Jason E. Hicken

Stanford University Unstructured (SU 2 ): An open-source integrated computational environment for multi-physics simulation and design

This paper describes the history, objectives, structure, and current capabilities of the Stanford University Unstructured (SU 2 ) tool suite. This computational analysis and design software collection is being developed to solve complex, multi-physics analysis and optimization tasks using arbitrary unstructured meshes, and it has been designed so that it is easily extensible for the solution of Partial Differential Equation-based (PDE) problems not directly envisioned by the authors. At its core, SU 2 is an open-source collection of C++ software tools to discretize and solve problems described by PDEs and is able to solve PDE-constrained optimization problems, including optimal shape design. Although the toolset has been designed with Computational Fluid Dynamics (CFD) and aerodynamic shape optimization in mind, it has also been extended to treat other sets of governing equations including potential flow, electrodynamics, chemically reacting flows, and several others. In our experience, capabilities for computational analysis and optimization have improved considerably over the past two decades. However, the ability to integrate the resulting software packages into coupled multi-physics analysis and design optimization solvers has remained a challenge: the variety of approaches chosen for the independent components of the overall problem (flow solvers, adjoint solvers, optimizers, shape parameterization, shape deformation, mesh adaption, mesh deformation, etc) make it difficult to (a) expand the range of applicability to situations not originally envisioned, and (b) to reduce the overall burden of creating integrated applications. By leveraging well-established object-oriented software architectures (using C++) and by enabling a common interface for all the necessary components, SU 2 is able to remove these barriers for both the beginner and the seasoned analyst. In this paper we attempt to describe our efforts to develop SU 2 as an integrated platform. In some senses, the paper can also be used as a software reference manual for those who might be interested in modifying it to suit their own needs. We carefully describe the C++ framework and object hierarchy, the sets of equations that can be currently modeled by SU 2 , the available choices for numerical discretization, and conclude with a set of relevant validation and verification test cases that are included with the SU 2 distribution. We intend for SU 2 to remain open source and to serve as a starting point for new capabilities not included in SU 2 today, that will hopefully be contributed by users in both academic and industrial environments.

Parallel Newton-Krylov Solver for the Euler Equations Discretized Using Simultaneous-Approximation Terms

0mesh continuity at block interfaces, accommodates arbitrary block topologies, and has low interblock-communication overhead. The resulting discrete equations are solved iteratively using an inexact-Newton method. At each Newton iteration, the linear system is solved inexactly using a Krylov-subspace iterative method, and both additive Schwarz and approximate Schur preconditioners are investigated. The algorithm is tested on the ONERA M6 wing geometry. We conclude that the approximate Schur preconditioner is an efficient alternative to the Schwarz preconditioner. Overall, the results demonstrate that the Newton–Krylov algorithm is very efficient: using 24 processors, a transonic flow on a 96-block, 1-million-node mesh requires 12 minutes for a 10-order reduction of the residual norm.

Induced-Drag Minimization of Nonplanar Geometries Based on the Euler Equations

The induced drag of several nonplanar configurations is minimized using an aerodynamic shape optimization algorithm based on the Euler equations. The algorithm is first validated using twist optimization to recover an elliptical lift distribution. Planform optimization reveals that an elliptical planform is not optimal when side-edge separation is present. Optimized winglet and box-wing geometries are found to have span efficiencies that agree well with lifting-line analysis, provided the bound constraints on the entire geometry are accounted for in the linear analyses. For the same spanwise and vertical bound constraints, a nonplanar split-tip geometry outperforms both the winglet and box-wing geometries, because it can more easily maximize the vertical extent at the tip. The performance of all the optimized geometries is verified using refined grids consisting of 88-152 million nodes.

Summation-by-parts operators and high-order quadrature

Summation-by-parts (SBP) operators are finite-difference operators that mimic integration by parts. The SBP operator definition includes a weight matrix that is used formally for discrete integration; however, the accuracy of the weight matrix as a quadrature rule is not explicitly part of the SBP definition. We show that SBP weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes. For diagonal weight matrices, the accuracy of SBP quadrature extends to curvilinear domains provided the Jacobian is approximated with the same SBP operator used for the quadrature. This quadrature has significant implications for SBP-based discretizations; in particular, the diagonal norm accurately approximates the L^2 norm for functions, and multi-dimensional SBP discretizations accurately approximate the divergence theorem.

Superconvergent Functional Estimates from Summation-By-Parts Finite-Difference Discretizations

Diagonal-norm summation-by-parts (SBP) operators can be used to construct time-stable high-order accurate finite-difference schemes. However, to achieve both stability and accuracy, these operators must use

A Simplified and Flexible Variant of GCROT for Solving Nonsymmetric Linear Systems

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Inverse problems for ODEs using contraction maps and suboptimality of the 'collage method'

-order accurate boundary closures when the interior scheme is

Multidimensional Summation-by-Parts Operators: General Theory and Application to Simplex Elements

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Dual consistency and functional accuracy

-order accurate. The boundary closure limits the solution to

Adjoint-based method for supersonic aircraft design using equivalent area distribution

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