Development of an integrated BEM approach for hot fluid structure interaction: BEST-FSI: Boundary Element Solution Technique for Fluid Structure Interaction

As part of the continuing effort at NASA LeRC to improve both the durability and reliability of hot section Earth-to-orbit engine components, significant enhancements must be made in existing finite element and finite difference methods, and advanced techniques, such as the boundary element method (BEM), must be explored. The BEM was chosen as the basic analysis tool because the critical variables (temperature, flux, displacement, and traction) can be very precisely determined with a boundary-based discretization scheme. Additionally, model preparation is considerably simplified compared to the more familiar domain-based methods. Furthermore, the hyperbolic character of high speed flow is captured through the use of an analytical fundamental solution, eliminating the dependence of the solution on the discretization pattern. The price that must be paid in order to realize these advantages is that any BEM formulation requires a considerable amount of analytical work, which is typically absent in the other numerical methods. All of the research accomplishments of a multi-year program aimed toward the development of a boundary element formulation for the study of hot fluid-structure interaction in Earth-to-orbit engine hot section components are detailed. Most of the effort was directed toward the examination of fluid flow, since BEM's for fluids are at a much less developed state. However, significant strides were made, not only in the analysis of thermoviscous fluids, but also in the solution of the fluid-structure interaction problem

A Method for Geometry Optimization in a Simple Model of Two-Dimensional Heat Transfer

This investigation is motivated by the problem of optimal design of cooling elements in modern battery systems. We consider a simple model of two-dimensional steady-state heat conduction described by elliptic partial differential equations and involving a one-dimensional cooling element represented by a contour on which interface boundary conditions are specified. The problem consists in finding an optimal shape of the cooling element which will ensure that the solution in a given region is close (in the least squares sense) to some prescribed target distribution. We formulate this problem as PDE-constrained optimization and the locally optimal contour shapes are found using a gradient-based descent algorithm in which the Sobolev shape gradients are obtained using methods of the shape-differential calculus. The main novelty of this work is an accurate and efficient approach to the evaluation of the shape gradients based on a boundary-integral formulation which exploits certain analytical properties of the solution and does not require grids adapted to the contour. This approach is thoroughly validated and optimization results obtained in different test problems exhibit nontrivial shapes of the computed optimal contours.Comment: Accepted for publication in "SIAM Journal on Scientific Computing" (31 pages, 9 figures

Smart FRP Composite Sandwich Bridge Decks in Cold Regions

INE/AUTC 12.0

Shingle 2.0: generalising self-consistent and automated domain discretisation for multi-scale geophysical models

The approaches taken to describe and develop spatial discretisations of the domains required for geophysical simulation models are commonly ad hoc, model or application specific and under-documented. This is particularly acute for simulation models that are flexible in their use of multi-scale, anisotropic, fully unstructured meshes where a relatively large number of heterogeneous parameters are required to constrain their full description. As a consequence, it can be difficult to reproduce simulations, ensure a provenance in model data handling and initialisation, and a challenge to conduct model intercomparisons rigorously. This paper takes a novel approach to spatial discretisation, considering it much like a numerical simulation model problem of its own. It introduces a generalised, extensible, self-documenting approach to carefully describe, and necessarily fully, the constraints over the heterogeneous parameter space that determine how a domain is spatially discretised. This additionally provides a method to accurately record these constraints, using high-level natural language based abstractions, that enables full accounts of provenance, sharing and distribution. Together with this description, a generalised consistent approach to unstructured mesh generation for geophysical models is developed, that is automated, robust and repeatable, quick-to-draft, rigorously verified and consistent to the source data throughout. This interprets the description above to execute a self-consistent spatial discretisation process, which is automatically validated to expected discrete characteristics and metrics.Comment: 18 pages, 10 figures, 1 table. Submitted for publication and under revie

Electro and magneto statics of topological insulators as modeled by planar, spherical and cylindrical θ\theta boundaries: Green function approach

The Green function (GF) method is used to analyze the boundary effects produced by a Chern Simons (CS) extension to electrodynamics. We consider the electromagnetic field coupled to a $\theta$ term that is piecewise constant in different regions of space, separated by a common interface $\Sigma$, the $\theta$ boundary, model which we will refer to as $\theta$ electrodynamics ($\theta$ ED). This model provides a correct low energy effective action for describing topological insulators (TI). In this work we construct the static GF in $\theta$ ED for different geometrical configurations of the $\theta$ boundary, namely: planar, spherical and cylindrical $\theta$ interfaces. Also we adapt the standard Green theorem to include the effects of the $\theta$ boundary. These are the most important results of our work, since they allow to obtain the corresponding static electric and magnetic fields for arbitrary sources and arbitrary boundary conditions in the given geometries. Also, the method provides a well defined starting point for either analytical or numerical approximations in the cases where the exact analytical calculations are not possible. Explicit solutions for simple cases in each of the aforementioned geometries for $\theta$ boundaries are provided. The adapted Green theorem is illustrated by studying the problem of a point like electric charge interacting with a planar TI with prescribed boundary conditions. Our generalization, when particularized to specific cases, is successfully compared with previously reported results, most of which have been obtained by using the methods of images.Comment: 24 pages, 4 figures, accepted for publication in PRD. arXiv admin note: text overlap with arXiv:1511.0117

Soliton approach to the noisy Burgers equation: Steepest descent method

The noisy Burgers equation in one spatial dimension is analyzed by means of the Martin-Siggia-Rose technique in functional form. In a canonical formulation the morphology and scaling behavior are accessed by mean of a principle of least action in the asymptotic non-perturbative weak noise limit. The ensuing coupled saddle point field equations for the local slope and noise fields, replacing the noisy Burgers equation, are solved yielding nonlinear localized soliton solutions and extended linear diffusive mode solutions, describing the morphology of a growing interface. The canonical formalism and the principle of least action also associate momentum, energy, and action with a soliton-diffusive mode configuration and thus provides a selection criterion for the noise-induced fluctuations. In a ``quantum mechanical'' representation of the path integral the noise fluctuations, corresponding to different paths in the path integral, are interpreted as ``quantum fluctuations'' and the growth morphology represented by a Landau-type quasi-particle gas of ``quantum solitons'' with gapless dispersion and ``quantum diffusive modes'' with a gap in the spectrum. Finally, the scaling properties are dicussed from a heuristic point of view in terms of a``quantum spectral representation'' for the slope correlations. The dynamic eponent z=3/2 is given by the gapless soliton dispersion law, whereas the roughness exponent zeta =1/2 follows from a regularity property of the form factor in the spectral representation. A heuristic expression for the scaling function is given by spectral representation and has a form similar to the probability distribution for Levy flights with index $z$.Comment: 30 pages, Revtex file, 14 figures, to be submitted to Phys. Rev.

Cascade of minimizers for a nonlocal isoperimetric problem in thin domains

For \Omega_\e=(0,\e)\times (0,1) a thin rectangle, we consider minimization of the two-dimensional nonlocal isoperimetric problem given by \inf_u E^{\gamma}_{\Omega_\e}(u) where E^{\gamma}_{\Omega_\e}(u):= P_{\Omega_\e}(\{u(x)=1\})+\gamma\int_{\Omega_\e}\abs{\nabla{v}}^2\,dx and the minimization is taken over competitors u\in BV(\Omega_\e;\{\pm 1\}) satisfying a mass constraint \fint_{\Omega_\e}u=m for some $m\in (-1,1)$. Here P_{\Omega_\e}(\{u(x)=1\}) denotes the perimeter of the set $\{u(x)=1\}$ in \Omega_\e, \fint denotes the integral average and $v$ denotes the solution to the Poisson problem -\Delta v=u-m\;\mbox{in}\;\Omega_\e,\quad\nabla v\cdot n_{\partial\Omega_\e}=0\;\mbox{on}\;\partial\Omega_\e,\quad\int_{\Omega_\e}v=0. We show that a striped pattern is the minimizer for \e\ll 1 with the number of stripes growing like $\gamma^{1/3}$ as $\gamma\to\infty.$ We then present generalizations of this result to higher dimensions.Comment: 20 pages, 2 figure

Generalized linear sampling method for elastic-wave sensing of heterogeneous fractures

A theoretical foundation is developed for active seismic reconstruction of fractures endowed with spatially-varying interfacial condition (e.g.~partially-closed fractures, hydraulic fractures). The proposed indicator functional carries a superior localization property with no significant sensitivity to the fracture's contact condition, measurement errors, and illumination frequency. This is accomplished through the paradigm of the $F_\sharp$-factorization technique and the recently developed Generalized Linear Sampling Method (GLSM) applied to elastodynamics. The direct scattering problem is formulated in the frequency domain where the fracture surface is illuminated by a set of incident plane waves, while monitoring the induced scattered field in the form of (elastic) far-field patterns. The analysis of the well-posedness of the forward problem leads to an admissibility condition on the fracture's (linearized) contact parameters. This in turn contributes toward establishing the applicability of the $F_\sharp$-factorization method, and consequently aids the formulation of a convex GLSM cost functional whose minimizer can be computed without iterations. Such minimizer is then used to construct a robust fracture indicator function, whose performance is illustrated through a set of numerical experiments. For completeness, the results of the GLSM reconstruction are compared to those obtained by the classical linear sampling method (LSM)