On robust discretization methods for poroelastic problems: Numerical examples and counter-examples

Bertrand, Fleurianne; Brodbeck, Maximilian; Ricken, Tim

doi:10.1016/j.exco.2022.100087

Cited by 6 publications

(4 citation statements)

References 30 publications

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“…We utilize a one-dimensional form of the TPM, which can be derived from the full set of equations of a two-phasic, incompressible, and isothermal continuum while neglecting volume forces, for example described by Bertrand et al 71 . These equations are given as: with (vertical) displacement u , pore fluid pressure p , vertical position z , time t , Lamé constants and , and Darcy permeability k .…”

Section: Methodsmentioning

confidence: 99%

“…1 . The reference solution was obtained by simulating of a two-dimensional column with the finite element method as described by Bertrand et al 71 . The simulation was done using FEniCS 72 with the DOLFINx solver 73 with 1000 elements in and 100 elements perpendicular to the consolidation direction for with and , and 1000 time steps for .…”

Section: Methodsmentioning

confidence: 99%

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Affine transformations accelerate the training of physics-informed neural networks of a one-dimensional consolidation problem

Mandl,

Mielke,

Seyedpour

et al. 2023

Sci Rep

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Physics-informed neural networks (PINNs) leverage data and knowledge about a problem. They provide a nonnumerical pathway to solving partial differential equations by expressing the field solution as an artificial neural network. This approach has been applied successfully to various types of differential equations. A major area of research on PINNs is the application to coupled partial differential equations in particular, and a general breakthrough is still lacking. In coupled equations, the optimization operates in a critical conflict between boundary conditions and the underlying equations, which often requires either many iterations or complex schemes to avoid trivial solutions and to achieve convergence. We provide empirical evidence for the mitigation of bad initial conditioning in PINNs for solving one-dimensional consolidation problems of porous media through the introduction of affine transformations after the classical output layer of artificial neural network architectures, effectively accelerating the training process. These affine physics-informed neural networks (AfPINNs) then produce nontrivial and accurate field solutions even in parameter spaces with diverging orders of magnitude. On average, AfPINNs show the ability to improve the $${\mathscr {L}}_2$$ L 2 relative error by $$64.84\%$$ 64.84 % after 25,000 epochs for a one-dimensional consolidation problem based on Biot’s theory, and an average improvement by $$58.80\%$$ 58.80 % with a transfer approach to the theory of porous media.

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Affine transformations accelerate the training of physics-informed neural networks of a one-dimensional consolidation problem

Mandl,

Mielke,

Seyedpour

et al. 2023

Sci Rep

Self Cite

View full text Add to dashboard Cite

show abstract

“…Direct approximations from the C 0 continuous field quantities lead to element-wise discontinuous dual quantities, which are quite inaccurate (see e.g. [21]). Beside these inaccuracies, reliable pressure approximations are challenging, if small permeabilities are considered.…”

Section: Equilibrationmentioning

confidence: 99%

“…Beside these inaccuracies, reliable pressure approximations are challenging, if small permeabilities are considered. As analysed in [22], spurious pressure oscillations depend strongly on the inf-sub stability of the involved finite element spaces and the spacial resolution of the problem. Local spacial refinements are required.…”

Section: Equilibrationmentioning

confidence: 99%

Adaptive strategies based on equilibration for the Biot equations

Bertrand

Brodbeck

2023

Proc Appl Math and Mech

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The hypercircle theorem leads to a posteriori error estimates for the three-field variational formulation of the Biot problem involving displacements, total pressure and fluid pressure. Based thereon, adaptive strategies can be constructed. In particular, the error estimator is derived on the basis of H(div)-conforming reconstructions of stress and flux approximations while the symmetry of the reconstructed stress is allowed to be satisfied only weakly. The reconstructions can therefore be performed locally on a set of vertex patches. The local nature of the reconstruction leads to two different possible adaptive strategies. In this paper, we review these two different adaptive strategies for the Biot problem. Numerical experiments underline the differences between the two strategies.

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A mesh‐in‐element method for the theory of porous media

Maike,

Schröder,

Bluhm

et al. 2024

Numerical Meth Engineering

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While direct homogenisation approaches such as the FE method are subject to the assumption of scale separation, the mesh‐in‐element (MIEL) approach is based on an approach with strong scale coupling, which is based on a discretization with finite elements. In this contribution we propose a two‐scale MIEL scheme in the framework of the theory of porous media (TPM). This work is a further development of the MIEL method which is based on the works of the authors A. Ibrahimbegovic, R.L. Taylor, D. Markovic, H.G. Matthies, R. Niekamp (in alphabetical order); where we find the physical and mathematical as well as the software coupling implementation aspects of the multi‐scale modeling of heterogeneous structures with inelastic constitutive behaviour, see for example, [Eng Comput, 2005;22(5‐6):664‐683.] and [Eng Comput, 2009;26(1/2):6‐28.]. Within the scope of this contribution, the necessary theoretical foundations of TPM are provided and the special features of the algorithmic implementation in the context of the MIEL method are worked out. Their fusion is investigated in representative numerical examples to evaluate the characteristics of this approach and to determine its range of application.

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On robust discretization methods for poroelastic problems: Numerical examples and counter-examples

Cited by 6 publications

References 30 publications

Affine transformations accelerate the training of physics-informed neural networks of a one-dimensional consolidation problem

Affine transformations accelerate the training of physics-informed neural networks of a one-dimensional consolidation problem

Adaptive strategies based on equilibration for the Biot equations

A mesh‐in‐element method for the theory of porous media

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