Higher order tensor inversion is possible for even order. This is due to the fact that a tensor group endowed with the contracted product is isomorphic to the general linear group of degree n. With these isomorphic group structures, we derive a tensor SVD which we have shown to be equivalent to well-known canonical polyadic decomposition and multilinear SVD provided that some constraints are satisfied. Moreover, within this group structure framework, multilinear systems are derived and solved for problems of high-dimensional PDEs and large discrete quantum models. We also address multilinear systems which do not fit the framework in the least-squares sense. These are cases when there is an odd number of modes or when each mode has distinct dimension. Numerically we solve multilinear systems using iterative techniques, namely, biconjugate gradient and Jacobi methods.

A parallel high-order Discontinuous Galerkin (DG) method is used to solve the compressible Navier-Stokes equations on an overset mesh framework. The DG solver has many capabilities including: hp-adaption, curved cells, and support for hybrid, mixed element meshes. Combining these capabilities with overset grids allows the DG solver to be used in problems with bodies moving in relative motion to be combined in a near-body off-body solver strategy. The overset implementation is constructed to preserve the design accuracy of the baseline DG discretization. Multiple simulations are carried out to validate the accuracy and performance of the overset DG solver. These simulations demonstrate the capability of the high-order DG solver to handle complex geometry, hp-adaption, and large scale parallel simulations in an overset framework.

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A parallel high-order Discontinuous Galerkin (DG) method is used to solve the compressible Navier-Stokes equations in an overset mesh framework. The DG solver has many capabilities including: hp-adaption, curved cells, support for hybrid, mixed-element meshes, and moving meshes. Combining these capabilities with overset grids allows the DG solver to be used in problems with bodies in relative motion and in a near-body offbody solver strategy. The overset implementation is constructed to preserve the design accuracy of the baseline DG discretization. Multiple simulations are carried out to validate the accuracy and performance of the overset DG solver. These simulations demonstrate the capability of the high-order DG solver to handle complex geometry and large scale parallel simulations in an overset framework.

Blade-resolved numerical simulations of wind energy applications using full blade and tower models are presented. The computational methodology combines solution technologies in a multi-mesh, multi-solver paradigm through a dynamic overset framework. The coupling of a finite volume solver and a high-order, hp-adaptive finite element solver is utilized. Additional technologies including in-situ visualization and atmospheric microscale modeling are incorporated into the analysis environment. Validation of the computational framework is performed on the National Renewable Energy Laboratory (NREL) 5MW baseline wind turbine, the unsteady aerodynamics experimental NREL Phase VI turbine, and the Siemens SWT-2.3-93 wind turbine. The power and thrust results of all single turbine simulations agree well with low-fidelity model simulation results and field experiments when available. Scalability of the computational framework is demonstrated using 6, 12, 24, 48, and 96 wind turbine setups including the 48 turbine wind plant known as Lillgrund. The largest case consisting of 96 wind turbines and a total of 385 overset grids are run on 44,928 cores at a weak scaling efficiency of 86%. Demonstration of the coupling of atmospheric microscale and Computational Fluid Dynamics (CFD) solvers is presented using the National Center for Atmospheric Research (NCAR) Weather Research and Forecasting Model (WRF) solver and the NREL Simulator fOr Wind Farm Applications (SOWFA) solver.