Hao Tian

In this paper, we introduce a novel bond-based peridynamic model that utilizes a Gaussian kernel function. Previous peridynamic models, when directly discretized, have exhibited a lack of asymptotically compatibility with their corresponding local elastic solutions. Additionally, these models have faced challenges in accurately computing the volume of intersecting regions between the horizon of the material point and its neighboring cells. These difficulties in numerical simulations within peridynamics have spurred numerous efforts to develop corrective numerical methods. While such corrective methods have addressed certain issues, they remain complex to formulate and computationally intensive. Instead of addressing these challenges through modified numerical discretization, this paper presents a novel approach: bond-based peridynamics with a Gaussian kernel . This model replaces the bounded kernel function commonly used in existing peridynamic models with an unbounded Gaussian kernel function. Through direct meshfree discretization, we demonstrate that the solution of the proposed model , without the need for compatibility correction or volume correction, aligns with the corresponding local elastic solution. Furthermore, we derive the relevant material parameters based on energy equivalence to an elastic continuum model. Building upon this foundation, we propose a damage model and a linearized version of the proposed peridynamic model. We validate our proposed model through simulations of 2D smooth problems, demonstrating its convergence and accuracy. Finally, we assess the applicability of our approach to realistic scenarios by predicting displacements caused by a 2D plate with pre-existing cracks and replicating high-velocity impact results from the Kalthoff-Winkler experiments.

In this paper, we introduce a novel bond-based peridynamic model that utilizes a Gaussian kernel function. Previous peridynamic models, when directly discretized, have exhibited a lack of asymptotically compatibility with their corresponding local elastic solutions. Additionally, these models have faced challenges in accurately computing the volume of intersecting regions between the horizon of the material point and its neighboring cells. These difficulties in numerical simulations within peridynamics have spurred numerous efforts to develop corrective numerical methods. While such corrective methods have addressed certain issues, they remain complex to formulate and computationally intensive.Instead of addressing these challenges through modified numerical discretization, this paper presents a novel approach: bond-based peridynamics with a Gaussian kernel . This model replaces the bounded kernel function commonly used in existing peridynamic models with an unbounded Gaussian kernel function. Through direct meshfree discretization, we demonstrate that the solution of the proposed model , without the need for compatibility correction or volume correction, aligns with the corresponding local elastic solution. Furthermore, we derive the relevant material parameters based on energy equivalence to an elastic continuum model. Building upon this foundation, we propose a damage model and a linearized version of the proposed peridynamic model. We validate our proposed model through simulations of 2D smooth problems, demonstrating its convergence and accuracy. Finally, we assess the applicability of our approach to realistic scenarios by predicting displacements caused by a 2D plate with pre-existing cracks and replicating high-velocity impact results from the Kalthoff-Winkler experiments.

A shallow water equations-based model with an improved local time-stepping (LTS) scheme is developed for modeling coastal hydrodynamics across multiple scales, from large areas to detailed local regions. To enhance the stability of the shallow water model for long-duration simulations and at larger LTS gradings, a prediction-correction method using a single-layer interface that couples coarse and fine time discretizations is adopted. The proposed scheme improves computational efficiency with an acceptable additional computational burden and ensures accurate conservation of time truncation errors in a discrete sense. The model performance is verified with respect to conservation and computational efficiency through two idealized tests: the spreading of a drop of shallow water and a tidal flat/channel system. The results of both tests demonstrate that the improved LTS scheme maintains precision as the LTS grading increases, preserves conservation properties, and significantly improves computational efficiency with a speedup ratio of up to 2.615. Furthermore, we applied the LTS scheme to simulate tides at grid scales of 40,000 m to 200 m for a portion of the Northwest Pacific. The proposed model shows promise for modeling cross-scale hydrodynamics in complex coastal and ocean engineering problems.

Existing nonlocal diffusion models are predominantly classified into two categories: bond-based models, which involve a single-fold integral and usually simulate isotropic diffusion, and state-based models, which contain a double-fold integral and can additionally prototype anisotropic diffusion. While bond-based models exhibit computational efficiency, they are somewhat limited in their modeling capabilities. In this paper, we develop a novel bond-based nonlocal diffusion model with matrix-valued coefficients in non-divergence form. Our approach incorporates the coefficients into a covariance matrix and employs the multivariate Gaussian function with truncation to define the kernel function, and subsequently model the nonlocal diffusion process through the bond-based formulation. We successfully establish the well-posedness of the proposed model along with deriving some of its properties on maximum principle and mass conservation. Furthermore, an efficient linear collocation scheme is designed for numerical solution of our model. Comprehensive experiments in two and three dimensions are conducted to showcase application of the proposed nonlocal model to both isotropic and anisotropic diffusion problems and to demonstrate numerical accuracy and effective asymptotic compatibility of the proposed collocation scheme.

The non-locality property of peridynamic poses significant computational challenges, especially for problems with discontinuous solutions (e.g., cracks). An efficient matrix-structure-based fast (MSBF) algorithm is proposed for the meshfree method (MF) in solving the linearized bond-based peridynamic models on a rectangular Cartesian computational domain with a uniform mesh. Taking full advantage of the intrinsic Teoplitz structure of the block-diagonal stiffness matrix, the fast Fourier transform (FFT) algorithm is utilized to significantly reduce the number of floating point operations (FLOPS) count to achieve a substantial speed up at fine mesh resolutions than the classical matrix-vector multiplication (MVMP) algorithm. Both methods are applied to multi-dimensional static and dynamic problems, including general boundary conditions and unsteady crack propagation, with good results. The MF-MSBF algorithm demonstrates its substantial computational efficiency with greatly reduced CPU times and smaller memory footprint over the MF-MVMP algorithm in multi-dimensional peridynamic simulations with fine meshes.

In this paper, we present a dual-horizon nonlocal diffusion model, in which the influence area at each point consists of a standard sphere horizon and an irregular dual horizon and its geometry is determined by the distribution of the varying horizon parameter. We prove the mass conservation and maximum principle of the proposed nonlocal model, and establish its well-posedness and convergence to the classical diffusion model. Noticing that the dual horizon-related term in fact vanishes in the corresponding variational form of the model, we then propose a finite element discretization for its numerical solution, which avoids the difficulty of accurate calculations of integrals on irregular intersection regions between the mesh elements and the dual horizons. Various numerical experiments in two and three dimensions are also performed to illustrate the usage of the variable horizon and demonstrate the effectiveness of the numerical scheme. Nonlocal diffusion, dual-horizon, maximum principle, finite element discretization, asymptotically compatible. 65R20, 65N35, 74S05 COPYRIGHT: © Global Science Press