MULTI-SCALE NUMERICAL SIMULATION OF A TSUNAMI USING MESH ADAPTIVE
METHODS
Joe Wallwo k,1,2 Ma hew Piggo ,2Da id Ham,2Hila y Welle 3.
1Ma hema ics o Plane Ea h Cen e o Doc o al T aining, 2Impe ial College London, 3Uni e si y o Reading.
INTRODUCTION
In 2011 a majo ea hquake caused he T¯
ohoku sunami whose
leading wa e s uck he Japanese coas nea Fukushima jus en
minu es a e i s genesis. Th ough e icien nume ical 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 a isk.
MAIN OBJECTIVES
IIn es iga e a enues opened by mesh adap i i y, including guidance
by adjoin equa ions, as in goal-based adap i i y.
IImplemen a mesh adap i e algo i hm using he ini e elemen
me hod (FEM) so wa e p o ided by Fi ed ake. Expe imen wi h
di e en a pos e io i e o es ima es.
IRun mesh adap i e simula ions o a ealis ic sunami and make
accu acy and e iciency compa isons be ween app oaches.
NON-ROTATIONAL SHALLOW WATER EQUATIONS
Fo a egion o ocean Ω⊂Ò2, de ine luid eloci y u:Ω→Ò2, ee
su ace displacemen η:Ω→Òand ba hyme y b:Ω→Ò.
Linea ising abou he su ace a es ¯
η=0 gi es
∂u
∂
+g+η=0,∂η
∂
+ + · (bu)=0,(1)
o g a i a ional accele a ion g=9.81m s−2. Deno e q:=(u,η).
MESH ADAPTIVE PROCESS
Fo piecewise linea (Ð1) app oxima ion, he Taylo emainde heo em
p o ides an a p io i e o es ima ion esul
=γ T|H| ,γ=cons .,(2)
upon which o base an aniso opic adap i e algo i hm. We
‘ econs uc ’ he Hessian Ho a ield ela ed o he luid low, such as
he ee su ace displacemen o luid speed, using a double L2
p ojec ion. Based on (2), a new mesh is gene a ed by modi ying he
Hessian o ensu e symme ic posi i e de ini eness. This p o ides a
me ic ield, which dic a es how he mesh is adap ed ac oss he domain.
blanIn he iso opic case, me ic ield alues a e diagonal ma ices
whose diagonal en ies a e he co esponding alues o 1
2.
GOAL-ORIENTED ADAPTIVITY
Fo a spa ial egion A⊂Ω, conside he objec i e unc ional
J(q)=∫Tend
Ts a ∬A
η(x,y, )dxdyd .(3)
Using (3) we may conside he ee su ace displacemen nea o
Fukushima’s Daiichi nuclea powe plan , say. Fi ed ake’s au oma ic
di e en ia ion unc ionali y allows he use o ob ain disc e e adjoin
solu ions o (1) associa ed wi h (3) in a “disc e ise- hen-di e en ia e”
ype app oach [Gunzbu ge , 2002]. blank space blank space blank spa
spa Goal-o ien ed adap i i y seeks o es ablish a ini e elemen
disc e isa ion (in his case, a mesh) which enables us o sol e (1) wi h
ou e o in e alua ing (3) being below some ole ance >0. Tha is,
|J(q) − J(qh)| <.(4)
We i s sol e (1) on a ela i ely coa se mesh, om which adjoin
solu ions may be ex ac ed and hence a pos e io i e o es ima o s
cons uc ed. The main e o es ima o conside ed in his wo k is he
dual-weigh ed esidual
Eh=hRh(qh),λhiL2(Ω),(5)
whe e Rhdeno es he (s ong) esidual o med by aking he LHS o (1)
as a 3-componen ec o unc ional and (qh,λh) o m ou app oxima ion
o he p imal-dual pai . In he inal un, (1) is sol ed adap i ely, wi h
(local) e o es ima es indica ing which egions o mesh should be
coa sened o e ined. The simples way o in eg a ing es ima o s is o
c ea e an iso opic me ic h ough scaling he iden i y ma ix by (5).
RESULTS
The quali y o ou app oxima ions may be assessed using (4). An
‘exac ’ objec i e unc ional alue o (3) is ob ained by i e a ing o e
inc easingly well- esol ed meshes un il con e gence is a ained o h ee
signi ican igu es, as displayed in Figu e 1. An objec i e alue
J(q)=2.44 ×1013 is achie ed on a 196,560 elemen mesh.
104105
#Elemen s
2.425
2.430
2.435
2.440
2.445
2.450
2.455
2.460
Objec i e unc ional J(u, , η) = RTend
Ts a RAηdxd
×1013
Figu e: Mesh ‘boo s apping’.
We compa e (aniso opic) adap ion
o he ee su ace Hessian (‘simple
adap i i y’) agains (iso opic)
‘goal-based’ adap i i y, which
inco po a es adjoin da a o o m
e o indica o s using (5). The
mixed space Ð1DG −Ð2 is used
h oughou .
Coa se Medium Fine Simple adap i e Goal-based
|J(q)−J(qh)|
|J(q)| 0.67% 0.34% 0.15% 1.51% 0.20%
Mean elemen coun 8,782 20,724 81,902 11,141 11,506
Mean un ime (s)11.7 38.1 131.7 405.2 7700.8
The in e pola ion in ol ed in o ming esiduals Rh(qh) o he e o
es ima o s (5) inc eases he cos o he goal-based algo i hm. Ins ead,
we could sol e local bounda y alue p oblems on each elemen , as in
he elemen esidual me hod [Ainswo h & Oden, 1997].
Figu e: Ini ial condi ion blank
space bl[Sai o e al., 2011]. Figu e: ‘Simple adap i e’ mesh. Figu e: ‘Goal-based’ mesh.
CONCLUSIONS
IThe 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 when only he o wa d
p oblem (1) 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 e ms o
accu acy and and is compe i i e ega ding elemen coun .
IWhils e o and elemen coun a e educed, he e is p og ess o be
made in educing he un ime o he mesh adap i e simula ions.
FURTHER RESEARCH
The elemen esidual me hod should be implemen ed, o p o ide a
mo e compe i i e un ime o he goal-based algo i hm. Fu he
esea ch will include inco po a ion o -adap i i y, allowing ‘mesh
mo emen ’. O he ocean modelling applica ions will be conside ed o
mesh adap i i y, such as s o m su ges and Gul S eam sepa a ion.
REFERENCES
Ainswo h & Oden (1997). A pos e io i e o es ima ion in ini e elemen analysis.
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¯
ohoku-Oki ea hquake, Japan:
In e sion analysis based on dispe si e sunami simula ions.
Gi Hub gi hub.com/jwallwo k23
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