Abstract

CIMNE coffee talks

The role of the Orthogonal Sub-Grid Scales - Variational Multi-Scale (OSGS-VMS) method in describing the Burgers "turbulence" phenomena

by Camilo Bayona (CIMNE)
    
Wednesday, May 31st, 2017 - 15h   
O.C. Zienkiewicz Room, C1, Campus Nord UPC, Barcelona
 
Abstract
The Orthogonal Sub-Grid Scales-Variational Multi-Scale (OSGS-VMS) method can accurately approximate Burgers "turbulence" (a highly simplified version of the incompressible Navier–Stokes turbulence), only with the addition of purely dissipative numerical terms. The analysis of a one-dimensional Burgers equation in the Fourier space allows to clarify the scale dependence of the numerical diffusion introduced by the OSGS-VMS method. In particular, results can be contrasted with Direct Numerical Simulations (DNS) and with a spectral Large Eddy Simulation (LES).

https://web.cimne.upc.edu/groups/publicacions/mails/2017/coffee-talks/coffee-talks-2017-05-31.html

Abstract

This paper exploits the concept of orthogonal sub-grid scales to stabilize the behaviour of mixed linear/linear simplicial elements (triangles and tetrahedra) in incompressible or nearly incompressible situations. Both incompressible elastic and J2-plastic constitutive behaviours have been considered. The different assumptions and approximations used to derive the method are exposed. Implementation and computational aspects are also discussed, showing that a robust application of the proposed formulation is feasible. Numerical examples show that the elements derived are free of volumetric locking and spurious oscillations of the pressure, and that the results obtained compare favourably with those obtained with the Q1P0 quadrilateral.

Abstract

This paper exploits the concept of stabilized finite element methods to formulate stable mixed stress/displacement and strain/displacement finite elements for the solution of nonlinear solid mechanics problems. The different assumptions and approximations used to derive the methods are exposed. The proposed procedure is very general, applicable to 2D and 3D problems. Implementation and computational aspects are also discussed, showing that a robust application of the proposed formulation is feasible. Numerical examples show that the results obtained compare favorably with those obtained with the corresponding irreducible formulation.

Abstract

In this paper, a stabilized finite element method to deal with incompressibility in solid mechanics is presented. Both elastic and J2-plastic constitutive behavior have been considered. A mixed formulation involving pressure and displacement fields is used and a continuous linear interpolation is considered for both fields. To circumvent the Babuška–Brezzi condition a stabilization technique based on the orthogonal sub-scale method is introduced. The main advantage of the method is the possibility of using linear triangular or tetrahedral finite elements, which are easy to generate for real industrial applications. Results are compared with standard Galerkin and Q1P0 mixed formulations in either elastic or elasto-plastic incompressible problems.

Abstract

This paper deals with the computational modeling and sub-grid scale stabilization of incompressibility and convection in the numerical simulation of the material flow around the probe tool in a friction stir welding (FSW) process. Within the paradigmatic framework of the multiscale stabilization methods, suitable pressure and convective derivative of the temperature sub-grid scale stabilized coupled thermomechanical formulations have been developed using an Eulerian description. Norton-Hoff and Sheppard-Wright thermo-rigid-viscoplastic constitutive material models have been considered. Constitutive equations for the sub-grid scale models have been proposed and an approximation of the sub-grid scale variables has been given. In particular, algebraic sub-grid scale (ASGS) and orthogonal sub-grid scale (OSGS) methods for mixed velocity, pressure and temperature P1/P1/P1 linear elements have been considered. Furthermore, it has been shown that well known classical stabilized formulations, such as the Galerkin least-squares (GLS) for incompressible (or quasi-incompressible) problems or the Streamline Upwind/Petrov-Galerkin (SUPG) method for convection dominant problems, can be recovered as particular cases of the multiscale stabilization framework considered. Using a product formula algorithm for the solution of the coupled thermomechanical problem, the resulting algebraic system of equations has been solved using a staggered procedure in which a mechanical problem, defined by the linear momentum balance equation, under quasi-static conditions, and the incompressibility equation, is solved first at constant temperature. Then a thermal problem, defined by the energy balance equation, is solved keeping constant the mechanical variables, i.e. velocity and pressure. The computational model has been implemented in an enhanced version of the finite element software COMET, developed by the authors at the International Center for Numerical Methods in Engineering (CIMNE). Two numerical examples have been considered. The first one deals with the numerical simulation of a coupled thermomechanical flow in a 2D rectangular domain. Steady-state and transient conditions have been considered. The goal of this numerical example has been the comparison between different sub-grid scale stabilization methods for the velocity and temperature equations. In particular, using a GLS stabilization method for the pressure equation, a comparison between SUPG and OSGS convective stabilization methods has been performed. Additionally, using a SUPG stabilization method for the temperature equation, a comparison between GLS and OSGS pressure stabilization methods has been done. The second example deals with the 3D numerical simulation of a representative FSW process. Numerical results obtained have been compared with experimental results available in the literature. A good agreement on the temperature distribution has been obtained and predicted peak temperatures compare well, both in value and position, with the experimental results available.

Abstract

In this paper we propose stabilized finite element methods for both Stokes' and Darcy's problems that accommodate any interpolation of velocities and pressures. Apart from the interest of this fact, the important issue is that we are able to deal with both problems at the same time, in a completely unified manner, in spite of the fact that the functional setting is different. Concerning the stabilization formulation, we discuss the effect of the choice of the length scale appearing in the expression of the stabilization parameters, both in what refers to stability and to accuracy. This choice is shown to be crucial in the case of Darcy's problem. As an additional feature of this work, we treat two types of stabilized formulations, showing that they have a very similar behavior
 

Abstract

Variational multiscale methods lead to stable finite element approximations of the NavierStokes equations, both dealing with the indefinite nature of the system (pressure stability) and the velocity stability loss for high Reynolds numbers. These methods enrich the Galerkin formulation with a sub-grid component that is modelled. In fact, the effect of the sub-grid scale on the captured scales has been proved to dissipate the proper amount of energy needed to approximate the correct energy spectrum. Thus, they also act as effective large-eddy simulation turbulence models and allow to compute flows without the need to capture all the scales in the system. In this article, we consider a dynamic sub-grid model that enforces the sub-grid component to be orthogonal to the finite element space in L 2 sense. We analyze the long-term behavior of the algorithm, proving the existence of appropriate absorbing sets and a compact global attractor. The improvements with respect to a finite element Galerkin approximation are the long-term estimates for the sub-grid component, that are translated to effective pressure and velocity stability. Thus, the stabilization introduced by the sub-grid model into the finite element problem is not deteriorated for infinite time intervals of computation. 

Abstract

In this paper we present stabilized finite element methods to discretize in space the monochromatic radiation transport equation. These methods are based on the decomposition of the unknowns into resolvable and subgrid scales, with an approximation for the latter that yields a problem to be solved for the former. This approach allows us to design the algorithmic parameters on which the method depends, which we do here when the discrete ordinates method is used for the directional approximation. We concentrate on two stabilized methods, namely, the classical SUPG technique and the orthogonal subscale stabilization. A numerical analysis of the spatial approximation for both formulations is performed, which shows that they have a similar behavior: they are both stable and optimally convergent in the same mesh-dependent norm. A comparison with the behavior of the Galerkin method, for which a non-standard numerical analysis is done, is also presented.

Abstract

In this work we present a general error estimator for the finite element solution of solid mechanics problems based on the Variational Multiscale method. The main idea is to consider a rich model for the subgrid scales as an error estimator. The subscales are considered to belong to a space orthogonal to the finite element space (Orthogonal Subgrid Scales) and we take into account their contribution both in the element interiors and on the element boundaries (Subscales on the Element Boundaries). A simple analysis shows that the upper bound for the obtained error estimator is sharper than in other error estimators based on the Variational Multiscale Method. Numerical examples show that the proposed error estimator is an accurate approximation for the energy norm error and can be used both in simple linear constitutive models and in more complex non-linear cases.

Abstract

A numerical approximation for the one‐dimensional Burgers equation is proposed by means of the orthogonal subgrid scales–variational multiscale (OSGS‐VMS) method. We evaluate the role of the variational subscales in describing the Burgers “turbulence” phenomena. Particularly, we seek to clarify the interaction between the subscales and the resolved scales when the former are defined to be orthogonal to the finite‐dimensional space. Direct numerical simulation is used to evaluate the resulting OSGS‐VMS energy spectra. The comparison against a large eddy simulation model is presented for numerical discretizations in which the grid is not capable of solving the small scales. An accurate approximation to the phenomena of turbulence is obtained with the addition of the purely dissipative numerical terms given by the OSGS‐VMS method without any modification of the continuous problem.