Uncertain loading and quantifying maximum energy concentration within composite structures

Abstract

We introduce a systematic method for identifying the worst case load among all boundary loads of fixed energy. Here the worst case load is defined to be the one that delivers the largest fraction of input energy to a prescribed subdomain of interest. The worst case load is identified with the first eigenfunction of a suitably defined eigenvalue problem. The first eigenvalue for this problem is the maximum fraction of boundary energy that can be delivered to the subdomain. We compute worst case boundary loads and associated energy contained inside a prescribed subdomain through the numerical solution of the eigenvalue problem. We apply this computational method to bound the worst case load associated with an ensemble of random boundary loads given by a second order random process. Several examples are carried out on heterogeneous structures to illustrate the method.

Multiscale-Spectral GFEM and optimal oversampling

“…However for A-harmonic extensions of boundary hat functions having 301 nodes as support we see from Figure 8 that the support of this function penetrates further into in ω * . This is the phenomenon of penetration where A-harmonic extensions of boundary data with less oscillation have a gradient that penetrates further into ω * , than A-harmonic extensions of more oscillatory boundary data, see [8] and [31]. In light of the exponential decay of the n-width eigenfunctions in the energy norm over ω it becomes clear that the space spanned by the lower n-width modes are generated by linear combinations of A-harmonic extensions of coarse boundary data, ie., boundary hat functions of large support.…”

“…When the number of boundary hat functions is reduced from N to M by defining boundary hat functions with larger support then the computational time for constructing S mi ω * i is reduced by a factor of M N . The time for computing each entry of the matrices (34) and (35) For example, if the support of the hat functions is increased from 1 to 3 nodes, then the total number of boundary value problems (31) to solve is M = N/2 and the time to fill the spectral matrices is about 1/4 the time as with single node hat functions. We illustrate these observations with the following example.…”

“…. , where k represents the number of nodes in the support of the hat function used as boundary data for problem (31).…”

A low-rank Schwarz method for radiative transport equation with heterogeneous scattering coefficient

“…Previously, several algorithms that make use of PDE compression and random sampling ideas have been proposed, mostly in the context of elliptic PDEs [5,8,12,18,[27][28][29]31]. A more systematic investigation of PDE compression appears recently in our previous work [7], where compressed PDE solution spaces are related to low rank structure of the matrix formed by the Green's functions.…”

Special Issue: Predictive multiscale materials modeling

“…• There are a key number of modeling challenges to be addressed. Amongst those included devising methods and protocols for addressing rare events and extreme value statistics such as those giving rise to materials failure [5].…”