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When using the traditional multilayered shell element to perform material nonlinearity analysis of large structures, the computation time is often overwhelming because of the repeated updating and factorization of the large-size global stiffness matrix. This paper aims to provide an efficient and general solution to such problems. To this end, a novel flat multilayered shell model is proposed by adding additional inelastic degrees of freedom (IDOFs) to describe the inelastic behavior of elements and adopting the approximate Woodbury formula that is an efficient solution method based on matrix perturbation theory as a solver. In this way, the entire nonlinear solution process only needs to factorize a sparse matrix called approximate Schur complement, whose dimension is consistent with the IDOF number, instead of the global stiffness matrix. To avoid the performance degradation of the proposed approach due to the activation of too many IDOFs when performing material nonlinearity analysis on large reinforced concrete (RC) structures, an improved concrete constitutive model and a two-level sparseness strategy are further developed. These measures are helpful to numerically reduce the IDOF number of the RC multilayered shell model, thus saving the computational overhead of factorizing and constructing the approximate Schur complement matrix. The accuracy and efficiency of the proposed scheme are verified by numerical examples. [ABSTRACT FROM AUTHOR]

Conventional reinforced concrete shear walls are widely used as primary lateral-force-resisting members owing to their high in-plane stiffness. However, some of these walls have been severely damaged in past earthquakes because of their limited energy dissipation capacity and ductility, thus requiring laborious repairs or even demolition. Self-centering rocking walls have recently been developed and proven effective for reducing the damage and residual displacement of structures after earthquakes. Because self-centering rocking walls are usually constructed by releasing constraints at the foundation, their contribution to structural stiffness may be limited, and they are very vulnerable to vibration. To overcome these limitations, a shear wall with variable phases of deformation was proposed; this wall consists of a rocking bearing, low-strength concrete zones, and buckling-restrained rebars. With increasing seismic intensity, the proposed wall first exhibits the shear–bending deformation mode of conventional reinforced concrete shear walls, which transitions into the rocking deformation mode of self-centering rocking walls. Cyclic loading tests were conducted to investigate the seismic performance of the proposed wall. The results demonstrated that the presented concept of the mechanism transformation is feasible in practice, and compared with conventional shear walls, the proposed wall displays improved seismic performance in terms of the damage mechanism, lateral resistance degradation, and deformation capacity. Moreover, a spring–truss model and a performance prediction model were also developed to predict the seismic performance of the proposed wall. Validating the predictions of the analytical models against the test results confirmed the accuracy of the developed models. [ABSTRACT FROM AUTHOR]

Large deformation analyses of structures are of great importance to the evaluation of structural performance under extreme environmental loads, but currently available methods are time-consuming because of the requirement of factorizing large-scale matrices. The inelasticity-separated finite-element method (IS FEM), which can keep the global stiffness matrix unchanged and uses the Woodbury formula as the solver, was presented recently to provide a highly efficient tool for local material nonlinear analysis. To extend the high efficiency advantage of the IS FEM to large deformation analyses, in which the material nonlinearity may be nonlocal and the geometric nonlinearity should be considered, this paper proposes a novel numerical solution scheme by incorporating the updated Lagrangian (UL) formulation into the IS FEM framework. Within this scheme, a Woodbury approximation method (WAM) is introduced as an efficient solver, in which the changing global stiffness matrix is approximated as a constant matrix within a short time period, and a linear equation related to the Schur complement matrix is solved by the combined approximations (CA) method. To eliminate the additional error induced by the approximation, an adaptive iteration strategy (AIS) is presented, in which the approximation error involved in WAM solution is evaluated based on energy norm concept, and the global stiffness matrix is required to be updated adaptively according to the calculated error. The high efficiency and accuracy of the proposed method are finally demonstrated by the time complexity analysis and numerical examples. [ABSTRACT FROM AUTHOR]

Nonlinear analyses of structures under dynamic excitation are becoming increasingly important in structural design and performance evaluation, but large computational effort is the main factor that limits their application. Because nonlinearity is commonly confined within small regions, many studies have been devoted to improving the efficiency of solving such local nonlinear problems by maintaining the structural stiffness elasticity and simulating the effects of local nonlinearity through fictitious nonlinear forces or local modification of the elastic structural response. For dynamic analysis, the nature of the stiffness matrix determines the formulation of the widely used Rayleigh damping; as a result, these local nonlinear analysis methods often use elastic stiffness-based Rayleigh damping models. However, such a damping model can generate unexpected artificial damping forces for inelastic systems and consequently produce inaccurate or even invalid results. Although a tangent stiffness–based Rayleigh damping model has been proven to be a reasonable basis for performing highly accurate dynamic analyses, this type of damping model has difficulty achieving direct compatibility with local nonlinear analysis methods. The present research focuses on the implementation of a tangent stiffness–based Rayleigh damping model for use in efficient local nonlinear analysis methods developed based on the Woodbury formula, which can calculate the structural inelastic behavior by updating the elastic solution rather than by updating the stiffness. By representing tangent stiffness–based Rayleigh damping as a low-rank perturbation to the elastic stiffness-based damping matrix, this study derives a modified dynamic Woodbury formula in which an additional influence coefficient is introduced to reflect the effects of local nonlinearity on the structural damping properties. Moreover, to overcome the potential solution difficulty caused by abrupt changes in the damping forces in certain steps and further improve the efficiency of the proposed method, a variable time step solution scheme that can meet the computational requirements of the Woodbury formula is presented. Verifications demonstrate the validity and high efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Nonlinearity problems in engineering structures primarily involve material nonlinearity and geometric nonlinearity. Significant effort has gone toward developing accurate and efficient models and methods to simulate these nonlinear behaviors of structures. Although advanced hardware technology has greatly enhanced the computational performance of nonlinear analyses, many researchers still pay extra attention to developing more efficient numerical solution methods with the emergence of complex large-scale structures. The inelasticity-separated finite element method (IS FEM), as an efficient algorithm, is suitable for solving local material nonlinearity problems by keeping the global stiffness matrix unchanged throughout the whole computational process such that the Woodbury formula can be used as an effective tool. However, this procedure does not obviously improve the computational efficiency for structures with large deformation because widely distributed geometric nonlinearity commonly occurs throughout entire structures rather than in local domains. Furthermore, the global stiffness matrix, which is equal to the sum of the initial stiffness matrix and geometric stiffness matrix, changes in real time. This study proposes an efficient Woodbury-CA hybrid (WCH) method by incorporating the combined approximations (CA) method into the framework of the IS FEM to obtain the response of engineering structures with hybrid nonlinear behaviors (both material and geometric nonlinearities) under external loads. Within this framework, the solution of linear equations in the Woodbury formula, which is related only to the geometric nonlinear behaviors of structures, can be obtained by employing the CA method; two other global stiffness matrices, which are used to formulate the Schur complement matrices, are approximated as constant matrices for small periods of time. Additional error induced by these approximations can be eliminated by updating the global stiffness matrix when the optimal adaptive criterion (AC) used for evaluating the difference between the exact solution and approximate solution is not satisfied. Additionally, the time complexity theory is used to evaluate the computational efficiency of the proposed method; the results show that the WCH method has outstanding advantages over the conventional finite element method (FEM). The proposed method is validated against the FEM results via two different numerical examples and has greater potential for solving local material and global geometric nonlinearity problems. [ABSTRACT FROM AUTHOR]

In mathematics, the Woodbury formula is an efficient solution method for low-rank modifications that has been utilized by many researchers for the implementation of structural analyses with local material nonlinearity. The advantages of this method in local nonlinearity include the ability to avoid updating of the global stiffness matrix and to limit factorization to a matrix with a small dimension, which is known as the Schur complement. However, this matrix is generally dense, and its dimension depends on the scale of the nonlinear domains. When the condition of local nonlinearity is not satisfied, the problem becomes high-rank modifications and the Woodbury formula becomes inefficient. To overcome the limitation of the Woodbury formula and extend its high-efficiency advantage to more-general situations, an improved Woodbury method is proposed in which the dense Schur complement matrix is approximated using a banded and sparse matrix based on Saint Venant's principle. To eliminate the error caused by this approximation and minimize its adverse effect on iterative calculations, a displacement modification process was developed in terms of the tangent response of the structure so that the iterative rate of the proposed method can be accelerated. Moreover, an adaptive iterative strategy was established to further improve the computational performance of the proposed scheme. A numerical example demonstrates that the proposed scheme can be implemented more efficiently than the classical approach for the nonlinear analysis of structures. [ABSTRACT FROM AUTHOR]

As an exact method, the Woodbury formula is used to solve local material nonlinearity problems in structural analysis, in which the calculation and factorization of the global stiffness matrix are avoidable compared to the direct method used in the conventional finite-element method (FEM). This study uses the time-complexity theory to evaluate the efficiency of the Woodbury formula. The results show that the time complexity increases as the number of the inelastic degrees of freedom (IDOF) increases, and the Woodbury formula is more efficient than the LDLT factorization method only when nonlinearity appears within local, small regions. For example, for a structure with a total of 10,000 degrees of freedom (DOF), the efficiency of the Woodbury formula is higher than the LDLT method when the number of IDOF is less than 10% of the total DOF (1,000). Although the nonlinearity only occurring in small partial domains is common for most engineering structures, this low efficiency threshold still limits the Woodbury formula application for some structures with large part of nonlinear regions. To extend this efficiency threshold value, a Woodbury approximation method (WAM) is proposed that incorporates the idea of a combined approximations (CA) approach into the framework of the Woodbury formula, in which the reduced-basis method and binomial series expansion are used to solve the system of linear equations whose scale depends on the number of IDOF. The accuracy considerations for the approximate solution are discussed and the convergence criterion for error evaluation is presented. Moreover, the time-complexity analysis indicates that the limit on the efficiency is enhanced greatly by the proposed WAM under the comparable computational accuracy to the Woodbury formula, such as expanding above percentage of 10 to 70%. Finally, a numerical example is presented to prove that the WAM can provide accurate results with high efficiency, and thus, has greater potential for solving nonlinear problems. [ABSTRACT FROM AUTHOR]

Material nonlinearity analyses are widely used to determine the safety of structural components or engineering structures. Although advanced computer hardware technology has considerably improved the computational performance of such analyses, large and complex emerging structures and expensive computational process still attract the attention of researchers toward finding more efficient and accurate numerical solution methods. This paper combines an inelasticity-separated (IS) concept with the finite-element method (FEM) to establish a novel and efficient framework (IS-FEM) for structures with material nonlinear behavior that only occurs within certain small local domains. The IS concept presented in this paper begins by decomposing the strain on a nonlinear material into its linear-elastic and inelastic components and runs through the whole framework. Based on the principle of virtual work, a novel governing equation for structures with an IS form is derived by treating the decomposed inelastic strain as additional degrees of freedom. Moreover, the changing stiffness matrix in the classical FEM is expressed as the sum of the invariant global linear elastic stiffness matrix and another changing inelastic stiffness matrix with a small rank, and it represents the material nonlinearity of local domains so that the efficient solution method can be applied to perform nonlinear analyses via the mathematical Sherman–-Morrison–Woodbury (SMW) formula. Because the unchanging global stiffness matrix is used throughout the whole nonlinear analysis and the computational effort only focuses on a small dimension matrix that represents the local inelastic behavior, the efficiency of the proposed IS-FEM is improved greatly. The proposed method is validated against the results of classical FEM via three separate numerical examples and its value and potential for use in any material nonlinearity analyses are also demonstrated. [ABSTRACT FROM AUTHOR]

Accessibility has long been considered as an important factor that influences activity generation, especially nonwork activities. However, the conclusions on whether greater accessibility leads to more travel have been mixed. One reason for the conflicting results is that individual heterogeneity is not taken into account in the previous location-based accessibility measures. This paper fills this gap by operationalizing a person-based space-time accessibility (STA) measure based on the individual's spatial-temporal constraints. A discrete-continuous model of shopping activity participation and duration is developed incorporating this STA measure in order to investigate the influences of accessibility on activity generation. Using data from a global positioning system (GPS)--based activity travel survey from the Shangdi area of Beijing, China, this paper confirms that accessibility remains a significant positive effect on shopping activity generation. This implies that an individual's spatial-temporal constraints should be taken into consideration to explore the effect of land use on travel behavior. [ABSTRACT FROM AUTHOR]