A multipoint stress mixed finite element method for elasticity on quadrilateral grids
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Mathematics↗
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A coupled multipoint stress–multipoint flux mixed finite element method for the Biot system of poroelasticity
In this work, we present a mixed finite element method for a five-field formulation of the Biot system of poroelasticity that reduces to a cell-centered pressure–displacement system on simplicial and quadrilateral grids. A mixed stress–displacement–rotation formulation for elasticity with weak stress symmetry is coupled with a mixed velocity–pressure Darcy formulation. The spatial discretization is based on combining the multipoint stress mixed finite element (MSMFE) method for elasticity and the multipoint flux mixed finite element (MFMFE) method for Darcy flow. It uses the lowest order Brezzi–Douglas–Marini mixed finite element spaces for the poroelastic stress and Darcy velocity, piecewise constant displacement and pressure, and continuous piecewise linear or bilinear rotation. A vertex quadrature rule is applied to the velocity, stress, and stress–rotation bilinear forms, which block-diagonalizes the corresponding matrices and allows for local velocity, stress, and rotation elimination. This leads to a cell-centered positive-definite system for pressure and displacement at each time step. We perform error analysis for the semidiscrete and fully discrete formulations, establishing first order convergence for all variables in their natural norms. The numerical tests confirm the theoretical convergence rates and illustrate the locking-free property of the method.
42 ENGINEERING↗
Efficient solvers for hybridized three-field mixed finite element coupled poromechanics
We consider a mixed hybrid finite element formulation for coupled poromechanics. A stabilization strategy based on a macro-element approach is advanced to eliminate the spurious pressure modes appearing in undrained/incompressible conditions. The efficient solution of the stabilized mixed hybrid block system is addressed by developing a class of block triangular preconditioners based on a Schur-complement approximation strategy. Robustness, computational efficiency and scalability of the proposed approach are theoretically discussed and tested using challenging benchmark problems on massively parallel architectures.
42 ENGINEERING↗
Droplet formation simulation using mixed finite elements
Droplet formation happens in finite time due to the surface tension force. The linear stability analysis is useful to estimate the size of a droplet but fails to approximate the shape of the droplet. This is due to a highly nonlinear flow description near the point where the first pinch-off happens. A one-dimensional axisymmetric mathematical model was first developed by Eggers and Dupont [“Drop formation in a one-dimensional approximation of the Navier–Stokes equation,” J. Fluid Mech. 262, 205–221 (1994)] using asymptotic analysis. This asymptotic approach to the Navier–Stokes equations leads to a universal scaling explaining the self-similar nature of the solution. Numerical models for the one-dimensional model were developed using the finite difference [Eggers and Dupont, “Drop formation in a one-dimensional approximation of the Navier–Stokes equation,” J. Fluid Mech. 262, 205–221 (1994)] and finite element method [Ambravaneswaran et al., “Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops,” Phys. Fluids 14, 2606–2621 (2002)]. The focus of this study is to provide a robust computational model for one-dimensional axisymmetric droplet formation using the Portable, Extensible Toolkit for Scientific Computation. The code is verified using the Method of Manufactured Solutions and validated using previous experimental studies done by Zhang and Basaran [“An experimental study of dynamics of drop formation,” Phys. Fluids 7, 1184–1203 (1995)]. The present model is used for simulating pendant drops of water, glycerol, and paraffin wax, with an aspiration of extending the application to simulate more complex pinch-off phenomena.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
A GPU-Enabled Higher Order Mixed Finite Element Method for Pulsed Power ICF Target Design
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Physics - Plasma physics↗
Algebraic Multigrid for mixed finite element discretizations of the Stokes Equations.
Abstract not provided.
Tuminaro, Raymond↗
Generalized Korn’s inequalities for piecewise $H¹$ and $H²$ vector fields
The purpose of this paper is to construct a new class of discrete generalized Korn’s inequalities for piecewise $H^1$ vector fields and piecewise $H^2$ vector fields in three-dimensional space. The resulting Korn’s inequalities are different from the standard Korn’s inequalities, as they involve the trace-free symmetric gradient operator, in place of the usual symmetric gradient operator. Furthermore, it is anticipated that the new generalized Korn’s inequalities will be useful for the analysis of a broad range of finite element methods, including mixed finite element methods and discontinuous Galerkin methods.
97 MATHEMATICS AND COMPUTING↗
Data-driven Whitney forms for structure-preserving control volume analysis
Control volume analysis models physics via the exchange of generalized fluxes between subdomains. Here, we introduce a scientific machine learning framework adopting a partition of unity architecture to identify physically-relevant control volumes, with generalized fluxes between subdomains encoded via Whitney forms. The approach provides a differentiable parameterization of geometry which may be trained in an end-to-end fashion to extract reduced models from full field data while exactly preserving physics. The architecture admits a data-driven finite element exterior calculus allowing discovery of mixed finite element spaces with closed form quadrature rules. An equivalence between Whitney forms and graph networks reveals that the geometric problem of control volume learning is equivalent to an unsupervised graph discovery problem. The framework is developed for manifolds in arbitrary dimension, with examples provided for H(div) problems in $\mathbb{R}$ establishing convergence and structure preservation properties. Finally, we consider a lithium-ion battery problem where we discover a reduced finite element space encoding transport pathways from high-fidelity microstructure resolved simulations. The approach reduces the 5.89M finite element simulation to 136 elements while reproducing pressure to under 0.1% error and preserving conservation.
97 MATHEMATICS AND COMPUTING↗
A fourth-order phase-field fracture model: Formulation and numerical solution using a continuous/discontinuous Galerkin method
Modeling crack initiation and propagation in brittle materials is of great importance to be able to predict sudden loss of load-carrying capacity and prevent catastrophic failure under severe dynamic loading conditions. Second-order phase-field fracture models have gained wide adoption given their ability to capture the formation of complex fracture patterns, e.g. via crack merging and branching, and their suitability for implementation within the context of the conventional finite element method. Higher-order phase-field models have also been proposed to increase the regularity of the exact solution and thus increase the spatial convergence rate of its numerical approximation. However, they require special numerical techniques to enforce the necessary continuity of the phase field solution. In this paper, we derive a fourth-order phase-field model of fracture in two independent ways; namely, from Hamilton’s principle and from a higher-order micromechanics-based approach. The latter approach is novel, and provides a physical interpretation of the higher-order terms in the model. In addition, we propose a continuous/discontinuous Galerkin (C/DG) method for use in computing the approximate phase-field solution. This method employs Lagrange polynomial shape functions to guarantee -continuity of the solution at inter-element boundaries, and enforces the required regularity with the aid of additional variational and interior penalty terms in the weak form. Finally, the phase-field equation is coupled with the momentum balance equation to model dynamic fracture problems in hyper-elastic materials. Two benchmark problems are presented to compare the numerical behavior of the C/DG method with mixed finite element methods.
42 ENGINEERING↗
A continuum framework for coupled solid deformation–fluid flow through anisotropic elastoplastic porous media
We present a continuum framework for coupled solid deformation–fluid flow in anisotropic elastoplastic porous media. A thermodynamic formulation of the coupled processes gives rise to an anisotropic Biot tensor that is a function of drained elastic tangent moduli tensor of the solid skeleton and the intrinsic bulk modulus of the solid constituent. Two effective stress measures emerge from the formulation, namely, σ', which is energy-conjugate to the elastic strain, and σ'', which is energy-conjugate to the plastic strain. For the special case of transverse isotropy that is commonly encountered in natural rocks, the Biot tensor can be expressed in terms of its normal and tangential components to the bedding plane, along with a microstructure tensor. Apart from its thermodynamic consistency, an advantage of this new formulation is that standard mixed finite element formulation can be employed to discretize the domain and solve initial boundary-value problems. We conduct plane strain simulations of coupled solid deformation–fluid flow in a transversely isotropic porous medium to demonstrate the impacts of material anisotropy, stress history, and the Biot tensor on the system response.
42 ENGINEERING↗
Berkeley Lab Finite Element Framework (BELFEM) v0.1
The software program, referred to as BELFEM, is a specialized finite element code designed for the magnetodynamic modeling of high-temperature superconducting (HTS) tapes. It incorporates novel mixed finite element formulations, particularly the h-ϕ-formulation with thin-shell simplification, to efficiently simulate larger geometries. This methodology is extremely promising for predicting the electrodynamic performance of HTS tapes used in superconducting cables and magnets, offering the benefit of reduced computational cost. Compared to similar technologies like COMSOL Multiphysics and GetDP, BELFEM's performance benchmarking indicates superior efficiency in its thin-shell implementation. In the future, it will also support features like thermal coupling and inter-tape current sharing, enhancing its utility in research and development, particularly in nuclear fusion applications. The intent is to develop BELFEM as a robust and efficient tool for the HTS community, contributing to the analysis and design of superconducting cables and magnets.
Messe, Christian↗
Improving the accuracy of discretisations of the vector transport equation on the lowest-order quadrilateral Raviart-Thomas finite elements
Within finite element models of fluids, vector-valued fields such as velocity or momentum variables are commonly discretised using the Raviart-Thomas elements. However, when using the lowest-order quadrilateral Raviart-Thomas elements, standard finite element discretisations of the vector transport equation typically have a low order of spatial accuracy. This paper describes two schemes that improve the accuracy of transporting such vector-valued fields on two-dimensional curved manifolds. The first scheme that is presented reconstructs the transported field in a higher-order function space, where the transport equation is then solved. The second scheme applies a mixed finite element formulation to the vector transport equation, simultaneously solving for the transported field and its vorticity. In this work, an approach to stabilising this mixed vector-vorticity formulation is presented that uses a Streamline Upwind Petrov-Galerkin (SUPG) method. These schemes are then demonstrated, along with their accuracy properties, through some numerical tests. Two new test cases are used to assess the transport of vector-valued fields on curved manifolds, solving the vector transport equation in isolation. The improvement of the schemes is also shown through two standard test cases for rotating shallow-water models.
97 MATHEMATICS AND COMPUTING↗
Physics-preserving enriched Galerkin method for a fully-coupled thermo-poroelasticity model
This paper proposes a new numerical method for a fully-coupled, quasi-static thermo-poroelasticity model in a unified enriched Galerkin (EG) method framework. In our method, the mechanics sub-problem is solved using a locking-free EG method, and the flow and heat sub-problems are solved using a locally-conservative EG method. The proposed method offers mass and energy conservation properties with much lower costs than other methods with the same properties, including discontinuous Galerkin methods and mixed finite element methods. The well-posedness and optimal a priori error estimates are carefully derived. Here, several numerical tests confirm the theoretical optimal convergence rates and the mass and energy conservation properties of the new method.
15 GEOTHERMAL ENERGY↗
Approximate inverse-based block preconditioners in poroelasticity
We focus on the fully implicit solution of the linear systems arising from a three-field mixed finite element approximation of Biot’s poroleasticity equations. The objective is to develop algebraic block preconditioners for the efficient solution of such systems by Krylov subspace methods. In this work, we investigate the use of approximate inverse-based techniques to decouple the native system of equations and obtain explicit sparse approximations of the Schur complements related to the physics-based partitioning of the unknowns by field type. Here, the proposed methods are tested in various numerical experiments including real-world applications dealing with petroleum and geotechnical engineering.
54 ENVIRONMENTAL SCIENCES↗
Anisotropic elastoplastic response of double-porosity media
Here we present a continuum framework to simulate fluid flow through anisotropic elastoplastic media with double porosity. Two effective stress measures σ' and δ" emerge from the thermodynamic formulation, which are energy-conjugate to the elastic and plastic components of strain, respectively. Both effective stress measures can be expressed as a combination of the total Cauchy stress σ and the average pore pressure $\bar{p}$ in the two pore scales. In the effective stress for elasticity, $\bar{p}$ is scaled with a rank-2 Biot tensor, whereas the effective stress for plasticity follows the Terzaghi form in which $\bar{p}$ is scaled by the Kronecker delta. The Biot tensor and storage coefficients are derived as functions of elasticity parameters and porosities. A mixed finite element formulation is introduced to discretize the domain and solve initial boundary-value problems. A stabilization scheme is employed on equal-order interpolation for both displacement and pressure fields throughout the entire range of drainage responses. Numerical simulations reproduce the hydromechanical response of Opalinus shale in one-dimensional consolidation tests throughout the range of primary and secondary consolidation under different external loads. Numerical simulations of the consolidation of a rectangular domain subjected to a strip load demonstrate the efficacy of the proposed stabilization scheme, as well as illustrate the impacts of stress history, mass transfer, and different pore systems on the hydromechanical response.
42 ENGINEERING↗
A robust framework for frictional fault contact in geological formations using a stabilized augmented Lagrangian approach
Numerical simulations are essential to evaluate the performance and safety of engineered subsurface systems such as geological carbon storage sites, enhanced geothermal fields, and oil and gas reservoirs. A key challenge lies in accurately modeling the frictional contact behavior along fault surfaces. This problem involves inequality constraints that arise from the physics of frictional slip, requiring specialized numerical methods to handle the resulting highly nonlinear and path-dependent behavior. Here, in this work, we address this challenge using an Augmented Lagrangian Method (ALM) implemented via the Uzawa algorithm. The formulation employs mixed finite element spaces, combining low-order piecewise linear displacements within the 3D domain cells with piecewise constant tractions defined on the fault surfaces. Furthermore, to ensure stability and satisfy the inf-sup condition, the discrete displacement space is enriched with face bubble functions on both sides of the contact interfaces. This approach offers several advantages over other stabilization techniques that rely on additional terms, and it integrates naturally in the Uzawa framework.
58 GEOSCIENCES↗
Energy based fracture initiation criterion for strain-crystallizing rubber-like materials with pre-existing cracks
Fracture prediction is indispensable for polymers, like rubbers, which have a broad range of applications mainly due to their high extensibility. The phenomenon known as strain-induced crystallization further contributes to the fracture toughness of certain rubbers. Here in this study, a criterion based on internal bond energy, incorporating the effects of crystallization, is proposed to predict fracture initiation in rubber-like materials with pre-existing cracks. First, a multi-scale mechanical model is developed for characterizing the behavior of rubber when subjected to both uniaxial and biaxial deformation states. At the microscale, both the amorphous and crystalline chain segments are modeled as elastic in order to consider the energy contribution by the molecular bond distortions. This internal energy is considered along with the entropic and crystalline free energy for each chain. In the chain model, the effects of loading condition and the relative orientation of a chain on its crystallinity are taken into account. At the macroscopic scale, an existing crystallinity distribution function is adapted and a mixed finite element formulation with an augmented Lagrangian multiplier is utilized to impose the incompressibility constraint. A non-affine maximal advance path constraint based homogenization model is utilized for bridging the two scales. Its potential to account for anisotropy in the stretched network compels the model to be preferable due to its physical significance, for the purpose of fracture modeling. The rigidity of the crystallites is accounted for by proposing a crystallite distortion energy in addition to the critical bond dissociation energy, for fracture initiation to occur. The model is validated by comparison with existing experimental results for both crystallizing and non-crystallizing rubbers. In addition to its potential to predict the material behavior when subjected to uniaxial and biaxial loading, the capability of the model to quantitatively estimate the effect of crystallization on fracture initiation is also verified.
36 MATERIALS SCIENCE↗
A Domain-Decomposed A-ϕ Formulation Based on Lagrange Multipliers for Low-Frequency Problems
A domain-decomposed A-ϕ formulation based on Lagrange multipliers is proposed to simulate low-frequency elec- tromagnetic problems. This method partitions the computational domain into smaller subdomains, allowing each subdomain to be independently formulated using Lagrange multipliers as Dirichlet boundary conditions, while ensuring continuity of the fields across the interfaces. A mixed finite element method, utilizing both vector and scalar basis functions, is employed to discretize the formulation, resulting in a global system to be solved. The proposed method is validated using TEAM Problem 7 at 50 Hz, demonstrating its effectiveness in handling complex geometries and addressing the low-frequency breakdown issues commonly encountered in traditional finite element methods.
Hossain, Amzad↗