Nonlocal Operator Method for Solving Partial Differential Equations: State-of-the-Art Review and Future Perspectives

Abstract

The nonlocal operator method (NOM) is based on nonlocal theory and employs nonlocal operators of integral form to replace the local partial differential operators. NOM naturally bridges models of different length scales and enables also the natural solution of problems with continuous to discontinuous solutions as they occur in the case of material failure. It also provides a natural framework for complex multifield problems. It is based on a variational principle or weighted residual method and only requires the definition of associated energy potential. As the NOM does not require any shape functions as ’traditional methods’ such as FEM, IGA or meshfree methods, its implementation is significantly simplified. It has been successfully applied to the solution of several partial differential equations (PDEs). This paper aims to provide a comprehensive description of the NOM together with a review of its major applications for the solution of PDEs for challenging engineering problems. Finally, we give some potential future research direction in the area of methods based on nonlocal operators.

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