Mixed Finite Element Research Articles

A numerical experiment on the stability and convergence characteristics of some splitting mixed-finite element methods for solving the incompressible Navier-Stokes equations

A numerical experiment on the stability and convergence characteristics of some splitting mixed-finite element methods for solving the incompressible Navier-Stokes equations

An online generalized multiscale approximation of the multipoint flux mixed finite element method

In this paper, we develop an online generalized multiscale approximation of the multipoint flux mixed finite element method (MFMFE) for Darcy flow in highly heterogeneous porous media, which can handle full tensor permeability fields and unstructured grids with local velocity elimination. Here, for illustration of the proposed method, we will mainly consider the situation that the porous medium is filled with fractures, which usually results in unstructured grids, and the extension to the situation that the permeability is represented as a full tensor is straightforward. From the fine-grid discretization by the MFMFE method, we derive the symmetric and positive definite discrete bilinear form for both the matrix and fracture pressure, and thereby obtain the corresponding discrete weak formulation that only related to pressure. In the offline stage, we compute offline basis functions which contain the important local multiscale information of each coarse-grid block to form the initial multiscale space. In the online stage, we construct and add online basis functions to enrich the multiscale space and consequently improve the accuracy of the multiscale solution. Each offline basis function and each online basis function contains multiscale information both for matrix and fracture. We give the theoretical analysis for the convergence of the online enrichment, which shows that more sufficient initial basis functions lead to faster convergence rates. A series of numerical examples that either the permeability is a full tensor or the porous medium is filled with fractures are presented to show the performance of the multiscale method.

Unfitted mixed finite element methods for elliptic interface problems

AbstractIn this article, new unfitted mixed finite elements are presented for elliptic interface problems with jump coefficients. Our model is based on a fictitious domain formulation with distributed Lagrange multiplier. The relevance of our investigations is better seen when applied to the framework of fluid‐structure interaction problems. Two finite element schemes with piecewise constant Lagrange multiplier are proposed and their stability is proved theoretically. Numerical results compare the performance of those elements, confirming the theoretical proofs and verifying that the schemes converge with optimal rates.

An enriched mixed finite element model for the simulation of microseismic and acoustic emissions in fractured porous media with multi‐phase flow and thermal coupling

AbstractA novel mixed enriched finite element model is developed for coupled non‐linear thermo‐hydro‐mechanical simulation of fractured porous media with three‐phase flow and thermal coupling. Simulation of induced acoustic emission (AE) and microseismic emission (ME) due to tensile fracturing and shear slip instability of pre‐existing fracture interfaces is carried out and the numerical results of the emitted signals are analysed. The mathematical model is based on the generalized Biot's theory for coupled interaction of solid and fluid phases. A computationally robust non‐linear solver is developed to handle the severe non‐linearities arising from fluid saturations, relative permeabilities of fluids, constitutive models of interfaces and convective thermal coupling. To model pre‐existing natural fractures and faults, discrete fracture propagation and nucleation of cracks (micro‐cracking) independently of the original mesh topology, a local Partition‐of‐Unity (PU) finite element method, namely, the Phantom Node Method (PNM) is implemented. The cohesive fracture modelling scheme is implemented to account for the non‐linear behaviour of fracturing and localization, and to rectify the non‐physical stress singularity condition at the fracture tip. Effects of different system parameters on fracturing, shear‐slip instability and the associated induced AEs and MEs are investigated through various numerical results.

A comprehensive analysis of in-plane functionally graded plates using improved first-order mixed finite element model

In this study, the mechanical behaviors of the in-plane functionally graded plate (IFG) are comprehensively explored via a finite element (FE) model. The current FE model is developed base on improved first-order shear deformation theory (FSDT) in conjunction with the mixed FE formulation. The volume fraction as well as the material properties change smoothly along the width and length of the plates for six different patterns. A simple improved transverse shear stress is applied to modify its distribution across the thickness of the IFG plates. Hence, the transverse shear stress equals zeros on the lower and upper surfaces of the IFG plates. Besides, no shear correction factor is needed for calculation of shear strain energy. A four-node mixed plate element, IMQ4, is established via mixed FE formulation for mechanical analysis of the IFG plates. The accuracy and fast convergence rate are validated via some examples. A comprehensive parametric analysis is provided to demonstrate the roles of several parameters including the material gradient index, the side-to-thickness ratio, aspect ratio, and boundary conditions on the mechanical behaviors of the IFG plates.

Galerkin mixed finite element method for parabolic p-biharmonic equation with memory term

Galerkin mixed finite element method for parabolic p-biharmonic equation with memory term

Efficient algorithm based on two-grid method for semiconductor device problem

We study a three-step two-grid algorithm for solving semiconductor device equations that is discretized by the standard finite element method for the concentration equation and the mixed finite element method for the electric potential equation. The main procedures involve solving the original nonlinear equations on the coarse grid to obtain the initial approximate solution, as well as the linearized equations on the fine grid to obtain a modified solution; furthermore, the calculation on the fine grid is corrected. Error estimation for the method shows that the coarse grid can be much coarser than the fine grid, and the two-grid method still preserves asymptotically optimal approximation. Numerical experiments validate the reliability and effectiveness of the algorithm.

On augmented finite element formulations for the Navier–Stokes equations with vorticity and variable viscosity

In this work, we propose and analyse an augmented mixed finite element method for solving the Navier–Stokes equations describing the motion of incompressible fluid. The model is written in terms of velocity, vorticity, and pressure, and takes into account non-constant viscosity and no-slip boundary conditions. The weak formulation of the method includes least-squares terms that arise from the constitutive equation and the incompressibility condition. We discuss the theoretical and practical implications of using augmentation in detail. Additionally, we use fixed–point strategies to show the existence and uniqueness of continuous and discrete solutions under the assumption of sufficiently small data. The method is constructed using any compatible finite element pair for velocity and pressure, as dictated by Stokes inf-sup stability, while for vorticity, any generic discrete space of arbitrary order can be used. We establish optimal a priori error estimates and provide a set of numerical tests in 2D and 3D to illustrate the behaviour of the discretisations and verify their theoretical convergence rates. Overall, this method provides an efficient and accurate solution for simulating fluid flow in a wide range of scenarios.

Block constrained pressure residual preconditioning for two-phase flow in porous media by mixed hybrid finite elements

This work proposes an original preconditioner that couples the Constrained Pressure Residual (CPR) method with block preconditioning for the efficient solution of the linearized systems of equations arising from fully implicit multiphase flow models. This preconditioner, denoted as Block CPR (BCPR), is specifically designed for Lagrange multipliers-based flow models, such as those generated by Mixed Hybrid Finite Element (MHFE) approximations. An original MHFE-based formulation of the two-phase flow model is taken as a reference for the development of the BCPR preconditioner, in which the set of system unknowns comprises both element and face pressures, in addition to the cell saturations, resulting in a 3×3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3\ imes 3$$\\end{document} block-structured Jacobian matrix with a 2×2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\ imes 2$$\\end{document} inner pressure problem. The CPR method is one of the most established techniques for reservoir simulations, but most research focused on solutions for Two-Point Flux Approximation (TPFA)-based discretizations that do not readily extend to our problem formulation. Therefore, we designed a dedicated two-stage strategy, inspired by the CPR algorithm, where a block preconditioner is used for the pressure part with the aim at exploiting the inner 2×2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\ imes 2$$\\end{document} structure. The proposed preconditioning framework is tested by an extensive experimentation, comprising both synthetic and realistic applications in Cartesian and non-Cartesian domains.

Computational Multiscale Methods for Nondivergence-Form Elliptic Partial Differential Equations

Abstract This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence form.

Stabilized mixed finite element method for a quasistatic Maxwell viscoelastic model

This paper considers a stabilized mixed finite element method (MFE) for a quasistatic Maxwell viscoelastic model based on the L2(Ω)×H1(Ω) variational framework. The spatial discretization uses arbitrary degree piecewise polynomials Pl−Pk(l≥0,k≥1) to approximate stress and velocity. The temporal discretization in the fully discrete method adopts a backward Euler difference scheme. We give a unified analysis to show the stability of the semi-discrete and fully discrete solutions and derive the corresponding priori error estimates. Numerical examples are provided to support the theoretical analysis.

Superconvergence analysis of the bilinear‐constant scheme for two‐dimensional incompressible convective Brinkman–Forchheimer equations

AbstractIn this article, a low order conforming mixed finite element method is proposed and investigated for two‐dimensional convective Brinkman–Forchheimer equations. Based on the special properties of the bilinear‐constant finite element pair on the rectangular mesh and the careful treatment of the nonlinear terms, the superclose error estimates for velocity in ‐norm and pressure in ‐norm are obtained. Then, in terms of interpolation post‐processing technique, the global superconvergence results are derived. Finally, some numerical experiments are carried out to demonstrate the correctness of the theoretical findings.

Two-Grid Method for a Fully Discrete Mixed Finite Element Solution of the Time-Dependent Schrödinger Equation

We study the backward Euler fully discrete mixed finite element method for the time-dependent Schrödinger equation; the error result of the mixed finite element solution is obtained in the L2-norm with order O(τ+hk+1). Then, a two-grid method is presented with a backward Euler fully discrete scheme. Using this method, we solve the original problem on a much coarser grid and solve elliptic equations on a fine grid. In addition, the error of the two-grid solution is also obtained in the L2-norm with order O(τ+hk+1+Hk+2). The numerical experiment is provided to demonstrate the efficiency of the algorithm.

A mixed finite element approach for a factional viscoelastic wave propagation in-time-domain

A mixed finite element approach for a factional viscoelastic wave propagation in-time-domain

Modeling the flexoelectric effect around the tip of nano-cracks using a collocation MFEM

Due to the stress concentration and large strain gradient around the crack tip, strong flexoelectric effect would be produced there. Meanwhile, the flexoelectric effect could conversely affect the mechanical fields in materials, thus influence the fracture property of materials. To model the impact of flexoelectricity on crack tip fields and the electromechanical behaviors of materials, intensive mesh refinement is needed around the crack tip. Thus, an efficient numerical method is desired. In this paper, the flexoelectric effect around the tip of nano-cracks is simulated by using a collocation mixed finite element method (MFEM). Our collocation MFEM is based on C0 continuous approximation and does not involve additional degrees of freedom (DOFs), so that it is quite efficient, which has been verified in our previous study (Tian et al., 2021). Using the collocation MFEM, we simulate the electric potential and field distributions around the crack tip in flexoelectric materials, and the influences of the direct flexoelectric effect on the crack tip opening displacement and J-integral are also studied. Numerical results indicate that the flexoelectric effect around the crack tip would be enhanced by increasing the tensile loading and crack length. Besides, the direct flexoelectric effect around the crack tip enhances the local strength of materials, and plays an important role in fracture mechanics of materials.

A pressure-robust mixed finite element method for the coupled Stokes–Darcy problem

A pressure-robust numerical method for the coupled Stokes–Darcy problem is proposed. We use the continuous piecewise linear polynomial space combined with the lowest order Raviart–Thomas space to discretize the Stokes equation in the fluid region and the lowest order Raviart–Thomas elements to discretize the Darcy equation in the porous media domain. We also perform a divergence-free reconstruction of the test function for the right-hand term. Our method has good properties of convergence, pressure-robustness and local mass conservation in the sense of projection. A discrete inf-sup condition and error estimates for the numerical scheme are demonstrated, the error of velocity is independent of viscosity and pressure. Finally, the theoretical analysis is validated with several examples, demonstrating the superiority of the pressure-robust scheme.

Large displacements and rotations of a local linear elastic rod

AbstractThe Local Linear Timoshenko (LLT) model for the planar motion of a rod that undergoes flexure, shear and extension, was recently derived in Van Rensburg et al. (2021). In this paper we present an algorithm developed for this model. The algorithm is based on the mixed finite element method, and projections into finite dimensional subspaces are used for dealing with nonlinear forces and moments. The algorithm is used for an investigation into elastic waves propagated in the LLT rod. Interesting properties of the LLT rod include the increased propagation speed of elastic waves when compared to the linear Timoshenko beam, and the appearance of buckled states or equilibrium solutions for compressed LLT beams. It is also shown that the LLT rod is applicable to large displacements and rotations for a wide range of slender elastic objects; from beams to highly slender flexible rods.

Coupled mixed finite element and finite volume methods for a solid velocity-based model of multidimensional sedimentation

In this paper we introduce and analyze a model of sedimentation based on a solid velocity formulation. A particular feature of the governing equations is given by the fact that the velocity field is non-divergence free. We introduce extra variables such as the pseudostress tensor relating the velocity gradient with the pressure, thus leading to a mixed variational formulation consisting of two systems of equations coupled through their source terms. A result of existence and uniqueness of solutions is shown by means of a fixed-point strategy and the help of the Babuška–Brezzi theory and Banach theorem. Additionally, we employ suitable finite dimensional subspaces to approximate both systems of equations via associated mixed finite element methods. The well-posedness of the resulting coupled scheme is also treated via a fixed-point approach, and hence the discrete version of the existence and uniqueness result is derived analogously to the continuous case. The above is then combined with a finite volume method for the transport equation. Finally, several numerical results illustrating the performance of the proposed model and the full numerical scheme, and confirming the theoretical rates of convergence, are presented.

Bifurcation Analysis and Steady-State Patterns in Reaction–Diffusion Systems Augmented with Self- and Cross-Diffusion

In this article, a study of long-term behavior of reaction–diffusion systems augmented with self- and cross-diffusion is reported, using an augmented Gray–Scott system as a generic example. The methodology remains general, and is therefore applicable to some other systems. Simulations of the temporal model (nonlinear parabolic system) reveal the presence of steady states, often associated with energy dissipation. A Newton method based on a mixed finite element method is provided, in order to directly evaluate the steady states (nonlinear elliptic system) of the temporal system, and validated against its solutions. Linear stability analysis using Fourier analysis is carried out around homogeneous equilibrium, and using spectral analysis around nonhomogeneous ones. For the latter, the spectral problem is solved numerically. A multiparameter bifurcation is reported. Original steady-state patterns are unveiled, not observable with linear diffusion only. Two key observations are: a dependency of the pattern with the initial condition of the system, and a dependency on the geometry of the domain.

Modeling the Interaction Between Inclusions and Nanocracks in Flexoelectric Solids

Abstract Natural defects such as nano inclusions and nanocracks are inevitable in dielectric materials. When materials are subjected to mechanical loading, the strain gradient around crack tips and inclusions would become large and induce significant flexoelectric fields. In contrast to classical crack–inclusion problems, the interactions between these flexoelectric fields may locally change the electromechanical behaviors of materials and result in some interesting phenomena. To better understand the crack–inclusion interactions in flexoelectric solids, in this work, we use a collocation mixed finite element method to model and analyze the flexoelectric fields around the crack tip and inclusion. On the basis of the J-integral, we analyze how the flexoelectric effect affect the interaction energy between nanocracks and nearby nano inclusions. This work proposes a new coupling mechanism in crack–inclusion problems and may inspire future experiments in flexoelectric solids.