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Journal ArticleDOI

A configurational force for adaptive re-meshing of gradient-enhanced poromechanics problems with history-dependent variables

TL;DR: In this article, a mesh-adaption framework that employs a multi-physical configurational force and Lie algebra to capture multiphysical responses of fluid-infiltrating geological materials while maintaining the efficiency of the computational models is introduced.
About: This article is published in Computer Methods in Applied Mechanics and Engineering.The article was published on 2019-12-01 and is currently open access. It has received 16 citations till now. The article focuses on the topics: Poromechanics.

Summary (5 min read)

1. Introduction

  • The path-dependent responses of geological materials, such as clay, sedimentary rock, limestone and crystalline rock are inherently anisotropic and size-dependent.
  • These material characteristics can be captured, for example, via nonlocal models by incorporating the internal microstructure (e.g., [1–5]) or gradient plasticity models (e.g., [6–11]).
  • To meet the demand of resolution to resolve the gradient of the state variables, a refinement strategy that combines the advantages of the configurational force (as a re-meshing criterion) and the Lie algebra transfer (as projector of history-dependent variables) to establish equilibrium upon remeshing is presented (Section 3).

2. Micromorphic regularization of the critical state plasticity

  • The authors present the micromorphic regularization for critical state plasticity models derived to capture the plastic dilatancy and pore collapse in geomaterials under different confining pressures.
  • This material force balance equation penalizes the discrepancy between the global field variable (α) and the local internal variable (ᾱ), while simultaneously regularizing the global field variable via introducing of a Laplacian term in the material force balance equation.
  • The key point in the local aspect is that the two variables are linked by a penalty term, in which the coupling is achieved by updating the internal variable (ᾱ) through the Newton’s iteration.
  • While the previous works mostly apply this method to von Mises-type elasto-plasticity by selecting the local equivalent plastic strain as ᾱ (e.g., [18–21,24]), the authors present how this approach is applied to Cam-Clay type plasticity models.
  • In the global aspect, the global and local variables are resolved together via the modified Helmholtz equation which introduces size effect associated with a plastic length scale.

2.2. Introducing regularization via a local–global operator split

  • Note that lp not only constitutes as the characteristic length of the materials, but also prevents mesh sensitivity upon strain localization [11,36–38].
  • The authors introduce gradient dependence to the local preconsolidation pressure p′c such that p′c = ᾱ.
  • The authors adopt the operator-split approach to solve the numerical solution incrementally.
  • Following this step, the authors then obtain the incremental update of the micromorphic variable α for the micromorphic regularization.

3. Property-preserving adaptive re-meshing

  • The authors introduce a mesh adaption procedure designed to enhance computational efficiency of the finite element simulations by leveraging the nonlocality of the gradient-enhanced plasticity model in the preceding section.
  • The authors note that this procedure can be associated with other material models (local or nonlocal) as well.
  • While increasing the numbers of degrees of freedom uniformly is an obvious remedy to improve the quality of finite element solutions (e.g., [39]), this approach demands significant computational resource and is inefficient to capture the localization of deformation upon material bifurcation.
  • Therefore, an adaptive mesh refinement algorithm that can identify the critical regions (i.e. shear, compaction, and dilation bands in sand, clay, and rocks) is often found to be more efficient for the strain localization problems (e.g., [41–43]).
  • Here their focus lies on how to properly perform the mesh refinement on problems involving historydependent materials characterized by gradient-dependent plastic flow.

3.1. Configurational force based refinement criteria

  • The first key necessary ingredient for an adaptive meshing scheme is a suitable criteria for mesh refinement.
  • This problem is closely related to how to identify the critical regions or singularities, such as strain localization and fracture, where the accuracy of the finite element approximation is diminished.
  • Traditionally, significant effort has been devoted to deriving the error estimates for finite element solutions, where the major focus was on posteriori error estimators, for example, the pioneering work by Babuvška and Rheinboldt [44] and a series of works after [45].
  • For adaptive mesh refinement, the authors utilize discrete configurational forces introduced by finite element discretization [53,57].
  • Among the various derivations for configurational force balance, the authors used the relation of invariance properties, specifically, the translational invariance followed by Buggisch et al. [59], Mueller et al. [55].

3.1.1. Configurational poromechanics for fully saturated porous media

  • The configurational force has been used as a mesh refinement criterion (e.g., [58,60,61]) for more than a decade.
  • The authors introduce this new derivation of configurational force for the fully saturated porous media by deriving the configurational force based on the translational invariance of a control volume [55].
  • The energy of the apparent solid skeleton therefore reads [62–64], W = Wsolid + Wfluid, (12) where Wsolid denotes the free energy stored in solid skeleton, and Wfluid is contribution of free energy from the bulk fluid to the solid–fluid interface.
  • Notice that (14) is specific for Cam-Clay models in which the plastic dilatancy introduces a coupling between the plastic work done due to the plastic deviatoric and volumetric strain and hence greatly simplify the expression of the plastic work.
  • This can be also considered as the special case of dynamic condition that is a plane stationary wave or standing wave condition.

3.1.2. Configurational force in idealized conditions

  • For simplicity, the authors may first consider the fully drained state in which no pore pressure is concerned.
  • Naturally, the last two terms in (24) are ignored, which results in only the energy–momentum tensor as, I = Σ = Wsolid I − (∇u)T · σ ′. (25) As Wsolid denotes the effective stain energy by assuming the drained limit state, this equation (25) is consistent with the solid only condition.
  • In terms of poromechanics theory, the authors then consider the undrained limit in which the pore-fluid remains trapped inside pores.
  • Note that D = ∇λa indicates the gradient of the test function or the shape function.
  • The discrete material forces are obtained from the existing solutions at the equilibrium state.

3.2. Transfer operation of internal variables via Lie algebra

  • Generally speaking, re-meshing for simulations dealing with path-dependent materials is more complicated than the path-independent counterpart.
  • It is therefore not difficult to imagine that an inaccurate extrapolation of the state variables could both cause significant errors and make it impossible to establish the new equilibrium in the new mesh.
  • The authors compute the spectral decomposition of the tensor and obtain the logarithms of the rotation and stretch component separately.
  • Fig. 1 describes the concept of the projection procedure.
  • As the source fields, or internal variables, are located at the integration, these values are projected on the nodal points by L2 projection (34) before the mesh is refined.

3.3. Reestablishment of equilibrium after mesh refinement

  • Once the authors recover the prime variables and internal variables onto the new mesh, the re-establishment of equilibrium completes the adaptive mesh refinement process.
  • If the remapping procedure recovers the exact states in the refined mesh from the original mesh condition, for example, the iteration will be unnecessary to reestablish the equilibrium.
  • Besides, the loss of ellipticity in governing equations leads to mesh dependent results under the strain localization problems.
  • Because the regularization is introduced via the micromorphic approach, therefore, their strategy is to obtain the state variables of the new mesh as close as those in the mesh before the refinement.
  • The recovery of scalar and tensor variables including prime fields is described in the preceding section (Section 3.2).

4. Balance principles

  • In this section the authors present balance principles and their finite element formulations for fluid-infiltrating materials associated with the micromorphic regularization.
  • Started with the coupled equations for fluid-saturated porous media (e.g., [30]) excluding any geometric nonlinearity effect, the micromorphic balance equation is considered as an additional governing equation in order to regularize the given systems.
  • For completeness, the authors briefly review the governing equations first.
  • The authors then describe spatial and temporal discretizations followed by the solution strategy to resolve a system of nonlinear equations.

4.1. Governing equations

  • The authors employ a u-p formulation for the saturated poromechanics problems in the infinitesimal regime, which includes the momentum and mass balance equations [79,80].
  • Due to the nonlocal effect introduced into the constitutive law, the modified Helmholtz equation is added as an additional governing equation.
  • As described in the preceding section, α and ᾱ indicate the global micromorphic and the local internal variables, respectively (Section 2).

4.2. Variational form

  • The authors present the weak form of the given boundary value problem for the finite element implementation.
  • For implementation, the authors introduce a temporal discretization using the backward Euler scheme.
  • Temporal discretizations and their linearization are straightforward, which are not described in this work for brevity.

4.3. Operator-split solver and solution strategies

  • The standard Galerkin form of the governing equations are discretized using only one set of basis functions.
  • The operator-split solution strategy is adopted to solve the problem.
  • Therefore, the finite element discretization of (35) to (37) and the corresponding linearization are straightforward because no couplings need to be constructed under the micromorphic setting.
  • Note that the equal-order finite element approach is adopted, where the authors use the stabilization scheme to resolve inf–sup deficiency under undrained condition by White and Borja [30].

5. Numerical examples

  • The authors present numerical examples to show the performance of the proposed remeshing algorithm associated with the gradient-enhanced anisotropic Cam-Clay models.
  • Firstly, the authors design 2D plane compression tests in which the mechanical and hydraulic responses are obtained by changing the bedding plane orientations.
  • The authors further analyze the influence of the nonlocal micromorphic energy (φm in (17)) on the configurational force associated with mesh refinement.
  • Second, the authors investigate how the configurational force criteria that are established by different poromechanics assumptions (Section: 3.1.2) affect the numerical simulations when the concentrated strain localization zone (as in the cases of shear banding) and sharp pore pressure gradient (as in the cases of injection or extraction in porous media of low effective permeability) exist.
  • The validation of the local anisotropic Cam-Clay model with experimental results, the computational efficiency of adaptive meshing simulations, and the performance of micromorphic regularization for mesh independence are included in Appendices A, B, and C, respectively.

5.1. Effect of anisotropy and nonlocal micromorphic energy

  • The authors firstly demonstrate how the adaptive mesh-refinement helps to capture the strain localization in anisotropic numerical specimens.
  • The authors note that the plastic length scale lp is selected in the order of mesh size for regularization purposes via the micromorphic setting.
  • The influence of horizontal bedding can also be identified from the pore pressure distribution in Fig. 4(a), in which the excess pore pressure is developed around the bottom region of the specimen while the drained top condition is adopted.
  • As can be seen in Fig. 5(a) and (b), however, the influence of the nonlocal micromorphic energy terms on configurational force is hardly observed.

5.2. Configurational poromechanics and anisotropy in flow

  • The authors test different refinement criteria for the multiphysical poromechanics problems described in Section 3.1.
  • The authors then evaluate the configurational force with different assumptions followed by Section 3.1.2, that is, Case A considers the effective stress under the drained limit condition (25); Case B considers the total stress under the undrained limit assumption (27); Case C considers the pressure dissipation in addition to the total stress (28).
  • When the flux was initiated, the refinement region is wider in Case C compared to Case B (Fig. 9(a)).
  • The mesh was not refined because only the contribution by Wsolid is used as refinement criterion for Case A. Likewise in Fig. 10(c) of Case A, the magnitude of configurational force due to D f is presented, which indicates the regions where sharp pressure gradient is expected (injection well surface and low permeability layer).
  • Therefore, in Fig. 10(a), (b), and (c) of Case A, the authors can identify the patterns of accumulated configurational force by Wsolid, Wsolid +.

5.3. Mesh refinement and configurational force

  • In their final example, the authors introduce an idealized vertical cut problem, a common boundary value problem in geotechnical engineering, to test the capacity of the refinement criteria to handle the demanding cases that require consecutive mesh refinements.
  • The schematics of the numerical test are depicted in Fig. 12.
  • The initial preconsolidation pressure p′c is set to −1000 kPa, and no gravity acceleration is assumed.
  • Fig. 13(b) presents how the multiple mesh refinement is performed.
  • To investigate how the configurational force dissipates with respect to the number of mesh refinement, the authors select a specific point located inside the strain localization (Fig. 14(b)).

6. Conclusion

  • The authors develop a mathematical framework that employs the micromorphic regularization to establish a nonlocal anisotropic critical state plasticity model.
  • An adaptive mesh refinement procedure that employs configurational poromechanics theory to generate refinement criteria and Lie algebra mapping to deal with tensorial history variables are established to resolve the sharp displacement and pore pressure gradients that may occur for the boundary value problems common in geotechnical and petroleum engineering and geological disposal.
  • Concepts from poromechanics theory (i.e., effective stress, total stress, and pressure dissipation) is incorporated in the derivation of the configurational force such that the resultant refinement criteria may account for hydromechanical coupling effect exhibited in fluid-infiltrating porous materials.
  • By taking account of the contributions from the additional terms that lead to the gradient dependent plastic flow, the authors present a derivation of configurational force that is suitable to be used as a remeshing criterion for both two-phase Boltzmann continua and the higher-order counterparts.
  • In addition, a Lie algebra internal variable mapping is used such that the history-dependent behaviors for the new configuration can be captured in the new equilibrium state.

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Citations
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01 Oct 2004
TL;DR: In this paper, the Mechanism based Strain Gradient (MSG) plasticity is proposed to analyze the non-uniform deformation behavior in micro/nano scale.
Abstract: Recent experiments have shown the 'size effects' in micro/nano scale. But the classical plasticity theories can not predict these size dependent deformation behaviors because their constitutive models have no characteristic material length scale. The Mechanism - based Strain Gradient(MSG) plasticity is proposed to analyze the non-uniform deformation behavior in micro/nano scale. The MSG plasticity is a multi-scale analysis connecting macro-scale deformation of the Statistically Stored Dislocation(SSD) and Geometrically Necessary Dislocation(GND) to the meso-scale deformation using the strain gradient. In this research we present a study of nano-indentation by the MSG plasticity. Using W. D. Nix and H. Gao’s model, the analytic solution(including depth dependence of hardness) is obtained for the nano indentation , and furthermore it validated by the experiments.

295 citations

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TL;DR: In this paper, a unique anisotropic double porosity elastoplastic framework was developed to describe the coupled solid deformation-fluid flow in the transversely isotropic fissured rocks.

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TL;DR: In this paper, a regularized constitutive law for the slip weakening/strengthening at different loading rates and temperature regimes is proposed to capture the coupling between the bulk and interfacial plasticity.

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TL;DR: An automated meta-modeling game where two competing AI agents systematically generate experimental data to calibrate a given constitutive model and to explore its weakness, in order to improve experiment design and model robustness through competition is introduced.

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TL;DR: A hybrid model/model-free data-driven approach to solve poroelasticity problems and introduces a hybridized model in which either the solid and the fluid solver can switch from a model-based to aModel-free approach depending on the availability and the properties of the data.

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Cites background from "A configurational force for adaptiv..."

  • ..., 2013), clay (Borja et al., 1997; Bryant and Sun, 2019; Na et al., 2019), sand (Cameron and Carter, 2009) and bone (Cowin, 1999) can be captured quite adequately with the existing state-of-the-art models (Borja, 2013b) such that the error of a well-calibrated *Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027....

    [...]

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TL;DR: In this paper, the authors investigated the hypothesis that localization of deformation into a shear band may be considered a result of an instability in the constitutive description of homogeneous deformation.
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Q1. What are the contributions in "A configurational force for adaptive re-meshing of gradient-enhanced poromechanics problems with history-dependent variables" ?

The authors introduce a mesh-adaption framework that employs a multi-physical configurational force and Lie algebra to capture multiphysical responses of fluid-infiltrating geological materials while maintaining the efficiency of the computational models. To resolve sharp gradients of both displacement and pore pressure, the authors introduce an energy-estimate-free re-meshing criterion by extending the configurational force theory to consider the energy dissipation due to the fluid diffusion and the gradient-dependent plastic flow. Then, the principal values and directions are projected onto smooth fields interpolated by the basis function of the finite element space via the Lie-algebra mapping. Their numerical results indicate that this Lie algebra operator in general leads to a new trial state closer to the equilibrium than the ones obtained from the tensor component mapping approach.