Computational Modeling of Wave Propagation in a Geophysical Domain

Transcription

1 Except fom the Poceedings of the COMSOL Confeence 008 Boston Computational Modeling of Wave Popagation in a Geophysical Domain Jason McKenna 1 Mihan McKenna 1 Segei Yushanov Jeffey Compton and Kyle Koppenhoefe * 1 US Amy Engineeing Reseach and Development Cente ACES of Columbus LLC *Coesponding autho: 750 Coss Pointe Road Suite E Gahanna OH 40 yle@acescolumbus.com Abstact: The popagation of shea (S) and compession (P) waves within the eath allows geologists to tac seismic events and to identify subteanean stuctue. Highly specialized geological based compute pogams developed have been instumental in detemining the location and chaacteistics of natual phenomena (e.g. eathquaes) and man-made activity (e.g. nuclea-blast tests). Use of these intenally developed pogams equies egula maintenance and the eliance on supecomputes limits boad accessibility. This pape sees to demonstate that commecially available softwae unning on destop computational esouces can povide accuate solutions to an impotant subset of poblems associated with wave popagation in the geophysical domain. The wo pesented hee uses COMSOL Multiphysics to solve the equilibium equations fo a time-vaying system using the finite element method. This wo focuses on developing a benchma solution of a homogeneous half-space loading with an impact and develops a geneal closed-fom solution against which to compae the computational esults. These esults show the ability to esolve both S and P wave acoss the computational domain. Thus COMSOL Multiphysics unning on destop computational esouces povides sufficiently accuate esults fo citical geophysical wave popagation poblems. Keywods: Geophysics shea wave pessue wave seismic. 1. Intoduction Scientists and enginees that see to undestand elastic wave popagation in geological stuctues typically conside the geneic poblem of a seismic wave that is geneated at a souce popagates though a media and is measued at a eceive. Some eseaches ae concened with natual souces e.g. eathquaes while othes focus on man-made souces e.g. explosions. Some who study these aeas see to undestand the natue of the souce while othes use well chaacteized souces and see to chaacteize the medium. These eseaches may be seeing to pedict futue eathquaes locate natual esouces o identify and locate specific human activities. Howeve in all these cases the oveaching physics of elastic wave popagation in a solid medium emain the same. Seismic waves popagating in bul mateial do so as eithe compessional waves P whee mateial tanslates in the diection of wave popagation and shea waves S whee mateial tanslates pependicula to the diection of wave popagation. P-waves tavel faste than S-waves. Anothe impotant class of waves is suface waves; these waves develop due to an enegy concentation nea the Eath s suface and consequently popagate in two dimensions. As such suface waves decay as while body waves decay as due to thei popagation in thee-dimensions. Thus enegy measued at a nea-suface eceive located a significant distance fom the souce is dominated by suface waves. Typically these types of waves ae efeed to as eithe Love o Rayleigh waves. Love waves ae shea waves that have been polaized in the hoizontal diection (paallel to the Eath s suface) while Rayleigh waves ae a mixtue of P waves and S waves that have been polaized in the vetical diection. Geophysics studies typically use specialized compute softwae unning on supecomputes to simulate wave popagation in geophysical domains. This wo demonstates that a finite element based softwae pacage that is commecially available COMSOL Multiphysics solves these wave popagation poblems using eadily available destop computational hadwae.

2 Repot Documentation Page Fom Appoved OMB No Public epoting buden fo the collection of infomation is estimated to aveage 1 hou pe esponse including the time fo eviewing instuctions seaching existing data souces gatheing and maintaining the data needed and completing and eviewing the collection of infomation. Send comments egading this buden estimate o any othe aspect of this collection of infomation including suggestions fo educing this buden to Washington Headquates Sevices Diectoate fo Infomation Opeations and Repots 115 Jeffeson Davis Highway Suite 104 Alington VA Respondents should be awae that notwithstanding any othe povision of law no peson shall be subject to a penalty fo failing to comply with a collection of infomation if it does not display a cuently valid OMB contol numbe. 1. REPORT DATE OCT 008. REPORT TYPE. DATES COVERED to TITLE AND SUBTITLE Computational Modeling of Wave Popagation in a Geophysical Domain 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) US Amy Engineeing Reseach and Development CenteGeotechnical & Stuctues Laboatoy909 Halls Fey RoadVicsbugMS PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR S ACRONYM(S) 1. DISTRIBUTION/AVAILABILITY STATEMENT Appoved fo public elease; distibution unlimited 1. SUPPLEMENTARY NOTES Pesented at the 008 COMSOL Confeence Boston 9-11 Oct Boston MA 11. SPONSOR/MONITOR S REPORT NUMBER(S) 14. ABSTRACT The popagation of shea (S) and compession (P) waves within the eath allows geologists to tac seismic events and to identify subteanean stuctue. Highly specialized geological based compute pogams developed have been instumental in detemining the location and chaacteistics of natual phenomena (e.g. eathquaes) and man-made activity (e.g. nuclea-blast tests). Use of these intenally developed pogams equies egula maintenance and the eliance on supecomputes limits boad accessibility. This pape sees to demonstate that commecially available softwae unning on destop computational esouces can povide accuate solutions to an impotant subset of poblems associated with wave popagation in the geophysical domain. The wo pesented hee uses COMSOL Multiphysics to solve the equilibium equations fo a time-vaying system using the finite element method. This wo focuses on developing a benchma solution of a homogeneous half-space loading with an impact and develops a geneal closed-fom solution against which to compae the computational esults. These esults show the ability to esolve both S and P wave acoss the computational domain. Thus COMSOL Multiphysics unning on destop computational esouces povides sufficiently accuate esults fo citical geophysical wave popagation poblems. 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified Same as Repot (SAR) 18. NUMBER OF PAGES 1 19a. NAME OF RESPONSIBLE PERSON

3 Standad Fom 98 (Rev. 8-98) Pescibed by ANSI Std Z9-18

4 . Method This wo uses the Stuctual Mechanics Module available in COMSOL Multiphysics to develop computational models fo a ange of poblems. These models incease with complexity anging fom a simple point souce in an infinite solid to a volume souce that epesents expeimentally measue foces applied in a layeed half-space..1 Point Souce in an Infinite Media Initially this wo focuses on modeling a point souce located in an infinite domain. To povide a point of compaison fo the finite element models an analytical solution fo this poblem was developed and is descibed as follows. Using the notation u ij to epesent the displacement in the i diection due to a concentated point foce specified as f ( x t; ξ) = f 0 ( t) δ ( x ξ) e j applied in the j diection at the point ξ the displacements ove the domain may be detemined by solving the following equation; ( t) 4πρ u x ij ( δ ) / β i j ij = f 0 / α ( t ) d i j f t 0 α α ( δ ) i j β whee ρ is the mateial density = x ξ is the distance fom the souce xi ξ i = ae the diection cosines of vecto and λ μ α = c P = ρ and μ β = c S = : ρ epesent the speeds of pessue and shea waves espectively. In these equations λ μ ae Lame elastic constants ij f 0 t β This wo epesents the typical souce foce as: at f 0 ( t) = H ( t) = a te whee H () t is the Heaviside step function a is a paamete contolling duation and amplitude of the souce fo additional infomation see [1]. Thee finite element models of a spheical domain loaded by a point souce wee developed in COMSOL Multiphysics. The volume was epesented in thee-dimensions using a quate symmetic finite element model. To educe the computational cost of this model twodimensions models whee constucted using an axisymmetic and a plane stain fomulation.. Point Souce in a Semi-Infinite Media To extend this wo to include suface wave a solution to Lamb s poblem of a souce loading at the suface of a semi-infinite domain is developed. Again a closed fom solution was developed as a point of compaison fo the finite element solution of this poblem. In this closed fom solution the vetical stess acting on the fee suface is denoted as σ ; whee foce and ( t) πa σ ( t) f = denotes the total applied a denotes the adius of the cicle ove which the stess is applied. Fom this loading the vetical displacement is denoted w ( t) and hoizontal displacements ae denoted u ( t). Thus the solution fo a special case of Lamb s poblem may be witten as w( t) σ α G( ) df = () π μ β β d u t dt R( ) whee t = / β

5 ( ) G 0 < δ π 6 96 = π 6 48 π /8 > ( 5) ( ) ( 5) ( ) ( 5) ( / 4 / 4) ( 4) 1 < < δ < < 1 R () 0 < δ 6Κ 16 6 = 6Κ / 16 6 π Peceding 4 ( ) 18Π( 8 ) ( 6 4 ) Π[ ( 0 1 ) ] ( 6 4 ) ( 0 1 ) ] Π 8 ( 6 4 ) Π ( 0 1 ) ( ) ( 6 4 ) Π ( 0 1 ) > and δ = α / β = / = ( β / )t = α / β 1 / α / β 1. [ ] The Κ ( ) and ( n) [ ] Π ae complete elliptic integals of the fist type and thid types espectively. This solution poves valid fo Poisson s atio of ν = 0.5. A solution fo abitay ν is given in [] An axisymmetic model of a semi-infinite domain was developed in COMSOL Multiphysics to solve Lamb s poblem. Fo both the closed-fom and finite element solution the focing function is descibed as π f( t) = hcos t fo T T t. T Fo the finite element analyses conducted in this wo the peiod of the loading is 10 μs ( T = 10μs ). 1 δ < < 1 1< < Figue 1. Compaison of vetical displacements fo a thee-dimensional finite element solution to a closed-fom solution fo point loading in an infinite domain.. Compaison with Expeimental Data The next level of complexity fo this wo is to epesent an actual geophysical domain ove which expeimental data was taen. The model developed peviously in COMSOL Multiphysics was modified to include measued pessue wave speeds (c p ) and shea wave speeds (c s ) and to epesent the actual focing function applied the suface. Ove a depth of thity metes ten mateial layes that epesent expeimentally measued values of c p and c s values wee included in the model. The pessue wave velocity inceased fom appoximately 650 m/s to 000 m/s with inceasing depth. The shea wave velocities inceased fom 00 m/s to 600 m/s ove this ange of depths.. Discussion.1 Point Souce in an Infinite Media Eo! Refeence souce not found. shows the compaison of vetical displacements ove a adial line fom the cente of the sphee. The esults show the tansmission of the wave as it moves though the infinite domain. These esults show the ability of COMSOL Multiphysics to epesent accuately the wave except fo mino vaiations at the pea of the wave. These

6 diffeences may be due to the mesh size used in this model. show the ability of the finite element analysis to epesent accuately the wave popagation geneated fom a volume souce though a semiinfinite domain. The finite element esults show the lowe magnitude faste moving P wave that pecedes the Rayleigh wave. The finite element analyses accuately calculate the aival time and fequency content of this pulse. Howeve the finite element analyses unde pedict the magnitude of the velocity by appoximately 0%.. Effects of Model Details Figue. Compaison of adial displacements fo a thee-dimensional finite element solution to closedfom solution fo point loading in an infinite domain. Figue compaes the esults developed fo an axisymmetic model of an infinite domain. These esults show simila accuacy compaed to the thee-dimension esults shown in Eo! Refeence souce not found.. Nea the pea of the wave the axisymmetic model shows geate accuacy than the thee-dimensional model. This impoved accuacy may be due to the small finite element size used in the axisymmetic models. This wo also used a plane stain fomulation to assess the ability to epesent a point souce using a plane stain fomulation. Figue clealy demonstates that the plane stain fomulation povides a poo appoximation to a point souce. The plane stain fomulation tansfoms a point souce into a line souce. Thus the computed wave fom changes damatically between the thee-dimensional and plane stain solutions. These esults indicate that an axisymmetic fomulation povides a bette method to educe the computational cost of this poblem than a plane stain fomulation.. Point Souce in a Semi-Infinite Media Figue 4 compaes esults developed fom a finite element analysis conducted using COMSOL Multiphysics with the analytical solution developed in Section.. These esults To compae with expeimental data a finite element model was constucted that includes the vaiation of the soil popeties as descibed in Section.. In Figue 5 the effects of the vaiable wave speed ae shown compaed to a model that has homogenous mateial popeties equal to the fist laye in the vaiable model. The homogeneous model shows clealy defined P and Rayleigh waves while the layeed model shows the wave eflections that developed due to the vaiations in mateial popeties though the depth of the model. The shote aival time fo the layeed model epesents one inteesting featue of this compaison: the inceased wave speed below the suface of the model geneates a wave that eflects off a subsuface laye and aives bac at the suface in less time than the P wave taveling in the fist laye of the model..4 Compaison with Expeimental Data Figue 6 compaes the adial velocity measued duing a well-contolled seies of expeiments with esults fom the finite element analyses. The magnitude of the velocities has been nomalized by the pea magnitude duing the duation of the pulse. This nomalization was done to povide a compaison using the two pimay metics of inteest in this wo: aival times and fequency content. Pediction of the magnitude of the wave is a seconday consideation. These esults show a easonable ageement between the finite element esults and expeimental data fo the pimay factos of inteest.

7 Figue. Compaison of vetical displacements fo thee-dimensional and plane stain finite element solution. Figue 5. Effect of vaying wave speed though layes in the computational model (R=0 m). 1.5 Nomalized Radial Velocity Expeimental Data FEA Time (s) Figue 4. Compaison of analytical solution and finite element solution fo a nea-suface point located at 100 m fom souce ( T = 10μs ). Figue 6. Compaison of finite element esults with expeimental data (R=0 m). The time specified on the odinate is elative to the stat of the expeiment.

8 4. Summay and Conclusions The esults pesented in this pape demonstate the capabilities of a COMSOL Multiphysics to solve wave popagation poblems in finite geophysical domains. The analyses wee conducted using computational hadwae that is eadily available on the destop. To maximize the use of destop hadwae this wo examines methods fo educing the poblem size. Fo the point souce poblems consideed hee axisymmetic modeling povides significantly moe accuate esults compaed with plane stain modeling techniques. This wo initially focuses on solving poblems in a unifom infinite space o infinite half-space fo which analytical solutions exist. Afte showing good ageement with these solutions additional complexity is added to epesent available expeimental data. The modeling methods developed in this wo again show good ageement with data developed. This wo shows the stong effect of including the vaiation of wave speed though the top thity metes of the eath. By including this vaiation in mateial popeties the wave aival time and fequency content agee with expeimental data. Thus this wo demonstates that COMSOL Multiphysics povides a useful tool fo pedicting wave popagation using commecially available finite element softwae with destop computing hadwae. 8. Refeences 1. Pujol J. Elastic Wave Popagation and Geneation in Seismology Cambidge Univesity Pess 00.. Mooney H. M. Some Numeical Solutions fo Lamb s Poblem Bulletin of Seismological Society of Ameica Vol. 64 No. pp Apil 1974

9 Pesented at the COMSOL Confeence 008 Boston Computational Modeling of Wave Popagation in a Geophysical Domain Jason McKenna1 Mihan McKenna1 S.Y. Yushanov J.S. Compton K.C. Koppenhoefe 1 U.S. Amy Engineeing Reseach and Development Cente ACES of Columbus LLC

10 Oveview Objective: Demonstate the capability of COMSOL Multiphysics to accuately solve wave popagation poblems in geophysics Motivation: Reduce eliance on custom softwae and supecomputes by obtaining solution using commecially available softwae on high-end destop compute Appoach: Develop closed-fom solutions fo Point souce in an infinite body Point souce on the suface of a semi-infinite body (Lamb s poblem) Develop using solid mechanics module w/ COMSOL Same fomulation as acoustics module Thee-dimensional Axisymmetic Plane stain Compaison w/ expeimental data Hamme blow on suface

11 Closed Fom Solution - Displacement Elastic Wave in an Infinite Body Semi-Infinite Body (Lamb s Poblem- Fixed Poisson atio ) = β β δ α α δ πρ β α t f t f d t f t u ij j i j i ij j i ij 0 0 / / 0 4 x () β β α μ π σ β = = d R G dt df t u w t t / > < < < < < = π π δ π δ / / 4 / G () [ ] [ ] > < < Π Π Π Κ < < Π Π Π Κ < = π δ δ 4 Peceding / R

12 Point Foce Solution D Long Duation Loading Quate Symmetic Model N DOF =

13 Compaison to Analytical Solution D Path Long Duation Loading

14 Point Foce Solution D Axisymmetic Plane Stain D Model N DOF =

15 Compaison of D and Axisymmetic Vetical Displacement

16 Compaison to Analytical Solution - Axisymmetic Path Long Duation Loading

17 Compaison of D and Plane Stain D solution Plane stain solution Time = 8 Suface: y-displacem ent [m] Min: -1.7<-17 M-.x: 7. 66c - 1S xlo ~ 0.8!: -N... :I!: 0.6 ~ Ql cu 1..1 c "' 0 " Ql N 0. ia E z Plane stain solution -0.4 L--~--~--~--~--~--~--~--~-~ Ac-length [m) Advanced Computational & Engineeing Sevices A C

18 Suface Wave Poblem R Shot Duation Loading D Model R = 00 m N DOF = ms f π () t = h cos t T T T t

19 Suface Wave Poblem Vetical Velocity Analysis FEA P wave R = 100 m Rayleigh

20 Hamme Data t=0.17 s Distance = 0 m V = 118 m/s

21 Compaison w/ Exp Data 7.5 ms (4%) Pimay Metics aival time and fequency content

22 Summay Closed-fom solution developed fo Elastic wave in infinite media Elastic wave in semi-infinite media Computation models developed using Solid Mechanics Module Thee-dimensional Two-dimensional Axisymmetic Plane Stain not sufficiently accuate fo point souce Compaison with analytical solutions and expeimental data Ageement with aival time and fequency content

23 Conclusions COMSOL Multiphysics povides a sufficient level of accuacy fo the poblems of inteest COMSOL Multiphysics povides a commecially available tool that can solve wave popagation poblems on destop computing esouces