Element differential method for free and forced vibration analysis for solids

Abstract

Abstracts In this work a new strong-form numerical method, element differential method (EDM), is proposed to perform free and forced vibration analysis of elastodynamic problems. The present method establishes the global algebraic system equations directly based on the strong form of the equilibrium equations without using any variational principles or energy principles. In this method, the isoparametric elements with a node inside them are utilized to discrete the geometries. And the direct differentiation of their shape functions is used to characterize the geometry and physical variables. A novel collocation technique is then proposed to generate the system of equations, in which the dynamic equilibrium equations are collocated only at the internal nodes of elements, and the traction equilibrium equations are collocated at the interface and outer surface nodes. Compared with the standard finite element method, no integrals are involved to form the coefficients of the system and a reduced and lamped mass matrix can be directly obtained from the density properties of the target problem. Since the mass term only exists at the internal nodes of elements, the dynamic coupling of final system equations of the structure can be reduced, which will greatly save the computational resources. Numerical examples about free and forced 2D and 3D dynamic problems are given to demonstrate the correctness and efficiency of the proposed method.