Abstract

This work explores the use of stabilized finite element formulations for the incompressible Navier-Stokes equations to simulate turbulent flow problems. Turbulence is a challenging problem due to its complex and dynamic nature and its simulation if further complicated by the fact that it involves fluid motions at vastly different length and time scales, requiring fine meshes and long simulation times. A solution to this issue is turbulence modeling, in which only the large scale part of the solution is retained and the effect of smaller turbulent motions is represented by a model, which is generally dissipative in nature.

In the context of finite element simulations for fluids, a second problem is the apparition of numerical instabilities. These can be avoided by the use of stabilized formulations, in which the problem is modified to ensure that it has a stable solution. Since stabilization methods typically introduce numerical dissipation, the relation between numerical and physical dissipation plays a crucial role in the accuracy of turbulent flow simulations. We investigate this issue by studying the behavior of stabilized finite element formulations based on the Variational Multiscale framework and on Finite Calculus, analyzing the results they provide for well-known turbulent problems, with the final goal of obtaining a method that both ensures numerical stability and introduces physically correct turbulent dissipation.

Given that, even with the use of turbulence models, turbulent flow problems require significant computational resources, we also focused on programming and parallel implementation aspects of finite element codes, and in particular in ensuring that our solver can perform efficiently on distributed memory architectures and high-performance computing clusters.

Finally, we have developed an adaptive mesh refinement technique to improve the quality of unstructured tetrahedral meshes, again with the goal of enabling the simulation of large turbulent flow problems. This technique combines an error estimator based on Variational Multiscale principles with a simple refinement procedure designed to work in a distributed memory context and we have applied it to the simulation of both turbulent and non-Newtonian flow problems.

Abstract

In this paper, an adjoint-based error estimation and mesh adaptation framework is developed for the compressible inviscid flows. The algorithm employs the Finite Calculus (FIC) scheme for the numerical solution of the flow and discrete adjoint equations in the context of the Galerkin finite element method (FEM) on triangular grids. The FIC scheme treats the instabilities normally generated in the numerical solution of the fluid equations through adding two stabilization terms, called streamline term and transverse term, to the original central-based discretized formulation. The non-linear system of equations resulting from the flow problem is solved implicitly using a damped Newton’s method accompanied with the exact Jacobian matrix. A defect corrected scheme is implemented to iteratively solve the linear system of equations related to the adjoint problem benefiting from the transpose of the Jacobian matrix. At each iteration, the linear systems of equations resulting from the fluid and adjoint problems are solved using a preconditioned GMRES method. Having calculated the error of a specified output functional locally, an h-refinement methodology based on the element subdivision is performed to refine the candidate elements. The quality of the numerical results proves the capability of the presented approach for the adjoint-based error estimation and mesh adaptation problems in different flow regimes.

Abstract

A new implicit stabilized formulation for the numerical solution of the compressible NavierStokes equations is presented. The method is based on the Finite Calculus (FIC) scheme using the Galerkin finite element method (FEM) on triangular grids. Via the FIC formulation, two stabilization terms, called streamline term and transverse term, are added to the original conservation equations in the space-time domain. The non-linear system of equations resulting from the spatial discretization is solved implicitly using a damped Newton method benefiting from the exact Jacobian matrix. The matrix system is solved at each iteration with a preconditioned GMRES method. The efficiency of the proposed stabilization technique is checked out in the solution of 2D inviscid and laminar viscous flow problems where appropriate solutions are obtained especially near the boundary layer and shock waves. The work presented here can be considered as a follow up of a previous work of the authors [24]. In that paper, the stabilized Galerkin FEM based on the FIC formulation was derived for the Euler equations together with an explicit scheme. In the present paper, the extension of this work to the Navier-Stokes equations using an implicit scheme is presented.

Abstract

This paper aims at the development of a new stabilization formulation based on the Finite Calculus (FIC) scheme for solving the Euler equations using the Galerkin finite element method (FEM) on unstructured triangular grids. The FIC method is based on expressing the balance of fluxes in a space-time domain of finite size. It is used to prevent the creation of instabilities typically present in numerical solutions due to the high convective terms and sharp gradients. Two stabilization terms, respectively called streamline term and transverse term, are added via the FIC formulation to the original conservative equations in the spacetime domain. An explicit fourth order Runge-Kutta scheme is implemented to advance the solution in time. The presented numerical test examples for inviscid flows prove the ability of the proposed stabilization technique for providing appropriate solutions especially near shock waves. Although the derived methodology delivers precise results with a nearly coarse mesh, a mesh refinement technique is coupled to the solution process for obtaining a suitable mesh particularly in the high gradient zones.

Abstract

We present a new stabilized finite element (FEM) formulation for incompressible flows based on the Finite Increment Calculus (FIC) framework (Oñate, 1998). In comparison to existing FIC approaches for fluids, this formulation involves a new term in the momentum equation, which introduces non-isotropic dissipation in the direction of velocity gradients. We also follow a new approach to the derivation of the stabilized mass equation, inspired by recent developments for quasi-incompressible flows (Oñate et al., 2014). The presented FIC-FEM formulation is used to simulate turbulent flows, using the dissipation introduced by the method to account for turbulent dissipation in the style of implicit large eddy simulation.

Abstract

The aim of this work is to design monotonicity-preserving stabilized finite element techniques for transport problems as a blend of linear and nonlinear (shock-capturing) stabilization. As linear stabilization, we consider and analyze a novel symmetric projection stabilization technique based on a local Scott--Zhang projector. Next, we design a weighting of the aforementioned linear stabilization such that, when combined with a finite element discretization enjoying a discrete maximum principle (usually attained via nonlinear stabilization), it does not spoil these monotonicity properties. Then, we propose novel nonlinear stabilization schemes in the form of an artificial viscosity method where the amount of viscosity is proportional to gradient jumps at either finite element boundaries or nodes. For the nodal scheme, we prove a discrete maximum principle for time-dependent multidimensional transport problems. Numerical experiments support the numerical analysis and we show that the resulting methods provide excellent results. In particular, we observe that the proposed nonlinear stabilization techniques do an excellent job eliminating oscillations around shocks