Abstract: Nonlocal theories have been successfully employed in many settings, such as fracture modeling, image processing, biology, sand pile formation, and more. Nonlocal models use integral operators which take into account the cumulative behavior in the neighborhood of a point, and thus they allow solutions or domains to be discontinuous. On the other hand, nonlocal models based on integro-differential equations have fundamentally different mathematical structure and properties compared to their local PDE-based counterparts, requiring new and novel analytical and numerical techniques to analyze and utilize them. In this minisymposium we will focus on recent advances in nonlocal theories, including theoretical results (well-posedness, regularity, and asymptotic behavior of solutions), numerical methods and analysis (such as convergence and conditioning), and practical application (modeling, simulation, engineering analysis) related to nonlocal theories that have appeared in fracture modeling (peridynamics), nonlocal diffusion, biology, image processing, and more.
Organizers:
Saturday, October 19th, 2019 at 16:40 - 18:00 (268 Carver Hall) | |||
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16:40 - 17:00 | Stewart Silling, Sandia National Laboratories |
The Mechanics of Unstable Peridynamic Materials | |
An elastic peridynamic material with a nonconvex strain energy density function can possess "imaginary wave speeds" and is therefore unstable in the sense of Hadamard. However, such materials prove to be useful in modeling important phenomena such as crack nucleation. This talk will discuss the solution of initial value problems in such materials, especially the exponential growth of initial data over time. This unstable growth can be used to compute the incubation time for crack nucleation as well as a rate effect on the critical strain for material failure. These useful features of the dynamics of unstable peridynamic materials are a consequence of the nonlocality in the model and are not present in the local theory. |
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17:00 - 17:20 | Michael Parks, Sandia National Laboratories |
On Neumann-type Boundary Conditions for Nonlocal Models | |
Peridynamics is a nonlocal reformulation of continuum mechanics that is suitable for representing fracture and failure. For practical engineering applications, precise application of boundary conditions is essential. However, nonlocal boundary conditions (sometimes called volume constraints) of Neumann type remain poorly understood. We limit our discussion to nonlocal diffusion models, reviewing existing approaches to nonlocal Neumann-type boundary conditions, and presenting a new approach for nonlocal boundary conditions of Neumann type. |
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17:20 - 17:40 | Kaushik Dayal, Carnegie Mellon University |
Spatially-nonlocal Time Derivatives and Multibody Interactions in Peridynamics | |
Peridynamics is a nonlocal continuum model to describe the mechanics of bodies that can potentially undergo fracture. I will discuss changes to the framework, in particular the possibility of spatial nonlocality in the time derivatives, and the relation between classical models and peridynamic models through a bond-level deformation gradient. |
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17:40 - 18:00 | Petronela Radu, University of Nebraska - Lincoln |
Nonlocal Helmholtz-Hodge Decompositions | |
Nonlocal operators that have appeared in a variety of physical models satisfy identities and enjoy a range of properties similar to their classical counterparts. In this presentation I will discuss Helmholtz-Hodge type decompositions for two-point vector fields in three components that have zero nonlocal curls, zero nonlocal divergence, and a third component which is (nonlocally) curl-free and divergence-free. The results obtained incorporate different nonlocal boundary conditions, thus being applicable in a variety of settings. |
Sunday, October 20th, 2019 at 10:20 - 11:40 (268 Carver Hall) | |||
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10:20 - 10:40 | Ratna Khatri, George Mason University |
Role of Fractional Laplacian in Inverse Problems | |
In this work, we will discuss the applications of fractional Laplacian in two kinds of inverse problems. At first we introduce an external source identification problem with fractional partial differential equation (PDE) as constraints. Our motivation to introduce this new class of inverse problems stems from the fact that the classical PDE models only allow the source/control to be placed on the boundary or inside the observation domain where the PDE is fulfilled. Our new approach allows us to place the source/control outside and away from the observation domain. The second problem is motivated by imaging science where we propose to use the fractional Laplacian as a regularizer to improve the reconstruction quality. In addition, inspired by residual neural networks, we create a bilevel optimization neural network (BONNet) to learn the optimal regularization parameters, like the strength of regularization and the exponent of fractional Laplacian. As our model problem, we consider tomographic reconstruction and show an improvement in the reconstruction quality, especially for limited data, via fractional Laplacian regularization. |
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10:40 - 11:00 | Siavash Jafarzadeh, University of Nebraska - Lincoln |
Efficient Computations for Nonlocal Models: A Fourier Spectral Method with Volume Penalization | |
We introduce an efficient numerical method for nonlocal models, in particular for peridynamics, based on Fourier spectral methods and Brinkman volume penalization. The proposed method reduces the computational complexity to O(NlogN), from O(N^2) with the conventional methods such as the meshfee discretization with Gaussian quadrature or the Finite element methods. In our spectral method, Fourier transformation untangles the convolution integral into a simple multiplication in the Fourier space. The major cost of this scheme then is the Fourier transform which is performed via FFT algorithms at a cost of O(NlogN). While the assumption of periodicity of the computational domain is necessary to allow for the Fourier transformation, a penalization technique is adopted to apply arbitrary volume constraints, and therefore, to impose arbitrary boundary conditions. The spectral method introduced removes the limitation of periodicity required by regular spectral methods, and leads to remarkable efficiency gains. We present 1D, 2D, and 3D peridynamic diffusion examples to demonstrate convergence to analytical solutions and speed-up compared to other discretization methods normally used for nonlocal models. |
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11:00 - 11:20 | Fatih Celiker, Wayne State University |
Convergence and Asymptotic Compatibility of Higher Order Collocation Methods for Nonlocal Problems | |
We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics, a nonlocal formulation of continuum mechanics. We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation. A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero. We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible. We find that, for polynomial degrees greater than or equal to two, there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for nonlocal diffusion are asymptotically compatible. We verify these findings through extensive numerical experiments. This is joint work with Burak Aksoylu and George Gazonas. |
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11:20 - 11:40 | Jeremy Trageser, Sandia National Laboratories |
Anisotropic Plane Strain and Stress Models in Linearized Bond-based Peridynamics | |
This presentation concerns anisotropic two-dimensional and planar elasticity models within the frameworks of classical linear elasticity and the bond-based peridynamic theory of solid mechanics. We begin by reviewing corresponding models from the classical theory of linear elasticity. We further present innovative formulations for peridynamic plane strain and plane stress, which are obtained using direct analogies of the classical planar elasticity assumptions, and we specialize these formulations to a variety of material symmetry classes. The uniqueness of the presented peridynamic plane strain and plane stress formulations in this work is that we directly reduce three-dimensional models to two-dimensional formulations, as opposed to matching two-dimensional peridynamic models to classical plane strain and plane stress formulations. This results in significant computational savings, while retaining the dynamics of the original three-dimensional bond-based peridynamic problems under suitable assumptions. |