Element Galerkin Research Articles

Contribution of Unsteadiness to Solid Fuel Burning Characteristics in a Scramjet Combustor

The contributions of unsteadiness to solid fuel combustion are investigated numerically and validated against new experiments in a deep cavity scramjet. Large-eddy simulations discretized with discontinuous Galerkin (DG) elements are solved with a flamelet manifold approach in three dimensions. A multiphase model that incorporated thermal decomposition inside the foam layer is coupled with stagnation flow flames to determine the combustion manifold and regression rate. The approach accurately models small-scale experiments of convective burning over solid fuel. The inclusion of the manifold into the DG code features two innovations: a polynomial fitting of the pressure to reduce the interpolation dimensions and the coefficient of determination to include non-monotonic manifolds in DG schemes. Model validation was performed using pressure and averaged regression rate data, both of which showed strong agreement with experiments. Three-dimensional pressure modes in the cavity support a substantial increase in regression rates and a broadening of the peak due to oscillations of the impingement point. The majority of fuel is pyrolyzed at the shear-layer reattachment point in a stagnation flow boundary layer. The fuel in the cavity is pyrolyzed by conductive heat transfer from the main shear layer. Poor combustion is observed in the expanding fuel section.

Reduced Element for Adaptive Finite Element Analysis of First-Order IVP with Built-in Error Estimator in Maximum Norm

This paper proposes a novel yet simple approach to the adaptive finite element (FE) analysis of the first-order Initial Value Problems (IVPs) in the maximum norm by introducing the reduced element technique. In the present approach, the FE solution uh of the conventional Galerkin element of degree m + 1 is decomposed into two parts: a reduced solution uRh from the reduced element of degree m obtained by ignoring the highest degree term of uh, and a built-in point-wise error estimator εRh directly given by the ignored term. Since the end node solutions of the reduced element are inherited from the full order element, it gains O(h2m+2) accuracy and achieves a nodal/element accuracy ratio as high as two, which greatly enhances its adaptive capability regarding solving IVPs on long time domains. The related error analysis is addressed and a complete adaptivity algorithm is given. Typical numerical examples of both linear and nonlinear IVPs of both single and systems of equations are presented to show the validity and effectiveness of the proposed approach.

A new optimal error analysis of a mixed finite element method for advection–diffusion–reaction Brinkman flow

AbstractThis article deals with the error analysis of a Galerkin‐mixed finite element methods for the advection–reaction–diffusion Brinkman flow in porous media. Numerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest‐order Galerkin‐mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest‐order Raviart–Thomas element, the lowest‐order Nédélec edge element and piece‐wise constant discontinuous Galerkin element are used for the velocity, vorticity and pressure, respectively. The existing error estimate of this lowest‐order finite element method is only for all variables in spatial direction, which is not optimal for the concentration variable. This paper focuses on a new and optimal error estimate of a linearized backward Euler Galerkin‐mixed FEMs, where the second‐order accuracy for the concentration in spatial directions is established unconditionally. The key to our optimal error analysis is a new negative norm estimate for Nédélec edge element. Moreover, based on the computed numerical concentration, we propose a simple one‐step recovery technique to obtain a new numerical velocity, vorticity and pressure with second‐order accuracy. Numerical experiments are provided to confirm our theoretical analysis.

Convergence of adaptive two-grid weak Galerkin finite element methods for semilinear elliptic differential equations

In this paper, we investigate the convergence of an adaptive two-grid weak Galerkin (ATGWG) finite element method for second order semilinear elliptic partial differential equations (PDEs). First, we propose an ATGWG method and then prove that the sum of the energy error and the error estimator of ATGWG method between two consecutive adaptive loops is a contraction. The weak Galerkin (WG) elements (Pj(T),Pℓ(∂T),RTj(T)) (Wang and Ye, 2013) are studied in this paper and numerical experiments based on the lowest order case with j=l=0 are provided to support the theoretical results.

Two‐order superconvergence for a weak Galerkin method on rectangular and cuboid grids

AbstractThis article introduces a particular weak Galerkin (WG) element on rectangular/cuboid partitions that uses th order polynomial for weak finite element functions and th order polynomials for weak derivatives. This WG element is highly accurate with convergence two orders higher than the optimal order in an energy norm and the norm. The superconvergence is verified analytically and numerically. Furthermore, the usual stabilizer in the standard weak Galerkin formulation is no longer needed for this element.

Condensed Galerkin element of degree m for first-order initial-value problem with O(h2m+2) super-convergent nodal solutions

Condensed Galerkin element of degree m for first-order initial-value problem with O(h2m+2) super-convergent nodal solutions

Discontinuous Galerkin methods through the lens of variational multiscale analysis

In this article, we present a theoretical framework for integrating discontinuous Galerkin methods in the variational multiscale paradigm. Our starting point is a projector-based multiscale decomposition of a generic variational formulation that uses broken Sobolev spaces and Lagrange multipliers to accommodate the non-conforming nature at the boundaries of discontinuous Galerkin elements. We show that existing discontinuous Galerkin formulations, including their penalty terms, follow immediately from a specific choice of multiscale projector. We proceed by defining the “fine-scale closure function”, which captures the closure relation between the remaining fine-scale term in the discontinuous Galerkin formulation and the coarse-scale solution via a single integral expression for each basis function in the coarse-scale test space. We show that the projectors that correspond to discontinuous Galerkin methods lead to fine-scale closure functions with more compact support and smaller amplitudes compared to the fine-scale closure function of the classical (conforming) finite element method. This observation provides a new perspective on the natural stability of discontinuous Galerkin methods for hyperbolic problems, and may open the door to rigorously designed variational multiscale based fine-scale models that are suitable for DG methods.

A Two-Grid Combined Mixed Finite Element and Discontinuous Galerkin Method for an Incompressible Miscible Displacement Problem in Porous Media

A Two-Grid Combined Mixed Finite Element and Discontinuous Galerkin Method for an Incompressible Miscible Displacement Problem in Porous Media

Modeling of failure at the interface of ductile materials by applying the cohesive discontinuous Galerkin method

In this study, the failure behavior at the interface of ductile materials is investigated. In order to capture the degradation of the tractions at the interface, a cohesive zone (CZ) model is applied. The choice of the type of the CZ approach, i.e. either intrinsic or extrinsic, brings about different drawbacks. The former includes an elastic regime at the interface prior to the failure, which can result in numerical difficulties whereas the latter necessitates the re-meshing of the structure during crack propagation. In order to overcome these problems, the incomplete interior penalty Galerkin variant of the discontinuous Galerkin (DG) method is applied both at the interface and in the bulk instead of the standard conforming finite element method. In addition, the application of the DG method enables to use nonmatching meshes in the discretized model. To treat the bulk, an elastoplastic material model with isotropic hardening as well as different hardening rules for small strains is incorporated into the DG framework. Two numerical examples are computed to study the convergence behavior of the new cohesive discontinuous Galerkin (CDG) method in comparison to that of the conventional models. The new CDG method outperforms the conventional CZ continuous Galerkin elements in the presence of locking effects as well as hanging nodes.

Effect of Heat source on an unsteady MHD free convection flow of Casson fluid past a vertical oscillating plate in porous medium using finite element analysis

The present research paper aims to study the effect Heat source on an MHD Casson fluid through a vertical fluctuating porous plate. Dimensional non-linear coupled differential equations transformed into dimensional less by introducing similarity variables. The transformed nondimensional ordinary differential equations of velocity, temperature is solved by Galerkin element technique. The influence of flow parameters like Casson fluid ( γ ) , Permeability (K), magnetic number (M), Phase angle ( ω t ) , Prandtl number (Pr), other physical quantities on velocity along with temperature profiles is explained in detailed via graphs. Skin friction, Nusselt number is also explained through tables. Moreover, with increasing the heat source parameter the result in the velocity and temperature increases.

A Simple Discretization of the Vector Dirichlet Energy

AbstractWe present a simple and concise discretization of the covariant derivative vector Dirichlet energy for triangle meshes in 3D using Crouzeix‐Raviart finite elements. The discretization is based on linear discontinuous Galerkin elements, and is simple to implement, without compromising on quality: there are two degrees of freedom for each mesh edge, and the sparse Dirichlet energy matrix can be constructed in a single pass over all triangles using a short formula that only depends on the edge lengths, reminiscent of the scalar cotangent Laplacian. Our vector Dirichlet energy discretization can be used in a variety of applications, such as the calculation of Killing fields, parallel transport of vectors, and smooth vector field design. Experiments suggest convergence and suitability for applications similar to other discretizations of the vector Dirichlet energy.

Locking‐free interface failure modeling by a cohesive discontinuous Galerkin method for matching and nonmatching meshes

SummaryIn this work, modeling of brittle failure of the interface for a linear elastic material is presented. The idea is to integrate a novel extrinsic cohesive zone model into the incomplete interior penalty Galerkin variant of the discontinuous Galerkin (DG) method. As a result, the initial stiffness in the prefailure regime is omitted without having to remesh the crack path during the crack propagation. The interface model is used in combination with different discretization techniques, including matching and nonmatching meshes. This is possible due to the DG method's weak continuity constraint. Moreover, the locking problem in the bulk is cured by the application of a reduced Gaussian integration scheme on the boundary terms. The performance of the new cohesive discontinuous Galerkin elements with different integration schemes is compared with one of the standard intrinsic cohesive models. Due to the elimination of locking, crack initiation at the interface can be realistically displayed.

Limiting and divergence cleaning for continuous finite element discretizations of the MHD equations

This work introduces a new type of constrained algebraic stabilization for continuous piecewise-linear finite element approximations to the equations of ideal magnetohydrodynamics (MHD). At the first step of the proposed flux-corrected transport (FCT) algorithm, the Galerkin element matrices are modified by adding graph viscosity proportional to the fastest characteristic wave speed. At the second step, limited antidiffusive corrections are applied and divergence cleaning is performed for the magnetic field. The limiting procedure developed for this stage is designed to enforce local maximum principles, as well as positivity preservation for the density and thermodynamic pressure. Additionally, it adjusts the magnetic field in a way which penalizes divergence errors without violating conservation laws or positivity constraints. Numerical studies for 2D test problems are performed to demonstrate the ability of the proposed algorithms to accomplish this task in applications to ideal MHD benchmarks.

Superconvergence of the gradient approximation for weak Galerkin finite element methods on nonuniform rectangular partitions

This article presents a superconvergence for the gradient approximation of the second order elliptic equation discretized by weak Galerkin finite element methods on nonuniform rectangular partitions. The result shows a convergence of O(hr), 1.5≤r≤2, for the numerical gradient obtained from the lowest order weak Galerkin element consisting of piecewise linear and constant functions. For this numerical scheme, the optimal order of error estimate is O(h) for the gradient approximation. The superconvergence reveals a superior performance of the weak Galerkin finite element methods. Some computational results are included to numerically validate the superconvergence theory.

Element Free Galerkin Post-processing Technique Based Error Estimator for Elasticity Problems

The study present a Mesh Free based post-processing technique for asymptotically (upper) bounded error estimator for Finite Element Analyses of elastic problems. The proposed technique uses Galerkin Element Free procedure for recovery of the displacement derivatives over a patch of nodes in radial domains. The radial nodes patches are used to construct the trial shape functions utilizing the moving least-squares (MLS) techniques. The proposed technique has been tested on three benchmark elastic problems discretized using 4-node quadrilateral elements. The recovered nodal stresses are utilized to calculate the error in finite element solution in energy norm. The study also demonstrates adaptive analysis application of proposed error estimator. The performance of proposed error estimator based on mesh independent node patches has been compared with that of mesh dependent node patches based Zienkiewicz-Zhu (ZZ) error estimator on structured and unstructured mesh. The improved results of the proposed error estimator in terms of convergence rate and effectivity are obtained. It is shown that present study incorporates the superiority of the Mesh Free Galerkin method into finite element analysis environment.

Operator splitting and discontinuous Galerkin methods for advection-reaction-diffusion problem. Application to plant root growth

The time dependant advection-reaction-diffusion equation is used in the C-Root model to simulate root growth. This equation can also be applied in many others applications in life sciences. In this context the unknown is related to densities and one of the important property of the problem is that the solution is non-negative for positive initial conditions. One of the difficulty at the discrete level is to preserve the positivity of the approximated solution during the simulations. In this work we solved the model using Discontinuous Galerkin elements combined with an operator splitting technique. The DG method is briefly presented then we motivated the use of the operator splitting technique by doing some numerical experiments. Those experiments showed that the same time approximation scheme may not be suitable for all the operators of the model. We validated our implementation of the splitting technique in a simple test case. Then we performed a simulation of a plagiotropic root of Eucalyptus.

A stable enriched Galerkin element for the Stokes problem

We propose a stable element for the divergence operator that approximates the velocity by continuous linear polynomials plus piecewise constants and the pressure by piecewise constants. A uniform inf–sup condition is obtained for conforming meshes in two or three dimensions. The resulting method belongs to the class of enriched Galerkin methods, and is applied to the solution of a Stokes system. A priori error estimates in the energy norm and in the L2 norm are derived. Extensions to the Navier–Stokes system are presented.

A locking free Reissner-Mindlin element with weak Galerkin rotations

A locking free finite element method is developed for the Reissner-Mindlin equations in their primary form. In this method, the transverse displacement is approximated by continuous piecewise polynomials of degree $k+1$ and the rotation is approximated by weak Galerkin elements of degree $k$ for $k≥1$ . A uniform convergence in thickness of the plate is established for this finite element approximation. The numerical examples demonstrate locking free of the method.

Geometrically nonlinear crystal plasticity implemented into a discontinuous Galerkin element formulation

AbstractSingle crystal viscoplasticity, with a regularization technique for the power law, is presented and implemented into a discontinuous Galerkin (DG) framework. Although single crystal plasticity has been extensively studied, its examination with the regularization method in combination with a DG formulation leads to a numerically efficient and robust model. The performance of the DG framework in crystal viscoplasticity is shown by an example. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Investigation of a locking‐free hybrid discontinuous Galerkin element that is very easy to implement into FE‐codes

AbstractIn this work, a hybrid discontinuous Galerkin (dG) quadrilateral element formulation is investigated. In hybrid formulations, the elements' interior is treated independently from the skeleton (the element boundaries). The global degrees of freedom (dofs) are located on the element corners. This makes the implementation into existing finite element codes simple. The dofs in the element interior are condensed out. The deformation gradient is taken homogeneous within the element instead of using bilinear shape funcitons. The result is a very simple formulation and numerical implementation. The element has been shown to be free of volumetric locking and shear locking. Moreover, the mesh convergence is close to well‐known formulations like the Q1E4‐, Q1P0‐ or Q1SP‐element. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)