Finite Element Numerical Scheme Research Articles

Multiscale formulation for materials composed by a saturated porous matrix and solid inclusions

Despite all the progress achieved in the characterization of heterogeneous materials by using multiscale paradigms based on the Representative Volume Element concept (RVE), there are still many aspects that demand ongoing development. We mention, for instance, in-homogeneous media with internal micro-structure comprising a mixture of components that require a dissimilar number/character of primary fields to describe their physical behavior. Micro-structures constituted by a saturated porous matrix (described through the pair displacement/pore pressure fields) endowed with impermeable non-porous solid inclusions (described only by the displacement field) are a typical example among many other practical applications. This new level of heterogeneity, between the set of primal descriptors for each micro-scale constituent, claims detailed revisions and novel adjustments in multiscale RVE-based theories.In this work, we present a first-order RVE-based formulation to address with the aforementioned phenomenology at a smaller length scale. At the macro-scale, the well-known poromechanics theory is preserved, where the constitutive response is provided by homogenization of the corresponding micro-scale problem. Two important attributes feature the model: (i) it is encompassed within the general framework posited by the “Method of Multiscale Virtual Power (MMVP)”, this leads to a variationally consistent methodology; (ii) our formulation preserves the objectivity with respect to the RVE-size (an indispensable but uncommon characteristic in RVE-based multiscale approaches for saturated porous media), due to it is based in a recent contribution of the authors (the so-called Selective Order Expansion technique “SOE”).The formulation is implemented following the squared Finite Element (FE2) numerical scheme. The potentialities of the model are demonstrated through a detailed series of numerical examples, including rigorous comparisons against the Direct Numerical Simulation (DNS) procedure.

Unconditionally stable fully‐discrete finite element numerical scheme for active fluid model

AbstractIn this paper, we propose a linear, decoupled, unconditionally stable fully‐discrete finite element scheme for the active fluid model, which is derived from the gradient flow approach for an effective non‐equilibrium free energy. The developed scheme is employed by an implicit‐explicit treatment of the nonlinear terms and a second‐order Gauge–Uzawa method for the decoupling of computations for the velocity and pressure. We rigorously prove the unique solvability and unconditional stability of the proposed scheme. Several numerical tests are presented to verify the accuracy, stability, and efficiency of the proposed scheme. We also simulate the self‐organized motion under the various external body forces in 2D and 3D cases, including the motion direction of active fluid from disorder to order. Numerical results show that the scheme has a good performance in accurately capturing and handling the complex dynamics of active fluid motion.

Weak Galerkin method for the Navier-Stokes equation with nonlinear damping term

<abstract><p>The primary focus of this research was to investigate the weak Galerkin (WG) finite element method for the Navier-Stokes equations with damping. We established the weak Galerkin finite element numerical scheme and demonstrated the existence and uniqueness of the weak Galerkin numerical solution. Additionally, optimal errors estimates for the velocity and pressure were obtained. Eventually, numerous numerical examples were reported to validate the theoretical analysis.</p></abstract>

Error estimates with low-order polynomial dependence for the fully-discrete finite element invariant energy quadratization scheme of the Allen–Cahn equation

In this paper, for the Allen–Cahn equation, we obtain the error estimate of the temporal semi-discrete scheme, and the fully-discrete finite element numerical scheme, both of which are based on the invariant energy quadratization (IEQ) time-marching strategy. We establish the relationship between the [Formula: see text]-error bound and the [Formula: see text]-stabilities of the numerical solution. Then, by converting the numerical schemes to a form compatible with the original format of the Allen–Cahn equation, using mathematical induction, the superconvergence property of nonlinear terms, and the spectrum argument, the optimal error estimates that only depends on the low-order polynomial degree of [Formula: see text] instead of [Formula: see text] for both of the semi and fully-discrete schemes are derived. Numerical experiment also validates our theoretical convergence analysis.

On a chemotaxis-Navier-Stokes system with Lotka-Volterra competitive kinetics: Theoretical and numerical analysis

This work is devoted to the theoretical and numerical analysis of a two-species chemotaxis-Navier-Stokes system with Lotka-Volterra competitive kinetics in a bounded domain of Rd, d=2,3. First, we study the existence of global weak solutions and establish a regularity criterion which provides sufficient conditions to ensure the strong regularity of the weak solutions. After, we propose a finite element numerical scheme approximating the continuous model, for which we study the well-posedness and derive some uniform estimates for the discrete variables. We present a convergence analysis proving optimal error estimates and show some numerical simulations in agreement with our theoretical analysis and theoretical results reported in previous works.

Fully-decoupled, energy stable second-order time-accurate and finite element numerical scheme of the binary immiscible Nematic-Newtonian model

We consider the numerical approximation of a two-phase complex fluid flow model that couples the immiscible liquid crystal fluid component immersed in the ambient free flow field, where the free interface motion between the two fluids is simulated by using the phase-field approach via the energy variational method. The model is a highly nonlinear coupling system, which is composed of the Navier–Stokes equations for the flow field, the Cahn–Hilliard equations for the free moving interface, and the constitutive equation for the nematic liquid crystal. This paper proposes an effective full-discrete finite element numerical scheme to solve the model and makes the scheme not only have the characteristics of linearity, second-order time accuracy, unconditional energy stability, decoupling structure but also only need to solve a few constant-coefficient elliptic equations at each time step. The unconditional energy stability of the scheme and the detailed implementation process are further given. Various numerical experiments including the well-known “beads-on-string” phenomena are also simulated to illustrate the effectiveness of the scheme.

A novel second-order time accurate fully discrete finite element scheme with decoupling structure for the hydrodynamically-coupled phase field crystal model

In this article, we construct a fully discrete finite element numerical scheme with linearity, decoupling, unconditional energy stability, and second-order time accuracy for the Navier-Stokes coupled phase-field crystal model. The key idea is based on the design of several auxiliary ODEs, combined with the finite element method for spatial discretization, the projection method for the Navier-Stokes equations, and the IEQ type method for the nonlinear potentials. At each time step, by using the nonlocal splitting technique, only a few decoupled elliptic constant-coefficient equations need to be solved. We further prove that the developed scheme is unconditionally energy stable, and the detailed implementation process is given as well. To verify the effectiveness of the developed scheme, various numerical experiments are carried out, including the crystal growth under the action of shear flow and the sedimentation process of a large number of particles.

Fully-discrete, decoupled, second-order time-accurate and energy stable finite element numerical scheme of the Cahn-Hilliard binary surfactant model confined in the Hele-Shaw cell

We consider the numerical approximation of the binary fluid surfactant phase-field model confined in a Hele-Shaw cell, where the system includes two coupled Cahn-Hilliard equations and Darcy equations. We develop a fully-discrete finite element scheme with some desired characteristics, including linearity, second-order time accuracy, decoupling structure, and unconditional energy stability. The scheme is constructed by combining the projection method for the Darcy equation, the quadratization approach for the nonlinear energy potential, and a decoupling method of using a trivial ODE built upon the “zero-energy-contribution” feature. The advantage of this scheme is that not only can all variables be calculated in a decoupled manner, but each equation has only constant coefficients at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Various numerical examples are further carried out to prove the effectiveness of the scheme, in which the benchmark Saffman-Taylor fingering instability problems in various flow regimes are simulated to verify the weakening effects of surfactant on surface tension.

Fully-discrete finite element numerical scheme with decoupling structure and energy stability for the Cahn–Hilliard phase-field model of two-phase incompressible flow system with variable density and viscosity

We construct a fully-discrete finite element numerical scheme for the Cahn–Hilliard phase-field model of the two-phase incompressible flow system with variable density and viscosity. The scheme is linear, decoupled, and unconditionally energy stable. Its key idea is to combine the penalty method of the Navier–Stokes equations with the Strang operator splitting method, and introduce several nonlocal variables and their ordinary differential equations to process coupled nonlinear terms. The scheme is highly efficient and it only needs to solve a series of completely independent linear elliptic equations at each time step, in which the Cahn–Hilliard equation and the pressure Poisson equation only have constant coefficients. We rigorously prove the unconditional energy stability and solvability of the scheme and carry out numerous accuracy/stability examples and various benchmark numerical simulations in 2D and 3D, including the Rayleigh–Taylor instability and rising/coalescence dynamics of bubbles to demonstrate the effectiveness of the scheme, numerically.

A Linearised θ Numerical Scheme for the Vibrations of Inextensible Beams

A linearised finite element numerical scheme for the vibration of inextensible beams is developed. The proposed scheme is based on the methodology introduced by S. Bartels [15] and satisfies a linearised form of the inextensibility constraint. The time m arching procedure is based on repeated use of the theta-parameter integration quadrature. Three parameters are introduced in total and appropriately selected such that the energy conservation features are improved compared to the Bartels algorithm while the inextensibility constraint is satisfied as accurately as possible. Cubic Hermite polynomials are employed for the spatial discretisation. The Bartels algorithm is retrieved as a special case. Several numerical experiments are presented demonstrating the theoretically predicted enhanced inextensibility mimicking and optimum values of the method parameters are identified.

Error estimate of a decoupled numerical scheme for the Cahn–Hilliard–Stokes–Darcy system

Abstract We analyze a fully discrete finite element numerical scheme for the Cahn–Hilliard–Stokes–Darcy system that models two-phase flows in coupled free flow and porous media. To avoid a well-known difficulty associated with the coupling between the Cahn–Hilliard equation and the fluid motion, we make use of the operator-splitting in the numerical scheme, so that these two solvers are decoupled, which in turn would greatly improve the computational efficiency. The unique solvability and the energy stability have been proved in Chen et al. (2017, Uniquely solvable and energy stable decoupled numerical schemes for the Cahn–Hilliard–Stokes–Darcy system for two-phase flows in karstic geometry. Numer. Math., 137, 229–255). In this work, we carry out a detailed convergence analysis and error estimate for the fully discrete finite element scheme, so that the optimal rate convergence order is established in the energy norm, i.e., in the $\ell ^{\infty } (0, T; H^1) \cap \ell ^2 (0, T; H^2)$ norm for the phase variables, as well as in the $\ell ^{\infty } (0, T; H^1) \cap \ell ^2 (0, T; H^2)$ norm for the velocity variable. Such an energy norm error estimate leads to a cancelation of a nonlinear error term associated with the convection part, which turns out to be a key step to pass through the analysis. In addition, a discrete $\ell ^2 (0;T; H^3)$ bound of the numerical solution for the phase variables plays an important role in the error estimate, which is accomplished via a discrete version of Gagliardo–Nirenberg inequality in the finite element setting.

An Energy Stable Finite Element Scheme for the Three-Component Cahn–Hilliard-Type Model for Macromolecular Microsphere Composite Hydrogels

In this article, we present and analyze a finite element numerical scheme for a three-component macromolecular microsphere composite (MMC) hydrogel model, which takes the form of a ternary Cahn–Hilliard-type equation with Flory–Huggins–deGennes energy potential. The numerical approach is based on a convex–concave decomposition of the energy functional in multi-phase space, in which the logarithmic and the nonlinear surface diffusion terms are treated implicitly, while the concave expansive linear terms are explicitly updated. A mass lumped finite element spatial approximation is applied, to ensure the positivity of the phase variables. In turn, a positivity-preserving property can be theoretically justified for the proposed fully discrete numerical scheme. In addition, unconditional energy stability is established as well, which comes from the convexity analysis. Several numerical simulations are carried out to verify the accuracy and positivity-preserving property of the proposed scheme.

Numerical simulations of Kuramoto–Sivashinsky equation in reaction-diffusion via Galerkin method

Finite element methods have been frequently employed in seeking the numerical solutions of PDEs. In this study, a Galerkin finite element numerical scheme is constructed to explore numerical solutions of the generalized Kuramoto–Sivashinsky (gKS) equation. A quartic trigonometric tension (QTT) B-spline function is adapted as base of the Galerkin technique. The incorporation of B-spline Galerkin in space discretization generates the time-dependent system. Then, the use of Crank–Nicolson time integration algorithm to this system gives the wholly discretized scheme. The efficiency of the method is tested over several initial boundary value problems. In addition, the stability of the computational scheme is analyzed by considering Von Neumann technique. The computational results obtained by the suggested scheme are simulated and compared with the commonly existing numerical findings.

SsPeristaltic activity in an asymmetric inclined channel with inertial forces under the inducement of magnetic field: Finite Element Method

This manuscript is presented to investigate peristaltic phenomenon of a flow driven peristaltically in an asymmetric inclined channel under the influence of externally applied magnetic field without employing the assumption of long wavelength and smaller Reynolds number assumptions which permit the effects of dominant inertial forces and to solve the Navier-Stokes equations in full form. The numerical results are obtained with help of finite element numerical scheme with Galerkin’s residual procedure. A comparative analysis for the obtained numerical data is performed in view of already reported studies and a favorable accuracy of results is noticed. The effects of other involved parameters are also presented in suitable graphs. Trapping and pumping phenomena is also discussed. It is found that trapped bolus shape is increased with increment of Hartmann number which tends to move towards central region of asymmetric channel and is predicted that trapped bolus will disappears at large Reynolds number. The pressure in peristaltic pumping region is enhanced by increasing angle of inclination.

Numerical approximation of the stochastic Cahn\u2013Hilliard equation near the sharp interface limit

We consider the stochastic Cahn–Hilliard equation with additive noise term varepsilon ^gamma g, {dot{W}} (gamma >0) that scales with the interfacial width parameter varepsilon . We verify strong error estimates for a gradient flow structure-inheriting time-implicit discretization, where varepsilon ^{-1} only enters polynomially; the proof is based on higher-moment estimates for iterates, and a (discrete) spectral estimate for its deterministic counterpart. For gamma sufficiently large, convergence in probability of iterates towards the deterministic Hele–Shaw/Mullins–Sekerka problem in the sharp-interface limit varepsilon rightarrow 0 is shown. These convergence results are partly generalized to a fully discrete finite element based discretization. We complement the theoretical results by computational studies to provide practical evidence concerning the effect of noise (depending on its ’strength’ gamma ) on the geometric evolution in the sharp-interface limit. For this purpose we compare the simulations with those from a fully discrete finite element numerical scheme for the (stochastic) Mullins–Sekerka problem. The computational results indicate that the limit for gamma ge 1 is the deterministic problem, and for gamma =0 we obtain agreement with a (new) stochastic version of the Mullins–Sekerka problem.

Numerical quasiconformal transformations for electron dynamics on strained graphene surfaces.

The dynamics of low-energy electrons in general static strained graphene surface is modelled mathematically by the Dirac equation in curved space-time. In Cartesian coordinates, a parametrization of the surface can be straightforwardly obtained, but the resulting Dirac equation is intricate for general surface deformations. Two different strategies are introduced to simplify this problem: the diagonal metric approximation and the change of variables to isothermal coordinates. These coordinates are obtained from quasiconformal transformations characterized by the Beltrami equation, whose solution gives the mapping between both coordinate systems. To implement this second strategy, a least-squares finite-element numerical scheme is introduced to solve the Beltrami equation. The Dirac equation is then solved via an accurate pseudospectral numerical method in the pseudo-Hermitian representation that is endowed with explicit unitary evolution and conservation of the norm. The two approaches are compared and applied to the scattering of electrons on Gaussian shaped graphene surface deformations. It is demonstrated that electron wave packets can be focused by these local strained regions.

Decoupled, Linear, and Unconditionally Energy Stable Fully Discrete Finite Element Numerical Scheme for a Two-Phase Ferrohydrodynamics Model

We consider in this paper numerical approximations of a phase field model for two-phase ferrofluids, which consists of the Navier--Stokes equations, the Cahn--Hilliard equation, the magnetostatic e...

Numerical treatment of Gray-Scott model with operator splitting method

<p style='text-indent:20px;'>This article focuses on the numerical solution of a classical, irreversible Gray Scott reaction-diffusion system describing the kinetics of a simple autocatalytic reaction in an unstirred ow reactor. A novel finite element numerical scheme based on B-spline collocation method is developed to solve this model. Before applying finite element method, "strang splitting" idea especially popularized for reaction-diffusion PDEs has been applied to the model. Then, using the underlying idea behind finite element approximation, the domain of integration is partitioned into subintervals which is sought as the basis for the B-spline approximate solution. Thus, the partial derivatives are transformed into a system of algebraic equations. Applicability and accuracy of this method is justified via comparison with the exact solution and calculating both the error norms <inline-formula><tex-math id="M1">\begin{document}$ L_2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ L_\infty $\end{document}</tex-math></inline-formula>. Numerical results arising from the simulation experiments are also presented.

Modeling SMA-enabled soft deployable structures for kirigami/origami reflectors

The synergic combination of smart soft composites with kirigami/origami principles leads to self-deployable systems. To date, the development of soft deployable structures has largely been an empirical process. Focusing on the recently developed shape memory alloy (SMA)-based soft deployable structures, this paper describes an analytical model and a finite element (FE) numerical scheme to investigate deformation and deployment performance of this system. The study provides insights into the working principles of these soft deployable structures by understanding how the system deforms as a function of the SMA recovery strain for a number of geometrical parameters. The FE method was further used to simulate the fabrication and deployment process of three types of kirigami/origami reflectors. The realization of a two-step scheme, i.e., predeformation of the deployable frame and then the concurrent deformation of both frame and kirigami/origami films, is detailed herein. This makes our simulation capable of capturing the features of SMA-based soft deployable structures observed in experiments, and also renders the prediction versatile and manufacturing-oriented. We expect the work would enhance the understanding of soft deployable structures, and may open the door for design and fabrication of soft robotics and machines.

Numerical Solutions for Time-Fractional Cancer Invasion System With Nonlocal Diffusion

This article studies the existence and uniqueness of a weak solution of the time-fractional cancer invasion system with nonlocal diffusion operator. Existence and uniqueness results are ensured by adapting the Faedo-Galerkin method and some a priori estimates. Further, finite element numerical scheme is implemented for the considered system. Finally, various numerical computations are performed along with the convergence analysis of the scheme.