Continuous Galerkin Formulations Research Articles

Use of algebraic dual spaces in domain decomposition methods for Darcy flow in 3D domains

In this work we use algebraic dual spaces with a domain decomposition method to solve the Darcy equations. We define the broken Sobolev spaces and their finite dimensional counterparts. A global trace space is defined that connects the solution between the broken spaces. Use of algebraic dual spaces results in a sparse, metric-free representation of the incompressibility constraint, the pressure gradient term, and on the continuity constraint between the sub domains. To demonstrate this, we solve two test cases: (i) a manufactured solution case, and (ii) an industrial benchmark reservoir modelling problem SPE10. The results demonstrate that the dual spaces can be used for domain decomposition formulation, and despite having more unknowns, requires less simulation time compared to the continuous Galerkin formulation, without compromising on the accuracy of the solution.

An enriched immersed finite element method for interface problems with nonhomogeneous jump conditions

This article presents the first higher degree immersed finite element (IFE) method with proven optimal convergence for elliptic interface problems with nonhomogeneous jump conditions. It also gives the first analysis for the condition numbers of the resulting systems including the optimal upper bounds with respect to the mesh size and its robustness with respect to small-cut interface elements. In this method, jump conditions are approximated optimally by basic IFE and enrichment IFE which are piecewise pth degree polynomial functions constructed by solving local Cauchy problems on interface elements. The proposed IFE method is based on a discontinuous Galerkin formulation on interface elements and a continuous Galerkin formulation on non-interface elements.

Finite element modeling of spherical indentation in a poro‐elasto‐plastic medium via step displacement loading

AbstractA hydromechanically coupled finite element method (FEM) algorithm following a mixed continuous Galerkin formulation for displacement and pore pressure is adopted for modeling of spherical indentation in a poro‐elasto‐plastic medium. The fully saturated porous medium is assumed to be isotropic and elasto‐perfectly plastic, obeying a Drucker‐Prager yield criterion with an associative or non‐associative flow rule. The Newton‐Raphson method with the tangent stiffness scheme is employed to deal with plasticity in the solid skeleton. A stabilization scheme, which permits equal‐order interpolation for the displacement and pore pressure fields and suppresses pore pressure oscillation in the incompressible or nearly incompressible limit, is incorporated in this FEM algorithm. The FEM algorithm is extensively benchmarked with poroelastic analytical solutions to the problems of Terzaghi, Mandel, Cryer, and De Leeuw and an analytic solution for one‐dimensional poro‐elasto‐plastic consolidation. Numerical simulations of poroelastic spherical indentation via step displacement loading are conducted to show that the normalized indentation force as a function of dimensionless time has only relatively weak dependence on material properties through a single derived parameter . Such universality is shown for three types of surface drainage conditions. For indentation in a poro‐elasto‐plastic medium, it is shown that even though plasticity occurs immediately at the undrained limit, if cohesion is within a certain range, there is no plastic strain accumulation during the transient period and the normalized force relaxation behavior could be approximated as poroelastic.

An immersed boundary method based on domain decomposition

A novel immersed boundary method based on a domain decomposition approach is proposed in the context of a finite element discretisation method. It is applicable to incompressible flows past rigid, deforming, or moving bodies. In this method, unlike most immersed boundary methods, strong boundary conditions are imposed in the regions of the computational domain that are occupied by the structure. In order to achieve this, the proposed formulation decomposes the computational domain by splitting the finite element test functions into solid and fluid parts. In the continuous Galerkin formulation, this produces a smeared representation of the fluid-structure interface. The absence of an immersed boundary forcing term implies that the method itself has no influence on the CFL stability criterion. Furthermore, the stiffness matrix in the momentum equation is sparser than compared with other forcing immersed boundary methods, and symmetry and positive-definiteness of the Laplacian operator in the pressure equation is preserved. As shown in this paper, stability and accurate imposition of boundary conditions make the method promising for high Reynolds number flows. The method is applied to the simulations of two-dimensional laminar flow over stationary and moving cylinders, as well as a moderately high Reynolds number flow past an aerofoil. Good results are obtained when compared with those from previous experimental and numerical studies.

Enriched Galerkin finite elements for coupled poromechanics with local mass conservation

Robust and efficient discretization methods for coupled poromechanical problems are critical to address a wide range of problems related to civil infrastructure, energy resources, and environmental sustainability. In this work, we propose a new finite element formulation for coupled poromechanical problems that ensures local (element-wise) mass conservation. The proposed formulation draws on the so-called enriched Galerkin method, which augments piecewise constant functions to the classical continuous Galerkin finite element method. These additional degrees of freedom allow us to obtain a locally conservative and nonconforming solution for the pore pressure field. The enriched and continuous Galerkin formulations are compared in several numerical examples ranging from a benchmark consolidation problem to a complex problem that involves plastic deformation due to unsaturated flow in a heterogeneous porous medium. The results of these examples show not only that the proposed method provides local mass conservation, but also that local mass conservation can be crucial to accurate simulation of deformation processes in fluid-infiltrated porous materials.

A Multi-Domain Spectral IPDG Method for Helmholtz Equation with High Wave Number

This paper is concerned with a multi-domain spectral method, based on an interior penalty discontinuous Galerkin (IPDG) formulation, for the exterior Helmholtz problem truncated via an exact circular or spherical Dirichlet-to-Neumann (DtN) boundary condition. An effective iterative approach is proposed to localize the global DtN boundary condition, which facilitates the implementation of multi-domain methods, and the treatment for complex geometry of the scatterers. Under a discontinuous Galerkin formulation, the proposed method allows to use polynomial basis functions of different degree on different subdomains, and more importantly, explicit wave number dependence estimates of the spectral scheme can be derived, which is somehow implausible for a multi-domain continuous Galerkin formulation.

A Discontinuous/continuous Galerkin method for modeling of interphase damage in fibrous composite systems

A Discontinuous Galerkin (DG) interface treatment embedded in a Continuous Galerkin formulation is presented for simulating the progressive debonding of bi-material interfaces. The method seamlessly tracks the progression from perfect adhesion to interface softening and finally to complete separation without resorting to node-to-node springs or deletion of connectivities. While the formulation is inspired by an augmented Lagrangian approach, it does not introduce multiplier fields to enforce the continuity conditions, resulting in a pure displacement method amenable to traditional symmetric positive-definite solvers. The study of fibrous composites serves as the application in the present work. The constitutive behavior of the interface is modeled within an energetic framework from which the initiation criterion and softening response emanate. Numerical simulations are conducted for various loading cases on a fiber-matrix unit cell that highlight the performance of the method on crude meshes.

Adaptive Finite Element Simulation of Incompressible Flows by Hybrid Continuous-Discontinuous Galerkin Formulations

In this work we design hybrid continuous-discontinuous finite element spaces that permit discontinuities on nonmatching element interfaces of nonconforming meshes. Then we develop an equal-order stabilized finite element formulation for incompressible flows over these hybrid spaces, which combines the element interior stabilization of SUPG-type continuous Galerkin formulations and the jump stabilization of discontinuous Galerkin formulations. Optimal stability and convergence results are obtained. For the adaptive setting, we use a standard error estimator and marking strategy. Numerical experiments show the optimal accuracy of the hybrid algorithm for both uniformly and adaptively refined nonconforming meshes. The outcome of this work is a finite element formulation that can naturally be used on nonconforming meshes, as discontinuous Galerkin formulations, while keeping the much lower CPU cost of continuous Galerkin formulations.

Convergence Estimates for a Linear Backward Euler Scheme for the Saturation Equation

In a previous work, stability and consistency results were established for a linearized Euler scheme for the saturation equation. In this paper we continue the mathematical analysis of the scheme, in preparation for its numerical treatment in a future work. We use the regularity results, obtained previously, to establish error estimates in L 2 () for the linear scheme. This work is done with the degenerate nature of the saturation equation in mind, but it is also valid for the non degenerate case like the concentration equation. We show that, if the regularization parameter  and the spatial discretization parameter h are carefully chosen in terms of the time stepping parameter t , the convergence is at least of order O((t)  ) for some determined  >0 . Examples of choices of  and h are given. We also establish a new (at our knowledge) regularity result for the continuous Galerkin formulation of the Saturation Equation and a new regularity result for the linear scheme. 2000 Mathematics Subject Classification. 35Q35, 65M06, 65M15, 65M60. 0)

A combined nodal continuous–discontinuous finite element formulation for the Maxwell problem

Continuous Galerkin formulations are appealing due to their low computational cost, whereas discontinuous Galerkin formulation facilitate adaptative mesh refinement and are more accurate in regions with jumps of physical parameters. Since many electromagnetic problems involve materials with different physical properties, this last point is very important. For this reason, in this article we have developed a combined cG–dG formulation for Maxwell’s problem that allows arbitrary finite element spaces with functions continuous in patches of finite elements and discontinuous on the interfaces of these patches. In particular, the second formulation we propose comes from a novel continuous Galerkin formulation that reduces the amount of stabilization introduced in the numerical system. In all cases, we have performed stability and convergence analyses of the methods. The outcome of this work is a new approach that keeps the low CPU cost of recent nodal continuous formulations with the ability to deal with coefficient jumps and adaptivity of discontinuous ones. All these methods have been tested using a problem with singular solution and another one with different materials, in order to prove that in fact the resulting formulations can properly deal with these problems.

A new domain decomposition method with overlapping patches for ultrascale simulations: Application to biological flows

We address the failure in scalability of large-scale parallel simulations that are based on (semi-)implicit time-stepping and hence on the solution of linear systems on thousands of processors. We develop a general algorithmic framework based on domain decomposition that removes the scalability limitations and leads to optimal allocation of available computational resources. It is a non-intrusive approach as it does not require modification of existing codes. Specifically, we present here a two-stage domain decomposition method for the Navier–Stokes equations that combines features of discontinuous and continuous Galerkin formulations. At the first stage the domain is subdivided into overlapping patches and within each patch a C 0 spectral element discretization (second stage) is employed. Solution within each patch is obtained separately by applying an efficient parallel solver. Proper inter-patch boundary conditions are developed to provide solution continuity, while a Multilevel Communicating Interface (MCI) is developed to provide efficient communication between the non-overlapping groups of processors of each patch. The overall strong scaling of the method depends on the number of patches and on the scalability of the standard solver within each patch. This dual path to scalability provides great flexibility in balancing accuracy with parallel efficiency. The accuracy of the method has been evaluated in solutions of steady and unsteady 3D flow problems including blood flow in the human intracranial arterial tree. Benchmarks on BlueGene/P, CRAY XT5 and Sun Constellation Linux Cluster have demonstrated good performance on up to 96,000 cores, solving up to 8.21B degrees of freedom in unsteady flow problem. The proposed method is general and can be potentially used with other discretization methods or in other applications.

Frequency-Domain Linearized Euler Model for Turbomachinery Noise Radiation Through Engine Exhaust

A numerical model for the exhaust noise radiation problem is presented. In the model, it is assumed that an incoming wave is propagating through the exhaust nozzle, or the fan duct, and radiating outside. The near-field propagation is based on the solution of the linearized Euler equations in the frequency domain: for each wave number, a linearized Euler problem is solved using a finite element method on unstructured grids for arbitrarily shaped axisymmetric geometries. The frequency-domain approach enables the suppression of the Kelvin-Helmholtz instability waves. Moreover, each single calculation, limited to a single frequency, is well suited to the exhaust noise radiation problem in which the incoming wave can be treated as a superposition of elementary duct modes. To reduce the memory requirements, a continuous Galerkin formulation with linear triangular and quadrangular elements is employed and the global matrix inversion is performed with a direct solver based on a parallel memory distributed multifrontal algorithm for sparse matrices. The acoustic near field is then radiated in the far field using the formulation of Ffowcs Williams and Hawkings. Numerical calculations for a validation test case, the Munt problem, and two turbomachinery configurations are compared with analytical solutions and experimental data.

Stability/dispersion analysis of the discontinuous Galerkin linearized shallow-water system

The frequency or dispersion relation for the discontinuous Galerkin mixed formulation of the 1-D linearized shallow-water equations is analysed, using several basic DG mixed schemes. The dispersion properties are compared analytically and graphically with those of the mixed continuous Galerkin formulation for piecewise-linear bases on co-located grids. Unlike the Galerkin case, the DG scheme does not exhibit spurious stationary pressure modes. However, spurious propagating modes have been identified in all the present discontinuous Galerkin formulations. Numerical solutions of a test problem to simulate fast gravity modes illustrate the theoretical results and confirm the presence of spurious propagating modes in the DG schemes. Copyright © 2005 John Wiley & Sons, Ltd.