A High Precision Direct Integration Scheme Based on Variational Principle and Its Applications

. Dynamics response of systems to impact or loading may be effectively treated by direct integration. However, it is often difficult to select the time-step of integration properly, especially in the case which the system is badly stiff. High Precision Direct integration based on variational principle is given (HPD-VP) for homogeneous systems and HHPD-VP method for the nonhomogeneous systems are given. This method not only takes the advantage of variational principle formula, which is much precise and is stiff A-stable, but also can avoid the truncation error of the computer. For the large systems, especially, the systems with different frequency or the stiff systems, our methods are stable, accurate and efficient. Numerical experiments show the convergence order of the scheme derived from the variational principle, and is much precise and is effective in engineering.


Introduction
Dynamic response is important for engineering.Such responses are analysed by means of direct integration methods, such as the Newmark-α, Wilson-θ or Houbolt schemes.When using such methods, the time-step size must be carefully selected, especially for some stiff equations.As a result, these direct integration schemes require a very s mall time -step and are generally timeconsuming and costly.Zhong [1] proposed High Precise Direct (HPD) method for homogeneous linear timeinvariant dynamic systems.HPD is more accurate than traditional R-K method and Newmark method, which is very valuable in engineering applications.For nonhomogeneous systems, Zhong [2] also established HPD-L algorith m, in wh ich the non-homogeneous term is simulated by piecewise linearization within a time step.
Meanwhile, Lin [3] HPD-F method, wh ich is based on Fourier expansion, is also proposed.However, HPD-L and HPD-F have to co mpute the inverse matrix.Zhou [4] [5] and Gu [6] proposed another improved HPD method independently, wh ich transformed the non-ho mogeneous term into homogeneous by expanding dimensions.Shi [7] and Fu [8] established HHPD-L and HHPD-C, which are based on Legendre and Chebyshev Serials, respectively.
Fro m the viewpoint of co mputation, finite element (FE) method is also a useful tool to discretize the time domain of structural dynamic problems.Early development on variational for t ime integration have been introduced by Argyris and Scharpf [9].Another approach to discrete time is discontinuous Galerkin method, which was first proposed by Reed and Hill [10], due to the flexible shape function, DG has been widely used in many areas of computation.Cockburn [11] gives LDG method, which is t ime-stepping.The t ime discontinuous Galerkin method leads to stable, higherorder accurate finite element methods, which is shown in Delfour [12].Gottlieb [13] g ives the unified discontinuous Galerkin framework for t ime integration, which show some special discontinuous Galerkin method equal to finite difference methods.[14] presented an improved predictor or mult i corrector solution algorith m based on the concept of time discontinuous Galerkin (TDG) FE method to compute the dynamic response of linear structures using the Gauss -Seidel method in an iterative way.
We propose HPD-VP method for the homogeneous systems and HHPD-VP method for the nonhomogeneous systems, which base on the variant princip le, and can lead the stable schemes.Firstly, HPD-VP and HHPD-VP are highly precise, co mpared to the tradit ional method, such Newmark-α, Wilson-θ, Runge-Kutta and so on.Secondly, HPD-VP and HHPD-VP are stiff A-stable, for the large systems especially, the stiff equations, our methods are reliable, and can simulate the mix frequency systems with large time step.

A Novel Variational Principle Approach
A novel variational principle approach for solving the following in itial value prob lem of linear differential equation.
The Eq. ( 1) can be technically (see [4] which is the homogeneous ODE.For Eq. ( 2), consider a partit ion of a closed interval [0, T] : and ℎ its length ℎ ≔ ‫ݐ‬ ାଵ − ‫ݐ‬ Establish the variational principle method for the Eq. ( 2).To describe the variational principle method, consider the following notation.The space of polyno mials of degree q or less on the interval ‫ܫ‬ .
and the space contains the piecewise polynomials to be U = ቄܷ: ܷ | ௧ ∈ P { ‫ܫ‬ }ቅ In the variational principle method, the trail function is continuous within each time interval ‫ܫ‬ .At each node ‫ݐ‬ , the limit ing values of numerical appro ximat ions from the left and the right are usually different.
Thus, we have Denote

The Higher Precise Integral Based on Variant Principle
For the first interval length ℎ = [ ‫ݐ‬ ଵ − ‫ݐ‬ ] , we can equally divide it into 2 ே , so the each length of subdivision is ߬ = ℎ 2 ே ൗ , and N is the parameter of HPD-VP method.The following shows how to computer ‫܅‬൫ ෩ ℎ൯.

Convergence of HPD-VP
Assume the error of Eq. ( 6) is ߳, then, ߳ is subject to

PROOF
For the initial value problem Eq. ( 2), it has the precise solution and the HPD-VP method has the approximation, where 8) and Eq. ( 9), the error ߳ should be

The systems with different frequencies
For the multi-frequency systems, the canonical equations are stiff equation.[15] shows courant number should satisfy ℎ‫ݓ‬ ≈ 1 where h is the length of time step, and w is the frequency of the systems , wh ich means for the high-frequency system, the traditional methods have to choose the tiny time step ℎ, vice versa.Consider the Hamiltonian system with d ifferent frequency signals.
The canonical equations We solve the Hamiltonian system with HPD-VP and some existing methods, such as FSJS3 [16], RKN [17], SPRKN [18] and time finite element method [19].The data in Tab. 1 are the the maximum of absolute errors with in ‫ݐ‬ ∈ [ ‫,ݏ0‬ ‫ݏ006‬ ] .As shown in Tab. 1 HPD-VP method can simu late the high and low frequency with the big time step.We show the error at different time, with figures as fo llo ws.As shown in Fig. 1, HPD-VP is accurate at different time with the relative large time step, however, when the step is larger, SPRKN, FSFS and TFE method are not stable, and lead to very large errors.

Spring-mass system
We consider the solution of the 3 degree-of-freedo m spring system [20].For simp licity, all the three masses are assumed to be unity.The stiffness of the left spring is assumed to be 10 7 , while the stiffness of the right spring is unity.In addition, the displacement of the left point mass is set to be sin ‫,ݐ2‬ while the displacements of the other two masses are initially equal to zero.Hence, the equilibriu m equations of this spring-mass system are given by Where ‫ݎ‬ ଵ is the reaction force at the left point mass.The 3 degree-of-freedom spring system shown in Fig. 2. Obviously, Eq. ( 10) is a stiff system.Since the displacement of the left node (i.e.‫ݑ‬ ଵ ) is known, Eq. ( 10) can be rewritten as with the initial values ‫ݑ‬ ଶ (0) = 0 and ‫ݑ‬ ଷ (0) = 0. Let ‫‬ = sin ‫ݐ2‬ ‫ݒ‬ ଶ = ‫̇ݑ‬ଶ ‫ݒ‬ ଷ = ‫̇ݑ‬ଷ ‫ݍ‬ = ‫̇‬ Then rewrite Eq. ( 11) by expanding dimensions as with the initial values ‫ݑ‬ ଶ ( 0 ) = 0 , ‫ݑ‬ ଷ ( 0 ) = 0, ‫‬ ( 0 ) = 0, ‫ݒ‬ ଶ (0) = 2 , ‫ݒ‬ ଷ (0) = 0, ‫ݍ‬ ( 0 ) = 2, We solve Eq. ( 11) by using Euler t ime integration method, Newmark−α [21], Wilson-β [22] with t ime step h = 0.0005, and HPD-VP method with time step h = 0.5.Fig. 3, Fig. 4 are the displacement response and Fig. 5, Fig. 6 are the velocity response.The exact solutions are obtained by the mode superposition method.We can get the conclusion that HHPD-VP method can highly increase the accuracy with big time step.

Conclusion
HPD-VP and HHPD-VP methods base on the variant principle, which are stiff A-stable methods, for large systems, our methods are stable, efficient and accurate.Especially for the systems with mix-frequency or the stiff system, traditional methods cannot simulate both high and low frequency with large time step, our method can solve this problem.HHPD-VP method avoid the computation of the inverse of matrix.For the dynamic systems, our method are reliable.The numerical experiments testify the results.

DOI: 10
.1051/ C Owned by the authors, published by EDP Sciences, 201

Figure 1 .
Figure 1.The error at different time by different method with different time step

Figure 2 .
Figure 2. Model problem of three degrees of freedom spring system

Figure 3 .
Figure 3. Displacement(right) and displacement error(left) of node 2 for various methods different time step

Figure 4 .Figure 5 .
Figure 4. Displacement(right) and displacement error(left) of node 3 for various methods different time step

Figure 6 .
Figure 6.Velocity(right) and velocity error(left) of node 3 for various methods.