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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Adapting Code Verification Methodology to Model Form

Code verification is an essential part of credibility analysis for computational models. It assesses whether the mathematical model is implemented correctly into the code and whether the numerical methods behave consistently, and is done before solution verification and validation. Robust guidance for code verification exists in the literature. However, there is no known, concise guide for selecting the approach based on the model form that also presents an overview of the common elements. This document was written to address this gap as an accessible reference for beginning a code-verification effort.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

A Poisson equation method for prescribing fully developed non-Newtonian inlet conditions for computational fluid dynamics simulations in models of arbitrary cross-section

Prescribing inlet boundary conditions for computational fluid dynamics (CFD) simulations of internal flow in complex geometries such as anatomical vascular models is challenging. In the absence of patient-specific inlet velocity data, a common approach for long blood vessels is to assume that the inlet flow is fully developed. In vessels of irregular cross section, however, prescribing fully developed conditions is complicated due to the lack of a general closed-form analytical solution. In this study, we develop a simple Poisson equation method for prescribing fully developed inlet conditions for the flow of either Newtonian or non-Newtonian fluids in CFD models of arbitrary cross-section. We first derive the generalized Poisson equation for fully developed flow of a non-Newtonian fluid and we then develop and verify a methodology for numerically computing the solution on any planar boundary domain. In addition, we develop a simple extension of the method for prescribing a non-orthogonal inlet velocity that represents fully developed flow from an upstream tube that is connected to the CFD inlet at a non-orthogonal angle. This may be used to investigate a common source of uncertainty in CFD simulations of internal flow that is due to a lack of information concerning the exact streamwise flow direction at the inlets. Comparison to several Newtonian and non-Newtonian benchmark verification solutions shows the method to be extremely accurate. As a practical demonstration case, we use the method to prescribe fully developed conditions on multiple non-circular inlets for the non-Newtonian flow of blood in a patient-specific model of the inferior vena cava (IVC). Finally, we further demonstrate the utility of the method by performing a sensitivity study using the patient-specific IVC model, wherein we investigate the influence of inlet velocity flow direction on the non-Newtonian IVC hemodynamics. Given its simplicity and computational efficiency, the method is shown to be far superior to alternative approaches for prescribing fully developed inlet conditions in such complicated geometries. In conclusion, to facilitate the adoption of our Poisson equation method, we have distributed our OpenFOAM source code and the associated test cases from this study as open-source software.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Validation of SAS4A/SASSYS-1 for predicting steady-state single-phase natural circulation

The validation of system analysis codes for nuclear reactor systems is required for the development and application of these computational tools. Designed as a comprehensive system analysis code for advanced nuclear reactors, SAS4A/SASSYS-1 requires validation of its physics model for capturing single-phase natural circulation behavior. To support the validation of SAS4A/SASSYS-1, high-precision experiments are performed capturing steady-state single-phase natural circulation on a scaled facility with comprehensive instrumentation. In this work, dedicated tests are performed quantifying the critical modeling parameters, and a single-phase natural circulation benchmark dataset is obtained with well-documented uncertainty and comprehensive facility description. The validation is then performed against the dataset examining the capability of SAS4A/SASSYS-1 in simulating steady-state single-phase natural circulation. The experimental facility is modeled in the candidate code. Solution verification is performed using Richardson-extrapolation-based estimators which quantify and restrict numerical errors from discretization. Input uncertainty provided by the benchmark dataset is forward propagated through the candidate code, quantifying the output uncertainty in a Monte Carlo approach. The composition of the output uncertainty is also quantified through a variance-based sensitivity analysis. With the uncertainty quantified for each individual condition, a detailed comparison between the simulation results and experimental data is performed covering the whole dataset. The results show consistent agreement for all primary parameters. The current validation activity provides a valuable benchmark dataset for the validation of system analysis codes in capturing single-phase natural circulation and demonstrates satisfactory prediction capability of SAS4A/SASSYS1 for steady-state single-phase natural circulation.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2023

Advanced two-phase subchannel method via non-linear iteration

A fast-running, robust two-phase flow, sub-channel model is presented based on non-linear solution of the steady-state subchannel fluid flow equations. The drift-flux model solves for conservation of liquid and vapor mass, mixture energy, and axial and transverse mixture momentum as part of an efficient planar marching scheme and nonlinear, nested outer and inner iteration. Here, models based on mechanistic subcooled boiling, two-phase turbulent void mixing, and drift are included. Solution verification and mesh convergence studies were performed for modern GE 10 × 10 fuel geometry and are shown to have excellent convergence behavior. Run time performance for a 50 axial mesh model showed 2.2 seconds on a single CPU core to tightly converge all 3D distributions (flow, void, pressure) for the GE 10 × 10 fuel geometry, supporting its efficient use within the Virtual Environment for Reactor Applications boiling water reactor framework.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2025

CFDverify

Estimating the discretization error of computational fluid dynamics (CFD) or other scientific codes as part of solution verification is often a non-trivial part of the analysis process. Methods can be complicated and may include assumptions/qualifications that need to be checked during analysis. CFD analysts, therefore, are likely to make errors when trying to conduct this necessary analysis on their own in not knowing about the best method for their problem, not correctly implementing a method, or in not have diagnostic tools to determine if the method was correctly applied.

Weinmeister, Justin [Oak Ridge National Laboratory↗
DOE OSTI2022

NucMesh: nuclear reactor geometry creation and mesh generation module in NEMoSys

NucMesh is a parameterized geometry and mesh generator for nuclear reactors developed within the Nuclear Energy Modeling System NEMoSys at Illinois Rocstar. NEMoSys is a platform developed for mesh generation, adaptive refinement, and solution verification. NucMesh is implemented to be generalized and extensible with a robust computer aided design engine and multiple mesh generation algorithms for unstructured triangular, quad-dominant, and structured quadrilateral meshing. In this paper, we present the geometric and meshing features of NucMesh. Geometrically objects are constructed bottom-to-top and overlaps are addressed automatically. A sophisticated object tracking algorithm prevents data from being lost for segmented objects. We discuss the primitive objects of circle and polygons that constitute the module and show how they are used with example inputs. Arrays of primitives and arrays of arrays are utilized to build large assemblies of objects. The concept of saved objects is discussed to demonstrate how repetitive objects can be reused easily and augmented in place. Three dimensional meshes can be obtained through mesh extrusion where all materials and side sets are extended to three dimensions. We show that side sets can be defined nearly anywhere within the geometry and can then be applied to the mesh. Finally, example reactor meshes are demonstrated for the Idaho National Laboratory Advanced Test Reactor and Los Alamos National Lab Empire reactor, both of which use control drums that NucMesh handles easily. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2023

Predictive Capability Maturity Model Demonstration for Cylindrical Cavity Coupling Using Gemma in the Next Generation Workflow

The predictive capability maturity model (PCMM) uses the expert elicitation process to generate credibility evidence for a particular analysis. To ensure Gemma has the capability to efficiently produce this credibility evidence, next generation workflows (NGW) are created for the solution verification, calibration/validation, and input uncertainty quantification portions of the PCMM assessment. These workflows are then used on the Higgins cylinder problem, which is representative of applications involving external-to-internal electromagnetic field coupling through a slot. The uncertainties calculated using these workflows are then used to calculate the validation comparison error and the validation uncertainty for the model following the American Society of Mechanical Engineers (ASME) verification and validation (V&V) 20 standard. These workflows will enable analysts to iterate each element of PCMM more efficiently than if completed without using a NGW workflow. An example of this iterative process is shown in Section 7.2.

42 ENGINEERING↗
DOE OSTI2025

The Fluid Dynamics Uncertainty Quantification Challenge Problem: XFOIL vs. MFOIL

Uncertainty quantification (UQ) has become more critical in aerospace engineering due to the growing dependence on computational tools for design optimization and performance analyses of aerospace vehicles. Even though the significance of UQ in assessing the credibility of computational analyses is well recognized, its costs and complexity impede its integration into standard practices, particularly in computational fluid dynamics (CFD) and other fluid analyses. This paper presents a UQ study for low-fidelity computational aerodynamics analyses with XFOIL and mfoil (i.e., the MATLAB version of XFOIL with several implementation modifications); these tools are utilized widely in both research and education. The main contributions of this paper are as follows: 1) improved precision in quantifying the uncertainty of the baseline Monte Carlo results used to benchmark surrogate modeling techniques for UQ, 2) quantification of the effect of the implementation differences between XFOIL and mfoil on solution quantities of interest (QoIs), such as lift and pitching moment coefficients, and 3) development of an open-source UQ library for use with XFOIL and mfoil, which has educational values and helps promote UQ for fluid analyses with aerospace applications. Results and discussions revolve around cases 1-4 of the challenge problem posed by the AIAA Fluid Dynamics Technical Committee’s Uncertainty Quantification Discussion Group (UQDG). In case 3, this work employs CFDverify, an open-source solution verification software, to quantify the discretization error and evaluate the extrapolated QoIs based on the grid convergence index (GCI). This UQ study differentiates itself from previous studies in the rigor of handling baseline Monte Carlo uncertainty and in including mfoil, which is a more accessible alternative to XFOIL. Finally, despite the growing computing power, low-fidelity computational tools remain valuable, such as for aerodynamic shape optimization at Mach numbers below 0.65 and low-to-mid Reynolds numbers.

Lay, Aidan S [University of Tennessee, Knoxville (↗
DOE OSTI2021

Robust verification of stochastic simulation codes

We introduce a robust verification tool for computational codes, which we call Stochastic Robust Extrapolation based Error Quantification (StREEQ). Unlike the prevalent Grid Convergence Index (GCI) [1] method, our approach is suitable for both stochastic and deterministic computational codes and is generalizable to any number of discretization variables. Building on ideas introduced in the Robust Verification [2] approach, we estimate the converged solution and orders of convergence with uncertainty using multiple fits of a discretization error model. In contrast to Robust Verification, we perform these fits to many bootstrap samples yielding a larger set of predictions with smoother statistics. Here, bootstrap resampling is performed on the lack-of-fit errors for deterministic code responses, and directly on the noisy data set for stochastic responses. This approach lends a degree of robustness to the overall results, capable of yielding precise verification results for sufficiently resolved data sets, and appropriately expanding the uncertainty when the data set does not support a precise result. For stochastic responses, a credibility assessment is also performed to give the analyst an indication of the trustworthiness of the results. Furthermore, this approach is suitable for both code and solution verification, and is particularly useful for solution verification of high-consequence simulations..

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Total metals, carbon, nitrogen & anion concentration data; Slate River & East River floodplains, Crested Butte, CO; May 2022-October 2022

This data package includes processed and undiluted measurements for metal, total carbon, total nitrogen, and anion concentrations from pore water (groundwater) and surface water samples from the Slate River and East River floodplains of Crested Butte, CO, focus field sites for the SLAC Floodplain Hydro-Biogeochemistry SFA. The data was generated as part of the work targeting the overarching research question for the SLAC SFA: How do ubiquitous subsurface interfaces mediate molecular-scale biogeochemical processes and groundwater quality in floodplains and watersheds? Samples were collected between May and October of 2022. These measurements were all recorded at the Arizona Laboratory for Emerging Contaminants (ALEC) at the University of Arizona located in Tucson, AZ. Groundwater samples were extracted from a network of installed rhizon (Rhizosphere Research Products, part no. 19.60.21F, 0.6 micrometer mesh size) and piezometer wells within the river floodplain. All water samples were shaded from sun exposure during extraction from the subsurface and preserved at 4C until measured at ALEC.Analysis by ICP-MS (metals):Measurements for total metals were made on the Agilent 7700x ICP-MS (for total metals) – Agilent Technologies, Santa Clara, CA. The analytical QA/QC protocol was adapted from US EPA Method 200.8 for analysis by ICP-MS. Calibration standards were prepared from multi-element stock solutions (SPEX Certiprep, Metuchen, NJ). Calibration curves include at least 7 points with correlation coefficients > 0.995. The QC protocol includes a continuing calibration blank (CCB), a continuing calibration verification (CCV) solution and at least one quality control sample (QCS) to be analyzed just after calibration and again after every 12 samples and at the completion of the run. The QCS solutions are from an independent source, such as NIST SRM 1643e - Trace elements in water, or QCS solutions from High Purity Standards (Charleston, SC). Acceptable QC responses must be between 90 and 110% of the certified value. Lastly, a suitable internal standard (usually Rh, In, Ga or Ge) is added using on-line addition into the sample line and mixing tee.Analysis by Shimadzu TOC-L (TOC/TN):The TOC-L system is a combustion technique where liquid samples are injected and combusted into CO2 for carbon detection by non-dispersive infrared (NDIR) and NO for detection by chemiluminescence. A calibration curve using five standard solutions between 0.1 and 7 ppm for carbon and 0.05 and 3.5 ppm for nitrogen is made for each type of measurement with a linearity >0.99. All samples, standards, and QC’s are prepared in 24mL scintillation vials that have been baked for 4hrs at 475 Cº and made using RO water (18.2mΩ). QC’s include a calibration blank check (CCB), continuing calibration check (CCC), and a certified reference material check (CRM). All QC’s are within ±10% error and are run before and after each batch of samples. Samples are diluted and rerun if any measurement concentrations are above the highest standard.Analysis by Ion Chromatography (Anions):The instrument used is the Thermo Scientific Dionex ICS-6000 using AS+AG22 column set for anion analysis with sodium carbonate eluent. A calibration curve using five standard solutions between 5 and 250 umol/L is made with a linearity >0.99. Standards and QC’s are prepared in 15mL polypropylene conical tubes, pipetted along with the samples into 1.5mL polypropylene vials. Dilutions are made using RO water (18.2mΩ). QC’s include a calibration blank check (CCB), continuing calibration check (CCC), and a certified reference material check (CRM). All QC’s are within ±10% error and are run before and after each batch of samples. Samples are diluted and rerun if any measurement concentrations are above the highest standard.All files are in csv format.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Total metals & anion concentration data; Slate River floodplain, Crested Butte, CO; May 2020-September 2020

This data package includes processed and undiluted measurements for metal and anion concentrations from pore water (groundwater) samples from the Slate River floodplain of Crested Butte, CO, a focus field site for the SLAC Floodplain Hydro-Biogeochemistry SFA. The data was generated as part of the work targeting the overarching research question for the SLAC SFA: How do ubiquitous subsurface interfaces mediate molecular-scale biogeochemical processes and groundwater quality in floodplains and watersheds? Samples were collected between May and September of 2020. These measurements were all recorded at the Arizona Laboratory for Emerging Contaminants (ALEC) at the University of Arizona located in Tucson, AZ. Groundwater samples were extracted from a network of installed rhizon (Rhizosphere Research Products, part no. 19.60.21F, 0.6 micrometer mesh size) and piezometer wells within the river floodplain. All water samples were shaded from sun exposure during extraction from the subsurface and preserved at 4C until measured at ALEC.Analysis by ICP-MS:Measurements for total metals were made on the Agilent 7700x ICP-MS (for total metals) – Agilent Technologies, Santa Clara, CA.The analytical QA/QC protocol was adapted from US EPA Method 200.8 for analysis by ICP-MS. Calibration standards were prepared from multi-element stock solution (Sigma-Aldrich Multielement standard solution for ICP, St. Louis, MO) using matrix matched to sample solutions (either 2% HCl or HNO3 from AriStar Plus,grade acids from VWR Scientific). Calibration curves include at least 7 points with correlation coefficients > 0.995. The QC protocol includes a continuing calibration blank (CCB), a continuing calibration verification (CCV) solution and at least one quality control sample (QCS) to be analyzed just after calibration and again after every 12 samples and at the completion of the run. The QCS solutions are from an independent source, such as NIST SRM 1643e - Trace Elements in Water, or QCS solutions from High Purity Standards (Charleston, SC). Acceptable QC responses must be between 90 and 110% of the certified value. An internal standard (Rh) is added via on-line addition into the sample line using a mixing tee.Analysis by Ion Chromatography (Anions):The protocol follows Method 4110 in Standard Methods for Examination of Water and Wastewater.The instrument used is the Thermo Scientific Dionex ICS-6000 using AS+AG22 column set for anion analysis with isocratic method using sodium carbonate eluent. Detection is by chemical suppression of eluent conductivity. Quality control solutions and mixed analyte standards purchased from Inorganic Ventures, Christiansburg, VA.All files are in csv format.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

On the Verification of Deep Reinforcement Learning Solution for Intelligent Operation of Distribution Grids

Capabilities of deep reinforcement learning (DRL) in obtaining fast decision policies in high dimensional and stochastic environments have led to its extensive use in operational research, including the operation of distribution grids with high penetration of distributed energy resources (DER). However, the feasibility and robustness of DRL solutions are not guaranteed for the system operator, and hence, those solutions may be of limited practical value. This paper proposes an analytical method to find feasibility ellipsoids that represent the range of multi-dimensional system states in which the DRL solution is guaranteed to be feasible. Empirical studies and stochastic sampling determine the ratio of the discovered to the actual feasible space as a function of the sample size. In addition, the performance of logarithmic, linear, and exponential penalization of infeasibility during the DRL training are studied and compared in order to reduce the number of infeasible solutions

Hosseini, Mohammad Mehdi↗
DOE OSTI2021

Computational Evaluation of Thermal Response of Open-Cell Foam With Circular Pore (Computional Evaluation of Thermal Response of Open-Cell Foam with Circular Pore)

The evaluation of effective material properties in heterogeneous materials (e.g., composites or multicomponent structures) critically relevant to a wide spectrum of applications, including nuclear power, electronic packaging, flame retardants, hypersonics, and gas turbine power. The work described in this paper is centered around the numerical assessment of the thermal behavior of porous materials obtained from finite element thermal modeling and simulation. Here, two-dimensional, steady state analyses were performed on unit cells with centered, circular pores using a second order accurate Galerkin finite element method (FEM). The effective thermal conductivities of the porous systems were examined, encompassing a range of porosities from 4.9% to 60.1%. The geometries of the models were generated based on ordered circular pores for each modeled porosity level. The system response quantity (SRQ) under investigation was the dimensionless effective thermal conductivity across the unit cell. The dimensionless effective thermal conductivity was compared across all simulated cases, producing a trend between porosity and effective thermal conductivity. In the presented investigation, the method of manufactured solutions (MMS) was used to perform code verification, and the grid convergence index (GCI) was employed to estimate discretization uncertainty as solution verification. Code verification concluded an approximately second order accurate Galerkin FEM solver. It was found that the introduction of porosity to the unit cell material structure reduces effective thermal conductivity, as anticipated. Numerical results obtained in this study are compared to an analytical solution and to a sample of empirical data.

36 MATERIALS SCIENCE↗
DOE OSTI2026

ORNL Report for the Preparation and Verification of a U-233 Spike Solution

In response to a Letter Request from the International Atomic Energy Agency to the United States Support Program, the NNSA’s Nuclear Reference Material Program (NRMP) tasked the Material Signatures and Isotopic Standards (MSIS) group at Oak Ridge National Laboratory to prepare 50 units of a U-233 CRM standard with a unit size of approximately 10 µg of U-233.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2025

Code associated with Publication “Analytic Solutions and Field-Scale Application for Verification of Coupled Thermo-Hydro-Mechanical Processes in Subsurface Fractured Media”

As part of a submitted paper, which is collection of previously published analytical solutions to coupled thermo-hydro-mechanical problems in subsurface flow and transport, we have prepared a collection of python scripts to compute and plot those analytic solutions. All code to be released implements existing methods; there are no novel algorithms nor any major innovations to existing software.

Hyman, Jeffrey↗
DOE OSTI2023

A fast Fourier transform-based solver for elastic micropolar composites

This work presents a spectral micromechanical formulation for obtaining the full-field and homogenized response of elastic micropolar composites. The algorithm relies on a coupled set of convolution integral equations for the micropolar strains, where periodic Green’s operators associated with a linear homogeneous reference medium are convolved with functions of the Cauchy and couple stress fields that encode the material’s heterogeneity, as well as any potential material nonlinearity. Such convolution integral equations take an algebraic form in the reciprocal Fourier space that can be solved iteratively. In this vein, the fast Fourier transform (FFT) algorithm is leveraged to accelerate the numerical solution, resulting in a mesh-free formulation in which the periodic unit cell representing the heterogeneous material can be discretized by a regular grid of pixels in two dimensions (or voxels in three dimensions). For verification, the numerical solutions obtained with the micropolar FFT solver are compared with analytical solutions for a matrix with a dilute circular inclusion subjected to plane strain loading. The developed computational framework is then used to study length-scale effects and effective (micropolar) moduli of composites with various topological configurations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Sensitivity analysis of a layered piezoelectric system using ZFEM

The complex variable finite element method (ZFEM) is a numerical technique which aims to find the partial derivatives of the independent variables with respect to variation in dependent parameters declared in the physics. This is done by combining the complex Taylor series expansion within the weak formulation of the governing equation in a coupled system of linear equations forming a complex valued block matrix given by the Cauchy–Riemann matrix representation. In this work, two-dimensional linear first-order elements have been implemented in ZFEM to predict the design derivatives of the mechanical displacement field and the voltage potential field for a layered piezoelectric system in a steady-state study with Dirichlet boundary condition applied at the top and bottom edges of the geometry. This approach allows the standard FEM solution to quantify the sensitivity of the mechanical displacement and voltage potential fields with respect to small variations in the material properties through the information obtained from the computation of the derivatives. The domain is formed by a layered body with PZT-4 and PZT-5 stacked together. For result verification, the numerical solution obtained with ZFEM was compared to results from a commercial FEM package and the solution from the imaginary part was compared to the exact solution of a well-known benchmark problem. In conclusion, comparison of the results showed good agreement for both the real and imaginary parts of the solution and the largest sensitivities were found in PZT-5 specifically in C 13 , C 33 , and ε 33 .

42 ENGINEERING↗