A modified implicit–explicit integration scheme: an application to elastoplasticity problems

Abstract

Based in the implicit-explicit integration scheme (IM-EX) proposed by Oliver ([5]), in this paper, a modified version of the IMPL-EX scheme is presented. Likewise the standard IMPL-EX scheme (see also [4]), the proposed scheme consists of using two integration scheme in the same time step: an standard implicit (Backward-Euler) scheme furnishing "implicit" values of stresses σ n+1 (ε n+1) in terms of the current strains (more details in Sect. 4) and an explicit integration scheme furnishing other "explicit" values of the stresses σ n+1 (ε n+1) (also to be detailed in Sect. 4 below). In [4] it is argued that, since the extrapolated values of the plastic multiplier ˜ n+1 are obtained in terms of the implicit integrated values, n and n−1 , one can expect that the well-known stability properties of the implicit integration procedures are being inherited by their explicit extrapolations. Likewise, the same argument can be made in favor of the proposed scheme. Another property pointed out in [4], that is automatically inherited by the Modified IMPL-EX, is the robustness (stability) of the standard IMPL-EX integration scheme. To explain that, in [4] it is said that, the IMPL-EX "can be understood as composed of a prediction stage, at the end of time step n, followed by a linear correction, characterized by the constant and positive definite operator" C tan (σ n+1), "during time step n + 1". Such property derives in an algorithmic stiffness matrix always positive definite resulting that "no numerical instabilities should be expected at any time and the robustness of the analysis should be dramatically increased". Also, it is said in [4] that "since the structural tangent stiffness matrix is now constant and well conditioned, the Newton-Raphson procedure should converge in just one iteration per time step".

An adaptive concurrent multiscale model for concrete based on coupling finite elements

“…In principle, any stress integration algorithm could be adopted to update the stresses. The implicit-explicit (IMPL-EX) integration scheme [45,46] was adopted in this work. This scheme has been proved as very efficient and robust to integrate the constitutive tensile damage mode adopted in this work.…”

On the use of finite elements with a high aspect ratio for modeling cracks in quasi-brittle materials

“…In order to avoid numerical convergence problems related to the strain softening regime, an implicit-explicit (IMPL-EX) integration scheme proposed by Oliver et al [50] is used for the integration of the stress-strain relation. This method has been widely tested for the integration of constitutive models based on damage mechanics [51] and plasticity theory [51,56].…”

A robust method for the simulation of quasi-brittle materials

“…a damage parameter, to determine a predicted consistent tangent matrix. More recently, Prazeres et al (2015) proposed the so-called 'modified IMPEX approach' for elasto-plastic models. Both the SLA approach and the IMPEX or modified IMPEX are robust solution techniques, but these methods do require further development if they are to be applicable to problems that include multiple materials and several nonlinear processes.…”