An asymptotically compatible treatment of traction loading in linearly elastic peridynamic fracture

Yu, Yue; You, Huaiqian; Trask, Nathaniel

doi:10.1016/j.cma.2021.113691

Cited by 35 publications

(33 citation statements)

References 55 publications

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“…In this work, without loss of generality, we consider the Dirichlet condition B I = I, where I is the identity operator. Other types of conditions, e.g., Neumann [40,42,46,47], Robin [43,48] or periodic [32], are also compatible with our learning algorithm.…”

Section: The Linear Peridynamic Solid (Lps) Modelmentioning

confidence: 85%

“…The LPS model has known well-posedness properties under certain assumptions [24]. This section summarizes the mathematical formulation for the LPS model and illustrates a meshfree discretization [39][40][41][42][43][44].…”

Section: The Linear Peridynamic Solid (Lps) Modelmentioning

confidence: 99%

“…A meshfree discretization of the LPS model. Given a collection of material points X = {x i } i=1,2,...,Np , we numerically evaluate L K (u) by employing the meshfree, particle discretization introduced in [40], which features ease of implementation and efficiency. At each material point x i , we adopt the following quadrature rule to approximate the integral in L K in (3.4), which we now denote by L h K .…”

Section: The Linear Peridynamic Solid (Lps) Modelmentioning

confidence: 99%

“…where V denotes the space of functions which should be integrated exactly. Following [40], in this work we take V := q(y) = p(y) |y−x| 3 | p ∈ P 5 (R 2 ) such that B δ (x) q(y)dy < ∞ and P 5 (R 2 ) denotes the space of quintic polynomials .…”

Section: The Linear Peridynamic Solid (Lps) Modelmentioning

confidence: 99%

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A data-driven peridynamic continuum model for upscaling molecular dynamics

You,

Yu,

Silling

et al. 2021

Preprint

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Nonlocal models, including peridynamics, often use integral operators that embed lengthscales in their definition. However, the integrands in these operators are difficult to define from the data that are typically available for a given physical system, such as laboratory mechanical property tests. In contrast, molecular dynamics (MD) does not require these integrands, but it suffers from computational limitations in the length and time scales it can address. To combine the strengths of both methods and to obtain a coarse-grained, homogenized continuum model that efficiently and accurately captures materials' behavior, we propose a learning framework to extract, from MD data, an optimal Linear Peridynamic Solid (LPS) model as a surrogate for MD displacements. To maximize the accuracy of the learnt model we allow the peridynamic influence function to be partially negative, while preserving the well-posedness of the resulting model. To achieve this, we provide sufficient well-posedness conditions for discretized LPS models with sign-changing influence functions and develop a constrained optimization algorithm that minimizes the equation residual while enforcing such solvability conditions. This framework guarantees that the resulting model is mathematically well-posed, physically consistent, and that it generalizes well to settings that are different from the ones used during training. We illustrate the efficacy of the proposed approach with several numerical tests for single layer graphene. Our two-dimensional tests show the robustness of the proposed algorithm on validation data sets that include thermal noise, different domain shapes and external loadings, and discretizations substantially different from the ones used for training.

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Section: The Linear Peridynamic Solid (Lps) Modelmentioning

confidence: 85%

Section: The Linear Peridynamic Solid (Lps) Modelmentioning

confidence: 99%

Section: The Linear Peridynamic Solid (Lps) Modelmentioning

confidence: 99%

Section: The Linear Peridynamic Solid (Lps) Modelmentioning

confidence: 99%

See 2 more Smart Citations

A data-driven peridynamic continuum model for upscaling molecular dynamics

You,

Yu,

Silling

et al. 2021

Preprint

Self Cite

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show abstract

“…Moreover, most existing numerical methods utilize spatial discretization and thus are prone to the curse of dimensionality. Newer alternatives such as phase-field [1][2][3][4] and peridynamics [5][6][7][8][9][10] models have been able to predict the outcome of carefully controlled experiments; however, the complexity of those models comes at a substantially higher computational cost. Furthermore, operational conditions and material properties in the field can have a significant deviation from those in a controlled laboratory environment, which impairs the reliability of the computed results.…”

Section: Introductionmentioning

confidence: 99%

A physics-informed variational DeepONet for predicting the crack path in brittle materials

Goswami,

Yin,

et al. 2021

Preprint

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Failure trajectories, identifying the probable failure zones, and damage statistics are some of the key quantities of relevance in brittle fracture applications. High-fidelity numerical solvers that reliably estimate these relevant quantities exist but they are computationally demanding requiring a high resolution of the crack. Moreover, independent intensive simulations need to be carried out even for a small change in domain parameters and/or material properties. Therefore, fast and generalizable surrogate models are needed to alleviate the computational burden but the discontinuous nature of fracture mechanics presents a major challenge to developing such models. We propose a physics-informed variational formulation of DeepONet (V-DeepONet) for brittle fracture analysis. V-DeepONet is trained to map the initial configuration of the defect to the relevant fields of interests (e.g., damage and displacement fields). Once the network is trained, the entire global solution can be rapidly obtained for any initial crack configuration and loading steps on that domain. While the original DeepONet is solely data-driven, we take a different path to train the V-DeepONet by imposing the governing equations in variational form and we also use some labelled data. We demonstrate the effectiveness of V-DeepOnet through two benchmarks of brittle fracture, and we verify its accuracy using results from high-fidelity solvers. Encoding the physical laws and also some data to train the network renders the surrogate model capable of accurately performing both interpolation and extrapolation tasks, considering that fracture modeling is very sensitive to fluctuations. The proposed hybrid training of V-DeepONet is superior to state-of-the-art methods and can be applied to a wide array of dynamical systems with complex responses.

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Data‐driven nonlocal model for fragmentation in the crushing of solids

Silling

2024

Num Anal Meth Geomechanics

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A technique is proposed for reproducing particle size distributions in three‐dimensional simulations of the crushing and comminution of solid materials. The method is designed to produce realistic distributions over a wide range of loading conditions, especially for small fragments. In contrast to most existing methods, the new model does not explicitly treat the small‐scale process of fracture. Instead, it uses measured fragment distributions from laboratory tests as the basic material property that is incorporated into the algorithm, providing a data‐driven approach. The algorithm is implemented within a nonlocal peridynamic solver, which simulates the underlying continuum mechanics and contact interactions between fragments after they are formed. The technique is illustrated in reproducing fragmentation data from drop weight testing on sandstone samples.

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An asymptotically compatible treatment of traction loading in linearly elastic peridynamic fracture

Cited by 35 publications

References 55 publications

A data-driven peridynamic continuum model for upscaling molecular dynamics

A data-driven peridynamic continuum model for upscaling molecular dynamics

A physics-informed variational DeepONet for predicting the crack path in brittle materials

Data‐driven nonlocal model for fragmentation in the crushing of solids

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