近场动力学与有限元方法耦合求解热传导问题

To accurately model discontinuous problems with cracks is one important topic in computational mechanics. It is very difficult to solve discontinuous problems using continuum mechanics methods based on partial differential equations. However, peridynamics (PD), a non-local theory based on integral equations, has great advantages in solving these problems. In this paper, a new method is proposed to solve heat conduction problems with cracks using coupled PD and finite element method (FEM). This method has both the advantage of the computational efficiency of FEM and the advantage of PD in solving discontinuous problems. The computational domain can be partitioned into two regions, PD region and FEM region. The region containing the crack is modeled by PD, and the other region is modeled by FEM. Application of the coupling scheme proposed in this paper is simple and convenient, since there is no need to introduce an overlapping region between PD region and FEM region. As for the coupling approach, the PD particle is connected non-locally to all particles (PD particles and finite element nodes) within its horizon, whereas the finite element node interacts with other nodes in the finite element manner. The heat conduction matrices of FEM and the matrices of the interaction between PD particles are combined to be a global heat conduction matrix. The Guyan reduction method is used to further reduce the computational cost. The temperature fields of a one-dimensional bar and a two-dimensional plate obtained by the coupling approach are compared with classical solutions. Results show that the proposed coupling method is accurate and efficient. The coupling scheme can be extended to solve crack propagation problems with the thermo-mechanical load.