# A Time Integration Method Based on Galerkin Weak Form for Nonlinear Structural Dynamics
## 1 Introduction  
The step-by-step time integration method is the most commonly used numerical method for transient analysis of nonlinear dynamics, and can in general be classified into an explicit and implicit scheme. For the explicit integration algorithm, it is not required to solve coupled equations in each time step, and this method is more suitable for determining solutions of wave propagation problems. While for the implicit algorithm, although the requirement for solutions of coupled equations in each time step makes it expensive, it still has been widely used in solid mechanics and structural engineering because of the superior numerical stability. In fact, it may often occur that an implicit algorithm of unconditional stability in linear problems fails to give a stable response for some nonlinear problems. For example, significant instability may appear in the trapezoidal rule [1] when it is used for the evaluation of large deformation problems and long time range responses [2; 3; 4; 5]. For linear dynamics, the stability of an integration algorithm can be estimated by spectral analysis, whereas in nonlinear dynamics spectral analysis is only one of the requirements to remain stable, and the conservation or decay of total energy within a time step interval should be the very sufficient condition for effective solutions [3]. From this point of view, the time integration method can be classified into three categories: One of numerical dissipation, one of enforced conservation of energy and one of algorithmic conservation of energy. Firstly, the algorithm belonging to the numerical dissipation category uses numerical dissipation to achieve energy conservation or decay. For example, numerical damping of high-order-frequency can be introduced into the Newmark method with \(\gamma>1/2\), however, in this case, the method only possesses an accuracy of the first order. For another example, in the HHT-\(\alpha\)[6] method and generalized-\(\alpha\)[7] method, the numerical accuracy has been improved to the second order and controllable numerical damping has been introduced as well. For such methods, the suitable selection of parameters turns out to be the key to achieving effective energy decay. Secondly,the algorithm of enforced conservation of energy was first proposed by [PERSON] et al. [8]. It is formed by introducing Lagrange multipliers into the trapezoidal integral method, and then achieves the target to satisfy energy conservation. However, the Newton-Raphson iteration may sometimes fail in the end [9]. The third kind of algorithm with algorithmic conservation of energy [10] was first proposed by [PERSON] and [PERSON] [11]. Its basic idea comes from the adjustment of the mid-point rule, which can realize energy conservation by expressing the stress tensor of the midpoint of the time interval in terms of the average of the stress tensors at the initial time and the end time of that time interval [12; 13; 14; 15]. Besides of the finite difference method, this is one of the main numerical methods for initial value problems (IVPs). In recent years, many alternative finite element methods (FEMs) for IVPs have been developed [16], which are as well-known as time FEM. According to the different ways of construction, the time FEM can be roughly classified into three kinds: (1) Constructed by [PERSON] variational principle, which transforms the initial-boundary value problem into the equivalent boundary value problem (BVP) by means of Laplace positive and inverse transformation [17; 18]; (2) Constructed by [PERSON]'s variational principle or [PERSON]'s law of variation [19; 20; 21], in which [PERSON] [22; 23], [PERSON] [24], and [PERSON] [25] have made important achievements successively and; (3) Constructed by the weighted residual method, and is what we use in the present paper. For time FEMs established by the weighted residual method, the trial functions commonly used involve Lagrange interpolation polynomials [26], linear/quadratic polynomials [27; 28], Taylor expansion [29; 30], or Taylor formulae [31; 32].