A Unified Library of Nonlinear Solution Schemes

Leon, Sofie E.; Paulino, Gláucio H.; Pereira, Anderson; Menezes, Ivan F. M.; Lages, Eduardo Nobre

doi:10.1115/1.4006992

Cited by 63 publications

(39 citation statements)

References 117 publications

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“…We note that the sign of

normalΔ {boldu}_{p}_{1}^{i - 1} \cdot normalΔ {boldu}_{p}_{1}^{i}

only changes immediately after load limit points. Therefore, the generalized displacement control method captures both load and displacement limit points …”

Section: Computational Implementationmentioning

confidence: 99%

“…Therefore, the generalized displacement control method captures both load and displacement limit points. 43,45 For the cohesive traction-separation relationship, in this study, a bilinear softening model is utilized because it has been widely used to predict fracture behaviors of plain concrete. 46,47 A bilinear softening model can be defined using four fracture parameters: cohesive strength ( max ), total fracture energy (G F ), initial fracture energy (G f ), and kink point ratio ( ), 48 as shown in Figure 8.…”

Section: Computational Implementationmentioning

confidence: 99%

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Removing mesh bias in mixed‐mode cohesive fracture simulation with stress recovery and domain integral

Choi

Park

2019

Numerical Meth Engineering

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Summary To remove mesh bias and provide an accurate crack path representation in mixed‐mode investigation, a novel stress recovery technique is proposed in conjunction with a domain integral and element splits. Based on a domain integral and stress recovery technique, a maximum strain energy release rate is estimated to determine a crack path direction. Then, for a given crack path direction, continuum elements are split, and a cohesive surface element is adaptively inserted. One notes that the proposed stress recovery technique provides a more accurate stress field than a standard stress evaluation procedure. The proposed computational framework is verified and validated by solving mode‐I and mixed‐mode examples. Computational results demonstrate that the domain integral with the stress recovery accurately evaluates a crack path, even with a lower‐quality mesh and under a biaxial stress state. Furthermore, the cohesive surface element approach, with the element split in conjunction with the stress recovery and the domain integral, predicts mixed‐mode fracture behaviors while removing mesh bias in the crack path representation. Additionally, the condition numbers of stiffness matrices are within the same order of magnitude during cohesive fracture simulation.

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“…We note that the sign of

normalΔ {boldu}_{p}_{1}^{i - 1} \cdot normalΔ {boldu}_{p}_{1}^{i}

only changes immediately after load limit points. Therefore, the generalized displacement control method captures both load and displacement limit points …”

Section: Computational Implementationmentioning

confidence: 99%

Section: Computational Implementationmentioning

confidence: 99%

Removing mesh bias in mixed‐mode cohesive fracture simulation with stress recovery and domain integral

Choi

Park

2019

Numerical Meth Engineering

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“…As shown in Figure 3, u p is a vector tangent to the equilibrium path in the displacement space. According to its definition in (8), u p points in the direction of the load factor gradient, 35 specifying the direction along which evolves. In order to trace the equilibrium path without returning to the previously converged point, the incremental displacements Δu in two consecutive prediction phases need to point in the positive direction, so that we conclude the following:…”

Section: Prediction Phasementioning

confidence: 99%

A stiffness parameter and truncation error criterion for adaptive path following in structural mechanics

Maghami

Schillinger

2019

Numerical Meth Engineering

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Summary We explore a truncation error criterion to steer adaptive step length refinement and coarsening in incremental‐iterative path following procedures, applied to problems in large‐deformation structural mechanics. Elaborating on ideas proposed by Bergan and collaborators in the 1970s, we first describe an easily computable scalar stiffness parameter whose sign and rate of change provide reliable information on the local behavior and complexity of the equilibrium path. We then derive a simple scaling law that adaptively adjusts the length of the next step based on the rate of change of the stiffness parameter at previous points on the path. We show that this scaling is equivalent to keeping a local truncation error constant in each step. We demonstrate with numerical examples that our adaptive method follows a path with a significantly reduced number of points compared to an analysis with uniform step length of the same fidelity level. A comparison with Abaqus illustrates that the truncation error criterion effectively concentrates points around the smallest‐scale features of the path, which is generally not possible with automatic incrementation solely based on local convergence properties.

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“…Para tal, utiliza-se um processo incremental-iterativo que consiste de duas etapas (LEON et al, 2011): 1) A partir da última configuração de equilíbrio da estrutura, seleciona-se um subincremento de força (definido como o subincremento de força inicial - (0) ), que satisfaça alguma equação de restrição imposta ao problema. Após a seleção desse parâmetro determina-se o incremento inicial do vetor de coordenadas nodais d (0) ; e 2) na segunda etapa de solução, procura-se por meio de uma estratégia de continuação corrigir a solução incremental inicialmente proposta na etapa anterior, com o objetivo de restaurar o equilíbrio da estrutura.…”

Section: -Método De Comprimento De Arco Linearunclassified

Métodos iterativos de terceira e quarta ordem associados à técnica de comprimento de arco linear

Souza¹,

Castelani²,

Shirabayashi³

et al. 2017

Cieng

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RESUMOTécnicas de continuação acopladas ao esquema iterativo de Newton-Raphson são bastante utilizadas na análise por Elementos Finitos de problemas de estruturas com comportamento não linear, cujas soluções para uma dada precisão são obtidas por meio da resolução de sistemas de equações não lineares. Neste artigo, são desenvolvidos algoritmos baseados nos métodos de Potra-Pták, Ponto Médio e Chun, associados à técnica de Comprimento de Arco Linear, para a solução de problemas de treliças planas e espaciais com não linearidade geométrica, cujos caminhos de equilíbrio apresentam pontos limites de força e deslocamento. As análises não lineares são efetuadas por meio do método dos Elementos Finitos Posicional. Os resultados numéricos alcançados evidenciam o melhor desempenho dos códigos computacionais implementados, em comparação com as análises feitas com os métodos tradicionais de Newton-Raphson Padrão e Modificado e de Broyden, quanto ao tempo de processamento, números totais de passos de força e iterações acumuladas até a convergência para a solução, número médio de iterações por passo de força e estimador médio da taxa de convergência. Palavras-chave: treliça, Elementos Finitos Posicional, Comprimento de Arco Linear, não linearidade geométrica, trajetória de equilíbrio. ABSTRACTPath-following techniques coupled with the Newton-Raphson iterative scheme are widely used in the Finite Element analysis of structures problems with nonlinear behavior, whose solutions for a given precision are obtained by solving a nonlinear equations systems. In this paper we develop algorithms based on the Potra-Pták, Midpoint and Chun methods, associated with a Linear Arc-Length technique, for the solution of problems of plane and space trusses with geometric nonlinear, whose equilibrium paths present force and displacement limits points. Nonlinear analyses are performed using the Positional Finite Element method. The numerical results achieved show the best performance of the implemented computer codes, in comparison with the analyses made with the Modified and Standard Newton-Raphson and Broyden traditional methods, as to the processing time, total numbers of load steps, accumulated iterations until convergence to the solution, average number of iterations by load step and average convergence rate estimator. Keywords: truss, Positional Finite Element, Linear Arc Length, geometric nonlinear, equilibrium path. -INTRODUÇÃOTreliça é um sistema estrutural eficiente que pode sustentar carregamentos consideráveis com uma quantidade menor de materiais. Desde o início do seu uso comercial, esse sistema tem sido cada vez mais popular, especialmente em grandes áreas abertas com poucos ou nenhum suporte intermediário, como ilustrado na Figura 1. Aplicações bem-sucedidas de sistemas estruturais em forma de treliça abrangem estádios, edifícios públicos, centros de exposições, hangares de avião e pontes suspensas (SEÇER, 2009).As treliças quando submetidas a grandes esforços exibem comportamento não linear e problemas de instabilidade podem sur...

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A Unified Library of Nonlinear Solution Schemes

Cited by 63 publications

References 117 publications

Removing mesh bias in mixed‐mode cohesive fracture simulation with stress recovery and domain integral

Removing mesh bias in mixed‐mode cohesive fracture simulation with stress recovery and domain integral

A stiffness parameter and truncation error criterion for adaptive path following in structural mechanics

Métodos iterativos de terceira e quarta ordem associados à técnica de comprimento de arco linear

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