Название: Computational Methods Based on Peridynamics and Nonlocal Operators: Theory and Applications
Автор: Timon Rabczuk, Huilong Ren, Xiaoying Zhuang
Издательство: Springer
Год: 2023
Страниц: 327
Язык: английский
Формат: pdf (true)
Размер: 10.29 MB

This book provides an overview of computational methods based on peridynamics and nonlocal operators and their application to challenging numerical problems which are difficult to deal with traditional methods such as the finite element method, material failure being “only” one of them. The authors have also developed a higher-order nonlocal operator approaches capable of solving higher-order partial differential equations on arbitrary domains in higher-dimensional space with ease. This book is of interest to those in academia and industry.

Feynman once said, calculus is the language of God. Calculus uses the partial differential derivatives (PDE) and integrals to account for various physical phenomena. On one hand, it is well known that many physical problems or physical theories are formulated concisely by partial differential equations. No matter how complicated the physical phenomena appear, the PDEs to describe the mechanism are just one/several lines of equations. Mathematically, PDEs are a combinational result of partial differential derivatives of different orders. Partial differential derivatives are defined at a point without size, in this sense, the PDEs model can be viewed as a local model. On the other hand, integral expression is defined in a finite domain, which consists of infinite points. In this sense, the integral can be viewed as a nonlocal model. Or we can say the local model corresponds to differential equations while the nonlocal model is related to integral equations. Physically, our world contains not just local models but also a lot of nonlocal models, for example, the universal gravitation, Quantum entanglement and social network. Nonlocal viewpoints offer us a new perspective to understand the world. The role of integral equations in solving physical problems is underestimated for a long time. It is time to view the world non-locally.

Among the nonlocal models, Peridynamics (PD) attracts much attention in the field of fracture mechanics. One key feature of PD is the nonlocality, which is quite different from the ideas in conventional methods such as FEM and meshless methods. However, conventional PD suffers from problems such as constant horizon, explicit algorithm, hourglass mode. In this book, by examining the nonlocality with scrutiny, we proposed several new concepts such as Dual-Horizon (DH) in PD, Dual-Support (DS) in smoothed particle hydrodynamics (SPH), nonlocal operators and operator energy functional. The conventional PD (SPH) is incorporated in the DH-PD (DS-SPH), which can adopt an inhomogeneous discretization and inhomogeneous support domains. The DH-PD (DS-SPH) can be viewed as some fundamental improvement on the conventional PD (SPH). Dual formulation of PD and SPH allows h-adaptivity while satisfying the conservation of linear momentum, angular momentum and energy.




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