Mixed Finite Element Research Articles

A Quantitative Insight Into the Role of Skip Connections in Deep Neural Networks of Low Complexity: A Case Study Directed at Fluid Flow Modeling

Abstract Deep feed-forward networks, with high complexity, backpropagate the gradient of the loss function from final layers to earlier layers. As a consequence, the “gradient” may descend rapidly toward zero. This is known as the vanishing gradient phenomenon that prevents earlier layers from benefiting from further training. One of the most efficient techniques to solve this problem is using skip connection (shortcut) schemes that enable the gradient to be directly backpropagated to earlier layers. This paper investigates whether skip connections significantly affect the performance of deep neural networks of low complexity or whether their inclusion has little or no effect. The analysis was conducted using four Convolutional Neural Networks (CNNs) to predict four different multiscale basis functions for the mixed Generalized Multiscale Finite Element Method (GMsFEM). These models were applied to 249,375 samples. Three skip connection schemes were added to the base structure: Scheme 1 from the first convolutional block to the last, Scheme 2 from the middle to the last block, and Scheme 3 from the middle to the last and the second-to-last blocks. The results demonstrate that the third scheme is most effective, as it increases the coefficient of determination (R2) value by 0.0224–0.044 and decreases the Mean Squared Error (MSE) value by 0.0027–0.0058 compared to the base structure. Hence, it is concluded that enriching the last convolutional blocks with the information hidden in neighboring blocks is more effective than enriching using earlier convolutional blocks near the input layer.

Primal and mixed finite element formulations for the relaxed micromorphic model

The classical Cauchy continuum theory is suitable to model highly homogeneous materials. However, many materials, such as porous media or metamaterials, exhibit a pronounced microstructure. As a result, the classical continuum theory cannot capture their mechanical behaviour without fully resolving the underlying microstructure. In terms of finite element computations, this can be done by modelling the entire body, including every interior cell. The relaxed micromorphic continuum offers an alternative method by instead enriching the kinematics of the mathematical model. The theory introduces a microdistortion field, encompassing nine extra degrees of freedom for each material point. The corresponding elastic energy functional contains the gradient of the displacement field, the microdistortion field and its Curl (the micro-dislocation). Therefore, the natural spaces of the fields are [H1]3 for the displacement and [H(curl)]3 for the microdistortion, leading to unusual finite element formulations. In this work we describe the construction of appropriate finite elements using Nédélec and Raviart–Thomas subspaces, encompassing solutions to the orientation problem and the discrete consistent coupling condition. Further, we explore the numerical behaviour of the relaxed micromorphic model for both a primal and a mixed formulation. The focus of our benchmarks lies in the influence of the characteristic length Lc and the correlation to the classical Cauchy continuum theory.

A mixed finite element method for a class of evolution differential equations with p-Laplacian and memory

We present a new mixed finite element method for a class of parabolic equations with p-Laplacian and nonlinear memory. The applicability, stability and convergence of the method are studied. First, the problem is written in a mixed formulation as a system of one parabolic equation and a Volterra equation. Then, the system is discretized in the space variable using the finite element method with Lagrangian basis of degree r≥1. Finally, the Cranck-Nicolson method with the trapezoidal quadrature is applied to discretize the time variable. For each method, we establish existence, uniqueness and regularity of the solutions. The convergence order is found to be dependent on the parameter p on the p-Laplacian in the sense that it decreases as p increases.

Static bending and buckling analysis of bi-directional functionally graded porous plates using an improved first-order shear deformation theory and FEM

The static bending and buckling behaviors of bi-directional functionally graded (BFG) plates with porosity are investigated in this paper. An improved first-order shear deformation theory with an assuming parabolic distribution shear stresses is developed to describe the displacement, strain, and stress fields of the plates. The significant novelty of the proposed theory is that the transverse shear stresses equal to zero at two free surfaces of the BFG plates. Therefore, no shear correction factor is required as in other first-order shear deformation theory. A four-node quadrilateral plate element (IMQ4) is developed based on the improved first-order shear deformation theory, mixed finite element method (FEM) and Hamilton's principle for analysis of BFG plates. Several comparison studies are provided to demonstrate the precision and robustness of the proposed plate element IMQ4. Then the proposed plate element, IMQ4, is employed to analyze the bending and buckling responses of the BFG plates. Some new numerical results on the flexural and buckling behaviors of BFG plates are achieved via a deep parametric study.

A comprehensive mechanical response and dynamic stability analysis of elastically restrained bi-directional functionally graded porous microbeams in the thermal environment via mixed finite elements

A comprehensive mechanical response and dynamic stability analysis of elastically restrained bi-directional functionally graded porous microbeams in the thermal environment via mixed finite elements

Direct serendipity and mixed finite elements on convex polygons

We construct new families of direct serendipity and direct mixed finite elements on general planar, strictly convex polygons that are H1 and H(div) conforming, respectively, and possess optimal order of accuracy for any order. They have a minimal number of degrees of freedom subject to the conformity and accuracy constraints. The name arises because the shape functions are defined directly on the physical elements, i.e., without using a mapping from a reference element. The finite element shape functions are defined to be the full spaces of scalar or vector polynomials plus a space of supplemental functions. The direct serendipity elements are the precursors of the direct mixed elements in a de Rham complex. The convergence properties of the finite elements are shown under a regularity assumption on the shapes of the polygons in the mesh, as well as some mild restrictions on the choices one can make in the construction of the supplemental functions. Numerical experiments on various meshes exhibit the performance of these new families of finite elements.

A posteriori error estimates for Biot system using a mixed discretization for flow

We first derive convergence and a priori stability, and next reliability and efficiency of a posteriori error indicators for a Biot poroelastic model coupled with an elastic model in R3, solved by a continuous Galerkin scheme (CG) for the displacement and a mixed finite element scheme for the flow. The coupled system is decoupled by a fixed stress splitting algorithm. The numerical implementation of the residual based error indicators is simple, even for the mixed discretization, but at the expense of two suboptimal bounds. The scheme is tested on two benchmark problems via several numerical experiments.

Mixed Generalized Multiscale Finite Element Method for flow problem in thin domains

In this paper, we construct a class of Mixed Generalized Multiscale Finite Element Methods for the approximation on a coarse grid for an elliptic problem in thin two-dimensional domains. We consider the elliptic equation with homogeneous boundary conditions on the domain walls. For reference solution of the problem, we use a Mixed Finite Element Method on a fine grid that resolves complex geometry on the grid level. To construct a lower dimensional model, we use the Mixed Generalized Multiscale Finite Element Method, which is based on some multiscale basis functions for velocity fields. The construction of the basis functions is based on the local snapshot space that takes all possible flows on the interface between coarse cells into account. In order to reduce the size of the snapshot space and obtain the multiscale approximation, we solve a local spectral problem to identify dominant modes in the snapshot space. We present a convergence analysis of the presented multiscale method. Numerical results are presented for two-dimensional problems in three testing geometries along with the errors associated to different numbers of the multiscale basis functions used for the velocity field. Numerical investigations are conducted for problems with homogeneous and heterogeneous properties respectively.

Delamination Criteria for SiN/TiW Interfaces Tested by Cross-Sectional Nanoidentation and by Four Point Bending

Interfaces between different material layers are potentially weakest links of microelectronic devices. Therefore, reliable criteria for delamination of structures are required as prerequisite of design optimization. In the present article, delamination of adjacent material layers is studied experimentally and by FEM simulation. An interface between SiN and TiW was delaminated by Cross-Sectional Nanoindentation (CSN) and by Four Point Bending. The adhesion energy observed during CSN is determined by FEM simulation using a method of equivalent loads, which compares CSN with a peel test. In a second approach, delamination during Four Point Bending is simulated by Mixed Finite Elements based on Strain Gradient Elasticity. The Mixed Elements were implemented in ABAQUS with user subroutine UEL. In this approach, the delamination criterion can be defined with use of a critical elastic energy density. The accordance between experiments and theoretical models is demonstrated.

A Nitsche mixed extended finite element method for the biharmonic interface problem

In this paper, a Nitsche mixed extended finite element method based on Ciarlet–Raviart formulation is proposed to discretize the biharmonic interface problem with interface unfitted meshes. We present a discrete mixed approximate formulation based on the piecewise quadratic continuous finite element space. By adding a stabilization procedure, we get the discrete inf–sup condition and prove a suboptimal a priori error estimate for the discrete problem. If the solution of the dual problem satisfies the additional regularity condition, then an optimal a priori error estimate for stream variable under piecewise H1-norm is obtained. It is shown that all results are uniform with respect to the mesh size, and the location of the interface relative to the mesh. Finally, numerical experiments are carried out to demonstrate our theoretical results.

A Mixed Finite Element Method for Stationary Magneto-Heat Coupling System with Variable Coefficients.

In this article, a mixed finite element method for thermally coupled, stationary incompressible MHD problems with physical parameters dependent on temperature in the Lipschitz domain is considered. Due to the variable coefficients of the MHD model, the nonlinearity of the system is increased. A stationary discrete scheme based on the coefficients dependent temperature is proposed, in which the magnetic equation is approximated by Nédélec edge elements, and the thermal and Navier–Stokes equations are approximated by the mixed finite elements. We rigorously establish the optimal error estimates for velocity, pressure, temperature, magnetic induction and Lagrange multiplier with the hypothesis of a low regularity for the exact solution. Finally, a numerical experiment is provided to illustrate the performance and convergence rates of our numerical scheme.

The Dimensionality Reduction of Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors for the Unsteady Stokes Equation

By means of a proper orthogonal decomposition (POD) to cut down the dimensionality of unknown finite element (FE) solution coefficient vectors in the Crank–Nicolson (CN) mixed FE (CNMFE) method for two-dimensional (2D) unsteady Stokes equations in regard to vorticity stream functions, a reduced dimension recursive-CNMFE (RDR-CNMFE) method is constructed. In this case, the RDR-CNMFE method has the same FE basis functions and accuracy as the CNMFE method. The existence, stability, and errors of RDR-CNMFE solutions are analyzed by matrix analyzing, resulting in very simple theory analysis. Some numerical tries are used to check on the validity of the RDR-CNMFE method. The RDR-CNMFE method has second-order time accuracy and few unknowns so as to be able to shorten CPU runtime and retard the error cumulation in simulation calculating process, and improve real-time calculating accuracy.

An optimal error estimate of the BDF-Galerkin FEM for the incompressible MHD system

In this article, a linearized backward differential formula type scheme for the numerical approximations of the magnetohydrodynamics equations is analyzed. The proposed fully discretized numerical scheme, which is comprised of a third order backward differential formula for time derivative terms, extrapolated treatments in linearization for nonlinear terms and mixed finite element approximation for spatial discretization. Based on the temporal-spatial error splitting technique, we demonstrate the almost unconditional optimal error estimate in L2 norm for the fluid velocity and the magnetic by getting boundedness of numerical solutions in certain norms. Finally, several numerical tests are carried out to verify the theoretical results.

A robust fully mixed finite element model for flow and transport in unsaturated fractured porous media

A fully mixed finite element (MFE) model is developed for nonlinear flow and transport in unsaturated fractured porous media with matrix-fracture and fracture-fracture fluid and mass exchanges. The model is based on the discrete fracture matrix (DFM) approach and assumes cross-flow equilibrium in the fractures. The MFE method is employed for the spatial discretization of both flow and transport on the 2D-matrix elements as well as on the 1D-fracture elements. An upwind scheme is employed to avoid unphysical oscillations in the case of advection dominant transport. The temporal discretization is performed using high-order time integration methods and efficient automatic time-stepping schemes via the MOL.Two test problems dealing with flow and mass transport in saturated and unsaturated fractured porous media are simulated to show the validity of the new model by comparison against (i) a 1D-2D Comsol finite element model and (ii) a 2D-2D Discontinuous Galerkin (DG) model where both fractures and matrix continua are discretized with small 2D mesh elements. The robustness and efficiency of the developed 1D-2D MFE model are then investigated for a challenging problem dealing with infiltration of contaminated water into an initially dry soil involving a fracture network.The new model yields stable results for advection-dominated and advection-dispersion transport configurations. Further, the results of the 1D-2D MFE model are in very good agreement with those of the 2D-2D DG model for both configurations. The simulation of infiltration of contaminated water into a dry fractured soil shows that the 1D-2D MFE model is within 15 times more efficient than the 2D-2D DG model, which confirms the high benefit of using robust and efficient DFM models for the simulation of flow and transport in fractured porous media.

An innovative application of deep learning in multiscale modeling of subsurface fluid flow: Reconstructing the basis functions of the mixed GMsFEM

In multiscale modeling of subsurface fluid flow in heterogeneous porous media, standard polynomial basis functions are replaced by multiscale basis functions. For instance, to produce such functions in the mixed Generalized Multiscale Finite Element Method (mixed GMsFEM), a number of Partial Differential Equations (PDEs) must be solved, which requires a considerable overhead. Thus, it makes sense to replace PDE solvers with data-driven methods, given their great capabilities and general acceptance in the recent decades. Convolutional Neural Networks (CNNs) automatically perform feature engineering, and they also need fewer parameters via defining two-dimensional convolutional filters without reducing the quality of models. This is why four distinct CNN models were developed to predict four different multiscale basis functions for the mixed GMsFEM in the present study. These models were applied to 249,375 samples, with the permeability field as the only input. The statistical results indicate that the AMSGrad optimization algorithm with a coefficient of determination (R2) of 0.8434–0.9165 and Mean Squared Error (MSE) of 0.0078–0.0206 performs slightly better than Adam with an R2 of 0.8328–0.9049 and MSE of 0.0109–0.0261. Graphically, all models precisely follow the observed trend in each coarse block. This work could contribute to the distribution of pressure and velocity in the development of oil/gas fields. Looking at this work as an image (matrix)-to-image (matrix) regression problem, the constructed data-driven-based models may have applications beyond reservoir engineering, such as hydrogeology and rock mechanics.

Consistent pressure Poisson splitting methods for incompressible multi-phase flows: eliminating numerical boundary layers and inf-sup compatibility restrictions

For their simplicity and low computational cost, time-stepping schemes decoupling velocity and pressure are highly popular in incompressible flow simulations. When multiple fluids are present, the additional hyperbolic transport equation in the system makes it even more advantageous to compute different flow quantities separately. Most splitting methods, however, induce spurious pressure boundary layers or compatibility restrictions on how to discretise pressure and velocity. Pressure Poisson methods, on the other hand, overcome these issues by relying on a fully consistent problem to compute the pressure from the velocity field. Additionally, such pressure Poisson equations can be tailored so as to indirectly enforce incompressibility, without requiring solenoidal projections. Although these schemes have been extended to problems with variable viscosity, constant density is still a fundamental assumption in existing formulations. In this context, the main contribution of this work is to reformulate consistent splitting methods to allow for variable density, as arising in two-phase flows. We present a strong formulation and a consistent weak form allowing standard finite element spaces. For the temporal discretisation, backward differentiation formulas are used to decouple pressure, velocity and density, yielding iteration-free steps. The accuracy of our framework is showcased through a wide variety of numerical examples, considering manufactured and benchmark solutions, equal-order and mixed finite elements, first- and second-order stepping, as well as flows with one, two or three phases.

Numerical investigations of degenerate equations for fluid flow and reactive transport in clogging porous media

Structural changes of the pore space and clogging phenomena are inherent to many porous media applications. However, numerical investigations of the related flow and transport problem remain challenging due to the degeneration of the hydrodynamic parameter porosity, and in turn of the permeability and diffusion tensor. A remedy is to use appropriate scalings of the unknowns velocity, pressure and species' concentration in terms of the porosity. This enables us to conduct numerical simulations for the combined, degenerating flow and transport problem. As discretization method, lowest order mixed finite elements are used and stability of the numerical scheme is shown. Under certain additional regularity assumptions, convergence (with optimal order) can be proven. Our numerical results confirm that optimal convergence is obtained for the transformed variables whereas the non-transformed variables might not converge.

Fully implicit local time-stepping methods for advection-diffusion problems in mixed formulations

This paper is concerned with numerical solution of transport problems in heterogeneous porous media. A semi-discrete continuous-in-time formulation of the linear advection-diffusion equation is obtained by using a mixed hybrid finite element method, in which the flux variable represents both the advective and diffusive flux, and the Lagrange multiplier arising from the hybridization is used for the discretization of the advective term. Based on global-in-time and nonoverlapping domain decomposition, we propose two implicit local time-stepping methods to solve the semi-discrete problem. The first method uses the time-dependent Steklov-Poincaré type operator and the second uses the optimized Schwarz waveform relaxation (OSWR) with Robin transmission conditions. For each method, we formulate a space-time interface problem which is solved iteratively. Each iteration involves solving the subdomain problems independently and globally in time; thus, different time steps can be used in the subdomains. The convergence of the fully discrete OSWR algorithm with nonmatching time grids is proved. Numerical results for problems with various Peclét numbers and discontinuous coefficients, including a prototype for the simulation of the underground storage of nuclear waste, are presented to illustrate the performance of the proposed local time-stepping methods.

Solution of Steady Incompressible MHD Problems with Quasi-Least Square Method

A quasi-least-squares (QLS) mixed finite element method (MFE) based on the L2-inner product is utilized to solve an incompressible magnetohydrodynamic (MHD) model. These models are associated with the three unknown terms, i.e., fluid velocity, fluid pressure, and magnetic field. For the MHD-based models, common theories and algorithms for approximation of the solutions are not always applicable because of the choice of the functional spaces during the utilization of the weak formulation. It is well known that the spaces used for the approximation of the different unknowns, e.g., the spaces for the unknowns, cannot be chosen independently for the variational formulation, and may have to satisfy strict stability conditions such as the inf-sup, or Ladyzhenskaya–Babuska–Brezzi (LBB) condition. The dependency of the selection of the spaces for the unknowns are critical and always not applicable for some pair of unknowns. Because of this, the numerical or theoretical solutions must have to face some stability issue. The proposed scheme (L2-inner product) is introduced to circumvent this deficiency of the conditions (inf-sup or LBB) and obtained a well-posed solution theoretically. The model equations are nonlinear and highly coupled with the combination of Navier–Stokes and Maxwell relations. First, these nonlinear models are made linear around a specific state wherein the modified system represents an algebraic equation in a first-order symmetric form. Secondly, a direct iteration technique is applied to solve the nonlinearities and obtain a theoretical convergent rate for a general initial guess. Theoretical results show that only a single parameter with a single initial guess is sufficient to establish the well-posedness of the solution.

A Pressure Projection Stabilized Mixed Finite Element Method for a Stokes Hemivariational Inequality

A Pressure Projection Stabilized Mixed Finite Element Method for a Stokes Hemivariational Inequality