Some practical procedures for the solution of nonlinear finite element equations

Bathe, Klaus‐Jürgen; Cimento, Arthur P.

doi:10.1016/0045-7825(80)90051-1

Cited by 278 publications

(89 citation statements)

References 19 publications

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“…We therefore resort to numerical solutions, and use a fully implicit finite-difference scheme with variable time steps to simulate spatially distributed values of the pressure head and moisture content. The quasi-Newton Broyden-Fletcher-Goldfarb-Shanno (BFGS) method [40,41] was used to solve numerically in time Eq.…”

Section: Governing Equationmentioning

confidence: 99%

The role of uncertainty in bedrock depth and hydraulic properties on the stability of a variably-saturated slope

Gomes

Vrugt

Vargas

et al. 2017

Computers and Geotechnics

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a b s t r a c tWe investigate the uncertainty in bedrock depth and soil hydraulic parameters on the stability of a variably-saturated slope in Rio de Janeiro, Brazil. We couple Monte Carlo simulation of a threedimensional flow model with numerical limit analysis to calculate confidence intervals of the safety factor using a 22-day rainfall record. We evaluate the marginal and joint impact of bedrock depth and soil hydraulic uncertainty. The mean safety factor and its 95% confidence interval evolve rapidly in response to the storm events. Explicit recognition of uncertainty in the hydraulic properties and depth to bedrock increases significantly the probability of failure.

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Section: Governing Equationmentioning

confidence: 99%

The role of uncertainty in bedrock depth and hydraulic properties on the stability of a variably-saturated slope

Gomes

Vrugt

Vargas

et al. 2017

Computers and Geotechnics

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“…However, in most cases such algorithm will yield an inaccurate solution or give rise to numerical instability. Discussion on the implementation of nonlinear algorithm is vast in the literature of finite element analysis [1,2,3]. The numerical behavior of various nonlinear solvers is not within present work's scope.…”

Section: Introductionmentioning

confidence: 99%

On Newton-Raphson formulation and algorithm for displacement based structural dynamics problem with quadratic damping nonlinearity

Koh

Yasreen

2017

MATEC Web Conf.

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Abstract. Quadratic damping nonlinearity is challenging for displacement based structural dynamics problem as the problem is nonlinear in time derivative of the primitive variable. For such nonlinearity, the formulation of tangent stiffness matrix is not lucid in the literature. Consequently, ambiguity related to kinematics update arises when implementing the time integration-iterative algorithm. In present work, an Euler-Bernoulli beam vibration problem with quadratic damping nonlinearity is addressed as the main source of quadratic damping nonlinearity arises from drag force estimation, which is generally valid only for slender structures. Employing Newton-Raphson formulation, tangent stiffness components associated with quadratic damping nonlinearity requires velocity input for evaluation purpose. For this reason, two mathematically equivalent algorithm structures with different kinematics arrangement are tested. Both algorithm structures result in the same accuracy and convergence characteristic of solution.

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“…The Newton-Raphson method is famous for its rapid convergence but is known to fail at points (limit points) on the equilibrium path where the Jacobian (tangent stiffness) is singular or nearly singular. Bathe and Cimento [4] highlight some of the problems with the Newton-Raphson method and present various forms of the method that involve accelerations or line searches to maintain convergence during the solution process.…”

Section: Introductionmentioning

confidence: 99%

An accelerated incremental algorithm to trace the nonlinear equilibrium path of structures

Saffari

Mirzai

Mansouri

2012

Lat. Am. j. solids struct.

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This paper deals with the convergence acceleration of iterative nonlinear methods. An effective iterative algorithm, named the three-point method, is applied to nonlinear analysis of structures. In terms of computational cost, each iteration of the three-point method requires three evaluations of the function. In this study the effective functions have been proposed to accelerate the convergence process. The proposed method has a convergence order of eight, and it is important to note that its implementation does not require the computation of higher order derivatives compared to most other methods of the same order. To trace the equilibrium path beyond the limit point, a normal flow algorithm is implemented into a computer program. The three-point method is applied as an inner step in the normal flow algorithm. The procedure can be used for structures with complex behavior, including: unloading, snap-through, elastic post-buckling and inelastic post-buckling analyses. Several numerical examples are given to illustrate the efficiency and performance of the new method. Results show that the new method is comparable with the well-known existing methods and gives better results in convergence speed.

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Some practical procedures for the solution of nonlinear finite element equations

Cited by 278 publications

References 19 publications

The role of uncertainty in bedrock depth and hydraulic properties on the stability of a variably-saturated slope

The role of uncertainty in bedrock depth and hydraulic properties on the stability of a variably-saturated slope

On Newton-Raphson formulation and algorithm for displacement based structural dynamics problem with quadratic damping nonlinearity

An accelerated incremental algorithm to trace the nonlinear equilibrium path of structures

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