NUMERICAL TSUNAMI SIMULATION USING GOAL-ORIENTED MESH ADAPTIVITY
Nicolas Ba al,1Da id Ham,1Ma hew Piggo ,1Joe Wallwo k,1,2 Hila y Welle 3.
1Impe ial College London, 2Ma hema ics o Plane Ea h Cen e o Doc o al T aining, 3Uni e si y o Reading.
INTRODUCTION
In shallow wa e sunami modelling and haza d mi iga ion he e is a
clea objec i e: o minimise he e o acc ued in app oxima ing he ee
su ace displacemen nea o coas al egions o impo ance. An adjoin
app oach enables inco po a ion o such egions in o a mesh adap i e
algo i hm, p o iding an e icien simula ion mee ing he objec i e abo e.
SHALLOW WATER EQUATIONS
Fo a egion o ocean Ω⊂Ò2, de ine he luid eloci y - ee su ace
displacemen uple q=(u,η) : Ω→Ò3, ba hyme y b:Ω→Òand
iscosi y νh:Ω→Ò. Fo g a i a ional accele a ion g>0,
∂u
∂ +(u·+)u+g+η+ D
z×u= + ·(νh(+u+(+u)T)),
∂η
∂ + + ·((η+b)u)=0.
(1)
ANISOTROPIC MESH ADAPTIVITY
This mesh adap i e p ocess is d i en by Riemannian me ic ields
M:Ω→Òn×nin nD. A each x∈Ω, he eigenpai s o Mdesc ibe he
ans o ma ion om Euclidean space in o he space de ined by M.
(1,0)
(0,1)
M1
2
M−1
2
(h1,0)
(0,h2)
Å2=(Ò2,I2) (Ò2,M)
Gi en a Ð1 me ic de ined on a mesh, aniso opic mesh adap i i y
enables he gene a ion o a new mesh using node inse ion,node
dele ion,edge swapping and node mo emen (in he 2D case).
Aniso opic mesh adap i i y is implemen ed in The is using PRAgMATIc
(PaRallel Aniso opic Mesh Adap i e ToolkI ).
GOAL-ORIENTED MESH ADAPTIVITY
Fo a spa ial egion A⊂Ω, conside he objec i e unc ional
J(q)=ZTend
Ts a "A
η(x,y, )dxdyd .(2)
In he case o he T¯
ohoku sunami, conside Aas a egion su ounding
Fukushima’s Daiichi nuclea powe plan , say.
Using (2) we may o mula e he associa ed adjoin shallow wa e
equa ions. Pyadjoin enables he gene a ion o adjoin solu ions in a
“disc e ise- hen-di e en ia e” ype app oach [Gun02]. Goal-o ien ed
mesh adap i i y seeks a mesh upon which we minimise e o in
e alua ion o (2):
min
H`J(q)−J(qh)`.(3)
Ha ing sol ed (1) on a ela i ely coa se mesh, adjoin solu ions a e
backed ou by pyadjoin . Va ious a pos e io i e o es ima o s may be
cons uc ed, such as he dual-weigh ed esidual (DWR). On each
elemen DWR gi es e o indica o s
EK
h= ¥Rh(qh),λh¦L2(K),K∈ H .(4)
He e Rhdeno es he (s ong) esidual o med by in e p e ing (1) as a
3-componen ec o unc ional equal o ze o and (qh,λh) o m ou
p imal-dual app oxima ion uple. In he inal un, he indica o s (4) a e
used o cons uc a me ic by Ð1 in e pola ion and scaling o he iden i y
ma ix, using which (1) is sol ed adap i ely.
CODE
In o de o un a mesh adap i e DWR simula ion o any ealis ic o
idealis ic sunami, all he use need do is speci y he ini ial mesh,
ba hyme y, ini ial and bounda y condi ions, Co iolis coe icien o choice
and any pa ame e s which di e om he de aul se ings.
Figu e: A sample sc ip o un a mesh adap i e DWR simula ion o he T¯
ohoku sunami.
RESULTS
Quali y o an app oxima ion is measu ed using ela i e e o in objec i e
unc ional e alua ion. An ‘exac ’ alue o (2) is ob ained using a high
esolu ion ixed mesh un.
Figu e: Ini ial condi ion blank
space bl[SI11].
Figu e: 50km adius o
impo ance su ounding
Fukushima.
Figu e: Mesh adap ed using
DWR indica o s.
CONCLUSIONS
IThe adjoin p oblem allows guidance o he mesh adap i e p ocess,
o e ing addi ional a pos e io i in o ma ion. This is pa icula ly use ul
i only a small numbe o egions a e o in e es o impo ance.
IWhils e o and elemen coun may be educed, he e is p og ess o
be made in educing CPU ime.
FURTHER RESEARCH
IConside mesh mo emen in he con ex o sunami modelling.
IFo mula e a hyb id h -adap i e me hod.
IIn es iga e mesh adap i i y in o he ocean modelling applica ions,
such as s o m su ges and Gul S eam sepa a ion.
REFERENCES
Gunzbu ge (2002), Pe spec i es in low con ol and op imiza ion.
Sai o e al. (2011), Tsunami sou ce o he 2011 T¯
ohoko-Oki ea hquake,
Japan: in e sion analysis based on dispe si e sunami simula ions.
Gi Hub gi hub.com/jwallwo k23
Fi ed ake i ed akep ojec .o g
The is he isp ojec .o g
This wo k could no ha e been done wi hou he gene ous suppo o
