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Register a new userA unified, stable and accurate meshfree framework for peridynamic correspondence modeling. Part I: core methods
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Added by
Masoud Behzadinasab
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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The overarching goal of this work is to develop an accurate, robust, and stable methodology for finite deformation modeling using strong-form peridynamics (PD) and the correspondence modeling framework. We adopt recently developed methods that make use of higher-order corrections to improve the computation of integrals in the correspondence formulation. A unified approach is presented that incorporates the reproducing kernel (RK) and generalized moving least square (GMLS) approximations in PD to obtain higher-order gradients. We show, however, that the improved quadrature rule does not suffice to handle correspondence-modeling instability issues. In Part I of this paper, a bond-associative, higher-order core formulation is developed that naturally provides stability. Numerical examples are provided to study the convergence of RK-PD, GMLS-PD, and their bond-associat
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The overarching goal of this work is to develop an accurate, robust, and stable methodology for finite deformation modeling using strong-form peridynamics (PD) and the correspondence modeling framework. We adopt recently developed methods that make use of higher-order corrections to improve the computation of integrals in the correspondence formulation. A unified approach is presented that incorporates the reproducing kernel (RK) and generalized moving least square (GMLS) approximations in PD to obtain higher-order gradients. We show, however, that the improved quadrature rule does not suffice to handle correspondence-modeling instability issues. In Part I of this paper, a bond-associative, higher-order core formulation is developed that naturally provides stability. Numerical examples are provided to study the convergence of RK-PD, GMLS-PD, and their bond-associat
In this work, we study the reproducing kernel (RK) collocation method for the peridynamic Navier equation. We first apply a linear RK approximation on both displacements and dilatation, then back-substitute dilatation, and solve the peridynamic Navier equation in a pure displacement form. The RK collocation scheme converges to the nonlocal limit and also to the local limit as nonlocal interactions vanish. The stability is shown by comparing the collocation scheme with the standard Galerkin scheme using Fourier analysis. We then apply the RK collocation to the quasi-discrete peridynamic Navier equation and show its convergence to the correct local limit when the ratio between the nonlocal length scale and the discretization parameter is fixed. The analysis is carried out on a special family of rectilinear Cartesian grids for the RK collocation method with a designated kernel with finite support. We assume the Lam{e} parameters satisfy $lambda geq mu$ to avoid adding extra constraints on the nonlocal kernel. Finally, numerical experiments are conducted to validate the theoretical results.
We introduce a hybrid method to couple continuous Galerkin finite element methods and high-order finite difference methods in a nonconforming multiblock fashion. The aim is to optimize computational efficiency when complex geometries are present. The proposed coupling technique requires minimal changes in the existing schemes while maintaining strict stability, accuracy, and energy conservation. Results are demonstrated on linear and nonlinear scalar conservation laws in two spatial dimensions.
As a nonlocal extension of continuum mechanics, peridynamics has been widely and effectively applied in different fields where discontinuities in the field variables arise from an initially continuous body. An important component of the constitutive model in peridynamics is the influence function which weights the contribution of all the interactions over a nonlocal region surrounding a point of interest. Recent work has shown that in solid mechanics the influence function has a strong relationship with the heterogeneity of a materials micro-structure. However, determining an accurate influence function analytically from a given micro-structure typically requires lengthy derivations and complex mathematical models. To avoid these complexities, the goal of this paper is to develop a data-driven regression algorithm to find the optimal bond-based peridynamic model to describe the macro-scale deformation of linear elastic medium with periodic heterogeneity. We generate macro-scale deformation training data by averaging over periodic micro-structure unit cells and add a physical energy constraint representing the homogenized elastic modulus of the micro-structure to the regression algorithm. We demonstrate this scheme for examples of one- and two-dimensional linear elastodynamics and show that the energy constraint improves the accuracy of the resulting peridynamic model.
We prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits that the DG approximation is constructed from polynomial basis functions and that integrals are approximated with high-order accurate Legendre-Gauss-Lobatto quadrature. The theoretical discussion serves to re-contextualize stable filtering results for finite difference methods into the DG setting. It is shown that the stability of the filtering is equivalent to a particular contractivity condition borrowed from the analysis of so-called transmission problems. As such, the temporal stability proof relies on the fact that the underlying spatial discretization of the problem possesses a semi-discrete bound on the solution. Numerical tests are provided to verify and validate the underlying theoretical results.
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