Recent & Upcoming Talks

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.

In this talk, I provide an overview of my research in surrogate model error estimation and code verification for examples in computational mechanics. For the first part of this talk, I present an approach to engineer features that, with machine-learning regression, can accurately predict the error incurred by reduced-order models and other approximations to parameterized systems of nonlinear equations. In the second part of this talk, I discuss our approaches to code verification for hypersonic flow in thermochemical nonequilibrium.

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.

Code verification is an important step towards establishing the credibility of the results of computational physics codes by assessing whether the underlying numerical methods have been correctly implemented. The discretization of differential, integral, or integro-differential equations necessarily incurs some truncation error, and thus the approximate solutions produced from the discretized equations will incur an associated discretization error. If the solution to the problem is known, a measure of the discretization error may be evaluated directly from the approximate solution. The code may be verified by examining the rate at which the error decreases as the discretization is refined, thereby verifying the observed order of accuracy of the discretization scheme is the expected order of accuracy. Electromagnetic surface integral equations yield many code-verification challenges due to the various sources of numerical error and their possible interactions. These error sources arise from the use of basis functions to approximate the solution, quadrature to numerically integrate, and planar elements to represent curved surfaces. In this work, we provide approaches to separately measure the numerical errors arising from these different error sources using manufactured solutions. We demonstrate the effectiveness of these approaches for cases with and without coding errors.

In this talk, I provide an overview of my research in surrogate model error estimation and code verification. For the first part of this talk, I present an approach to engineer features that, with machine-learning regression, can accurately predict the error incurred by reduced-order models and other approximations to parameterized systems of nonlinear equations. In the second part of this talk, I discuss novel approaches to code verification for hypersonics, ablation, and electromagnetics. These approaches include manufactured solutions, methods for algebraic terms, nonintrusive manufactured solutions, and methods for integral equations with singularities.

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.

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 work, 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.

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 work, 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.

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 work, 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.

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 work, 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).

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 work, 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.

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.