established through a piecewise approach.The principle of virtual work can be used to construct the governing equation of the proposed inelasticity-separated S-element in SBFEM coordinates,the results show that both the scaled boundary element and finite element have same form,the same solution process for finite element model is also applicable for scaled boundary element.In terms of the expression form of element model,the finite element can be considered as a scaled boundary element with fewer edges or faces such that the finite element can also be directly regarded as a special scaled boundary element to improve the element calculation accuracy.This idea is also applicable to polyhedral scaled boundary element and higher order scaled boundary elements,indicating that the proposed inelasticity-separated scaled boundary finite element method has a wide range application field.（3）Based on the Woodbury approximation method（WAM）,an improved Woodbury approximation approach（IWAA）is proposed such that the efficient solution for nonlocal material nonlinearity of large-scale refined model can be realized.The IWAA directly uses the original form of matrix K_inf to calculate the basis vector of combined approximation approach,while the reduced basis matrix can be calculated by the transition variables generated in the calculation process of the basis vector and orthogonalized transformation matrix of the basis vector can be directly constructed by the reduced basis matrix.The entire solution process of the IWAA during a certain iteration step was replaced by few back-substitutions of the global initial stiffness and the matrices and vectors productions,which greatly simplifies the solution process of WAM and reduces the requirement of computer storage space,and has the same calculation accuracy and convergence speed with WAM.Time complexity comparison analysis indicate that the time complexity of the IWAA has a linear relationship with the inelastic degrees of freedom,and the larger of structure scale,the computational efficiency is higher than the traditional variable stiffness method.Moreover,when the refined model reaches a certain scale,the IWAA can maintain its efficiency advantage for nonlocal or even global nonlinear problems.（4）Combined the viscous-spring artificial boundary and inelasticity-separated method,an efficient nonlinear analysis method for soil-structure dynamic interaction is proposed.Based on the viscous-spring artificial boundary condition,the inelastcity-separated equation of motion for soil-structure dynamic interaction is constructed and the Newmark-βmethod is adopted to integrate the equation of motion such that the global dynamic govering equation can be eastblished.It is evident that the form of the dynamic governing equation is the same as the dynamic governing equation of fix boundary model,the difference between them is that need to add the spring and damper matrix to the near-field soil-structure system of the initial stiffness matrix and damping matrix thus that the sparse bandwidth characteristics of effective stiffness matrix is retained.The proposed method retains the advantage of viscous-spring artificial boundary and is suitable for layered or arbitrary foundation.