Identification of Elastic Modulus using Extended Kalman Filter Finite Element Method

(2007 )

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Identification of Elastic Modulus using Extended Kalman Filter Finite Element Method

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Yusuke KATO

1

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2

2.1

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xk+1 = fk(xk) +Gkwk (1) yk = hk(xk) +vk (2)

]]c,xk^_¯()*?K,fk ^_K+,Vï,Gk

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1. [ ˆP_0|−1] = [Px0] , {ˆx_0|−1}={¯x0} 2. {ˆx_k+1|k}= [Fk]{ˆxk|k}

3. [Kk] = [ ˆPk|k−1][Hk]^T([Rvk] + [Hk][ ˆPk|k−1][Hk]^T)⁻¹ 4. [ ˆPk|k] = ([I]−[Kk][Hk])[ ˆPk|k−1]

5. {ˆxk|k}={ˆxk|k−1}+ [Kk]({yk} −[Hk]{ˆxk|k−1}) 6. [ ˆP_k+1|k] = [Fk][ ˆP_k|k][Fk]^T+ [Gk][Qwk][Gk]^T

=h, ^õ>?Fk, Hk ^{_? Vï^3 4{}, Fk =∂fk

∂xk

x=ˆxk

, Hk=∂hk

∂xk

x=x^∗k

] ]c, ˆxk|k ^{_¶· ¸m¡}y0,^@,yk ^{xz{D^!?}

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u⁽ⁿ⁺¹⁾_iβ =u⁽ⁿ⁾_iβ + ˙u⁽ⁿ⁾_iβ ∆t + 1

2u¨⁽ⁿ⁾_iβ ∆t² (4) + β∆t²(¨u⁽ⁿ⁺¹⁾_iβ −u¨⁽ⁿ⁾_iβ )

˙

u⁽ⁿ⁺¹⁾_iβ = u˙⁽ⁿ⁾_iβ + ¨u⁽ⁿ⁾_iβ ∆t+ ∆tγ(¨u⁽ⁿ⁺¹⁾_iβ −u¨⁽ⁿ⁾_iβ ),(5)

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˙

ukβ, ^aOukβ ^{Bœv\3‘’Jä>U' v}.

¨

uⁿ⁺¹_kβ =−S⁻¹_iαkβAiαkβu¨ⁿkβ+Biαkβu˙ⁿkβ (6) +Kiαkβuⁿ_kβ−ˆΓiα

2.3

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» ¼B? VïHk ^{U' v}. Hk =−S⁻¹_iαkβ

∂Siαkβ

∂S u¨ⁿ⁺¹_kβ +∂Aiαkβ

∂E u¨ⁿ_kβ (7) +∂Biαkβ

∂E u˙ⁿ_kβ+∂Kiαkβ

∂E uⁿ_kβ

Žc_,^{¶·®xf íB¶·è{D^"\Dƒ}, ^—

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xk+1=Ixk (8)

2.4 $%&'()

1). [Γ0] = [v0],{ˆ¨u−1}={uˆ¨0} 2). u˙n un by eq.(4), eq.(5)

3). [Kn] = [Γn][Hn]^T([Rn] + [Hn][Γk][Hn]^T)⁻¹ 4). [Pn] = ([I]−[Kn][Hn])[Γn]

5). [Γn+1] = [I][Pn]

6). {E^∗n}={E_n−1^∗ }+ [Kn]({yn−[Hn]{¨u^∗_n−1}) 7). {E_n+1^∗ }={E_n^∗}

If {x^∗_k+1} − {x^∗k}< ǫ, go to next time cycle.

else if return to 2).

3

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31;^3‘’¬43

-3 -2 -1 0 1 2 3

0 20 40 60 80 100 120 140 160 180 200 Accerelation [m/sec2]

Cycle number

observation data with noise

32;P oint1 ^{BÓ \}x^JV?1ü

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5

0 20 40 60 80 100 120 140 160 180 200 Accerelation [m/sec2]

Cycle number

observation data with noise

33;P oint1 ^{BÓ \}y^JV?1ü

-8 -6 -4 -2 0 2 4 6

0 20 40 60 80 100 120 140 160 180 200 Accerelation [m/sec2]

Cycle number

observation data with noise

34;P oint1 ^{BÓ \}z^JV?1ü

/; ^{—˜™ š??~®^¤Î®}E[N/m²]

Layer ^{?~® ¤ÎQ}

Layer1 5.0×10⁵ 1.0×10⁵ Layer2 5.0×10⁵ 8.0×10⁵ Layer3 5.0×10⁵ 1.0×10⁶ Layer4 5.0×10⁵ 1.4×10⁶

35;^{¶·¿ 3}

2; ^{2m(BÓ \ y¶·¿?O ^š}

Case ^{y¶·¿ ¶·¿š}

Case1 1 1

Case2 1,2 2

Case3 1,2,3 3

Case4 1,2,3,4 4

3.2 + ,

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400000 500000 600000 700000 800000 900000 1e+06 1.1e+06 1.2e+06 1.3e+06 1.4e+06

0 500 1000 1500 2000 2500 3000 3500 Elastic modulus [N/m2]

Cycle number Layer1 Layer2 Layer3 Layer4

36; Case1 BÓ \—˜™š?í!

400000 500000 600000 700000 800000 900000 1e+06 1.1e+06 1.2e+06 1.3e+06 1.4e+06

0 500 1000 1500 2000 2500 3000 3500 4000 Elastic modulus [N/m2]

Cycle number Layer1 Layer2 Layer3 Layer4

37; Case2 BÓ \—˜™š?í!

200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06

0 1000 2000 3000 4000 5000 6000 7000 Elastic modulus [N/m2]

Cycle number Layer1 Layer2 Layer3 Layer4

38; Case3 BÓ \—˜™š?í!

0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06

0 1000 2000 3000 4000 5000 6000 7000 Elastic modulus [N/m2]

Cycle number Layer1 Layer2 Layer3 Layer4

39; Case4 BÓ \—˜™š?í!

2;2m(BÓ \—˜™š? â[%]

Layer1 Layer2 Layer3 Layer4

Case1 96 92 95 0

Case2 98 97 99 0

Case3 99 99 98 42

Case4 99 99 99 99

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\? |U" \Dƒ, ^{»¼?1g? 2m(B¬Ói89}

v\.

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Case ^?~®

Case1 1.0×10⁵ Case2 1.0×10⁶ Case3 1.0×10⁷ Case4 1.0×10⁸

”• ,^{¶· ¿?O} ,^šŠ_G89/?Case2^UyW

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Uhih¦„]^|z, ^{? ~®BÓ\Aݍm}

m‹h´_AݍÒTU"\& xd\.

4;2m(BÓ \—˜™š? â[%]

Layer1 Layer2 Layer3 Layer4

Case1 78 94 97 0

Case2 97 97 99 0

Case3 99 99 98 0

500000 600000 700000 800000 900000 1e+06 1.1e+06 1.2e+06 1.3e+06 1.4e+06

0 200 400 600 800 1000

Elastic modulus [N/m2]

Cycle number

10⁵ 10⁶ 10⁷

10; Case1Case3

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1 R.E.Kalman, A New Approach to Linear Filtering and Prediction Problems, Trans.ASME,J. Basic Eng., vol182D, no.1,34-45,(1960)

3 A.Murakami and T.Hasegawa, Back Analysis by Kalman Filter Finite Elements and a Determina- tion of Optimal Observed Points Location. Jour- nals of the Japan Society of Civil Engineers Vol.388,227- 235,(1987)

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