Mul i-scale nume ical simula ion o a sunami
using mesh adap i e me hods
An explo a ion o aniso opic mesh adap i i y
Joe Wallwo k
Ma hema ics o Plane Ea h CDT
Supe ised by Ma hew Piggo and Da id Ham
[email p o ec ed]
Abs ac
Mesh adap i e me hods a e ypically ca ego ised as ei he h-adap i e o -adap i e. In wo dimensions, he o me
in ol e ope a ions al e ing he numbe o mesh deg ees o eedom by he inse ion o dele ion o elemen edges,
while he la e hold bo h he numbe o deg ees o eedom and he mesh opology ixed and edis ibu e mesh
en i ies ( e ices, edges and elemen s) geome ically. Aniso opic mesh adap i i y seeks o inco po a e aspec s o
bo h h- and -adap i i y, p o iding a hyb id (h ) app oach. As such, aniso opic mesh adap i i y bene i s om he
h-adap i e abili y o comple ely egene a e a mesh be o e i gains angled nodes, as well as he -adap i e abili y o
allow deg ees o eedom o ollow ea u es o luid low, such as a sunami wa e. A hyb id mesh adap i e app oach
is ideal o sunami p oblems, since he impo an ea u es we would like o accu a ely esol e a e clus e ed in a
ela i ely small egion o ocean, which i sel mo es as ime p og esses. A s andalone ini e elemen shallow wa e
sol e is cons uc ed o sol ing sunami modelling p oblems, along wi h an aniso opic mesh adap i i y lib a y
capable o adap ion bo h o ields ela ed o he low (such as luid speed) and as guided by adjoin solu ion da a.
In oduc ion
In 2011 a sunami caused by a majo ea hquake lead o eno mous damage w eaked on he coas o
Fukushima, Japan, wi h leading sunami wa es eaching he Japanese coas a e jus en minu es.
Th ough highly e icien nume ical sunami simula ions, su icien wa ning could be p o ided in u-
u e scena ios, allowing e acua ion and damage mi iga ion in coas al a eas de e mined mos a isk.
Main Objec i es
1. Build a s andalone shallow wa e sol e , using he au oma ed ini e elemen me hod code w i ing
so wa e p o ided by Fi ed ake.
2. In es iga e a enues opened by mesh adap i i y, including guidance o he mesh adap i e p ocess
using adjoin equa ions, as in goal-based adap i i y.
3. Implemen a mesh adap i e algo i hm wi hin he s andalone shallow wa e sol e . Expe imen
wi h di e en me hods o Hessian econs uc ion, no malisa ion and me ic g ada ion.
4. Run mesh adap i e simula ions o a ealis ic sunami case s udy and make nume ical compa isons
be ween app oaches, in e ms o bo h accu acy and e iciency.
Non- o a ional shallow wa e equa ions
We linea ise abou a la su ace ¯
η, ypically aken as ze o. Fo an ocean domain Ω⊂R2, de ine luid
eloci y u: Ω →R2, ee su ace displacemen η: Ω →Rand ba hyme y b: Ω →R. Then
∂u
∂ +g∇η=0,∂η
∂ +∇·((¯
η+b)u)=0,(1)
whe e g= 9.81m s−2deno es g a i a ional accele a ion.
Model e i ica ion expe imen s we e ca ied ou on a ealis ic ocean domain displayed in Figu e 1,
using he ini ial condi ion shown in Figu e 2. These expe imen s showed e y li le di e ence in he
app oxima ions made by (1) and he co esponding nonlinea o o a ional coun e pa s.
Figu e 1: P oblem domain. Figu e 2: Ini ial condi ion [2]. Figu e 3: Gauge loca ions [2].
Adjoin shallow wa e equa ions
Co espondingly, o adjoin a iables (λu, λη):Ω→R3, we ha e
−∂λu
∂ −b∇λη=∂J∗
∂u,−∂λη
∂ −g∇·λu=∂J∗
∂η ,(2)
o an objec i e unc ional, J. Fo a spa ial egion o impo ance A⊂Ω, his is gi en by
J(u, η) = ZTend
Ts a ZA
η(x, y, ) dxdyd . (3)
The objec i e (3) allows us o conside he ee su ace displacemen nea o Daiichi nuclea powe
plan , say, he eby e alua ing he ex en o which damage is caused on he Japanese coas .
Mesh adap i e p ocess
Sol e
H0,q0
E o model
Remeshe
In e polan
A imes ep
k, we s a
wi h mesh
Hkand
solu ion qk.
Hi,qi
Hi+1,qi
Hi+1,qi+1
Fo piecewise linea (P1) app oxima ion, he
Taylo emainde heo em e o esul
=γ T|H| ,(4)
p o ides an e o es ima e upon which o base
an adap i e algo i hm. We compu e he Hes-
sian o a ield ela ed o he luid low, such as
ee su ace displacemen o luid speed. Spe-
cial econs uc ion is equi ed, such as double
L2p ojec ion. Based on (4), a new mesh is
gene a ed by modi ying he Hessian o ob ain
an e o me ic, which dic a es he way a mesh
is adap ed ac oss he domain.
Resul s
Fi e meshing app oaches we e conside ed: inc easingly e ined ixed meshes (i), (ii) and (iii); a ‘sim-
ple adap i e’ app oach (i ), adap ing o he ee su ace Hessian; and a goal-based adap i e app oach
( ), inco po a ing adjoin solu ion in o ma ion. Time se ies a wo o sho e p essu e gauges shown in
Figu e 3 a e displayed in Figu e 4 and 5, along wi h e o s, un imes and mean e ex coun s.
0 5 10 15 20 25
Time elapsed (mins)
−2
−1
0
1
2
3
4
5
F ee su ace (m)
Gauge measu emen
Mesh app oach (i)
Mesh app oach (ii)
Mesh app oach (iii)
Mesh app oach (i )
Mesh app oach ( )
Figu e 4: Timese ies a gauge P02.
0 5 10 15 20 25
Time elapsed (mins)
−1
0
1
2
3
4
5
F ee su ace (m)
Gauge measu emen
Mesh app oach (i)
Mesh app oach (ii)
Mesh app oach (iii)
Mesh app oach (i )
Mesh app oach ( )
Figu e 5: Timese ies a gauge P06.
No m ype Coa se Medium Fine Simple adap i e Goal-based
L1no m 0.398 0.291 0.321 0.315 0.255
P02 L2no m 0.480 0.331 0.355 0.351 0.326
L∞no m 1.190 0.788 0.780 0.659 1.035
L1no m 0.324 0.128 0.111 0.163 0.106
P06 L2no m 0.460 0.172 0.136 0.225 0.152
L∞no m 1.368 0.547 0.390 0.785 0.680
Mean e ex coun 3,126 17,086 97,343 5,896 3,756
Mean un ime 39.9 s 239.3 s ≈4 m 1701.4 s ≈28 m 1602.8 s ≈27 m 2383.4 s ≈40 m
The goal-based app oach i s sol es (2) on a ela i ely low esolu ion, ixed mesh. Then (1) is sol ed
adap i ely, using adjoin solu ion da a o es ablish poin wise a pos e io i e o indica o s. This es-
ablishes which egions a e signi ican and can be ed in o adap i e p ocedu e. ‘Simple adap i e’ and
goal-based meshes a e displayed in Figu es 6 and 7, espec i ely.
Figu e 6: ‘Simple adap i e’ app oach. Figu e 7: ‘Goal-based’ app oach.
Conclusions
1. Aniso opic mesh adap i i y p o ides a means o main aining bo h a low e ex coun du ing
sunami modelling and a su icien ly well- esol ed egion su ounding impo an dynamics.
2. The adjoin p oblem can be use ul o guiding he mesh adap i e p ocess, o e ing addi ional in o -
ma ion han in he case whe e only he o wa d p oblem is conside ed. Fo he T¯
ohoku sunami,
he goal-based app oach ou -pe o ms he ‘simple adap i e’ me hod in accu acy and e ex coun .
Fu he Resea ch
I will u he esea ch aniso opic mesh adap i i y as a PhD p ojec , including a mo e ho ough inco -
po a ion o -adap i i y in o he app oach desc ibed he e. I will conside a numbe o o he applica-
ions o mesh adap i i y in he geosciences, such as s o m su ges and Gul S eam sepa a ion. Whils
my MRes compu a ions we e pe o med on a s andalone sol e o my own design, he PhD wo k will
be pa o he The is p ojec , c ea ed o sol ing coas al, nea -es ua ine and ocean FEM p oblems.
Re e ences
[1] Powe e al. Adjoin a pos e io i e o measu es o aniso opic mesh op imisa ion. 2006.
[2] Sai o e al. Tsunami sou ce o he 2011 Tohoku-Oki ea hquake, Japan: In e sion analysis based
on dispe si e sunami simula ions. 2011.
[3] Da is & LeVeque. Adjoin me hods o guiding adap i e mesh e inemen in sunami modelling.
2016.
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