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๷Ӵେֶߍ਺ֶڭҭࣨ

֓ཁ. ίϯΫϦʔτݐ଄෺ʹର͢Δtomographicͳඇഁյݕٕࠪज़͸ະ։ൃͰ͋Δͱ

͍͏ͷ͕ݱঢ়Ͱ͋Δ͕ɼۙ೥ɺࡾాلߦ(৬ۀೳྗ։ൃ૯߹େֶߍ)^{ͱߨԋऀͷڠಇʹΑ} ΓɼίϯΫϦʔτݐ଄෺ʹର͢Δacoustic CT։ൃͷΞΠσΞ͕ఏҊ͞Ε͍ͯΔɻຊߨ ԋͰ͸ɼίϯΫϦʔτݐ଄෺ʹର͢Δacoustic CT։ൃͷΞΠσΞΛ঺հ͠ɼ͜ͷ໰୊

ʹର͢Δ૭Radonม׵ͷԠ༻ʹؔͯٞ͠࿦͢Δɻ

Development of ultrasonic CT for concrete structures and application of window functions

Takashi Takiguchi

^∗

∗

National Defense Academy of Japan

Abstract. There having been developed no tomographic non-destructive testing tech-nique for concrete structures for the time being, N. Mita and the speaker proposed an idea to develop an acoustic CT for concrete structures. In this talk, we introduce their idea and discuss application of the windowed Radon transform for development of an acoustic CT for concrete structures.

1. ͸͡Ίʹ

ίϯΫϦʔτݐ଄෺ʹର͢Δ

tomographic

ͳඇഁյݕٕࠪज़͸ະ։ൃͰ͋Δͱ͍͏ͷ͕

ݱঢ়Ͱ͋Δ͕ɼۙ೥ɺ༷ʑͳख๏͕ఏҊ͞Ε࢝Ί͍ͯΔɻࡾాلߦ

(

৬ۀೳྗ։ൃ૯߹େ

ֶߍ

)

ͱߨԋऀ͸ɼ

[2, 3]

ʹ͓͍ͯίϯΫϦʔτݐ଄෺ʹର͢Δ௒Ի೾

CT

։ൃͷΞΠσΞ ΛఏҊ͍ͯ͠ΔɻຊߨԋͰ͸ɼίϯΫϦʔτݐ଄෺ʹର͢Δ௒Ի೾

CT

։ൃͷΞΠσΞΛ

঺հ͠ɼ͜ͷ໰୊ʹର͢Δ૭

Radon

ม׵ͷԠ༻ʹؔͯٞ͠࿦͢Δɻ

2. ίϯΫϦʔτͷجૅ஌ࣝ

ຊઅͰ͸ɼຊߨԋͰඞཁͱͳΔίϯΫϦʔτͷجૅ஌ࣝʹ͍ͭͯ؆୯ʹઆ໌͢Δɻ ఆٛ2.1 (ίϯΫϦʔτ) ίϯΫϦʔτ͸ɺηϝϯτ

(cement: C)

ɺਫ

(W)

࠭

(fine aggregate

or sand: S)

ɼ࠭ར

(coarse aggregate or gravel: G)

ͷࠞ߹෺Ͱ͋Δɻ

஫ҙ2.1

(i)

ηϝϯτ+^ਫ→^{ηϝϯτϖʔετ}

(cement paste)

(ii)

ηϝϯτ+ਫ+࠭

(

ηϝϯτϖʔετ+࠭

)

→Ϟϧλϧ

(mortar) (iii)

ίϯΫϦʔτ

(concrete)

͸

“

Ϟϧλϧ+^࠭ར

”

ͱཧղͰ͖Δɻ

ຊઅͰ͸ɼҎԼͷओுʹج͖ͮɼίϯΫϦʔτͱ͸Կ͔Λઆ໌͢Δɻ ओு2.1

The concrete materials are artificial megaliths.

ओு

2.1

͸

[2, 3]

ʹԙ͍ͯओு͞Εͨ΋ͷͰ͋Δ͕ɼҎԼͷٞ࿦ʹج͍͍ͮͯΔɻྫ

͑͹

, Valley Temple (BC2500?)

΍

Parthenon

ਆ఼

(BC447-432)

͸ੴ଄ݐ଄෺Ͱ͋Γɼங ޙ

2000

೥Ҏ্ܦͬͨݱࡏͰ΋ɼͦͷߏ଄Λอ͍ͬͯΔɻͱ͜Ζ͕ɼϩʔϚͷ

Colosseum

(AD70-80)

ͷ֎น͸ੴ଄Ͱ͋Δ͕ɼ಺น͸ɼ࿃נͷഁยɾੴɾ࠭ɾփ౳ͰຒΊΒΕ͍ͯΔɻ

͜Ε͸ɼϩʔϚͰ͸े෼ͳڊੴ͕ೖखͰ͖ͳ͔ͬͨͨΊͰ͋Δ͕ɺ

Colosseum

ͷ಺นʹ༻

͍ΒΕͨૉࡐ

(Roman or promitive concrete)

͸

2000

೥ۙ͘ڊੴͷ୅ସ෺ͱͯ͠ͷ໾ׂΛ Ռ͖ͨͯͨ͠ͱ͍͑Δɻ

ίϯΫϦʔτ͕ڊੴͷ୅ସ෺ʹͳΔʹ͸ɼ

(a) Concrete is not tough to tensile strength.

(b) Concrete easily gets cracks in and on itself.

౳ͷ໰୊఺Λվળͤ͞ͳ͚Ε͹ͳΒͳ͍͕ɼ

(

α

) Concrete is easily made and shaped in any form because of its fluidity before it gets hard.

(

β

) The cost of concrete is very cheap.

౳ɼڊੴΑΓ༏Εͨ఺΋ଟ͍ɻܽ఺

(a)

Λิ͏ͨΊʹ։ൃ͞Εͨͷ͕మےίϯΫϦʔτ

(RC)

Ͱ͋Δɻ

• మͱίϯΫϦʔτͱͷ೤๲ு܎਺͸΄΅ಉ͡Ͱ͋Δ

• ηϝϯτͷΞϧΧϦੑ͕మےΛࡺ͔ΒकΔ

౳ͷཧ༝͔ΒɼమےίϯΫϦʔτ͕։ൃ͞Εͨ౰࣌͸ɼమےίϯΫϦʔτ͸ɼ

“

ح੻ͷૉ

ࡐ

”

ɼ

“

Ӭٱૉࡐ

”

౳ͱݺ͹Ε͍͕ͯͨɼޙʹͦΕΒ͕ݬӨͰ͋ͬͨ͜ͱ͕Θ͔ͬͨɻݱࡏ

͸ɼ

RC

ݐ଄෺ͷ௕ण໋Խͱͦͷҡ࣋؅ཧ͕Πϯϑϥͷҡ࣋؅ཧɾ࠶੔උʹ͓͍ͯॏཁͳ

՝୊ͷҰͭͱͳ͍ͬͯΔɻ

3. ίϯΫϦʔτݐ଄෺ʹର͢Δ CT

લઅͰड़΂ͨ௨Γɼ

RC

ݐ଄෺ͷ௕ण໋Խͱͦͷҡ࣋؅ཧ͸ඇৗʹॏཁͳ໰୊Ͱ͋Δ͕ɼ ίϯΫϦʔτݐ଄෺ʹର͢Δ

tomographic

ͳඇഁյݕࠪ๏͸ཱ֬͞Ε͍ͯͳ͍ͱ͍͏ͷ͕

ݱঢ়Ͱ͋Δɻۙ೥ɼ༷ʑͳओு͕ఏҊ͞Ε͍ͯΔ͕ɼࡾాͱߨԋऀ͸

[2, 3]

_ʹ͓͍ͯ

,

_௒ Ի೾΍ి࣓ύϧεԻڹΛ༻͍ͨ

acoustic CT scan

ͷ։ൃΛఏҊ͍ͯ͠ΔɻຊઅͰ͸ɼ

[2, 3]

ͰఏҊ͞ΕͨΞΠσΞΛ঺հ͢Δɻ

൴Β͸ɼ·ͣɼ࣮ݧΛ௨ͯ͡

,

ҎԼͷੑ࣭Λ֬ೝͨ͠ɻ

ੑ࣭3.1

•

We can conclude that for the test pieces of the length less than 1200mm, there is no decay of the acoustic velocity from the viewpoint of its first arriving time.

•

The first arrival wave of the ultrasonic one takes the fastest route in the cement paste, the mortar and the concrete.

͜ΕΒͷੑ࣭Λ༻͍ͯɼίϯΫϦʔτݐ଄෺ʹର͢Δ

acoustic CT

։ൃͷͨΊʹ͸࣍ͷ

໰୊Λղܾ͢Ε͹ྑ͍͜ͱΛओு͍ͯ͠Δɻ

໰୊3.1 (Problem for non-destructive inspection for concrete structure)

Let

Ω ⊂ R³

be a domain and f (x) be the propagation speed of the ultrasonic wave at the point x

∈Ω

. For

α, β ∈ ∂Ω

, we denote by

γα,β

a route from

α

to

β

contained in

Ω

. In this case, reconstruct

f (x) ( x

∈Ω

) out of the data

(3.1) min

γα,β

γα,β

1

/

f (x)d

γ,

for

∀α, β∈∂Ω

.

ੑ࣭

3.1

ʹ͋ΔΑ͏ʹ͜ͷ໰୊͸௕͕͞

1

.

2m

ҎԼͷݐ଄෺Λର৅ʹ͍ͯ͠Δ͕ɼమے ߴ૚ݐ଄෺ͷபɼߴ଎ಓ࿏ڮྊʹ͓͚Δ͔ͿΓίϯΫϦʔτͷҟৗௐࠪ౳ɼଟ͘ͷ໰୊΁

ͷԠ༻͕ݕ౼͞Ε͍ͯΔɻৄ͘͠͸

[3]

Λࢀর͞Ε͍ͨɻ

4. ి࣓ύϧεԻڹ૷ஔ

લઅͰ঺հͨ͠

acoustic CT

Ͱ͸ɼ௒Ի೾૷ஔ΋ి࣓ύϧε૷ஔ΋Իݯͱड৴૷ஔͷ૊

Έ߹ΘͤͰ༻͍ΒΕ͓ͯΓɼݪཧతʹ྆ऀͷҧ͍͸ແ͍͕ɼطଘͷ૷ஔΛ༻͍ͨ৔߹ɼి

࣓ύϧε૷ஔͷํ͕؍ଌͷਫ਼౓͕௿͍ɻ໪࿦ɼ؍ଌਫ਼౓͸૷ஔͷਫ਼౓ʹґଘ͢ΔͨΊɼҰ

֓ʹ͸݁࿦෇͚͸ग़དྷͳ͍͕ɼి࣓ύϧε૷ஔʹΑΔ؍ଌਫ਼౓ͷ௿͞ͷݪҼ͸ɼύϧεԻ

ൃੜͷݪཧʹґΔͱ͜Ζ͕গͳ͘ͳ͍Α͏ʹײ͡ΒΕΔɻ

ి࣓ύϧεʹΑΔԻ೾ൃੜ͸ҎԼͷΑ͏ͳݪཧʹج͍͍ͮͯΔɻ

Fig. 1. ి࣓ύϧεԻൃੜͷݪཧ

ਤ

1

ʹ͓͍ͯ

,

ਤͷԖ௚ํ๏ʹ࣓৔Λ͔͚ɼίΠϧʹύϧεిྲྀΛྲྀ͢͜ͱʹΑΓ

,

૷ஔ

಺ͷۚଐ

(

మ

)

͕࣓࿪ޮՌ

(magneto-striction e

ff

ect)

ʹΑΓ௒ߴԻΛൃੜ͢Δɻैͬͯɼి

࣓ύϧεԻ͸௒Ի೾Ͱ͋Γɼͦͷप೾਺Λௐ੔͢Δٕज़΋ਐΜͰ͍Δ͕ɼۚଐͷͲͷ෦෼

͔ΒԻΛൃ͍ͯ͠Δ͔͸ݫີʹ͸ௐ΂ΒΕ͍ͯͳ͍ɻ໪࿦ɺి࣓ύϧεԻڹ૷ஔ಺ͷۚଐ

͸ඍখͰ͋ΔͨΊɼେ͖ͳޡࠩ͸ੜ͍ͯ͡ͳ͍ͱࢥΘΕΔ͕ɼ΋͠ɼ؍ଌຖʹԻݯͷ৔ॴ

͕มΘ͍ͬͯΕ͹ɼ؍ଌ஍ͷਫ਼౓ʹେ͖ͳӨڹΛ༩͍͑ͯΔ͸ͣͰ͋Δɻ

ຊߨԋͰ͸ɼమےίϯΫϦʔτए͘͠͸మےϞϧλϧʹରͯ͠ɼ௚઀ి࣓ύϧεΛͿͭ

͚Δ͜ͱʹΑΓɼమےͷͲ͔͜ΒԻ͕ग़͍ͯΔ͔Λௐ΂Δ໰୊Λߟ͑Δɻ

໰୊4.1 మےίϯΫϦʔτɼए͘͠͸మےϞϧλϧɼʹର͢Δి࣓ύϧεԻڹʹԙ͍ͯɼ Իݯ఺ΛશͯಛఆͤΑɻ

͜ͷ໰୊͕ղܾ͢Ε͹ɼ಺෦஫ೖΛࢪ͞ͳ͍

RC

ݐ଄෺ʹର͢ΔԻڹ

PET(Positron

Emission Tomography)

΁ͷԠ༻͕ՄೳʹͳΔɻຊߨԋͰ͸ɼҎԼͷ

3

఺Λ঺հ͢Δ͜ͱ

Λओ໨ඪͱ͢Δɻ

• ໰୊

4.1

͕࣍અͰ঺հ͢Δ੾அ૭

X

ઢม׵ͱີ઀ͳؔ܎Λ࣋ͭ͜ͱΛࣔ͢͜ͱɻ

• ز͔ͭͷ૝ఆʹରͯ͠໰୊

4.1

͕ղܾՄೳͰ͋Δ͜ͱΛࣔ͠ɼͦͷղΛ༩͑Δ͜ͱɻ

• ^໰୊

4.1

ͷղੳ݁ՌΛ༻͍ͯɼ಺෦஫ೖΛࢪ͞ͳ͍

RC

ݐ଄෺ʹର͢ΔԻڹ

PET

։ൃ͕ՄೳͰ͋Δ͜ͱɻ

5. ૭ Radon ม׵

ຊઅͰ͸ɼલઅͷ࠷ޙʹఏىͨ͠໰୊

4.1

ʹؔ࿈͢Δੵ෼ม׵Λ঺հ͢Δɻຊઅͷओ

໨ඪ͸੾அ૭

X

ઢม׵ͷఆٛΛ༩͑Δ͜ͱͰ͋Δ͕ɼ·ͣ͸ɼͦͷجʹͳ͍ͬͯΔɼ૭

Radon

ม׵

(

૭

X

ઢม׵

)

ͷఆٛΛ෮श͢Δɻ૭

Radon

ม׵

(

૭

X

ઢม׵

)

͸

Radon

ม׵

(X

ઢม׵

)

ʹ૭ؔ਺Λֻ͚ͨੵ෼ม׵Ͱ͋Δ͕ɼͦͷఆٛ͸ҎԼͰ༩͑ΒΕΔɻ

ఆٛ5.1 (૭Radonม׵)

Let 1

≤

d

≤

n. The d dimensional Windowed Radon Transform R

h

f (x

,

A) of a function f defined on

Rⁿ

is defined by

(5.1) R

h

f (x

,

A)

≡

R^d

h(t) f (x

+

At)dt

,

where h is a function defined on

R^d

and A

∈ {

A

=

vRJ J :

R^d → Rⁿ

(inclusion)

,

R

∈

S O(n)

/

S O(n

−

d)

,

v

>

0

}

. h means the complex conjugate of h. We write R ¯

h

f (x

,

A)

=

f

h

(x

,

A).

ఆٛ5.2 (૭Xઢม׵)

For a function f defined on

Rⁿ

we call

(5.2) X

_h

f (x

, ξ

)

≡

R

h(t) f (x

+

t

ξ

)dt

,

the Windowed X-ray Transform (WXT) with window h, where h is a function defined on

R

and

ξ ∈Rⁿ

. We adopt the notation

ξ=

v

ω,

v

>

0

, ω ∈

S

ⁿ⁻¹

.

͜ΕΒͷม׵ʹର͢Δੑ࣭ͳͲ͸ɼ

[4]

Λࢀর͞Ε͍ͨɻ

ৄ͘͠͸ߨԋʹͯઆ໌͢Δ͕ɼ໰୊

4.1

ʹ͓͍ͯɼ࠷΋୯७ͳ৔߹ʹ͸ɼҎԼͷٯ໰୊

Λղܾ͢Ε͹ྑ͍͜ͱ͕͍͑Δɻ

໰୊5.1 R^{2্Ͱఆٛ͞Εͨؔ਺}

f (x)

,

x

∈R^{2 ͕ط஌Ͱ͋Δ࣌}

,

؍ଌ஋

(5.3)

_∞

a

f (y

+

t

θ

)dt

͔Β

a

Λ෮ݩͤΑɻୠ͠

,

θ∈

S

¹,

y

⊥θͱ͢Δɻ

(5.3)

͸੾அ

X

ઢม׵ͱݺ͹ΕΔม׵Ͱ͋Δ͕ɼ໰୊

4.1

ʹؔͯ͠͸ɼ͜ͷม׵͕੾அ

૭

X

ઢม׵ʹରԠ͢Δࣄ͕Θ͔Δɻৄ͘͠͸ߨԋʹͯઆ໌͢Δɻ

6. ͓ΘΓʹ

ຊߨԋʹԙ͍ͯɼి࣓ύϧεԻڹʹ͓͚ΔԻݯͷಛఆʹؔ͢Δ໰୊Λఏى͕ͨ͠ɼ͜ͷ

໰୊͸੾அ૭

X

ઢม׵ʹ͓͚Δ੾அ఺Λ෮ݩ͢Δ໰୊

5.1

ͱਂ͍ؔ܎Λ͍࣋ͬͯΔɻ໰

୊

4.1

ͷݚڀΛൃలͤ͞ΔͱɼίϯΫϦʔτݐ଄෺ʹର͢ΔԻڹ

PET

ͷ։ൃ͕Մೳʹͳ Δɻࠓޙɺ࣮ݧͱཧ࿦ͷ྆ऀͷཱ৔͔Β͜ΕΒͷ໰୊ͷݚڀΛਐΊɼίϯΫϦʔτݐ଄෺

ͷඇഁյݕࠪ๏ͷ͞ΒͳΔൃలʹܨ͍͛ͨɻ

ँࣙ

ຊݚڀ͸ɼ

JSPS

Պֶݚڀඅ

(C)26400184

ͷॿ੒Λड͚͍ͯ·͢ɻ

ࢀߟจݙ

[1] Mita, N., Horie, Y. and Matsudome, S. : Fundamental study on estimation of com-pressive strength of cement mortar by pulsed electromagnetic force acoustic method, Architectural Institute of Japan

76

(662) (2011) , pp. 721–727.

[2] Mita, N. and Takiguchi, T. : Basic propertiers of concrete and its non destructive test-ing, MI Research

2, “Collaboration between theory and practice in inverse problems”,

(2015), pp. 117-137.

[3] ——- : Development of ultrasonic tomography for concrete structures, MI Research

5,

“Mathematical backgrounds and future progress of practical inverse problems”, (2016), pp. 205-233.

[4] Takiguchi, T. : On invertibility of the windowed Radon transform, J. Math. Sci., Univ.

Tokyo,

2

(1995) pp. 621-636.

ୌޱ ޹ࢤ

(

๷Ӵେֶߍ਺ֶڭҭࣨ

)

˟

239-8686

ਆಸ઒ݝԣਢլࢢ૸ਫ

1-10-20

E-mail:

[email protected]

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