Revisiting finite difference and finite element methods applied to structural mechanics within enriched continua

Challamel, Noël; Picandet, Vincent; Collet, Bernard; Michelitsch, Thomas; Elishakoff, Isaac; Wang, Chen

doi:10.1016/j.euromechsol.2015.03.003

Cited by 34 publications

(19 citation statements)

References 45 publications

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“…Such quasi-continuum models take into account higher order terms for the diffusion operator together with the characteristic length of the system to approach, in a continuum way, the underlying discrete nature of the problem, see [26] for a review. It is worth noting that similar models were also proposed recently in the field of phase transitions [27] and in structural mechanics [28,29].…”

Section: Introductionsupporting

confidence: 56%

Discretization-dependent model for weakly connected excitable media

2018

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Pattern formation has been widely observed in extended chemical and biological processes. Although the biochemical systems are highly heterogeneous, homogenized continuum approaches formed by partial differential equations have been employed frequently. Such approaches are usually justified by the difference of scales between the heterogeneities and the characteristic spatial size of the patterns. Under different conditions, for example, under weak coupling, discrete models are more adequate. However, discrete models may be less manageable, for instance, in terms of numerical implementation and mesh generation, than the associated continuum models. Here we study a model to approach discreteness which permits the computer implementation on general unstructured meshes. The model is cast as a partial differential equation but with a parameter that depends not only on heterogeneities sizes, as in the case of quasicontinuum models, but also on the discretization mesh. Therefore, we refer to it as a discretization-dependent model. We validate the approach in a generic excitable media that simulates three different phenomena: the propagation of action membrane potential in cardiac tissue, in myelinated axons of neurons, and concentration waves in chemical microemulsions.

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Section: Introductionsupporting

confidence: 56%

Discretization-dependent model for weakly connected excitable media

2018

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“…It is worth noting that the node displacements in the elastic part, u + Ã i , differ slightly from the continuous elastic solution, as already shown in case of linear load distribution [37]. From Equation (19), the yield tip displacements (displacements of the n th node), l y,P , obtained for b P = 1 and l y,Q , obtained for b Q = 1, are…”

Section: Exact Displacement Solution In the Hardening Regimementioning

confidence: 67%

“…In fact, Eringen [18] introduced a stress gradient model with a length scale calibrated from axial lattice wave dispersion characteristics (high-frequency calibration). This stress gradient model can be also rigorously justified from a rational expansion of the pseudo-differential operator for low-frequency calibration or even in the static range [19,20]. The static behaviour of such nonlinear elastic lattices and its associated continua has been probably less studied.…”

Section: Introductionmentioning

confidence: 99%

Nonlocality of one-dimensional bilinear hardening–softening elastoplastic axial lattices

Picandet

Challamel

2019

Mathematics and Mechanics of Solids

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The static behaviour of an elastoplastic axial lattice is studied in this paper through both discrete and nonlocal continuum analyses. The elastoplastic lattice system is composed of piecewise linear hardening–softening elastoplastic springs connected between each other via nodes, loaded by concentrated tension forces. This inelastic lattice evolution problem is ruled by some difference equations, which are shown to be equivalent to the finite difference formulation of a continuous elastoplastic bar problem under distributed tension load. Exact solutions of this inelastic discrete problem are obtained from the resolution of this piecewise linear difference equations system. Localization of plastic strain in the first elastoplastic spring, connected to the fixed end, is observed in the softening range. A continuous nonlocal elastoplastic theory is then built from the lattice difference equations using a continualization process, based on a rational asymptotic expansion of the associated pseudo-differential operators. The continualized lattice-based model is equivalent to a distributed nonlocal continuous elastoplastic theory coupled to a cohesive elastoplastic model, which is shown to capture efficiently the scale effects of the reference axial lattice. The hardening–softening localization process of the nonlocal elastoplastic continuous model strongly depends on the lattice spacing, which controls the size of the nonlocal length scales. An analogy with the one-dimensional lattice system in bending is also shown. The considered microstructured elastoplastic beam is a Hencky bar-chain connected by elastoplastic rotational springs. It is shown that the length scale calibration of the nonlocal model strongly depends on the degree of the difference equations of each lattice model (namely axial or bending lattice). These preliminary results valid for one-dimensional systems allow possible future developments of new nonlocal elastoplastic models, including two- or even three-dimensional elastoplastic interactions.

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“…Owing to the large number of factors that influence the structures analyzed in this study, whose behavior is governed by partial non-linear differential equations, the exact response cannot be computed analytically, affirm Awrejcewicz et al (2014). Challamel et al (2015a) noted that, in numerical approaches, unsolvable mathematical equations of continuous systems are often converted into finite numbers of variables associated with discrete equivalent systems to reduce the complexity of the problem. According to Challamel et al (2015b), the possibility of solving complex continuous systems using a computer makes the discretization approach particularly advantageous over continuous methods.…”

Section: Introductionmentioning

confidence: 99%

Analytical determination of the vibration frequencies and buckling loads of slender reinforced concrete towers

Wahrhaftig

Silva

Brasil

2019

Lat. Am. j. solids struct.

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This study focused on improving the design of slender structures with reinforced concrete (RC) telecommunication towers as the main application. Analytical procedure based on Rayleigh's method to compute the first natural vibration frequency and the critical buckling load was development. All the nonlinearities present in the system were considered, in addition to the soil-structure interaction and the variation of the geometric properties along the length of the structure. The geometric nonlinearity and imperfections of the tower structure were computed as functions of the axial load using a geometric stiffness matrix. Further, the material nonlinearity was accounted for by reducing the flexural stiffness. As concrete structures exhibit viscoelasticity, creep was calculated using the Eurocode 2 model. The soilstructure interaction was modeled as a set of distributed springs. To validate the proposed method, the first frequency and critical buckling load were compared with those yielded by FEM simulations. The frequency results were in good agreement with those of the FEM simulations, indicating that the proposed method is sufficiently accurate for use in engineering design applications and easy to implement. On the other hand, the buckling load results obtained using the proposed method and FEM differed significantly, motivating further investigation.

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Revisiting finite difference and finite element methods applied to structural mechanics within enriched continua

Cited by 34 publications

References 45 publications

Discretization-dependent model for weakly connected excitable media

Discretization-dependent model for weakly connected excitable media

Nonlocality of one-dimensional bilinear hardening–softening elastoplastic axial lattices

Analytical determination of the vibration frequencies and buckling loads of slender reinforced concrete towers

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