TY  - JOUR
A1  - Zhang, Yongzheng
A1  - Ren, Huilong
T1  - Implicit implementation of the nonlocal operator method: an open source code
JF  - Engineering with computers
N2  - In this paper, we present an open-source code for the first-order and higher-order nonlocal operator method (NOM) including a detailed description of the implementation. The NOM is based on so-called support, dual-support, nonlocal operators, and an operate energy functional ensuring stability. The nonlocal operator is a generalization of the conventional differential operators. Combined with the method of weighed residuals and variational principles, NOM establishes the residual and tangent stiffness matrix of operate energy functional through some simple matrix without the need of shape functions as in other classical computational methods such as FEM. NOM only requires the definition of the energy drastically simplifying its implementation. The implementation in this paper is focused on linear elastic solids for sake of conciseness through the NOM can handle more complex nonlinear problems. The NOM can be very flexible and efficient to solve partial differential equations (PDEs), it’s also quite easy for readers to use the NOM and extend it to solve other complicated physical phenomena described by one or a set of PDEs. Finally, we present some classical benchmark problems including the classical cantilever beam and plate-with-a-hole problem, and we also make an extension of this method to solve complicated problems including phase-field fracture modeling and gradient elasticity material.
KW  - Strukturmechanik
KW  - Nonlocal operator method
KW  - Operator energy functional
KW  - Implicit
KW  - Dual-support
KW  - Variational principle
KW  - Taylor series expansion
KW  - Stiffness matrix
Y1  - 2022
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20220216-45930
UR  - https://link.springer.com/article/10.1007/s00366-021-01537-x
VL  - 2022
SP  - 1
EP  - 35
PB  - Springer
CY  - London
ER  - 
TY  - JOUR
A1  - Ren, Huilong
A1  - Zhuang, Xiaoying
A1  - Oterkus, Erkan
A1  - Zhu, Hehua
A1  - Rabczuk, Timon
T1  - Nonlocal strong forms of thin plate, gradient elasticity, magneto-electro-elasticity and phase-field fracture by nonlocal operator method
JF  - Engineering with Computers
N2  - The derivation of nonlocal strong forms for many physical problems remains cumbersome in traditional methods. In this paper, we apply the variational principle/weighted residual method based on nonlocal operator method for the derivation of nonlocal forms for elasticity, thin plate, gradient elasticity, electro-magneto-elasticity and phase-field fracture method. The nonlocal governing equations are expressed as an integral form on support and dual-support. The first example shows that the nonlocal elasticity has the same form as dual-horizon non-ordinary state-based peridynamics. The derivation is simple and general and it can convert efficiently many local physical models into their corresponding nonlocal forms. In addition, a criterion based on the instability of the nonlocal gradient is proposed for the fracture modelling in linear elasticity. Several numerical examples are presented to validate nonlocal elasticity and the nonlocal thin plate.
KW  - Bruchmechanik
KW  - Elastizität
KW  - Peridynamik
KW  - energy form
KW  - weak form
KW  - peridynamics
KW  - variational principle
KW  - explicit time integration
Y1  - 2021
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20211207-45388
UR  - https://link.springer.com/article/10.1007/s00366-021-01502-8
VL  - 2021
SP  - 1
EP  - 22
ER  -