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Aspects of finite element formulations for the coupled problem of poroelasticity based on a canonical minimization principle

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Abstract

This work presents a new finite element treatment of the coupled problem of Darcy–Biot-type fluid transport in porous media undergoing large deformations, that is free from any stabilization techniques. The formulation bases on an incremental two-field minimization principle that is constrained by the equation of continuity for the fluid mass content and determines at a given state the deformation and the fluid mass flux vector. The big advantage of the minimization formulation over classical saddle point principles of poroelasticity is the omission of the inf-sup condition—a condition that makes the construction of stable and computationally efficient finite element formulations difficult. Due to the \(H(\hbox {Div}, {\mathcal B}_0)\) variational structure of the minimization principle on the fluid side, lowest order Raviart–Thomas elements are used for the conforming approximation of the fluid mass flux. Furthermore, a standard nodal-based element using bilinear interpolation for both fields combined with a reduced numerical integration of the (volumetric) coupling term is analyzed and used for the solution of the minimization principle. Representative numerical examples demonstrate the performance of the proposed finite element designs of the minimization principle and clearly underline advantages over finite element formulations of the classical two-field saddle point principle formulated in deformation and fluid potential.

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Notes

  1. At that time this element was not called like that and Taylor and Hood [59] published their work four years after Sandhu and Wilson [55].

  2. Note, that since the dissipation potential \(\widehat{\phi }\) defined in (20) is a homogeneous function of degree two, the dissipation potential functional D is half the dissipation in the overall body.

  3. Note, that the solid grains forming the matrix still can be modelled as incompressible by setting Biot’s coefficient to \(b=1\).

  4. The restriction \({\varvec{v}}\in {\varvec{H}}^1_0({\mathcal B})\) is in line with the additive split (48)\(_1\) of the displacement field into an extension \({\varvec{u}}_D \in {\varvec{H}}^1({\mathcal B})\) that fulfills the Dirichlet boundary condition and \({\varvec{u}}_0 \in {\varvec{H}}^1_0({\mathcal B})\) that is zero on the Dirichlet boundary.

  5. A fourth-order tensor is called positive definite if the inequality is satisfied for all second-order tensors \({\varvec{B}}\ne {\varvec{0}}\) at all points \({\varvec{x}}\in {\mathcal B}\). Alternatively, one can define positive definitness in the following way: there exists a constant \(A_0>0\) such that the inequality holds for all second-order tensors \({\varvec{B}}\ne {\varvec{0}}\) at all points \({\varvec{x}}\in {\mathcal B}\). This definition is equivalent to the previous one if as well as \({\varvec{B}}\) are continuous. Note that if has minor symmetries we have to restrict \({\varvec{B}}\) to the set of symmetric second-order tensors \({\varvec{B}}= {\varvec{B}}^T\).

  6. The index \(k \in {\mathcal N}_0\) indicates the polynomial degree of the normal component of the fluid flux on the element edges.

  7. For \({\varvec{H}}^1({\mathcal B}_0)\)-conformity, piecewise polynomial vector functions have to satisfy continuity in every component across element boundaries, i.e. \({\varvec{H}}^1({\mathcal B}_0) \subset H(\hbox {Div}, {\mathcal B}_0)\).

  8. Finite Element Shapes. Associated with a node I of a two-dimensional finite element \({\mathcal B}_{0}^e\), the interpolation matrix has the form

    figure a

    in terms of the shape function \(N^I\) and its derivatives.

  9. Note, that the mixed variables (126) do not arise in the presented reduced integration formulation. Latter is fully governed by the algebraic minimization principle (66) in terms of the potential (80). Hence, the reduced integration approach is not constrained by the discrete inf-sup condition which is algebraic in nature.

  10. We use an Enhanced-Assumed-Strain element with a strain enhancement of the standard \(\hbox {Q}_1\) element with two vectorial parameters \(\underline{{\varvec{\alpha }}}_I^e \in {\mathbb {R}}^2\), \(I=1,2\), per element, see Simo and Armero [56].

  11. Another possibility to overcome locking could be the increase of the polynomial degree of the finite element shape functions (\(\hbox {Q}_1\hbox {--}\hbox {RT}_1\), \(\hbox {Q}_2\hbox {--}\hbox {RT}_1\),...). Due to the higher computational costs this approach is not further investigated.

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Acknowledgements

Steffen Mauthe and Stephan Teichtmeister thank their late advisor Prof. Christian Miehe for his guidance, intensive mentorship and unlimited support. The anonymous reviewers’ detailed comments and suggestions are highly acknowledged. Moreover, we thank the German Research Foundation (DFG) for funding this work within SFB 1313, Research Project B.01.

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  1. Institute of Applied Mechanics, Chair I, University of Stuttgart, Pfaffenwaldring 7, 70569, Stuttgart, Germany

    S. Teichtmeister, S. Mauthe & C. Miehe

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Teichtmeister, S., Mauthe, S. & Miehe, C. Aspects of finite element formulations for the coupled problem of poroelasticity based on a canonical minimization principle. Comput Mech 64, 685–716 (2019). https://doi.org/10.1007/s00466-019-01677-4

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