research article theory and implementation of a two-step...

Research ArticleTheory and Implementation of a Two-StepUnconditionally Stable Explicit Integration Algorithm forVibration Analysis of Structures

Li Changqing,1 Jie Junping,1 Jiang Lizhong,1,2 and T. Y. Yang3,4

1School of Civil Engineering, Central South University, Changsha, Hunan 410075, China2National Engineering Laboratory for High Speed Railway Construction, Changsha 410075, China3International Joint Research Laboratory of Earthquake Engineering, Tongji University, 1239 Siping Road, Shanghai 2000092, China4Department of Civil Engineering, The University of British Columbia, 2329 West Mall, Vancouver, BC, Canada V6T 1Z4

Correspondence should be addressed to Li Changqing; lcq [email protected]

Received 29 June 2016; Revised 2 October 2016; Accepted 1 November 2016

Academic Editor: Mahmoud Bayat

Copyright © 2016 Li Changqing et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Time history analysis is becoming the routine process to quantify the response of the structure under dynamic loads. In this paper, anovel two-step unconditionally stable explicit integration algorithm, named Unconditional Stable Two-Step Explicit DisplacementMethod (USTEDM), is proposed for vibration analysis of structure. USTEDM is unconditionally stable, requires low memory,produces no overshoot, and is third order accurate. The accuracy and efficiency of USTEDM are presented and compared withother commonly used integration algorithms. The result shows that the proposed algorithm has superior performance and can beused efficiently in solving vibration response of civil engineering structure.

1. Introduction

With the advance of computer hardware, more advancedcomputer software is being used to solve the response ofthe structure under dynamic loads. Hence, more advancednumerical methods have been developed to efficiently andaccurately predict the dynamic response of the structure. Inthe past,multiple implicit [1–7] and explicit [8–12] integrationalgorithms have been developed, whose relationship [13] anddesign methods are comprehensively analyzed [14–16]. Theexplicit method has high computational efficiency, while theimplicit method has higher stable. The commonly referencedexplicit algorithm is Central DifferenceMethod (CDM) (ref),CDM is easy to use, but it is only conditionally stable.Many new explicit methods which are unconditionally stable[17–19] have been developed. Though these methods areunconditionally stable, they require multiple steps which arenot as efficient as the CDM. The paper presents a novel2-step explicit integration method, named UnconditionalStable 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. Theperformance of USTEDM is compared with other explicitalgorithms. The result shows that USTEDM has superiorperformance and can be used efficiently in solving vibrationresponse of civil engineering structures.

2. Derivation of the Two-StepIntegration Algorithm

Equation (1) shows the generalized dynamic equations ofmotion of a structure:

𝑚𝑎 (𝑡) + 𝑐V (𝑡) + 𝑘𝑢 (𝑡) = 𝑓 (𝑡) , (1)

where 𝑎(𝑡), V(𝑡), 𝑢(𝑡) are the acceleration, velocity, and dis-placement of the structure, respectively.𝑚, 𝑐, 𝑘 are the mass,damping, and stiffness of the structure, respectively. 𝑓(𝑡) isexternal applied force.

Hindawi Publishing CorporationShock and VibrationVolume 2016, Article ID 2831206, 8 pageshttp://dx.doi.org/10.1155/2016/2831206

2 Shock and Vibration

In this paper, a two-step integration algorithm, as pre-sented in (2), is proposed to solve the dynamic equation aspresented in (1):

𝑢𝑡+2Δ𝑡 = 𝑓 (𝑢𝑡+Δ𝑡, 𝑢𝑡, 𝑓𝑡+2Δ𝑡, 𝑓𝑡+Δ𝑡, 𝑓𝑡, 𝑚, 𝑐, 𝑘, Δ𝑡) , (2)where 𝑢𝑡+2Δ𝑡 is the displacement at time step 𝑡 + 2Δ𝑡 and Δ𝑡is the integration time step. Similarly, 𝑢𝑡+Δ𝑡 and 𝑢𝑡 are thedisplacement at time 𝑡 + Δ𝑡 and 𝑡, while 𝑓𝑡+2Δ𝑡, 𝑓𝑡+Δ𝑡 and 𝑓𝑡are the force at 𝑡 + 2Δ𝑡, 𝑡 + Δ𝑡, and 𝑡, respectively.

It should be noted that (2) as presented above has physicalunits, where the displacement could be in the unit of “meter”or “feet,” while the mass can be in the unit of “kg” or “slugs.”To make the equation dimensionless, (2) can be rewritten inform of dimensionless quantity as

𝑢𝑡+2Δ𝑡𝑢𝑡+Δ𝑡 =𝑢𝑡+2Δ𝑡𝑢𝑡+Δ𝑡 (1,

𝑢𝑡𝑢𝑡+Δ𝑡 ,𝑓𝑡+2Δ𝑡Δ𝑡2𝑚𝑢𝑡+Δ𝑡 ,

𝑓𝑡+Δ𝑡Δ𝑡2𝑚𝑢𝑡+Δ𝑡 ,𝑓𝑡Δ𝑡2𝑚𝑢𝑡+Δ𝑡 ,

1, 𝑐Δ𝑡𝑚 , 𝑘Δ𝑡2𝑚 , 1) .(3)

To construct function 𝑢𝑡+2Δ𝑡/𝑢𝑡+Δ𝑡 at the right hand of(3), the linear combination of 𝑢𝑡+2Δ𝑡/𝑢𝑡+Δ𝑡 with the argumentsincluding 1, (𝑢𝑡/𝑢𝑡+Δ𝑡)(𝑓𝑡+2Δ𝑡Δ𝑡2/𝑚𝑢𝑡+Δ𝑡), 𝑓𝑡+Δ𝑡Δ𝑡2/𝑚𝑢𝑡+Δ𝑡and 𝑓𝑡Δ𝑡2/𝑚𝑢𝑡+Δ𝑡, can be used:

𝑟1 𝑢𝑡+2Δ𝑡𝑢𝑡+Δ𝑡 = 𝑟2 + 𝑟3𝑢𝑡𝑢𝑡+Δ𝑡 + 𝑟4

𝑓𝑡+2Δ𝑡Δ𝑡2𝑚𝑢𝑡+Δ𝑡 + 𝑟5𝑓𝑡+Δ𝑡Δ𝑡2𝑚𝑢𝑡+Δ𝑡

+ 𝑟6 𝑓𝑡Δ𝑡2𝑚𝑢𝑡+Δ𝑡 ,(4)

where 𝑟𝑖 (𝑖 = 1, . . . , 6) are undetermined coefficients.The effects of the dimensionless arguments 𝑐Δ𝑡/𝑚, 𝑘Δ𝑡2/𝑚that represent the structure’s material properties to thedisplacements can be expressed by replacing the coefficients𝑟𝑖 (𝑖 = 1, 2, 3) with linear combination of constant term1, 𝑐Δ𝑡/𝑚, 𝑘Δ𝑡2/𝑚:

𝑟1 = 𝑙1 + 𝑙2 𝑐Δ𝑡𝑚 + 𝑙3 𝑘Δ𝑡2𝑚 ,𝑟2 = 𝑙4 + 𝑙5 𝑐Δ𝑡𝑚 + 𝑙6 𝑘Δ𝑡2𝑚 ,𝑟3 = 𝑙7 + 𝑙8 𝑐Δ𝑡𝑚 + 𝑙9 𝑘Δ𝑡2𝑚 ,

(5)

where 𝑙𝑖 (𝑖 = 1, . . . , 9) are coefficients.Substituting (5) into (4) yields

(𝑙1 + 𝑙2 𝑐Δ𝑡𝑚 + 𝑙3 𝑘Δ𝑡2𝑚 ) 𝑢𝑡+2Δ𝑡𝑢𝑡+Δ𝑡= (𝑙4 + 𝑙5 𝑐Δ𝑡𝑚 + 𝑙6 𝑘Δ𝑡2𝑚 )+ (𝑙7 + 𝑙8 𝑐Δ𝑡𝑚 + 𝑙9 𝑘Δ𝑡2𝑚 ) 𝑢𝑡𝑢𝑡+Δ𝑡 + 𝑙10

𝑓𝑡+2Δ𝑡Δ𝑡2𝑚𝑢𝑡+Δ𝑡+ 𝑙11𝑓𝑡+Δ𝑡Δ𝑡2𝑚𝑢𝑡+Δ𝑡 + 𝑙12

𝑓𝑡Δ𝑡2𝑚𝑢𝑡+Δ𝑡 ,

(6)

where 𝑙𝑖 (𝑖 = 1, . . . , 12) are coefficients to be determined and𝑙10 = 𝑟4, 𝑙11 = 𝑟5, 𝑙12 = 𝑟6.Equation (6) can be simplified to be

(𝑙1𝑚 + 𝑙2𝑐Δ𝑡 + 𝑙3𝑘Δ𝑡2) 𝑢𝑡+2Δ𝑡= (𝑙4𝑚 + 𝑙5𝑐Δ𝑡 + 𝑙6𝑘Δ𝑡2) 𝑢𝑡+Δ𝑡+ (𝑙7𝑚 + 𝑙8𝑐Δ𝑡 + 𝑙9𝑘Δ𝑡2) 𝑢𝑡 + 𝑙10𝑓𝑡+2Δ𝑡Δ𝑡2+ 𝑙11𝑓𝑡+Δ𝑡Δ𝑡2 + 𝑙12𝑓𝑡Δ𝑡2.

(7)

In order to decrease the number of undetermined coeffi-cients 𝑙𝑖, (7) can be simplified by dividing 𝑙1 at both sides:

(𝑚 + 𝑏1𝑐Δ𝑡 + 𝑏2𝑘Δ𝑡2) 𝑢𝑡+2Δ𝑡= (𝑏3𝑚 + 𝑏4𝑐Δ𝑡 + 𝑏5𝑘Δ𝑡2) 𝑢𝑡+Δ𝑡+ (𝑏6𝑚 + 𝑏7𝑐Δ𝑡 + 𝑏8𝑘Δ𝑡2) 𝑢𝑡 + 𝑏9𝑓𝑡+2Δ𝑡Δ𝑡2+ 𝑏10𝑓𝑡+Δ𝑡Δ𝑡2 + 𝑏11𝑓𝑡Δ𝑡2,

(8)

where 𝑏𝑖 = (𝑙𝑖+1/𝑙1) (𝑖 = 1, . . . , 11) are undetermined coeffi-cients.

Equation (8) is the preliminary form of USTEDM.

3. Determination of Coefficients

3.1. Accuracy Analysis. The accuracy of USTEDM can beexamined using convergence test. Equation (9) shows theTaylor series expansion of the displacement, velocity, andacceleration of the structure at time step of 𝑡 + 2Δ𝑡, 𝑡 + Δ𝑡,respectively.

𝑢(𝑡+Δ𝑡) = 𝑢(𝑡) + V(𝑡)𝑑𝑡 + 12𝑎(𝑡)𝑑𝑡2 + 16 �̇�(𝑡)𝑑𝑡3 + ⋅ ⋅ ⋅ ,V(𝑡+Δ𝑡) = V(𝑡) + 𝑎(𝑡)𝑑𝑡 + 12 �̇�(𝑡)𝑑𝑡2 + 16 �̈�(𝑡)𝑑𝑡3 + ⋅ ⋅ ⋅ ,𝑎(𝑡+Δ𝑡) = 𝑎(𝑡) + �̇�(𝑡)𝑑𝑡 + 12 �̈�(𝑡)𝑑𝑡2 + 16 ...𝑎(𝑡)𝑑𝑡3 + ⋅ ⋅ ⋅ ,𝑢(𝑡+2Δ𝑡) = 𝑢(𝑡) + 2V(𝑡)𝑑𝑡 + 2𝑎(𝑡)𝑑𝑡2 + 43 �̇�(𝑡)𝑑𝑡3 + ⋅ ⋅ ⋅ ,V(𝑡+2Δ𝑡) = V(𝑡) + 2𝑎(𝑡)𝑑𝑡 + 2�̇�(𝑡)𝑑𝑡2 + 43 �̈�(𝑡)𝑑𝑡3 + ⋅ ⋅ ⋅ ,𝑎(𝑡+2Δ𝑡) = 𝑎(𝑡) + 2�̇�(𝑡)𝑑𝑡 + 2�̈�(𝑡)𝑑𝑡2 + 43 ...𝑎(𝑡)𝑑𝑡3 + ⋅ ⋅ ⋅ .

(9)

Similarly, the dynamic equilibriums at 𝑡 + 2Δ𝑡, 𝑡 + Δ𝑡, 𝑡are expressed in

𝑓𝑡+2Δ𝑡 = 𝑚𝑎(𝑡+2Δ𝑡) + 𝑐V(𝑡+2Δ𝑡) + 𝑘𝑢(𝑡+2Δ𝑡),𝑓𝑡+Δ𝑡 = 𝑚𝑎(𝑡+Δ𝑡) + 𝑐V(𝑡+Δ𝑡) + 𝑘𝑢(𝑡+Δ𝑡),𝑓𝑡 = 𝑚𝑎(𝑡) + 𝑐V(𝑡) + 𝑘𝑢(𝑡).

(10)

Shock and Vibration 3

Substituting (9) into (10) yields

𝑓𝑡+2Δ𝑡 = 𝑚 (𝑎(𝑡) + 2�̇�(𝑡)𝑑𝑡 + ⋅ ⋅ ⋅)+ 𝑐 (𝑢(𝑡) + 2V(𝑡)𝑑𝑡 + ⋅ ⋅ ⋅)+ 𝑘 (𝑢(𝑡) + 2V(𝑡)𝑑𝑡 + ⋅ ⋅ ⋅) ,

𝑓𝑡+Δ𝑡 = 𝑚 (𝑎(𝑡) + �̇�(𝑡)𝑑𝑡 + ⋅ ⋅ ⋅) + 𝑐 (V(𝑡) + 𝑎(𝑡)𝑑𝑡 + ⋅ ⋅ ⋅)+ 𝑘 (𝑢(𝑡) + V(𝑡)𝑑𝑡 + ⋅ ⋅ ⋅) ,

𝑓𝑡 = 𝑚𝑎(𝑡) + 𝑐V(𝑡) + 𝑘𝑢(𝑡).

(11)

Substituting (9)–(11) into (8) yields

(𝑚 + 𝑏1𝑐Δ𝑡 + 𝑏2𝑘Δ𝑡2) 𝜀𝑢𝑡+2Δ𝑡= (𝑚 + 𝑏1𝑐Δ𝑡 + 𝑏2𝑘Δ𝑡2) 𝑢(𝑡+2Δ𝑡)− (𝑏3𝑚 + 𝑏4𝑐Δ𝑡 + 𝑏5𝑘Δ𝑡2) 𝑢(𝑡+Δ𝑡)− (𝑏6𝑚 + 𝑏7𝑐Δ𝑡 + 𝑏8𝑘Δ𝑡2) 𝑢(𝑡) − 𝑏9𝑓𝑡+2Δ𝑡Δ𝑡2− 𝑏10𝑓𝑡+Δ𝑡Δ𝑡2 − 𝑏11𝑓𝑡Δ𝑡2

= 𝑑0 + 𝑑1Δ𝑡 + 𝑑2Δ𝑡2 + 𝑑3Δ𝑡3 + ⋅ ⋅ ⋅ ,

(12)

where

𝑑0 = (1 − 𝑏3 − 𝑏6) 𝑢(𝑡)𝑚,𝑑1 = (2 − 𝑏3) V(𝑡)𝑚 + (𝑏1 − 𝑏4 − 𝑏7) 𝑢(𝑡)𝑐,𝑑2 = (2 − 0.5𝑏3 − 𝑏9 − 𝑏10 − 𝑏11) 𝑎(𝑡)𝑚

+ (2𝑏1 − 𝑏4 − 𝑏9 − 𝑏10 − 𝑏11) V(𝑡)𝑐+ (𝑏2 − 𝑏5 − 𝑏8 − 𝑏9 − 𝑏10 − 𝑏11) 𝑢(𝑡)𝑘,

𝑑3 = (43 − 16𝑏3 − 2𝑏9 − 𝑏10) �̇�(𝑡)𝑚+ (2𝑏1 − 0.5𝑏4 − 2𝑏9 − 𝑏10) 𝑎(𝑡)𝑐+ (2𝑏2 − 𝑏5 − 2𝑏9 − 𝑏10) V(𝑡)𝑘,

𝑑4 = (23 − 124𝑏3 − 2𝑏9 − 12𝑏10) �̈�(𝑡)𝑚+ (43𝑏1 − 16𝑏4 − 2𝑏9 − 12𝑏10) �̇�(𝑡)𝑐+ (2𝑏2 − 12𝑏5 − 2𝑏9 − 12𝑏10) 𝑎(𝑡)𝑘,

𝑑5 = ( 415 − 1120𝑏3 − 43𝑏9 − 16𝑏10) ...𝑎(𝑡)𝑚+ (23𝑏1 − 124𝑏4 − 43𝑏9 − 16𝑏10) �̈�(𝑡)𝑐+ (43𝑏2 − 16𝑏5 − 43𝑏9 − 16𝑏10) �̇�(𝑡)𝑘 ⋅ ⋅ ⋅ .

(13)

USTEDM having first-order accuracy means

limΔ𝑡→0

𝜀𝑢𝑡+2Δ𝑡Δ𝑡 = 0. (14)

Combining (12) and (14) leads to

limΔ𝑡→0

(𝑑0Δ𝑡 + 𝑑1 + 𝑑2Δ𝑡 + 𝑑3Δ𝑡2 + . . .)

= limΔ𝑡→0

((𝑚 + 𝑏1𝑐Δ𝑡 + 𝑏2𝑘Δ𝑡2) 𝜀𝑢𝑡+2Δ𝑡Δ𝑡 )= 𝑚 limΔ𝑡→0

(𝜀𝑢𝑡+2Δ𝑡Δ𝑡 ) = 0.(15)

Equation (15) is satisfied only when

𝑑0 = 𝑑1 = 0. (16)

Combining (16) and (13) yields

1 − 𝑏3 − 𝑏6 = 0,2 − 𝑏3 = 0,

𝑏1 − 𝑏4 − 𝑏7 = 0.(17)

Similarly, (8) has second-order accuracy only when 𝑑0 =𝑑1 = 𝑑2 = 0. This means that

1 − 𝑏3 − 𝑏6 = 0,2 − 𝑏3 = 0,

𝑏1 − 𝑏4 − 𝑏7 = 0,2 − 0.5𝑏3 − 𝑏10 − 𝑏9 − 𝑏11 = 0,2𝑏1 − 𝑏4 − 𝑏9 − 𝑏10 − 𝑏11 = 0,

−𝑏9 + 𝑏2 − 𝑏8 − 𝑏11 − 𝑏5 − 𝑏10 = 0.

(18)

Solving (18) yields

𝑏1 = 1 − 𝑏7,𝑏2 = 1 + 𝑏8 + 𝑏5,𝑏3 = 2,𝑏4 = 1 − 2𝑏7,𝑏6 = −1,𝑏11 = 1 − 𝑏10 − 𝑏9.

(19)

This shows that USTEDM is second order accurate.


Similarly, (8) has third-order accuracy only when 𝑑0 =𝑑1 = 𝑑2 = 0. This means that

𝑏1 = 0.5,𝑏2 = −𝑏8,𝑏3 = 2,𝑏4 = 0,𝑏5 = −2𝑏8 − 1,𝑏6 = −1,𝑏7 = 0.5,𝑏9 = 𝑏11,𝑏10 = −2𝑏11 + 1,𝑏11 = 𝑏11.

(20)

Equation (8) has fourth-order accuracy when 𝑑0 = 𝑑1 =𝑑2 = 𝑑3 = 𝑑4 = 0, which can be satisfied when

𝑏1 = 0.5,𝑏2 = −𝑏8,𝑏3 = 2,𝑏4 = 0,𝑏5 = −2𝑏8 − 1,𝑏6 = −1,𝑏7 = 0.5,𝑏9 = 𝑏11,𝑏10 = −2𝑏11 + 1,

− 124𝑏3 − 2𝑏9 − 12𝑏10 + 23 = 0,43𝑏1 − 16𝑏4 − 2𝑏9 − 12𝑏10 = 0,2𝑏2 − 12𝑏5 − 2𝑏9 − 12𝑏10 = 0.

(21)

Equations (21) are contradictory equations, which showsthat two-step displacementmethod expressed by (8) can havethird-order accuracy at most, and (20) are adopted in thefollowing derivation.

3.2. Stability Analysis. According to Lax theorem, (8) isconvergent 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

{𝑢𝑡+2Δ𝑡𝑢𝑡+Δ𝑡 }

= [[𝑏3𝑚 + 𝑏4𝑐Δ𝑡 + 𝑏5𝑘Δ𝑡2𝑚 + 𝑏1𝑐Δ𝑡 + 𝑏2𝑘Δ𝑡2

𝑏6𝑚 + 𝑏7𝑐Δ𝑡 + 𝑏8𝑘Δ𝑡2𝑚 + 𝑏1𝑐Δ𝑡 + 𝑏2𝑘Δ𝑡21 0]]{𝑢𝑡+Δ𝑡𝑢𝑡 }

+ {{{𝑏9𝑓𝑡+2Δ𝑡Δ𝑡2 + 𝑏10𝑓𝑡+Δ𝑡Δ𝑡2 + 𝑏11𝑓𝑡+Δ𝑡Δ𝑡2𝑚 + 𝑏1𝑐Δ𝑡 + 𝑏2𝑘Δ𝑡20

}}}.

(22)

Amplification matrix 𝐴 in (22) determining (8)’s stabilityis

𝐴 = [𝐴11 𝐴12𝐴21 𝐴22]

= [[𝑏3𝑚 + 𝑏4𝑐Δ𝑡 + 𝑏5𝑘Δ𝑡2𝑚 + 𝑏1𝑐Δ𝑡 + 𝑏2𝑘Δ𝑡2

𝑏6𝑚 + 𝑏7𝑐Δ𝑡 + 𝑏8𝑘Δ𝑡2𝑚 + 𝑏1𝑐Δ𝑡 + 𝑏2𝑘Δ𝑡21 0]].(23)

Substituting 𝑐 = 2𝜉𝜔𝑚, 𝑘 = 𝑚𝜔2, 𝜔Δ𝑡 = Ω (𝜉-dampratio, 𝜔-natural frequency) into (23) yields

𝐴 = [𝐴11 𝐴12𝐴21 𝐴22]

= [[𝑏3 + 2𝑏4𝜉Ω + 𝑏5Ω21 + 2𝑏1𝜉Ω + 𝑏2Ω2

𝑏6 + 2𝑏7𝜉Ω + 𝑏8Ω21 + 2𝑏1𝜉Ω + 𝑏2Ω21 0]].

(24)

The characteristic equation of matrix 𝐴 can be written as

det (𝐴 − 𝜆𝐼) = 𝜆2 − 2𝐴1𝜆 + 𝐴2 = 0, (25)

where 𝐼 is the identity matrix, 𝜆 denotes eigenvalue of𝐴, and𝐴1 = 12 tr (𝐴) = 𝐴11 + 𝐴222 = 12 𝑏3 + 2𝑏4𝜉Ω + 𝑏5Ω

2

1 + 2𝑏1𝜉Ω + 𝑏2Ω2 ,𝐴2 = |𝐴| = −𝑏6 + 2𝑏7𝜉Ω + 𝑏8Ω21 + 2𝑏1𝜉Ω + 𝑏2Ω2 .

(26)

The stability of (8) depends on spectral radius 𝜌(𝐴) ofmatrix 𝐴 and (8) is stable only when

𝜌 (𝐴) ⩽ 1. (27)

Because 𝜌(𝐴) is determined by 𝐴1, 𝐴2, the stabilitydomain of (8) can be expressed by functions 𝐴1 and 𝐴2. Insimilar way, the study of stability made by Hilber and Hughes[20] is performed. Firstly, the boundary line of stabilitydomain of (8) where 𝜌(𝐴) = 1 is to be derived. 𝜌(𝐴) = 1means that

|𝜆|max = 1, (28)

where |𝜆|max is the maximum amplitude of 𝜆. Complex value𝜆 whose amplitude is 1 can be expressed by 𝜆 = 𝑒𝑖𝜑, where


𝑖 = √−1 and 𝜑 is argument of 𝜆. Substituting 𝜆 = 𝑒𝑖𝜑 into(25) yields

𝑒𝑖2𝜑 − 2𝐴1𝑒𝑖𝜑 + 𝐴2 = 0. (29)

The identities including 𝑒𝑖𝜑 = cos𝜑 + 𝑖 sin𝜑, cos 2𝜑 =2 cos2𝜑 − 1, sin 2𝜑 = 2 sin𝜑 cos𝜑 are used and (29) can berewritten as

(2 cos𝜑 (cos𝜑 − 𝐴1) + 𝐴2 − 1)+ 𝑖 (2 sin𝜑 (cos𝜑 − 𝐴1)) = 0. (30)

Equation (30) is satisfied only when

2 cos𝜑 (cos𝜑 − 𝐴1) + 𝐴2 − 1 = 0,2 sin𝜑 (cos𝜑 − 𝐴1) = 0. (31)

The boundary lines of stability domain of (8) can bedetermined by (31) when 𝜑 ∈ [0, 2𝜋]. It can be derived thatwhen 0 < 𝜑 < 𝜋, 𝜋 < 𝜑 < 2𝜋, (31) are satisfied when

𝐴1 = cos𝜑,𝐴2 = 1. (32)

When 𝜑 = 0, 𝜑 = 2𝜋, (31) are satisfied when

1 − 2𝐴1 + 𝐴2 = 0. (33)

When 𝜑 = 𝜋, (31) are satisfied when

1 + 2𝐴1 + 𝐴2 = 0. (34)

The boundary of stability domain of (8) is then made upwith three lines described by (32), (33), and (34), respectively,in 𝐴1-𝐴2 plane as shown in Figure 1. Because 𝜌(𝐴) iscontinuous function of 𝐴1𝐴2 and when 𝐴1 = 𝐴2 =0, 𝜌(𝐴) = 0 < 1, the stability domain of (8) is the inner partof the triangle as shown in Figure 1 and can be expressed by

−1 ≤ 𝐴2 ≤ 1,−12 (1 + 𝐴2) ≤ 𝐴1 ≤ 12 (1 + 𝐴2) .

(35)

3.3. Final Form of Two-Step Displacement Method withThird-Order Accuracy. Combining (20) and (8) yields

(𝑚 + 0.5𝑐Δ𝑡 − 𝑏8𝑘Δ𝑡2) 𝑢𝑡+2Δ𝑡= (2𝑚 + (−2𝑏8 − 1) 𝑘Δ𝑡2) 𝑢𝑡+Δ𝑡+ (−1𝑚 + 0.5𝑐Δ𝑡 + 𝑏8𝑘Δ𝑡2) 𝑢𝑡 + 𝑏11𝑓𝑡+2Δ𝑡Δ𝑡2+ (−2𝑏11 + 1) 𝑓𝑡+Δ𝑡Δ𝑡2 + 𝑏11𝑓(𝑡)Δ𝑡2.

(36)

Combining (20), (26), and (35) yields, after simplification,the stability domain of (36):

1 − 𝑏8Ω2 ⩾ 0,1 + 𝜉Ω − 𝑏8Ω2 > 0,

4 + (−4𝑏8 − 1)Ω2 ⩾ 0.(37)

(1, 1)

0.5

A2

A1

(−1, 1)

−0.5

−1

𝜌 < 1 𝜌 > 1

Figure 1: Stability domain of (8).

According to (36) and (37), the two-step displacementmethods satisfying different requirements can be obtained.The explicitness ofmethod can be obtained only when 𝑏11 = 0and the unconditional stability can be obtained when (37) arealways satisfiedwhenΩ ∈ [0,∞). It can be found that (37) arealways satisfied only when

𝑏8 ⩽ −0.25. (38)

Substituting 𝑏8 = −0.25, 𝑏11 = 0 into (36) yields(𝑚 + 0.5𝑐Δ𝑡 + 0.25𝑘Δ𝑡2) 𝑢𝑡+2Δ𝑡= (2𝑚 − 0.5𝑘Δ𝑡2) 𝑢𝑡+Δ𝑡+ (−𝑚 + 0.5𝑐Δ𝑡 − 0.25𝑘Δ𝑡2) 𝑢𝑡 + 𝑓𝑡+Δ𝑡Δ𝑡2.

(39)

Equation (39) shows the final form of USTEDM, whichis an unconditionally stable two-step explicit integrationalgorithm. USTEDM is third order accurate. The analysisshows that USTEDM has very similar numerical attributes asNewmark’s average acceleration method. Figure 2 shows thatthe spectral radius of amplification matrix of USTEDM is 1,which is the same as that of Newmark’s average accelerationmethod.

Figure 3 shows that the numerical damping of USTEDMis zero, which is the same as that of Newmark’s averageacceleration method. Figure 4 shows the relative perioderror of USTEDM is minimal compared with other methodsincluding Houbolt Method, Wilson-𝜃Method (𝜃 = 1.4), andGeneralized 𝛼 Method (𝛼𝑚 = 0.3, 𝛼𝑓 = 0.35). Though withsimilar numerical attributes as Newmark’s average accelera-tion method, USTEDM is explicit and has great computationadvantage over the implicit Newmark’s average accelerationmethod.

The applicability of the USTEDM for nonlinear problemis also investigated. Compared with other unconditional


USTEDMAverage accelerationWilson-𝜃 (𝜃 = 1.4)

Houbolt

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

𝜌(A

)

100 101 102 103 10410−1

Ω

G-𝛼 (𝛼m = 0.3, 𝛼f = 0.35)

Figure 2: Comparison of spectral radius.


Houbolt

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Num

eric

al d

ampi

ng ra

tio

0.05 0.1 0.15 0.2 0.25 0.3 0.350Δt/T

G-𝛼 (𝛼m = 0.3, 𝛼f = 0.35)

Figure 3: Comparison of numerical damping ratio.

stable explicit algorithms [17–19], whose main calculationamount includes at least solving two-reverse matrix ofnondiagonal matrix for displacement and velocity, respec-tively, USTEDM shows higher calculation efficiency sinceits main calculation amount includes solving only one-reverse matrix of nondiagonal matrix for displacement. Inaddition, USTEDM shows minimal memory requirements,since its calculation only involves displacement when onlydisplacement-related nonlinearity occurs. This shows thatUSTEDM can be viewed as a complementary or improvedform of conditional stable explicit CDM.


Houbolt

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Relat

ive p

erio

d er

ror

0.05 0.1 0.15 0.2 0.25 0.3 0.350Δt/T

G-𝛼 (𝛼m = 0.3, 𝛼f = 0.35)

Figure 4: Comparison of relative period error.

3.4. Implementing USTEDM for Nonlinear Problem. The pro-cedure to use USTEDM is similar to that of CDM and isoutlined as follows.

(A) Initial input and calculation:

(1) Input the initial stiffnessmatrix𝐾0, massmatrix𝑀, and damping matrix C.(2) Input initial displacement vector {𝑢0} and initial

velocity vector {V0}, solving initial accelerationvector {𝑎0}: {𝑎0} = 𝑀−1({𝑓0} − 𝐾0{𝑢0} − 𝐶{V0}).

(3) Select time step duration Δ𝑡.(4) Calculate

{𝑢−Δ𝑡} = {𝑢0} − {V0} Δ𝑡 + 12 {𝑎0} Δ𝑡2. (40)

(B) For each time step (from time point 𝑡 to time point𝑡 + Δ𝑡):(1) based on known {𝑢𝑡}, {𝑢𝑡−Δ𝑡},𝐾𝑡, and outer force

vector {𝑓𝑡}, calculate the “effective” load vector{𝑓𝑡+Δ𝑡};{𝑓𝑡+Δ𝑡} = (2𝑀 − 0.5𝐾𝑡Δ𝑡2) {𝑢𝑡}

+ (−𝑀 + 0.5𝐶Δ𝑡 − 0.25𝐾𝑡Δ𝑡2) {𝑢𝑡−Δ𝑡}+ {𝑓𝑡} Δ𝑡2

(41)

(2) form “effective”massmatrix: �̂� = 𝑀+0.5𝐶Δ𝑡+0.25𝐾𝑡Δ𝑡2;(3) calculate {𝑢𝑡+Δ𝑡} = �̂�−1{�̂�𝑡+Δ𝑡};


u2 u200u1

m2 m200m1

k2 k200k1

· · ·

Figure 5: 200-DOF spring-mass system.

Table 1: Total time consumption (sec).

USTEDM Houbolt Average acceleration0.4825 353.0519 292.1621

(4) if necessary, calculate velocity vector and accel-eration vector at time point 𝑡 + Δ𝑡:

{V𝑡+Δ𝑡} = 3 {𝑢𝑡+Δ𝑡} − 4 {𝑢𝑡} + {𝑢𝑡−Δ𝑡}2Δ𝑡 ,{𝑎𝑡+Δ𝑡} = 𝑀−1 (𝑓𝑡+Δ𝑡 − 𝐾𝑡+Δ𝑡 {𝑢𝑡+Δ𝑡} − 𝐶 {V𝑡+Δ𝑡}) .

(42)

3.5. Numerical Example. In this example [7, 17], the forcedvibration response of an 200-degree-of-freedom spring-masssystem as shown in Figure 5 is examined using the proposednumerical method. The structural properties of this systemare assumed to be 𝑚𝑖 = 100 kg and 𝑘𝑖 = 107[1 − (𝑢𝑖 −𝑢𝑖−1)2]N/m in which 𝑖 = 1, 2, . . . , 200. This system is sub-jected to a ground acceleration of 10 sin(𝜋𝑡). The lowestand 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 Δ𝑡 = 0.001 s is obtained from theNewmark’s average acceleration method and is selected to bethe exact solution. Displacement responses versus time of thelargest degree of freedom are plotted in Figure 6. Analysis isaccomplished by considering time step duration to be equalto 0.02 s. The displacement errors are presented in Figure 7.It can be seen that here USTEDM provides the most accuratedisplacement solutions, compared with Houbolt Method andcommonly used Newmark’s average acceleration method.More importantly, without needs of iteration calculation, theunconditionally stable explicit USTEDM has much less timeconsumption of calculation, compared with those uncondi-tionally stable implicit methods, which are listed in Table 1.

4. Conclusions

Anovel two-step unconditionally stable explicit displacementmethod, named Unconditional Stable Two-Step Explicit Dis-placement Method (USTEDM), is proposed in this paper.USTEDM has third-order accuracy and is unconditionallystable. The conclusions mainly include the following:

(1) The novel way of constructing direct time integrationmethods from the start point of dimensional analysis

Exact solutionUSTEDMNewmark average acceleration methodHoubolt method

1 2 3 4 5 6 7 8 9 100Time (sec)

u200

(m)

−8

−6

−4

−2

02468

Figure 6: Displacement responses of 200th DOF system.

USTEDMNewmark average acceleration methodHoubolt method

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Erro

r u200

(m)

2 4 6 8 100Time (sec)

Figure 7: Comparison of displacement errors of 200-DOF system.

is applicable, and it is believed that it will contributeto improvement of numerical calculation in structuraldynamics.

(2) With no prerequisite difference assumption of veloc-ity and acceleration, the derivation of USTEDM iscompletely different from any documented one. Withunconditional stability, third-order accuracy, highercalculation efficiency, and lower storage requirement,USTEDM is promising in solving vibrations of struc-tures, especially those with displacement-dependentnonlinearity.

The zero numerical damping of USTEDM is its short-coming, and the introduction of numerical damping will bestudied.


Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors would like to acknowledge the partial supportof this work from Project Research on Parallel AdaptiveMethod in Solving Seismic Response of High Speed RailwayBridge supported by National Natural Science Foundationof China (51308555) and Hunan Provincial Natural ScienceFoundation of China (14JJ3015). The research was supportedby the Postdoctoral Science Foundation of Central SouthUniversity and the Strategic Pilot Project of Central SouthUniversity.

References

[1] J. C. Houbolt, “A recurrence matrix solution for the dynamicresponse of aircraft in gusts,” Journal of the AeronauticalSciences, vol. 17, no. 9, pp. 540–550, 1950.

[2] N. M. Newmark, “A method for computation of structuraldynamics,” Journal of EngineeringMechanics (ASCE), vol. 85, pp.67–94, 1959.

[3] E. L. Wilson, I. Farhoomand, and K. J. Bathe, “Nonlineardynamic analysis of complex structures,” Earthquake Engineer-ing and Structural Dynamics, vol. 1, no. 3, pp. 241–252, 1973.

[4] T. C. Fung, “Third order complex-time-step methods fortransient analysis,”ComputerMethods in AppliedMechanics andEngineering, vol. 190, no. 22-23, pp. 2789–2802, 2001.

[5] K.-J. Bathe, “Conserving energy and momentum in nonlineardynamics: a simple implicit time integration scheme,” Comput-ers and Structures, vol. 85, no. 7-8, pp. 437–445, 2007.

[6] K.-J. Bathe and G. Noh, “Insight into an implicit time integra-tion scheme for structural dynamics,”Computers and Structures,vol. 98-99, pp. 1–6, 2012.

[7] A. Akbar Gholampour,M. Ghassemieh, andM. Karimi-Rad, “Anewunconditionally stable time integrationmethod for analysisof nonlinear structural dynamics,” Journal of AppliedMechanics,vol. 80, no. 2, pp. 111–121, 2013.

[8] W.-M. Zhai, “Two simple fast integration methods for large-scale dynamic problems in engineering,” International Journalfor Numerical Methods in Engineering, vol. 39, no. 24, pp. 4199–4214, 1996.

[9] G. M. Hulbert and J. Chung, “Explicit time integration algo-rithms for structural dynamics with optimal numerical dissipa-tion,” Computer Methods in AppliedMechanics and Engineering,vol. 137, no. 2, pp. 175–188, 1996.

[10] S.-Y. Chang, “An explicit method with improved stabilityproperty,” International Journal for Numerical Methods in Engi-neering, vol. 77, no. 8, pp. 1100–1120, 2009.

[11] G. Noh andK.-J. Bathe, “An explicit time integration scheme forthe analysis of wave propagations,” Computers and Structures,vol. 129, pp. 178–193, 2013.

[12] C. Yang, S. Xiao, L. Lu, andT. Zhu, “Two dynamic explicitmeth-ods based on double time steps,” Proceedings of the Institution ofMechanical Engineers, Part K: Journal of Multi-body Dynamics,vol. 228, no. 3, pp. 330–337, 2014.

[13] C. Li and L. Jiang, “Explicit concomitance of implicit methodto solve vibration equation,” Journal of Earthquake Engineeringand Engineering Vibration, vol. 11, no. 2, pp. 269–272, 2012.

[14] J. Har and K. K. Tamma, Computational Dynamics of Particles,Materials and Structures, John Wiley & Sons, 2012.

[15] S. U. Masuri, A. Hoitink, X. Zhou, and K. K. Tamma, “Algo-rithms by design: a new normalized time-weighted residualmethodology and design of a family of energy–momentumconserving algorithms for non-linear structural dynamics,”International Journal for Numerical Methods in Engineering, vol.79, no. 9, pp. 1094–1146, 2009.

[16] K. K. Tamma, J. Har, X. Zhou, M. Shimada, and A. Hoitink, “Anoverview and recent advances in vector and scalar formalisms:space/time discretizations in computational dynamics—a uni-fied approach,” Archives of Computational Methods in Engineer-ing, vol. 18, no. 2, pp. 119–283, 2011.

[17] S.-Y. Chang, “Improved explicit method for structural dynam-ics,” Journal of Engineering Mechanics, vol. 133, no. 7, pp. 748–760, 2007.

[18] C. Kolay and J. M. Ricles, “Development of a family of uncon-ditionally stable explicit direct integration algorithms with con-trollable numerical energy dissipation,” Earthquake Engineeringand Structural Dynamics, vol. 43, no. 9, pp. 1361–1380, 2014.

[19] C. Kolay and J. M. Ricles, “Assessment of explicit and semi-explicit classes of model-based algorithms for direct integrationin structural dynamics,” International Journal for NumericalMethods in Engineering, vol. 107, no. 1, pp. 49–73, 2016.

[20] H. M. Hilber and T. J. Hughes, “Collocation, dissipation and[overshoot] for time integration schemes in structural dynam-ics,” Earthquake Engineering & Structural Dynamics, vol. 6, no.1, pp. 99–117, 1978.

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