arXiv subjects

Brian A. Freno

Publications and source records attributed to Brian A. Freno.

14 recordsLinked to original sources

arXiv2026

Code-Verification Techniques for Particle-in-Cell Simulations with Direct Simulation Monte Carlo Collisions

Particle-in-cell methods with stochastic collision models are commonly used to simulate collisional plasma dynamics, with applications ranging from hypersonic flight to semiconductor manufacturing. Code verification of such methods is challenging due to the interaction between the spatial- and temporal-discretization errors, the statistical sampling noise, and the stochastic nature of the collision algorithm. In this paper, we introduce our code-verification approaches to apply the method of manufactured solutions to plasma dynamics, and we derive expected convergence rates for the different sources of discretization and statistical error. For the particles, we incorporate the method of manufactured solutions into the equations of motion. We manufacture the particle distribution function and inversely query the cumulative distribution function to obtain known particle positions and velocities at each time step. In doing so, we avoid modifying the particle weights, eliminating risks from potentially negative weights or modifications to weight-dependent collision algorithms. For the collision algorithm, we average independent outcomes at each time step and we derive a corresponding manufactured source term for the velocity change for each particle. By having known solutions for the particle positions and velocities, we are able to compute the error in these quantities directly instead of attempting to compute differences in distribution functions. These approaches are equally valid for particle-in-cell simulations with Monte Carlo collisions and direct simulation Monte Carlo simulations of neutral gas flows. We demonstrate the effectiveness of our approaches in three dimensions for different couplings between the particles and field, with and without binary elastic collisions, and with and without coding errors.

physics.comp-ph↗

arXiv2025

A Note on the Convergence of Symmetric Triangle Quadrature Rules

Symmetric polynomial quadrature rules for triangles are commonly used to efficiently integrate two-dimensional domains in finite-element-type problems. While the development of such rules focuses on the maximum degree a given number of points can exactly integrate, smooth integrands are generally not polynomials of finite degree. Therefore, for such integrands, one needs to balance integration accuracy and computational cost. A natural approach to this balance is to choose the number of points such that the convergence rate with respect to the mesh size $h$ matches that of the other properties of the scheme, such as the planar or curved triangles that approximate the geometry or the basis functions that approximate the solution. In general, it is expected that a quadrature rule capable of integrating polynomials up to degree $d$ yields an integration error that is $\mathcal{O}(h^p)$, where $p=d+1$. However, as we describe in this paper, for symmetric triangle quadrature rules, when $d$ is even, $p=d+2$; therefore, for a $p^\text{th}$-order-accurate quadrature rule, fewer quadrature points are necessary, reducing the time required for matrix assembly in finite-element-type problems. This reduction in cost is modest for local differential operators that yield sparse matrices but appreciable for global integral operators that yield dense matrices. In this paper, we briefly summarize the details of symmetric triangle quadrature rules, discuss error implications for quadrature rules for one dimension and triangles, and we provide numerical examples that support our observation that polynomials that exactly integrate even maximum degrees converge faster than the conventional expectation for sequences of regular meshes.

math.NA↗

arXiv2025

Code-Verification Techniques for an Arbitrary-Depth Electromagnetic Slot Model

Electromagnetic slot models are employed to efficiently simulate electromagnetic penetration through openings in an otherwise closed electromagnetic scatterer. Such models, which incorporate varying assumptions about the geometry of the openings, are typically coupled with electromagnetic surface integral equations that model electromagnetic scattering. In this paper, we introduce novel code-verification approaches and build upon our previously developed methodologies to assess the correctness of the numerical implementation of an arbitrary-depth slot model. Through these approaches, we measure the convergence rates of the different interacting sources of numerical error and demonstrate the impact of various factors on these rates for several cases.

physics.comp-ph↗

arXiv2024

Manufactured Solutions for an Electromagnetic Slot Model

The accurate modeling of electromagnetic penetration is an important topic in computational electromagnetics. Electromagnetic penetration occurs through intentional or inadvertent openings in an otherwise closed electromagnetic scatterer, which prevent the contents from being fully shielded from external fields. To efficiently model electromagnetic penetration, aperture or slot models can be used with surface integral equations to solve Maxwell's equations. A necessary step towards establishing the credibility of these models is to assess the correctness of the implementation of the underlying numerical methods through code verification. Surface integral equations and slot models yield multiple interacting sources of numerical error and other challenges, which render traditional code-verification approaches ineffective. In this paper, we provide approaches to separately measure the numerical errors arising from these different error sources for the method-of-moments implementation of the electric-field integral equation with a slot model. We demonstrate the effectiveness of these approaches for a variety of cases.

physics.comp-ph↗

arXiv2023

Code-Verification Techniques for the Method-of-Moments Implementation of the Magnetic-Field Integral Equation

For computational physics simulations, code verification plays a major role in establishing the credibility of the results by assessing the correctness of the implementation of the underlying numerical methods. In computational electromagnetics, surface integral equations, such as the method-of-moments implementation of the magnetic-field integral equation, are frequently used to solve Maxwell's equations on the surfaces of electromagnetic scatterers. These electromagnetic surface integral equations yield many code-verification challenges due to the various sources of numerical error and their possible interactions. In this paper, we provide approaches to separately measure the numerical errors arising from these different error sources. We demonstrate the effectiveness of these approaches for cases with and without coding errors.

physics.comp-ph↗

arXiv2023

Code-Verification Techniques for the Method-of-Moments Implementation of the Combined-Field Integral Equation

Code verification plays an important role in establishing the credibility of computational simulations by assessing the correctness of the implementation of the underlying numerical methods. In computational electromagnetics, the numerical solution to integral equations incurs multiple interacting sources of numerical error, as well as other challenges, which render traditional code-verification approaches ineffective. In this paper, we provide approaches to separately measure the numerical errors arising from these different error sources for the method-of-moments implementation of the combined-field integral equation. We demonstrate the effectiveness of these approaches for cases with and without coding errors.

physics.comp-ph↗

arXiv2022

Code Verification for Practically Singular Equations

The method-of-moments implementation of the electric-field integral equation (EFIE) yields many code-verification challenges due to the various 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. To address these singular integrals, an approach was previously presented wherein both the solution and Green's function are manufactured. Because the arising equations are poorly conditioned, they are reformulated as a set of constraints for an optimization problem that selects the solution closest to the manufactured solution. In this paper, we demonstrate how, for such practically singular systems of equations, computing the truncation error by inserting the exact solution into the discretized equations cannot detect certain orders of coding errors. On the other hand, the discretization error from the optimal solution is a more sensitive metric that can detect orders less than those of the expected convergence rate.

math.NA↗

arXiv2022

Nonintrusive Manufactured Solutions for Non-Decomposing Ablation in Two Dimensions

Code verification is a necessary step towards establishing credibility in computational physics simulations. It is used to assess the correctness of the implementation of the numerical methods within the code, and it is a continuous part of code development. Code verification is typically performed using exact and manufactured solutions. However, exact solutions are often limited, and manufactured solutions generally require the invasive introduction of an artificial forcing term within the source code, such that the code solves a modified problem for which the solution is known. The equations for some physics phenomena, such as non-decomposing ablation, yield infinite analytic solutions, but the boundary conditions may eliminate these possibilities. For such phenomena, however, we can manufacture the terms that comprise the boundary conditions to obtain exact solutions. In this paper, we present a nonintrusive method for manufacturing solutions for non-decomposing ablation in two dimensions, which does not require the addition of a source term.

physics.comp-ph↗

arXiv2021

Code-Verification Techniques for the Method-of-Moments Implementation of the Electric-Field Integral Equation

The method-of-moments implementation of the electric-field integral equation yields many code-verification challenges due to the various 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 paper, we provide approaches to separately assess the numerical errors arising from the use of basis functions to approximate the solution and the use of quadrature to approximate the integration. Through these approaches, we are able to verify the code and compare the error from different quadrature options.

physics.comp-ph↗

arXiv2021

Manufactured Solutions for the Method-of-Moments Implementation of the Electric-Field Integral Equation

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. In this paper, we provide an approach through which we can apply the method of manufactured solutions to isolate and verify the solution-discretization error. We accomplish this by manufacturing both the surface current and the Green's function. Because the arising equations are poorly conditioned, we reformulate them as a set of constraints for an optimization problem that selects the solution closest to the manufactured solution. We demonstrate the effectiveness of this approach for cases with and without coding errors.

physics.comp-ph↗

arXiv2020

Characterization and Integration of the Singular Test Integrals in the Method-of-Moments Implementation of the Electric-Field Integral Equation

In this paper, we characterize the logarithmic singularities arising in the method of moments from the Green's function in integrals over the test domain, and we use two approaches for designing geometrically symmetric quadrature rules to integrate these singular integrands. These rules exhibit better convergence properties than quadrature rules for polynomials and, in general, lead to better accuracy with a lower number of quadrature points. We demonstrate their effectiveness for several examples encountered in both the scalar and vector potentials of the electric-field integral equation (singular, near-singular, and far interactions) as compared to the commonly employed polynomial scheme and the double Ma--Rokhlin--Wandzura (DMRW) rules, whose sample points are located asymmetrically within triangles.

physics.comp-ph↗

arXiv2020

Code-Verification Techniques for Hypersonic Reacting Flows in Thermochemical Nonequilibrium

The study of hypersonic flows and their underlying aerothermochemical reactions is particularly important in the design and analysis of vehicles exiting and reentering Earth's atmosphere. Computational physics codes can be employed to simulate these phenomena; however, verification of these codes is necessary to certify their credibility. To date, few approaches have been presented for verifying codes that simulate hypersonic flows, especially flows reacting in thermochemical nonequilibrium. In this paper, we present our code-verification techniques for verifying the spatial accuracy and thermochemical source term in hypersonic reacting flows in thermochemical nonequilibrium. We demonstrate the effectiveness of these techniques on the Sandia Parallel Aerodynamics and Reentry Code (SPARC).

physics.comp-ph↗

arXiv2020

Symmetric Triangle Quadrature Rules for Arbitrary Functions

Despite extensive research on symmetric polynomial quadrature rules for triangles, as well as approaches to their calculation, few studies have focused on non-polynomial functions, particularly on their integration using symmetric triangle rules. In this paper, we present two approaches to computing symmetric triangle rules for singular integrands by developing rules that can integrate arbitrary functions. The first approach is well suited for a moderate amount of points and retains much of the efficiency of polynomial quadrature rules. The second approach better addresses large amounts of points, though it is less efficient than the first approach. We demonstrate the effectiveness of both approaches on singular integrands, which can often yield relative errors two orders of magnitude less than those from polynomial quadrature rules.

math.NA↗

arXiv2019

Machine-learning error models for approximate solutions to parameterized systems of nonlinear equations

This work proposes a machine-learning framework for constructing statistical models of errors incurred by approximate solutions to parameterized systems of nonlinear equations. These approximate solutions may arise from early termination of an iterative method, a lower-fidelity model, or a projection-based reduced-order model, for example. The proposed statistical model comprises the sum of a deterministic regression-function model and a stochastic noise model. The method constructs the regression-function model by applying regression techniques from machine learning (e.g., support vector regression, artificial neural networks) to map features (i.e., error indicators such as sampled elements of the residual) to a prediction of the approximate-solution error. The method constructs the noise model as a mean-zero Gaussian random variable whose variance is computed as the sample variance of the approximate-solution error on a test set; this variance can be interpreted as the epistemic uncertainty introduced by the approximate solution. This work considers a wide range of feature-engineering methods, data-set-construction techniques, and regression techniques that aim to ensure that (1) the features are cheaply computable, (2) the noise model exhibits low variance (i.e., low epistemic uncertainty introduced), and (3) the regression model generalizes to independent test data. Numerical experiments performed on several computational-mechanics problems and types of approximate solutions demonstrate the ability of the method to generate statistical models of the error that satisfy these criteria and significantly outperform more commonly adopted approaches for error modeling.

math.NA↗