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Results for “residual minimization”

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 OSTI2023

Error Estimates of Residual Minimization Using Neural Networks for Linear PDES

We propose an abstract framework for analyzing the convergence of least-squares methods based on residual minimization when feasible solutions are neural networks. With the norm relations and compactness arguments, we derive error estimates for both continuous and discrete formulations of residual minimization in strong and weak forms. The formulations cover recently developed physicsinformed neural networks based on strong and variational formulations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Low synchronization Gram–Schmidt and generalized minimal residual algorithms

The Gram–Schmidt process uses orthogonal projection to construct the A = QR factorization of a matrix. When Q has linearly independent columns, the operator P = I - Q(QTQ)-1QT defines an orthogonal projection onto Q⊥. In finite precision, Q loses orthogonality as the factorization progresses. A family of approximate projections is derived with the form P = I - QTQT, with correction matrix T. When T = (QTQ)-1, and T is triangular, it is postulated that the best achievable orthogonality is $\mathcal{O}(ε)\mathcal{K}(A)$. We present new variants of modified (MGS) and classical Gram–Schmidt algorithms that require one global reduction step. An interesting form of the projector leads to a compact WY representation for MGS. In particular, the inverse compact WY MGS algorithm is equivalent to a lower triangular solve. Our main contribution is to introduce a backward normalization lag into the compact WY representation, resulting in a $\mathcal{O}(ε)\mathcal{K}[r_0, AV_m])$ stable Generalized Minimal Residual Method (GMRES) algorithm that requires only one global reduce per iteration. Finally, further improvements in performance are achieved by accelerating GMRES on GPUs.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

An adaptive discontinuous Petrov-Galerkin method for the Grad-Shafranov equation

In this work, we propose and develop an arbitrary-order adaptive discontinuous Petrov--Galerkin (DPG) method for the nonlinear Grad--Shafranov equation. An ultraweak formulation of the DPG scheme for the equation is given based on a minimal residual method. The DPG scheme has the advantage of providing more accurate gradients compared to conventional finite element methods, which is desired for numerical solutions to the Grad--Shafranov equation. The numerical scheme is augmented with an adaptive mesh refinement approach, and a criterion based on the residual norm in the minimal residual method is developed to achieve dynamic refinement. Nonlinear solvers for the resulting system are explored and a Picard iteration with Anderson acceleration is found to be efficient to solve the system. Finally, the proposed algorithm is implemented in parallel on MFEM using a domain-decomposition approach, and our implementation is general, supporting arbitrary order of accuracy and general meshes. Furthermore, numerical results are presented to demonstrate the efficiency and accuracy of the proposed algorithm.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Preconditioned least‐squares Petrov–Galerkin reduced order models

Abstract In this article, we introduce a methodology for improving the accuracy and efficiency of reduced order models (ROMs) constructed using the least‐squares Petrov–Galerkin (LSPG) projection method through the introduction of preconditioning. Unlike prior related work, which focuses on preconditioning the linear systems arising within the ROM numerical solution procedure to improve linear solver performance, our approach leverages a preconditioning matrix directly within the minimization problem underlying the LSPG formulation. Applying preconditioning in this way has the potential to improve ROM accuracy for several reasons. First, preconditioning the LSPG formulation changes the norm defining the residual minimization, which can improve the residual‐based stability constant bounding the ROM solution's error. The incorporation of a preconditioner into the LSPG formulation can have the additional effect of scaling the components of the residual being minimized to make them roughly of the same magnitude, which can be beneficial when applying the LSPG method to problems with disparate scales (e.g., dimensional equations, multi‐physics problems). Importantly, we demonstrate that an “ideal preconditioned” LSPG ROM (a ROM in which the preconditioner is the inverse of the Jacobian of its corresponding full order model) emulates projection of the full order model solution increment onto the reduced basis. This quantity defines a lower bound on the error of a ROM solution for a given reduced basis. By designing preconditioners that approximate the Jacobian inverse—as is common in designing preconditioners for solving linear systems—it is possible to obtain a ROM whose error approaches this lower bound. The proposed approach is evaluated on several mechanical and thermo‐mechanical problems implemented within the Albany HPC code and run in the predictive regime, with prediction across material parameter space. We demonstrate numerically that the introduction of simple Jacobi, Gauss‐Seidel, and ILU preconditioners into the proper orthogonal decomposition/LSPG formulation reduces significantly the ROM solution error, the reduced Jacobian condition number, the number of nonlinear iterations required to reach convergence, and the wall time (thereby improving efficiency). Moreover, our numerical results reveal that the introduction of preconditioning can deliver a robust and accurate solution for test cases in which the unpreconditioned LSPG method fails to converge.

Lindsay, Payton↗
DOE OSTI2024

A comprehensive review of dwell time optimization methods in computer-controlled optical surfacing

Dwell time plays a vital role in determining the accuracy and convergence of the computer-controlled optical surfacing process. However, optimizing dwell time presents a challenge due to its ill-posed nature, resulting in non-unique solutions. To address this issue, several well-known methods have emerged, including the iterative, Bayesian, Fourier transform, and matrix-form methods. Despite their independent development, these methods share common objectives, such as minimizing residual errors, ensuring dwell time's positivity and smoothness, minimizing total processing time, and enabling flexible dwell positions. This paper aims to comprehensively review the existing dwell time optimization methods, explore their interrelationships, provide insights for their effective implementations, evaluate their performances, and ultimately propose a unified dwell time optimization methodology.

36 MATERIALS SCIENCE↗
DOE OSTI2023

MTA Removable Beam Absorber

Once commissioned, the secondary production beamline in the MeV Test Area (MTA) will require a removable beam absorber for continuous beam operation. A 4"x 4" x 4" copper core encased within steel will absorb the full 15 Hz 400 MeV Fermilab Linac primary proton beam. However the residual activation will constrain work activities in the hall, so a 6” marble casing is proposed entombing the core. Marble has minimal residual activation. This presentation describes the initial concept and planned ANSYS analysis.

Blas, Efren↗
DOE OSTI2022

Damage Tolerance Assessment of Laser Clad Repairs of Coarse Grain Ti-6Al-4V

Background: Laser cladding offers a promising approach to repairing aviation components, with the ability to restore the geometric shape and the static and fatigue strength. Objective: The damage tolerance of a coarse grain Ti-6Al-4 V material is assessed via three sets of samples: (i) baseline, (ii) laser clad repaired, and (iii) laser cladding followed by heat treatment. Method: The specimens were subjected to fatigue loading using a marker band technique, in which cracks initiated from pre-defined laser slots and the small crack growth rates were measured by quantitative fractography. Additional characterization was performed on these samples, including hardness measurements, microstructure analysis, and multimodal residual stress analyses. Results: The region of the specimen undergoing laser cladding exhibited higher hardness values and a compressive residual stress, based on the slitting method. The samples subjected to heat treatment after cladding displayed negligible differences in grain size and minimal residual stress relaxation. Conclusions: The results demonstrated significant variability, with data demonstrating that cracks in the laser clad region (exhibiting the largest compressive residual stress) grew faster than cracks in the substrate region of the sample or in the baseline cases. Finally, these counterintuitive results are rationalized with additional analyses from synchrotron-based energy dispersive X-ray diffraction and high energy X-ray diffraction, demonstrating significant spatial variability in the residual elastic strains in the millimeter-sized prior β grains within the material.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Numerical algorithms for water waves with background flow over obstacles and topography

Abstract We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

Ambrose, David M.↗
DOE OSTI2021

A Segregated Approach for Modeling the Electrochemistry in the 3-D Microstructure of Li-Ion Batteries and Its Acceleration Using Block Preconditioners

Abstract Battery performance is strongly correlated with electrode microstructure. Electrode materials for lithium-ion batteries have complex microstructure geometries that require millions of degrees of freedom to solve the electrochemical system at the microstructure scale. A fast-iterative solver with an appropriate preconditioner is then required to simulate large representative volume in a reasonable time. In this work, a finite element electrochemical model is developed to resolve the concentration and potential within the electrode active materials and the electrolyte domains at the microstructure scale, with an emphasis on numerical stability and scaling performances. The block Gauss-Seidel (BGS) numerical method is implemented because the system of equations within the electrodes is coupled only through the nonlinear Butler–Volmer equation, which governs the electrochemical reaction at the interface between the domains. The best solution strategy found in this work consists of splitting the system into two blocks—one for the concentration and one for the potential field—and then performing block generalized minimal residual preconditioned with algebraic multigrid, using the FEniCS and the Portable, Extensible Toolkit for Scientific Computation libraries. Significant improvements in terms of time to solution (six times faster) and memory usage (halving) are achieved compared with the MUltifrontal Massively Parallel sparse direct Solver. Additionally, BGS experiences decent strong parallel scaling within the electrode domains. Last, the system of equations is modified to specifically address numerical instability induced by electrolyte depletion, which is particularly valuable for simulating fast-charge scenarios relevant for automotive application.

25 ENERGY STORAGE↗
DOE OSTI2025

A detailed study of pre-heating effects in electron beam melting powder bed fusion process

Metal-based additive manufacturing processes, such as powder bed fusion with electron beam (PBF-EB) process, also referred to as electron beam melting (EBM), can produce high-density parts with minimal residual stresses due to the uniform and coherent preheating of the powder bed. However, understanding and controlling the multiple stages of preheating is required to enable the production of high-quality, consistent parts of various materials. This work presents a large-scale, multi-layer, three-dimensional numerical analysis focused on studying the preheating stages for predicting thermal history during the PBF-EB process. The model follows a continuous multi-stage cyclic process, that incorporates all the main stages of the PBF-EB process for 316 L stainless steel. This includes the gradual deposition of a new powder layer, the first and second preheating levels of the powder bed, and the energy deposition during melting (excluding the actual melt-pool behavior simulation). The model employs an adaptive time-scaling approach that automatically adjusts the energy deposition for each solution time-increment. This allows for localized changes in time-resolution over an otherwise computationally expensive multi-layer procedure. The material property variations are also taken into account, with an emphasis on the subtle irreversible changes in powder effective thermal conductivity after the two requisite preheating stages of the powder bed. This effect is studied using simplified conductivity models from the literature for partially sintered powder, validated by a dedicated experiment and numerical simulation. The large-scale model is then used to estimate the actual temperatures during first and second preheating levels for 316 L steel, which is not yet fully supported commercially for PBF-EB. Model predictions are corroborated by experiments, using and analyzing IR images, taken at the completion of each layer by the machine’s built-in infrared camera. The current model also incorporates a qualitative assessment for the effects of conductivity change during pre-heating, as well as evaluates the applicability of the time-scaling approach.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Hybrid eigensolvers for nuclear configuration interaction calculations

We examine and compare several iterative methods for solving large-scale eigenvalue problems arising from nuclear structure calculations. In particular, we discuss the possibility of using block Lanczos method, a Chebyshev filtering based subspace iterations and the residual minimization method accelerated by direct inversion of iterative subspace (RMM-DIIS) and describe how these algorithms compare with the standard Lanczos algorithm and the locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm. Although the RMM-DIIS method does not exhibit rapid convergence when the initial approximations to the desired eigenvectors are not sufficiently accurate, it can be effectively combined with either the block Lanczos or the LOBPCG method to yield a hybrid eigensolver that has several desirable properties. We will describe a few practical issues that need to be addressed to make the hybrid solver efficient and robust.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Influence of silicon infiltration conditions on microstructure and mechanical properties in binder jet 3D printed SiC

Atmosphere, temperature, and hold time were varied for silicon infiltrated binder jet 3D printed SiC preforms. A two-step infiltration process optimized infiltration. The first step at 1450 °C under vacuum facilitated improved wetting. The second step at elevated temperature in argon advanced infiltration while suppressing vaporization. Variation of the second-step temperature (1550–1750 °C) and hold time revealed that 1550 °C for 1 h in argon following initial vacuum treatment yielded the highest silicon uptake ratio (∼96 %) with minimal residual porosity and strengths of 150–180 MPa. Flexural strength was either stable or improved with elevated temperatures up to 1000°C. In contrast, higher temperatures or extended hold times led to increased silicon vaporization and pores. The optimized process was applied to formulations with extra carbon to enhance SiC content and interconnectedness through reaction bonding, which corresponded to increased flexural strength. These findings provide guidance for silicon infiltration in additively manufactured SiC.

Yoon, Bola [ORNL] (ORCID:0000000260875373)↗
DOE OSTI2024

Additive manufacturing of AISI M2 tool steel by binder jetting (BJ): Investigation of microstructural and mechanical properties

The presented research demonstrates for the first time the successful processing of AISI M2 tool steel by binder jetting, a promising additive manufacturing technique capable of producing complex shapes with minimal residual stresses and isotropic properties. The optimal printing parameters were explored by varying processing parameters such as the binder saturation (45 %–105 %), binder set time (0 to 10 s), targeted bed temperature (50–60 °C), oscillator (2600–2750 rpm), recoater (20–28 mm/s), and roller speeds (200–300 rpm). Microstructural characterization and evaluation of mechanical properties of binder jetted parts were performed using x-ray diffraction (XRD), scanning electron microscopy (SEM), and energy dispersive spectroscopy (EDS) to study their chemical composition, powder morphology, microstructure, carbide morphologies, relative density, hardness, compressive strength, and ductility. Two powder sizes (5 and 10 μm) were used, and sintering was performed at varying temperatures (1270, 1280, and 1300 °C) and durations (60 and 120 min), followed by a furnace, air, and water cooling. An optimum hardness of ~970 HV was obtained when parts were sintered at 1270 °C for 60 min, followed by water quenching. Impressive compressive strength of ~ 3580 MPa was observed in the sample sintered at 1280 °C for 60 min duration, followed by air cooling. Furnace-cooled parts showed the highest density of ~95 %, whereas the relative density of air- and water-cooled parts varied between ~91 to 93.50 %, respectively. The microstructure of sintered samples revealed the formation of M 6 C stable carbide, M 2 C metastable carbide, MC as a secondary carbide, and a-Fe matrix, which contributed to the observed increase in mechanical properties.

36 MATERIALS SCIENCE↗