Annual Report 2008.pdf - SAMSI

“Hierarchical Reconstruction for DG and Finite Volume Schemes”
(Joint with Chi-Wang Shu, Eitan Tadmor and Mengping Zhang)
Motivated by the moment limiter of Biswas, Devine and Flaherty [Appl. Numer. Math. 14
(1994)], we develop a general non-oscillatory hierarchical reconstruction (HR) procedure for
removing the spurious oscillations in the high degree polynomial of a cell computed by the
central DG scheme. This procedure is fully multidimensional and can be applied to any shapes of
cells at least in theory. HR uses the most compact stencil and thus fitts very well with DG.
Further more, it doesn't need any charcteristic decomposition even for very high order, such as
5th order, even though there will be small overshoots/undershoots for very high order when there
are interactions of discontinuities. HR can be applied to central and finite volume schemes as
well resulting in a new finite volume approach. These finite volume schemes can be used for
unstructured meshes with essentially no restriction on meshes, and can be done without
characteristic decomposition. We will demonstrate through numerical experiments the
effectiveness of the HR.
Miao-jung Yvonne Ou
University of Central Florida
Department of Mathematics
mou@mail.ucf.edu
“De-Homogenization: Mathematics for Retrieving Microstructural Information from
Macroscopic Properties”
The effective (macroscopic) properties of a composite depend both on the properties of its
constituents and on how they are arranged, i.e. the microstructure. In this poster, we present a
mathematical formulation of retrieving microstructural information from effective properties of
composite materials at different frequencies/temperatures. The core of this formulation is the
integral representation formula (IRF) for effectively properties. The IRF is a Stieltjes type
integral which includes the dependence on microstructure in the positive Borel measure of the
integral. The moments of this measure provide a natural hierarchy of microstructural information
in the sense of refining bounds on the effective properties as presented in the papers by Kantor &
Bergman and Golden & Papanicolaou. More precisely, the N-th moment is related to the (N+1)-
point correlation function of the microstructure. For the case of dielectric composites of two
isotropic constituents, the moments can be reconstructed from effective dielectric permittivites of
different frequencies by using the inversion method proposed in our recent papers. Remarkably,
despite the fact that the reconstruction of the Borel measure is a very ill-posed problem, the
reconstruction of moments is very stable, as is shown in the analysis in our papers. The
reconstructed moments can be further used to compute the effective properties of other
frequncies. Most importantly, the volume fraction of one constituent is equal to the zeroth
moment. The problem of retrieving information from moments of higher orders is open.
Generalization of this scheme to composite of viscoelastic materials will also be presented in the
poster. This requires the derivation of a suitable IRF by using theory of several complex
variables. Another application of this scheme can be to recover the property of one constituent
given both the property of the other and the microstructure.
* Some of the contents presented in this poster are joint work with E.Cherkaev and D. Zhang.