Standard Galerkin Formulation Research Articles

Adaptive Multi-level Algorithm for a Class of Nonlinear Problems

Abstract In this article, we propose an adaptive mesh-refining based on the multi-level algorithm and derive a unified a posteriori error estimate for a class of nonlinear problems. We have shown that the multi-level algorithm on adaptive meshes retains quadratic convergence of Newton’s method across different mesh levels, which is numerically validated. Our framework facilitates to use the general theory established for a linear problem associated with given nonlinear equations. In particular, existing a posteriori error estimates for the linear problem can be utilized to find reliable error estimators for the given nonlinear problem. As applications of our theory, we consider the pseudostress-velocity formulation of Navier–Stokes equations and the standard Galerkin formulation of semilinear elliptic equations. Reliable and efficient a posteriori error estimators for both approximations are derived. Finally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed.

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.

Towards model order reduction for fluid-thermal analysis

The authors develop basic components of a parametric model-order reduction (pMOR) procedure targeting high Rayleigh-number buoyancy-driven flows. The pMOR is based on Galerkin formulation of the governing Boussinesq equations using eigenfunction bases derived from proper orthogonal decomposition. The advantages of pMOR over parametric interpolation are demonstrated for quantities of interest (QOIs) with linear and nonlinear dependencies on the solution in low Rayleigh-number cases. Constraint-base and Leray-type regularizations are shown to offer significant advantages over the standard reduced-order Galerkin formulation in a variety of examples including 3D Rayleigh–Bénard flow at Rayleigh number 107 and turbulent flow in a half-pipe at Reynolds number 5300.

Three-dimensional dispersion analysis and stabilized finite element methods for acoustics

Galerkin/least-squares and Galerkin gradient/least-squares stand out among several approaches designed to improve the numerical solution accuracy and counteract the pollution effect by adding terms to the standard Galerkin formulation. These added terms are multiplied by a ‘stability parameter’, which must be properly defined. In this paper, an original three-dimensional dispersion analysis is performed for the Helmholtz equation, together with the determination of the three-dimensional stability parameters for structured and unstructured meshes. Numerical experiments show the relative efficiency of the proposed methods for solving acoustic problems arising from industry.

Modeling of high pressure arc-discharge with a fully-implicit Navier–Stokes stabilized finite element flow solver

This work addresses the modeling of high pressure electric discharge in an arc-heated wind tunnel. The combined numerical solution of Poisson’s equation, radiative transfer equations, and the set of Favre-averaged thermochemical nonequilibrium Navier–Stokes equations allows for the determination of the electric, radiation, and flow fields, accounting for their mutual interaction. Semi-classical statistical thermodynamics is used to determine the plasma thermodynamic properties, while transport properties are obtained from kinetic principles with the Chapman–Enskog method. A multi-temperature formulation is used to account for thermal non-equilibrium. Finally, the turbulence closure of the flow equations is obtained by means of the Spalart–Allmaras model, which requires the solution of an additional scalar transport equation. A Streamline upwind Petrov–Galerkin stabilized finite element formulation is employed to solve the Navier–Stokes equation. The electric field equation is solved using the standard Galerkin formulation. A stable formulation for the radiative transfer equations is obtained using the least-squares finite element method. The developed simulation framework has been applied to investigate turbulent plasma flows in the 20 MW Aerodynamic Heating Facility at NASA Ames Research Center. The current model is able to predict the process of energy addition and re-distribution due to Joule heating and thermal radiation, resulting in a hot central core surrounded by colder flow. The use of an unsteady three-dimensional treatment also allows the asymmetry due to a dynamic electric arc attachment point in the cathode chamber to be captured accurately. The current work paves the way for detailed estimation of operating characteristics for arc-heated wind tunnels which are critical in testing thermal protection systems.

On mesh restrictions to satisfy comparison principles, maximum principles, and the non-negative constraint: Recent developments and new results

ABSTRACTThis article concerns mesh restrictions that are needed to satisfy several important mathematical properties—maximum principles, comparison principles, and the nonnegative constraint—for a general linear second-order elliptic partial differential equation. We critically review some recent developments in the field of discrete maximum principles, derive new results, and discuss some possible future research directions in this area. In particular, we derive restrictions for a three-node triangular (T3) element and a four-node quadrilateral (Q4) element to satisfy comparison principles, maximum principles, and the nonnegative constraint under the standard single-field Galerkin formulation. Analysis is restricted to uniformly elliptic linear differential operators in divergence form with Dirichlet boundary conditions specified on the entire boundary of the domain. Various versions of maximum principles and comparison principles are discussed in both continuous and discrete settings. In the literature, it is well-known that an acute-angled triangle is sufficient to satisfy the discrete weak maximum principle for pure isotropic diffusion. Herein, we show that this condition can be either too restrictive or not sufficient to satisfy various discrete principles when one considers anisotropic diffusivity, advection velocity field, or linear reaction coefficient. Subsequently, we derive appropriate restrictions on the mesh for simplicial (e.g., T3 element) and nonsimplicial (e.g., Q4 element) elements. Based on these conditions, an iterative algorithm is developed to construct simplicial meshes that preserve discrete maximum principles using existing open source mesh generators. Various numerical examples based on different types of triangulations are presented to show the pros and cons of placing restrictions on a computational mesh. We also quantify local and global mass conservation errors using representative numerical examples and illustrate the performance of metric.

Large-Scale Optimization-Based Non-negative Computational Framework for Diffusion Equations: Parallel Implementation and Performance Studies

It is well-known that the standard Galerkin formulation, which is often the formulation of choice under the finite element method for solving self-adjoint diffusion equations, does not meet maximum principles and the non-negative constraint for anisotropic diffusion equations. Recently, optimization-based methodologies that satisfy maximum principles and the non-negative constraint for steady-state and transient diffusion-type equations have been proposed. To date, these methodologies have been tested only on small-scale academic problems. The purpose of this paper is to systematically study the performance of the non-negative methodology in the context of high performance computing (HPC). PETSc and TAO libraries are, respectively, used for the parallel environment and optimization solvers. For large-scale problems, it is important for computational scientists to understand the computational performance of current algorithms available in these scientific libraries. The numerical experiments are conducted on the state-of-the-art HPC systems, and a single-core performance model is used to better characterize the efficiency of the solvers. Our studies indicate that the proposed non-negative computational framework for diffusion-type equations exhibits excellent strong scaling for real-world large-scale problems.

A Smoothed Particle Galerkin Formulation for Extreme Material Flow Analysis in Bulk Forming Applications

This paper presents a new particle formulation for extreme material flow analyses in the bulk forming applications. The new formulation is first established by an introduction of a smoothed displacement field to the standard Galerkin formulation to eliminate zero-energy modes in conventional particle methods. The discretized system of linear equations is consistently derived and integrated using a direct nodal integration scheme. The linear formulation is next extended to the large deformation quasi-static analysis of inelastic materials. As quasi-static Lagrangian simulation proceeds in the severe deformation range, the analysis method is switched to explicit dynamics formulation and an adaptive Lagrangian kernel approach is preformed to reset the reference configuration and maintain the injective deformation mapping at the particles. Both nonconvex and convex meshfree approximations are investigated in this study. Several numerical benchmarks are provided to demonstrate the effectiveness and accuracy of the proposed method.

Zonal Detached-Eddy Simulation of Turbulent Unsteady Flow over Iced Airfoils

This paper presents a multiscale finite-element formulation for the second mode of zonal detached-eddy simulation. The multiscale formulation corrects the lack of stability of the standard Galerkin formulation by incorporating the effect of unresolved scales to the grid (resolved) scales. The stabilization terms arise naturally and are free of user-defined stability parameters. Validation of the method is accomplished via the turbulent flow over tandem cylinders. The boundary-layer separation, free shear-layer rollup, vortex shedding from the upstream cylinder, and interaction with the downstream cylinder are well reproduced. Good agreement with experimental measurements gives credence to the accuracy of zonal detached-eddy simulation in modeling turbulent separated flows. A comprehensive study is then conducted on the performance degradation of ice-contaminated airfoils. NACA 23012 airfoil with a spanwise ice ridge and Gates Learjet Corporation-305 airfoil with a leading-edge horn-shape glaze ice are selected for investigation. Appropriate spanwise domain size and sufficient grid density are determined to enhance the reliability of the simulations. A comparison of lift coefficient and flowfield variables demonstrates the added advantage that the zonal detached-eddy simulation model brings to the Spalart–Allmaras turbulence model. Spectral analysis and instantaneous visualization of turbulent structures are also highlighted via zonal detached-eddy simulation.

Isogeometric rotation-free analysis of planar extensible-elastica for static and dynamic applications

Finite deformations of planar slender beams for which shear strain can be neglected are described by the extensible- elastica model, where the strain-displacement relation is geometrically exact and the Biot stress–strain relation is linear. However, if the formulation is expressed in terms of displacements without rotation, the kinematics are described by a partial differential equation involving a fourth-order spatial operator, which cannot be approximated by the classical $${\mathcal {C}}^0$$ -continuous FE method in the standard Galerkin framework. In this work, we propose the spatial approximation of such high-order PDE by means of NURBS-based isogeometric analysis (IGA) which allows the use of globally high-order continuous basis functions. The employed IGA approach possesses three advantages: first, it facilitates the encapsulation of the exact geometric representation of the beams in the spatial approximation with fewer discrete points, especially useful for curved structures; second, it allows the discretization of high-order spatial operators; and third, it provides an efficient numerical solution of the discrete problem by using a limited number of degrees of freedom since the employed standard Galerkin formulation does not require rotational degrees of freedom. Yet this approach has not been directly compared to appropriate analytical solutions. To this end, we compare and validate numerical results from FE with the closed-form solutions for a set of static beam problems, including a newly derived solution for an initially curved beam, based on the extensible-elastica theory, by estimating the convergence orders of the errors. We also highlight the advantages of this formulation with the numerical solution of three dynamic problems: the swinging of a pinned beam, the propagation of solitons (nonlinear waves) in post-buckled beams, and snap-through buckling of a pinned beam that is axially buckled before transverse loading.

Modeling and simulation of morphology variation during the solidification of polymer melts with amorphous and semi-crystalline phases

The solidification of polymer melts in practical processing such as extrusion, injection molding and blow molding can significantly influence the inner structure and performance of final products. The investigation of its mechanism has both scientific and industrial interests. In the study, the three-dimensional mathematical model is developed for the simulation of morphology variation in the solidification of polymer melts with amorphous and semi-crystalline phases. The amorphous phase is simulated as the finite extensible nonlinear elastic dumbbell with a peterlin closure approximation (FENE-P) fluid and the semi-crystalline phase is approximated as rigid rods that grow and oriented in the flow field. The model of amorphous phase and semi-crystalline phase are coupled through the stress and momentum balance and the feedback of crystallinity to the system relaxation time. The evolution of crystallization kinetics process are described by using a set of Schneider equation that discriminating the relative roles of the thermal and the flow effect on the crystallization behavior. With the standard Galerkin formulation adopted as basic computational framework, the discrete elastic viscous stress splitting algorithm in cooperating with the streamline upwinding approach serves as a relatively robust numerical scheme by using penalty finite element–finite difference simulation with a decoupled solving algorithm. The proposed mathematical model and numerical method have been successfully applied to the investigation of solidification of polymer melts in the extrusion process. The variations of orientation and crystallization morphology during the solidification process are further discussed. Copyright © 2014 John Wiley & Sons, Ltd.

Multiscale finite element method applied to detached-eddy simulation for computational wind engineering

A multiscale finite element method is applied to the Spalart-Allmaras turbulence model based detached-eddy simulation (DES). The multiscale arises from a decomposition of the scalar field into coarse (resolved) and fine (unresolved) scales. It corrects the lack of stability of the standard Galerkin formulation by modeling the scales that cannot be resolved by a given spatial discretization. The stabilization terms appear naturally and the resulting formulation provides effective stabilization in turbulent computations, where reaction-dominated effects strongly influence near-wall predictions. The multiscale DES is applied in the context of high-Reynolds flow over the Commonwealth Advisory Aeronautical Council (CAARC) standard tall building model, for both uniform and turbulent inflows. Time-averaged pressure coefficients on the exterior walls are compared with experiments and it is demonstrated that DES is able to resolve the turbulent features of the flow and accurately predict the surface pressure distributions under atmospheric boundary layer flows.

Conservative Residual Distribution Method for Viscous Double Cone Flows in Thermochemical Nonequilibrium

AbstractA multi-dimensionally upwind conservative Residual Distribution algorithm for simulating viscous axisymmetric hypersonic flows in thermo-chemical nonequilibrium on unstructured grids is presented and validated in the case of the complex flow-field over a double cone configuration. The resulting numerical discretization combines a state-of-the-art nonlinear quasi-monotone second order blended scheme for distributing the convective residual and a standard Galerkin formulation for the diffusive residual. The physical source terms are upwinded together with the convective fluxes. Numerical results show an excellent agreement with experimental measurements and available literature.

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

SUMMARYThe paper introduces a methodology to compute strict upper and lower bounds for linear‐functional outputs of the exact solutions of the advection–diffusion–reaction equation. The bounds are computed using implicit a posteriori error estimators from stabilized finite element approximations of the exact solution. The new methodology extends the a posteriori error estimates yielding bounds for the standard Galerkin formulation to be able to obtain bounds for stabilized formulations. This methodology is combined with both hybrid‐flux and flux‐free techniques for error assessment. The application to stabilized formulations provides sharper estimates than when applied to Galerkin methods. The best results are found in combination with the flux‐free technique. Copyright © 2012 John Wiley & Sons, Ltd.

A NUMERICAL APPROACH STUDYING THE EFFECTS OF PRECIPITATION SCAVENGING ON STEADY-STATE DISPERSION OF AIR POLLUTANTS

The present paper demonstrates the applicability of finite element method of weighted residuals to study the effects of precipitation scavenging through raindrops on the steady-state dispersion of air pollutants in the atmosphere under realistic variable wind and diffusivity profiles. Here, the washout of pollutants by rain droplets is considered and the whole analysis for the pollutant concentration is carried out in two phases viz. gaseous phase and droplet phase. The model solutions are obtained by using local basis and asymmetric quadratic weighting functions which can provide a promising alternative to the standard Galerkin formulation for problems with advective and diffusive terms. The results of the model reveal that the precipitation scavenging by rain is quite effective in cleaning the polluted atmosphere. Anticipating the need of pollutant concentration in rain drops regarding acid precipitation (or acid rain), the concentration of the absorbed pollutant in the droplet phase are also analyzed. So, the present study provides an understanding on the concentration distribution of pollutant in gaseous and droplet phases, under the effects of variable wind and diffusivity profiles.

Multiscale finite element method applied to the Spalart–Allmaras turbulence model for 3D detached-eddy simulation

Employing the advection–diffusion–reaction equation as a model problem we present a multiscale method that yields a stabilized finite element formulation for Reynolds-Averaged Navier–Stokes (RANS) based turbulence models. The multiscale method arises from a decomposition of the scalar field into coarse (resolved) and fine (unresolved) scales. Modeling of the unresolved scales corrects the lack of stability of the standard Galerkin formulation. The proposed method possesses superior properties like that of the Streamline Upwind/Petrov–Galerkin (SUPG) method and the Galerkin/Least-Squares (GLS) method. The stabilization terms appear naturally and the resulting formulation provides effective stabilization in turbulent computations where reaction-dominated effects strongly influence the boundary layer prediction. A family of 2D and 3D elements is developed and comparison of the proposed method with the SUPG method is presented. The multiscale formulation is then applied to the Spalart–Allmaras turbulence model [1] in FENSAP-ICE [2–7] for detached-eddy simulation (DES). Numerical results obtained by the proposed method are compared with experimental and DNS results for backward-facing step problem at Reynolds number of 5000.

Efficient Iterative Solvers for Stochastic Galerkin Discretizations of Log-Transformed Random Diffusion Problems

We consider the numerical solution of a steady-state diffusion problem where the diffusion coefficient is the exponent of a random field. The standard stochastic Galerkin formulation of this problem is computationally demanding because of the nonlinear structure of the uncertain component of it. We consider a reformulated version of this problem as a stochastic convection-diffusion problem with random convective velocity that depends linearly on a fixed number of independent truncated Gaussian random variables. The associated Galerkin matrix is nonsymmetric but sparse and allows for fast matrix-vector multiplications with optimal complexity. We construct and analyze two block-diagonal preconditioners for this Galerkin matrix for use with Krylov subspace methods such as the generalized minimal residual method. We test the efficiency of the proposed preconditioning approaches and compare the iterative solver performance for a model problem posed in both diffusion and convection-diffusion formulations.

Three-dimensional modeling of problems in poro-elasticity via a mixed least-squares method using linear tetrahedral elements

In a previous publication we developed a new mixed least-squares method for poro-elasticity. The approximate solution was obtained via a minimization of a least-squares functional, based upon the equations of equilibrium, the equations of continuity and weak forms of the constitutive relationships for elasticity and Darcy flow. The formulation involved four independent types of variables: displacements, stresses, pore pressures and velocities. All of them were approximated by linear continuous triangles. Encouraged by the computational results, obtained from the two-dimensional implementation of the method, we extended our formulation to three dimensions. In this paper we present numerical examples for the performance of continuous linear tetrahedra within the context of the mixed least-squares method. The initial results suggest that the method works well in the nearly and entirely incompressible limits for elasticity. For poro-elasticity, the obtained pore pressures are stable without exhibiting the oscillations, which are observed when the standard Galerkin formulation is used. Copyright © 2010 John Wiley & Sons, Ltd.

Dynamic finite element analysis of a gradient elastic bar with micro-inertia

We study the dynamic behavior of a first strain gradient elastic bar with micro-inertia by means of the finite element method. The partial differential equation describing the motion of the bar expressed in terms of displacement is of fourth order with respect to the spatial variable, therefore, for the standard Galerkin formulation, Hermite elements are required. Consistent mass matrices are employed. The results are validated by comparison with some special cases where the exact solutions are derivable. Dispersion relations for the longitudinal waves are also derived. The effect of micro-inertia in the dynamic response of the bar is analyzed and comparisons are made with the classical elastic case. It is found that the micro-inertia parameter considerably affects the dynamic response of the bar and the dispersion characteristics of longitudinal waves.

Stabilized Finite Element Methods for the Schrödinger Wave Equation

This paper presents two stabilized formulations for the Schrödinger wave equation. First formulation is based on the Galerkin/least-squares (GLS) method, and it sets the stage for exploring variational multiscale ideas for developing the second stabilized formulation. These formulations provide improved accuracy on cruder meshes as compared with the standard Galerkin formulation. Based on the proposed formulations a family of tetrahedral and hexahedral elements is developed. Numerical convergence studies are presented to demonstrate the accuracy and convergence properties of the two methods for a model electronic potential for which analytical results are available.