Theory and Implementation of a Two-Step Unconditionally Stable Explicit Integration Algorithm for Vibration Analysis of Structures

1School of Civil Engineering, Central South University, Changsha, Hunan 410075, China 2National Engineering Laboratory for High Speed Railway Construction, Changsha 410075, China 3International Joint Research Laboratory of Earthquake Engineering, Tongji University, 1239 Siping Road, Shanghai 2000092, China 4Department of Civil Engineering, The University of British Columbia, 2329 West Mall, Vancouver, BC, Canada V6T 1Z4


Introduction
With the advance of computer hardware, more advanced computer software is being used to solve the response of the structure under dynamic loads.Hence, more advanced numerical methods have been developed to efficiently and accurately predict the dynamic response of the structure.In the past, multiple implicit [1][2][3][4][5][6][7] and explicit [8][9][10][11][12] integration algorithms have been developed, whose relationship [13] and design methods are comprehensively analyzed [14][15][16].The explicit method has high computational efficiency, while the implicit method has higher stable.The commonly referenced explicit algorithm is Central Difference Method (CDM) (ref), CDM is easy to use, but it is only conditionally stable.Many new explicit methods which are unconditionally stable [17][18][19] have been developed.Though these methods are unconditionally stable, they require multiple steps which are not as efficient as the CDM.The paper presents a novel 2-step explicit integration method, named Unconditional Stable Two-Step Explicit Displacement Method (USTEDM), that is used to solve the dynamic response of a structure.USTEDM is efficient to use and unconditionally stable.The performance of USTEDM is compared with other explicit algorithms.The result shows that USTEDM has superior performance and can be used efficiently in solving vibration response of civil engineering structures.

Derivation of the Two-Step Integration Algorithm
Equation (1) shows the generalized dynamic equations of motion of a structure:  () + V () +  () =  () , where (), V(), () are the acceleration, velocity, and displacement of the structure, respectively., ,  are the mass, damping, and stiffness of the structure, respectively.() is external applied force.

Shock and Vibration
In this paper, a two-step integration algorithm, as presented in (2), is proposed to solve the dynamic equation as presented in (1): where  +2Δ is the displacement at time step  + 2Δ and Δ is the integration time step.Similarly,  +Δ and   are the displacement at time  + Δ and , while  +2Δ ,  +Δ and   are the force at  + 2Δ,  + Δ, and , respectively.
Equation ( 8) is the preliminary form of USTEDM.

Determination of Coefficients
3.1.Accuracy Analysis.The accuracy of USTEDM can be examined using convergence test.Equation (9) shows the Taylor series expansion of the displacement, velocity, and acceleration of the structure at time step of  + 2Δ,  + Δ, respectively.

Stability Analysis.
According to Lax theorem, ( 8) is convergent if and only if it is stable.The stability of (8) can be analyzed through studying a single degree-of-freedom (SDOF) system.For a SDOF system, (8) can be rewritten as Amplification matrix  in ( 22) determining ( 8)'s stability is Substituting  = 2,  =  2 , Δ = Ω (-damp ratio, -natural frequency) into (23) yields The characteristic equation of matrix  can be written as where  is the identity matrix,  denotes eigenvalue of , and The stability of (8) depends on spectral radius () of matrix  and ( 8) is stable only when Because () is determined by  1 ,  2 , the stability domain of (8) can be expressed by functions  1 and  2 .In similar way, the study of stability made by Hilber and Hughes [20] is performed.Firstly, the boundary line of stability domain of (8) where () = 1 is to be derived.() = 1 means that where || max is the maximum amplitude of .Complex value  whose amplitude is 1 can be expressed by  =   , where Shock and Vibration 5  = √ −1 and  is argument of .Substituting  =   into (25) yields The identities including   = cos  +  sin , cos 2 = 2 cos 2  − 1, sin 2 = 2 sin  cos  are used and (29) can be rewritten as (2 cos  (cos  −  1 ) +  2 − 1) Equation ( 30) is satisfied only when The boundary lines of stability domain of ( 8) can be determined by (31) when  ∈ [0, 2].It can be derived that when 0 <  < ,  <  < 2, (31) are satisfied when When  = 0,  = 2, (31) are satisfied when When  = , (31) are satisfied when The boundary of stability domain of ( 8) is then made up with three lines described by (32), (33), and (34), respectively, in  1 - 2 plane as shown in Figure 1.Because () is continuous function of  1  2 and when  1 =  2 = 0, () = 0 < 1, the stability domain of ( 8) is the inner part of the triangle as shown in Figure 1 and can be expressed by (35)

Final Form of Two-
Step Displacement Method with Third-Order Accuracy.Combining ( 20) and ( 8) yields Combining ( 20), (26), and (35) yields, after simplification, the stability domain of (36): According to (36) and ( 37), the two-step displacement methods satisfying different requirements can be obtained.The explicitness of method can be obtained only when  11 = 0 and the unconditional stability can be obtained when (37) are always satisfied when Ω ∈ [0, ∞).It can be found that (37) are always satisfied only when Substituting  8 = −0.25, 11 = 0 into (36) yields Equation (39) shows the final form of USTEDM, which is an unconditionally stable two-step explicit integration algorithm.USTEDM is third order accurate.The analysis shows that USTEDM has very similar numerical attributes as Newmark's average acceleration method.Figure 2 shows that the spectral radius of amplification matrix of USTEDM is 1, which is the same as that of Newmark's average acceleration method.
Figure 3 shows that the numerical damping of USTEDM is zero, which is the same as that of Newmark's average acceleration method.Figure 4 shows the relative period error of USTEDM is minimal compared with other methods including Houbolt Method, Wilson- Method ( = 1.4), and Generalized  Method (  = 0.3,   = 0.35).Though with similar numerical attributes as Newmark's average acceleration method, USTEDM is explicit and has great computation advantage over the implicit Newmark's average acceleration method.
The applicability of the USTEDM for nonlinear problem is also investigated.Compared with other unconditional stable explicit algorithms [17][18][19], whose main calculation amount includes at least solving two-reverse matrix of nondiagonal matrix for displacement and velocity, respectively, USTEDM shows higher calculation efficiency since its main calculation amount includes solving only onereverse matrix of nondiagonal matrix for displacement.In addition, USTEDM shows minimal memory requirements, since its calculation only involves displacement when only displacement-related nonlinearity occurs.This shows that USTEDM can be viewed as a complementary or improved form of conditional stable explicit CDM.(1) based on known {  }, { −Δ },   , and outer force vector {  }, calculate the "effective" load vector { +Δ }; (2) form "effective" mass matrix: M = +0.5Δ+0.25  Δ 2 ; (3) calculate { +Δ } = M−1 { f+Δ }; (42)

Numerical Example.
In this example [7,17], the forced vibration response of an 200-degree-of-freedom spring-mass system as shown in Figure 5 is examined using the proposed numerical method.The structural properties of this system are assumed to be   = 100 kg and   = 10 7 [1 − (  −  −1 ) 2 ] N/m in which  = 1, 2, . . ., 200.This system is subjected to a ground acceleration of 10 sin().The lowest and highest natural frequency is 2.62 rad/s and 632.4 rad/s, respectively [17].According to the highest natural frequency, a time step duration of Δ = 0.001 s is obtained from the Newmark's average acceleration method and is selected to be the exact solution.Displacement responses versus time of the largest degree of freedom are plotted in Figure 6.Analysis is accomplished by considering time step duration to be equal to 0.02 s.The displacement errors are presented in Figure 7.
It can be seen that here USTEDM provides the most accurate displacement solutions, compared with Houbolt Method and commonly used Newmark's average acceleration method.More importantly, without needs of iteration calculation, the unconditionally stable explicit USTEDM has much less time consumption of calculation, compared with those unconditionally stable implicit methods, which are listed in Table 1.

Conclusions
A novel two-step unconditionally stable explicit displacement method, named Unconditional Stable Two-Step Explicit Displacement Method (USTEDM), is proposed in this paper.USTEDM has third-order accuracy and is unconditionally stable.The conclusions mainly include the following: (1) The novel way of constructing direct time integration methods from the start point of dimensional analysis is applicable, and it is believed that it will contribute to improvement of numerical calculation in structural dynamics.
(2) With no prerequisite difference assumption of velocity and acceleration, the derivation of USTEDM is completely different from any documented one.With unconditional stability, third-order accuracy, higher calculation efficiency, and lower storage requirement, USTEDM is promising in solving vibrations of structures, especially those with displacement-dependent nonlinearity.
The zero numerical damping of USTEDM is its shortcoming, and the introduction of numerical damping will be studied.

Figure 4 :
Figure 4: Comparison of relative period error.