OBMeshfree: An optimization-based meshfree solver for nonlocal diffusion and peridynamics models

We present OBMeshfree, an Optimization-Based Meshfree solver for compactly supported nonlocal integro-differential equations (IDEs) that can describe material heterogeneity and brittle fractures. OBMeshfree is developed based on a quadrature rule calculated via an equality constrained least square problem to reproduce exact integrals for polynomials. As such, a meshfree discretization method is obtained, whose solution possesses the asymptotically compatible convergence to the corresponding local solution. Moreover, when fracture occurs, this meshfree formulation automatically provides a sharp representation of the fracture surface by breaking bonds, avoiding the loss of mass. As numerical examples, we consider the problem of modeling both homogeneous and heterogeneous materials with nonlocal diffusion and peridynamics models. Convergences to the analytical nonlocal solution and to the local theory are demonstrated. Finally, we verify the applicability of the approach to realistic problems by reproducing high-velocity impact results from the Kalthoff-Winkler experiments. Discussions on possible immediate extensions of the code to other nonlocal diffusion and peridynamics problems are provided. OBMeshfree is freely available on GitHub.Comment: For associated code, see https://github.com/youhq34/meshfree_quadrature_nonloca

An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞ (Ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence

An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞(Ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence

A Physics-Guided Neural Operator Learning Approach to Model Biological Tissues from Digital Image Correlation Measurements

We present a data-driven workflow to biological tissue modeling, which aims to predict the displacement field based on digital image correlation (DIC) measurements under unseen loading scenarios, without postulating a specific constitutive model form nor possessing knowledges on the material microstructure. To this end, a material database is constructed from the DIC displacement tracking measurements of multiple biaxial stretching protocols on a porcine tricuspid valve anterior leaflet, with which we build a neural operator learning model. The material response is modeled as a solution operator from the loading to the resultant displacement field, with the material microstructure properties learned implicitly from the data and naturally embedded in the network parameters. Using various combinations of loading protocols, we compare the predictivity of this framework with finite element analysis based on the phenomenological Fung-type model. From in-distribution tests, the predictivity of our approach presents good generalizability to different loading conditions and outperforms the conventional constitutive modeling at approximately one order of magnitude. When tested on out-of-distribution loading ratios, the neural operator learning approach becomes less effective. To improve the generalizability of our framework, we propose a physics-guided neural operator learning model via imposing partial physics knowledge. This method is shown to improve the model's extrapolative performance in the small-deformation regime. Our results demonstrate that with sufficient data coverage and/or guidance from partial physics constraints, the data-driven approach can be a more effective method for modeling biological materials than the traditional constitutive modeling.This is a pre-print of the article You, Huaiqian, Quinn Zhang, Colton J. Ross, Chung-Hao Lee, Ming-Chen Hsu, and Yue Yu. "A Physics-Guided Neural Operator Learning Approach to Model Biological Tissues from Digital Image Correlation Measurements." arXiv preprint arXiv:2204.00205 (2022). DOI: 10.48550/arXiv.2204.00205. Copyright 2022 The Authors. Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0). Posted with permission

A partitioned coupling framework for peridynamics and classical theory : analysis and simulations

We develop and analyze a concurrent framework for coupling peridynamics and the corresponding classical elasticity theory, with applications to the numerical simulations of damage problems. In this framework, the peridynamic model and the elastic model are solved separately and coupled with a partitioned approach. In the region where material failure is expected to initiate, we employ the peridynamic theory. In the rest of the problem domain, the material is modeled by the classical elasticity theory. On the peridynamic–classical theory interface, there is a transition region where the two subdomains overlap. The two solvers communicate by exchanging proper boundary conditions at the peridynamic–classical theory interface, which enables a modular software implementation. We analyze different coupling strategies on a 1D simplified problem and obtain expressions for the optimal reduction factor (convergence rate index). The selection of optimal coupling parameters is verified with numerical experiments, where we demonstrate that the optimal Robin coefficient from 1D simplified problem analysis can be extrapolated to more complicated problems, including cases with damage. Both the analysis and the numerical results suggest that the optimal Robin boundary condition on the classical theory side combined with a Dirichlet boundary condition with Aitken relaxation rule on the peridynamic side would be the most robust choice. Comparing with the commonly employed Dirichlet interface conditions, the optimal Robin boundary condition together with Aitken relaxation accelerates the coupling convergence rate by 10 times. With the developed optimal coupling strategy, we also numerically demonstrate the coupling framework’s asymptotic convergence to the local solution and its capability to capture crack initiation and growth in 2D problems340905931CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO - CNPQCOORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIOR - CAPES140501/2009-6PDEE - 5379/10-5Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. Yue Yu would like to acknowledge support from the National Science Foundation under awards DMS 1620434. Fabiano F. Bargos and Marco L. Bittencourt would like to acknowledge support from CNPq (grant 140501/2009-6), CAPES (grant PDEE - 5379/10-5). Huaiqian You was partially supported by the National Science Foundation under awards DMS 1620434. Michael L. Parks acknowledges support from the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4). George E. Karniadakis would like to acknowledge support from NIH (grant U01HL116323 “Multi-scale, Multiphysics Model of Thrombus Biomechanics in Aortic Dissection”). This work also used resources of the “Centro Nacional de Processamento de Alto Desempenho em São Paulo (CENAPAD-SP). https://www.cenapad.unicamp.br/diversos/guia/guia.shtml