Abstract : The past decade has seen a rapid growth in the development of nonlocal mathematical models. Nonlocal modeling is now being used in applications including continuum mechanics and fracture mechanics, anomalous diffusion and advection diffusion, and other fields. This minisymposium seeks to bring together mathematicians and domain scientists from different disciplines working on nonlocal modeling and is intended to serve as an international forum for the state of the art in the modeling, analysis, and numerical aspects of nonlocal models.
Organizer(s) : Patrick Diehl, Pablo Seleson, Robert Lipton, Qiang Du
Abstract : The most intuitive applications of nonlocal modeling arise when long-range interactions, such as electrostatic fields, are present in a physical system. However, nonlocal descriptions are also produced by the homogenization or coarse-graining of heterogeneous, small-scale systems. In this talk, it is shown that the coarse-graining of molecular systems or of local, elastic, heterogeneous systems leads to the peridynamic nonlocal linear momentum balance. Examples demonstrate the discovery of nonlocal material models applicable to a coarse-grained description.
[00366] Wellposedness, regularity, and convergence of nonlocal solutions to classical counterparts
Format : Talk at Waseda University
Author(s) :
Petronela Radu (University of Nebraska-Lincoln)
Abstract : The successful employment of nonlocal models in a variety of applications relies on a deep understanding of mathematical properties and analysis of the underlying integral operators and associated systems of equations. In this talk I will present some recent results on nonlocal frameworks systems based on some existing, as well as newly introduced, nonlocal operators. The studies include a series of results on nonlocal versions of integration by parts theorems, boundary conditions (both, Dirichlet and Neumann), Helmholtz-Hodge type decompositions, as well as convergence of operators to their classical equivalents as the interaction horizon vanishes.
[00370] Coupling of an atomistic model and peridynamic model using an extended Arlequin framework
Author(s) :
Jieqiong Zhang (Northwest University )
Fei Han (Dalian University of Technology)
Abstract : A general nonlocal coupling technique between an atomistic (AM) model and the bond-based peridynamic (PD) model is proposed, based on the Arlequin framework. This technique applies the complementary weight function and constraint conditions to transmit energies through the overlapping region between the AM and PD regions. We extend the original Arlequin framework to discrete cases by redefining constraint conditions by the peridynamic differential operator, which enables the interpolation and corresponding derivative of scattered data. Besides, the preconditioning of calibration for the PD effective micromodulus is implemented to guarantee the equilibrium of energy. One-dimensional benchmark tests investigate the coupling effects influenced by several key factors, including the coupling length, weight function, grid size and horizon in the PD model, and constraint conditions. Two- and three-dimensional numerical examples are provided to verify the applicability and effectiveness of this coupling model. Results illustrate this AM–PD coupling model takes the mutual advantages of the computational efficiency of PD model and the accuracy of AM model, which provides a flexible extension of the Arlequin framework to couple particle methods.
[00445] Local and nonlocal energy-based coupling models
Format : Talk at Waseda University
Author(s) :
Julio D. Rossi (Buenos Aires Univ.)
Abstract : In this talk we will present two different ways of coupling a local operator with a nonlocal one in such a way that the resulting equation is related to an energy functional.
In the first strategy the coupling is given via source terms in the equation and in the
second one a flux condition in the local part appears.
For both models we prove existence and uniqueness of a solution that is obtained
via direct minimization of the related energy functional.
In the second part of this talk we extend these ideas to deal with local/nonlocal elasticity models in which we couple classical local elasticity with nonlocal peridynamics.
joint work with G. Acosta and F. Bersetche.
[01086] Machine-learning based coupling of local and nonlocal models
Format : Talk at Waseda University
Author(s) :
Patrick Diehl (LSU)
Noujoude Nader (LSU)
Serge Prudhomme (PolyMTL)
Abstract : This talk will present a machine-learning coupling approach for local and nonlocal models. We will identify when to switch two the coupled system and where to place the nonlocal region within the local region. We will present some one-dimensional and two-dimensional examples to showcase the applicability of the approach.
[01235] Nonlocal Neural Operators for Learning Complex Physical Systems with Momentum Conservation
Format : Online Talk on Zoom
Author(s) :
Yue Yu (Lehigh University)
Abstract : Neural operators have recently become popular tools for learning responses of complex physical systems. Nevertheless, their applications neglects the intrinsic preservation of fundamental physical laws. Herein, we introduce a novel integral neural operator architecture, to learn physical models with conservation laws of linear and angular momentums automatically guaranteed. As applications, we demonstrate our model in learning complex material behaviors from both synthetic and experimental datasets, and show that our models achieves state-of-the-art accuracy and efficiency.
[03023] A Numerical Study of the Peridynamic Differential Operator Discretization of Incompressible Navier-Stokes Problems
Format : Online Talk on Zoom
Author(s) :
Burak Aksoylu (Texas A&M University-San Antonio)
Fatih Celiker (Wayne State University)
Abstract : We study the incompressible Navier-Stokes equations using the Projection Method. The applications of interest are the classical channel flow problems such as Couette, shear, and Poiseuille. In addition, we consider the Taylor-Green vortex and lid-driven cavity applications. For discretization, we use the Peridynamic Differential Operator (PDDO). The main emphasis of the paper is the performance of the PDDO as a discretization method under these flow problems. We present a careful numerical study with quantifications and report convergence tables with convergence rates. We also study the approximation properties of the PDDO and prove that the $N$-th order PDDO approximates polynomials of degree at most $N$ exactly. As a result, we prove that the PDDO discretization guarantees the zero row sum property of the arising system matrix.
[03546] An efficient peridynamics-based coupling method for composite fracture
Format : Talk at Waseda University
Author(s) :
Zihao Yang (Northwestern Polytechnical University)
Abstract : In this talk, we will introduce a peridynamics-based statistical multiscale framework and related numerical algorithms to predict the fracture of composite structure with randomly distributed particles. The heterogeneities of composites, including the shape, spatial distribution and volume fraction of particles, are characterized within the representative volume elements, and their impact on structure failure are extracted as two types of peridynamic parameters, namely, statistical critical stretch and equivalent micromodulus. Two- and three-dimensional numerical examples illustrate the validity, accuracy and efficiency of the proposed method.
[03632] Nonlocal Boundary Value Problems with Local Boundary Conditions
Author(s) :
James Scott (Columbia University)
Abstract : We state and analyze classical boundary value problems for nonlocal operators. The model takes its horizon parameter to be spatially dependent, vanishing near the boundary of the domain. We show the variational convergence of solutions to the nonlocal problem with mollified Poisson data to the solution of the localized classical Poisson problem with $H^{-1}$ data as the horizon uniformly converges to zero. Several classes of boundary conditions are considered.
[03718] On the optimal control of a linear peridynamic model
Format : Talk at Waseda University
Author(s) :
Tadele Mengesha (University of Tennessee Knoxville )
Abner Salgado (University of Tennessee Knoxville )
Joshua Siktar (University of Tennessee Knoxville )
Abstract : We present a result on a non-local optimal control problem involving a linear, bond-based peridynamics model.
In addition to proving existence and uniqueness of solutions to our problem, we investigate
their behavior as the horizon parameter, which controls the degree of nonlocality, approaches
zero. We then study a finite element-based discretization of this problem, its convergence, and the
so-called asymptotic compatibility as the discretization parameter and the horizon parameter
vanish simultaneously.
[03800] Nonlocal half-ball vector operators and their applications to nonlocal variational problems
Author(s) :
Xiaochuan Tian (UC San Diego)
Zhaolong Han (UC San Diego)
Abstract : Motivated by the growing interests in nonlocal models, and particularly peridynamics, we present a nonlocal vector calculus framework defined using the half-ball gradient, divergence, and curl operators. Theoretical developments of the nonlocal half-ball vector operators include nonlocal vector identities, nonlocal Poincare inequality on bounded domains, and Bourgain-Brezis-Mironescu type compactness results. As a result, well-posedness of nonlocal variational problems can be obtained, and the applications include nonlocal convection-diffusion problems and the peridynamics correspondence model. In particular, we illustrate that the new peridynamics correspondence model defined by the half-ball vector operator is energy stable which removes the known zero-energy mode instability issue of peridynamics correspondence models.
[05210] CabanaPD: A meshfree GPU-enabled peridynamics code for exascale fracture simulations
Format : Online Talk on Zoom
Author(s) :
Pablo Seleson (Oak Ridge National Laboratory)
Sam Reeve (Oak Ridge National Laboratory)
Abstract : Peridynamics is a nonlocal reformulation of classical continuum mechanics suitable for material failure and damage simulation, which has been successfully demonstrated as an effective tool for the simulation of complex fracture phenomena in many applications. However, the nonlocal nature of peridynamics makes it highly computationally expensive, compared to classical continuum mechanics, which often hinders large-scale fracture simulations. In this talk, we will present ongoing efforts to develop CabanaPD, a meshfree GPU-enabled peridynamics code for large-scale fracture simulations. CabanaPD is built on top of two main libraries: Kokkos and Cabana, both developed throughout the Exascale Computing Project (ECP). CabanaPD is performance-portable and exascale-capable, and it is designed to run on U.S. Department of Energy’s supercomputers, including the newly deployed Frontier, which is the first exascale machine and today's top supercomputer worldwide.
