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A vorticity-based mixed formulation for the unsteady Brinkman–Forchheimer equations
... Experimental observations, summarized mathematically as Darcy's law, govern the momentum equations in such scenarios. However, Darcy's law proves insufficient under conditions of Reynolds numbers greater than one or high porosity [2,8,31,32]. To address this limitation, an extension to the traditional Darcy's law is introduced as the Brinkman term, specifically designed to account for transitional flow between boundaries [6,7]. ...
... Assumption 3.1. There exists a positive real number ρ such that for all h and for every E ∈ Ω h : (1) the ratio between the shortest edge e min and the diameter h E of E is greater than ρ, i.e., emin h E > ρ; (2) E is star-shaped with respect to a ball of radius ρh E and center x E ∈ E. ...
... (High Reynolds number). As mentioned in [2,9,37], the emergence of the Forcheheimier term overcomes the inaccuracies caused by large Reynolds numbers in fluid simulations. In order to verify the robustness with respect to parameter ν, we fix the parameters α = 1, F = 1, p = 4 and choose different values for ν to verify the convergence of the problem. ...
... The method is robust with respect to the Brinkman parameter. More recently, a vorticity-based mixed variational formulation is analyzed in [2], where the velocity, vorticity, and pressure are the main unknowns of the system. Existence and uniqueness of a weak solution, as well as stability bounds are derived by employing classical results on nonlinear monotone operators. ...
... The Brinkman model [9] incorporates both viscous and permeability effects, enabling precise simulations of fluid movement in diverse environments, including highly porous media. On the other hand, many applications of interest involve Furthermore, unlike [4], [3], or [2], our method does not require any augmentation process. It is also important to mention that another novelty and advantage of the present work is that it generalizes the model studied in [2] by including the nonlinear convective term and an additional time-derivative term, thus considering viscoelastic flows. ...
... On the other hand, many applications of interest involve Furthermore, unlike [4], [3], or [2], our method does not require any augmentation process. It is also important to mention that another novelty and advantage of the present work is that it generalizes the model studied in [2] by including the nonlinear convective term and an additional time-derivative term, thus considering viscoelastic flows. ...
In this paper, we propose and analyze a mixed formulation for the Kelvin-Voigt-Brinkman-Forchheimer equations for unsteady viscoelastic flows in porous media. Besides the velocity and pressure, our approach introduces the vorticity as a further unknown. Consequently, we obtain a three-field mixed variational formulation, where the aforementioned variables are the main unknowns of the system. We establish the existence and uniqueness of a solution for the weak formulation, and derive the corresponding stability bounds, employing a fixed-point strategy, along with monotone operators theory and Schauder theorem. Afterwards, we introduce a semidiscrete continuous-in-time approximation based on stable Stokes elements for the velocity and pressure, and continuous piecewise polynomial spaces for the vorticity. Additionally, employing backward Euler time discretization, we introduce a fully discrete finite element scheme. We prove well-posedness, derive stability bounds, and establish the corresponding error estimates for both schemes. We provide several numerical results verifying the theoretical rates of convergence and illustrating the performance and flexibility of the method for a range of domain configurations and model parameters.
... The formulation of viscous flow equations using vorticity, velocity and pressure has been used and analysed (in terms of solvability of the continuous and discrete formulations and deriving error estimates) extensively in, e.g., [1,2,7,8,6,9,14,24,28,29,58]. Methods based on vorticity formulations are useful for visualisation of rotational flows and they are convenient when dealing with rotation-based boundary conditions. ...
In this paper, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. The proposed method is based on the use of parameter-weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis of the proposed method and derive optimal error estimates in appropriate norms. These error estimates are shown to be robust in the case of large Lam\'e parameters and small permeability and storativity coefficients. To illustrate the effectiveness of the proposed method, we provide a few representative numerical examples, including convergence verification, poroelastic channel flow simulation, and test the robustness of block-diagonal preconditioners with respect to model parameters.
In this article, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence‐conforming filtration fluxes. The proposed method is based on the use of parameter‐weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis of the proposed method and derive optimal error estimates in appropriate norms. These error estimates are shown to be robust in a variety of regimes, including the case of large Lamé parameters and small permeability and storativity coefficients. To illustrate the effectiveness of the proposed method, we provide a few representative numerical examples, including convergence verification and testing of robustness of block‐diagonal preconditioners with respect to model parameters.
We propose and analyze a new mixed finite element method for the nonlinear problem given by the coupling of the steady Brinkman--Forchheimer and double-diffusion equations. Besides the velocity, temperature, and concentration, our approach introduces the velocity gradient, the pseudostress tensor, and a pair of vectors involving the temperature/concentration, its gradient and the velocity, as further unknowns. As a consequence, we obtain a fully mixed variational formulation presenting a Banach spaces framework in each set of equations. In this way, and differently from the techniques previously developed for this and related coupled problems, no augmentation procedure needs to be incorporated now into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Banach theorem, combined with classical results on nonlinear monotone operators and the Babusska-Brezzi theory in Banach spaces, are applied to prove the unique solvability of the continuous and discrete systems. Appropriate finite element subspaces satisfying the required discrete inf-sup conditions are specified, and optimal a priori error estimates are derived. Several numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method.
This paper deals with the finite element approximation of the Darcy-Brinkman-Forchheimer equation, involving a porous media with spatially-varying porosity, with mixed boundary condition such as inhomogeneous Dirichlet and traction boundary conditions. We first prove that the considered problem has a unique solution if the source terms are small enough. The convergence of a Taylor-Hood finite element approximation using a finite element interpolation of the porosity is then proved under similar smallness assumptions. Some optimal error estimates are obtained if the solution to the Darcy-Brinkman-Forchheimer model are smooth enough. We end this paper by providing a fixed-point method to solve the discrete non-linear problems and with some numerical experiments to make more precise the smallness assumptions on the source terms and to illustrate the theoretical convergence results.
In this brief note, we introduce a non-symmetric mixed finite element formulation for Brinkman equations written in terms of velocity, vorticity, and pressure with non-constant viscosity. The analysis is performed by the classical Babuška–Brezzi theory, and we state that any inf–sup stable finite element pair for Stokes approximating velocity and pressure can be coupled with a generic discrete space of arbitrary order for the vorticity. We establish optimal a priori error estimates, which are further confirmed through computational examples.
p>The FEniCS Project is a collaborative project for the development of
innovative concepts and tools for automated scientific computing,
with a particular focus on the solution of differential equations by
finite element methods. The FEniCS Projects software consists of a
collection of interoperable software components, including DOLFIN,
FFC, FIAT, Instant, UFC, UFL, and mshr. This note describes the new
features and changes introduced in the release of FEniCS
version 1.5.</p
We develop a loosely coupled fluid-structure interaction finite element solver based on the Lie operator splitting scheme. The scheme is applied to the interaction between an incompressible, viscous, Newtonian fluid, and a multilayered structure, which consists of a thin elastic layer and a thick poroelastic material. The thin layer is modeled using the linearly elastic Koiter membrane model, while the thick poroelastic layer is modeled as a Biot system. We prove a conditional stability of the scheme and derive error estimates. Theoretical results are supported with numerical examples. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2014
In this paper we consider the continuous piecewise linear finite element approximation of the following problem: Given p in (1,∞ ) , f, and g, find u such that - nabla \cdot (\vertnabla u{\vert^{p - 2}}nabla u) = fquad {\... ... Omega subset {R^2},quad u = gquad {text{on}} partial Omega . The finite element approximation is defined over {Omega ^h} , a union of regular triangles, yielding a polygonal approximation to Omega . For sufficiently regular solutions u, achievable for a subclass of data f, g, and Omega , we prove optimal error bounds for this approximation in the norm {W^{1,q}}({Omega ^h}),q = p for p 2 , under the additional assumption that {Omega ^h} subseteq Omega . Numerical results demonstrating these bounds are also presented.
In this work, we study the Brinkman-Forchheimer equations driven under slip
boundary conditions of friction type. We prove the existence and uniqueness of
weak solutions by means of regularization combined with the Faedo-Galerkin
approach. Next we discuss the continuity of the solution with respect to
Brinkman's and Forchheimer's coefficients. Finally, we show that the weak
solution of the corresponding stationary problem is stable.
The Crouzeix-Raviart triangular finite elements are inf-sup stable for the Stokes equations for any mesh with at least one interior vertex. This result affirms a conjecture of Crouzeix-Falk from 1989 for p = 3. Our proof applies to any odd degree p >= 3 and concludes the overall stability analysis: Crouzeix-Raviart triangular finite elements of degree p in two dimensions and the piecewise polynomials of degree p - 1 with vanishing integral form a stable Stokes pair for all positive integers p.
We propose and analyze a new mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Besides the velocity, our approach introduces the velocity gradient and a pseudostress tensor as further unknowns. As a consequence, we obtain a three-field Banach spaces-based mixed variational formulation, where the aforementioned variables are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation, and derive the corresponding stability bounds, employing classical results on nonlinear monotone operators. We then propose a semidiscrete continuous-in-time approximation on simplicial grids based on the Raviart–Thomas elements of degree k≥0 for the pseudostress tensor and discontinuous piecewise polynomials of degree k for the velocity and the velocity gradient. In addition, by means of the backward Euler time discretization, we introduce a fully discrete finite element scheme. We prove well-posedness and derive the stability bounds for both schemes, and under a quasi-uniformity assumption on the mesh, we establish the corresponding error estimates. We provide several numerical results verifying the theoretical rates of convergence and illustrating the performance and flexibility of the method for a range of domain configurations and model parameters.
We propose and analyse an augmented mixed finite element method for the Oseen equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and homogeneous Dirichlet boundary condition for the velocity. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition, and we show that it satisfies the hypotheses of the Babuška-Brezzi theory. Repeating the arguments of the continuous analysis, the stability and solvability of the discrete problem are established. The method is suited for any Stokes inf-sup stable finite element pair for velocity and pressure, while for vorticity any generic discrete space (of arbitrary order) can be used. A priori and a posteriori error estimates are derived using two specific families of discrete subspaces. Finally, we provide a set of numerical tests illustrating the behaviour of the scheme, verifying the theoretical convergence rates, and showing the performance of the adaptive algorithm guided by residual a posteriori error estimation.
In this paper, we propose and analyze a uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem. Our formulation is based on velocity gradient-velocity-pressure and the resulting scheme can be flexibly applied to fairly general polygonal meshes. We relax the tangential continuity for velocity, which is the key ingredient in achieving the uniform robustness. We present well-posedness and error analysis for both the semi-discrete scheme and the fully discrete scheme, and the theories indicate that the error estimates for velocity are independent of pressure. Several numerical experiments are presented to confirm the theoretical findings.
The Crouzeix-Raviart triangular finite elements are inf \inf - sup \sup stable for the Stokes equations for any mesh with at least one interior vertex. This result affirms a conjecture of Crouzeix-Falk from 1989 for p = 3 p=3 . Our proof applies to any odd degree p ≥ 3 p\ge 3 and concludes the overall stability analysis: Crouzeix-Raviart triangular finite elements of degree p p in two dimensions and the piecewise polynomials of degree p − 1 p-1 with vanishing integral form a stable Stokes pair for all positive integers p p .
We propose and analyse a mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Our approach is based on the introduction of a pseudostress tensor related to the velocity gradient and pressure, leading to a mixed formulation where the pseudostress tensor and the velocity are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation in a Banach space setting, employing classical results on nonlinear monotone operators and a regularization technique. We then present well posedness and error analysis for semidiscrete continuous-in-time and fully discrete finite element approximations on simplicial grids with spatial discretization based on the Raviart–Thomas spaces of degree k for the pseudostress tensor and discontinuous piecewise polynomial elements of degree k for the velocity and backward Euler time discretization. We provide several numerical results to confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method for a range of model parameters.
In this work we present and analyse a mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by the Navier--Stokes and the Darcy--Forchheimer equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers--Joseph--Saffman law. We consider the standard mixed formulation in the Navier--Stokes domain and the dual-mixed one in the Darcy--Forchheimer region, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The well-posedness of the problem is achieved by combining a fixed-point strategy, classical results on nonlinear monotone operators and the well-known Schauder and Banach fixed-point theorems. As for the associated Galerkin scheme we employ Bernardi--Raugel and Raviart--Thomas elements for the velocities, and piecewise constant elements for the pressures and the Lagrange multiplier, whereas its existence and uniqueness of solution is established similarly to its continuous counterpart, using in this case the Brouwer and Banach fixed-point theorems, respectively. We show stability, convergence, and a priori error estimates for the associated Galerkin scheme. Finally, we report some numerical examples confirming the predicted rates of convergence, and illustrating the performance of the method.
In this paper, we consider numerical simulation of wormhole propagation with the Darcy–Brinkman–Forchheimer model. Matrix acidization in carbonate reservoirs is a widely practiced technique in the product enhancement of the oil and gas reservoir. A wormhole, i.e. a flow channel with high porosity, is generated during reactive dissolution of carbonates by the action of the injected acid. In the wormhole forming process, the porosity changes non-uniformly in space, and it even becomes close to unity in the central regions of a wormhole. The Darcy–Brinkman–Forchheimer model accounts for both the porous media and clear fluid area, so it can be used to model wormhole propagation perfectly. This model, however, strongly depends on porosity. Therefore, the time schemes for solving the evolutionary equation of porosity have a significant effect on accuracy and stability of the numerical simulation for wormhole propagation. We propose a semi-analytic time scheme, which solves the porosity equation analytically at each time step for given acid concentration. The proposed numerical method can improve the accuracy and stability of numerical simulation significantly. For theoretical analysis of the proposed time scheme for the wormhole simulation, we first reconstruct the analytical functions of porosity to analyze the time error of the porosity, and on the basis of error estimates of porosity, we employ a coupled analysis approach to achieve the estimates of pressure, velocity and solute concentration. The time error estimates for velocity, pressure, concentration and porosity are obtained in different norms. Finally, numerical results are provided to verify the effectiveness of the proposed scheme.
In this work, we propose and analyze the pressure stabilization method for the unsteady incompressible Brinkman-Forchheimer equations. We present a time discretization scheme which can be used with any consistent finite element space approximation. Second-order error estimate is proven. Some numerical results are also given.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017
In this work, we study the Brinkman-Forchheimer equations for unsteady flows. We prove the continuous dependence of the solution on the Brinkman's and Forchheimer's coefficients as well as the initial data and external forces. Next, we propose and study a perturbed compressible system that approximate the Brinkman-Forchheimer equations. Finally, we propose a time discretization of the perturbed system by a semi-implicit Euler scheme and next a lowest-order Raviart-Thomas element is applied for spatial discretization. Some numerical results are given.
This paper deals with the analysis of a new augmented mixed finite element method in terms of vorticity, velocity and pressure, for the Brinkman problem with non-standard boundary conditions. The approach is based on the introduction of Galerkin least-squares terms arising from the constitutive equation relating the aforementioned unknowns, and from the incompressibility condition. We show that the resulting augmented bilinear form is continuous and elliptic which, thanks to the Lax-Milgram Theorem, and besides proving the well-posedness of the continuous formulation, ensures the solvability and stability of the Galerkin scheme with any finite element subspace of the continuous space. In particular, Raviart-Thomas elements of any order k≥0 for the velocity field, and piecewise continuous polynomials of degree k + 1 for both the vorticity and the pressure, can be utilized. A priori error estimates and the corresponding rates of convergence are also given here. Next, we derive two reliable and efficient residual-based a posteriori error estimators for this problem. The ellipticity of the bilinear form together with the local approximation properties of the Clément interpolation operator are the main tools for showing the reliability. In turn, inverse inequalities and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of the method, confirming the properties of the estimators, and showing the behaviour of the associated adaptive algorithms, are reported. This article is protected by copyright. All rights reserved.
-This is a short presentation of the freefem++ software. In Section 1, we recall most of the characteristics of the software, In Section 2, we recall how to to build the weak form of a partial differential equation (PDE) from the strong form. In the 3 last sections, we present different examples and tools to illustrated the power of the software. First we deal with mesh adaptation for problems in two and three dimension, second, we solve numerically a problem with phase change and natural convection, and the finally to show the possibilities for HPC we solve a Laplace equation by a Schwarz domain decomposition problem on parallel computer.
The paper is devoted to a general finite element approximation of the solution of the Stokes equations for an incompressible viscous fluid. Both conforming and nonconforming finite-element methods are studied and various examples of simplistic elements well suited for the numerical treatment of the incompressibility condition are given. Optimal error estimates are derived in the energy norm and in the L**2-norm.
We prove the stability for the approximation of the stationary Stokes equations by means of piecewise continuous velocities of degree k+1 and piecewise continuous pressures of degree k for k≥1. The necessary and sufficient condition required on the triangulation is that it contains at least three triangles. The theorem is compared with previous results.
The Brinkman–Forchheimer equations for non-slow flow in a saturated porous medium are analyzed. It is shown that the solution depends continuously on changes in the Forchheimer coefficient, and convergence of the solution of the Brinkman–Forchheimer equations to that of the Brinkman equations is deduced, as the Forchheimer coefficient tends to zero. The next result establishes continuous dependence on changes in the Brinkman coefficient. Following this, a result is proved establishing convergence of a solution of the Brinkman–Forchheimer equations to a solution of the Darcy–Forchheimer equations, as the Brinkman coefficient (effective viscosity) tends to zero. Finally, upper and lower bounds are derived for the energy decay rate which establish that the energy decays exponentially, but not faster than this.
A finite difference method is presented for solving the 3D Navier–Stokes equations in vorticity–velocity form. The method involves solving the vorticity transport equations in ‘curl-form’ along with a set of Cauchy–Riemann type equations for the velocity. The equations are formulated in cylindrical co-ordinates and discretized using a staggered grid arrangement. The discretized Cauchy–Riemann type equations are overdetermined and their solution is accomplished by employing a conjugate gradient method on the normal equations. The vorticity transport equations are solved in time using a semi-implicit Crank–Nicolson/Adams–Bashforth scheme combined with a second-order accurate spatial discretization scheme. Special emphasis is put on the treatment of the polar singularity. Numerical results of axisymmetric as well as non-axisymmetric flows in a pipe and in a closed cylinder are presented. Comparison with measurements are carried out for the axisymmetric flow cases. Copyright © 2003 John Wiley & Sons, Ltd.
We consider the solution of both symmetric and unsymmetric systems of sparse linear equations. A new parallel distributed memory multifrontal approach is described. To handle numerical pivoting efficiently, a parallel asynchronous algorithm with dynamic scheduling of the computing tasks has been developed. We discuss some of the main algorithmic choices and compare both implementation issues and the performance of the LDLT and LU factorizations. Performance analysis on an IBM SP2 shows the efficiency and the potential of the method. The test problems used are from the Rutherford–Boeing collection and from the PARASOL end users.
The finite element discretisation technique is used to effect a solution of the Navier- Stokes equations. Two methods of formulation are presented, and a comparison of the effeciency of the methods, associated with the solution of particular problems, is made. The first uses velocity and pressure as field variables and the second stream function and vorticity. It appears that, for contained flow problems the first formulation has some advantages over previous approaches using the finite elemental method[1,2].
In this paper we consider modifications to Darcy's equation wherein the drag
coefficient is a function of pressure, which is a realistic model for
technological applications like enhanced oil recovery and geological carbon
sequestration. We first outline the approximations behind Darcy's equation and
the modifications that we propose to Darcy's equation, and derive the governing
equations through a systematic approach using mixture theory. We then propose a
stabilized mixed finite element formulation for the modified Darcy's equation.
To solve the resulting nonlinear equations we present a solution procedure
based on the consistent Newton-Raphson method. We solve representative test
problems to illustrate the performance of the proposed stabilized formulation.
One of the objectives of this paper is also to show that the dependence of
viscosity on the pressure can have a significant effect both on the qualitative
and quantitative nature of the solution.
We prove continuous dependence of solutions of the Brinkman–Forchheimer equations on the Brinkman and Forchheimer coefficients in H1 norm.
A Simple Introduction to the Mixed Finite Element Method
- G N Gatica
G.N. Gatica, A Simple Introduction to the Mixed Finite Element Method. Theory and Applications, in: Springer Briefs in Mathematics,
Springer, Cham, 2014.
- M S Alnaes
- J Blechta
- J Hake
- A Johansson
- B Kehlet
- A Logg
- C Richardson
- J Ring
- M E Rognes
- G N Wells
M.S. Alnaes, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M.E. Rognes, G.N. Wells, The FEniCS
project version 1.5, Arch. Numer. Softw. 3 (2015) 9-23.