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 引用本文: 刘硕, 方国东, 王兵, 付茂青, 梁军. 近场动力学与有限元方法耦合求解热传导问题[J]. 力学学报, 2018, 50(2): 339-348.
Liu Shuo, Fang Guodong, Wang Bing, Fu Maoqing, Liang Jun. STUDY OF THERMAL CONDUCTION PROBLEM USING COUPLED PERIDYNAMICS AND FINITE ELEMENT METHOD[J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(2): 339-348.
 Citation: Liu Shuo, Fang Guodong, Wang Bing, Fu Maoqing, Liang Jun. STUDY OF THERMAL CONDUCTION PROBLEM USING COUPLED PERIDYNAMICS AND FINITE ELEMENT METHOD[J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(2): 339-348.

## STUDY OF THERMAL CONDUCTION PROBLEM USING COUPLED PERIDYNAMICS AND FINITE ELEMENT METHOD

• 摘要: 求解含裂纹等不连续问题一直是计算力学的重点研究课题之一,以偏微分方程为基础的连续介质力学方法处理不连续问题时面临很大的困难. 近场动力学方法是一种基于积分方程的非局部理论,在处理不连续问题时有很大的优越性. 本文提出了求解含裂纹热传导问题的一种新的近场动力学与有限元法的耦合方法. 结合近场动力学方法处理不连续问题的优势以及有限元方法计算效率高的优势,将求解区域划分为两个区域,近场动力学区域和有限元区域. 包含裂纹的区域采用近场动力学方法建模,其他区域采用有限元方法建模. 本文提出的耦合方案实施简单方便,近场动力学区域与有限元区域之间不需要设置重叠区域. 耦合方法通过近场动力学粒子与其域内所有粒子(包括近场动力学粒子和有限元节点)以非局部方式连接,有限元节点与其周围的所有粒子以有限元方式相互作用. 将有限元热传导矩阵和近场动力学粒子相互作用矩阵写入同一整体热传导矩阵中,并采用Guyan缩聚法进一步减小计算量. 分别采用连续介质力学方法和近场动力学方法对一维以及二维温度场算例进行模拟,结果表明,本文的耦合方法具有较高的计算精度和计算效率. 该耦合方案可以进一步拓展到热力耦合条件下含裂纹材料和结构的裂纹扩展问题.

Abstract: 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.

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