Abstract: \begin{document} We have developed fast implementations of B-spline/NURBS based finite element solvers, written using PETSc. PETSc is frequently used in software packages to leverage its optimized and parallel implementation of solvers, however we also are using PETSc data structures to assemble the linear systems. These structures in PETSC (called DA’s) were originally intended for the parallel assembly of linear systems resulting from finite differences. We have reworked this structure for linear systems resulting from isogeometric analysis based on tensor product spline spaces. The result of which is the PetIGA framework for solving problems using isogeometric analysis which is scalable and greatly simplified over previous solvers. Our infrastructure has also allowed us to develop scalable solvers for a variety of problems. We have chosen to pursue nonlinear time dependent problems~\cite{PetIGAp, PetIGAc}, such as: \begin{itemize} \item Cahn-Hilliard \item Navier-Stokes-Korteweg \item Variational Multiscale for Navier-Stokes \item Diffusive Wave Approximation to Shallow Water Equations \item Phase-Field Crystal (PFC) equation and its time integration \item Divergence-conforming B-spline modeling of nanoparticle suspensions \end{itemize} We also have solvers for an assortment of linear problems: Poisson, elasticity, Helmholtz, thin shells, advection-diffusion, and diffusion-reaction. All solvers are written to be inherently parallel and run on anything from a laptop to a supercomputer such as Shaheen, KAUST’s IBM-BlueGeneP supercomputer. In this presentation we will focus on new time integration techniques for phase-field modeling which are energy stable and allow for stable linearizations of the underlying non-linear model~\cite{PFC}. \begin{thebibliography}{99} \setlength{\parskip}{0pt} \bibitem{PetIGAp} N. Collier, L. Dalcin, and V.M. Calo, ``PetIGA: High-Performance Isogeometric Analysis,'' submitted, 2013. \bibitem{PetIGAc} L. Dalcin and N. Collier, ``PetIGA: A framework for high performance Isogeometric Analysis,'' https://bitbucket.org/dalcinl/petiga/, 2013 \bibitem{PFC} P. Vignal, L. Dalcin, D.L. Brown, N. Collier, and V.M. Calo, ``Energy-stable time-discretizations for the phase-field crystal equation,'' in preparation, 2014. \end{thebibliography}

Abstract: In this paper, we present a multi-frontal solver algorithm for the adaptive finite element method expressed by graph grammar productions. The graph grammar productions construct first the binary elimination tree, and then process frontal matrices stored in distributed manner in nodes of the elimination tree. The solver is specialized for a class of one, two and three dimensional h refined meshes whose elimination tree has a regular structure. In particular, this class contains all one dimensional grids, two and three dimensional grids refined towards point singularities, two dimensional grids refined in an anisotropic way towards edge singularity as well as three dimensional grids refined in an anisotropic way towards edge or face singularities. In all these cases, the structure of the elimination tree and the structure of the frontal matrices are similar. The solver is implemented within the Galois environment, which allows parallel execution of graph grammar productions. We also compare the performance of the Galois implementation of our graph grammar based solver with the MUMPS solver

Abstract: For the simulation of wave propagation problems, it is necessary to truncate the computational domain. Perfectly Matched Layers are often employed for that purpose, especially in high contrast layered materials where absorbing boundary conditions are difficult to design. In here, we define a Perfectly Matched Layer that automatically adjusts its parameters without any user interaction. The user only has to indicate the desired decay in the surrounding layer. With this Perfectly Matched Layer, we show that even in the most complex scenarios where the material contrast properties are as high as sixteen orders of magnitude, we do not introduce numerical reflections when truncating the domain, thus, obtaining accurate solutions.

Abstract: In this paper we present a theoretical proof of linear computational cost and complexity for a recently developed direct solver driven by hypergraph grammar productions. The solver is specialized for computational meshes with point singularities in two and three dimensions. Linear complexity is achieved due to utilizing the special structure of such grids. We describe the algorithm and estimate the exact computational cost on an example of a two-dimensional mesh containing a point singularity. We extend this reasoning to the three dimensional meshes. Numerical results fully support our theoretical estimates.

Abstract: In a dynamic world, the process of prioritizing where to invest limited conservation resources is extremely complex. It needs to incorporate information on features (species, or landforms), planning units, ongoing or predicted future threats, and the costs and effectiveness of potential conservation actions. Extended research has been conducted on the spatial and temporal conservation prioritization using software tools such as Marxan, C-Plan, and Zonation to aid managers in their decision-making process. However, these tools are limited in various ways in addressing the full complexity of day-to-day management decisions. Some tools fail to consider variation in: land values in space and time; multiple threats and their spatio-temporal variations; multiple conservation actions applied to individual areas; the feasibility, effectiveness, and varying costs of actions; and the dynamic nature of biodiversity responses in space and time. Optimizing such a multi-dimensional system is a large challenge in complexity mathematics. What is needed is a new software tool that builds on current approaches, but allows for more realistic scenarios as described above, developed and parameterised in close collaboration with managers. This includes the modification of existing tools and the creation of new algorithms. The new software will be trialled in conservation planning exercises for islands in north-western Western Australia and the Great Barrier Reef. The current approaches mostly exploit simulated annealing as it was proven the fastest and sufficiently efficient for problems which do not need the best solution. The new software, however, intends to include sub-models on threats, costs, and contribution of action on individual islands. We are examining the option of use constraint programming to incorporate these sub-models into the decision process, with desirable time resolution.