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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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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints

The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Modeling approaches for addressing unrelaxable bound constraints with unconstrained optimization methods

We explore novel approaches for solving nonlinear optimization problems with unrelaxable bound constraints, which must be satisfied before the objective function can be evaluated. Our method reformulates the unrelaxable bound-constrained problem as an unconstrained optimization problem that is amenable to existing unconstrained optimization methods. The reformulation relies on a domain warping to form a merit function; the choice of the warping determines the level of exactness with which the unconstrained problem can be used to find solutions to the bound-constrained problem, as well as key properties of the unconstrained formulation such as smoothness. We develop theory when the domain warping is a multioutput sigmoidal warping, and we explore the practical elements of applying unconstrained optimization methods to the formulation. Here, we develop an algorithm that exploits the structure of the sigmoidal warping to guarantee that unconstrained optimization algorithms applied to the merit function will find a stationary point to the desired tolerance.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A Scalable Interior‐Point Gauss–Newton Method for PDE‐Constrained Optimization With Bound Constraints

Here, we present a scalable approach to solve a class of partial differential equation (PDE)‐constrained optimization problems with bound constraints. This approach utilizes a robust full‐space interior‐point (IP)‐Gauss–Newton optimization method. To cope with the poorly‐conditioned IP‐Gauss–Newton saddle‐point linear systems that need to be solved approximately, once per optimization step, we propose two spectrally related preconditioners. These preconditioners leverage the limited informativeness of data in regularized PDE‐constrained optimization problems. A block Gauss–Seidel preconditioner is proposed for the GMRES‐based solution of the IP‐Gauss–Newton linear systems. It is shown, for a large‐class of PDE‐ and bound‐constrained optimization problems, that the spectrum of the block Gauss–Seidel preconditioned IP‐Gauss–Newton matrix is asymptotically independent of discretization and is not impacted by the ill‐conditioning that notoriously plagues interior‐point methods. We exploit symmetry of the IP‐Gauss–Newton linear systems and propose a regularization and log‐barrier Hessian preconditioner for the preconditioned conjugate gradient (PCG)‐based solution of the equivalent IP‐Gauss–Newton–Schur complement linear systems. The eigenvalues of the block Gauss–Seidel preconditioned IP‐Gauss–Newton matrix, that are not equal to one, are identical to the eigenvalues of the regularization and log‐barrier Hessian preconditioned Schur complement matrix. The scalability of the approach is demonstrated on two example problems. The numerical solution of these optimization problems is shown to require a discretization independent number of IP‐Gauss–Newton linear solves. Furthermore, the linear systems are solved in a discretization and IP ill‐conditioning independent number of preconditioned Krylov subspace iterations. The parallel scalability of the preconditioner, achieved via algebraic multigrid component solvers when applicable, and the aforementioned algorithmic scalability permits a parallel scalable means to compute solutions of a large class of PDE‐ and bound‐constrained problems.

PDE-constrained optimization↗
DOE OSTI2015

A trust region algorithm for PDE-constrained optimization with bound constraints using reduced order modeling

This paper presents a novel trust region algorithm that relies on proper orthogonal de-composition techniques to construct accurate reduced order models during optimization.The algorithm samples high fidelity snapshots to compute the POD functions that are used to generate reduced order models. The reduced order models are used to replace the computationally intensive high fidelity finite element evaluations during optimization.The proposed algorithm employs a trust region framework to detect loss in predictive ac-curacy in the reduced order model and automatically update the POD functions during optimization. The trust region framework allows the algorithm to use sound mathematical metrics to effectively improve the accuracy and robustness of the reduced order models during optimization. The algorithm also employs a projected gradient algorithm to model bound constraints and compute optimal and feasible controls.This paper also presents an accurate Hessian formulation for topology optimization problems. The proposed trust region framework relies on a quadratic model to update the control. This quadratic model needs reliable second order derivative information to predict the behavior of the objective function within a suitable trust region. If a nonlinear Hessian formulation is used, the computational effort increases due to additional finite element evaluations. The proposed linear Hessian formulation reduces the computational effort and enables the calculation of the second order derivative information without additional finite element model evaluations. Examples in topology optimization are presented to demonstrate the applicability of the proposed algorithm and linear Hessian formulation for large-scale PDE-constrained optimization.

bound constraints↗
DOE OSTI2022

The Age of Evapotranspiration: Lower‐Bound Constraints From Distributed Water Fluxes Across the Continental United States

Abstract Unlike streamflow, which can be sampled in aggregate at the catchment outlet, evapotranspiration (ET) is spatially dispersed, challenging large‐scale age estimation. Here, we introduce an approach for constraining the age of ET via mass balance and present the minimum flux‐weighted age of ET across the continental United States using distributed, publicly available water flux data sets. The lower‐bound constraint on ET age can be calculated by assuming that ET is preferentially sourced from the most recent precipitation through a last‐in, first‐out algorithm. From 2012 to 2017, ET was at least several months old across large areas of the western continental United States, including in Mediterranean and (semi‐)arid climate zones and shrub and evergreen needleleaf plant communities. The primary limitation of this approach is that it provides only a minimum flux‐weighted average age to satisfy the mass balance of outgoing fluxes; true ET fluxes are composed of distributions of ages and may be composed of much older water. The primary advantage of the approach is that flux time series of precipitation and ET are sufficient to constrain ET age, and model parameterization is unnecessary. ET ages can be used to validate tracer‐aided and modeling approaches and inform studies of biogeochemistry, water‐rock interactions, and plant water sourcing under drought.

Hahm, W. J.↗
DOE OSTI2020

A Survey of Constrained Gaussian Process: Approaches and Implementation Challenges

Gaussian process regression is a popular Bayesian framework for surrogate modeling of expensive data sources. As part of a larger effort in scientific machine learning, many recent works have incorporated physical constraints or other a priori information within Gaussian process regression to supplement limited data and regularize the behavior of the model. We provide an overview and survey of several classes of Gaussian process constraints, including positivity or bound constraints, monotonicity and convexity constraints, differential equation constraints provided by linear PDEs, and boundary condition constraints. We compare the strategies behind each approach as well as the differences in implementation, concluding with a discussion of the computational challenges introduced by constraints.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Incorporating physical constraints into Gaussian process surrogate models (LDRD Project Summary)

This report summarizes work done under the Laboratory Directed Research and Development (LDRD) project titled "Incorporating physical constraints into Gaussian process surrogate models?' In this project, we explored a variety of strategies for constraint implementations. We considered bound constraints, monotonicity and related convexity constraints, Gaussian processes which are constrained to satisfy linear operator constraints which represent physical laws expressed as partial differential equations, and intrinsic boundary condition constraints. We wrote three papers and are currently finishing two others. We developed initial software implementations for some approaches. This report summarizes the work done under this LDRD.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Symmetry Parameter Constraints from a Lower Bound on Neutron-matter Energy

We propose the existence of a lower bound on the energy of pure neutron matter (PNM) on the basis of unitary-gas considerations. We discuss its justification from experimental studies of cold atoms as well as from theoretical studies of neutron matter. We demonstrate that this bound results in limits to the density-dependent symmetry energy, which is the difference between the energies of symmetric nuclear matter and PNM. In particular, this bound leads to a lower limit to the volume symmetry energy parameter S {sub 0}. In addition, for assumed values of S {sub 0} above this minimum, this bound implies both upper and lower limits to the symmetry energy slope parameter L , which describes the lowest-order density dependence of the symmetry energy. A lower bound on neutron-matter incompressibility is also obtained. These bounds are found to be consistent with both recent calculations of the energies of PNM and constraints from nuclear experiments. Our results are significant because several equations of state that are currently used in astrophysical simulations of supernovae and neutron star mergers, as well as in nuclear physics simulations of heavy-ion collisions, have symmetry energy parameters that violate these bounds. Furthermore, below the nuclear saturation density, the bound on neutron-matter energies leads to a lower limit to the density-dependent symmetry energy, which leads to upper limits to the nuclear surface symmetry parameter and the neutron-star crust–core boundary. We also obtain a lower limit to the neutron-skin thicknesses of neutron-rich nuclei. Above the nuclear saturation density, the bound on neutron-matter energies also leads to an upper limit to the symmetry energy, with implications for neutron-star cooling via the direct Urca process.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1978

An algorithm for optimal structural design with frequency constraints

The paper presents a finite element method for minimum weight design of structures with lower-bound constraints on the natural frequencies, and upper and lower bounds on the design variables. The design algorithm is essentially an iterative solution of the Kuhn-Tucker optimality criterion. The three most important features of the algorithm are: (1) a small number of design iterations are needed to reach optimal or near-optimal design, (2) structural elements with a wide variety of size-stiffness may be used, the only significant restriction being the exclusion of curved beam and shell elements, and (3) the algorithm will work for multiple as well as single frequency constraints. The design procedure is illustrated with three simple problems.

Kiusalaas, J.↗
NASA NTRS1996

Pattern Search Algorithms for Bound Constrained Minimization

We present a convergence theory for pattern search methods for solving bound constrained nonlinear programs. The analysis relies on the abstract structure of pattern search methods and an understanding of how the pattern interacts with the bound constraints. This analysis makes it possible to develop pattern search methods for bound constrained problems while only slightly restricting the flexibility present in pattern search methods for unconstrained problems. We prove global convergence despite the fact that pattern search methods do not have explicit information concerning the gradient and its projection onto the feasible region and consequently are unable to enforce explicitly a notion of sufficient feasible decrease.

Lewis, Robert Michael↗
DOE OSTI2026

Feature Engineering and Ensemble Methods for Imbalanced ICS Intrusion Detection: Pipeline Audit and Constrained Evaluation

Industries are becoming increasingly connected and are more vulnerable to cyberattacks due to the widened attack surface. Industrial Control Systems (ICS) are among the most critical sectors that malicious actors can target, as such attacks can cause significant operational disruption and physical damage. It is imperative to detect such attacks as early as possible. This paper evaluates constraint-conditioned optimistic performance estimates for traditional ML models in ICS intrusion detection (i.e., estimates obtained under contiguous, non-shuffled temporal evaluation without test-set alteration, but with pre-split feature engineering that may introduce temporal leakage, due to dataset constraints). Our findings are threefold. First, we quantify how iterative feature engineering affects tree-based ensemble performance and examine how pipeline decisions (split strategy, sampling scope, and cleaning policy) can inflate or reduce reported IDS results under constraint-bound evaluation. Second, we compare intrinsic class-imbalance handling across ensemble models. Third, under our current pipeline constraints (including pre-split feature engineering), CatBoost achieves the best performance on Water Storage Tank (accuracy: 0.9831, class-1 F1: 0.9682), while Light- GBM achieves the best performance on Gas Pipeline (accuracy: 0.9618, class-1 F1: 0.9086).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

A memory-distributed quasi-Newton solver for nonlinear programming problems with a small number of general constraints

We approach the problem of parallelizing state-of-the-art nonlinear programming optimization algorithms.Specifically, we focus on parallelizing quasi-Newton interior-point methods that use limited-memory secant Hessian approximations. Such interior-point methods are known to have better convergence properties and to be more effective on large-scale problems than gradient-based and derivative-free optimization algorithms. We target nonlinear and potentially nonconvex optimization problems with an arbitrary number of bound constraints and a small number of general equality and inequality constraints on the optimization variables. These problems occur for example in the form of optimal control, optimal design, and inverse problems governed by ordinary or partial differential equations, whenever they are expressed in a “reduced-space” optimization approach. We introduce and analyze the time and space complexity of a decomposition method for solving the quasi-Newton linear systems that leverages the fact that the quasi-Newton Hessian matrix has a small number of dense blocks that border a low-rank update of a diagonal matrix. This enables an efficient parallelization on memory-distributed computers of the iterations of the optimization algorithm, a state-of-the-art filter line-search interior-point algorithm by Wächter et. al. We illustrate the efficiency of the proposed method by solving structural topology optimization problems on up to 4608 cores on a parallel machine.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2012

Structural Analysis and Optimization of a Composite Fan Blade for Future Aircraft Engine

This report addresses the structural analysis and optimization of a composite fan blade sized for a large aircraft engine. An existing baseline solid metallic fan blade was used as a starting point to develop a hybrid honeycomb sandwich construction with a polymer matrix composite face sheet and honeycomb aluminum core replacing the original baseline solid metallic fan model made of titanium. The focus of this work is to design the sandwich composite blade with the optimum number of plies for the face sheet that will withstand the combined pressure and centrifugal loads while the constraints are satisfied and the baseline aerodynamic and geometric parameters are maintained. To satisfy the requirements, a sandwich construction for the blade is proposed with composite face sheets and a weak core made of honeycomb aluminum material. For aerodynamic considerations, the thickness of the core is optimized whereas the overall blade thickness is held fixed so as to not alter the original airfoil geometry. Weight is taken as the objective function to be minimized by varying the core thickness of the blade within specified upper and lower bounds. Constraints are imposed on radial displacement limitations and ply failure strength. From the optimum design, the minimum number of plies, which will not fail, is back-calculated. The ply lay-up of the blade is adjusted from the calculated number of plies and final structural analysis is performed. Analyses were carried out by utilizing the OpenMDAO Framework, developed at NASA Glenn Research Center combining optimization with structural assessment.

Coroneos, Rula M.↗
DOE OSTI2025

A simple introduction to the SiMPL method for density-based topology optimization

We introduce a novel method for solving density-based topology optimization problems: Sigmoidal Mirror descent with a Projected Latent variable (SiMPL). The SiMPL method (pronounced as “the simple method”) optimizes a design using only first-order derivative information of the objective function. The bound constraints on the density field are enforced with the help of the (negative) Fermi–Dirac entropy, which is also used to define a non-symmetric distance function called a Bregman divergence on the set of admissible designs. This Bregman divergence leads to a simple update rule that is further simplified with the help of a so-called latent variable. Because the SiMPL method involves discretizing the latent variable, it produces a sequence of pointwise-feasible iterates, even when high-order finite elements are used in the discretization. Numerical experiments demonstrate that the method outperforms other popular first-order optimization algorithms. In conclusion, to outline the general applicability of the technique, we include examples with (self-load) compliance minimization and compliant mechanism optimization problems.

Calculus of Variations and Optimization↗
DOE OSTI2017

Improved measurements of turbulence in the hot gaseous atmospheres of nearby giant elliptical galaxies

Here, we present significantly improved measurements of turbulent velocities in the hot gaseous haloes of nearby giant elliptical galaxies. Using deep XMM–NewtonReflection Grating Spectrometer ( RGS ) observations and a combination of resonance scattering and direct line broadening methods, we obtain well bounded constraints for 13 galaxies. Assuming that the turbulence is isotropic, we obtain a best-fitting mean 1D turbulent velocity of 110 km s -1 . This implies a typical 3D Mach number ~0.45 and a typical non-thermal pressure contribution of ~6 per cent in the cores of nearby massive galaxies. The intrinsic scatter around these values is modest – consistent with zero, albeit with large statistical uncertainty – hinting at a common and quasi-continuous mechanism sourcing the velocity structure in these objects. Using conservative estimates of the spatial scales associated with the observed turbulent motions, we find that turbulent heating can be sufficient to offset radiative cooling in the inner regions of these galaxies (<10 kpc, typically 2–3 kpc). The full potential of our analysis methods will be enabled by future X-ray micro-calorimeter observations.

79 ASTRONOMY AND ASTROPHYSICS↗