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CSPlib is an open source software library for analyzing general ordinary differential equation (ODE) systems and detailed chemical kinetic ODE/DAE systems. It relies on the computational singular perturbation (CSP) method for the analysis of these systems.
PARET was originally created in 1969 at what is now Idaho National Laboratory (INL), to analyze reactivity insertion events in research and test reactor cores cooled by light or heavy water, with fuel composed of either plates or pins. The use of PARET is also appropriate for fuel assemblies with curved fuel plates when their radii of curvatures are large with respect to the fuel plate thickness. The PARET/ANL version of the code has been developed at Argonne National Laboratory (ANL) under the sponsorship of the U.S. Department of Energy/NNSA since the inception of the Reactor Conversion Program. Since Reduced Enrichment for Research and Test Reactors (RERTR) began in 1978, PARET/ANL has been benchmarked to experimental data including SPERT testing, and used to determine the expected transient behavior of a large number of reactors regardless of enrichment both inside and outside the Reactor Conversion Program. This document provides the pertinent information for the use of PARET/ANL Version 7.6. PARET/ANL models the various fueled regions of a reactor core as channels. Each of these channels consists of a single flat fuel plate/pin (including cladding and, optionally, a gap) with water coolant on each side. In slab geometry the coolant channels for a given fuel plate are of identical dimensions (mirror symmetry), but they can be of different thickness in each channel. There can be many channels, but each channel is independent and coupled only through reactivity feedback effects to the whole core. The time-dependent differential equations that represent the system are replaced by an equivalent set of finite-difference equations in space and time, which are integrated numerically. PARET/ANL uses fundamentally the same numerical scheme as RELAP5 for the time-integration of the point-kinetics equations. The one-dimensional thermal-hydraulic model includes temperature-dependent thermal properties of the solid materials, such as heat capacity and thermal conductivity, as well as the transient heat production and heat transfer from the fuel meat to the coolant. Temperature- and pressure-dependent thermal properties of the coolant such as enthalpy, density, thermal conductivity, and viscosity are also used in determining parameters such as friction factors and heat transfer coefficients. The code first determines the steady-state solution for the initial state. Then the solution of the transient is obtained by integration in time and space. Multiple heat transfer, DNB and flow instability correlations are available. The code was originally developed to model reactors cooled by an open loop, which was adequate for rapid transients in pool-type cores. An external loop model appropriate for Miniature Neutron Source Reactors (MNSR’s) was also added to PARET/ANL to model natural circulation within the vessel, heat transfer from the vessel to pool and heat loss by evaporation from the pool. PARET/ANL also contains models for decay heat after shutdown, control rod reactivity versus time or position, time-dependent pump flow, and loss-of-flow event with flow reversal as well as logic for trips on period, power, and flow. Feedback reactivity effects from coolant density changes and temperature changes are represented by tables. Feedback reactivity from fuel heat-up (Doppler Effect) is represented by a four-term polynomial in powers of fuel temperature. Photo-neutrons produced in beryllium or in heavy water may be included in the point-kinetics equations by using additional delayed neutron groups.
Stochastic modelling approaches are presented to capture random effects at multiple time and length scales. Random processes that occur at the microscale produce nondeterministic effects at the macroscale. Here we present three stochastic modeling approaches that describe random processes at microscopic length scales and map these processes to the macroscopic length scale. The first stochastic modeling approach is based upon a particle based numerical technique to solve a Stochastic Differential Equation (SDE) using an arbitrary diffusion process to capture random processes at the microstructural level. The second approach prescribes a Probability Density Function (PDF) for the drift and diffusion of the random variable derived using the forward and backward Kolmogorov equations. This method requires mean and drift evolution PDF transport equations. The third approach is the coupling of multiple random variables which are dependent on each other. The relationship of the PDFs and a coupling function, known as a copula, produces a Joint Probability Density Function (JPDF). These stochastic modeling approaches are implemented into a Multiple Component (MC) shock physics computational code and used to model statistical fracture and reactive flow applications.
The coming massive parallelism of exascale computing presents a pressing challenge for the many DOE simulations of time-dependent partial differential equations (PDEs), which typically use traditional sequential time stepping methods. Since this traditional approach is inherently serial, it presents a sequential bottleneck when moving to exascale computing, because future performance gains will come through greater concurrency, not faster clock speeds. Thus, the goal of this work is to research parallelism in time, i.e., methods that compute multiple time values simultaneously, not sequentially. The focus will be on hyperbolic and chaotic problems of interest to DOE, with the goal of enabling scalable simulations of time-dependent hyperbolic and chaotic problems on future architectures. The chosen methodology for solving these problems parallel-in-time is multigrid, because multigrid (when it works) is a powerful, optimal, and scalable solver for discretized PDEs. Multigrid is already commonly used in many DOE simulations for scalably and optimally solving space-only PDE problems. The areas of hyperbolic and chaotic problems are chosen because of their relevance to problems of programmatic interest to DOE. However, these problems are also well-known to be difficult for parallelin-time methods, with the most common method, parareal, diverging in many cases. The current stateof-the-art for parallel-in-time at LLNL is the multigrid reduction in time (MGRIT) XBraid package, which also struggles for such problems, while still showing some improvement over parareal. In summary, new methods are needed for an efficient parallel-in-time scheme for hyperbolic and chaotic problems, and this work shall research promising new multigrid methods in this area. In particular, we take inspiration from the Least Squares Shadowing (LSS by Wang) approach for solving chaotic problems. Here, an optimization approach is able to find “well-conditioned” shadow trajectories/solutions to the original “ill-conditioned” chaotic problem. Thus, the new multigrid methods researched here also arise in an optimization context.
Nonlocal models provide a much-needed predictive capability for important Sandia mission applications, ranging from fracture mechanics for nuclear components to subsurface flow for nuclear waste disposal, where traditional partial differential equations (PDEs) models fail to capture effects due to long-range forces at the microscale and mesoscale. However, utilization of this capability is seriously compromised by the lack of a rigorous nonlocal interface theory, required for both application and efficient solution of nonlocal models. To unlock the full potential of nonlocal modeling we developed a mathematically rigorous and physically consistent interface theory and demonstrate its scope in mission-relevant exemplar problems.
Projection-based model order reduction allows for the parsimonious representation of full order models (FOMs), typically obtained through the discretization of a set of partial differential equations (PDEs) using conventional techniques (e.g., finite element, finite volume, finite difference methods) where the discretization may contain a very large number of degrees of freedom. As a result of this more compact representation, the resulting projection-based reduced order models (ROMs) can achieve considerable computational speedups, which are especially useful in real-time or multi-query analyses. One known deficiency of projection-based ROMs is that they can suffer from a lack of robustness, stability and accuracy, especially in the predictive regime, which ultimately limits their useful application. Another research gap that has prevented the widespread adoption of ROMs within the modeling and simulation community is the lack of theoretical and algorithmic foundations necessary for the “plug-and-play” integration of these models into existing multi-scale and multi-physics frameworks. This paper describes a new methodology that has the potential to address both of the aforementioned deficiencies by coupling projection-based ROMs with each other as well as with conventional FOMs by means of the Schwarz alternating method [41]. Leveraging recent work that adapted the Schwarz alternating method to enable consistent and concurrent multiscale coupling of finite element FOMs in solid mechanics [35, 36], we present a new extension of the Schwarz framework that enables FOM-ROM and ROM-ROM coupling, following a domain decomposition of the physical geometry on which a PDE is posed. In order to maintain efficiency and achieve computation speed-ups, we employ hyper-reduction via the Energy-Conserving Sampling and Weighting (ECSW) approach [13]. We evaluate the proposed coupling approach in the reproductive as well as in the predictive regime on a canonical test case that involves the dynamic propagation of a traveling wave in a nonlinear hyper-elastic material.
This manual describes the use of the Portable, Extensible Toolkit for Scientific Computation (PETSc) and the Toolkit for Advanced Optimization (TAO) for the numerical solution of partial differential equations and related problems on high-performance computers. PETSc/TAO is a suite of data structures and routines that provide the building blocks for the implementation of large-scale application codes on parallel (and serial) computers. PETSc uses the MPI standard for all distributed memory communication.
Multiphysics and analytical calculations were conducted for a heat exchanger with passive, natural circulation flow. A glycol/water working fluid convects the heat to a dimpled heat exchanger shell, which subsequently transfers the heat to the soil, which acts as the ultimate heat sink. Because the system is fully-passive, it is not subject to the expenses, maintenance, and mechanical breakdowns associated with moving parts. Density, heat capacity, and thermal conductivity material properties were measured for various soil samples, and subsequently included as input for the soil heat conduction model. The soil model was coupled to a computational fluid dynamics (CFD) heat exchanger model that included the dynamic Smagorinsky large eddy simulation and k- omega turbulence models. The analysis showed that the fluid dynamics and heat transfer models worked properly, albeit at a slow pace. Nevertheless, the coupled CFD/heat conduction simulation ran long enough to determine a key parameter—the amount of heat conducted from the heat exchanger to the ground. This unique performance value, along with experimental data, was used as input for stand-alone, fast-running CFD models, as well as boundaries to obtain solutions to partial differential equations for soil heat conduction.
Optimal mitigation planning for highly disruptive contingencies to a transmission-level power system requires optimization with dynamic power system constraints, due to the key role of dynamics in system stability to major perturbations. We formulate a generalized disjunctive program to determine optimal grid component hardening choices for protecting against major failures, with differential algebraic constraints representing system dynamics (specifically, differential equations representing generator and load behavior and algebraic equations representing instantaneous power balance over the transmission system). We optionally allow stochastic optimal pre-positioning across all considered failure scenarios, and optimal emergency control within each scenario. This novel formulation allows, for the first time, analyzing the resilience interdependencies of mitigation planning, preventive control, and emergency control. Using all three strategies in concert is particularly effective at maintaining robust power system operation under severe contingencies, as we demonstrate on the Western System Coordinating Council (WSCC) 9-bus test system using synthetic multi-device outage scenarios. Towards integrating our modeling framework with real threats and more realistic power systems, we explore applying hybrid dynamics to power systems. Our work is applied to basic RL circuits with the ultimate goal of using the methodology to model protective tripping schemes in the grid. Finally, we survey mitigation techniques for HEMP threats and describe a GIS application developed to create threat scenarios in a grid with geographic detail.
Cryocooler conduction cooled devices can experience significant cooldown time due to lower available cooling capacity compares to convection cooled devices. Therefore, the cooldown time is an important design parameter for conduction cooled devices. This article introduces a framework developed in Python for calculating the cooldown profiles and cooldown time of cryocooler conduction-cooled devices such as superconducting magnets and accelerator cavities. The cooldown time estimation problem is essentially a system of ordinary first-order differential equations comprising the material properties (temperature dependent thermal conductivity and specific heat capacity) of the components intertwined with the prevailing heat transfer channels (conduction, radiation, and heat flow across pressed contacts) and the cryocooler capacity. The formulation of this ODE system is first presented. This ODE system is then solved using the in-built Python library odeint. A case study is presented comprising a small cryocooler conduction-cooled copper stabilized niobium-titanium magnet. The case study is supplemented with the Python script enabling the reader to simply tweak the device design parameters and optimize the design from the point of view of slow/fast cooldown.
Constrained optimization in the context of time dependent, partial differential equations (PDE) leads to a symmetric, block-tridiagonal system of nonlinear equations that must be solved repeatedly in an iterative solution strategy. The blocks represent spatial discretization, while the connection between the blocks represents a forward and backward integration in time. The focus of this project is to apply a parallel-in-time (PiT) solution technique to the large block-triangular system.
Self-powering sensors and networks are a reality. The ability to extract ambient energy from the surroundings to power electronic devices has a profound impact on the realization of smart adaptable sensor networks. In this study, a magnetically coupled dual spring and magnet design has been investigated to improve the efficiency and performance bandwidth of vibration energy harvesting (VEH) sensors. Using numerical models based on traditional systems of coupled ordinary differential equations (ODE), an optimized design was developed and compared to experimental measurements. Numerical and empirical results show good agreement. Results show improvement in the bandwidth over an equivalent linear system and corresponding improvement in output power conversion efficiency. The increased bandwidth allows improved conversion sensitivity and enhanced power harvesting capabilities. This operational bandwidth coincides with the expected input spectrum for in situ applications.
This manual describes the use of the Portable, Extensible Toolkit for Scientific Computation (PETSc) and the Toolkit for Advanced Optimization (TAO) for the numerical solution of partial differential equations and related problems on high-performance computers. PETSc/TAO is a suite of data structures and routines that provide the building blocks for the implementation of large-scale application codes on parallel (and serial) computers. PETSc uses the MPI standard for all distributed memory communication. PETSc/TAO includes a large suite of parallel linear solvers, nonlinear solvers, time integrators, and opti mization that may be used in application codes written in Fortran, C, C++, and Python (via petsc4py; see Getting Started). PETSc provides many of the mechanisms needed within parallel application codes, such as parallel matrix and vector assembly routines. The library is organized hierarchically, enabling users to employ the level of abstraction that is most appropriate for a particular problem. By using techniques of object-oriented programming, PETSc provides enormous flexibility for users. PETSc is a sophisticated set of software tools; as such, for some users it initially has a much steeper learning curve than packages such as MATLAB or a simple subroutine library. In particular, for individuals without some computer science background, experience programming in C, C++, python, or Fortran and experience using a debugger such as gdb or lldb, it may require a significant amount of time to take full advantage of the features that enable efficient software use. However, the power of the PETSc design and the algorithms it incorporates may make the efficient implementation of many application codes simpler than “rolling them” yourself. For many tasks a package such as MATLAB is often the best tool; PETSc is not intended for the classes of problems for which effective MATLAB code can be written. There are several packages, built on PETSc, that may satisfy your needs without requiring directly using PETSc. We recommend reviewing these packages functionality before starting to code directly with PETSc. PETSc can be used to provide a “MPI parallel linear solver” in an otherwise sequential, or OpenMP parallel code. This approach cannot provide extremely large improvements in the application time by utilizing large numbers of MPI processes but can still improve the performance. Certainly all parts of a previously sequential code need not be parallelized but the matrix generation portion must be parallelized to expect true scalability to large numbers of MPI processes. See PCMPI for details on how to utilize the PETSc MPI linear solver server. Since PETSc is under continued development, small changes in usage and calling sequences of routines will occur. PETSc has been supported for twenty-five years; see mailing list information on our website for information on contacting support.
Solar photovoltaic (PV) installations have experienced unprecedented growth in the United States. PV will become not only an energy producer but also a necessary provider of ancillary services at multiple timescales. Conventional methods to simulate power systems operations - such as long-term production simulation (which typically considers schedules from hours to minutes by using an optimization framework) and short-term transient studies (which simulate dynamics from seconds to sub-seconds using state variables and differential equations) - are not sufficient for studying the multiple-timescale variation of solar generation and its impact on system reliability. Long-term system economics and short-term system dynamics are highly coupled, particularly when the penetration level of renewable generation is extremely high, because the uncertainty and variability of solar generation will impact both power system steady-state and dynamic performance. This project helps meet and exceed the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy Solar Energy Technologies Office goal of systems integration by directly addressing this stability and reliability challenge for power grid planning and operation. We have developed a temporally comprehensive, closed-loop simulation model, named Multi-timescale Integrated Dynamics and Scheduling (MIDAS), that seamlessly simulates power system operations from economic scheduling (day-ahead to hours) to dynamic response analysis (seconds to sub-seconds). For schedules with very high levels of inverter-based resources (IBRs), up to and including 100%, the stability of grid controls has been evaluated through electromagnetic transient (EMT) simulations and power-hardware-in-the-loop (PHIL) simulations of key transient events at key schedule points. Specifically, MIDAS provides: 1) a closed-loop simulation framework for simulating timescales from economic scheduling to dynamic stability analysis; 2) machine learning-based stability assessment; 3) EMT modeling and analysis for large-scale power systems; 4) MIDAS PHIL test bed. We worked with Hawaii Electric Companies to apply the MIDAS study framework to a Maui grid study. The entire island's transmission system was modeled in detail - from a yearly scheduling model, to a second-level frequency dynamic model, down to a sub-second-scale EMT model to address critical stability issues. The project demonstrated how MIDAS can help system planners and operators assess system reliability and stability while the power grid is marching toward a high-renewable, high-IBR future. In this Maui grid study, we found that 100% instantaneous IBR operation is achievable in EMT simulation and PHIL testing, and grid planners and operators might need new analysis/simulation tools to assess grid reliability and stability in the scheduling stage. MIDAS will bring Maui and other systems closer to 100% clean and stable energy futures. (In this study, we examined transient stability. Other topics necessary for 100% IBR operation, such as protection and resource adequacy, were not examined.)
This work presents a novel method for generating electricity price scenarios from statistical properties of past electricity prices using a hybrid statistical and reduced-form stochastic model. Previous work in applying stochastic differential equations (SDE) to model electricity prices has focused on daily average prices. To extend stochastic price generation methods to hourly or sub-hourly pricing, we address several weaknesses in the state-of-the-art: (1) we replace the mean-reversion component of the SDE with an ARIMA process that is better able to characterize the daily and weekly trends; (2) we extend the price-spike, or jump process to account for conditional probabilities of price spikes occurring in consecutive time steps by replacing the traditional Poisson process for modeling jumps with a generalized point process model inspired by brain neuron models; and (3) we replace the traditional method of estimating spike intensity with empirical variance with a Markov process based on observed price spike intensity transitions. The method is demonstrated with electricity prices from the US ERCOT market and a use-case example is provided for bidding an energy storage unit into the day-ahead and real-time energy markets of ERCOT using stochastic optimization methods. Results show that the the synthetic price model out performs a (naive) persistence forecast model by resulting in 24% to 47% more in profits over 168 simulated days.
Finite element methods approximate the solution to a partial differential equation (PDE) on a mesh consisting of many elements. In general, using more, smaller elements results in a lower error in the approximation. However, it is often possible to lower the error substantially by only refining, or decreasing the size of, the few elements in the mesh that have the highest error. Adaptive mesh refinement (AMR) is a process that selectively refines regions of a mesh with high error to achieve a desired accuracy in as few degrees of freedom (DOFs) as possible. AMR is favorable compared to uniform refinement, which refines all elements of the mesh equally, because it can often achieve the same accuracy without wasting extra computation time on refinement of elements that already have low error. However, it is difficult to know which elements to refine. In this report we explore ways to choose which elements to refine such that we minimize both the resulting error and the cumulative DOFs used in computation. In particular, we introduce a Pareto-front learning algorithm that trains a policy to give the optimal refinement actions to minimize the cumulative DOFs used to achieve a given target error. Such a policy is useful because it can be deployed on many different problem types where different accuracy levels are desired. Furthermore, training a single policy for a range of target errors allows us to use transfer learning to reduce the required training time.
The Newton, Gauss–Newton and Levenberg–Marquardt methods all use the first derivative of a vector function (the Jacobian) to minimise its sum of squares. When the Jacobian matrix is ill-conditioned, the function varies much faster in some directions than others and the space of possible improvement in sum of squares becomes a long narrow ellipsoid in the linear model. This means that even a small amount of nonlinearity in the problem parameters can cause a proposed point far down the long axis of the ellipsoid to fall outside of the actual curved valley of improved values, even though it is quite nearby. This paper presents a differential equation that ‘follows’ these valleys, based on the technique of geodesic acceleration, which itself provides a 2 nd order improvement to the Levenberg–Marquardt iteration step. Higher derivatives of this equation are computed that allow n th order improvements to the optimisation methods to be derived. These higher-order accelerated methods up to 4 th order are tested numerically and shown to provide substantial reduction of both number of steps and computation time.
The work for the subcontract was situated in the area of parallel-in-time integration for hyperbolic partial differential equations (PDEs). Parallel-in-time integration is an active area of research due to its ability to enable faster numerical simulations for applications throughout many areas of science. Over the past two decades, much progress has been made in this area; however, this progress has largely been limited to diffusion-dominated PDEs, with some recent success in scalar linear hyperbolic PDEs. Given the ubiquity of numerical simulations of hyperbolic PDEs throughout the sciences, in particular, nonlinear hyperbolic systems, there is a strong need to develop efficient parallel-in-time techniques for hyperbolic problems beyond simple scalar and linear cases, which is the main topic of this subcontract. The main focus of the work was to further develop and perfect coarse-grid operators for the Multigrid Reduction-inTime (MGRIT) method applied to hyperbolic PDEs that were recently proposed in PhD thesis, based on a modified semi-Lagrangian approach.