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We develop a computational method based on an Eulerian field called the "reference map", which relates the current location of a material point to its initial. The reference map can be discretized to permit finite-difference simulation of…

Soft Condensed Matter · Physics 2009-05-07 Ken Kamrin , Jean-Christophe Nave

In this work we present a mass conservative numerical scheme for two-phase flow in porous media. The model for flow consists on two fully coupled, non-linear equations: a degenerate parabolic equation and an elliptic equation. The proposed…

Numerical Analysis · Mathematics 2017-05-02 Florin Adrian Radu , Kundan Kumar , Jan Martin Nordbotten , Iuliu Sorin Pop

Pseudo-parabolic equations have been used to model unsaturated fluid flow in porous media. In this paper it is shown how a pseudo-parabolic equation can be upscaled when using a spatio-temporal decomposition employed in the…

Analysis of PDEs · Mathematics 2018-10-09 Arthur J. Vromans , Fons van de Ven , Adrian Muntean

In this article we present a goal-oriented adaptive finite element method for a class of subsurface flow problems in porous media, which exhibit seepage faces. We focus on a representative case of the steady state flows governed by a…

Numerical Analysis · Mathematics 2021-01-12 Ben Ashby , Cassiano Bortolozo , Alex Lukyanov , Tristan Pryer

We consider a porous medium being saturated with a pore fluid (Biot's theory). The fluid is assumed as incompressible. It is shown that the general integral of the elastic and pressure equations can be written in form of a time dependent…

Materials Science · Physics 2007-05-23 F. Tzschichholz

The fully-implicit time discretization (i.e. the backward Euler formula) is applied to compressible nonlinear dynamical models of thermo-viscoelastic solids in the Eulerian description, i.e. in the actual deforming configuration, formulated…

Numerical Analysis · Mathematics 2025-12-09 Tomáš Roubíček

Accretion and ablation, i.e. the addition and removal of mass at the surface, is important in a wide range of physical processes including solidification, growth of biological tissues, environmental processes, and additive manufacturing.…

Analysis of PDEs · Mathematics 2021-11-22 S. Kiana Naghibzadeh , Noel Walkington , Kaushik Dayal

We propose a sharp-interface model for a hyperelastic material consisting of two phases. In this model, phase interfaces are treated in the deformed configuration, resulting in a fully Eulerian interfacial energy. In order to penalize large…

Analysis of PDEs · Mathematics 2024-02-16 Katharina Brazda , Martin Kružík , Fabian Rupp , Ulisse Stefanelli

In this paper, we consider the flow of an incompressible fluid in a deformable porous solid. We present a mathematical model using the framework offered by the theory of interacting continua. In its most general form, this framework…

Numerical Analysis · Computer Science 2013-02-27 D. Z. Turner , K. B. Nakshatrala , M. J. Martinez

This paper develops and benchmarks an immersed peridynamics method to simulate the deformation, damage, and failure of hyperelastic materials within a fluid-structure interaction framework. The immersed peridynamics method describes an…

Numerical Analysis · Mathematics 2023-10-19 Keon Ho Kim , Amneet P. S. Bhalla , Boyce E. Griffith

Growth occurs in a wide range of systems ranging from biological tissue to additive manufacturing. This work considers surface growth, in which mass is added to the boundary of a continuum body from the ambient medium or from within the…

Analysis of PDEs · Mathematics 2021-05-27 Kiana Naghibzadeh , Noel Walkington , Kaushik Dayal

In this paper, we develop a discretization for the non-linear coupled model of classical Darcy-Forchheimer flow in deformable porous media, an extension of the quasi-static Biot equations. The continuous model exhibits a generalized…

Numerical Analysis · Mathematics 2021-05-24 Jakub Wiktor Both , Jan Martin Nordbotten , Florin Adrian Radu

This work develops and analyzes a variational-monolithic unfitted finite element formulation of a linear fluid-structure interaction problem in Eulerian coordinates with a fixed interface. The overall discretization is based on a backward…

Numerical Analysis · Mathematics 2026-03-27 Stefan Frei , Tobias Knoke , Marc C. Steinbach , Anne-Kathrin Wenske , Thomas Wick

In this work, an efficient physics-constrained deep learning model is developed for solving multiphase flow in 3D heterogeneous porous media. The model fully leverages the spatial topology predictive capability of convolutional neural…

Geophysics · Physics 2021-05-21 Bicheng Yan , Dylan Robert Harp , Bailian Chen , Rajesh Pawar

Hysteresis in the pressure-saturation relation in unsaturated porous media, which is due to surface tension on the liquid-gas interface, exhibits strong degeneracy in the resulting mass balance equation. Solutions to such degenerate…

Analysis of PDEs · Mathematics 2025-02-19 Chiara Gavioli , Pavel Krejčí

Mechanical modelling of poroelastic media under finite strain is usually carried out via phenomenological models neglecting complex micro-macro scales interdependency. One reason is that the mathematical two-scale analysis is only…

Computational Engineering, Finance, and Science · Computer Science 2021-05-07 Hamidreza Dehghani , Andreas Zilian

Knowledge of the underlying mechanisms of multiphase flow dynamics in porous media is crucial for optimizing subsurface engineering applications like geological carbon sequestration. However, studying the micro-mechanisms of multiphase…

Fluid Dynamics · Physics 2025-08-01 Quanwei Dai , Kang Duan , Chung-Yee Kwok

Biot's consolidation model is a classical model for the evolution of deformable porous media saturated by a fluid and has various interdisciplinary applications. While numerical solution methods to solve poroelasticity by typical schemes…

Numerical Analysis · Mathematics 2026-05-21 Stephan B. Lunowa , Barbara Wohlmuth

Soft porous materials, such as biological tissues and soils, are exposed to periodic deformations in a variety of natural and industrial contexts. The detailed flow and mechanics of these deformations have not yet been systematically…

Fluid Dynamics · Physics 2023-06-30 Matilde Fiori , Satyajit Pramanik , Christopher W. MacMinn

In this paper, we propose a linear and monolithic finite element method for the approximation of an incompressible viscous fluid interacting with an elastic and deforming plate. We use the arbitrary Lagrangian-Eulerian (ALE) approach that…

Numerical Analysis · Mathematics 2023-01-13 Sebastian Schwarzacher , Bangwei She , Karel Tuma