Isogeometric Analysis: Toward Integration of CAD and FEA

by J. Austin Cottrell, Thomas J.R. Hughes, Yuri Bazilevs

There has already been considerable work on using IA for structural problems, and some problems in fluid dynamics for incompressible flow, but I have not come across any works on compressible flows.

GeoPdes is a matlab/octave software for isogeometric analysis which was recently announced on the NA Digest mailing list

Quote:

GeoPDEs is a suite of free software tools for research on IGA, being developed and maintained as a cooperation of IMATI, Universita' di Pavia and Politecnico di Milano. It has been implemented in Octave, and is fully compatible with Matlab. GeoPDEs has been written with two main objectives. First, to serve as an entry point for beginners interested in understanding the basic issues involved in the implementation of an IGA code. Second, to provide a rapid prototyping and testing tool for new IGA algoritmhs and methods. The software is available at http://geopdes.sourceforge.net |

Let be normal to a cell face and whose magnitude is equal to face area. Let be the conserved vector. The finite volume update equation using forward Euler time discretization is

Here is a normal vector pointing from current cell "j" into the neighbouring cell "k". Note that the conserved variable Q is updated in the global Cartesian coordinate frame.

As an example, the Rusanov flux would be defined as

Here, we have used the definition

where is the velocity vector, etc., and

with being speed of sound. Note that there are many other ways to define . ]]>

Elmer has a good gui and has many physical models already available. There is a good amount of manuals and tutorials available. To get all of them, check out the code from svn. It also has some ability to program your own solvers but this is not well documented. It is a good aid for teaching numerical methods. Windows executable can be downloaded from their website. There is also a Mac binary made by someone else. But for Linux, you will have to compile it yourself.

http://www.csc.fi/english/pages/elmer

deal.ii is a FEM library which is very powerful in the sense that it is a general PDE solver. To use it you have to write your own code using deal.ii libraries. They have the best tutorials I have seen, which takes you step by step from simple problems to more complex ones. It also has discontinuous finite elements. I am currently working through the tutorials. It has very rudimentary grid generation capacity and you will have to use an external grid generator. One limitation of deal.ii is that it supports only quadrilateral/hexahedral elements.

http://www.dealii.org ]]>

SIAM J. Numer. Anal. Volume 45, Issue 6, pp. 2671-2696 (2007)

http://dx.doi.org/10.1137/060665117

This paper gives a notion of consistent adjoint discretizations for DG methods. However the definition seems to be restricted to smooth solutions only. For the case of a conservation law, they show that boundary conditions of the primal problem must be appropriate for the corresponding adjoint discretization to be consistent. The DG method requires the use of a numerical flux function. It is common among many CFD codes to use the same numerical flux function for computing the boundary fluxes, with the help of ghost data which incorporates the boundary conditions. The authors show that this does not lead to consistent adjoint discretizations. One must use a boundary flux which is consistent with the exact boundary conditions. Without knowing this, I have always followed the correct approach, i.e., I never use the numerical flux function for boundary flux computation. In the case of Euler equations, the boundary flux contains only the pressure contribution. Hence one can extrapolate the pressure from the interior cell upto the boundary face, and then compute the flux using the exact analytical definition of flux, i.e.,

for the 2-D case. ]]>

http://math.tifrbng.res.in/~praveen ]]>

http://link.aip.org/link/?SNA/48/882/1&agg=rss

Convergence of Linearized and Adjoint Approximations for Discontinuous Solutions of Conservation Laws. Part 1: Linearized Approximations and Linearized Output Functionals

SIAM Journal on Numerical Analysis on 6/29/10

Mike Giles and Stefan Ulbrich

This paper analyzes the convergence of discrete approximations to the linearized equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. A simple modified LaxFriedrichs discretization is used on a uniform grid, and a key point is that the numerical smooth ... [SIAM J. Numer. Anal. 48, 882 (2010)] published Tue Jun 29, 2010.

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http://link.aip.org/link/?SNA/48/905/1&agg=rss

Convergence of Linearized and Adjoint Approximations for Discontinuous Solutions of Conservation Laws. Part 2: Adjoint Approximations and Extensions

Mike Giles and Stefan Ulbrich

This paper continues the convergence analysis in [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 882904] of discrete approximations to the linearized and adjoint equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. We consider a simple m ... [SIAM J. Numer. Anal. 48, 905 ]]>