Improved Woodbury Solution Method for Nonlinear Analysis with High-Rank Modifications Based on a Sparse Approximation Approach

Yu, Ding‐Hao; Li, Gang; Li, Hong‐Nan

doi:10.1061/(asce)em.1943-7889.0001530

Cited by 15 publications

(29 citation statements)

References 32 publications

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“…40 Thus, in each step, only small number of entries in these two matrices corresponding to the inelastic DOFs that are newly activated in the current step should be calculated, so that the complete recalculation of these two matrices is not required. 40,43,48 From the above illustration, it can be seen that the main computational cost of the Woodbury formula invests only in the factorization of the small-dimensional Schur complement matrix instead of the recalculation and factorization of the large-dimensional global stiffness matrix. This is actually the main source of the highly efficient computational capacity of the Woodbury formula for local material nonlinear problems.…”

Section: Presentation Of the Woodbury Solution Methods For Local Mate...mentioning

confidence: 99%

“…Because of the local material nonlinearity in the structure, the value of m can be much smaller than the dimension of the global stiffness of structure n (i.e., m << n ). Through the condensing of the inelastic DOFs in Equation (), the following relation can be derived 39,40 :

(\underset{n \times n}{\underset{︸}{K_{e}}} - \underset{n \times m}{\underset{︸}{K^{'}}} \underset{m \times m}{\underset{︸}{{boldK}_{p}^{''}^{- 1}}} \underset{m \times n}{\underset{︸}{{boldK}^{'}^{T}}}) \underset{n \times 1}{\underset{normalΔ boldX}{true}} = \underset{n \times 1}{\underset{normalΔ boldF}{true}}

…”

Section: Presentation Of the Woodbury Solution Methods For Local Mate...mentioning

confidence: 99%

“…As a result, the dimension of the Schur complement can be reduced to a relatively small level. Although in such case the results of the Woodbury formula per iteration become an approximation of the structural tangent response, the additional error caused by this approximation can actually be minimized by adopting a suitable error correction approach or by appropriately increasing the number of iterations 40,43 …”

Section: Efficient Woodbury Solution Schemementioning

confidence: 99%

“…This is mainly because in such case, using the Woodbury formula can avoid the repeated calculation and factorization of the global tangent stiffness matrix of the structure during nonlinear analysis and requires solving only a very small-dimensional Schur complement system representing local nonlinearity, 37,39 thus improving the computational efficiency greatly. More recently, by extending the deformation decomposition concept in the force analogy method 41,42 into finite element theory, Li et al 39,40,43 proposed an efficient local nonlinearity analysis method, which is called the inelasticity-separated finite element method (IS-FEM), to provide a unified way of adopting the Woodbury formula to depict the nonlinear evolution of general structures. To date, the Woodbury formula is mainly used to solve material nonlinear problems because this type of nonlinearity is often localized.…”

Section: Introductionmentioning

confidence: 99%

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A Woodbury solution method for efficient seismic collapse analysis of space truss structures based on hybrid nonlinearity separation

2021

Earthq Engng Struct Dyn

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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.

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Section: Presentation Of the Woodbury Solution Methods For Local Mate...mentioning

confidence: 99%

(\underset{n \times n}{\underset{︸}{K_{e}}} - \underset{n \times m}{\underset{︸}{K^{'}}} \underset{m \times m}{\underset{︸}{{boldK}_{p}^{''}^{- 1}}} \underset{m \times n}{\underset{︸}{{boldK}^{'}^{T}}}) \underset{n \times 1}{\underset{normalΔ boldX}{true}} = \underset{n \times 1}{\underset{normalΔ boldF}{true}}

…”

Section: Presentation Of the Woodbury Solution Methods For Local Mate...mentioning

confidence: 99%

Section: Efficient Woodbury Solution Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Woodbury solution method for efficient seismic collapse analysis of space truss structures based on hybrid nonlinearity separation

2021

Earthq Engng Struct Dyn

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“…When utilizing the SBFEM developed by Wolf and Song (2000) and Song and Wolf (2000) to investigate the structural responses of plates (Man et al, 2012, 2013, 2014; Xiang et al, 2014; Lin et al, 2018; Zhang et al, 2019, 2020a, 2020b), an arbitrary surface parallel with the top-plane is required to be discretized only, which helps to reduce the computational cost and increase the calculation efficiency. Comparing with the FEM (Fu et al, 2020; Li and Yu, 2018; Li et al, 2017a, 2018a, 2018b, 2018d, 2019a, 2020; Yu et al, 2018), the analytical formulations of the stiffness matrix along the thickness direction can be given out. Owing to these unique characteristics of the semi-analytical scheme and only discretization of the boundary, the SBFEM has been adopted in many engineering fields, such as modelling both the unbounded and bounded domains (Bazyar and Song, 2008; Birk et al, 2012; Chen et al, 2014; Song, 2009), voided materials (Sladek et al, 2015, 2016), parametrization (Arioli et al, 2019), weak discontinuities (Natarajan et al, 2020), the interaction of acoustic-structures (Li et al, 2018c; Liu et al, 2019b), uniform beams (Li et al, 2017b), cylindrical shells (Li et al, 2019b), and so on.…”

Section: Introductionmentioning

confidence: 99%

Free vibration analysis of functionally graded magneto-electro-elastic plates with in-plane material heterogeneity

Zhang

Fang

et al. 2020

Journal of Intelligent Material Systems and Structures

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A three dimensional elasticity analysis for the transverse free vibration characteristics of functionally graded magneto-electro-elastic plates based on the scaled boundary finite element method (SBFEM) incorporated with the precise integration algorithm (PIA) is presented. The material properties of magneto-electro-elastic plates are changing along the in-plane direction with arbitrary mathematical functions. In the proposed methodology, the strategies of only discretizing the in-plane surface and utilizing the two-dimensional spectral elements to construct diagonal coefficient matrices are adopted, which contributes to decreasing the calculation effort. The derivation process begins with the three dimensional governing equations of magneto-electro-elastic materials. Neither the plate mechanical kinematics nor invoking assumptions on the spatial distributions of electric and magnetic quantities are adopted. Built upon the introduced scaled boundary coordinates, the principle of virtual work and the technique of dual vectors, a first order ordinary differential SBFEM matrix equation for the in-plane functionally graded magneto-electro-elastic plates is obtained. Its general solution is analytically denoted as the matrix exponent. To improve the computation accuracy of the matrix exponent, the PIA is utilized to form the stiffness matrix. By virtue of the kinetic energy technique, it is convenient to construct the mass matrix of the in-plane functionally graded magneto-electro-elastic plates based on the SBFEM for the first time. Finally, comparisons of flexural frequency parameters with those from exact solutions and other numerical methods are provided. The accuracy, effectiveness, and versatility of the employed technique are validated. Moreover, additional numerical exercises are conducted to exhibit the influences of boundary conditions, material gradation functions, and aspect ratios on the free vibration behaviors of in-plane functionally graded magneto-electro-elastic rectangular, circular, and perforated plates.

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Efficient optimal seismic design method of passive energy dissipation systems based on the inelasticity‐separated finite element method

Luo

et al. 2023

Earthquake Engineering and Resilience

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Installing passive energy dissipation dampers in the main structure is an effective way to reduce structural seismic response. The traditional optimal design of passive energy dissipation systems requires a massive nonlinear time history analysis, which is an extremely inefficient process and, as a result, limits its application in practice. In this paper, an efficient optimal design method for dampers is proposed based on the inelasticity‐separated finite element method (IS‐FEM) and genetic algorithm. The IS‐FEM is a novel efficient structural nonlinear analysis method that can avoid the real‐time updating and decomposition of the structural global stiffness matrix in a traditional structure analysis and only requires a small‐scale Schur complementary matrix representing local nonlinear behavior being decomposed per iteration. To make the IS‐FEM available for the analysis of structures equipped with an energy dissipation system, the inelasticity‐separated governing equations of displacement‐based dampers are derived first. Then, an improved constrained optimal genetic algorithm based on population feasibility is presented to enhance the searching ability near the feasible domain boundary. Finally, an efficient damper optimization design method is proposed by combining the developed genetic algorithm with the proposed IS‐FEM‐based analysis procedure for passive energy dissipation structures. The accuracy and efficiency of the proposed method are verified by designing a set of metallic yield dampers to retrofit a steel structure.

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Improved Woodbury Solution Method for Nonlinear Analysis with High-Rank Modifications Based on a Sparse Approximation Approach

Cited by 15 publications

References 32 publications

A Woodbury solution method for efficient seismic collapse analysis of space truss structures based on hybrid nonlinearity separation

A Woodbury solution method for efficient seismic collapse analysis of space truss structures based on hybrid nonlinearity separation

Free vibration analysis of functionally graded magneto-electro-elastic plates with in-plane material heterogeneity

Efficient optimal seismic design method of passive energy dissipation systems based on the inelasticity‐separated finite element method

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