Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Lagrange multipliers”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Thermodynamically consistent semi-compressible fluids: a variational perspective

This paper presents (Lagrangian) variational formulations for single and multicomponent semi-compressible fluids with both reversible (entropy-conserving) and irreversible (entropy-generating) processes. Semi-compressible fluids are useful in describing low-Mach dynamics, since they are soundproof. These models find wide use in many areas of fluid dynamics, including both geophysical and astrophysical fluid dynamics. Specifically, the Boussinesq, anelastic and pseudoincompressible equations are developed through a unified treatment valid for arbitrary Riemannian manifolds, thermodynamic potentials and geopotentials. By design, these formulations obey the 1st and 2nd laws of thermodynamics, ensuring their thermodynamic consistency. This general approach extends and unifies existing work, and helps clarify the thermodynamics of semi-compressible fluids. To further this goal, evolution equations are presented for a wide range of thermodynamic variables: entropy density s, specific entropy η, buoyancy b, temperature T, potential temperature θ and a generic entropic variable χ; along with a general definition of buoyancy valid for all three semicompressible models and arbitrary geopotentials. Finally, the elliptic equation for the pressure perturbation (the Lagrange multiplier that enforces semi-compressibility) is developed for all three equation sets in the case of reversible dynamics, and for the Boussinesq/anelastic equations in the case of irreversible dynamics; and some discussion is given of the difficulty in formulating it for the pseudoincompressible equations with irreversible dynamics.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Dual-cone variational calculation of the two-electron reduced density matrix

The computation of strongly correlated quantum systems is challenging because of its potentially exponential scaling in the number of electron configurations. Variational calculation of the two-electron reduced density matrix (2-RDM) without the many-electron wave function exploits the pairwise nature of the electronic Coulomb interaction to compute a lower bound on the ground-state energy with polynomial computational scaling. Recently, a dual-cone formulation of the variational 2-RDM calculation was shown to generate the ground-state energy, albeit not the 2-RDM, at a substantially reduced computational cost, especially for higher $\textit{N}$-representability conditions such as the T2 constraint. Here we generalize the dual-cone variational 2-RDM method to compute not only the ground-state energy but also the 2-RDM. The central result is that we can compute the 2-RDM from a generalization of the Hellmann-Feynman theorem. Specifically, we prove that in the Lagrangian formulation of the dual-cone optimization the 2-RDM is the Lagrange multiplier. Further, we apply the method to computing the energies and properties of strongly correlated electrons—including atomic charges, electron densities, dipole moments, and orbital occupations—in an illustrative hydrogen chain and the nitrogen-fixation catalyst FeMoco. The dual variational computation of the 2-RDM with T2 or higher N -representability conditions provides a polynomially scaling approach to strongly correlated molecules and materials with significant applications in atomic and molecular and condensed-matter chemistry and physics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

2020 IEEE PES Innovative Smart Grid Technologies Europe (ISGT-Europe)

Recent proliferation of distributed energy sources in distribution or sub-transmission systems necessitates close monitoring of these three-phase power grids which typically operate under unbalanced loading conditions. Unlike the transmission systems where the network equations are commonly based on the positive sequence component models, a detailed three phase model will have to be used in implementing network applications for these systems. In the specific case of the state estimator, where measurement and parameter errors may bias the solution, bad data and parameter error detection algorithms should also be incorporated. Implementing the state estimator and error detection algorithms for three-phase systems impose additional computational burden and modifications to the state estimation code. This paper proposes a practical solution to avoid these issues by using synchronized phasor measurements and modal decoupling. The previously developed parameter error detection algorithm based on the normalized Lagrange multipliers (NLM) test is applied to the measurements independently in each mode in parallel, not only saving CPU time but also avoiding new code development for a three-phase estimator. Different parameter error scenarios are created and tested to verify the effectiveness of the proposed error detection approach.

Khalili, Ramtin↗
DOE OSTI2025

Optimal Control Strategy With Efficiency and Reliability Improvement for Offshore DC Microgrids

Offshore microgrids, due to their remote location and lack of external energy support, face significant challenges in wide-range load operation and maintenance. Consequently, efficiency and reliability are critical concerns for converters in offshore dc microgrids. This article presents an optimal control strategy aimed at enhancing both efficiency and reliability. A normalized nonlinear relationship between power loss and thermal stress of a paralleled converter is first established. Based on this, a dual-objective optimization function with an active weight function as well as a system overall performance index is established. The active weight function dynamically adjusts the control priority based on converter efficiency and switching device thermal stress. Then, the optimal power-sharing strategy is derived by the Lagrange multiplier method with the proposed optimal function. Additionally, to accommodate a wide load range, an optimal selection strategy for operating converter combinations is proposed, requiring only low-bandwidth communication. Experiment verification is given to validate the effectiveness of the proposed control strategy. The experiment results demonstrate that the proposed control strategy can improve the overall performance of offshore microgrids by optimizing efficiency and reliability.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2025

An Optimization-Based Coupling of Reduced Order Models with an Efficient Reduced Adjoint Basis Generation Approach

Optimization-based coupling (OBC) is an attractive alternative to traditional Lagrange multiplier approaches in multiple modeling and simulation contexts. However, application of OBC to time-dependent problems has been hindered by the computational cost of finding the stationary points of the associated Lagrangian, which requires primal and adjoint solves. This issue can be mitigated by using OBC in conjunction with computationally efficient reduced order models (ROMs). To demonstrate the potential of this combination, in this paper, we develop an optimization-based ROM-ROM coupling for a transient advection-diffusion transmission problem. We pursue the “optimize-then-reduce” path toward solving the minimization problem at each time step and solve reduced space adjoint system of equations, where the main challenge in this formulation is the generation of adjoint snapshots and reduced bases for the adjoint systems required by the optimizer. One of the main contributions of the paper is a new technique for an efficient adjoint snapshot collection for gradient-based optimizers in the context of optimization-based ROM-ROM couplings. In conclusion, we present numerical studies demonstrating the accuracy of the approach along with comparison between various approaches for selecting a reduced order basis for the adjoint systems, including decay of snapshot energy, average iteration counts, and timings.

coupled problems↗
DOE OSTI2025

Analysis of the SiMPL Method for Density-Based Topology Optimization

We present a rigorous convergence analysis of a new method for density-based topology optimization that provides pointwise bound-preserving design updates and faster convergence than other popular first-order topology optimization methods. Due to its strong bound preservation, the method is exceptionally robust, as demonstrated in numerous examples here and in the companion article [D. Kim et al., Struct. Multidiscip. Optim., 68 (2025), 74]. Furthermore, it is easy to implement with clear structure and analytical expressions for the updates. Our analysis covers two versions of the method, characterized by the employed line search strategies. We consider a modified Armijo backtracking line search and a Bregman backtracking line search. For both line search algorithms, our algorithm delivers a strict monotone decrease in the objective function and further intuitive convergence properties, e.g., strong and pointwise convergence of the density variables on the active sets, norm convergence to zero of the increments, convergence of the Lagrange multipliers, and more. In addition, the numerical experiments demonstrate apparent mesh-independent convergence of the algorithm. Here, we refer to the new algorithm as the SiMPL method (pronounced “simple”), which stands for Sigmoidal Mirror descent with a Projected Latent variable.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

High-dimensional maximum-entropy phase space tomography

Reconstructing 4D or 6D phase space distributions from 1D or 2D measurements is a challenging inverse problem encountered in particle accelerators. Entropy maximization is an established method to incorporate prior information in the reconstruction, but it is typically infeasible in high-dimensional spaces. In this paper, I review two recent approaches to high-dimensional entropy maximization. The first approach utilizes differentiable simulations and a class of generative models known as normalizing flows, whereas the second approach employs the method of Lagrange multipliers and Markov Chain Monte Carlo (MCMC) sampling. My aim is to provide a short explanation of each method using a common notation. I conclude by mentioning several unsolved problems in phase space tomography.

Hoover, Austin [ORNL] (ORCID:0000000153136962)↗
DOE OSTI2021

Optimization-based algorithms for nonlinear mechanics and frictional contact

An optimization-based strategy for solving nonlinear mechanics problems is proposed. In contrast to typical nonlinear equation solver algorithms that aim to find zeros in the residual force function, we minimize an energy (or energy-like) function to encourage solutions which are locally stable equilibria. These smooth and potentially non-convex objective functions are minimized using a preconditioned conjugate-gradient trust-region algorithm. Contact is formulated as an inequality constrained minimization problem, and is solved with an augmented Lagrangian algorithm. Friction is included in the approach via a regularized quasi-potential energy, and other dissipative behavior is included through the use of variational constitutive updates. Finally, to accelerate convergence rates for the Lagrange multipliers, we propose a novel multiplier update algorithm utilizing the Fischer-Burmeister function, and demonstrate super-linear solver convergence for some applications.

42 ENGINEERING↗
DOE OSTI2021

Fission Product Yield Data Adjustment in a Prototype Version of TSURFER

The TSURFER (Tool for Sensitivity/Uncertainty analysis of Response Functionals using Experimental Results) module of Oak Ridge National Laboratory’s (ORNL’s) SCALE code system has been updated to perform nuclear data adjustments for fixed-source irradiation/depletion problems. TSURFER uses a generalized linear least squares (GLLS) approach to consolidate a prior set of measured responses and corresponding calculated values to create the most self-consistent set of nuclear data. Traditionally, TSURFER adjustments have been performed for multigroup nuclear data such as reaction cross sections. In this work, TSURFER is expanded to perform adjustments to independent fission product yields and branching ratios that need equality constraints. To preserve equality constraints after the data adjustment procedure, an updated GLLS formulation includes a new Lagrange multiplier that forces data adjustment to sum to 0 for a given fission yield/branching ratio parent. A test problem illustrates that the newly updated TSURFER module satisfies the required constraint that adjustments for fission yield data sum to 0.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2023

Electrodynamic Shaker Capability Estimation Through Experimental Dynamic Substructuring [Thesis]

Electrodynamic shaker systems are an essential tool in shock and vibration testing of dynamic environments. However, the specific performance capability of these systems is difficult to characterize. The dynamics of the shaker itself, the device under test and the specific test configuration used all couple to create a dynamic response unique to each test. Poorly predicted limitations in shaker capability affect the ability to achieve test specifications, delay testing schedules, and create difficulties for choosing test equipment. To predict shaker capability for a specific test configuration prior to setup, a lumped parameter model of the shaker system and a modal model of a device under test was developed. These models were then analytically coupled using LaGrange multiplier frequency based substructuring to estimate their coupled frequency response functions. The coupled frequency response functions were used to predict electrical inputs required to meet a given test specification. These input requirements were finally compared to a validation test using the specification and setup. Input requirements estimated using the substructuring estimated frequency response functions showed significant error. However, results using an ideal frequency response function showed very little error. These results indicate that with a better method of experimental dynamic substructuring employed it would be possible to accurately predict shaker capability for a given test configuration prior to setup.

42 ENGINEERING↗
DOE OSTI2022

Implement and Test 3D Mortar Contact in BISON

We leverage the extension of the generation of mortar segment meshes to three dimensions in MOOSE’s framework to extend thermomechanical modeling capabilities to problems with three dimensions. A modular approach to gap heat transfer physics using the mortar finite element method was created and documented, mechanical contact was extended to three dimensions—including frictional behavior, performance and ease of use were improved, and steps towards scalability of solid mechanics problems involving contact were taken. Many of these new developments are demonstrated in the simulation of 3D light-water reactor (LWR) problems, where the thermomechanical interface problem is solved using the mortar finite element method. Usage of the mortar framework has improved convergence in 2D problems and has enabled employing friction in 3D problems, of which we show results of a short, local stack of 3D pellets. Consequently, the benefits of mortar in terms of solution convergence and quality are extended to three dimensions. Section 2 discusses fundamental developments that enabled the simulation of practical mortar problems in three dimensions and other general improvements, including the reduction of the derivative container size, the modification of dual basis computations when edge dropping (lack of secondary element projection) takes place, the improvement of conditioning when employing the VCP in-edge dropping conditions, and code usability and quality improvements. These latter code enhancements include the migration of tests using “old” mortar contact constraints to using dual mortar with a semi-smooth Newton solution strategy and the reuse of lower dimensional domains for straightforwardly setting up a mortar thermomechanical LWR problem, i.e. the MOOSE action is employed for mechanical contact and the thermal LWR action is employed to capture the gas conductance, contact, and radiation components of gap heat transfer physics. Independently of the mortar LWR thermal action, we developed a modular approach to gap heat transfer that resides in MOOSE and can be leveraged, e.g., in metallic fuel problems. This approach, whose code design based on MOOSE’s user objects to model specific physics was proposed by the maintenance activity, is detailed in Section 3. Based on the dual mortar finite element method, the frictional contact constraints were extended to three dimensions. A block sheared in two directions in and out of contact with a rigid plane is employed in Section 4 to show the way the approach handles changes in frictional states (e.g. stick to slip) within a competitive number of Newton iterations. Equations and numerical results on the use of the VCP with Cartesian Lagrange multipliers, whose combination enables their direct condensation, are described in Section 5.3. Two-dimensional and three-dimensional BISON LWR simulations are discussed in Section 6. Particularly, a stack of five eccentric pellets with a surface defect is simulated and the effect of pellet-cladding friction is assessed. Finally, conclusions are outlined in Section 7.

42 ENGINEERING↗
DOE OSTI2024

Average Incremental Cost Pricing for the AC Unit Commitment Problem [Rev. 1]

Unit Commitment (UC) problems that consider the Alternating Current (AC) model of the transmission network have long been considered intractable to solve at scale by the power system community. Recently, the Grid-Optimization (GO) Competition held by the Advanced Research Project Agency-Energy (ARPA-E) has facilitated the development of the first algorithms to solve large-scale ACUC problems. This new capability opens a path towards the explicit consideration of the AC transmission network model in UC problems used to clear day-ahead electricity markets. This calls for the analysis of electricity market structures that accommodate both the continuous non-linearity of the AC transmission network and the discrete non-linearity of the UC problem simultaneously. This paper serves as an initial effort to do so by proposing an Average Incremental Cost (AIC) pricing structure that is designed around the ACUC problem. In particular, an AIC one-pass pricing problem is proposed that represents a continuously constrained variant of the ACUC problem and allows for the computation of Locational Incremental Prices (LIPs) for both real and reactive power as the local optimal Lagrange multipliers of the power balance constraints. To avoid degeneracy, the pricing problem includes a small parameter ϵ > 0. Under certain assumptions market participants are shown to realize profit that converges to a non-negative value as ϵ approaches zero, practically ensuring profitability for small values of ϵ. We additionally provide many simple and important examples that provide intuition and insights into the proposed prices. Examples illustrate the basic concept of AIC pricing, the derived profitability results, the existence of multiple LIPs, the importance of including reactive power in the dispatch and pricing problems, the need for reactive power prices, and the improved incentives exhibited by LIPs as compared to traditional Locational Marginal Prices (LMPs). We additionally indicate many directions for future work including analysis of larger test cases.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2024

Determining SMEFT and PDF parameters simultaneously based on the CTEQ-TEA framework

The SM effective field theory (SMEFT) provides a model-independent and systematically improvable framework for new physics searches. In this talk, we outline our approach of simultaneously fitting SMEFT parameters and Probability Density Functions (PDFs) in an extension of the CT18 global analysis framework. To enhance the efficiency of our global fitting and Lagrange multiplier scans, we leverage machine-learning techniques. We focus on several representative operators relevant to top-quark pair production and jet production. Through this approach, we establish self-consistent limitations on the associated Wilson coefficients, and explore the correlations between these Wilson coefficients and the PDFs.

Shen, XiaoMin↗
NASA NTRS1967

Program for calculating optimum dimensions of alpha radioisotope capsules exposed to varying stress and temperature

A method and computer program were developed for calculating the creep and optimizing the dimensions of capsules filled with alpha-emitting radioisotopes. The method solves an integral equation that was developed assuming linear accumulation of partial creep lives and relating life to time-dependent stress and temperature using the Larson-Miller parameter. The computer program, CAPSUL, is written in Fortran language for the IBM 360/75 computer. The program makes a least squares fit of the creep life function using conventional constant stress, constant temperature creep data. Dimensions of capsules having maximum thermal power per unit of weight, volume, or area are calculated for a given creep life and pressure-temperature history using a numerical Lagrange Multiplier formulation. The program also calculates the life to a prescribed strain for capsules of given dimensions and pressure-temperature history. The method has been used to analyze creep data for the alloys 304 stainless steel, Hastelloy N, Cb-1% Zr, FS-85, and T-222.

J. P. Nichols↗
NASA NTRS1970

COPTRAN - A method of optimum communication systems design

Single set of mathematical expressions describes system cost and probability of error of data transmission in terms of four basic parameters in the link equation. A Lagrange multiplier sets up equations whose solutions yield the optimum values for system design considerations and weight and cost values.

Brinkman, K. L.↗