Mixed Finite Element Method Research Articles

A two-grid discretization method for nonlinear Schrödinger equation by mixed finite element methods

A two-grid discretization method for nonlinear Schrödinger equation by mixed finite element methods

Numerical simulations of a mixed finite element method for damped plate vibration problems

<abstract><p>The mixed finite element method can reduce the requirement for the smoothness of the finite element space and simplify the interpolation space for finite elements, and hence is especially effective in solving high order differential equations. In this work, we establish a mixed finite element scheme for the initial boundary conditions of damped plate vibrations and prove the existence and uniqueness of the solution of the semi-discrete and backward Euler fully discrete schemes. We use linear element approximation for the introduced intermediate variables, conduct the error analysis, and obtain the optimal order error estimate. We verify the efficiency and the accuracy of the mixed finite element scheme via numerical case studies and quantify the influence of the damping coefficient on the frequency and amplitude of the vibration.</p></abstract>

Broadband S-Parameter Simulation Based on the Mixed Finite-Element Method and Symmetric Lanczos Algorithm

In this paper, the symmetric Lanczos algorithm and mixed finite element method are combined to improve the efficiency and accuracy of S-parameter simulation in a broad frequency band. The mixed finite element method based on tree-cotree splitting is adopted to overcome the low-frequency (high-density) breakdown problem, thus ensuring the accuracy of low frequency (high integration density) results. The modified symmetric Lanczos algorithm for second order eigenvalue problem is employed for fast frequency sweep by one matrix inversion, then the field in the entire frequency band is obtained rapidly by solving the reduced order matrix equation. The accuracy of the high frequency results is improved by increasing the number of Lanczos vectors. Numerical simulations of two integrated passive filters (IPD filter) and an interconnect model are shown to verify the hybrid method along with commercial software HFSS.

A posteriori error estimates of mixed discontinuous Galerkin method for a class of Stokes eigenvalue problems

<abstract><p>For a class of Stokes eigenvalue problems including the classical Stokes eigenvalue problem and the magnetohydrodynamic Stokes eigenvalue problem a residual type a posteriori error estimate of the mixed discontinuous Galerkin finite element method using $ \mathbb{P}_{k}-\mathbb{P}_{k-1} $ element $ (k\geq 1) $ is studied in this paper. The a posteriori error estimators for approximate eigenpairs are given. The reliability and efficiency of the posteriori error estimator for the eigenfunction are proved and the reliability of the estimator for the eigenvalue is also analyzed. The numerical results are provided to confirm the theoretical predictions and indicate that the method considered in this paper can reach the optimal convergence order $ O(dof^{\frac{-2k}{d}}) $.</p></abstract>

A Posteriori Error Estimate of Two Grid Mixed Finite Element Methods for Semilinear Elliptic Equations

In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.

Superclose analysis of $H^1$-Galerkin mixed finite element methods combined with two-grid scheme for semilinear parabolic equations

Superclose analysis of $H^1$-Galerkin mixed finite element methods combined with two-grid scheme for semilinear parabolic equations

Error estimates of mixed finite elements combined with Crank-Nicolson scheme for parabolic control problems

<abstract><p>In this paper, a mixed finite element method combined with Crank-Nicolson scheme approximation of parabolic optimal control problems with control constraint is investigated. For the state and co-state, the order $ m = 1 $ Raviart-Thomas mixed finite element spaces and Crank-Nicolson scheme are used for space and time discretization, respectively. The variational discretization technique is used for the control variable. We derive optimal priori error estimates for the control, state and co-state. Some numerical examples are presented to demonstrate the theoretical results.</p></abstract>

Strictly Uniform Exponential Decay of the Mixed-FEM Discretization for the Wave Equation With Boundary Dissipation

Uniform preservation of stability in approximations of wave equations is a long-standing issue. In this paper, a one-dimensional wave equation with a partially reflective boundary is approximated using a first-order mixed finite element method. The multiplier method is used to prove that the approximated systems are exponentially stable with a decay rate independent of the mesh size. Upper bounds on the exponential decay are obtained in terms of the physical parameters.

A grad‐div stabilized method using the Jacobi iteration for the thermally coupled incompressible magnetohydrodynamic system

AbstractThis paper presents a grad‐div stabilization with the Jacobi iteration to the thermally coupled incompressible magnetohydrodynamic system, which avoids breakdown of solver and increase of computational time with increasing value of grad‐div stabilized parameter. The proposed method includes two steps: a fully discrete first‐order Euler semi‐implicit scheme based on the mixed finite element method is implemented for the considered problem in Step 1. Then, in Step 2, the Jacobi approximation is applied to the grad‐div operator whose each component will be decoupled. Moreover, we obtain theoretical analysis results of the presented method, both unconditional stability and convergence analysis. Finally, some numerical experiments are presented to show the accuracy and verify effectiveness of the proposed method.

Gradient theory of thermoelasticity for interface crack problems with a quasicrystal layer

The interface crack problem between two dissimilar materials under a thermal load is analyzed by the gradient elasticity theory since the size effect is expected in the very thin quasicrystal layer. The uncoupled thermoelasticity is considered here, where the thermal fields are independent on mechanical ones. The thermal transport can’t be well described by classical Fourier’s law in nano-sized structures. Therefore, a novel gradient theory is developed in such structures, where the second derivatives of temperature in the constitutive equation for the high-order heat flux are considered. Moreover, the phonon strain gradients are incorporated into the higher-grade continuum model. Decagonal quasicrystal material properties are considered here. Governing equations and corresponding boundary conditions are derived from the principle of virtual work. The mixed finite element method (MFEM) is developed for a general 2D boundary value problem. Numerical examples are presented to illustrate the phonon, phason and thermal response to external thermal loading within the considered model.

Application of novel refined mixed finite elementmethod in the analysis of composite laminated beams

This paper shows the application of refined finite element model developed by authors, which incorporate all the elastic equation that are, equilibrium equation, constitutive equation and kinematic equation to be satisfied explicitly at nodes. The present method considers the equilibrium equation for the refinement of the approximation function. Authors used the concept of mixed finite element model by choosing stress and displacement as the primary variable, along with the equilibrium equation as a refinement on approximation function. Ramtekkar [1] in his model restrict the arbitrary choosing of stress function by selecting the stress variation from displacement derived space, by imposing the elastic relations on the variation and then used the Minimum Potential Energy Principle to obtain the governing differential equation. In present work, authors satisfy the equilibrium equation at nodes also by considering the body forces as variables in the formulation which enables the model to satisfy the equilibrium equation explicitly at the node. The formulation is validated by the elastic beam problem solved by Pagano [2] for all three cases. Results obtained are very accurate and also shows faster convergence over the Ramtekkar [1] model.

A mixed finite element discretisation of linear and nonlinear multivariate splines using the Laplacian penalty based on biorthogonal systems

We consider a mixed finite element method for a linear multivariate spline using the Laplacian penalty. Our discretisation is based on biorthogonal systems leading to a very simple and efficient finite element scheme. We also extend our approach to a nonlinear case and describe a split Bregman iteration scheme for the resulting nonlinear equations. We apply our numerical schemes to remove the mixture of Gaussian and impulsive noise for some test images.•This paper presents a method of discretising a multivariate spline using a finite element method.•The method uses a biorthogonal system to achieve an efficient finite element method.•The method is extended to cover a discretisation scheme for a nonlinear case, including an adaptation of the split Bregman method for the nonlinear case.

Mixed convection of ferrofluids in a square enclosure with obstacles: Effect of iso‐perimetric shapes

AbstractMixed convection of ferrofluid in square configurations having various iso‐perimetric shaped obstacles has been investigated for fluid flow and heat transfer characteristics. The iso‐perimetric shapes are allowed to vary in size and tested at three different vertical positions subject to adiabatic and isothermal conditions. Simulations have been carried out for various choices of controlling parameters using the higher order mixed finite element method. In particular, the higher order and stable finite element pair of cubic and quadratic () approximations on the hybrid grid is utilized to achieve the discretization of the governing non‐dimensional equations of mass, momentum and energy. The adaptive Newtons method is chosen to solve the discrete system of nonlinear equations. The present solver has been tested against a well‐known open source finite element analysis tool (FEAT2) and published results in the literature. The impact of governing parameters is investigated in the ranges as nanoparticles volume fraction ), the Richardson number , the perimeter of the obstacle , and positions of the obstacles . It is concluded that an increase in the perimeter of the inner obstacles causes a weakness in the fluid flow while a good thermal enclosure is obtained. Moreover, the isothermal obstacles enhance the average Nusselt number more than the adiabatic obstacles in all cases.

An [formula omitted] weak Galerkin mixed finite element method for Sobolev equation

In this paper, we present a new H1 weak Galerkin mixed finite element method for the Sobolev equation which includes the exact solution u and the intermediate solution p. In the H1 weak Galerkin method, we adopt the discontinuous finite elements Pk/Pk for the approximation solution pair (uh,ph) on finite element partitions consisting of arbitrary shape of polygons which are WG shape regular. We give the semi-discrete and full-discrete formulations which are proven to be stable and parameter-free and possess the optimal error estimates. Numerical experiments show the efficiency of our methods.

Solving the Pure Neumann Problem by a Mixed Finite Element Method

Solving the Pure Neumann Problem by a Mixed Finite Element Method

Analysis of a new mixed FEM for stationary incompressible magneto-hydrodynamics

In this paper we propose and analyze a new mixed finite element method for a stationary magneto-hydrodynamic (MHD) model. The method is based on the utilization of a new dual-mixed formulation recently introduced for the Navier-Stokes problem, which is coupled with a classical primal formulation for the Maxwell equations. The latter implies that the velocity and a pseudostress tensor relating the velocity gradient with the convective term for the hydrodynamic equations, together with the magnetic field and a Lagrange multiplier related with the divergence-free property of the magnetic field, become the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree k for the aforementioned pseudostress tensor, discontinuous piecewise polynomial elements of degree k for the velocity, Nédélec elements of degree k for the magnetic field and Lagrange elements of degree k for the associated Lagrange multiplier. The analysis of the continuous and discrete problems are carried out by means of the Lax–Milgram lemma, the Banach–Nečas–Babuška and Banach fixed-point theorems, under a sufficiently small data assumption. In particular, the analysis of the discrete scheme requires a quasi-uniformity assumption on mesh. We also develop an a priori error analysis and show that the proposed finite element method is optimal convergent. Finally, some numerical results illustrating the good performance of the method are provided.

A new mixed finite-element method for H2 elliptic problems

Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of the different boundary conditions that arise naturally in a variational formulation. Our formulation is based on introducing the gradient of the solution as an explicit variable, constrained using a Lagrange multiplier. The essential boundary conditions are enforced weakly, using Nitsche's method where required. As a result, the problem is rewritten as a saddle-point system, requiring analysis of the resulting finite-element discretization and the construction of optimal linear solvers. Here, we discuss the analysis of the well-posedness and accuracy of the finite-element formulation. Moreover, we develop monolithic multigrid solvers for the resulting linear systems. Two and three-dimensional numerical results are presented to demonstrate the accuracy of the discretization and efficiency of the multigrid solvers proposed.

Modelling and Analysis of Thermal Vibrations of Coated Fiber-Reinforced Composite Cylindrical Thin Shells

The thermal vibration characteristics of fiber-reinforced composite (FRC) cylindrical thin shells (FRCCTSs) coated with functionally graded porous graphene platelet (FGPGP) are investigated in this work, which is based on a theoretical model constructed by a mixed analytical and finite element method. Firstly, the porosity distributions of the FGPGP coating are assumed to be uniform or nonuniform along thickness direction with four porous forms of coating being taken into account. Next, the displacement field functions along with the axial, circumferential, and transverse directions are assumed on the basis of Love’s first-order approximation theory. Furthermore, this coated thin shell is discretized by the four-node shell element method to calculate the mass and stiffness matrices, with the artificial spring technology being adopted to simulate arbitrary boundary conditions. After the frequency parameters and dynamic responses are successfully solved, the proposed model with and without coating material is roughly validated by comparing with literature results at different boundary conditions without considering the temperature effect. Meanwhile, by utilizing the natural frequencies and vibration responses measured via a thermal vibration experiment bench, the comprehensive verification is performed within a temperature range of 20–200∘C. Finally, parametric studies are undertaken to study the influences of boundary condition, porosity distribution of coating, fiber layup pattern, the predefined thickness ratio, and elastic modulus ratio on the corresponding thermal vibration properties.

A Preserving Precision Mixed Finite Element Dimensionality Reduction Method for Unsaturated Flow Problem

The unsaturated flow problem is of important applied background and its mixed finite element (MFE) method can be used to simultaneously calculate both water content and flux in soil, which is the most ideal calculation method. Nonetheless, it includes many unknowns. Thereby, herein we will employ the proper orthogonal decomposition (POD) to lower the dimension of unknown solution coefficient vectors in the MFE method for the unsaturated flow problem. Thus, we first examine the MFE method for the unsaturated flow problem and the existence and convergence of the classical MFE solutions. We then take advantage of the initial L MFE solution coefficient vectors to generate a set of POD basis vectors and utilize the most POD basis vectors to create the preserving precision MFE reduced-dimension (PPMFERD) format. Under the circumstances, the PPMFERD format has the same basis functions as the classical MFE format so that it can maintain the same accuracy as the classical MFE format, but it only includes a few unknowns, so it greatly lightens the calculating load, retards the accumulation of computing errors, saves CPU runtime, and improves the accuracy of the real-time calculation in the computational process. Next, we employ the analysis of matrices to demonstrate the existence and convergence of the PPMFERD solutions such that the theoretical analysis becomes very simple and elegant. Finally, we take advantage of some numerical simulations to check on the correctness of the PPMFERD method. It shows that the PPMFERD method is effective and feasible for simulating both water content and flux in unsaturated flow soil.

Energy norm analysis of exactly symmetric mixed finite elements for linear elasticity

We consider mixed finite element methods for linear elasticity for which the symmetry of the stress tensor is exactly satisfied. We derive a new quasi-optimal a priori error estimate uniformly valid with respect to the compressibility. For the a posteriori error analysis we consider the Prager-Synge hypercircle principle and introduce a new estimate uniformly valid in the incompressible limit. All estimates are validated by numerical examples.