Mixed Finite Element Procedure Research Articles

Hybrid mixed discontinuous Galerkin finite element method for incompressible miscible displacement problem

A new hybrid mixed discontinuous Galerkin finite element (HMDGFE) method is constructed for incompressible miscible displacement problem. In this method, the hybrid mixed finite element (HMFE) procedure is considered to solve the pressure and velocity equations, and a new hybrid mixed discontinuous Galerkin finite element procedure is constructed to solve the concentration equation with upwind technique. Compared with other traditional discontinuous Galerkin methods, the new method can reach a global system with less unknowns and sparser stencils. The consistency and conservation of the method are analyzed, the stability and optimal error estimates are also derived by the new technique. Meanwhile, numerical examples are provided to confirm the corresponding theoretical analysis.

Numerical analysis of incompressible wormhole propagation with mass-preserving characteristic mixed finite element procedure

In this article, we construct a new combined characteristic mixed finite element procedure to simulate the incompressible wormhole propagation. In this procedure, we use the classical mixed finite element method to solve the pressure equation and a modified mass-preserving characteristic finite element method for the solute transport equation, and solve the porosity function straightly by the given concentration. This combined method not only keeps mass balance globally but also preserves maximum principle for the porosity. We considered the corresponding convergence and derive the optimal L2-norm error estimate. Finally, we present some numerical examples to confirm theoretical analysis.

An expanded mixed finite element simulation for two-sided time-dependent fractional diffusion problem

In this paper, we consider a time-dependent diffusion problem with two-sided Riemann-Liouville fractional derivatives. By introducing a fractional-order flux as auxiliary variable, we establish the saddle-point variational formulation, based on which we employ a locally conservative mixed finite element method to approximate the unknown function, its derivative and the fractional flux in space and use the backward Euler scheme to discrete the time derivative, and thus propose a fully discrete expanded mixed finite element procedure. We prove the well-posedness and the optimal order error estimates of the proposed procedure for a sufficiently smooth solution. Numerical experiments are presented to confirm our theoretical findings.

A priori and a posteriori error estimates of H1‐Galerkin mixed finite element method for parabolic optimal control problems

SummaryIn this exposition, we study both a priori and a posteriori error analysis for the H1‐Galerkin mixed finite element method for optimal control problems governed by linear parabolic equations. The state and costate variables are approximated by the lowest order Raviart‐Thomas finite element spaces, whereas the control variable is approximated by piecewise constant functions. Compared to the standard mixed finite element procedure, the present method is not subject to the Ladyzhenskaya‐Babuska‐Brezzi condition and the approximating finite element spaces are allowed to be of different degree polynomials. A priori error analysis for both the semidiscrete and the backward Euler fully discrete schemes are analyzed, and convergence properties for the state variables and the control variable are obtained. In addition, L2(L2)‐norm a posteriori error estimates for the state and control variables and ‐norm for the flux variable are also derived.

Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation

In this article we adopt the saddle-point theoretical framework to analyze the conservative fractional diffusion equations. By introducing a fractional-order flux as auxiliary variable, we establish the well-posedness of the saddle-point variational formulation. Based on the formulation we propose a locally-conservative expanded mixed finite element procedure to approximate the unknown, its derivative and the fractional flux directly, and prove the existence and uniqueness of the mixed finite element solution. By proving that the negative fractional derivative spaces are equivalent to the negative fractional-order Sobolev spaces we derive the approximation capability of the standard projection operator in fractional Sobolev spaces for non-smooth functions. This leads an optimal-order error estimates in terms of the right-hand side both for sufficiently smooth solution and non-smooth solution. Numerical experiments are included to confirm our theoretical findings.

A mixed finite element procedure of gradient Cosserat continuum for second-order computational homogenisation of granular materials

A mixed finite element (FE) procedure of the gradient Cosserat continuum for the second-order computational homogenisation of granular materials is presented. The proposed mixed FE is developed based on the Hu---Washizu variational principle. Translational displacements, microrotations, and displacement gradients with Lagrange multipliers are taken as the independent nodal variables. The tangent stiffness matrix of the mixed FE is formulated. The advantage of the gradient Cosserat continuum model in capturing the meso-structural size effect is numerically demonstrated. Patch tests are specially designed and performed to validate the mixed FE formulations. A numerical example is presented to demonstrate the performance of the mixed FE procedure in the simulation of strain softening and localisation phenomena, while without the need to specify the macroscopic phenomenological constitutive relationship and material failure model. The meso-structural mechanisms of the macroscopic failure of granular materials are detected, i.e. significant development of dissipative sliding and rolling frictions among particles in contacts, resulting in the loss of contacts.

A new MMOCAA-MFE method for compressible miscible displacement in porous media

Combining the modified method of characteristics with adjusted advection with a splitting positive definite mixed element scheme, we establish a new mixed finite element procedure for solving compressible miscible displacement in porous media. This procedure can preserve the mass conservation globally, the coefficient matrix of the mixed system is symmetric positive definite and the flux equation is separated from the pressure equation. We analyse the convergence and give an optimal L2-norm error estimate. Finally we present some numerical results to confirm our theoretical analysis.

Superconvergence of fully discrete splitting positive definite mixed FEM for hyperbolic equations

In this article, we use a splitting positive definite mixed finite element procedure to solve the second‐order hyperbolic equation. We analyze the superconvergence property of the mixed element methods with discrete‐time approximation for the hyperbolic equation. Some numerical examples are presented to illustrate our theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 175–186, 2014

Computational multiscale methods for granular materials

The fine-scale heterogeneity of granular material is characterized by its polydisperse microstructure with randomness and no periodicity. To predict the mechanical response of the material as the microstructure evolves, it is demonstrated to develop computational multiscale methods using discrete particle assembly-Cosserat continuum modeling in micro- and macro- scales, respectively. The computational homogenization method and the bridge scale method along the concurrent scale linking approach are briefly introduced. Based on the weak form of the Hu-Washizu variational principle, the mixed finite element procedure of gradient Cosserat continuum in the frame of the second-order homogenization scheme is developed. The meso-mechanically informed anisotropic damage of effective Cosserat continuum is characterized and identified and the microscopic mechanisms of macroscopic damage phenomenon are revealed.

Two splitting positive definite mixed finite element methods for parabolic integro-differential equations

Two novel mixed finite element procedures are established for parabolic integro-differential equations, which can be split into two independent symmetric positive definite sub-schemes and do not need to solve a coupled system of equations without requiring the LBB consistency condition. The convergence analysis shows that both methods lead to the optimal order L2(Ω) norm error estimate for u and optimal H(div;Ω) norm error estimate for σ. A numerical example is presented to illustrate the theoretical analysis.

Superconvergence of H 1-Galerkin Mixed Finite Element Methods for Second-Order Elliptic Equations

We derive superconvergence result for H 1-Galerkin mixed finite element method for second-order elliptic equations over rectangular partitions. Compared to standard mixed finite element procedure, the method is not subject to the Ladyzhenskaya–Babŭska–Brezzi (LBB) condition and the approximating finite element spaces are allowed to be of different polynomial degrees. Superconvergence estimate of order 𝒪(h k+3), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas elements, is established for the flux via a postprocessing technique.

A splitting positive definite mixed finite element method for two classes of integro-differential equations

In this article, we establish a new mixed finite element procedure to solve the second-order hyperbolic and pseudo-hyperbolic integro-differential equations, in which the mixed element system is symmetric positive definite without requiring the LBB consistency condition. Convergence analysis shows that the method yields the approximate solutions with optimal accuracy in L2(Ω) norm for u and in H( div;Ω) norm for the flux σ. Numerical experiments are given to verify the theoretical results.

Superconvergence for a time‐discretization procedure for the mixed finite element approximation of miscible displacement in porous media

AbstractA time‐stepping procedure is established and analyzed for the problem of miscible displacement in porous medium. The fluid velocity based on mixed element method is post‐processed by interpolation along Gauss lines. Error analysis shows that this procedure has a superconvergence error bound O(h) with respect to the parameter hp, which is one order higher than the conventional Galerkin or mixed finite element procedures. Compared with the results in the references, the parameter constraint conditions in this article are obviously relaxed. © 2011Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012

A lagged implicit segregated data reconstruction procedure to treat open boundaries

The problem of treating open boundaries is still a challenging one. Applying fully developed condition is constrained to long enough domains. Without having enough physical evidence about what happens on boundaries, the domain extent could not be shortened and computational costs could not be reduced. From the advent of free (open) boundary conditions, they were confined to mixed finite element procedures. Recent works have extended their application to coupled finite volume solvers based on the shape function data reconstruction. A wider class of flow solvers available, however, rely on the segregated procedure where the velocity components and pressure are solved in succession. Moreover, many finite volume algorithms do not use the shape function reconstruction. In this work, by proposing a lagged implicit procedure, we have extended the application of the open boundary condition to these wider classes of flow solvers. The proposed extension is a combination of lagged implicit data reconstruction and overall mass conservation enforcement, which is easily applicable to any segregated and coupled flow solver. To validate the compatibility of this extension, benchmark problem of backward facing step is solved on successively truncated domains, where open boundary may pass through recirculation zones. Results show that the proposed extension works fine. For that problem, it reduced the computational domain length (and hence memory) by a factor of 4.6 and the required computational time by a factor of 21. Flow passing a cylinder is also solved which proves that the method could be applied to external flow problems as well.

A splitting positive definite mixed element method for second‐order hyperbolic equations

AbstractIn this article, we establish a new mixed finite element procedure, in which the mixed element system is symmetric positive definite, to solve the second‐order hyperbolic equations. The convergence of the mixed element methods with continuous‐ and discrete‐time scheme is proved. And the corresponding error estimates are given. Finally some numerical results are presented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Mixed Finite Element Analysis of Elastomeric Butt-Joints

This paper presents a mixed finite element formulation approximating large deformations observed in the analysis of elastomeric butt-joints. The rubber has been considered as nearly incompressible continuum obeying the Mooney/Rivlin (M/R) strain energy density function. The parameters of the model were determined by fitting the available from the literature uniaxial tension experimental data with the constitutive equation derived from the M/R model. The optimum value of the Poisson ratio is adjusted by comparing the experimentally observed diametral contraction of the model with that numerically obtained using the finite element method. The solution of the problem has been obtained utilizing the mixed finite element procedure on the basis of displacement/pressure mixed interpolation and enhanced strain energy mixed formulation. For comparison purposes, an axisymmetric with two-parameter M/R model and a three-dimensional (3D) with nine-parameters M/R model of the butt-joint are formulated and numerical results are illustrated concerning axisymmetric or general loading. For small strains the stress and/or strain distribution in the 2D axisymmetric butt-joint problem was compared with derived analytical solutions. Stress distributions along critical paths are evaluated and discussed.

Stochastic Modeling of Variably Saturated Transient Flow in Fractal Porous Media

This work presents the application of a Monte Carlo simulation method to perform an statistical analysis of transient variably saturated flow in an hypothetical random porous media. For each realization of the stochastic soil parameters entering as coefficients in Richards' flow equation, the pressure head and the flow field are computed using a mixed finite element procedure for the spatial discretization combined with a backward Euler and a modified Picard iteration in time. The hybridization of the mixed method provides a novel way for evaluating hydraulic conductivity on interelement boundaries. The proposed methodology can handle both large variability and fractal structure in the hydraulic parameters. The saturated conductivity Ks and the shape parameter αvg in the van Genuchten model are treated as stochastic fractal functions known as fractional Brownian motion (fBm) or fractional Gaussian noise (fGn). The statistical moments of the pressure head, water content, and flow components are obtained by averaging realizations of the fractal parameters in Monte Carlo fashion. A numerical example showing the application of the proposed methodology to characterize groundwater flow in highly heterogeneous soils is presented.

Hydrogeologic behavior of an alluvial aquifer, Salta Province, Argentina: Simulations of hydraulic conductivity field, groundwater flow, and chloride migration

This work was designed to analyze the hydrogeological behavior of an alluvial aquifer in the River Mojotoro basin site in the Province of Salta, Argentina. The study area presents coarse-grained sediments with high infiltration capacity. The hydraulic conductivity field is affected by the physical heterogeneity of the medium and a geostatistical method, kriging, was used to construct this field from local measurements. The groundwater flow velocity field is computed using a hybrid mixed finite-element method and small variations in the flow directions were determined. Given the existence of a point source of chloride in the region under study, the spatial distribution of the generated plume in the saturated zone is analyzed. The advection-diffusion equation describing the transport of chloride in vertical and horizontal cross-sectional planes of the aquifer is discretized employing a method of characteristics combined with a mixed finite-element procedure, which provides an efficient method to eliminate spurious numerical oscillations and handle the convective term in the equation. The migration of chloride shows patterns, which are affected by the litologic heterogeneities of the aquifer and the regional groundwater flow.

A computational modelling framework to predict macroscopic phenomena in solder joint formation

A computational model of solder joint formation and the subsequent cooling behaviour is described. Given the rapid changes in the technology of printed circuit boards, there is a requirement for comprehensive models of solder joint formation which permit detailed analysis of design and optimization options. Solder joint formation is complex, involving a range of interacting phenomena. This paper describes a model implementation (as part of a more comprehensive framework) to describe the shape formation (conditioned by surface tension), heat transfer, phase change and the development of elastoviscoplastic stress. The computational modelling framework is based upon mixed finite element and finite volume procedures, and has unstructured meshes enabling arbitrarily complex geometries to be analysed. Initial results for both through-hole and surface-mount geometries are presented.

On a mixed approach to the finite element solution of large strain elastoplastic problems

A mixed approach to large strain elastoplastic problems is presented in a somewhat different way to that usually used within the context of the additive split of the rate of deformation tensor into an elastic and plastic part. A non‐linear extended mixed variational equation, in which the Jacobian of the deformation gradient and the pressure part of the stress tensor appear as additional independent variables, is introduced. This equation is then linearized in the accordance with the Newton‐Raphson method to obtain the system of linear equations which represent the basis of the mixed finite element procedure. For the case of a bilinear isoparametric interpolation of the displacement field, and for piece‐wise constant pressure and Jacobian, simplified expressions, differing from similar expressions corresponding to mixed finite element implementations, are obtained. The effectiveness of the proposed mixed approach is demonstrated by means of two examples.