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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

A Domain-Decomposed A-ϕ Formulation Based on Lagrange Multipliers for Low-Frequency Problems

A domain-decomposed A-ϕ formulation based on Lagrange multipliers is proposed to simulate low-frequency elec- tromagnetic problems. This method partitions the computational domain into smaller subdomains, allowing each subdomain to be independently formulated using Lagrange multipliers as Dirichlet boundary conditions, while ensuring continuity of the fields across the interfaces. A mixed finite element method, utilizing both vector and scalar basis functions, is employed to discretize the formulation, resulting in a global system to be solved. The proposed method is validated using TEAM Problem 7 at 50 Hz, demonstrating its effectiveness in handling complex geometries and addressing the low-frequency breakdown issues commonly encountered in traditional finite element methods.

Hossain, Amzad↗
DOE OSTI2015

Application of parallel distributed Lagrange multiplier technique to simulate coupled Fluid-Granular flows in pipes with varying Cross-Sectional area

Fluid-granular flows are common phenomena in nature and industry. Here, an efficient computational technique based on the distributed Lagrange multiplier method is utilized to simulate complex fluid-granular flows. Each particle is explicitly resolved on an Eulerian grid as a separate domain, using solid volume fractions. The fluid equations are solved through the entire computational domain, however, Lagrange multiplier constrains are applied inside the particle domain such that the fluid within any volume associated with a solid particle moves as an incompressible rigid body. The particle–particle interactions are implemented using explicit force-displacement interactions for frictional inelastic particles similar to the DEM method with some modifications using the volume of an overlapping region as an input to the contact forces. Here, a parallel implementation of the method is based on the SAMRAI (Structured Adaptive Mesh Refinement Application Infrastructure) library.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Physics successfully implements Lagrange multiplier optimization

Optimization is a major part of human effort. While being mathematical, optimization is also built into physics. For example, physics has the Principle of Least Action; the Principle of Minimum Power Dissipation, also called Minimum Entropy Generation; and the Variational Principle. Physics also has Physical Annealing, which, of course, preceded computational Simulated Annealing. Physics has the Adiabatic Principle, which, in its quantum form, is called Quantum Annealing. Thus, physical machines can solve the mathematical problem of optimization, including constraints. Binary constraints can be built into the physical optimization. In that case, the machines are digital in the same sense that a flip–flop is digital. A wide variety of machines have had recent success at optimizing the Ising magnetic energy. We demonstrate in this paper that almost all those machines perform optimization according to the Principle of Minimum Power Dissipation as put forth by Onsager. Further, we show that this optimization is in fact equivalent to Lagrange multiplier optimization for constrained problems. We find that the physical gain coefficients that drive those systems actually play the role of the corresponding Lagrange multipliers.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

The two faces of mimetic Horndeski gravity: disformal transformations and Lagrange multiplier

We show that very general scalar-tensor theories of gravity (including, e.g., Horndeski models) are generically invariant under disformal transformations. However there is a special subset, when the transformation is not invertible, that yields new equations of motion which are a generalization of the so-called 'mimetic' dark matter theory recently introduced by Chamsedinne and Mukhanov. These conclusions hold true irrespective of whether the scalar field in the action of the assumed scalar-tensor theory of gravity is the same or different than the scalar field involved in the transformation. The new equations of motion for our general mimetic theory can also be derived from an action containing an additional Lagrange multiplier field. The general mimetic scalar-tensor theory has the same number of derivatives in the equations of motion as the original scalar-tensor theory. As an application we show that the simplest mimetic scalar-tensor model is able to mimic the cosmological background of a flat FLRW model with a barotropic perfect fluid with any constant equation of state.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Comparison of Real-Time Pressure Rail Selection Algorithms for the Hybrid Hydraulic Electric Architecture: Case Study on a Track Loader

Abstract The hybrid hydraulic electric architecture (HHEA) seeks to combine the high power/torque/force density of hydraulics with the efficiency of electric machines. A set of common pressure rails is used to provide a majority of the power and this power is modulated by small electric machines to provide precise control for the operator. The HHEA has been studied in previous work using off-line dynamic programming optimization to determine energy efficient pressure rail selections, but this approach requires drive cycle information apriori. A Lagrange multiplier method has also been investigated where a set of gains (Lagrange multipliers) are optimized off-line with the idea the these gains, once determined, could be used for real-time operation. In this work, three new real-time pressure rail selection algorithms that do not require future drive cycle information are investigated; greedy, torque minimizing, and thresholding. The greedy control is found to only use 1% more energy than the globally optimal dynamic programming solution; but a model of energy loss is required.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2019

Scalable flexibility control of distributed loads in a power grid

The technology described herein is generally directed towards a distributed optimization technology for the control of aggregation of distributed flexibility resource nodes that operates iteratively until a commanded power profile is produced by aggregated loads. The technology uses a distributed iterative solution in which each node solves a local optimization problem with local constraints and states, while using a global Lagrange multiplier that is based upon information from each other node. The global Lagrange multiplier is determined at an aggregation level using load-specific information that is obtained in a condensed form (e.g., a scalar) from each node at each iteration. The global Lagrange multiplier is broadcasted to the nodes for each new iteration. The technology provides an iterative, distributed solution to the network optimization problem of power tracking of aggregated loads.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2022

Scalable preconditioning for the stabilized contact mechanics problem

We present a family of preconditioning strategies for the contact problem in fractured and faulted porous media. We combine low-order continuous finite elements to simulate the bulk deformation with piecewise constant Lagrange multipliers to impose the frictional contact constraints. This formulation is not uniformly inf-sup stable and requires stabilization. We improve previous work by Franceschini et al. (2020) by introducing a novel jump stabilization technique that requires only local geometrical and mechanical properties. We then design scalable preconditioning strategies that take advantage of the block structure of the Jacobian matrix using a physics-based partitioning of the unknowns by field type, namely displacement and Lagrange multipliers. The key to the success of the proposed preconditioners is a pseudo-Schur complement obtained by eliminating the Lagrange multiplier degrees of freedom, which can then be efficiently solved using an optimal multigrid method. Numerical results, including complex real-world problems, are presented to illustrate theoretical properties, scalability and robustness of the preconditioner. A comparison with other approaches available in the literature is also provided.

58 GEOSCIENCES↗
DOE OSTI2016

Efficient sensitivity analysis method for chaotic dynamical systems

The direct differentiation and improved least squares shadowing methods are both developed for accurately and efficiently calculating the sensitivity coefficients of time averaged quantities for chaotic dynamical systems. The key idea is to recast the time averaged integration term in the form of differential equation before applying the sensitivity analysis method. An additional constraint-based equation which forms the augmented equations of motion is proposed to calculate the time averaged integration variable and the sensitivity coefficients are obtained as a result of solving the augmented differential equations. The application of the least squares shadowing formulation to the augmented equations results in an explicit expression for the sensitivity coefficient which is dependent on the final state of the Lagrange multipliers. The LU factorization technique to calculate the Lagrange multipliers leads to a better performance for the convergence problem and the computational expense. Numerical experiments on a set of problems selected from the literature are presented to illustrate the developed methods. The numerical results demonstrate the correctness and effectiveness of the present approaches and some short impulsive sensitivity coefficients are observed by using the direct differentiation sensitivity analysis method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Hybrid PN-SN Calculations with SAAF for the Multiscale Transport Capability in Rattlesnake

Two interface conditions, the Lagrange multiplier method and the upwinding method, for hybrid \pn-\sn calculations is proposed for the self-adjoint angular flux (SAAF) formulation of the transport equation using the continuous finite element method (FEM) for spatial discretization. These interface conditions are implemented in Rattlesnake, the radiation transport application built on MOOSE, for the on-going multiscale transport simulation effort at INL. For smoothing the solution at the interface for the Lagrange multiplier method, a method based on \sn Lagrange interpolation on the sphere is proposed. Numerical results indicate that the interface conditions give the expected convergence.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

An Algorithmic and Software Pipeline for Very Large Scale Scientific Data Compression with Error Guarantees

Efficient data compression is becoming increasingly critical for storing scientific data because many scientific applications produce vast amounts of data. This paper presents an end-to-end algorithmic and software pipeline for data compression that guarantees both error bounds on primary data (PD) and derived data, known as Quantities of Interest (QoI).We demonstrate the effectiveness of the pipeline by compressing fusion data generated by a large-scale fusion code, XGC, which produces tens of petabytes of data in a single day. We demonstrate that the compression is conducted by setting aside computational resources known as staging nodes, and does not impact the simulation performance. For efficient parallel I/O, the pipeline uses ADIOS2, which many codes such as XGC already use for their parallel I/O. We show that our approach can compress the data by two orders of magnitude while guaranteeing high accuracy on both the PD and the QoIs. Further, the amount of resources required by compression is a few percent of the resources required by simulation while ensuring that the compression time for each stage is less than the corresponding simulation time.This pipeline consists of three main steps. The first step decomposes the data using domain decomposition into small subdomains. Each subdomain is then compressed independently to achieve a high level of parallelism. The second step uses existing techniques that guarantee error bounds on the primary data for each subdomain. The third step uses a post-processing optimization technique based on Lagrange multipliers to reduce the QoI errors for data corresponding to each subdomain. The Lagrange multipliers generated can be further quantized or truncated to increase the compression level. All of the above characteristics of our approach make it highly practical to apply on-the-fly compression while guaranteeing errors on QoIs that are critical to the scientists.

Banerjee, Tania↗