Investigation of dynamic behavior of thin spherical shells Semiannual technical progress report, 1 Aug. 1966 - 31 Jan. 1967
Green function approach to vibration problem of thin spherical shells
THIN WALLED SHELL↗
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Results for “boundary value problems”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Estimates of the eigenvalues of operator arising in swelling pressure model
Swelling pressures from materials confined by structures can cause structural deformations and instability. Due to the complexity of interactions between expansive solid and solid-liquid equilibrium, the forces exerting on retaining structures from swelling are highly nonlinear. This work is our initial attempt to study a simplistic spectral problem based on the Euler-elastic beam theory and some simplistic swelling pressure model. In this work estimates of the eigenvalues of some initial/boundary value problem for nonlinear Euler-elastic beam equation are obtained.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Linearization of a Free Boundary Problem of Magnetohydrodynamics
We consider the linear conjugation problem for a magnetic field. Such a problem arises in the linearization of a free boundary problem with two fluids. We prove the unique solvability in the Sobolev–Slobodetskii spaces.
97 MATHEMATICS AND COMPUTING↗
Classical Solvability of the Radial Viscous Fingering Problem in a Hele–Shaw Cell with Surface Tension
We consider the one-phase problem on radial viscous fingering structures in a Hele–Shaw cell with surface tension. This problem is a nonlinear free-boundary problem for elliptic equations. Unlike the Stefan problem for the heat equation, we deal with a problem of hydrodynamic type. We establish the classical solvability of the one-phase Hele–Shaw problem with radial geometry by using the same method as that used for the Stefan problem and justifying the vanishing coefficient of the time-derivative in the parabolic equation.
97 MATHEMATICS AND COMPUTING↗
Analytical Solution for Three-Dimensional, Unsteady Heat Conduction in a Multilayer Sphere
An analytical solution has been obtained for the transient problem of three-dimensional multilayer heat conduction in a sphere with layers in the radial direction. The solution procedure can be applied to a hollow sphere or a solid sphere composed of several layers of various materials. In general, the separation of variables applied to 3D spherical coordinates has unique characteristics due to the presence of associated Legendre functions as the eigenfunctions. Moreover, an eigenvalue problem in the azimuthal direction also requires solution; again, its properties are unique owing to periodicity in the azimuthal direction. Therefore, extending existing solutions in 2D spherical coordinates to 3D spherical coordinates is not straightforward. In a spherical coordinate system, one can solve a 3D transient multilayer heat conduction problem without the presence of imaginary eigenvalues. A 2D cylindrical polar coordinate system is the only other case in which such multidimensional problems can be solved without the use of imaginary eigenvalues. The absence of imaginary eigenvalues renders the solution methodology significantly more useful for practical applications. The methodology described can be used for all three types of boundary conditions in the outer and inner surface of the sphere. Lastly, the solution procedure is demonstrated on an illustrative problem for which results are obtained.
36 MATERIALS SCIENCE↗
On Solutions to the Schwarz Problem in a Disk in the Three-Dimensional Case
We consider the Schwarz problem for first order elliptic systems in the plane. We obtain the conditions on three– and two-dimensional matrices J admitting nondiagonal Jordan form under which a solution to the Schwarz problem in an arbitrary disk exists and is unique in Hölder classes. We describe an algorithm for constructing the solution. Bibliography: 5 titles.
97 MATHEMATICS AND COMPUTING↗
Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability
In the hyperbolic research community, there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability. In the first part of the series, the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly. By applying this technique, the authors demonstrate that a pure continuous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way. In this work, we extend this investigation to the nonlinear case and focus on entropy conservation. Here, by switching to entropy variables, we provide an estimation of the boundary operators also for nonlinear problems, that guarantee conservation. In numerical simulations, we verify our theoretical analysis.
97 MATHEMATICS AND COMPUTING↗
Network Uncertainty Quantification for Analysis of Multi-Component Systems
To impact physical mechanical system design decisions and realize the full promise of high-fidelity computational tools, simulation results must be integrated at the earliest stages of the design process. This is particularly challenging when dealing with uncertainty and optimizing for system-level performance metrics, as full-system models (often notoriously expensive and time-consuming to develop) are generally required to propagate uncertainties to system-level quantities of interest. Methods for propagating parameter and boundary condition uncertainty in networks of interconnected components hold promise for enabling design under uncertainty in real-world applications. These methods avoid the need for time consuming mesh generation of full-system geometries when changes are made to components or subassemblies. Additionally, they explicitly tie full-system model predictions to component/subassembly validation data which is valuable for qualification. These methods work by leveraging the fact that many engineered systems are inherently modular, being comprised of a hierarchy of components and subassemblies that are individually modified or replaced to define new system designs. By doing so, these methods enable rapid model development and the incorporation of uncertainty quantification earlier in the design process. The resulting formulation of the uncertainty propagation problem is iterative. We express the system model as a network of interconnected component models, which exchange solution information at component boundaries. We present a pair of approaches for propagating uncertainty in this type of decomposed system and provide implementations in the form of an open-source software library. We demonstrate these tools on a variety of applications and demonstrate the impact of problem-specific details on the performance and accuracy of the resulting UQ analysis. This work represents the most comprehensive investigation of these network uncertainty propagation methods to date.
42 ENGINEERING↗
Construction of a Code Verification Matrix for Heat Conduction With Finite Element Code Applications
When establishing the pedigree of a simulation tool, code verification is used to ensure that the implemented numerical algorithm is a faithful representation of its underlying mathematical model. During this process, numerical results on various meshes are systematically compared to a reference analytic solution. The selection of analytic solutions can be a laborious process, as it is difficult to establish adequate code confidence without performing redundant work. In this report we address this issue by applying a physics-based process that establishes a set of reference problems. In this process, code simulation options are categorized and systematically tested, which ensures that gaps in testing are easily identified and addressed. The resulting problems are primarily intended for code verification analysis but may also be useful for comparison to other simulation codes, troubleshooting activities, or training exercises. The process is used to select fifteen code verification problems relevant for the one-dimensional steady-state heat conduction equation. These problems are applicable to a wide variety of simulation tools, but, in this work, a demonstration is performed using the finite element-based nuclear fuel performance code BISON. Convergence to the analytic solution at the theoretical rate is quantified for a selection of the problems, which establishes a baseline pedigree for the code. Not only can this standard set of conduction solutions be used for verification of other codes, but also the physics-based process for selecting problems can be utilized to quantify and expand testing for any simulation tool.
42 ENGINEERING↗
Multistage Stabilized Continuation for Indirect Optimal Control of Three-Dimensional Hypersonic Trajectories
This work presents the development of a multistaged stabilized continuation for the three-dimensional unpowered hypersonic trajectory planning problem using indirect optimal control methods. The stabilized continuation method is noniterative and guaranteed to terminate within a finite number of floating point operations, thereby making it well suited for onboard autonomous implementations. Here, we present a multistage formulation of the stabilized continuation scheme that involves starting with a “loose” integration tolerance during the first stage and ramping up toward a “strict” integration tolerance through subsequent stages. An important benefit of this approach is that even when the solution to the underlying optimal control problem is numerically unstable, such as with the hypersonic vehicle footprint generation problem, the stabilized continuation algorithm is shown to be successful in finding a solution while providing some additional insights into the underlying cause of the numerical instability.
33 ADVANCED PROPULSION SYSTEMS↗
Fully nonlinear δf gyrokinetics for scrape-off layer parallel transport
Edge plasmas present a few challenges for gyrokinetic simulations that are absent in tokamak cores. Among them are large fluctuation amplitudes and plasma-wall interactions in the open field line region. In this work, the widely used core turbulence code GENE, which employs a δf -splitting technique, is extended to simulate open systems with large electrostatic fluctuations. With inclusion and proper discretization of the parallel nonlinear term, it becomes equivalent to a full- f code and the δf -splitting causes no fundamental difficulty in handling large fluctuations. The loss of particles to the wall is accounted for by using a logical sheath boundary, which is implemented in the context of a finite-volume method. The extended GENE code is benchmarked for the well-established one-dimensional parallel transport problem in the scrape-off layer during edge-localized modes. The parallel heat flux deposited onto the divertor target is compared with previous simulation results and shows good agreement.
43 PARTICLE ACCELERATORS↗
Altitude/path-angle transitions in fuel-optimal problems for transport aircraft
Altitude/path angle transitions to fuel-optimal, energy climb and descent paths are examined for subsonic transport aircraft. These transitions are termed boundary layers in singular perturbation theory, the framework used herein to simplify solutions to the state-Euler system. The energy climb and descent paths provide equilibrium or stationary points for the boundary layer system, and thus for moderate transitions, presumed valid for the class of aircraft considered, the boundary layer system may be linearized. This simplification allows derivation of an explicit solution to the two-point boundary-value, boundary layer problem and, as a consequence, yields a nearly optimal control law, in feedback form, for the transitions. Numerical simulation results using the feedback control law are presented for a Boeing 737 airframe (NASA-ATOPS Vehicle) employing twin JT8D-7-7A engines.
Gracey, C.↗
Verification of a specialized hydrodynamic simulation code for modeling deflagration and detonation of high explosives
A specialized hydrodynamic simulation code has been developed and verified for the simulation of one-dimensional unsteady problems involving the detonation and deflagration of high explosives. To model all the relevant physical processes in these problems, a code is required to simulate compressible hydrodynamics, unsteady thermal conduction, and chemical reactions with complex rate laws. Several verification exercises are presented which test the implementation of these capabilities. The code also requires models for physics processes such as equations of state and conductivity for pure materials and mixtures as well as rate laws for chemical reactions. Additional verification tests are required to ensure that these models are implemented correctly. Though this code is limited in the types of problems it can simulate, its computationally efficient formulation allows it to be used in calibration studies for reactive burn models for high explosives. Furthermore, this study demonstrates how a series of verification tests can be used to ensure that the various physics processes needed to simulate complex phenomenon can be tested to ensure that they are correctly implemented.
97 MATHEMATICS AND COMPUTING↗
High-beta analytic equilibria in circular, elliptical, and D-shaped large aspect ratio axisymmetric configurations with poloidal and toroidal flows
The Grad-Shafranov-Bernoulli system of equations is a single fluid magnetohydrodynamical description of axisymmetric equilibria with mass flows. Using a variational perturbative approach [E. Hameiri, Phys. Plasmas 20, 024504 (2013)], analytic approximations for high-beta equilibria in circular, elliptical, and D-shaped cross sections in the high aspect ratio approximation are found, which include finite toroidal and poloidal flows. Assuming a polynomial dependence of the free functions on the poloidal flux, the equilibrium problem is reduced to an inhomogeneous Helmholtz partial differential equation (PDE) subject to homogeneous Dirichlet conditions. An application of the Green's function method leads to a closed form for the circular solution and to a series solution in terms of Mathieu functions for the elliptical case, which is valid for arbitrary elongations. In order to extend the elliptical solution to a D-shaped domain, a boundary perturbation in terms of the triangularity is used. A comparison with the code FLOW [L. Guazzotto et al., Phys. Plasmas 11(2), 604-614 (2004)] is presented for relevant scenarios.
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems
In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.
97 MATHEMATICS AND COMPUTING↗
Projection-Based Model Reduction for Coupled Conduction—Enclosure Radiation Systems
We report a projection-based reduced order model (pROM) methodology has been developed for transient heat transfer problems involving coupled conduction and enclosure radiation. The approach was demonstrated on two test problems of varying complexity. The reduced order models demonstrated substantial speedups (up to 185×) relative to the full order model with good accuracy (less than 3% L ∞ error). An attractive feature of pROMs is that there is a natural error indicator for the ROM solution: the final residual norm at each time-step of the converged ROM solution. Using example test cases, we discuss how to interpret this error indicator to assess the accuracy of the ROM solution. The approach shows promise for many-query applications, such as uncertainty quantification and optimization. The reduced computational cost of the ROM relative to the full-order model (FOM) can enable the analysis of larger and more complex systems as well as the exploration of larger parameter spaces.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Stability of Solutions of Initial Boundary-Balue Problems of Aerohydroelasticity
In designing structures and devices interacting with the flow of gas or liquid, it is necessary to solve the problems associated with the investigation of the stability required for their functioning and operational reliability. The definition of stability of an elastic body taken in the article corresponds to Lyapunov’s concept of stability of a dynamical system. On the base of the proposed nonlinear mathematical model the dynamic stability of the elastic aileron of the wing taking into account the incident subsonic flow of gas or liquid (in an ideal model of an incompressible environment) is investigated. Also a nonlinear mathematical model of the device relating to the vibration technique, which is intended for intensification of technological processes, for example, the process of mixing, is considered. The action of these devices is based on the oscillations of elastic elements during the gas or liquid flow. The dynamic stability of the elastic element located on one wall of the flow channel with the subsonic flow of gas or liquid (in an ideal model of a compressible environment) is investigated. Both models are described by a coupled nonlinear system of differential equations for the unknown functions — the potential of the gas velocity and deformation of the elastic element. On the basis of the construction of functionals, we obtain the sufficient conditions of the stability, the imposed restrictions on the free-stream velocity of the gas, the flexural stiffness of the elastic element, and other parameters of the mechanical system. The examples of construction of the stability regions for particular parameters of the mechanical system are presented.
42 ENGINEERING↗
Optimal Portfolio Management in a Modified Constant Elasticity of Variance Model
We investigate the optimal management of a portfolio consisting of a risk asset and a risk-free bond. The asset dynamics is specified by the M-CEV model. Assuming that the investor has a power utility function, we derive an explicit analytical portfolio-management formula that contains confluent hypergeometric functions. The asymptotic form of the proposed portfolio management strategy is obtained, as well as approximate formulas consisting only of elementary functions. We additionally present an application of our results in the context of algorithmic statistical-arbitrage strategies.
97 MATHEMATICS AND COMPUTING↗