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A method for studying collisionless tearing mode properties in cases where complicated orbit integrals must be evaluated is presented. The method is based on a quadratic form of a self-adjoint integrodifferential operator. The collisionless tearing instability in a non-Maxwellian neutral sheet is used to illustrate the advantages of the present method.
A methodology is developed and implemented to mitigate the lengthy software development cycle typically associated with constructing a discrete adjoint solver for aerodynamic simulations. The approach is based on a complex-variable formulation that enables straightforward differentiation of complicated real-valued functions. An automated scripting process is used to create the complex-variable form of the set of discrete equations. An efficient method for assembling the residual and cost function linearizations is developed. The accuracy of the implementation is verified through comparisons with a discrete direct method as well as a previously developed handcoded discrete adjoint approach. Comparisons are also shown for a large-scale configuration to establish the computational efficiency of the present scheme. To ultimately demonstrate the power of the approach, the implementation is extended to high temperature gas flows in chemical nonequilibrium. Finally, several fruitful research and development avenues enabled by the current work are suggested.
A methodology is developed and implemented to mitigate the lengthy software development cycle typically associated with constructing a discrete adjoint solver for aerodynamic simulations. The approach is based on a complex-variable formulation that enables straightforward differentiation of complicated real-valued functions. An automated scripting process is used to create the complex-variable form of the set of discrete equations. An efficient method for assembling the residual and cost function linearizations is developed. The accuracy of the implementation is verified through comparisons with a discrete direct method as well as a previously developed handcoded discrete adjoint approach. Comparisons are also shown for a large-scale configuration to establish the computational efficiency of the present scheme. To ultimately demonstrate the power of the approach, the implementation is extended to high temperature gas flows in chemical nonequilibrium. Finally, several fruitful research and development avenues enabled by the current work are suggested.
Quantum optimal control plays a crucial role in quantum computing by providing the interface between compiler and hardware. Solving the optimal control problem is particularly challenging for multi-qubit gates, due to the exponential growth in computational complexity with the system's dimensionality and the deterioration of optimization convergence. To ameliorate the computational complexity of time-integration, this paper introduces a multiple-shooting approach in which the time domain is divided into multiple windows and the intermediate states at window boundaries are treated as additional optimization variables. Further, this enables parallel computation of state evolution across time-windows, significantly accelerating objective function and gradient evaluations. Since the initial state matrix in each window is only guaranteed to be unitary upon convergence of the optimization algorithm, the conventional gate trace infidelity is replaced by a generalized infidelity that is convex for non-unitary state matrices. Continuity of the state across window boundaries is enforced by equality constraints. A quadratic penalty optimization method is used to solve the constrained optimal control problem, and an efficient adjoint technique is employed to calculate the gradients in each iteration. We demonstrate the effectiveness of the proposed method through numerical experiments on quantum Fourier transform gates in systems with 2, 3, and 4 qubits, noting a speedup of 80x for evaluating the gradient in the 4-qubit case, highlighting the method's potential for optimizing control pulses in multi-qubit quantum systems.
Approximate method of stability analysis for equilibrium in elastic continuums subjected to follower type surface traction
These are slides from a seminar given to the University of Mexico Nuclear Engineering Department. Whisper is a statistical analysis package developed to support nuclear criticality safety validation. It uses the sensitivity profile data for an application as computed by MCNP6 along with covariance files for the nuclear data to determine a baseline upper-subcritical-limit for the application. Whisper and its associated benchmark files are developed and maintained as part of MCNP6, and will be distributed with all future releases of MCNP6. Although sensitivity-uncertainty methods for NCS validation have been under development for 20 years, continuous-energy Monte Carlo codes such as MCNP could not determine the required adjoint-weighted tallies for sensitivity profiles. The recent introduction of the iterated fission probability method into MCNP led to the rapid development of sensitivity analysis capabilities for MCNP6 and the development of Whisper. Sensitivity-uncertainty based methods represent the future for NCS validation – making full use of today’s computer power to codify past approaches based largely on expert judgment. Validation results are defensible, auditable, and repeatable as needed with different assumptions and process models. The new methods can supplement, support, and extend traditional validation approaches.
Here, the concept of transport mediated through the dynamics of 'jumping' particles is used to develop an iterative method for obtaining steady-state solutions to the nonlocal transport equation in two dimensions (2D). The technique is self-adjoint and capable of correctly treating spatially non-uniform, asymmetric systems. An appropriate reduced version of the iteration method is used to compare with results obtained with a self-adjoint 1D transport matrix approach. The transport 'jump' probability distribution functions (pdfs) are based on Lévy α-stable distributions. The technique can handle the entire Lévy α-parameter range from one (Lorentz distributions) to two (Gaussian distributions). Cases with α = 2 (standard diffusion) are used to establish the validity of the iterative method. The capabilities of the iterative method are demonstrated by presenting examples from systems with various source configurations, boundary shapes, boundary conditions and spatial variations in parameters.
This paper presents a study that applied the Monte Carlo code SHIFT to calculate the radiological environment within a detailed Molten Salt Breeder Reactor (MSBR) model. It represents one of the first applications of the SHIFT code in stand-alone simulations for non-light water reactors to help demonstrate the code's potential uses in the design, licensing, and operation of advanced reactors. The radiological conditions of the MSBR were modeled when the reactor is at two different operation modes: normal full power and a drained state. The Forward Weighted-Consistent Adjoint Driven Importance Sampling (FW-CADIS) hybrid method in SHIFT was applied successfully to calculate the ex-core neutron and gamma dose rates for the MSBR at full power. Neutron and gamma dose rates within the drain cell were also calculated for the MSBR at the drained state by integrating the source terms obtained from an ORIGEN-S depletion calculation into the SHIFT simulation. The results indicate that in the drained state, the delayed gammas from the depleted fuel salt are the main contributors to the dose rates. (authors)
The spline scheme of Banks and Kappel (1979) is used as an aproximation method for calculating optimal feedback gains for control systems governed by functional differential equations. It is shown that the adjoints of the spline-based approximation scheme for delay equations do not converge strongly.
To solve optimization problems with parabolic PDE constraints, often methods working on the reduced objective functional are used. They are computationally expensive due to the necessity of solving both the state equation and a backward-in-time adjoint equation to evaluate the reduced gradient in each iteration of the optimization method. In this study, we investigate the use of the parallel-in-time method PFASST in the setting of PDE-constrained optimization. In order to develop an efficient fully time-parallel algorithm, we discuss different options for applying PFASST to adjoint gradient computation, including the possibility of doing PFASST iterations on both the state and the adjoint equations simultaneously. Here, we also explore the additional gains in efficiency from reusing information from previous optimization iterations when solving each equation. Numerical results for both a linear and a nonlinear reaction-diffusion optimal control problem demonstrate the parallel speedup and efficiency of different approaches.
Variance reduction is a key ingredient for solving radiation-protection problems with Monte-Carlo particle-transport codes. Many variance-reduction methods require the definition of an importance map and exhibit optimal performance if the importance map is given by the adjoint flux, the solution of the adjoint Boltzmann equation. This paper presents the implementation of the Consistent Adjoint-Driven Importance Sampling (CADIS) methodology via a coupling between the TRIPOLI-4 R Monte-Carlo particle-transport code and the IDT deterministic solver, Additionally, we describe the implementation of a new TRIPOLI-4 R score that makes it possible to estimate the adjoint flux during a direct Monte-Carlo calculation. These new features are expected to simplify the solution of difficult shielding problems. (authors)
The existence of a potential energy functional in the zero-Larmor-radius collisionless plasma theory of Kruskal & Oberman (Phys. Fluids, vol. 1, 1958 p. 275), Rosenbluth & Rostoker (Phys. Fluids, vol. 2, 1959, p. 23) allows us to derive easily sufficient conditions for linear stability. However, this kinetic magnetohydrodynamics (KMHD) theory does not have a self-adjointness property, making it difficult to derive necessary conditions. In particular, the standard methods to prove that an instability follows if some trial perturbation makes the incremental potential energy negative, which rely on the self-adjointness of the force operator or on the existence of a complete basis of normal modes, are not applicable to KMHD. This paper investigates KMHD linear stability criteria based on the time evolution of initial-value solutions, without recourse to the classic bounds or comparison theorems of Kruskal–Oberman and Rosenbluth–Rostoker for the KMHD potential energy. The adopted approach does not solve the kinetic equations by integration along characteristics and does not require that the particle orbits be periodic or nearly periodic. Most importantly, the investigation of a necessary condition for stability does not require the self-adjointness of the force operator or the existence of a complete basis of normal modes. Thus, it is thereby shown that stability in isothermal ideal-MHD is a sufficient condition for stability in KMHD and that, with a proviso on the long-time behaviour of oscillations about stable equilibria, stability in the double-adiabatic fluid theory, including the variation of the parallel fluid displacement, would be a necessary condition for stability in KMHD.
Flutter is a critical factor in designing and certifying aircraft. The linearized frequency-domain method offers a lower cost alternative to time-marching computational fluid dynamics for high-fidelity flutter analysis. In this work, adjoint-based sensitivities are added to a flutter analysis based on the linearized frequency-domain method to efficiently compute derivatives of flutter cost functions with respect to design variables or uncertain parameters. The derivation of the adjoint equations, which involve complications such as derivatives of a nonlinear generalized eigenvalue problem with complex-valued inputs and derivatives of the linearized Navier-Stokes equations, is provided. The implemented adjoint terms and derivatives are verified before demonstrating the approach for derivatives of flutter dynamic pressure with respect to Mach number for the AGARD 445.6 wing.
A new equilibrium thermodynamics analysis tool was built based on the CEA method using the OpenMDAO framework. The new tool provides forward and adjoint analytic derivatives for use with gradient based optimization algorithms. The new tool was validated against the original CEA code to ensure an accurate analysis and the analytic derivatives were validated against finite-difference approximations. Performance comparisons between analytic and finite difference methods showed a significant speed advantage for the analytic methods. To further test the new analysis tool, a sample optimization was performed to find the optimal air-fuel equivalence ratio, , maximizing combustion temperature for a range of different pressures. Collectively, the results demonstrate the viability of the new tool to serve as the thermodynamic backbone for future work on a full propulsion modeling tool.
This work describes the application of a control theory-based aerodynamic shape optimization method to the problem of supersonic aircraft design. The design process is greatly accelerated through the use of both control theory and a parallel implementation on distributed memory computers. Control theory is employed to derive the adjoint differential equations whose solution allows for the evaluation of design gradient information at a fraction of the computational cost required by previous design methods. The resulting problem is then implemented on parallel distributed memory architectures using a domain decomposition approach, an optimized communication schedule, and the MPI (Message Passing Interface) Standard for portability and efficiency. The final result achieves very rapid aerodynamic design based on higher order computational fluid dynamics methods (CFD). In our earlier studies, the serial implementation of this design method was shown to be effective for the optimization of airfoils, wings, wing-bodies, and complex aircraft configurations using both the potential equation and the Euler equations. In our most recent paper, the Euler method was extended to treat complete aircraft configurations via a new multiblock implementation. Furthermore, during the same conference, we also presented preliminary results demonstrating that this basic methodology could be ported to distributed memory parallel computing architectures. In this paper, our concern will be to demonstrate that the combined power of these new technologies can be used routinely in an industrial design environment by applying it to the case study of the design of typical supersonic transport configurations. A particular difficulty of this test case is posed by the propulsion/airframe integration.
This work describes the application of a control theory-based aerodynamic shape optimization method to the problem of supersonic aircraft design. The design process is greatly accelerated through the use of both control theory and a parallel implementation on distributed memory computers. Control theory is employed to derive the adjoint differential equations whose solution allows for the evaluation of design gradient information at a fraction of the computational cost required by previous design methods (13, 12, 44, 38). The resulting problem is then implemented on parallel distributed memory architectures using a domain decomposition approach, an optimized communication schedule, and the MPI (Message Passing Interface) Standard for portability and efficiency. The final result achieves very rapid aerodynamic design based on higher order computational fluid dynamics methods (CFD). In our earlier studies, the serial implementation of this design method (19, 20, 21, 23, 39, 25, 40, 41, 42, 43, 9) was shown to be effective for the optimization of airfoils, wings, wing-bodies, and complex aircraft configurations using both the potential equation and the Euler equations (39, 25). In our most recent paper, the Euler method was extended to treat complete aircraft configurations via a new multiblock implementation. Furthermore, during the same conference, we also presented preliminary results demonstrating that the basic methodology could be ported to distributed memory parallel computing architectures [241. In this paper, our concem will be to demonstrate that the combined power of these new technologies can be used routinely in an industrial design environment by applying it to the case study of the design of typical supersonic transport configurations. A particular difficulty of this test case is posed by the propulsion/airframe integration.
Uncertainty quantification and sensitivity analyses are a vital component for predictive modeling in the sciences and engineering. The adjoint approach to sensitivity analysis requires solving a primary system of equations and a mathematically related set of adjoint equations. The information contained in the equations can be combined to produce sensitivity information in a computationally efficient manner. Here in this work, sensitivity analyses are performed on systems described by flux-limited radiative diffusion using the adjoint approach. The sensitivities computed are shown to agree with standard perturbation theory and require significantly less computational time. The adjoint approach saves the computational cost of one forward solve per sensitivity, making the method attractive when multiple sensitivities are of interest.
A library is introduced that enables users to conduct Monte Carlo tallies on heterogeneous computing systems. The library can be built with or linked to a Monte Carlo transport code to provide tally capabilities for the project. The library provides a set of tools and an interface for constructing tallies that can be computed on CPUs or GPUs regardless of where the sample data was generated. The library can be incorporated into existing Monte Carlo transport codes at both the research and production level with minimal intrusion into the transport code. The tally library is capable of computing a variety of tallies including track-length histogram (mesh) tallies, collisionbased kernel density estimators, functional expansion tallies, eigenvalue sensitivity coefficients via iterated fission probability, and sensitivities of reaction rates or adjoint-weighted quantities via differential operator sampling. The parallelization methods are discussed and speedups are shown for 2-D and 3-D problems with comparisons to both the tally library CPU implementations as well as the tally capabilities in the host transport code. Performance results and examples are shown for MCNP, MCATK, and OpenMC.