Stabilized Finite Elements in FUN3D

A streamlined upwind Petrov–Galerkin (SUPG)–stabilized finite-element discretization has been implemented as a library into the FUN3D unstructured-grid flow solver. Motivation for the selection of this methodology is given, details of the implementation are provided, and the discretization for the interior scheme is verified for linear and quadratic elements by using the method of manufactured solutions. A methodology is also described for capturing shocks, and simulation results are compared with the finite-volume formulation that is currently the primary method employed for routine engineering applications. The finite-element methodology is demonstrated to be more accurate than the finite-volume technology, particularly on tetrahedral meshes where the solutions obtained using the finite-volume scheme can suffer from adverse effects caused by bias in the grid. Although no effort has been made to date to optimize computational efficiency, the finite-element scheme is competitive with the finite-volume scheme in terms of computer time to reach convergence.

In this paper enhancements to a stabilized finite-element overset scheme are presented. First, the recently proposed grid overlapping method referred to as elliptic hole cutting has been modified to be more suitable for parallel implementation. These modifications dramatically reduce the amount of searching and interpolation required to establish the overset grid network. The second enhancement entails combining the overset approach with an adaptive grid refinement capability. For stabilized finite-elements, the adaptive methodology must ensure that the state-variables remain continuous across element boundaries. Finally, the overset methodology has been extended to three-dimensions with moving domains. Order of accuracy is examined via the method of manufactured solutions for linear and quadratic elements. Overset grid results of a sinusoidally oscillating with and without dynamic hole cutting are compared with single grid. A wing-store separation simulation is carried out to demonstrate the overset capability of current methodology.

A high-order Petrov–Galerkin finite element scheme is used to compute turbulent flow over a NACA 0012 airfoil at a freestream Mach number of 0.15, an angle of attack of 10 deg, and a Reynolds number based on the airfoil chord of 6 million. Results are obtained on a series of grids available on the NASA Turbulence Modeling Resource Web site and are compared with reference solutions that have been obtained using the FUN3D and CFL3D finite volume solvers on meshes with as many as 14.6 million degrees of freedom. Forces, moments, pressure distributions, skin friction, and profiles of velocity and turbulence working variable for the Spalart–Allmaras turbulence model are compared between the finite element and the finite volume solutions. It is demonstrated that the finite element scheme shows similar results as the finite volume schemes for most of the comparisons, but demonstrates significantly less dissipation of the wake profiles downstream of the airfoil. It is shown that, when the same number of degrees of freedom is used in the simulations, solutions obtained with quadratic elements are more accurate than those obtained using linear elements with only minor increases in computational cost. Finally, initial results with an adjoint-based hp-adaptive methodology are obtained that further demonstrate that the high-order finite element framework can efficiently yield accurate results.

A large number of massive crystal-plasticity-finite-element (CPFE) simulations are performed and post-processed to reveal the effects of element type and mesh resolution on accuracy of predicted mechanical fields over explicit grain structures. A CPFE model coupled with Abaqus/Standard is used to simulate simple-tension and simple-shear deformations to facilitate such quantitative mesh sensitivity studies. A grid-based polycrystalline grain structure is created synthetically by a phase-field simulation and converted to interface-conformal hexahedral and tetrahedral meshes of variable resolution. Procedures for such interface-conformal mesh generation over complex shapes are developed. FE meshes consisting of either hexahedral or tetrahedral, fully integrated as linear or quadratic elements are used for the CPFE simulations. It is shown that quadratic tetrahedral and linear hexahedral elements are more accurate for CPFE modeling than linear tetrahedral and quadratic hexahedral elements. Furthermore, tetrahedral elements are more desirable due to fast mesh generation and flexibility to describe geometries of grain structures.

SUMMARY We present a new numerical method to solve the heterogeneous anelastic seismic wave equations with arbitrary high order of accuracy in space and time on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. Using the velocity–stress formulation provides a linear hyperbolic system of equations with source terms that is completed by additional equations for the anelastic functions including the strain history of the material. These additional equations result from the rheological model of the generalized Maxwell body and permit the incorporation of realistic attenuation properties of viscoelastic material accounting for the behaviour of elastic solids and viscous fluids. The proposed method relies on the finite volume (FV) approach where cell-averaged quantities are evolved in time by computing numerical fluxes at the element interfaces. The basic ingredient of the numerical flux function is the solution of Generalized Riemann Problems at the element interfaces according to the arbitrary high-order derivatives (ADER) approach of Toro et al., where the initial data is piecewise polynomial instead of piecewise constant as it was in the original first-order FV scheme developed by Godunov. The ADER approach automatically produces a scheme of uniformly high order of accuracy in space and time. The high-order polynomials in space, needed as input for the numerical flux function, are obtained using a reconstruction operator acting on the cell averages. This reconstruction operator uses some techniques originally developed in the Discontinuous Galerkin (DG) Finite Element framework, namely hierarchical orthogonal basis functions in a reference element. In particular, in this article we pay special attention to underline the differences as well as the points in common with the ADER-DG schemes previously developed by the authors, especially concerning the MPI parallelization of both methods. The numerical convergence analysis demonstrates that the proposed FV schemes provide very high order of accuracy even on unstructured tetrahedral meshes while computational cost for a desired accuracy can be reduced when applying higher order reconstructions. Applications to a series of well-acknowledged elastic and anelastic test cases and comparisons with analytic and numerical reference solutions, obtained by different well-established numerical methods, confirm the performance of the proposed method. Therefore, the development of the highly accurate ADER-FV approach for tetrahedral meshes including viscoelastic material provides a novel, flexible and efficient numerical technique to approach 3-D wave propagation problems including realistic attenuation and complex geometry.

Two-step multi-resolution reconstruction-based compact gas-kinetic scheme on tetrahedral mesh

The performances of various numerical schemes used to model hyperbolic/parabolic equations have been studied by the calculation of their numerical group velocities. Numerical experiments conducted with one dimensional linear and quadratic Lagrangian finite elements with a Crank‐Nicolson finite differencing in time confirm the results of the analysis. The group velocity analysis supplements the well‐known amplitude and phase portraits introduced by Leendertse1 and helps explain the occurrence and behaviour of numerical oscillations in both finite difference and finite element schemes.

Rotational test spaces for a fully-implicit FVM and FEM for the DNS of fluid-particle interaction

An Arbitrary-Lagrangian-Eulerian hybrid finite volume/finite element method on moving unstructured meshes for the Navier-Stokes equations

Wavelet methods for PDEs — some recent developments

Numerical simulations of flow problems with moving boundaries commonly require the solution of the fluid equations on unstructured and deformable dynamic meshes. In this paper, we present a unified theory for deriving Geometric Conservation La.ws (GCLs) for such problems. We consider several popular discretization methods for the spatial approximation of the flow equations including the Arbitrary Lagrangian-Eulerian (ALE) finite volume and finite element schemes, and space-time stabilized finite element formulations. We show that, except for the case of the space-time discretization method, the GCLs impose import ant constraints on the algorithms employed for time-integrating the semi-discrete equations governing the fluid and dynamic mesh motions. We address the impa.ct of theses constraints on the solution of coupled aeroelastic problems, and highlight the importance of the GCLs with an illustration of their effect on the computation of the transient aeroelastic response of a flat panel in transonic flow.

Continuous finite element methods for Reissner-Mindlin plate problem

Three-dimensional J-integral evaluation for finite strain elastic–plastic solid using the quadratic tetrahedral finite element and automatic meshing methodology

Stability and accuracy properties of finite element schemes for the shallow water equations have been investigated earlier (Gray and Lynch, 1977) for different time marching schemes. This chapter concentrates on two time marching procedures, the primitive equation leapfrog approximation and the wave equation formulation, in conjunction with various spatial discretizations. Consistent and lumped, linear and quadratic isoparametric Lagrangian finite elements as well as second and fourth order finite differences are studied on a uniform mesh with constant bathymetry in Section 5.2. It is shown that quadratic Lagrangian finite elements require the use of a smaller time step than linear elements to remain stable (Kinnmark and Gray, 1984c). This effect is more severe for the leapfrog method than for the wave equation formulation. For the two-dimensional cases considered, in Section 5.5, with equal node spacing, constant bathymetry and the Coriolis force neglected, the stability constraint is identical in form to the one-dimensional case with the square of the Courant number replaced by the sums of squares of the Courant number in x- and y-directions.

Disk structure on the performance of a rotating-disk dynamic filter: A case study on microalgae microfiltration