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DOE OSTI2022

String correlators on $\text{AdS}_3$: Analytic structure and dual CFT

We continue our study of string correlators on Euclidean \text{AdS}_3 AdS 3 with pure NS-NS flux. The worldsheet and spacetime correlators have a rich analytic structure, which we analyse completely for genus 0 four-point functions. We show that correlators exhibit a simple behaviour near their singularities. The spacetime correlators are meromorphic functions in the \mathrm{SL}(2,\mathbb{R}) S L ( 2 , ℝ ) -spins, whose pole structure is shown to agree with the prediction of a recent proposal for the dual \text{CFT}_2 CFT 2 . Moreover, we also compute the residues of the spacetime correlators for some of the poles exactly and find again a perfect match with the proposal for the dual \text{CFT}_2 CFT 2 , thereby checking the duality for some non-trivial four-point functions exactly. Our computations simplify drastically in the tensionless limit of \mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathbb{T}^4 A d S 3 × S 3 × 𝕋 4 where the behaviour near the poles gives in fact the exact answer. This paper is the third in a series with several installments.

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2024

Geometric surprises in the Python's lunch conjecture

A bulge surface, on a time reflection-symmetric Cauchy slice of a holographic spacetime, is a non-minimal extremal surface that occurs between two locally minimal surfaces homologous to a given boundary region. According to the python’s lunch conjecture of Brown et al., the bulge’s area controls the complexity of bulk reconstruction, in the sense of the amount of post-selection that needs to be overcome for the reconstruction of the entanglement wedge beyond the outermost extremal surface. We study the geometry of bulges in a variety of classical spacetimes, and discover a number of surprising features that distinguish them from more familiar extremal surfaces such as Ryu-Takayanagi surfaces: they spontaneously break spatial isometries, both continuous and discrete; they are sensitive to the choice of boundary infrared regulator; they can self-intersect; and they probe entanglement shadows, orbifold singularities, and compact spaces such as the sphere in AdS _p× S^q p × S q . These features imply, according to the python’s lunch conjecture, novel qualitative differences between complexity and entanglement in the holographic context. We also find, surprisingly, that extended black brane interiors have a non-extensive complexity; similarly, for multi-boundary wormhole states, the complexity pleateaus after a certain number of boundaries have been included.

Physics↗

DOE OSTI2022

Zero modes and matching for the twist-3 PDFs

The quasi-PDF approach, proposed by X. Ji in 2013, has made it possible to directly extract light-cone PDFs from lattice QCD. This approach relies on the extraction of matrix elements of space-like operators for fast-moving hadrons. Quasi-PDFs can be related to the light-cone PDFs through a perturbatively calculable matching coefficient. We explore the formalism of matching, for the very first time, for the twist-3 PDFs g_{T}(x) g T ( x ) , e(x) e ( x ) and h_{L}(x) h L ( x ) . In this work, we address the non-trivialities involved in the extraction of the matching coefficient due to the presence of (singular) zero-mode contributions.

Bhattacharya, Shohini↗

DOE OSTI2020

Analysis of the performance of different culvert boundary conditions in 2D shallow flow models

Culverts allow roads to safely traverse small streams or drainage ditches, and their proper design is critical to ensure a safe and reliable transportation network. A correct modelization of these hydraulic structures becomes crucial in the assessment of flood footprints or discharge peak estimation in a risk evaluation plan. The question of how to include culverts comes up frequently when assembling a hydraulic model that requires the presence of as many singular elements as possible. In this work, three different culvert integrations with the surface domain are studied and compared in the context of a 2D shallow water (SW) model. All of them are based on the Federal Highway Administration (FHWA) formulation for the culvert discharge estimation but differ in complexity and in the interaction with the numerical model for surface flow, some of them as internal boundary conditions. Several steady and unsteady validation test cases are presented and the numerical results are compared with the predictions from HEC-RAS 1D and HY-8 software. Furthermore, the culvert area, shape and their sensitivity to the 2D computational mesh is also analyzed.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

DOE Energy Frontier Research Centers Center for Direct Catalytic Conversion of Biomass to Biofuels (C3Bio)

New capabilities to predict, design and control the chemistries of carbon could answer a global imperative to transition from fossil-based to sustainable transportation fuels. While the use of inexpensive hydrocarbons has been an unparalleled achievement and enabler of economic prosperity for many nations, singular dependence upon crude oil has given rise to systemic vulnerabilities in climate, energy, economic, and national security. Lignocellulosic biomass, a renewable and carbon-neutral resource, has the potential to displace an estimated annual equivalent of three billion barrels of oil in the U.S. alone (National Research Council 2009, U. S. Department of Energy, 2011). However, biomass has only one-third the energy density of crude oil (Agrawal and Singh 2009, Richard 2010) and lacks petroleum’s versatility as a feedstock for fuels and chemicals. These limitations keep biomass conversion below the efficiency level needed for strategic impact while the scientific challenge of routing carbon from one molecular context to another remains unmet. In 2009, the Center for Direct Catalytic Conversion of Biomass to Biofuels (C3Bio) recognized the potential of chemical catalysis and fast pyrolysis to overcome such limitations by transforming the main components of biomass (cellulose, xylan, and lignin) from grasses and trees directly to liquid hydrocarbons and aromatic co-products. Enabled by the EFRC high-risk, high-reward approach to grand challenge science, C3Bio researchers have been key national players in disrupting the conventional paradigm of the cellulosic biorefinery into a new future of “no carbon left behind”—the full utilization of carbon from plant cell walls in energy-dense fuels (Fig. 1). We identified catalytic and fast-pyrolytic pathways that utilize cellulose, xylan and, most significantly, lignin. We developed catalytic processes that deoxygenate and transform monomers and isolated polymers into useful products and tested their use with intact biomass. We gained control of lignin synthesis within plants and initiated tailoring biomass to its end-use through the tools of plant molecular biology and genetic engineering. C3Bio breakthroughs have increased the energy density of biomass-derived substrates via catalytic and pyrolytic conversions into products such as benzoquinones, furfural and hydroxymethylfurfural, levoglucosan and levulinic acid, methoxypropylphenols and propylbenzene. Such advances in biomass conversion would not have been possible without simultaneous advances in analytical instrumentation and methodologies, and imaging technologies and applications. The legacy science developed by C3Bio enables design and control strategies for achieving a targeted product portfolio of fuel and chemical feedstocks from a diverse range of native and tailored biomass. Our research provides the knowledge base required for a bio-economy with product streams as diverse in functionality as those of the petrochemical industry. Coupling targeted computational modeling with experimentation, we achieved: (1) fundamental understanding of biopolymers and cell wall architecture assembly, (2) discovery of new chemistries that allow the development of highly selective pathways to fuels and desirable chemicals, and (3) an integrated systems-level understanding to control catalytic and pyrolytic pathways.

09 BIOMASS FUELS↗

DOE OSTI2020

Numerical methods for nonlocal and fractional models

Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDE models fail to adequately model observed phenomena or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article, we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis, and specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference, and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modeling and algorithmic extensions which serve to show the wide applicability of nonlocal modeling.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

The Nuclei Of Atoms In The Third Period Of The Periodic Table Of Elements (Monograph No.3)

Nuclei are shown for the elements sodium, magnesium, aluminum, silicon, phosphorus, sulphur, chlorine, and argon. Individual nucleons are not identified as protons or neutrons. Nucleons are illustrated using a sphere schematic of the wave functions of singular, linear, and tetrahedral nucleon arrangements. Nucleon arrangements in protium, deuteron, triton, helion, and alpha are shown as building blocks in the nuclei of elements in the third period of the periodic table of elements.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2020

Advanced Modular Sub-Atmospheric Hybrid Heat Engine (Final Report)

The Phase 1 Final Technical Report describes the results of the work completed during the “Advanced Modular Sub-Atmospheric Hybrid Heat Engine” project. A key part of the Phase 1 work was the completion of a thermodynamic cycle analysis for the MHHE at the selected module size. The hybrid heat engine has been developed as a modular unit (sized in the range of 500kW – 60MW) that can be used with modular coal or biomass gasifiers, with distributed power generation systems, with large power plants comprised of multiple generating units, and with natural gas compression stations. The MHHE will provide cleaner, more efficient, and lower cost generation with better load following capabilities than existing competing technologies with a singular generating source such as solar farm, gas turbine, or combustion engine. The drivers of the MHHE technology are: benefits of modular power generation (reduced equipment cost, construction cost and implementation time, connection ready on delivery, flexible scalability, serviceability), fuel flexibility, lowest cost power generation and reduced emissions.A logical progression of work and a clear path forward toward meeting the FOA goals and objectives have been established. Namely, a preliminary market analysis and primary fuel identification was completed first and then the modularity of the system was defined. The benefits of the proposed hybrid and modular heat engine were described when applied to modular coal gasifiers, distributed power generators, and larger power plants. Based on the market analysis, modularity, and chosen primary fuel, a conceptual design and layout of the hybrid heat engine was developed, analyzed, and characterized. The technology gaps already identified have been reviewed and expanded upon, and a test plan to address these gaps through bench scale testing in Phase 2 has been developed. Cost estimate methodology and considerations in support of a potential Phase 2 project have been described.

08 HYDROGEN↗

DOE OSTI2020

High Order Implicit Residual-Based Spatial Discretization Error Estimation for S N Neutron Transport

This work demonstrates our novel residual source spatial discretization error estimator (LeR/TEAD) for a DGFEM-1 discretization and assesses it along with two contemporary estimators, Ragusa and Wang's h -refinement estimator (RW) and Duo, Azmy, and Zikatanov's explicit residual-based estimator (DAZ), on a suite of Method of Manufactured Solutions (MMS) 2D problems and three realistic problem geometries. LeR/TE-AD is attractive because it directly estimates the local error in the angular flux, as opposed to a mere indicator of the error's behavior, on the same mesh and method order as the original numerical solution, thus typically being less computationally intensive than a refinement-based method. On the MMS suite, LeR/TE-AD consistently displayed a reduced performance versus its DGFEM-0 results in terms of accuracy and precision metrics, though it was not typically grossly inaccurate. This is attributed to the irregularities in the true solution across singular characteristics limiting the local accuracy of the numerical flux solution, leading to poor derivative approximations used in the residual approximations. The error transport problem then spreads the error in the residual to nearby cells, causing a greater degree of imprecision that did not afflict DAZ or RW. In testing the estimators on realistic problem geometries, however, LeR/TE-AD fared better. In practice, the true error is much larger in non-idealized geometries like in MMS, and a superlinear true solution means that RW and DAZ are not beneficially biased for DGFEM-1 error estimation. LeR/TE-AD was typically first or second in accuracy, primarily competing with RW, but the latter usually consumed 2-4 times the computational time as LeR/TE-AD, and requires a solution with four times as many unknowns. Furthermore, RW and LeR/TE-AD can be used to compute direct estimates of the error in any quantity of interest that is based on the angular ux solution, such as the fission rate density in a fuel pin, whereas DAZ requires a heuristic extension due to its norm-based nature.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Development of Thermal Tomography Imaging System for In-Service Nondestructive Evaluation of Additively Manufactured Components

36 MATERIALS SCIENCE↗

DOE OSTI2020

Pulsed Thermal Tomography Nondestructive Examination of Additively Manufactured Reactor Materials (Second Annual Progress Report)

Additive manufacturing (AM) is an emerging method for cost-efficient fabrication of nuclear reactor parts. AM of metallic structures for nuclear energy applications is currently based on laser powder bed fusion (LPBF) process, which can introduce internal material flaws, such as pores and anisotropy. Integrity of AM structures needs to be evaluated nondestructively because material flaws could lead to premature failures due to exposure to high temperature, radiation and corrosive environment in a nuclear reactor. Thermal tomography (TT) provides a capability for non-destructive evaluation of sub-surface defects in arbitrary size structures. Thermal tomography is a computational method for heat diffusion-based imaging of solids, which provides 3D visualization of data from flash thermography measurements. We investigate thermal tomography imaging and nondestructive evaluation of stainless steel and nickel super alloy metallic structures produced with laser powder bed fusion (LPBF) additive manufacturing (AM) process. Metallic structures produced with LPBF contain defects, and there are limited capabilities to evaluate these structures non-destructively. Thermal tomography reconstruction of 3D apparent spatial effusivity provides information about AM structure geometry and internal material flaws. We study performance of thermal tomography in imaging of metallic structures through COMSOL computer simulations of transient heat transfer, and through reconstruction of data obtained from experimental measurements. Reconstruction of internal defects is investigated using a stainless steel 316L specimen with flat bottom hole (FBH) indentations, and Inconel 718 plate produced with laser powder bed fusion (LPBF) method, which contains imprinted hemispherical shape low density regions containing non-sintered metallic powder. The FBH’s have the same sizes as the imprinted defects in the LPBF specimens, but offer better imaging contrast. Thermal tomography reconstructions provide visualizations of internal defects, and allow for estimation of their sizes and locations. Detection sensitivity of TT is limited by noises. We investigate separation of signal from noise in thermography images using several machine learning (ML) methods, including new spatio-temporal blind source separation (STBSS) and spatio-temporal sparse dictionary learning (STSDL) methods. Performance of the ML methods is benchmarked using thermography data obtained from imaging stainless steel 316L and Inconel 718 specimens produced LPBF method with imprinted calibrated porosity defects. The ML methods are ranked by F-score and execution runtime. Finally, we investigate TT of AM stainless steel 316L specimen with imprinted internal porosity defects using relatively low-cost, small form factor infrared (IR) camera based on uncooled micro bolometer detector. Sparse coding related K-means singular value decomposition (SVD) machine learning, image processing algorithms are developed to improve quality of TT images through removal of Additive white Gaussian noise without blurring the images. Following initial qualification of an AM component for deployment in a nuclear reactor, a compact TT can also be used for in-service nondestructive evaluation (NDE). With capability to perform in-service NDE of the AM component lifecycle, TT data can be used for development of a component digital twin.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2021

Computational Tools for Additive Manufacture of Tailored Microstructure and Properties

Additive manufacturing has the potential to revolutionize industrial hardware and unlock efficiency gains through the fabrication of geometries and architectures not possible by conventional processing. Currently most additive builds use a single set of process parameters (e.g. laser power and scan speed) which results in a part with a homogenous microstructure that provides a singular performance level. To move beyond this state, Raytheon Technologies Research Center worked to create a set of computational tools to track material evolution through each step of the additive process. Computational fluid dynamics and phase field models for microstructure evolution as a function of processing parameters, and crystal plasticity models fully coupling microstructure and mechanical properties for performance predictions were leveraged to establish a connection between additive parameters and the final microstructure. This framework was utilized to tailor spatially-varying mechanical properties in a part by appropriately controlling the microstructure evolution during the additive process. Specifically, a turbine blade was 3D printed from nickel superalloy IN718 using laser powder bed fusion with coarse grains in the airfoil section which experiences the highest temperatures and is creep limited while finer grains were printed in the root of the blade which experience higher stresses but at lower temperatures and is therefore fatigue limited. The benefit of being able to intentionally insert coarse grains in the high temperature region of the blade was showcased with a microstructure sensitive creep model that indicates longer creep life for coarser grains.

20 FOSSIL-FUELED POWER PLANTS↗

DOE OSTI2021

CSPlib - A Software Toolkit for the Analysis of Dynamical Systems and Chemical Kinetic Models

CSPlib is an open source software library for analyzing general ordinary differential equation (ODE) systems and detailed chemical kinetic ODE systems. It relies on the computational singular perturbation (CSP) method for the analysis of these systems. The software provides support for: General ODE models (gODE model class) for computing source terms and Jacobians for a generic ODE system; TChem model (ChemElemODETChem model class) for computing source term, Jacobian, other necessary chemical reaction data, as well as the rates of progress for a homogenous batch reactor using an elementary step detailed chemical kinetic reaction mechanism. This class relies on the TChem [2] library; A set of functions to compute essential elements of CSP analysis (Kernel class). This includes computations of the eigensolution of the Jacobian matrix, CSP basis vectors and co-vectors, time scales (reciprocals of the magnitudes of the Jacobian eigenvalues), mode amplitudes, CSP pointers, and the number of exhausted modes. This class relies on the Tines library; A set of functions to compute the eigensolution of the Jacobian matrix using Tines library GPU eigensolver; A set of functions to compute CSP indices (Index Class). This includes participation indices and both slow and fast importance indices.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

SU(n) and Quantum SU(n) Symmetries in Physical Systems [Slides]

Presence of SU(n) or other Lie group symmetry in a physical system is its powerful, usually underutilized property. In many cases it allows for finding analytical solutions to nonlinear differential equations describing this system. Power of the method is presented on diversified examples from mathematical physics: Lie-group symmetries in finding solutions of generalized, multidimensional theory of gravity; analytical Dirac–equation solutions for description of conducting polymers; stability of qubit states in quantum computers; spatial defects in condensed matter; reconstruction of 3D object from its 2D tomographic image; significant improvement of numerical solutions stability for Euler equations. The next question after obtaining such Lie group symmetric solution is: does a generalized solution with appropriate quantum group symmetry exists for the given physical system, and if yes what is the physical meaning of the deformation parameter q introduced by such solution. In many cases it can be identified. Any SU(n) solution is by its nature singular, assuming a perfect symmetry of the physical system discussed. Such solution gives a powerful insight to theoretical physics, yet the assumption may be too demanding for experimental applications. Deformation parameter q from a quantum group symmetry allows for a continuum of solutions, more applicable to experiments.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Modeling of Glycolate Destruction in the Recycle Collection Tank

The Savannah River Site’s DWPF is being upgraded with the introduction of the NG flowsheet. Glycolic acid has been shown to be a superior alternative to formic acid for sludge processing. The new flowsheet improves or maintains necessary parameters such as 1) reduction of mercury, 2) adjustment of feed rheology, 3) pH stability, and 4) adjustment of melter oxidation/reduction potential. Further, the use of glycolic acid virtually eliminates the potential for catalytic hydrogen generation in DWPF processing DWPF process condensates are collected and returned to the Savannah River Site (SRS) CSTF. The RCT collects off-gas condensate during chemical processing, vitrification, and other unit operations performed in DWPF and is the singular return vessel delivering recycle effluent back to CSTF. Each batch of recycle will have a small amount of glycolate from chemical processing and melter off-gas condensates. To avoid potential flammability issues due to thermolysis of glycolate in the CSTF, Savannah River National Laboratory (SRNL) provided to Savannah River Remediation (SRR) at their request a Task Technical and Quality Assurance Plan (TTQAP) to quantify and mitigate glycolate returns via DWPF’s recycle stream. The request included testing of a process to oxidize glycolate and other organic species that are responsible for hydrogen generation from thermolysis. Following that work SRR provided a Task Technical Request (TTR) that requested process modeling. In 2021 a TTQAP was issued to cover the modeling work. Modeling draws data from laboratory scale studies using chemical simulants and radioactive waste samples. Chemical kinetic modeling was performed to evaluate the feasibility of using sodium permanganate to destroy glycolate in the RCT. The results from the laboratory studies were summarized in a series of reports. Reference 9 is a report of lab scale processing of actual DWPF Slurry Mix Evaporate Condensate Tank (SMECT) and Offgas Condensate Tank (OGCT) samples in the SRNL Shielded Cells. Tests at caustic conditions demonstrated sodium permanganate was effective in converting glycolate to oxalate, and permanganate (Mn 7+ ) is reduced to manganate (Mn 6+ ) with no significant formation of carbon dioxide or carbonate. Equation (1) was found to best describe the observed reaction of glycolate with permanganate under nominal (60 to 145 mg/L in RCT) glycolate entrainment conditions.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

DOE OSTI2021

Recommendations for Distributed Energy Resource Patching

While computer systems, software applications, and operational technology (OT)/Industrial Control System (ICS) devices are regularly updated through automated and manual processes, there are several unique challenges associated with distributed energy resource (DER) patching. Millions of DER devices from dozens of vendors have been deployed in home, corporate, and utility network environments that may or may not be internet-connected. These devices make up a growing portion of the electric power critical infrastructure system and are expected to operate for decades. During that operational period, it is anticipated that critical and noncritical firmware patches will be regularly created to improve DER functional capabilities or repair security deficiencies in the equipment. The SunSpec/Sandia DER Cybersecurity Workgroup created a Patching Subgroup to investigate appropriate recommendations for the DER patching, holding fortnightly meetings for more than nine months. The group focused on DER equipment, but the observations and recommendations contained in this report also apply to DERMS tools and other OT equipment used in the end-to-end DER communication environment. The group found there were many standards and guides that discuss firmware lifecycles, patch and asset management, and code-signing implementations, but did not singularly cover the needs of the DER industry. This report collates best practices from these standards organizations and establishes a set of best practices that may be used as a basis for future national or international patching guides or standards.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

AOI [1]: Passive Wireless Sensor Systems Fabricated by Direct-Writing for Temperature and Health Monitoring of Energy Systems in Harsh-Environments (Final Report)

Researchers at West Virginia University (WVU) propose to demonstrate a wireless, high-temperature sensor system for monitoring the temperature and health of energy-system components. The active sensor and electronics for wireless communication will be composed entirely of an electroceramic materials (conductive ceramics) which are capable of withstanding the harsh-environments required for fossil energy-based technologies. This work will focus primarily on the fabrication and testing of temperature (thermocouples and thermistors) and health (strain/stress and crack propagation sensors) that function at extreme temperatures (up to 700-1700ºC). The electronics will accompany the high-temperature sensor, which will include a passive wireless communication circuit that allows the transmission of the data based on the LCR resonance principle to a near-by reader antenna. This passive transmission of data will reduce the need for interconnect wires near the active, and possibly rotating, energy-system component. The research will include process development to permit the 2D/3D direct-writing of the entire sensor and communication circuit onto the energy-system component. The direct-writing will be facilitated by using silicon-based polymer-derived precursors to form the high-temperature, stable electroceramic compositions. In addition, methods for direct-writing the circuit onto a fugitive carrier substrate will be completed, which will permit a “peel-and-stick”-like transfer of the sensor circuit to the energy-system component. This feature will allow the economical and precise placement of the sensor circuit onto components of various shapes and locations, without altering the geometry and active features of the manufactured component, or the removal (or decommissioning) of the component for installation. The proposed work will be directed at the following areas: 1) Investigation of phase formation, sintering/grain growth, and electrical properties of polymer-derived electroceramic composites; 2) Definition of processes to direct-write through ink-jet and robo-casting the polymer-derived electroceramic composites onto oxide and polymer surfaces; 3) Development of methods to form monolithic "peel-and-stick" preforms that will efficiently transfer the sensor circuit to ceramic surfaces after thermal treatment; 4) Design of passive wireless LCR circuits and receiver (reader) antennas for communication and testing at high temperatures; 5) Investigation of the passive wireless sensor system developed (and method of transferring sensor system) for temperature and stress/strain measurements on a SOFC repeat unit and a singular gas turbine blade prototype as example applications.

01 COAL, LIGNITE, AND PEAT↗

DOE OSTI2021

An Overview of Gemma FY2021 Verification Activities

Though the method-of-moments implementation of the electric-field integral equation plays an important role in computational electromagnetics, it provides many code-verification challenges due to the different sources of numerical error and their possible interactions. Matters are further complicated by singular integrals, which arise from the presence of a Green's function. In this report, we document our research to address these issues, as well as its implementation and testing in Gemma.

97 MATHEMATICS AND COMPUTING↗