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DOE OSTI2017

HFEM3D

Software solves the three-dimensional Poisson equation div(k(grad(u)) = f, by the finite element method for the case when material properties, k, are distributed over hierarchy of edges, facets and tetrahedra in the finite element mesh. Method is described in Weiss, CJ, Finite element analysis for model parameters distributed on a hierarchy of geometric simplices, Geophysics, v82, E155-167, doi:10.1190/GEO2017-0058.1 (2017). A standard finite element method for solving Poisson’s equation is augmented by including in the 3D stiffness matrix additional 2D and 1D stiffness matrices representing the contributions from material properties associated with mesh faces and edges, respectively. The resulting linear system is solved iteratively using the conjugate gradient method with Jacobi preconditioning. To minimize computer storage for program execution, the linear solver computes matrix-vector contractions element-by-element over the mesh, without explicit storage of the global stiffness matrix. Program output vtk compliant for visualization and rendering by 3rd party software. Program uses dynamic memory allocation and as such there are no hard limits on problem size outside of those imposed by the operating system and configuration on which the software is run. Dimension, N, of the finite element solution vector is constrained by the the addressable space in 32-vs-64 bit operating systems. Total storage requirements for the problem. Total working space required for the program is approximately 13*N double precision words.

Weiss, Chester J↗

DOE OSTI2016

Gradient elution moving boundary electrophoresis enables rapid analysis of acids in complex biomass-derived streams

Biomass conversion processes such as pretreatment, liquefaction, and pyrolysis often produce complex mixtures of intermediates that are a substantial challenge to analyze rapidly and reliably. To characterize these streams more comprehensively and efficiently, new techniques are needed to track species through biomass deconstruction and conversion processes. Here, we present the application of an emerging analytical method, gradient elution moving boundary electrophoresis (GEMBE), to quantify a suite of acids in a complex, biomass-derived streams from alkaline pretreatment of corn stover. GEMBE offers distinct advantages over common chromatography-spectrometry analytical approaches in terms of analysis time, sample preparation requirements, and cost of equipment. As demonstrated here, GEMBE is able to track 17 distinct compounds (oxalate, formate, succinate, malate, acetate, glycolate, protocatechuate, 3-hydroxypropanoate, lactate, glycerate, 2-hydroxybutanoate, 4-hydroxybenzoate, vanillate, p-coumarate, ferulate, sinapate, and acetovanillone). The lower limit of detection was compound dependent and ranged between 0.9 and 3.5 umol/L. Results from GEMBE were similar to recent results from an orthogonal method based on GCxGC-TOF/MS. Altogether, GEMBE offers a rapid, robust approach to analyze complex biomass-derived samples, and given the ease and convenience of deployment, may offer an analytical solution for online tracking of multiple types of biomass streams.

09 BIOMASS FUELS↗

NASA NTRS1992

Development of iterative techniques for the solution of unsteady compressible viscous flows

The development of efficient iterative solution methods for the numerical solution of two- and three-dimensional compressible Navier-Stokes equations is discussed. Iterative time marching methods have several advantages over classical multi-step explicit time marching schemes, and non-iterative implicit time marching schemes. Iterative schemes have better stability characteristics than non-iterative explicit and implicit schemes. In this work, another approach based on the classical conjugate gradient method, known as the Generalized Minimum Residual (GMRES) algorithm is investigated. The GMRES algorithm has been used in the past by a number of researchers for solving steady viscous and inviscid flow problems. Here, we investigate the suitability of this algorithm for solving the system of non-linear equations that arise in unsteady Navier-Stokes solvers at each time step.

Sankar, Lakshmi N.↗

DOE OSTI2018

Constructing the Pareto front for multi-objective Markov chains handling a strong Pareto policy approach

In this paper, we present a multi-objective solution of the Pareto front for a certain class of discrete-time ergodic controllable Markov chains. For solving the problem, we transform the original multi-objective problem into an equivalent nonlinear programming problem implementing the Lagrange principle. We then propose to adopt the Tikhonov’s regularization method to ensure the convergence of the cost functions to a unique point into the Pareto front. We show that Pareto policies are characterized as optimal policies. One of the fundamental problems related to the construction of the Pareto front is the existence and characterization of both, Pareto and strong Pareto policies. By introducing the Tikhonov’s regularizator, we ensure the existence of strong Pareto policies. This paper proposes a real solution to this problem. We formulate the original problem considering several constraints: (a) we employ the c-variable method for the introduction of linear constraints over the nonlinear problem and (b) restrict the cost functions, allowing points in the Pareto front to have a small distance from one another. The constraints imposed by the c-variable method make the problem computationally tractable and the restriction imposed by the small distance change ensures the continuation of the Pareto front. The resulting equation in this nonlinear system is an optimization problem for which the necessary and efficient condition of a minimum is solved using the projected gradient method. Moreover, we also prove the convergence conditions and compute the estimate rate of convergence of variables corresponding to the Lagrange principle and the Tikhonov’s regularization, respectively. We provide all the details needed to implement the proposed method in an efficient way. The usefulness of the method is successfully demonstrated by a numerical example.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

A deterministic method for PWR loading pattern optimization

A new deterministic approach to core loading optimization is presented. Selection of fuel enrichment and burnable poison (BP) arrangement in a pressurized water reactor (PWR) core are optimally determined subject to power peaking factor (PF) constraints. The deterministic approach is based on the method of Lagrange multipliers and the direct adjoining approach for the inequality power PF constraint. The optimality conditions are derived using calculus of variations resulting in the presence of jump conditions and a linear control Hamiltonian. The optimal control problems are addressed by simultaneously minimizing the Hamiltonian using the gradient method and performing Newton's method in different regions of the core to produce viable solutions for the controls. Without active control of the power profile in PWRs during depletion, our methodology is able to satisfy the power PF constraint at all times by proper loading selection at the beginning of cycle (BOC). This is achieved by promoting the proper burnup path forward in time during the calculation of the forward optimality conditions. Here this information is utilized in the adjoint burnup calculation, which acts to propagate the optimal control path back to the BOC during the calculation of the adjoint optimality conditions. The methodology has been developed into a new code DMCO (Deterministic Multi-Control Optimization) which can produce two-dimensional two-group depletion results in approximately seven minutes per optimization iteration on a personal computer. No application of heuristics is required and the code is capable of finding solutions from initial control distributions. A preliminary case study is presented for the Westinghouse AP600 first cycle fuel loading design which achieved an extended fuel cycle while satisfying power PF constraints.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

NASA NTRS1996

Fast Multigrid Techniques in Total Variation-Based Image Reconstruction

Existing multigrid techniques are used to effect an efficient method for reconstructing an image from noisy, blurred data. Total Variation minimization yields a nonlinear integro-differential equation which, when discretized using cell-centered finite differences, yields a full matrix equation. A fixed point iteration is applied with the intermediate matrix equations solved via a preconditioned conjugate gradient method which utilizes multi-level quadrature (due to Brandt and Lubrecht) to apply the integral operator and a multigrid scheme (due to Ewing and Shen) to invert the differential operator. With effective preconditioning, the method presented seems to require Omicron(n) operations. Numerical results are given for a two-dimensional example.

Oman, Mary Ellen↗

NASA NTRS1996

Quasi-Optimal Schwarz Methods for the Conforming Spectral Element Discretization

Fast methods are proposed for solving the system K(sub N)x = b resulting from the discretization of self-adjoint elliptic equations in three dimensional domains by the spectral element method. The domain is decomposed into hexahedral elements, and in each of these elements the discretization space is formed by polynomials of degree N in each variable. Gauss-Lobatto-Legendre (GLL) quadrature rules replace the integrals in the Galerkin formulation. This system is solved by the preconditioned conjugate gradients method. The conforming finite element space on the GLL mesh consisting of piecewise Q(sub 1) elements produces a stiffness matrix K(sub h) that is spectrally equivalent to the spectral element stiffness matrix K(sub N). The action of the inverse of K(sub h) is expensive for large problems, and is therefore replaced by a Schwarz preconditioner B(sub h) of this finite element stiffness matrix. The preconditioned operator then becomes B(sub h)(exp -l)K(sub N). The technical difficulties stem from the nonregularity of the mesh. Tools to estimate the convergence of a large class of new iterative substructuring and overlapping Schwarz preconditioners are developed. This technique also provides a new analysis for an iterative substructuring method proposed by Pavarino and Widlund for the spectral element discretization.

Casarin, Mario↗

DOE OSTI2016

Bioluminescence Tomography–Guided Radiation Therapy for Preclinical Research

Purpose: In preclinical radiation research, it is challenging to localize soft tissue targets based on cone beam computed tomography (CBCT) guidance. As a more effective method to localize soft tissue targets, we developed an online bioluminescence tomography (BLT) system for small-animal radiation research platform (SARRP). We demonstrated BLT-guided radiation therapy and validated targeting accuracy based on a newly developed reconstruction algorithm. Methods and Materials: The BLT system was designed to dock with the SARRP for image acquisition and to be detached before radiation delivery. A 3-mirror system was devised to reflect the bioluminescence emitted from the subject to a stationary charge-coupled device (CCD) camera. Multispectral BLT and the incomplete variables truncated conjugate gradient method with a permissible region shrinking strategy were used as the optimization scheme to reconstruct bioluminescent source distributions. To validate BLT targeting accuracy, a small cylindrical light source with high CBCT contrast was placed in a phantom and also in the abdomen of a mouse carcass. The center of mass (CoM) of the source was recovered from BLT and used to guide radiation delivery. The accuracy of the BLT-guided targeting was validated with films and compared with the CBCT-guided delivery. In vivo experiments were conducted to demonstrate BLT localization capability for various source geometries. Results: Online BLT was able to recover the CoM of the embedded light source with an average accuracy of 1 mm compared to that with CBCT localization. Differences between BLT- and CBCT-guided irradiation shown on the films were consistent with the source localization revealed in the BLT and CBCT images. In vivo results demonstrated that our BLT system could potentially be applied for multiple targets and tumors. Conclusions: The online BLT/CBCT/SARRP system provides an effective solution for soft tissue targeting, particularly for small, nonpalpable, or orthotopic tumor models.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

DOE OSTI2025

Improved Guarantees for Optimal Nash Equilibrium Seeking and Bilevel Variational Inequalities

We consider a class of hierarchical variational inequality (VI) problems that subsumes VI-constrained optimization and several other problem classes, including the optimal solution selection problem and the optimal Nash equilibrium (NE) seeking problem. Our main contribution is threefold. (i) We consider bilevel VIs with monotone and Lipschitz continuous mappings and devise a single-timescale iteratively regularized extragradient method, named IR-EG 𝚖,𝚖 . We improve the existing iteration complexity results for addressing both bilevel VI and VI-constrained convex optimization problems. (ii) Under the strong monotonicity of the outer-level mapping, we develop a method named IR-EG 𝚜,𝚖 and derive faster guarantees than those in (i). We also study the iteration complexity of this method under a constant regularization parameter. These results appear to be new for both bilevel VIs and VI-constrained optimization. (iii) To our knowledge, complexity guarantees for computing the optimal NE in nonconvex settings do not exist. Motivated by this lacuna, we consider VI-constrained nonconvex optimization problems and devise an inexactly projected gradient method, named IPR-EG, where the projection onto the unknown set of equilibria is performed using IR-EG 𝚜,𝚖 with a prescribed termination criterion and an adaptive regularization parameter. We obtain new complexity guarantees in terms of a residual map and an infeasibility metric for computing a stationary point. Here, we validate the theoretical findings using preliminary numerical experiments for computing the best and the worst NEs.

bilevel optimization↗

NASA NTRS1990

Parallel processors and nonlinear structural dynamics algorithms and software

Techniques are discussed for the implementation and improvement of vectorization and concurrency in nonlinear explicit structural finite element codes. In explicit integration methods, the computation of the element internal force vector consumes the bulk of the computer time. The program can be efficiently vectorized by subdividing the elements into blocks and executing all computations in vector mode. The structuring of elements into blocks also provides a convenient way to implement concurrency by creating tasks which can be assigned to available processors for evaluation. The techniques were implemented in a 3-D nonlinear program with one-point quadrature shell elements. Concurrency and vectorization were first implemented in a single time step version of the program. Techniques were developed to minimize processor idle time and to select the optimal vector length. A comparison of run times between the program executed in scalar, serial mode and the fully vectorized code executed concurrently using eight processors shows speed-ups of over 25. Conjugate gradient methods for solving nonlinear algebraic equations are also readily adapted to a parallel environment. A new technique for improving convergence properties of conjugate gradients in nonlinear problems is developed in conjunction with other techniques such as diagonal scaling. A significant reduction in the number of iterations required for convergence is shown for a statically loaded rigid bar suspended by three equally spaced springs.

Belytschko, Ted↗

NASA NTRS1984

The numerical solution of the Helmholtz equation for wave propagation problems in underwater acoustics

The Helmholtz Equation (-delta-K(2)n(2))u=0 with a variable index of refraction, n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. A numerical algorithm was developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range. The equation is discretized using the finite element method, thus allowing for the modeling of complicated geometrices (including interfaces) and complicated boundary conditions. A global radiation boundary condition is imposed at the far field boundary that is exact for an arbitrary number of propagating modes. The resulting large, non-selfadjoint system of linear equations with indefinite symmetric part is solved using the preconditioned conjugate gradient method applied to the normal equations. A new preconditioner is developed based on the multigrid method. This preconditioner is vectorizable and is extremely effective over a wide range of frequencies provided the number of grid levels is reduced for large frequencies. A heuristic argument is given that indicates the superior convergence properties of this preconditioner.

Bayliss, A.↗

NASA NTRS1990

Multigrid solution of unsteady Navier-Stokes equations using a pressure method

A multigrid relaxation method is applied to a pressure-based implicit procedure to solve unseady, incompressible Navier-Stokes equations. The present multigrid method is a Correction Scheme according to Brandt. This method is used to solve the scalar matrices resulting from the finite-volume formulation and uses flux averaging as the restriction operator. The accuracy and computational efficiency are demonstrated with a steady state driven cavity flow and an unsteady flow over a circular cylinder case. The results are compared with single grid results using the OrthoMin conjugate gradient method and experimental data.

Jiang, Y.↗

DOE OSTI2012

Development and Clinical Evaluation of a Three-Dimensional Cone-Beam Computed Tomography Estimation Method Using a Deformation Field Map

Purpose: To develop a three-dimensional (3D) cone-beam computed tomography (CBCT) estimation method using a deformation field map, and to evaluate and optimize the efficiency and accuracy of the method for use in the clinical setting. Methods and Materials: We propose a method to estimate patient CBCT images using prior information and a deformation model. Patients' previous CBCT data are used as the prior information, and the new CBCT volume to be estimated is considered as a deformation of the prior image volume. The deformation field map is solved by minimizing deformation energy and maintaining new projection data fidelity using a nonlinear conjugate gradient method. This method was implemented in 3D form using hardware acceleration and multi-resolution scheme, and it was evaluated for different scan angles, projection numbers, and scan directions using liver, lung, and prostate cancer patient data. The accuracy of the estimation was evaluated by comparing the organ volume difference and the similarity between estimated CBCT and the CBCT reconstructed from fully sampled projections. Results: Results showed that scan direction and number of projections do not have significant effects on the CBCT estimation accuracy. The total scan angle is the dominant factor affecting the accuracy of the CBCT estimation algorithm. Larger scan angles yield better estimation accuracy than smaller scan angles. Lung cancer patient data showed that the estimation error of the 3D lung tumor volume was reduced from 13.3% to 4.3% when the scan angle was increased from 60 Degree-Sign to 360 Degree-Sign using 57 projections. Conclusions: The proposed estimation method is applicable for 3D DTS, 3D CBCT, four-dimensional CBCT, and four-dimensional DTS image estimation. This method has the potential for significantly reducing the imaging dose and improving the image quality by removing the organ distortion artifacts and streak artifacts shown in images reconstructed by the conventional Feldkamp-Davis-Kress (FDK) algorithm.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

NASA NTRS1993

Two-dimensional radiative transfer in cloudy atmospheres - The spherical harmonic spatial grid method

A new two-dimensional monochromatic method that computes the transfer of solar or thermal radiation through atmospheres with arbitrary optical properties is described. The model discretizes the radiative transfer equation by expanding the angular part of the radiance field in a spherical harmonic series and representing the spatial part with a discrete grid. The resulting sparse coupled system of equations is solved iteratively with the conjugate gradient method. A Monte Carlo model is used for extensive verification of outgoing flux and radiance values from both smooth and highly variable (multifractal) media. The spherical harmonic expansion naturally allows for different levels of approximation, but tests show that the 2D equivalent of the two-stream approximation is poor at approximating variations in the outgoing flux. The model developed here is shown to be highly efficient so that media with tens of thousands of grid points can be computed in minutes. The large improvement in efficiency will permit quick, accurate radiative transfer calculations of realistic cloud fields and improve our understanding of the effect of inhomogeneity on radiative transfer in cloudy atmospheres.

Evans, K. F.↗

DOE OSTI2016

Nonlinear bioheat transfer models and multi-objective numerical optimization of the cryosurgery operations

Numerical computation of the three dimensional problem of the freezing interface propagation during the cryosurgery coupled with the multi-objective optimization methods is used in order to improve the efficiency and safety of the cryosurgery operations performing. Prostate cancer treatment and cutaneous cryosurgery are considered. The heat transfer in soft tissue during the thermal exposure to low temperature is described by the Pennes bioheat model and is coupled with an enthalpy method for blurred phase change computations. The finite volume method combined with the control volume approximation of the heat fluxes is applied for the cryosurgery numerical modeling on the tumor tissue of a quite arbitrary shape. The flux relaxation approach is used for the stability improvement of the explicit finite difference schemes. The method of the additional heating elements mounting is studied as an approach to control the cellular necrosis front propagation. Whereas the undestucted tumor tissue and destucted healthy tissue volumes are considered as objective functions, the locations of additional heating elements in cutaneous cryosurgery and cryotips in prostate cancer cryotreatment are considered as objective variables in multi-objective problem. The quasi-gradient method is proposed for the searching of the Pareto front segments as the multi-objective optimization problem solutions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1996

Numerical Modeling of Spray Combustion with an Unstructured-Grid Method

The present unstructured-grid method follows strictly the basic finite volume forms of the conservation laws of the governing equations for the entire flow domain. High-order spatially accurate formulation has been employed for the numerical solutions of the Navier-Stokes equations. A two-equation k-epsilon turbulence model is also incorporated in the unstructured-grid solver. The convergence of the resulted linear algebraic equation is accelerated with preconditioned Conjugate Gradient method. A statistical spray combustion model has been incorporated into the present unstructured-grid solver. In this model, spray is represented by discrete particles, rather than by continuous distributions. A finite number of computational particles are used to predict a sample of total population of particles. Particle trajectories are integrated using their momentum and motion equations and particles exchange mass, momentum and energy with the gas within the computational cell in which they are located. The interaction calculations are performed simultaneously and eliminate global iteration for the two-phase momentum exchange. A transient spray flame in a high pressure combustion chamber is predicted and then the solution of liquid-fuel combusting flow with a rotating cup atomizer is presented and compared with the experimental data. The major conclusion of this investigation is that the unstructured-grid method can be employed to study very complicated flow fields of turbulent spray combustion. Grid adaptation can be easily achieved in any flow domain such as droplet evaporation and combustion zone. Future applications of the present model can be found in the full three-dimensional study of flow fields of gas turbine and liquid propulsion engine combustion chambers with multi-injectors.

Shang, H. M.↗

DOE OSTI2016

SU-G-BRB-05: Automation of the Photon Dosimetric Quality Assurance Program of a Linear Accelerator

Purpose: To develop an automated method to calculate a linear accelerator (LINAC) photon radiation field size, flatness, symmetry, output and beam quality in a single delivery for flattened (FF) and flattening-filter-free (FFF) beams using an ionization chamber array. Methods: The proposed method consists of three control points that deliver 30×30, 10×10 and 5×5cm{sup 2} fields (FF or FFF) in a step-and-shoot sequence where the number of monitor units is weighted for each field size. The IC Profiler (Sun Nuclear Inc.) with 5mm detector spacing was used for this study. The corrected counts (CCs) were calculated and the locations of the maxima and minima values of the first-order gradient determined data of each sub field. Then, all CCs for each field size are summed in order to obtain the final profiles. For each profile, the radiation field size, symmetry, flatness, output factor and beam quality were calculated. For field size calculation, a parameterized gradient method was used. For method validation, profiles were collected in the detector array both, individually and as part of the step-and-shoot plan, with 9.9cm buildup for FF and FFF beams at 90cm source-to-surface distance. The same data were collected with the device (plus buildup) placed on a movable platform to achieve a 1mm resolution. Results: The differences between the dosimetric quantities calculated from both deliveries, individually and step-and-shoot, were within 0.31±0.20% and 0.04±0.02mm. The differences between the calculated field sizes with 5mm and 1mm resolution were ±0.1mm. Conclusion: The proposed single delivery method proved to be simple and efficient in automating the photon dosimetric monthly and annual quality assurance.

61 RADIATION PROTECTION AND DOSIMETRY↗

DOE OSTI2019

On the Convergence of the Inexact Running Krasnosel'skii-Mann Method

This letter leverages a framework based on averaged operators to tackle the problem of tracking fixed points associated with maps that evolve over time. In particular, this letter considers the Krasnosel'skii-Mann (KM) method in a settings where: 1) the underlying map may change at each step of the algorithm, thus leading to a 'running' implementation of the KM method, and 2) an imperfect information of the map may be available. An imperfect knowledge of the maps can capture cases where processors feature a finite precision or quantization errors, or the case where (part of) the map is obtained from measurements. The analytical results are applicable to inexact running algorithms for solving optimization problems, whenever the algorithmic steps can be written in the form of (a composition of) averaged operators; examples are provided for inexact running gradient methods and the forward-backward splitting method. Convergence of the average fixed-point residual is investigated for the non-expansive case; linear convergence to a unique fixed-point trajectory is showen in the case of inexact running algorithms emerging from contractive operators.

97 MATHEMATICS AND COMPUTING↗