Free element boundary integration method for solving heat conduction and mechanics problems

In this paper, a novel numerical method called the free element boundary integration method (FEBIM) is proposed for solving general engineering problems on the basis of the free element method (FrEM). The proposed method is a weak-form FrEM, which approximates physical quantities through isoparametric elements as in finite element method. The characteristic of FEBIM is that each used isoparametric element can be freely composed by the collocation node and its surrounding nodes. The main difference of FEBIM from FrEM is that the governing equations for internal nodes and the equilibrium equations for boundary points are transformed into boundary integrals in FEBIM. The source point of the local boundary integrals is the collocation node. As a result, FEBIM not only has the advantage of narrower bandwidth over FrEM, but also has higher computational precision, since the second-order derivative is not involved in the process of forming the equations. Numerical examples of heat transfer and solid mechanics problems are given in the paper to verify the correctness and stability of the proposed method.