Publications

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This report is the final report for the LDRD project "Fast and Robust Linear Solvers using Hierarchical Matrices". The project was a success. We developed two novel algorithms for solving sparse linear systems. We demonstrated their effectiveness on ill-conditioned linear systems from ice sheet simulations. We showed that in many cases, we can obtain near-linear scaling. We believe this approach has strong potential for difficult linear systems and should be considered for other Sandia and DOE applications. We also report on some related research activities in dense solvers and randomized linear algebra.

A mechanical model is introduced for predicting the initiation and evolution of complex fracture patterns without the need for a damage variable or law. The model, a continuum variant of Newton’s second law, uses integral rather than partial differential operators where the region of integration is over finite domain. The force interaction is derived from a novel nonconvex strain energy density function, resulting in a nonmonotonic material model. The resulting equation of motion is proved to be mathematically well-posed. The model has the capacity to simulate nucleation and growth of multiple, mutually interacting dynamic fractures. In the limit of zero region of integration, the model reproduces the classic Griffith model of brittle fracture. The simplicity of the formulation avoids the need for supplemental kinetic relations that dictate crack growth or the need for an explicit damage evolution law.

We will develop Malliavin estimators for Monte Carlo radiation transport by formulating the governing jump stochastic differential equation and deriving the applicable estimators that produce sensitivities for our equations. Efficient and effective sensitivity can be used for design optimization and uncertainty quantification with broad utilization for radiation environments. The technology demonstration will lower development risk for other particle-based simulation methods.

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The rise of low-power neuromorphic hardware has the potential to change high-performance computing; however much of the focus on brain-inspired hardware has been on machine learning applications. A low-power solution for solving partial differential equations could radically change how we approach large-scale computing in the future. The random walk is a fundamental stochastic process that underlies many numerical tasks in scientific computing applications. We consider here two neural algorithms that can be used to efficiently implement random walks on spiking neuromorphic hardware. The first method tracks the positions of individual walkers independently by using a modular code inspired by grid cells in the brain. The second method tracks the densities of random walkers at each spatial location directly. We present the scaling complexity of each of these methods and illustrate their ability to model random walkers under different probabilistic conditions. Finally, we present implementations of these algorithms on neuromorphic hardware.

Modeling material and component behavior using finite element analysis (FEA) is critical for modern engineering. One key to a credible model is having an accurate material model, with calibrated model parameters, which describes the constitutive relationship between the deformation and the resulting stress in the material. As such, identifying material model parameters is critical to accurate and predictive FEA. Traditional calibration approaches use only global data (e.g. extensometers and resultant force) and simplified geometries to find the parameters. However, the utilization of rapidly maturing full-field characterization techniques (e.g. Digital Image Correlation (DIC)) with inverse techniques (e.g. the Virtual Feilds Method (VFM)) provide a new, novel and improved method for parameter identification. This LDRD tested that idea: in particular, whether more parameters could be identified per test when using full-field data. The research described in this report successfully proves this hypothesis by comparing the VFM results with traditional calibration methods. Important products of the research include: verified VFM codes for identifying model parameters, a new look at parameter covariance in material model parameter estimation, new validation techniques to better utilize full-field measurements, and an exploration of optimized specimen design for improved data richness.

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Traditionally, material identification is performed using global load and displacement data from simple boundary-value problems such as uni-axial tensile and simple shear tests. More recently, however, inverse techniques such as the Virtual Fields Method (VFM) that capitalize on heterogeneous, full-field deformation data have gained popularity. In this work, we have written a VFM code in a finite-deformation framework for calibration of a viscoplastic (i.e. strain-rate dependent) material model for 304L stainless steel. Using simulated experimental data generated via finite-element analysis (FEA), we verified our VFM code and compared the identified parameters with the reference parameters input into the FEA. The identified material model parameters had surprisingly large error compared to the reference parameters, which was traced to parameter covariance and the existence of many essentially equivalent parameter sets. This parameter non-uniqueness and its implications for FEA predictions is discussed in detail. Lastly, we present two strategies to reduce parameter covariance – reduced parametrization of the material model and increased richness of the calibration data – which allow for the recovery of a unique solution.

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Here, we introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, nonlocal diffusion problem. We use the discrete solution to approximate the continuous solution. Our method is nonconforming and uses a localized Lagrange basis that is constructed out of radial basis functions. By verifying that certain inf-sup conditions hold, we demonstrate that both the continuous and discrete problems are well-posed, and also present numerical and theoretical results for the convergence behavior of the method. The stiffness matrix is assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, symmetric matrix. This then is used to find the discretized solution.

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In this paper, a nonlocal convection-diffusion model is introduced for the master equation of Markov jump processes in bounded domains. With minimal assumptions on the model parameters, the nonlocal steady and unsteady state master equations are shown to be well-posed in a weak sense. Finally, then the nonlocal operator is shown to be the generator of finite-range nonsymmetric jump processes and, when certain conditions on the model parameters hold, the generators of finite and infinite activity Lévy and Lévy-type jump processes are shown to be special instances of the nonlocal operator.

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We introduce a meshfree discretization for a nonlocal diffusion problem using a localized basis of radial basis functions. Our method consists of a conforming radial basis of local Lagrange functions for a variational formulation of a volume constrained nonlocal diffusion equation. We also establish an L2 error estimate on the local Lagrange interpolant. The stiffness matrix is assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, sparse, symmetric positive definite stiffness matrix. We demonstrate that both the continuum and discrete problems are well-posed and present numerical results for the convergence behavior of the radial basis function method. We explore approximating the solution to inhomogeneous differential equations by solving inhomogeneous nonlocal integral equations using the proposed radial basis function method.