2,152 research outputs found
Spacetime Slices and Surfaces of Revolution
Under certain conditions, a -dimensional slice of a
spherically symmetric black hole spacetime can be equivariantly embedded in
-dimensional Minkowski space. The embedding depends on a real parameter
that corresponds physically to the surface gravity of the black hole
horizon.
Under conditions that turn out to be closely related, a real surface that
possesses rotational symmetry can be equivariantly embedded in 3-dimensional
Euclidean space. The embedding does not obviously depend on a parameter.
However, the Gaussian curvature is given by a simple formula: If the metric is
written , then
\K_g=-{1/2}\phi''(r).
This note shows that metrics and occur in dual pairs, and that
the embeddings described above are orthogonal facets of a single phenomenon. In
particular, the metrics and their respective embeddings differ by a Wick
rotation that preserves the ambient symmetry.
Consequently, the embedding of depends on a real parameter. The ambient
space is not smooth, and is inversely proportional to the cone angle
at the axis of rotation. Further, the Gaussian curvature of is given
by a simple formula that seems not to be widely known.Comment: 15 pages, added reference
A Modular Approach to Large-scale Design Optimization of Aerospace Systems.
Gradient-based optimization and the adjoint method form a synergistic combination that enables the efficient solution of large-scale optimization problems. Though the gradient-based approach struggles with non-smooth or multi-modal problems, the capability to efficiently optimize up to tens of thousands of design variables provides a valuable design tool for exploring complex tradeoffs and finding unintuitive designs. However, the widespread adoption of gradient-based optimization is limited by the implementation challenges for computing derivatives efficiently and accurately, particularly in multidisciplinary and shape design problems. This thesis addresses these difficulties in two ways.
First, to deal with the heterogeneity and integration challenges of multidisciplinary problems, this thesis presents a computational modeling framework that solves multidisciplinary systems and computes their derivatives in a semi-automated fashion. This framework is built upon a new mathematical formulation developed in this thesis that expresses any computational model as a system of algebraic equations and unifies all methods for computing derivatives using a single equation. The framework is applied to two engineering problems: the optimization of a nanosatellite with 7 disciplines and over 25,000 design variables; and simultaneous allocation and mission optimization for commercial aircraft involving 330 design variables, 12 of which are integer variables handled using the branch-and-bound method. In both cases, the framework makes large-scale optimization possible by reducing the implementation effort and code complexity.
The second half of this thesis presents a differentiable parametrization of aircraft geometries and structures for high-fidelity shape optimization. Existing geometry parametrizations are not differentiable, or they are limited in the types of shape changes they allow. This is addressed by a novel parametrization that smoothly interpolates aircraft components, providing differentiability. An unstructured quadrilateral mesh generation algorithm is also developed to automate the creation of detailed meshes for aircraft structures, and a mesh convergence study is performed to verify that the quality of the mesh is maintained as it is refined. As a demonstration, high-fidelity aerostructural analysis is performed for two unconventional configurations with detailed structures included, and aerodynamic shape optimization is applied to the truss-braced wing, which finds and eliminates a shock in the region bounded by the struts and the wing.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111567/1/hwangjt_1.pd
Reconfigurable Model Execution in the OpenMDAO Framework
NASA's OpenMDAO framework facilitates constructing complex models and computing their derivatives for multidisciplinary design optimization. Decomposing a model into components that follow a prescribed interface enables OpenMDAO to assemble multidisciplinary derivatives from the component derivatives using what amounts to the adjoint method, direct method, chain rule, global sensitivity equations, or any combination thereof, using the MAUD architecture. OpenMDAO also handles the distribution of processors among the disciplines by hierarchically grouping the components, and it automates the data transfer between components that are on different processors. These features have made OpenMDAO useful for applications in aircraft design, satellite design, wind turbine design, and aircraft engine design, among others. This paper presents new algorithms for OpenMDAO that enable reconfigurable model execution. This concept refers to dynamically changing, during execution, one or more of: the variable sizes, solution algorithm, parallel load balancing, or set of variables-i.e., adding and removing components, perhaps to switch to a higher-fidelity sub-model. Any component can reconfigure at any point, even when running in parallel with other components, and the reconfiguration algorithm presented here performs the synchronized updates to all other components that are affected. A reconfigurable software framework for multidisciplinary design optimization enables new adaptive solvers, adaptive parallelization, and new applications such as gradient-based optimization with overset flow solvers and adaptive mesh refinement. Benchmarking results demonstrate the time savings for reconfiguration compared to setting up the model again from scratch, which can be significant in large-scale problems. Additionally, the new reconfigurability feature is applied to a mission profile optimization problem for commercial aircraft where both the parametrization of the mission profile and the time discretization are adaptively refined, resulting in computational savings of roughly 10% and the elimination of oscillations in the optimized altitude profile
TALOS: A toolbox for spacecraft conceptual design
We present the Toolbox for Analysis and Large-scale Optimization of
Spacecraft (TALOS), a framework designed for applying large-scale
multidisciplinary design optimization (MDO) to spacecraft design problems. The
framework is built using the Computational System Design Language (CSDL), with
abstractions for users to describe systems at a high level. CSDL is a compiled,
embedded domain-specific language that fully automates derivative computation
using the adjoint method. CSDL provides a unified interface for defining MDO
problems, separating model definition from low-level program implementation
details. TALOS provides discipline models for spacecraft mission designers to
perform analyses, optimizations, and trade studies early in the design process.
TALOS also provides interfaces for users to provide high-level system
descriptions without the need to use CSDL directly, which simplifies the
exploration of different spacecraft configurations. We describe the interfaces
in TALOS available to users and run analyses on selected spacecraft subsystem
disciplines to demonstrate the current capabilities of TALOS
Solution of Ordinary Differential Equations in Gradient-Based Multidisciplinary Design Optimization
A gradient-based approach to multidisciplinary design optimization enables efficient scalability to large numbers of design variables. However, the need for derivatives causes difficulties when integrating ordinary differential equations (ODEs) in models. To simplify this, we propose the use of the general linear methods framework, which unifies all Runge-Kutta and linear multistep methods. This approach enables rapid implementation of integration methods without the need to differentiate each one, even in a gradient-based optimization context. We also develop a new parallel time integration algorithm that enables vectorization across time steps. We present a set of benchmarking results using a stiff ODE, a non-stiff nonlinear ODE, and an orbital dynamics ODE, and compare integration methods. In a modular gradient-based multidisciplinary design optimization context, we find that the new parallel time integration algorithm with high-order implicit methods, especially Gauss-Legendre collocation, is the best choice for a broad range of problems
Dynamical study of the hyperextended scalar-tensor theory in the empty Bianchi type I model
The dynamics of the hyperextended scalar-tensor theory in the empty Bianchi
type I model is investigated. We describe a method giving the sign of the first
and second derivatives of the metric functions whatever the coupling function.
Hence, we can predict if a theory gives birth to expanding, contracting,
bouncing or inflationary cosmology. The dynamics of a string inspired theory
without antisymetric field strength is analysed. Some exact solutions are
found.Comment: 18 pages, 3 figure
Automated shape and thickness optimization for non-matching isogeometric shells using free-form deformation
Isogeometric analysis (IGA) has emerged as a promising approach in the field
of structural optimization, benefiting from the seamless integration between
the computer-aided design (CAD) geometry and the analysis model by employing
non-uniform rational B-splines (NURBS) as basis functions. However, structural
optimization for real-world CAD geometries consisting of multiple non-matching
NURBS patches remains a challenging task. In this work, we propose a unified
formulation for shape and thickness optimization of separately-parametrized
shell structures by adopting the free-form deformation (FFD) technique, so that
continuity with respect to design variables is preserved at patch intersections
during optimization. Shell patches are modeled with isogeometric
Kirchhoff--Love theory and coupled using a penalty-based method in the
analysis. We use Lagrange extraction to link the control points associated with
the B-spline FFD block and shell patches, and we perform IGA using the same
extraction matrices by taking advantage of existing finite element assembly
procedures in the FEniCS partial differential equation (PDE) solution library.
Moreover, we enable automated analytical derivative computation by leveraging
advanced code generation in FEniCS, thereby facilitating efficient
gradient-based optimization algorithms. The framework is validated using a
collection of benchmark problems, demonstrating its applications to shape and
thickness optimization of aircraft wings with complex shell layouts
Accelerating model evaluations in uncertainty propagation on tensor grids using computational graph transformations
Methods such as non-intrusive polynomial chaos (NIPC), and stochastic
collocation are frequently used for uncertainty propagation problems.
Particularly for low-dimensional problems, these methods often use a
tensor-product grid for sampling the space of uncertain inputs. A limitation of
this approach is that it encounters a significant challenge: the number of
sample points grows exponentially with the increase of uncertain inputs.
Current strategies to mitigate computational costs abandon the tensor structure
of sampling points, with the aim of reducing their overall count.
Contrastingly, our investigation reveals that preserving the tensor structure
of sample points can offer distinct advantages in specific scenarios. Notably,
by manipulating the computational graph of the targeted model, it is feasible
to avoid redundant evaluations at the operation level to significantly reduce
the model evaluation cost on tensor-grid inputs. This paper presents a
pioneering method: Accelerated Model Evaluations on Tensor grids using
Computational graph transformations (AMTC). The core premise of AMTC lies in
the strategic modification of the computational graph of the target model to
algorithmically remove the repeated evaluations on the operation level. We
implemented the AMTC method within the compiler of a new modeling language
called the Computational System Design Language (CSDL). We demonstrate the
effectiveness of AMTC by using it with the full-grid NIPC method to solve three
low-dimensional UQ problems involving an analytical piston model, a
multidisciplinary unmanned aerial vehicle design model, and a multi-point air
taxi mission analysis model, respectively. For all of the test problems, AMTC
reduces the model evaluation cost by between 50% and 90%, making the full-grid
NIPC the most efficacious method to use among the UQ methods implemented
Large-Scale Multidisciplinary Optimization of a Small Satellite’s Design and Operation
Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/140669/1/1.a32751.pd
A Mixed Integer Efficient Global Optimization Algorithm for the Simultaneous Aircraft Allocation-Mission-Design Problem
Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143026/1/6.2017-1305.pd
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