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The Woodbury formula is an efficient tool in mathematics to calculate low‐rank perturbation problems and has been applied to improve the computational efficiency of nonlinear seismic response analysis (NSRA) of structures with local nonlinearity. Using the Woodbury formula for NSRA can avoid the time‐consuming recalculation and factorization of the large‐dimensional global stiffness matrix of a structure by only solving a small‐dimensional Schur complement system representing local nonlinearity per iteration. Because the dimension of the Schur complement matrix is determined by the inelastic degree of freedom (IDOF) number, which represents the scale of local nonlinear regions, a small IDOF number is helpful for achieving the high‐efficiency advantage of the Woodbury formula. However, when performing NSRA for large‐scale structures, the IDOF number is usually relatively large, which contradicts the efficiency requirement of the Woodbury formula. To solve this problem and extend the advantage of the Woodbury formula to the NSRA of large‐scale structures, this paper first proposes a two‐stage IDOF number reduction method by eliminating the IDOFs that have insignificant effects on the results, and consequently, a variant Woodbury formula is derived. Because only the principal component in the Schur complement matrix is retained, the dimension of this matrix and the cost for factorizing it can be reduced significantly without losing accuracy, thus greatly improving the efficiency of the proposed method. Moreover, to reduce the additional computational time introduced by the IDOF number reduction procedure and to further improve the computational performance of the proposed method, an OpenMP parallel computational strategy is incorporated. Finally, the validity of the proposed method is verified by implementing incremental dynamic analysis for a large‐scale reinforced concrete structure. [ABSTRACT FROM AUTHOR]

Nonlinear analysis has become an indispensable means for structural performance evaluation, but is still an expensive process especially for large and complex structures. Although the computer technology has largely improve the performance of nonlinear analysis, more efficient and accurate methods still attract the attention of the researchers. The Woodbury nonlinear method which uses the Woodbury formula to solve the incremental displacement response is an efficient kind of method for local nonlinear analysis, but it is no longer applicable for problems with large part of nonlinearity. Although some approximate methods have been proposed to improve the solution of linear equations in Woodbury nonlinear method, the induced error cannot be effectively controlled which leads to decreasing the convergence rate of the iterative computation and offsetting the improved computational efficiency. To address this problem, this paper proposes the Inexact Newton Woodbury (INW) method for efficient nonlinear analysis, which comprehensively considers the efficiency of both the solution of the linear equations and iterative computation. In terms of the linear equations, the improved Woodbury method is presented, where the approximation method is combined with the Woodbury formula, considerably enhancing the solution efficiency of the linear equations and largely extending the application range of the Woodbury formula. Moreover, to ensure the convergence rate which is another important influence factor of the overall nonlinear analysis efficiency, the inexact Newton condition with a new proposed forcing term applicable for the INW method is employed to make the iterative computation with superlinear convergence rate. Then, the time complexity analysis is conducted to make a comprehensive evaluation of the proposed INW method and the traditional Finite Element Method (FEM) from the aspects of the linear equations solution and iterative computation. The results show that the INW method is more efficient than the FEM especially for problems with large part of nonlinearity. Finally, a simulation example is presented to validate the efficiency and accuracy of the INW method. The results demonstrate that the INW method has higher efficiency under comparable accuracy with the Woodbury nonlinear method so that it has significantly potential for large part of nonlinearity problems. CE Database Subject Headings: Engineering structures; finite element method; material nonlinearity; time complexity [ABSTRACT FROM AUTHOR]

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]

This study presents a fast algorithm for collapse behavior simulation of space truss structures under extreme earthquake excitation by introducing the Woodbury formula to efficiently solve the structural response caused by material and geometric nonlinearity (hybrid nonlinearity). The Woodbury formula, which is an efficient tool in mathematics for solving low‐rank perturbation problems, has successfully been used to improve the efficiency of local material nonlinear analysis but still has difficulties with seismic collapse analysis in which geometric nonlinearity should be considered. In this study, by implementing stiffness matrix decomposition according to the unchanged reference configuration, the effects of hybrid nonlinearity on the change in tangent stiffness of truss structures are uniformly formulated in the form of hybrid nonlinear perturbation to the reference elastic stiffness. Thus, a hybrid nonlinearity separated governing equation can be established, in which the hybrid nonlinear behaviors are depicted by the additional nonlinear degrees of freedom (NLDOFs) separated from the reference system. This allows for employing the Woodbury formula to perform seismic collapse analysis of space truss structures for avoiding the repeated updating of the global stiffness. To overcome the adverse effect of the large NLDOF number caused by the global characteristics of geometric nonlinearity on the efficiency advantages of the Woodbury formula during seismic collapse analysis, an element state judgment strategy and an adaptive restart mechanism are presented to activate only a small number of NLDOFs within critical local regions. The accuracy and efficiency of the proposed method are verified by two 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]

Summary: Evaluating the inelastic seismic response of structures accurately is of great importance in earthquake engineering and generally requires refined simulation, which is a time‐consuming process. Because the material nonlinearity generally occurs in a small part of the whole structure, many researches focus on taking advantage of this characteristic to improve the computational efficiency and the inelasticity‐separated finite element method (IS‐FEM) proposed recently provide a generic finite element formulation for solving this kind of problems efficiently. Although the fiber beam‐column element is widely used for the simulation of reinforced concrete (RC) framed structures, the inelastic deformation is often detected in a large part of the numerical model under earthquake excitation so that it is hard to achieve high efficient computation when applying the IS‐FEM to the inelastic response analysis of RC fiber models directly. In this paper, a new numerical scheme for seismic response analysis of RC framed structures model by fiber beam‐column element is proposed based on the IS‐FEM. To implement the RC fiber model for use in IS‐FEM and improve the computational performance of proposed scheme, a method of identifying the local domains with severe section inelasticity level is proposed and a modified Kent‐Park concrete material model is developed. Because the Woodbury formula is adopted as the solver, the global stiffness matrix can keep unchanged throughout the analysis and the main computational effort is only invested on a small matrix representing local inelastic behavior. The numerical examples demonstrate the validity and efficiency of the proposed scheme. [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]

• A inelasticity-separated RC frame model considering beam-column joint is presented. • The structural stiffness matrix can keep unchanged by adding some inelastic DOFs. • The proposed elements for bond slip and joint shear produce high accurate results. • The use of Woodbury formula improve computational efficiency greatly. Many previous investigations of reinforced concrete (RC) frame structures have demonstrated that beam-column joints may experience significant nonlinearities under earthquake excitations. However, most available analysis methods assume that the joint region remains rigid. Because joint failure may result in strength and stiffness losses and even induce structural collapse, it is difficult for an analytical model with a rigid joint assumption to obtain an accurate prediction of the structural nonlinear response. The behavior of a joint is mainly affected by the shear deformation of the joint core and bond-slip of beam longitudinal reinforcements in the joint. Although many different types of modeling approaches have been presented to effectively represent the behavior mechanism of beam-column joints, the introduction of joint models inevitably increases the complexity of the analytical model of the global structure and the effort of the solution process. Therefore, achieving highly efficient analysis on the premise of accurately predicting the seismic response of RC frame structures is still a topic that deserves investigation. This study presents an efficient analysis method for the accurate evaluation of the nonlinear response of RC frames considering the effect of beam-column joint deformation based on the concept of local inelasticity separation. The proposed method captures the bond-slip behavior at beam ends by adding inelastic rotation hinge mechanisms to the ends of the existing inelasticity-separated fiber beam-column model and uses a sliding hinge mechanism to simulate the inelastic shear response of the joint core. A small number of additional inelastic degrees of freedom that are separated from the global displacement degrees of freedom of the structure is introduced to describe both the inelastic rotation behavior and the inelastic deformation of the proposed bond-slip hinge mechanism and the sliding hinge mechanism. Consequently, the Woodbury formula can be utilized for nonlinear iterative solutions to avoid updating the global stiffness matrix, thus significantly improving the computational efficiency. The exactness of the proposed method is verified against experimental data, and its application is illustrated through a seismic response analysis example. [ABSTRACT FROM AUTHOR]