Kawahara Lab. 18th/Mar./2011

An Identification of Damping Parameters

Based on Blasting Waves

Using First-order Adjoint Equation Method

Shigenori MIKAME

Department of Civil Engineering, Chuo University,Kasuga 1-13-27, Bunkyo-ku, Tokyo 112-8551, Japan

Email : mikame@civil.chuo-u.ac.jp

Abstract

The purpose of this research is an identification of damping parameters of the ground usingfirst-order adjoint equation method. Practical application site is Iwatayama tunnel. The unknowndamping parameters are identified. In this research, we apply proportional damping model to ex-press damping of oscillation assuming elastic ground. The damping model is assumed as the sumof mass matrix and stiffness matrix multiplied by the proportional coefficients α0 and α1. In thispaper, the damping parameters α0 and α1 are identified. The parameter identification is consideredas a minimization problem to find the unknown parameters so as to minimize the performance func-tion. The performance function is defined by the square sum of discrepancy between the calculatedand observed velocity. The observed velocity is caused by blasting of the tunnel excavation, andobserved at the construction site. In order to determine the damping parameters minimized theperformance function, it is necessary to calculate the gradients with respect to the damping param-eters. Therefore, we introduce the extended performance function using the Lagrange multiplier.In order to minimize the performance function, the first variation of extended performance functionshould be zero. It is called stationary condition. Following the condition, the adjoint equation,terminal conditions, and gradients with respect to the damping parameters can be obtained. Then,applying the wighted gradient method as a minimization technique, the damping parameters min-imized performance function are found. In this research, to express geological characteristic of theground, linear equations are employed. As the basic equation, the equilibrium of stress equation,the strain-displacement equation and the stress-strain equation are used. To solve the basic andadjoint equations, the finite element method and the Newmark β method are applied as the spatialand temporal discretizations, respectively. As a computational model, the finite element model ofthe Iwatayama tunnel construction site in Japan is employed. We carry out following the numericalstudies. In case 1, we verify the identification method. In the verification, the observed velocity iscomputed by the dynamic analysis of application site. In the dynamic analysis, we set the dampingparameters as the target values. We verify whether the damping parameters are converged to thetarget values or not from the assumed initial values. In case 2, we carry out an identification basedon the blasting waves observed at the construction site. The observed velocity is caused by theblasting of the tunnel excavation. In this case, the damping parameters are unknown. The unknowndamping parameters can be successfully identified.

key words : Parameter identification, Iwatayama tunnel construction site,First-order adjoint method, Performance function,Damping parameters α0 and α1, Blasting waves.

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1 Introduction

Because Japan is mountainous country, massive excavation is needed on civil engineering construction suchas tunnel, dam, and road construction. In particular, civil engineering works that are close to residential areahave possibility of being greatly influence on surrounding environment and residents. For example, they arenoise, vibration, land subsidence, collapse of mountain side and so on. Therefore, It is indispensable to clarifygeological behaviors.

Related to the population growth and urbanization in Japan, artificial reclamation has been progressed tohilly area and tableland with massive excavation. For instance, we cut foot of mountains for reclamation.Such an area has disaster risk which the slope collapse of mountain side will occur and so on. Also, the civilengineering works are often carried out close to the residential area, because Japan have no large land. In thecircumstance, the civil engineering works can bring disaster. If accidents are occurred in civil engineering work,it is expected to be human suffering. Moreover, it causes delay of construction, increase of cost and so on. Thusit is important to preliminarily investigate geological condition for prevention of accidents.

As an example, the boring survey is used to investigate geological condition in case of tunnel construction.However, the boring survey needs improvement. The weaknesses of the investigation are to delay the construc-tion and to stop working at the tunnel face for surveys. Huge survey cost is needed and it is difficult to acquirearound the fault and the fracture zone. It is important to consider the geological investigation method presentedin this paper. The precision of investigation method should be accurate. It is necessary to be able to reducethe cost and do as common construction control. The common construction control indicates that to be able todo without interruption of construction. Technique of numerical analysis plays a large role with advancementof computer and analysis technique in civil engineering works in recent years. Then, the unknown parametersindicating geological conditions can be calculated by carrying out the inverse analysis of the geological behavior.In this study, geological survey using numerical analysis is proposed as one of these methods.

In this research, we identify the damping parameters based on the blasting waves using first-order adjointequation method. The phenomenon is assumed tunnel excavation. The oscillation is cased by blasting of thetunnel excavation. The data is observed at the construction site. Parameter identification is performed by theinverse analysis, and is considered to be minimization problem. We employ the Iwatayama tunnel constructionsite in Gifu prefecture in Japan as practical application site. The characteristic of the construction field is closeto the residential area and appointed as a dangerous area of slope collapse. Therefore, it is important to predictthe influence on surrounding area, which is the oscillation of construction and so on. This research is the firststep to put this geological survey using numerical analysis to practical use.

2 Basic Equation

In this paper, indicial notation and summation convention are used. The following equations (Eqs.(1)～(3))are used as the basic equations in order to analyze the elastic body. There are the equilibrium of stress equation,the strain-displacement equation and stress-strain equation as;

σij,j − ρbi − ρui = 0, (1)

εij =1

2(ui,j + uj,i), (2)

σij = Dijklεkl, (3)

where σij , bi, ui, ui, ρ and εkl are total stress, body force, displacement, acceleration, density and strain,respectively. Dijkl is called elastic coefficient matrix. It is expressed as follows;

Dijkl = λδijδkl + µ(δikδjl + δilδjk), (4)

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where δij is Kronecker’s delta, and λ and µ are Lame’s constants. These are written as follows;

λ =νE

(1 − 2ν)(1 + ν), (5)

µ =E

2(1 + ν), (6)

where E and ν are the elastic modulus and Poisson ratio, respectively.

3 Boundary Condition

The following boundary conditions are employed in the present analysis. The displacements ui are given onthe boundary Γ1, and the surface forces ti are given on the boundary Γ2.

ui = ui on Γ1, (7)

ti = σijnj = ti on Γ2, (8)

The initial conditions are given as follows;

ui = u0i at t = t0, (9)

ui = ˆu0i at t = t0. (10)

4 Discretization

4.1 Finite Element Method

The basic equations are discretized by the finite element method and expressed as follows;

Mαiβkuβk + Kαiβkuβk = Γαi, (11)

Considering the damping effect, Eq.(11) is written as follows;

Mαiβkuβk + Cαiβkuβk + Kαiβkuβk = Γαi, (12)

Each matrix can be expressed in the following form;

Mαiβk =

∫V

(ρ δikNαNβ)dV , (13)

Cαiβk = α0Mαiβk + α1Kαiβk, (14)

Kαiβk =

∫V

(Nα,jDijklNβ,l)dV , (15)

Γαi =

∫Γ2

(Nαti)dΓ2 −

∫V

(Nαρbi)dV , (16)

where Mαiβk, Cαiβk, Kαiβk, and Γαi are mass, damping, stiffness and load matrices, respectively, and Nα is theliner interpolation function of the finite element method.

The damping matrix Cαiβk is expressed as proportional damping model. Damping matrix Cαiβk is the sumof mass matrix Mαiβk and stiffness matrix Kαiβk which are multiplied by the proportional coefficient α0 andα1, respectively. In this paper, above damping parameters α0 and α1 are identified.

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4.2 Newmark β Method

In order to discretize in time, the Newmark β method is applied to the finite element equation. In theNewmark β method, the displacement and the velocity at time (n + 1) step are written as follows;

u(n+1)βk = u

(n)βk + ∆tu

(n)βk + ∆t2

1

2u

(n)βk + ∆t2β(u

(n+1)βk − u

(n)βk ), (17)

u(n+1)βk = u

(n)βk + ∆tu

(n)βk + ∆tγ(u

(n+1)βk − u

(n)βk ), (18)

where u(n+1)βk and u

(n+1)βk are the displacement and the velocity at time (n + 1), respectively. Time increment is

∆t, and β and γ are constants which are 0.25 and 0.5, respectively. The finite element equation at time (n + 1)is transformed into;

Gαiβku(n+1)βk = Γαi − Eαiβku

(n)βk − Fαiβku

(n)βk − Kαiβku

(n)βk , (19)

Each matrix can be expressed as;

Gαiβk = Mαiβk +1

2∆tCαiβk +

1

4∆t2Kαiβk, (20)

Eαiβk =1

2∆tCαiβk +

1

4∆t2Kαiβk, (21)

Fαiβk = Cαiβk + ∆tKαiβk, (22)

Calculating acceleration u(n+1)βk using Eq.(19) and substituting these into Eqs.(17) and (18), the displacement

u(n+1)βk and the velocity u

(n+1)βk can be obtained.

5 Performance Function

In this research, parameter identification is performed to find parameters making computed velocity be asclose as observed velocity. The performance function is defined by the square sum of discrepancies betweenthe computed and observed velocities. In case of the computed velocity is identical with the observed velocity,the performance function is converged to the minimum value. Namely, the parameter identification problemis considered to be the minimization problem to minimize performance function. Therefore, the performancefunction is employed as convergence index of identification. The performance function is written as follows;

J =1

2

∫ tf

t0

(uαi − ηαi)Wαiβk(uβk − ηβk)dt, (23)

where uαi and ηαi are calculated and observed velocities, respectively, and t0 and tf mean the initial and thefinal times, respectively.

6 First-order Adjoint Equation

The first-order adjoint equation method is applied in order to identify the damping parameters. The extendedperformance function using the Lagrange multiplier is introduced. The extended performance function is writtenas follows;

J∗ =1

2

∫ tf

t0

(uαi − ηαi)Wαiβk(uβk − ηβk)dt

+

∫ tf

t0

λαi(Γαi − Mαiβkuβk − Cαiβkuβk − Kαiβkuβk)dt, (24)

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where λαi is the Lagrange multiplier. In the inverse analysis, it is necessary to calculate the gradient of theperformance function to update an unknown parameter. Taking the first variation of the extended performancefunction, the gradients with respect to the damping parameters, the adjoint equation and terminal conditionscan be obtained. The first variation of the extended performance function δJ∗ is expressed as follows;

δJ∗ =

∫ tf

t0

(uαi − ηαi)Wαiβkδuβkdt

+

∫ tf

t0

δλαi(Γαi − Mαiβkuβk − Cαiβkuβk − Kαiβkuβk)dt

−

∫ tf

t0

λαi(Mαiβkδuβk + Cαiβkδuβk + Kαiβkδuβk)dt

−

∫ tf

t0

λαi(Mαiβkuβk)δα0dt

−

∫ tf

t0

λαi(Kαiβkuβk)δα1dt. (25)

Integrating by parts, Eq.(25) is transformed into the following form;

δJ∗ = {uαi(tf ) − ηαi(tf )}Wαiβkδuβk(tf ) − {uαi(t0) − ηαi(t0)}Wαiβkδuβk(t0)

− λαi(tf )Mαiβkδuβk(tf ) + λαi(t0)Mαiβkδuβk(t0)

+ λαi(tf )Mαiβkδuβk(tf ) − λαi(t0)Mαiβkδuβk(t0)

− λαi(tf )Cαiβkδuβk(tf ) + λαi(t0)Cαiβkδuβk(t0)

−

∫ tf

t0

{λαiMαiβk − λαiCαiβk + λαiKαiβk + (uαi − ηαi)Wαiβk}δuβkdt

−

∫ tf

t0

λαi(Mαiβkuβk)dtδα0

−

∫ tf

t0

λαi(Kαiβkuβk)dtδα1. (26)

Each terms of Eq.(26) should be zero in order to satisfy the following condition to minimize performance functionJ . It is called stationary condition;

δJ∗ = 0. (27)

Considering Eq.(27), the adjoint equation and terminal conditions can be obtained as follows;

−λαiMαiβk + λαiCαiβk − λαiKαiβk = (uαi − ηαi)Wαiβk, (28)

λαi(tf ) = 0, (29)

λαi(tf )Mαiβk = −{uαi(tf ) − ηαi(tf )}Wαiβk. (30)

Substituting terminal conditions Eq.(29) and Eq.(30) into Eq.(28), the terminal condition of acceleration iscalculated as follows;

λαi(tf )Mαiβk = −{uαi(tf ) − ηαi(tf )}Wαiβk + λαi(tf )Cαiβk. (31)

The gradients with respect to the damping parameters α0 and α1 can be obtained as follows;

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grad(J∗)α0=

∫ tf

t0

λαiMαiβkuβkdt, (32)

grad(J∗)α1=

∫ tf

t0

λαiKαiβkuβkdt. (33)

7 Weighted Gradient Method

The weighted gradient method is applied as the minimization technique. Modified performance function isintroduced as follows;

K(m) = J∗(m) +1

2

∫ tf

t0

(X(m+1)α − X(m)

α )W(m)αβ (X

(m+1)β − X

(m)β )dt, (34)

where Xα is identified parameters and W(m)αβ is the weighting function for the stability of computation. The

optimal condition of the modified performance function can be derived as follows;

δK(m) = 0. (35)

Then, the updated equation of the identified parameter at each iteration cycle is expressed as;

W(m)αβ X

(m+1)β = W

(m)αβ X

(m)β − grad(J∗(m))α. (36)

Using Eq.(36), the parameters are updated by the iterative calculations.

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8 Numerical Study

In this numerical study, we carried out the identification of damping parameters based on blasting waves atthe Iwatayama tunnel construction site. In case 1, the identification method is verified. We set the target valuesof damping parameters. As observed velocity, we employ the velocity which is computed by the dynamic analysiswith the target values. Then, the verification is carried out to obtain the computed parameters coincident withthe assumed target values. In case 2, the identification based on the actual blasting waves is carried out. Theobserved velocity is caused by the blasting of the tunnel excavation. In this case, the damping parameters areunknown. The unknown damping parameters are identified.

8.1 Iwatayama Tunnel

In this research, we employ the construction field of Iwatayama tunnel as application site. The constructionfield is located in Gifu prefecture in Japan. As a part of construction of Gifu eastern bypass, the tunnel is underconstruction. The length of the tunnel is planed 1001[m], and NATM method is employed as constructiontechnique. Figure 1 shows aerial view of Mt.Iwata and tunnel plan. Figure 2 shows the tunnel mouth which isunder constructing. The characteristic of the construction field is close to the residential area, and seems to bedangerous area of slope collapse as shown in Figure 3.

Figure 1: Aerial view of Mt.Iwata and tunnel plan

Figure 2: The tunnel mouth of Iwatayama tun-nel under construction

Figure 3: The surrounding area of construction field

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8.2 Computational Conditions

Figure 4 shows finite element mesh. The length, width and hight are about 200[m], 200[m] and 110[m],respectively. The total number of nodes and elements are 12722 and 69939, respectively. The mesh is assumedthat the tunnel was constructed about 70[m] from tunnel mouth as shown in Figure 5. The observation pointsare set at the mountain side shown by the blue dots in Figure 4. The points are about 80[m] away from thetunnel face. In this numerical study, boundary conditions are summarized in Table 1. The bottom is fixed in alldirections, and other sides are fixed only perpendicular direction. Figure 6 shows the external force loaded atthe tunnel face. Amplitude of external fore is set as momentary impact 1.0× 107[kN ], and loaded for 0.01[sec].Time increment 0.01[sec] is used for computation.

Figure 4: Finite element mesh

HH

HH

H

Displacement of the x direction Displacement of the y direction Displacement of the z direction

X-side Fixed Free Free

Y-side Free Fixed Free

Bottom Fixed Fixed Fixed

Table 1: Boundary conditions

Figure 5: Cross section Figure 6: Tunnel face and applied blasting force

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8.3 Case 1 Verification

In the verification, identification of damping parameters α0 and α1 are carried out. The verification showseffectiveness of the algorithm. We set initial and target values of damping parameters as shown in Table 2. Asthe observed velocity, we apply the velocity computed by the dynamic analysis using target values of dampingparameters. The parameters of the ground is listed in Table 3.

Figure 7 shows the history of the performance function. Figure 8 shows the variations of the dampingparameters α0 and α1, respectively. The performance function is converged to zero as in Figure 7. Bothdamping parameters are converged to the target values from the initial values which are arbitrarily assumed asin Figure 8.

HH

HH

H

Initial value Target value

α0 0.0100 0.0003

α1 0.0100 0.0050

Table 2: Computational conditions

PP

PP

PP

PP

Elastic Modulus [kN/m2] Density [kg/m3] Poisson Ratio

Values 5.0 × 107 2.0 × 103 0.26

Table 3: Parameters of rock

Iteration

Per

form

ance

func

tion

5 10 15 20 250

0.2

0.4

0.6

0.8

1

Figure 7: Variation of performance function

Iteration

Dam

ping

para

met

ers

0 5 10 15 20 250

0.002

0.004

0.006

0.008

0.01

Alpha0Alpha1

Figure 8: Variation of the both parameters

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8.4 Case 2 Practical application

We carry out an identification of damping parameters α0 and α1 based on the actual blasting waves. Thewaves are caused by the blasting of the tunnel excavation, and observed at the construction site. We set initialvalues of damping parameters as shown in Table 4. The parameters of the ground is listed in Table 5.

Figure 9 shows the history of the performance function. Figure 10 shows the variations of the dampingparameter α0 and α1. The performance function is converged to the steady value as shown in Figure 9. Thedamping parameters α0 and α1 are converged to the converged values 0.0012 and 0.0035 from the initial valuesas is in Figure 10. The comparisons between observed and computed velocity in the each direction is representedin Figures 11～13. It is seen that the computed velocity is almost identical to the observed velocity.

HH

HH

H

Initial value Target value

α0 0.0010 objective

α1 0.0010 objective

Table 4: Computational conditions

PP

PP

PP

PP

Elastic Modulus [kN/m2] Density [kg/m3] Poisson Ratio

Values 5.0 × 107 2.0 × 103 0.26

Table 5: Parameters of rock

Iteration

Per

form

ance

func

tion

10 20 30 400

0.2

0.4

0.6

0.8

1

Figure 9: Variation of performance function

Iteration

Dam

ping

par

amet

ers

0 10 20 30 400.001

0.0015

0.002

0.0025

0.003

0.0035

Alpha0Alpha1

Figure 10: Variation of the damping parameter α1

10

Time [sec]

Velo

cityi

nth

eX

dire

ctio

n[m

/sec

]

0 0.1 0.2 0.3 0.4

-0.0005

0

0.0005

Observed velocityComputed velocity

Figure 11: Comparison of velocity in the X direction

Time [sec]

Velo

cityi

nth

eY

dire

ctio

n[m

/sec

]

0 0.1 0.2 0.3 0.4

-0.0005

0

0.0005

Observed velocityComputed velocity

Figure 12: Comparison of velocity in the Y direction

Iteration

Velo

cityi

nth

eZ

dire

ctio

n[m

/sec

]

0 0.1 0.2 0.3 0.4

-0.0005

0

0.0005

Observed velocityComputed velocity

Figure 13: Comparison of velocity in the Z direction

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9 Conclusion

In conclusion, damping parameters both α0 and α1 can be identified based on the blasting waves at thesame time. We applied this identification technique to the Iwatayama tunnel construction site. The dampingparameters can be found so as to minimize the performance function. The damping parameters are convergedto the steady values from the initial values. The present method is shown to be effective for the determinationof the damping parameter. As the future work, we must improve in the computational accuracy of this method.

References

[1] R.Mahnken, E.Stein : ”Parameter Identification for Viscoplastic Models Based on Analytical Derivatives ofa Least-Squares Functional and Stability Investigations”, Int. J. Plasticity, Vol.12, No.4, pp.451-479, 1996.

[2] N.Koizumi and M.Kawahara : ”Parameter Identification Method for Determination of Elastic Modulus ofRock Based on Adjoint Equation and Blasting Wave Measurements” Int. J. Numer. Analy. Me th. Ge-omech.,2008.(In press)

[3] N. Koizumi and M. Kawahara : “Parameter Identification Method for Determination of Elastic Mod-ulus of Rock Based on Adjoint Equation Method and Blasting wave measurements“, (2009) Num-mer.Anal.Meth.Geomech, Vol.33, pp1513-1533.

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