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poroelasticity in infinite orthotropic materials with smooth interfaces via Green's functions

ARCHIVE OF APPLIED MECHANICS 2025年第5期95卷 1-15页

作者： Tariq, Muzammal Hameed Pan, Li-Hua Zhou, Yue-Ting Ren, Jingli Dang, Huayang Tongji Univ Sch Aerosp Engn & Appl Mech Shanghai 200092 Peoples R China AECC Hunan Aviat Powerplant Res Inst Zhuzhou 412002 Peoples R China Shanghai Inst Aircraft Mech & Control 100 Zhangwu Rd Shanghai 200092 Peoples R China Zhengzhou Univ Sch Math & Stat Zhengzhou 450001 Peoples R China Zhengzhou Univ Sch Mech & Engn Sci Zhengzhou 450001 Peoples R China

Green's functions are essential analytical tools in physics, providing key insights into the behavior of complex multiphase systems. This study examines poroelastic effects in infinite orthotropic materials with smooth contact interfaces using Green's functions, which play a significant role in flotation mattresses and rock-soil systems. By employing strict differential operator theory and potential theory, we present compact mono-harmonic general solutions to the governing equations that satisfy sixth-order homogeneous partial differential equations. Based on the obtained compact general solution, the 2D fundamental solution for a steady line fluid source in an infinite orthotropic poroelastic plane with smooth interfaces is derived using six newly introduced harmonic ***, ensuring their ease of use in practical applications. Numerical simulations, presented through non-dimensional contours and spatial visualization, reveal elliptical behavior, high-density levels near the fluid source, decay at a distance, higher-order singularities, zero-common tangents, and adherence to interface continuity conditions.

关键词： Two-phase Orthotropic materials General solution poroelasticity Green's functions Potential theory method

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A variationally-consistent hybrid equilibrium element formulation for linear poroelasticity

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2025年 434卷

作者： Lo Franco, Simona Parrinello, Francesco Borino, Guido Univ Palermo Dept Engn Viale Sci 8 I-90128 Palermo Italy

A poroelastic medium is defined as a continuous system in which the mechanical response arises from the interaction between a deformable elastic solid skeleton and a pressurised fluid, fully saturating the interconnected porous network. The coupled theory of Poromechanics is effectively employed to solve a broad class of problems spanning various fields, ranging from its original application in Geomechanics to addressing contemporary challenges in tissue Biomechanics. Besides the seminal theory introduced by Biot, several variational multi-field principles have been proposed, leading to various finite element formulations. Finite Element approaches are mostly based on a two-field discretisation involving the coupled solid displacement and interstitial fluid pressure as primary variables. However, this choice of the two-primary variables leads to a saddle-point problem, posing a challenge in achieving accurate solutions and thus making the use of stabilisation methods rather mandatory. This study proposes a novel variationally consistent Hybrid Equilibrium Element formulation, based on the complementary strain energy function. A two-field minimisation principle is formulated, with total Cauchy stress and interstitial fluid pressure serving as primary variables. The proposed variational principle provides a variationally consistent framework for designing inherently self-equilibrated Hybrid Equilibrium Elements, where the solid displacement field emerges as a Lagrangian variable to enforce a co-diffusivity constraint at element sides. The fulfilment of the inf-sup condition, typically challenging in developing stable and computationally efficient finite element formulations, is no longer required, a minimisation principle having been adopted, rather than the classical saddle-point principle. The proposed Hybrid Equilibrium Element approach offers a statically admissible solution even in the proximity of singularities or high space gradient stress distribution, providi

关键词： poroelasticity Variational principles Hybrid finite elements Mixed finite elements

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Modelling and simulation of anisotropic growth in brain tumours through poroelasticity: A study of ventricular compression and therapeutic protocols

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COMPUTATIONAL MECHANICS 2024年第5期74卷 1137-1169页

作者： Ballatore, Francesca Lucci, Giulio Giverso, Chiara Politecn Torino Dept Math Sci GL Lagrange Corso Duca Abruzzi 24 I-10129 Turin Italy Sapienza Univ Rome Dept Struct & Geotech Engn Via Eudossiana 18 I-00184 Rome Italy

Malignant brain tumours represent a significant medical challenge due to their aggressive nature and unpredictable locations. The growth of a brain tumour can result in a mass effect, causing compression and displacement of the surrounding healthy brain tissue and possibly leading to severe neurological complications. In this paper, we propose a multiphase mechanical model for brain tumour growth that quantifies deformations and solid stresses caused by the expanding tumour mass and incorporates anisotropic growth influenced by brain fibres. We employ a sharp interface model to simulate localised, non-invasive solid brain tumours, which are those responsible for substantial mechanical impact on the surrounding healthy tissue. By using patient-specific imaging data, we create realistic three-dimensional brain geometries and accurately represent ventricular shapes, to evaluate how the growing mass may compress and deform the cerebral ventricles. Another relevant feature of our model is the ability to simulate therapeutic protocols, facilitating the evaluation of treatment efficacy and guiding the development of personalized therapies for individual patients. Overall, our model allows to make a step towards a deeper analysis of the complex interactions between brain tumours and their environment, with a particular focus on the impact of a growing cancer on healthy tissue, ventricular compression, and therapeutic treatment.

关键词： Brain tumours poroelasticity Anisotropic growth Ventricular compression Therapeutic protocols Finite element method

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Matrix Solutions of Biot's poroelasticity in Saturated Multilayered Media

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JOURNAL OF ELASTICITY 2025年第1期157卷 1-35页

作者： Yue, Zhongqi Quentin Harbin Inst Technol Shenzhen Sch Sci Shenzhen 518055 Guangdong Peoples R China Univ Hong Kong Dept Civil Engn Hong Kong Peoples R China

This paper presents analytical formulations for systematically deriving the solutions of Biot's poroelasticity in saturated multilayered media of either full-space or halfspace extents. The number of the saturated multilayer media is either n+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n+2$\end{document} for full-space extent or n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n+1$\end{document} for halfspace extent, where n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n$\end{document} is a positive or zero integer. The applied loadings include the internal forces and liquid source for full-space and both internal and external loadings for halfspace region with eight cases of four boundary conditions. The mathematical tools for the formulations are classical and include the two-dimensional Fourier transform, the Hankel transform, Laplace transform as well as linear algebra. The solutions are expressed in matrix forms and each matrix is explicitly expressed with clear physical meaning and well-defined elements. The matrix solutions in the Fourier and Laplace transform domains are axially symmetric about the vertical axis. The internal and boundary conditions can be four-dimensional and the matrix solutions in the physical domain are also four-dimensional.

关键词： poroelasticity Heterogeneity Matrix Fourier transform Laplace transform Solution

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New stabilized mixed finite element methods for two-field poroelasticity with low permeability

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APPLIED MATHEMATICS AND COMPUTATION 2025年 494卷

作者： He, Linshuang Jing, Luru Feng, Minfu Sichuan Univ Sch Math Chengdu 610065 Peoples R China

In this paper, we develop new stabilized mixed finite element (MFE) methods for two-field Biot's model of poroelasticity. We employ the H(div)-conforming element and discontinuous element to approximate the displacement and pressure variables, and use the theta-scheme to discretize time. By adding the stabilization term based on polynomial pressure projection, the fully-discrete and stabilized MFE methods are obtained. Our methods work well for both inf-sup stable and unstable element pairs, and provide oscillation-free pressure solutions in heterogeneous materials with low-permeable layers or interfaces. These methods are also volumetric locking-free and locally mass-conservative. We establish optimal apriori error estimates and perform numerical examples, which show the uniform robustness of the proposed methods for low permeability.

关键词： poroelasticity Low permeability Stabilized method Mixed finite element method

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2-D SPH modelling of poroelasticity: u-w-p and u-p formulations with absorbing boundary conditions and volumetric locking treatments

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COMPUTERS AND GEOTECHNICS 2025年 179卷

作者： Yao, C. Fourtakas, G. Rogers, B. D. Lombardi, D. UNIV MANCHESTER Sch Engn MANCHESTER M13 9PL England Hartree Ctr STFC Daresbury Lab Keckwick Lane Warrington WA4 4AD England

This paper presents a new simulation approach for modelling strongly-coupled processes in saturated porous media using the meshless method smoothed particle hydrodynamics (SPH). This paper proposes new SPH formulations for the a-w-p and a-p versions of Biot's model that address key challenges including accurate boundary conditions and numerical stability. Using corrected SPH gradient and Laplacian operators both absorbing and explicit treatment of free-surface conditions are achieved. The improved standard viscous boundary is then introduced into SPH as an absorbing boundary for problems involving dynamic loading. The volumetric locking issue is observed and anew countermeasure, named volumetric strain diffusion, is proposed, based on the well-established B-bar method and density diffusion technique. The proposed formulations are then systematically examined with six classical poroelastic problems spanning quasi-static and dynamic ranges. The simulated results agree with the reference solutions and display the desired order of convergence. The a-w- p and a-p models perform similarly under slow to moderate loading, but show distinct behaviour under highly dynamic loading. This observation matches existing understanding, thereby demonstrating the correctness of the proposed formulations. The introduced absorbing boundary condition and volumetric locking treatment also show promising results.

关键词： Biot's theory SPH poroelasticity Absorbing boundary Volumetric locking DualSPHysics u-w-p andu-p formulations

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The zeta potential for cancellous bone is calculated by poroelasticity

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MECHANICS RESEARCH COMMUNICATIONS 2025年 146卷

作者： Yoon, Young June 312-28 Myungil Dong We Bldg 5F Seoul 05297 South Korea

The zeta potential of cancellous bone is estimated by using poroelasticity. In colloidal chemistry, the zeta potential refers to the electric potential at the interface of the double layer. It is the potential difference between the stational layer on the solid bone matrix and interstitial fluid. The cancellous bone porosity, with a porosity of around 70-80 %, generates the peak of zeta potential in the wave propagation. If the zeta potential is small, attractive forces may exceed repulsive forces, and the aggregation of charged particles may be disrupted. If the zeta potential is significant, the charged molecules are stable and tend to aggregate. Thus, cancellous bone porosity, which ranges from 70 to 80 %, is ideal for the stability of charged molecules because a more significant zeta potential is required in this porosity range.

关键词： poroelasticity Cancellous bone Zeta potential Wave propagation Mathematical modeling

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Multi-porous extension of anisotropic poroelasticity: Consolidation and related coefficients

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INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS 2024年第8期48卷 2179-2206页

作者： Adamus, Filip P. Healy, David Meredith, Philip G. Mitchell, Thomas M. Stanton-Yonge, Ashley UCL Dept Earth Sci London England Univ Aberdeen Sch Geosci Aberdeen Scotland

We propose the generalization of the anisotropic poroelasticity theory. At a large scale, a medium is viewed as quasi-static, which is the original assumption of Biot. At a smaller scale, we distinguish different sets of pores or fractures that are characterized by various fluid pressures, which is the original poroelastic extension of Aifantis. In consequence, both instantaneous and time-dependent deformation lead to fluid content variations that are different in each set. We present the equations for such phenomena, where the anisotropic properties of both the solid matrix and pore sets are assumed. Novel poroelastic coefficients that relate solid and fluid phases in our extension are proposed, and their physical meaning is determined. To demonstrate the utility of our equations and emphasize the meaning of new coefficients, we perform numerical simulations of a triple-porosity consolidation. These simulations reveal positive pore pressure transients in the drained behaviour of weakly connected pore sets, and these may result in the mechanical weakening of the material.

关键词： anisotropy fractures multiple-permeability multiple-porosity poroelasticity rock mechanics

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A stabilized total pressure-formulation of the Biot's poroelasticity equations in frequency domain: Numerical analysis and applications

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2024年 432卷

作者： Carcamo, Cristian Caiazzo, Alfonso Galarce, Felipe Mura, Joaquin Leibniz Inst Forsch Verbund Berlin eV WIAS Weierstr Inst Angew Anal & Stochast Berlin Germany Pontificia Univ Catolica Valparaiso Sch Civil Engn Valparaiso Chile Univ Tecn Federico Santa Maria Dept Mech Engn Santiago Chile

This work focuses on the numerical solution of the dynamics of a poroelastic material in the frequency domain. We provide a detailed stability analysis based on the application of the Fredholm alternative in the continuous case, considering a total pressure formulation of the Biot's equations. In the discrete setting, we propose a stabilized equal order finite element method complemented by an additional pressure stabilization to enhance the robustness of the numerical scheme with respect to the fluid permeability. Utilizing the Fredholm alternative, we extend the well-posedness results to the discrete setting, obtaining theoretical optimal convergence for the case of linear finite elements. We present different numerical experiments to validate the proposed method. First, we consider model problems with known analytic solutions in two and three dimensions. As next, we show that the method is robust for a wide range of permeabilities, including the case of discontinuous coefficients. Lastly, we show the application for the simulation of brain elastography on a realistic brain geometry obtained from medical imaging.

关键词： Biot poroelasticity Magnetic resonance elastography Stabilized finite element

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Analysis of two discontinuous Galerkin finite element methods for the total pressure formulation of linear poroelasticity model

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APPLIED NUMERICAL MATHEMATICS 2024年 204卷 60-85页

作者： He, Linshuang Guo, Jun Feng, Minfu Sichuan Univ Sch Math Chengdu 610064 Sichuan Peoples R China Chengdu Univ Informat Technol Coll Appl Math Chengdu 610225 Sichuan Peoples R China

In this paper, we develop two discontinuous Galerkin (DG) finite element methods to solve the linear poroelasticity in the total pressure formulation, where displacement, fluid pressure, and total pressure are unknowns. The fully-discrete standard DG and conforming DG methods are presented based on the discontinuous approximations in space and the implicit Euler discretization in time. Compared to the standard DG method with penalty terms, the conforming DG method removes all stabilizers and maintains conforming finite element formulation by utilizing weak operators defined over discontinuous functions. The two methods provide locally conservative solutions and achieve locking-free properties in poroelasticity. We also derive the well-posedness and optimal a priori error estimates, which show that our methods satisfy parameter-robustness with respect to the infinitely large Lam & eacute;constant and the nullconstrained specific storage coefficient. Several numerical experiments are performed to verify these theoretical results, even in heterogeneous porous media.

关键词： poroelasticity Total pressure Discontinuous Galerkin Weak Galerkin Locking-free Local mass conservation

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