Discontinuous Finite Element Formulation Research Articles

A continuum framework for modeling liquid-vapor interfaces out of local thermal equilibrium

Continuum numerical methods modeling liquid-vapor phase change processes typically assume local thermal equilibrium at the liquid-vapor interface – continuity of temperature at phase interfaces, and a relation between interfacial saturation pressure and temperature based on phase co-existence curves. Several standard macroscopic problems have been solved accurately by adhering to this assumption. However, in micro-scale and certain non-standard macro-scale applications, significant jumps in temperature are observed at liquid-vapor interfaces during phase change, and vapor pressure can be far from equilibrium values. In this paper, we present a locally discontinuous arbitrary Lagrangian Eulerian finite element formulation with temperature discontinuities at interfaces. A penalty-based approach is used to weakly enforce the temperature jump condition. Furthermore, we use kinetic-theory-based Schrage relationship to evaluate the rate of phase change. We apply our methodology to solve the problem of flowing vapor in a planar heat pipe. The flow rates predicted by our continuum simulations are in excellent agreement with recently published molecular dynamics simulation results of the same problem. Interestingly, accounting for temperature discontinuities leads to only a slight improvement in the prediction of mass fluxes as compared to the case when temperature continuity is assumed. This is in contrast to the large improvement in the prediction of temperature profiles and is a consequence of a weak dependence of the evaporation/condensation rates on the vapor temperature.

A locally discontinuous ALE finite element formulation for compressible phase change problems

The simulation of phase change processes that occur at high rates, like the collapse of a vapor bubble or the combustion of dense energetic materials, poses significant challenges that include strong discontinuities in select field variables at the interface, high-speed flows in at least one phase, significant role of compressibility, disparate phase rate and acoustic time scales and advection dominated processes. In this paper we present a finite element based method that addresses these challenges. The discretization is continuous everywhere except at the interface and it inherits its stability properties from both continuous and discontinuous finite element formulations like the SUPG and interior penalty methods. We track the evolution of the interface mesh and accommodate its motion in the volume by moving the mesh in accordance with an elastic analogy within an arbitrary eulerian lagrangian (ALE) framework. This motion is interspersed with a few select steps of mesh modification. We demonstrate that the proposed method has desirable discrete conservation properties and outline a proof for its stability. We also describe how this method is implemented in a finite element code within an implicit predictor-corrector time stepping scheme. Finally we apply this method to a series of phase change problems involving an energetic material, where we verify its implementation and demonstrate its utility.

A discontinuous Galerkin method for a new class of Green–Naghdi equations on simplicial unstructured meshes

In this paper, we introduce a discontinuous Finite Element formulation on simplicial unstructured meshes for the study of free surface flows based on the fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a new class of asymptotically equivalent equations, which have a simplified analytical structure, we consider a decoupling strategy: we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non-hydrostatic effects and we show that this source term can be computed through the resolution of scalar elliptic second-order sub-problems. The assets of the proposed discrete formulation are: (i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary order of approximation in space, (iii) the exact preservation of the motionless steady states, (iv) the preservation of the water height positivity, (v) a simple way to enhance any numerical code based on the nonlinear shallow water equations. The resulting numerical model is validated through several benchmarks involving nonlinear wave transformations and run-up over complex topographies.

Multiphase flow simulation through porous media with explicitly resolved fractures

AbstractAccurate simulation of multiphase flow in fractured porous media remains a challenge. An important problem is the representation of the discontinuous or near discontinuous behaviour of saturation in real geological formations. In the classical continuum approach, a refined mesh is required at the interface between fracture and porous media to capture the steep gradients in saturation and saturation‐dependent transport properties. This dramatically increases the computational load when large numbers of fractures are present in the numerical model. A discontinuous finite element method is reported here to model flow in fractured porous media. The governing multiphase porous media flow equations are solved in the adaptive mesh computational fluid dynamics codeIC‐FERSTon unstructured meshes. The method is based on a mixed control volume – discontinuous finite element formulation. This is combined with thePN+1DG‐PNDGelement pair, which has discontinuous (orderN+1) representation for velocity and discontinuous (orderN) representation for pressure. A number of test cases are used to evaluate the method's ability to model fracture flow. The first is used to verify the performance of the element pair on structured and unstructured meshes of different resolution. Multiphase flow is then modelled in a range of idealised and simple fracture patterns. Solutions with sharp saturation fronts and computational economy in terms of mesh size are illustrated.

Discontinuous finite element formulations for neutron transport in spherical geometry

We have developed the linear and quadratic Galerkin discontinuous finite element methods for the solution of both time-independent and time-dependent spherical geometry neutron transport problems. Discrete ordinates method is used for the angular discretization while the implicit method is utilized for temporal discretization in time-dependent problems. In order to assess the relative performance of the newly developed linear and quadratic discontinuous finite element spatial differencing methods relative to the previously developed linear discontinuous finite element and diamond difference discretizations, a computer code is developed and numerical solutions of the neutron transport equation for some benchmark problems are obtained. These numerical applications reveal that the newly developed quadratic discontinuous finite element method produces the most accurate results while the newly developed linear discontinuous finite element method follows as the second best discontinuous finite element method.

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.

Comparison of light transport models based on finite element and the discrete ordinates methods in view of optical tomography applications

Numerical tests are used to evaluate the accuracy of two finite element formulations associated with the discrete ordinates method for solving the radiative transfer equation: the Least Square and the Discontinuous Galerkin finite element formulations. The results show that the use of a penalization method to set the Dirichlet boundary conditions leads to a more accurate solution than the weakly type setting where the Least Square method is seen to be more sensitive. Convergence in mesh size shows that, while both methods give accurate results, the Discontinuous Galerkin formulation uses five times more degrees of freedom than the Least Square formulation, which may lead to large systems to handle when the number of mesh elements is large. The comparison of both methods using the S n and the T n angular quadratures has shown that the Discontinuous Galerkin gives more accurate solutions, as expected, for problems with strong discontinuities, but may exhibit some oscillations due to the Galerkin procedure. A last test featuring a collimated irradiation shows that both methods give the same accuracy due to the separation of the radiative intensity into transmitted and scattered components, which removes the discontinuities in the implementation of the boundary conditions.

Practical evaluation of five partly discontinuous finite element pairs for the non‐conservative shallow water equations

AbstractThis paper provides a comparison of five finite element pairs for the shallow water equations. We consider continuous, discontinuous and partially discontinuous finite element formulations that are supposed to provide second‐order spatial accuracy. All of them rely on the same weak formulation, using Riemann solver to evaluate interface integrals. We define several asymptotic limit cases of the shallow water equations within their space of parameters. The idea is to develop a comparison of these numerical schemes in several relevant regimes of the subcritical shallow water flow. Finally, a new pair, using non‐conforming linear elements for both velocities and elevation (P−P), is presented, giving optimal rates of convergence in all test cases. P−P1 and P−P1 mixed formulations lack convergence for inviscid flows. P−P2 pair is more expensive but provides accurate results for all benchmarks. P−P provides an efficient option, except for inviscid Coriolis‐dominated flows, where a small lack of convergence is observed. Copyright © 2009 John Wiley & Sons, Ltd.

Discontinuous element pressure gradient stabilizations for compressible Navier-Stokes equations based on local projections

A pressure gradient discontinuous finite element formulation for the compressible Navier-Stokes equations is derived based on local projections. The resulting finite element formulation is stable and uniquely solvable without requiring a B-B stability condition. An error estimate is obtained.

Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations

We present space- and space–time discontinuous Galerkin finite element (DGFEM) formulations for systems containing nonconservative products, such as occur in dispersed multiphase flow equations. The main criterium we pose on the weak formulation is that if the system of nonconservative partial differential equations can be transformed into conservative form, then the formulation must reduce to that for conservative systems. Standard DGFEM formulations cannot be applied to nonconservative systems of partial differential equations. We therefore introduce the theory of weak solutions for nonconservative products into the DGFEM formulation leading to the new question how to define the path connecting left and right states across a discontinuity. The effect of different paths on the numerical solution is investigated and found to be small. We also introduce a new numerical flux that is able to deal with nonconservative products. Our scheme is applied to two different systems of partial differential equations. First, we consider the shallow water equations, where topography leads to nonconservative products, in which the known, possibly discontinuous, topography is formally taken as an unknown in the system. Second, we consider a simplification of a depth-averaged two-phase flow model which contains more intrinsic nonconservative products.

A locally discontinuous enriched finite element formulation for acoustics

AbstractIn (Comput. Methods Appl. Mech. Eng. 2006, in press) we introduced a discontinuous Galerkin finite element method for Helmholtz equation in which continuity is relaxed locally in the interior of the element. The shape functions associated with interior nodes of the element are bilinear discontinuous bubbles, and the corresponding degrees of freedom can be eliminated at element level by static condensation yielding a global finite element method with the same connectivity of classical C0 Galerkin finite element approximations. Stability is provided by the discontinuous bubbles with appropriate choice of the stabilization parameters related to the weak enforcement of continuity inside each element. In the present work, departing from the stencil obtained by condensation of the bubble degrees of freedom, we build a new strategy for determining the optimal values of these parameters aiming at matching the exact wave number in two different directions. Stability and accuracy of the proposed formulation are demonstrated in several numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.

A discontinuous finite element method at element level for Helmholtz equation

A discontinuous finite element formulation is presented for Helmholtz equation. Continuity is relaxed locally in the interior of the element instead of across the element edges. The interior shape functions can be viewed as discontinuous bubbles and the corresponding degrees of freedom can be eliminated at element level by static condensation yielding a global matrix topologically equivalent to those of classical C 0 finite element approximations. The discontinuous bubbles are introduced to consistently add stability to the formulation, preserving the optimal rates of convergence and improving accuracy. A crucial point of the discontinuous formulation relies on the choice of the stabilization parameters related to the weak enforcement of continuity inside each element. Explicit values of these stabilization parameters minimizing the pollution effect are presented for uniform meshes. The accuracy and stability of the proposed formulation for bilinear shape functions are demonstrated in several numerical examples.

Discontinuous Finite-Element Phase-Field Modeling of Polycrystalline Grain Growth with Convection

This paper presents a discontinuous finite-element computational framework for the numerical modeling of polycrystalline microstructure formation during solidification processing, taking into account the effect of convection. An important aspect of the discontinuous formulation is that the interelement continuity is relaxed for all the field variables or more precisely the continuity across the elements is weakly enforced, rather than strongly enforced as done in conventional finite-element methods. Because of this local nature, the discontinuous finite-element method enjoys certain advantages over the traditional finite difference and finite-element methods, in particular, for large scale numerical simulations of physical phenomena in thermal and fluids systems. The solution procedure involves an element-by-element iterative sweep. The discontinuous finite-element formulation is presented for a fully coupled set of nonlinear equations including the Navier-Stokes equations, the phase-field equation, the thermal equation, and the crystal orientation equation, which describe the physical phenomena associated with solidification processing. Both single grain and polygrain models are considered. Numerical results are presented that illustrate the effect of convection on the microstructure formation during polycrystalline solidification.

Stabilized discontinuous finite element approximations for Stokes equations

In this paper, we derive two stabilized discontinuous finite element formulations, symmetric and nonsymmetric, for the Stokes equations and the equations of the linear elasticity for almost incompressible materials. These methods are derived via stabilization of a saddle point system where the continuity of the normal and tangential components of the velocity/displacements are imposed in a weak sense via Lagrange multipliers. For both methods, almost all reasonable pair of discontinuous finite element spaces can be used to approximate the velocity and the pressure. Optimal error estimate for the approximation of both the velocity of the symmetric formulation and pressure in L 2 norm are obtained, as well as one in a mesh-dependent norm for the velocity in both symmetric and nonsymmetric formulations.

A Discontinuous Finite-Element Formulation for Radiative Transfer in Axisymmetric Finite Cylindrical Enclosures and Coupling with Other Mode Heat Transfer

ABSTRACT This article presents a discontinuous finite-element formulation for numerical solution of internal thermal radiative transfer problems in axisymmetric cylindrical enclosures. While the computation of axisymmetric radiative transfer is essentially three-dimensional, an appropriate mapping procedure entailing the use of the axisymmetric and periodic conditions associated with the enclosures may be used to make it possible to compute the problem over a 2-D mesh only. This will result in a significant reduction in both computing time and memory storage requirement. Mathematical formulation and numerical implementation using the discontinuous Galerkin method for axisymmetric internal radiation heat transfer calculations are given. The procedures for incorporating the mapping in the discontinuous formulation to convert an essentially 3-D calculation into a 2-D calculation are discussed in detail. The computed results are given and compare well with the solutions reported in the literature using other methods. Examples include both nonscattering and scattering cases. The effects of both solid-angular and spatial discretization are discussed, and both the theory and results show that an even discretization of solid angle is important to ensure adequate numerical accuracy. The coupling of the discontinuous and conventional finite-element methods for mixed-mode heat transfer calculations, including swirling flow, conduction, natural convection, and thermal radiation in participating media, is illustrated in the last example.

A discontinuous finite element formulation for Helmholtz equation

A discontinuous finite element formulation for Helmholtz problems is presented with C0 continuity across inter-element boundaries enforced in a weak sense depending on two parameters β and λ whose choice is crucial for the performance of the proposed method. These parameters are chosen through one dimension numerical experiments aiming at minimizing the pollution error. Optimal values of these parameters are then adopted in more general situations. The accuracy and stability of the proposed formulation for linear and bilinear shape functions are demonstrated in several numerical examples in one and two dimensions.

Convergence analysis of a discontinuous finite element formulation based on second order derivatives

A new discontinuous Galerkin formulation is introduced for the elliptic reaction–diffusion problem that incorporates local second order distributional derivatives. The corresponding bilinear form satisfies both coercivity and continuity properties on the broken Hilbert space of H 2 functions. For piecewise polynomial approximations of degree p ⩾ 2, optimal uniform h and p convergence rates are obtained in the broken H 1 and H 2 norms. Convergence in L 2 is optimal for p ⩾ 3, if the computational mesh is strictly rectangular. If the mesh consists of skewed elements, then optimal convergence is only obtained if the corner angles satisfy a given regularity condition. For p = 2, only suboptimal h convergence rates in L 2 are obtained and for linear polynomial approximations the method does not converge.

A DISCONTINUOUS FINITE-ELEMENT FORMULATION FOR MULTIDIMENSIONAL RADIATIVE TRANSFER IN ABSORBING, EMITTING, AND SCATTERING MEDIA

This article presents a discontinuous finite-element formulation for the numerical solution of internal thermal radiation problems in multidimensional domains. In contrast with the conventional finite-element formulation, the discontinuous finite element is based on the idea of allowing the discontinuity of field variables across the internal interelement boundaries and thus is particularly useful for the integral-differential equations governing the thermal radiative transfer phenomena in emitting, absorbing, and scattering media. Mathematical formulation and numerical details using the discontinuous finite element method for internal radiation heat transfer calculations are given. Computational procedures are presented. A parallel computing algorithm exploiting the advantages of the localized discontinuous finite-element formulation and space/solid-angle discretization characteristics is presented. The computed results are given and compared with analytical solutions whenever available as reported in references. Examples include both nonscattering and scattering cases. Parallel computing performance is also evaluated for both 2-D and 3-D cases and results show that the parallel algorithm can be easily implemented within the framework of the discontinuous finite-element method. For the cases tested, the parallel scheme based on the solid-angle discretization gives a reduction in CPU time approaching to almost an idealized parallel CPU scenario.

A DISCONTINUOUS FINITE-ELEMENT FORMULATION FOR INTERNAL RADIATION PROBLEMS

This article presents a discontinuous finite-element formulation for internal thermal radiation problems. In contrast to the conventional finite-element formulation, the discontinuous algorithm permits the discontinuity of field variables across the internal interelement boundaries and is useful for integral-differential equations describing thermal radiation in absorbing/scattering media. Mathematical formulation and numerical implementation are given. The convergence rate and local mesh adaptivity are discussed. Two approaches for coupling of the discontinuous and conventional methods for mixed heat transfer calculations are presented. Numerical results are given for internal radiation and combined conduction/radiation problems and are compared with analytical solutions whenever available.

The discontinuous Petrov–Galerkin method for elliptic problems

We propose a novel discontinuous mixed finite element formulation for the solution of second-order elliptic problems. Fully discontinuous piecewise polynomial finite element spaces are used for the trial and test functions. The discontinuous nature of the test functions at the element interfaces allows to introduce new boundary unknowns that on the one hand enforce the weak continuity of the trial functions, and on the other avoid the need to define a priori algorithmic fluxes as in standard discontinuous Galerkin methods. Static condensation is performed at the element level, leading to a solution procedure based on the sole interface unknowns. The resulting family of discontinuous dual–primal mixed (DPM) finite element methods is presented in the one and two-dimensional cases. In the one-dimensional case, we show the equivalence of the method with implicit Runge–Kutta schemes of the collocation type exhibiting optimal behavior and we develop the stability analysis. Numerical experiments in one and two dimensions demonstrate the order accuracy of the new method, confirming the results of the analysis.