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  - Zhang, Yongzheng
T1  - Nonlocal dynamic Kirchhoff plate formulation based on nonlocal operator method
JF  - Engineering with Computers
N2  - In this study, we propose a nonlocal operator method (NOM) for the dynamic analysis of (thin) Kirchhoff plates. The nonlocal Hessian operator is derived based on a second-order Taylor series expansion. The NOM does not require any shape functions and associated derivatives as ’classical’ approaches such as FEM, drastically facilitating the implementation. Furthermore, NOM is higher order continuous, which is exploited for thin plate analysis that requires C1 continuity. The nonlocal dynamic governing formulation and operator energy functional for Kirchhoff plates are derived from a variational principle. The Verlet-velocity algorithm is used for the time discretization. After confirming the accuracy of the nonlocal Hessian operator, several numerical examples are simulated by the nonlocal dynamic Kirchhoff plate formulation.
KW  - Angewandte Mathematik
KW  - nonlocal operator method
KW  - nonlocal Hessian operator
KW  - operator energy functional
KW  - dual-support
KW  - variational principle
Y1  - 2022
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20220209-45849
UR  - https://link.springer.com/article/10.1007/s00366-021-01587-1
VL  - 2022
SP  - 1
EP  - 35
PB  - Springer
CY  - London
ER  -