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The vector form intrinsic finite element (VFIFE) method is a solution technique for nonlinear structural problems, which describes a continuous body using a set of particles instead of a mathematical function. Thus, a dynamic particle equation can be established by Newton's law of motion, and a viscous or kinetic damping can be introduced to obtain the steady state of the structure. This paper focuses mainly on the development of a stability condition regarding the explicit central difference method used in VFIFE to guarantee the system's convergence. The process is established and evaluated in combination with a dynamic relaxation method with kinetic damping and discrete control theory. Four numerical examples of structure nonlinear problems are used to verify the accuracy, stability and efficiency of the method.

This work applies the dual reciprocity boundary element method (DRBEM) to the transient analysis of two-dimensional elastodynamic problems. Adopting the elastostatic fundamental solution in the integral formulation of elastodynamics creates an inertial volume integral as well as the boundary ones. This volume integral is further transformed into a surface integral by invoking the reciprocal theorem. The analysis includes the quadratic three-noded boundary elements in the spatial domain. Importantly, the second-order ordinary differential equations in the time domain formulated by the DRBEM are solved using the time-discontinuous Galerkin finite element method. Particularly, both the displacement and velocity variables in the time domain are independently represented by quadratic interpolation functions that allow the unknown variables to be discontinuous at the discrete time levels. This method can filter out the spurious high modes and provide solutions with a fifth-order accuracy. Numerical examples are presented, confirming that the proposed method is more stable and accurate than widespread direct time integration algorithms, such as the Houbolt method.

An innovative time integration method that incorporates spurious high-frequency dissipation capability into the so called “high precision direct integration algorithm” is presented, and its numerical stability and accuracy is discussed. The integration algorithm is named “high precision” to emphasize its numerical capability in reaching computer hardware precision. The proposed procedure employs the well-known state space approach to solve the simultaneous ordinary differential equations, the exact solution of which contains an exponential matrix to be efficiently computed using the truncated Taylor series expansion together with the power-of-two algorithm. The proposed method, belonging to the category of explicit methods, is found to provide better accuracy than many other existing time integration methods, and the integration scheme remains numerically stable over a wide range of frequencies of engineering interest. This paper is also devoted to study numerical accuracy of the Precise Integration Method in solving forced vibration problems, particularly near resonance conditions. The numerically obtained transfer functions are then compared with the analytical exact solution to detect spurious resonance. Finally, numerical examples are used to illustrate its high performance in numerical stability and accuracy. The proposed method carries the merit that can be directly applied to solve momentum equations of motion with exactly the same procedure.Copyright © 2013 by ASME