Submi ed o Impe ial College London
and he Uni e si y o Reading in
ul ilmen o he equi emen s o he
Deg ee o Mas e by Resea ch
Mas e s Thesis
Mul i-scale nume ical simula ion o a
sunami using mesh adap i e
me hods
Joseph G ego y Wallwo k
Supe ised by
P o . Ma hew Piggo & D . Da id Ham.
F iday 25 h Augus , 2017
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 es 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, and is
explo ed in dep h he e. This app oach 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 aspec s 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 a la ge po ion o luid low, which
we would like o accu a ely esol e, is 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. By applying mesh adap i i y o shallow wa e p oblems, his wo k aims
o e icien ly gene a e nume ical solu ions o sunami wa e p opaga ion p oblems. The
case s udy o he sunami which s uck Fukushima, Japan, in 2011 is conside ed, whe ein
some leading sunami wa es eached he coas in jus en minu es. Nume ical esul s
indica e compu a ional cos can be educed, whils e aining a su icien ly high accu acy,
unde mesh adap i i y. Th ough es ablishing a highly e icien app oach o nume ical
sunami simula ion, su icien wa ning could be p o ided in u u e scena ios, allowing o
e acua ion and damage mi iga ion in hose coas al a eas de e mined o be mos a isk.
I he eby decla e ha his Mas e s hesis is en i ely my own wo k and has no p e iously
been submi ed as pa o ano he highe deg ee a any o he uni e si y o ins i u ion.
Whe e e ex e nal sou ces ha e been used, e e ences a e p o ided in he bibliog aphy.
Joseph G ego y Wallwo k,
Impe ial College London,
F iday 25 h Augus , 2017
ii
Con en s
Lis o Figu es 1
1 P elimina ies 2
1.1 Abb e ia ions.................................. 2
1.2 No a ion..................................... 2
1.3 No es on compu e esou ces . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 In oduc ion 4
3 Ma hema ical o mula ion 6
3.1 Shallowwa e equa ions ............................ 6
3.2 Fo wa dp oblem ................................ 8
3.3 Bounda ycondi ions .............................. 9
3.4 Adjoin p oblem................................. 10
3.5 Fini e elemen p oblem sol ing by compu e . . . . . . . . . . . . . . . . . 11
4 Me hodology and an idealised expe imen 12
4.1 Mesh-adap i ep ocess ............................. 12
4.2 E alua ion o h- and -adap i e app oaches . . . . . . . . . . . . . . . . . . 14
4.3 Timein eg a ion ................................ 15
4.4 One-dimensional sunami es p oblem . . . . . . . . . . . . . . . . . . . . 16
5 Aniso opic mesh adap i i y 18
5.1 Measu ingdis ance............................... 18
5.2 Gaugingmeshquali y.............................. 21
5.3 Me iccompu a ion............................... 22
5.4 Hessian eco e y ................................ 24
5.5 Adap ing o mul iple solu ion ields . . . . . . . . . . . . . . . . . . . . . . 25
5.6 Me icg ada ion ................................ 26
5.7 No es on he adap i e algo i hm code . . . . . . . . . . . . . . . . . . . . . 28
5.8 Adap i i ycode es s.............................. 29
5.9 Goal-based mesh adap i i y . . . . . . . . . . . . . . . . . . . . . . . . . . 31
iii
6 Tsunami applica ion 34
6.1 Compu a ional se up o he T¯ohoku sunami . . . . . . . . . . . . . . . . 34
6.2 Model e i ica ion................................ 36
6.3 Damagequan i ica ion ............................. 38
6.4 Modelcompa isons ............................... 39
6.5 Conclusions and u u e wo k . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Bibliog aphy 43
Acknowledgemen s
I g ea ly hank my supe iso s P o . Ma hew Piggo and D . Da id Ham o hei
con inued help, suppo and ad ice h oughou my unde going o his p ojec . They we e
always happy o explain concep s I didn’ ully unde s and and o o e sugges ions on
how I migh p oceed wi h my esea ch.
Especial hanks also go o D . Alexand os A dis, P o . Ta suhiko Sai o and in pa -
icula D . Nicolas Ba al o hei ex ensi e knowledge sha ing and esou ce p o iding.
Fu he hanks o D . S ephan K ame , D . Law ence Mi chell, D . E han Kuba ko,
Thomas Gibson and Simon Wa de o hei sound sugges ions ela ing o compu a ional
aspec s o he p ojec .
Las bu ce ainly no leas , hank you o D . Anna Radomska o he con inued
adminis a i e suppo and always being happy o answe ques ions and y o esol e
issues.
Lis o Figu es
2.1 “The G ea Wa e o Kanagawa” by Hokusai, ein e p e ed in he con ex
o adap i emeshing. .............................. 4
2.2 Tide and p essu e gauge loca ions nea o Fukushima, Japan. . . . . . . . . 5
3.1 Se up o he shallow wa e equa ions. . . . . . . . . . . . . . . . . . . . . . 7
4.1 The mesh-adap i e p ocess. . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Ba hyme y and ini ial condi ions o an idealised 1D sunami simula ion. . 16
4.3 Signi ican egions o ocean in an idealised 1D sunami simula ion. . . . . . 17
5.1 The mapping o an ellipse o he uni ci cle unde a me ic. . . . . . . . . . 19
5.2 Meshes adap ed wi h espec o wo di e en scala unc ions. . . . . . . . 29
5.3 Mesh adap i e ad ec ion and di usion o a Gaussian sou ce unde a con-
s an wind ield.................................. 30
5.4 Mesh adap i e shallow wa e simula ion ac oss a shel b eak discon inui y
wi haGaussiansou ce. ............................ 31
6.1 Domain geome y used in modelling he T¯ohoku sunami. . . . . . . . . . . 34
6.2 Ini ial ee su ace displacemen used in modelling he 2011 T¯ohoku sunami. 35
6.3 ‘Eyeball no m’ compa isons o he non- o a ional and o a ional, linea and
nonlinea s andalone sol e s on a ine mesh o he T¯ohoku p oblem. . . . . 36
6.4 Model e i ica ion using ime se ies o ee su ace measu emen s and p e-
dic ions a p essu e gauges o he coas o Japan. . . . . . . . . . . . . . . 38
6.5 Damage measu e quan i ica ion o he T¯ohoku sunami, alongside he spa-
ial egion o impo ance conside ed. . . . . . . . . . . . . . . . . . . . . . 39
6.6 A ime se ies compa ison o di e en meshing app oaches a p essu e gauges
o he coas o Japan, alongside e o no m alues o i e app oaches o
meshing in sunami modelling. . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.7 Meshes gene a ed a e 14 minu es o simula ion ime, wi h and wi hou
guidance by adjoin in o ma ion, o he T¯ohoku sunami p oblem. . . . . . 41
6.8 Time pe o mance analyses o a ious meshing app oaches o sunami
modelling he T¯ohoku sunami. . . . . . . . . . . . . . . . . . . . . . . . . 42
1
1. P elimina ies
1.1 Abb e ia ions
AMR: adap i e mesh e inemen .
DOF: deg ees o eedom.
FEM: ini e elemen me hod.
GIS: geog aphical in o ma ion sys em.
PDE: pa ial di e en ial equa ion.
SPD: symme ic posi i e-de ini e.
SW: shallow wa e .
SWEs: shallow wa e equa ions.
UFL: uni ied o m language.
UTM: uni e sal ans e se me ca o .
1.2 No a ion
The ollowing i ems es ablish all pieces o no a ion used in his p ojec which a e no
en i ely s anda d in he li e a u e.
•N0:= N∪{0}={0,1,2,3, . . . }deno es he na u al numbe s wi h ze o included.
•Fo unc ions :A→Rand u:A→Rnon A⊆Rn,
∂
∂u:= h∂
∂u1. . . ∂
∂uniT:A→Rn,wi h he g adien ∂
∂x=∇ . (1.1)
•Fo unc ions :A→Rmand u:A→Rnon A⊆Rn,
∂
∂u:=
∂ 1
∂u1. . . ∂ 1
∂un
.
.
.....
.
.
∂ m
∂u1. . . ∂ m
∂un
:A→Rm×n,(1.2)
p o ides he Jacobian ma ix o he ans o ma ion om a iables o a iables u.
The Jacobian ∂
∂x=J( ),o p o ides a common example.
•I a ma ix A∈Rn×nis posi i e de ini e hen we w i e A0.
2
•Gi en a me ic space (X, d), a subse U⊆X he eo has closu e
U={x∈X| ∀ > 0,∃y∈U|d(x, y)< },
in e io U◦=X (X U) and bounda y ∂U =U U◦.
•We deno e by In he n×niden i y ma ix, o n∈N.
•Meshes a e deno ed by Hand con ain h ee ypes o en i y: ( iangula ) elemen s
K⊂R2, e ices p∈R2and edges pq, which a e o en also conside ed as ec o s
in R2.
•Gi en some domain, Pkdeno es a deg ee-kpiecewise polynomial unc ion space
de ined o e he domain, whe e k∈N. The same no a ion is used in ec o and
scala cases, bu he dis inc ion is made clea .
1.3 No es on compu e esou ces
The majo i y o he compu e code used in his p ojec is w i en in Py hon, wi h in-
co po a ion o UFL in o de o deal wi h o ms. Addi ional compu e packages used a e
lis ed below, along wi h web links o hei download and associa ed documen a ion.
Fi ed ake: h p:// i ed akep ojec .
o g
GMSH: h p://gmsh.in o
GMT: h p://gm .soes .hawaii.edu
Pa a iew: h p://www.pa a iew.o g
Plo Digi ize : h p://plo digi ize .
sou ce o ge.ne
PRAgMaTIc: h ps://gi hub.com/
meshadap a ion/p agma ic
QGIS: h p://www.qgis.o g
QMESH: [A dis e al.,2017]
The is: h p:// he isp ojec .o g
In addi ion, we make use o wo geog aphic da abases, e e enced below.
GSHHG Global Sel -consis en , Hie a chical, High- esolu ion Geog aphy:
h p://www.soes .hawaii.edu/pwessel/gshhg/
GEBCO Gene al Ba hyme ic Cha o he Oceans: h p://www.gebco.ne
All Py hon code o his p ojec can be ound on Gi Hub, a he web add ess
h ps://gi hub.com/jwallwo k23/MResP ojec . The e can be ound a numbe o
Py hon sc ip s, whose oles a e desc ibed in Sec ion 5.7, along wi h di ec o ies con aining
da a om he abo e sou ces and also plo s.1
1E e y hing needed o un he code is included in he di ec o ies o he Gi Hub page, excep o he
mesh iles. This is o copy igh ing pu poses, bu he meshes can be p o ided upon eques , wi h he
pe mission o he QMESH de elope s.
3
2. In oduc ion
Figu e 2.1: In e p e a ion o an adap i ely meshes sunami. O iginal image “The G ea Wa e
o Kanagawa” by he Japanese a is Ka sushika Hokusai (1830-1833) edi ed by he au ho .
The wo d sunami de i es om Japanese, whe e ‘ su’ e e s o he ha bou and ‘nami’
e e s o a sea wa e [Lisi zin,1974]. This language is sugges i e o he se ious damage ha
hese wa es w eak upon hei a i al in o ha bou s ac oss Japan. In ecen and ancien
his o y, and in pa icula in Japan, sunamis ha e p o ed o be na u al disas e s wi h
de as a ing impac s on human ci ilisa ions. This kind o damaging sunami a e common
in he seismically ac i e Paci ic im nea Japan’s coas , and as a coun y comp ising a
collec ion o small islands, Japan’s coas al ci ies pa icula ly ulne able. One o he mos
memo able sunamis o ecen imes, and indeed he mos powe ul one o s ike Japan
on eco d, had i s epicen e o he coas o T¯ohoku and s uck many pa s o he coun y
in 2011, including he Fukushima egion.
The sunami caused 15,893 dea hs, much de as a ion o hund eds o housands o
homes and played a la ge pa in causing he le el 7 mel down o he Fukushima Dai-
ichi nuclea powe plan .1Al hough he mel down e en i sel caused no dea hs, much
dis up ion was caused; an eno mous clean-up ope a ion was equi ed, and he e was a
mass-e acua ion o o e 160,000 people who li ed wi hin a 20 km adius, whe ein he
yea ly dosage o adia ion was expec ed o each dange ous le els o 20 mS yea −1in he
yea s ollowing he acciden .2Only now, six yea s a e he inciden , is he adioac i i y
low enough o begin he clean-up, and s ill some obo s a e unable o handle he condi-
ions. P elimina y and la e epo s o he ea hquake, sunami and ollowing mel down
1De ails ob ained om he Na ional Police Agency o Japan:
h p://www.npa.go.jp/a chi e/keibi/biki/higaijokyo_e.pd .
2Fo u he de ails, see h p:// ukushimaon heglobe.com.
4
e en s include hose by [Kazama and Noda,2012], [Okada,2011] and [Simons e al.,2011],
wi h u he analyses a e p o ided by [Baba e al.,2017] and [Suzuki e al.,2012].
The e is an ob ious desi e o a oid such occu ences in he u u e, and he Japanese
go e nmen ha e e en mo ioned o e iew i s use o nuclea powe in i s ene gy g id due
o he high isk o hese ex eme e en s. Ye in he ligh o he eno mous p oblem posed
by clima e change, we should no be oo has y o abandon his ela i ely clean al e na i e
o ca bon dioxide-emi ing ossil uels which has he po en ial o gene a e as amoun s
o elec ici y. An al e na i e app oach o he p oblem is o make sunami ea ly-wa ning
sys ems much mo e e icien .
Figu e 2.2: Gauge lo-
ca ions nea Fukushima,
cou esy o [Sai o e al.,
2011]. The s a indica es
he ea hquake epicen e
and small ci cles indica e
a e shock loca ions.
This p oblem will be conside ed in his hesis, making use
o adap i e meshes in nume ical ocean modelling. Mesh adap-
i i y aims o disc e ise he domain o ocean o in e es in an
uns uc u ed way, his no only a ies in esolu ion spa ially,
bu also adap s empo ally. In his way, luid dynamics nea o
he sunami wa e can be esol ed mo e accu a ely, whils min-
imising he compu a ional e o expended on sol ing he low
equa ions in loca ions dis an om bo h he wa e and ulne able
coas al ci ilisa ions. The esul is an e icien and ye ela i ely
compu a ionally cheap app oach.
The ma hema ical model o which we apply he adap i e p o-
cess is p o ided by he shallow wa e equa ions, which a e well-
known o p o ide a good app oxima ion o la ge scale oceanic
luid dynamics. These equa ions may be sol ed using he ini e
elemen me hod which, unlike he ini e di e ence me hod, has
he ad an age o wo king well on uns uc u ed g ids.
The cen al aim o his p ojec is o cons uc an e ec i e and
e icien means o modelling he 2011 T¯ohoku sunami, which a i es a a solu ion wi hin
he 10 minu e ime ame be o e he wa es we e i s el , and which su icien ly accu a ely
esol es he coas al bound sunami wa e. Thus, by implemen ing such a me hod in he
u u e, ulne able egions may be gi en su icien ime o e acua e and damage may be
mi iga ed. Th oughou , we e e o he case s udy sunami as he T¯ohoku sunami.
The s uc u e o his hesis is as ollows. Chap e 3 ou lines he ma hema ical se up,
including equa ions and ini ial and bounda y condi ions used. Chap e 4 hen goes on
o desc ibe he solu ion app oach o sol ing p oblems o he o m desc ibed in Chap e
3 and ou lines in b ie he gene al p ocedu e o aniso opic mesh adap i i y, wi h u he
de ail p o ided in Chap e 5. Chap e s 4 and 5 also include plo s and esul s om he
non-adap i e and adap i e algo i hms applied o model sunami p oblems, espec i ely.
Finally, Chap e 6 uses he con en o he p eceding chap e s o apply aniso opic mesh
adap i i y o he T¯ohoku sunami case s udy, including some model e i ica ion and esul
compa isons wi h al e na i e solu ion app oaches.
5
4. Me hodology and an idealised
expe imen
In his chap e we ou line and e alua e he gene al concep o aniso opic mesh adap i i y,
which is explained in mo e de ail in Chap e 5, along wi h he amewo k wi hin which we
sol e PDEs using his p ocess. In addi ion, we conside a model 1D p oblem o in e p e
he adjoin p oblem concep , hin ing a how his could p o e use ul in guiding adap i i y.
4.1 Mesh-adap i e p ocess
A ixed mesh de ined on a domain Ω has a spa ially ixed se o DOFs. I hese DOFs a e
no sui ably dis ibu ed, o i he p oblem a ies empo ally, compu a ional ine iciencies
may become appa en . Whils aiming o a oid such ine iciencies, we would also like o
e ain a sui ably accu a e app oxima ion o he ue dynamics. As such, i makes sense
o conside a ine mesh whe e ou app oxima ion is deemed as poo and a coa se mesh
whe e i is al eady o a high quali y. This is p ecisely he app oach o mesh adap i i y.
O cou se, we gene ally do no know he ue e o made in ou app oxima ion. Hence,
o pe o m mesh adap i i y, we mus ha e a means o e o es ima ion.
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
Figu e 4.1: The mesh-adap i e p ocess.
Mo eo e , an adap i e mesh a ies
in ime as well as space, meaning
he mesh used may change ac oss
imes eps, as illus a ed in Figu e 4.1.
The sequence { k}k∈N⊆N0de e mines
hose imes eps in which he mesh is e-
gene a ed, which could also be selec ed
adap i ely, bu which is inc emen ed
uni o mly in his wo k.
The abili y o wo k wi h adap i e
meshes is pa icula ly a ac i e o
sunami modelling, whe e he egion o in e es (su ounding he coas -bound sunami
wa e) mo es a he apidly as ime p og esses. The ask is o ensu e bo h ha he pa h
o he sunami is su icien ly esol ed and also he pa o he domain con aining he wake
o he wa e becomes mo e coa sely meshed as i mo es away, educing compu a ional cos .
12
Addi ionally, we would like o mesh inely nea o loca ions o in e es , such as densely
popula ed a eas, busy ha bou s and nuclea powe plan s.
The mesh-adap i e p ocess con ains h ee c ucial s eps: e o es ima ion, mesh adap-
ion and in e pola ion, as desc ibed in he ollowing subsec ions.
4.1.1 E o es ima ion
In sol ing PDEs using FEM we o en use P1 app oxima ion, as o he ee su ace dis-
placemen in he SWEs in Chap e 3. Fo his unc ion space, FEM heo y as in [B enne
and Sco ,2007] p o ides he Taylo emainde heo em app oxima ion e o esul
A=γ T|H| ,(4.1)
whe e γ=O(1) is a scala , ∈Rnis a ec o co esponding o a di ec ion and magni ude
and His he Hessian o he solu ion ield. As such, i makes sense o conside e o
es ima es o he adap i e p ocess based on he Hessian, desc ibed in de ail in Chap e 5.
4.1.2 Mesh adap ion
Ha ing cons uc ed e o measu es, he nex ask is o modi y he mesh acco dingly,
p o iding he mesh adap ion. The e a e a ious la ou s o adap i i y, each wi h a di e en
app oach o his mesh adap ion s age. App oaches include p-adap i i y,h-adap i i y and
-adap i i y. We a e mainly in e es ed in he la e wo app oaches. Adap i e mesh
e inemen (AMR) is a popula ype o h-adap i i y. Fo a summa y o he app oaches
o AMR and p-adap i i y, see [Beh ens and Bade ,2009] and [Gio giani e al.,2013],
espec i ely. The cases o h- and -adap i i y a e e alua ed in Sec ion 4.2.
I is also possible o conside combina ions o some o hese app oaches, as discussed
in Sec ion 4.2. While he pu pose o he p e ious s age was o lag pa s o he mesh o
edi ing, he adap ion phase seeks o pe o m hese modi ica ions.
4.1.3 In e pola ion o solu ion
Once mesh adap ion has been implemen ed, he inal ask is o in e pola e he solu ion
om he ‘old mesh’ on o he ‘new mesh’, p o iding ano he oppo uni y o acc uing
e o . As such, cen al o h-adap i e me hods is he concep o in e pola ion e o
I=kq−Πhqk,in some no m k·k:X→R,(4.2)
whe e again X={ : Ω×(0, T]→R3} 3 q. In e pola ion e o is made upon app oxima -
ing he exac solu ion qby some in e polan Πhqo e he mesh, in ou case a Lag ange
in e polan . In e pola ion e o does no a ise in -adap i i y, o easons discussed in
Sec ion 4.2.
13
4.2 E alua ion o h- and -adap i e app oaches
As de ailed in Sec ion 5.2, h-adap i i y gauges mesh quali y ia a unc ional (5.15) and
adap s acco ding o a me ic. This SPD enso ield depends on he Hessian o he solu ion
ield, in acco dance wi h (4.1). In h-adap i i y, mesh en i ies may be c ea ed o des oyed,
hus al e ing he mesh opology, whils -adap i i y ixes he mesh opology, wi h en i ies
simply mo ing a ound he domain, as de e mined by ano he unc ional.
The h-adap i e p ocedu e epea edly e alua es he quali y unc ional ollowing a ious
geome ical and opological ope a ions upon he mesh, aiming owa ds i s minimisa ion,
whe eby he mesh has desi able p ope ies. His o ically, h-adap i i y is usually conside ed
on Ca esian meshes and he ‘h’ e e s o he hwhich usually deno es he size o elemen s
used in FEM. Al e ing mesh opology can equi e an excessi e amoun o in e pola ion and
pose an issue o ou p oblem, since he egion su ounding he sunami, which equi es
ine esolu ion, mo es wi h he low. Excessi e in e pola ion con ibu es o a heigh ened
compu a ional cos , which is comple ely agains ou mo i e o pu suing adap i e meshing
in o de o c ea e a mo e e icien algo i hm. This mo i a es conside ing he idea o a
mo ing mesh which is able o chase he wa e, lea ing coa se mesh in i s wake.
The app oach o -adap i i y is indeed o en e e ed o as mesh mo emen and can
p o ide a mo e e ec i e app oach o cases in which egions o inc eased esolu ion mo e
wi h he low. The ‘ ’ o -adap i i y comes om he eloca ion o he mesh nodes, which
can be achie ed in a numbe o ways, discussed in [Piggo e al.,2005]: a a ia ional
app oach makes use o he Eule -Lag ange equa ions; i is also possible o apply a so-
called mesh smoo hing algo i hm in o de o achie e mesh mo emen . As no ed in [Piggo
e al.,2005], he ac -adap i i y mo es he mesh wi h he low has he esul o educing
la ge anspo eloci ies, such as hose associa ed wi h he sunami. This means he e a e
ewe imes ep es ic ions han o applica ion o h-adap i i y, p o iding a clea nume ical
ad an age. On he o he hand, unlike wi h h-adap i i y whe e new nodes can be c ea ed,
-adap i e me hods hold he DOF coun cons an , po en ially leading o p oblems i he
dynamics o he p oblem a hand become a he complex. In hese si ua ions, i would
be ad an ageous o be able o ‘injec ’ mesh esolu ion in o a ce ain egion – some hing
which is no a ailable in -adap i i y. In addi ion, -adap i i y can p oduce p oblema ic,
so-called ‘ angled meshes’, whe eby wo o mo e elemen s o e lap.
Clea ly, he e a e disad an ages o he app oaches o bo h h- and -adap i i y, bu
i appea s ha many o hese a e complemen ed by co esponding ad an ages o he
o he app oach. As sugges ed abo e, i is bo h desi able and possible o combine he
wo, yielding a hyb id, h -adap i e me hod. One such app oach, as desc ibed in [Habashi
e al.,2000], is o inco po a e a mo ing mesh wi hin a mesh op imisa ion me hod such
as desc ibed ea lie . Al e na ely, h-adap i i y can be inco po a ed wi hin he s uc u e
o mo ing meshes on a local scale, as in [Lang e al.,2003], o ensu e elemen s a e no
poo ly shaped and nei he is he speci ied e o ole ance eached.
14
4.3 Time in eg a ion
As discussed in Chap e 3, solu ion o he SWEs equi es imes epping, wi h FEM sol es
a each i e a ion. Two candida e imes epping schemes a e desc ibed in he ollowing.
Fo he pu poses o hese analyses, conside a simple PDE p oblem
∂u
∂ = (u( ), ), u(0) = u0(4.3)
de ined on some domain Ω ⊆R, o a dependen a iable u: Ω →R, o cing e m
: Ω →Rand ini ial alue u0∈R. Suppose we ha e a uni o m ime disc e isa ion
{u(n)}n∈N0o he dependen a iable, wi h (cons an ) imes ep ∆ > 0.
One app oach is he implici Eule me hod, whe eby he alue o ua he cu en
imes ep depends on in o ma ion a ailable only a he cu en imes ep:
u(n+1) −u(n)
∆ = (u(n+1), (n+1)).(4.4)
Whils his equa ion is no as s aigh o wa d o sol e as wi h explici me hods, amoun ing
o sol ing a ma ix sys em, i has he ad an age o ha ing a la ge egion o absolu e
s abili y. Howe e his app oach is only O(∆ ) a a pa icula ime and does no
conse e ene gy. We would p e e o use a highe o de , conse a i e ime in eg a o .
In (4.4), he o cing unc ion is e alua ed only a cu en ime alues. Ano he
app oach is o e alua e a an in e media e ime, say hal way be ween he p e ious and
cu en imes eps. F om his we de i e he implici midpoin ule, gi en by
u(n+1) −u(n)
∆ = u(n+1) +u(n)
2, (n)+∆
2.(4.5)
Whils mo e compu a ionally expensi e han (4.4), his app oach has a global e o o
O(∆ 2), meaning he e o decays quicke as we conside smalle imes eps. Fu he , (4.5)
is a symplec ic in eg a o in he con ex o Hamil onian dynamics, so conse es ene gy.
Fo explici ime in eg a o s, he spa io- empo al disc e isa ion used mus sa is y he
Cou an -F ied ichs-Le y (CFL) condi ion (es ablished in [Cou an e al.,1928]),
c:= ∆ |u|
∆x+| |
∆y≤1,(4.6)
o s abili y. Since (4.4) and (4.5) a e implici in eg a o s, his condi ion can be b oken
wi hou incu ing penal ies o educed s abili y. Howe e , i is s ill use ul o make use o
(4.6) when choosing a imes ep leng h. We showed in Sec ion 3.1 ha he wa e speed o
a linea SW wa e is gi en by pg¯
h, whe e ¯
his he wa e dep h when a es . As such,
o he SWEs, |u|,| | ≤ √gbmax, whe e bmax deno es he maximal ba hyme y in Ω. In
selec ing a imes ep leng h, we should ensu e i is su icien ly less han he a io o he
minimum ole a ed elemen size and wice his uppe bound o he componen speeds.
15
4.4 One-dimensional sunami es p oblem
As a i s model SW p oblem, conside he idealised 1D sunami p oblem desc ibed in
[Da is and LeVeque,2016]. This p oblem conce ns p opaga ion o an ini ial su ace p o ile
caused by a sunami ac oss a simpli ied ocean domain o wid h 400 km. The domain
ba hyme y bhas a shel b eak discon inui y, as illus a ed in Sub igu e 4.2a:
b(x) = (200 m x≤50 km
4,000 m x > 50 km .(4.7)
Illus a ed in Sub igu e 4.2b, he ini ial condi ion used is gi en by
η0(x) = (0.4 sin (x−100,000)π
50,000 m 100 km ≤x≤150 km
0 m o he wise .(4.8)
As in Sec ion 3.3, (un ealis ic) impe meable bounda ies a e conside ed, whe eby u(4 ·
105) = u(0) = 0. This means wa es simply e lec o bo h he coas al and open ocean
bounda ies. The 1D o m o he SWEs used in [Da is and LeVeque,2016] ake an al e -
na i e linea o m
µ +g¯
h(x)ηx= 0, η +µx= 0,(4.9)
whe e µ=hu ep esen s momen um and we ha e (again) linea ised abou a la su ace
¯ηwi h ze o eloci y ¯u= 0. As wi h (3.3), (4.9) may be w i en in ma ix- ec o o m
q +A(x)qx=0,whe e A(x) = "0g¯
h(x)
1 0 #and q="µ
η#.(4.10)
In 1D sunami es .py, we also conside he adjoin p oblem. Gi en he equa ion o
(4.10), as ou lined in [Da is and LeVeque,2016], es by an app op ia ely sized ec o
unc ional λand in eg a e o e he spa io- empo al domain o ob ain
Z4·105
0Z4200
0
λT(q +Aqx) d dx= 0.(4.11)
(a) Ba hyme y (b) Fo wa d ini ialisa ion (c) Adjoin ini ialisa ion
Figu e 4.2: Ba hyme y and ‘ini ial’ condi ions used o he 1D sunami es p oblem.
16
In eg a ing by pa s in bo h space and ime,
Z4·105
0
(λTq)4200
0dx+ZT
0
(λTAq)4·105
0d −Z4·105
0Z4200
0
qT(λ + (ATλ)x) dx= 0.(4.12)
We conside an objec i e unc ional only a he end ime, wi h Ts a =Tend =T, so
he in eg and o he hi d e m o (4.12) con ains he adjoin equa ion o his p oblem,
λ + (AT(x)λ)x= 0,whe e λ=hλµληiT
.(4.13)
We en o ce an ‘ini ial condi ion’ o he adjoin p oblem, as a scaled indica o unc ion
λη,0(x)=0.4
1
[10 km,25 km], which app oxima es a co esponding del a unc ion, as illus-
a ed in Sub igu e 4.2c. Again, we impose he bounda y condi ion λu(4·105) = λu(0) = 0.
Since he adjoin equa ion is sol ed backwa ds in ime, we ans o m ime by 7→ 4200− ,
whence ∂ 7→ −∂ . Unde he same bounda y condi ions, (4.13) becomes
λ −(AT(x)λ)x= 0.(4.14)
As in [Da is and LeVeque,2016], we in e p e hese da a by es ablishing egions o
ocean whe e he o wa d and adjoin ee su ace solu ions ake nume ical alues wi h
magni ude abo e some ole ance, say 0.05m, wi h esul ing plo s displayed in Sub igu es
4.3a and 4.3b, espec i ely. The plo s include a dashed line ma king he loca ion, 50 km
o sho e, o he shel b eak discon inui y in ba hyme y. Signi ican egions o he o wa d
p oblem can be in e p e ed as a eas o ocean o which he ini ial condi ion will p opaga e,
whils he adjoin signi icance egions, on he o he hand, co espond o some app oxima-
ion o he ‘domain o dependence’ whe ein sunami wa e p opaga i e ac i i y will ha e
an e ec on he nea -coas al a ea o impo ance a he end- ime o he simula ion.
Ano he in e p e a ion may be gleaned by conside ing egions o ocean whe ein he
inne p oduc be ween o wa d and adjoin p oblem solu ions is signi ican , as displayed
in Sub igu e 4.3c. Thus, we gain in o ma ion conce ning which sunami ajec o ies a e
impo an o ou pa icula p oblem, gi ing a i s sugges ion o whe e mesh adap ion
should be pe o med. No ice he only impo an egions lie be ween coas and shel b eak.
(a) P imal solu ion η(b) Adjoin solu ion λη(c) Inne p oduc qTλ
Figu e 4.3: Regions o ocean whe e whe e o wa d and adjoin shallow wa e solu ions (and he
inne p oduc he eo ) ha e magni ude a leas 0.05 m o he 1D sunami es p oblem.
17
5. Aniso opic mesh adap i i y
When a mesh is egene a ed using aniso opic mesh adap i i y, he goal is o imp o e he
quali y o he wo s elemen o he mesh, in a local sense. In 2D, his is achie ed by a
pe o ming some combina ion o he ollowing ope a ions.
1. edge spli ing; 2. edge collapsing; 3. edge swapping; 4. node mo emen .
In he o me h ee ope a ions, he app oach embodies ha o h-adap i i y and in he la -
e ope a ion i embodies ha o -adap i i y, he eby p o iding a a ian o h -adap i i y.
An op imisa ion algo i hm loops o e he nodes o he mesh, and uses he abo e ope -
a ions o p oposes a new local mesh con igu a ion, based on e o measu es as men ioned
in Subsec ion 4.1.1. I ce ain c i e ia a e sa is ied, insis ing he mesh quali y is imp o ed
su icien ly ac oss meshes (so i is wo h he compu a ional e o associa ed wi h egene -
a ing he mesh), he new con igu a ion is accep ed. The means o measu ing mesh quali y
is desc ibed in Sec ion 5.2, wi h an al e na i e app oach ound in [Pain e al.,2001].
Fu he in icacies o he mesh adap i e p ocess a e also ou lined in his chap e .
5.1 Measu ing dis ance
Ascala p oduc is de ined as an SPD o m, which can be ep esen ed by an SPD ma ix
M, known as a me ic. Hence o h in his p ojec , we wo k solely in 2D and so all meshes
Hcan be unde s ood as subse s o R2. As such, o ou pu poses, a me ic p o ides a
map
h·,·iM:R2×R2→[0,∞),hx,yiM=xTMy,x,y∈R2.(5.1)
Euclidean space E2is ob ained by equipping a ec o space wi h such a scala p oduc .
Gi en he scala p oduc de ined by (5.1), we ob ain he co esponding no m by
k·kM:R2→[0,∞),kxkM=√xTMx,x∈R2.(5.2)
Leng hs o mesh edges pq ∈ H can be calcula ed using (5.2). As commen ed in [Ba al,
2015], angles be ween ec o s can also be compu ed in he usual way, wi h he angle
θ∈[0,2π) be ween wo ec o s x,y∈R2wi h espec o a me ic Mgi en by
cos θ=hx,yiM
kxkMkykM
.(5.3)
18
Fu he , gi en an elemen Ko he mesh has a ea |K|I2wi h espec o E2, i s co e-
sponding a ea in he me ic space de ined by Mcan be ound using he de e minan :
|K|M=pde (M)|K|I2.(5.4)
As Mis symme ic, i is o hogonally diagonalisable, wi h eigen alue decomposi ion
M=VTΛV="u1 1
u2 2#"λ10
0λ2#"u1u2
1 2#(5.5)
whe e λ1, λ2>0, and eigen ec o s u= [u1, u2]Tand = [ 1, 2]Ta e o hono mal.
(1,0)
(0,1)
M1
2
M−1
2
(h1,0)
(0, h2)
E2= (R2, I2) (R2, M)
Figu e 5.1: The mapping o an ellipse o he uni
ci cle unde a me ic.
Combining he abo e ing edien s, we
come o an al e na i e unde s anding o a
me ic Min e ms o geome y. Tha is,
Mcan be desc ibed in e ms o ellipses,
wi h i s eigen ec o s de e mining he axes
in which he ellipse is skewed and he in-
e se squa e oo o i s eigen alues de e -
mining he magni udes o skew along hese
axes. In he p ocess o mesh adap ion, we
seek o ob ain a uni o m mesh wi h espec o a me ic. Le h1and h2deno e he mag-
ni udes o he ellipse axes. Then, o he uni ec o e1= [1,0]T∈R2,
kh1e1k2
M=hh10iVTΛV"h1
0#=h2
1λ1= 1 ⇐⇒ λ1=1
h2
1
,(5.6)
using (5.5), and simila ly o he second s anda d uni ec o , e2= [0,1]T. The me ic
achie es i s pu pose o con olling he mesh adap ion p ocess by de ining he mesh edge
leng hs we desi e a each node o he mesh. In une wi h (5.6), o each eigenpai (λi, i)
o M, we ake he co esponding edge leng h in di ec ion i, as de e mined by
hi=1
√λi
.(5.7)
Figu e 5.1 depic s an ellipse wi h axes in di ec ions gi en by he eigen ec o s and
magni ude gi en by he in e se squa e eigen alues. Applica ion o he squa e oo M1
2o
he me ic hen yields a uni ci cle (wi h axes being he s anda d uni ec o s). He e he
ma ix squa e oo is well-de ined since all eigen alues a e posi i e, meaning we can ake
he squa e oo h ough he eigen alue decomposi ion,
M1
2=VTΛ1
2V=VTdiag(pλ1,pλ2)V. (5.8)
To con as , in iso opic mesh adap i i y, me ics Ma e no only SPD, bu diagonal,
19
aking he o m M= diag(h−2, h−2) o some edge leng h h > 0. As such, he iso opic
app oach allows only changes in he size o elemen s, and no hei o ien a ion.
The scala p oduc p o ided in he Euclidean case by (5.1) is cons an ac oss he do-
main, so dis ances a e measu ed using he same ellipse, ega dless o loca ion. As ema ked
in [Ba al,2015], i is a ac i e o be able o measu e dis ances in a way dependen on he
loca ion wi hin he domain. This is pa icula ly so o sunami p opaga ion modelling,
enabling us o conside mesh elemen s which ha e di e en sizes, using a ine mesh whe e
high esolu ion is equi ed and a coa se mesh in ‘less impo an ’ egions.
Fo spa ially dependen me ics, we conside Riemannian, a he han Euclidean, ge-
ome y. We now conside no jus one SPD ma ix M, bu a space M={Mx}x∈Ω he eo ,
de ined on all poin s o he domain. Locally, each o hese me ics de ines a scala p od-
uc when e alua ed on a poin x∈Ω. Using hese scala p oduc s, we can es ablish a
Riemannian me ic space, o which we can ex end no ions o dis ance and angle om he
Euclidean case. The ollowing ex ends de ini ions as in [Pain e al.,2001].
Fo an elemen Ko he mesh, conside one o i s edges e=pq ∈EK, wi h endpoin s
p,q∈R2. The leng h o ewi h espec o me ic Mis calcula ed analy ically as
`M(e) = Z1
0ppqTM(p+ pq)pq d ≈
k
X
i=1
ωippqTM(p+αipq)pq,(5.9)
using he no a ion o [Alauze ,2010]. While he s ic equali y in (5.9) p o ides a
con inuous measu e o dis ance along an edge, we need conside a disc e ised me ic
Mh={Mp}p∈H, as e alua ed a he e ices o he mesh H. Thus we conside a k-poin
quad a u e ule wi h weigh s {ωi}k
i=1 and Gauss poin s {αi}k
i=1, as in he app oxima e
equali y o (5.9). In P1 space, he e is linea a ia ion wi hin each elemen . As such, we
conside linea a ia ion o he disc e ised me ic along each edge o he mesh, meaning
i is su icien o calcula e leng h using one-poin quad a u e, as in [Piggo e al.,2005]:
`Me(e) = kpqkMe=ppqTMepq,whe e Me=1
2(Mp+Mq) (5.10)
deno es he edge-cen ed me ic, summed a he endpoin s o e. Tha is, α1=1
2is
he single Gauss poin . Ve ex-wise me ics a e calcula ed using in e pola ion be ween
meshes. Since M(x)0,∀x∈Ω, in pa icula Mp, Mq0. Applica ion o he de ini ion
o posi i e-de ini eness implies Mp+Mq0, meaning he edge-cen ed me ic is also
posi i e-de ini e, o each edge o he mesh.
Simila ly o (5.4), he a ea o an elemen Kin Riemannian me ic space M={Mp}p∈Ω
is gi en by he in eg al
|K|M=ZKpde (M(x, y)) dxdy. (5.11)
As ema ked in [Ba al,2015], his may be app oxima ed o i s o de using (5.4), wi h
20
M eplaced by Mapplied a he ba ycen e o K. Wi h he same modi ica ion, (5.3) can
be used o calcula e he angle be ween ec o s u1and u2in he Riemannian me ic space.
As al eady men ioned, he idea o me ic-based mesh adap ion, as i s in oduced
in [Geo ge e al.,1991], is o he Riemannian me ic space o be used in compu ing
geome ical quan i ies which a ange o he new mesh o be a uni mesh wi h espec o
his me ic space. This links wi h he ea lie in e p e a ion o mo ing om an ellipse o a
uni ci cle, in e ms o he me ic conside ed. As de ined in [Alauze and Loseille,2016],
Kis a uni elemen i each o i s sides ha e uni leng h wi h espec o he go e ning
me ic. Hence, K∈ H is a uni elemen i and only i Kis equila e al wi h all sides equal
o 1. By basic igonome y, such a iangle has a ea √3
4wi h espec o M, and hence
Euclidean a ea √3
4(de (M))−1
2. I we a e o conside uni meshes as consis ing pu ely o
uni elemen s, i is clea mos domains canno be illed pa icula ly well. Fo example,
in Euclidean space, whe e M=I2, a squa e canno be illed wi h equila e al iangles
wi hou a numbe o gaps. I can, howe e , be illed using igh angled iangles.
Due o he abo e, i is necessa y o us o elax he cons ain ha ou meshes con ain
only uni elemen s. Ins ead, as in [Ba al,2015], we conside a mesh which con ains
only quasi-uni elemen s, whose edges each ha e leng h in he ange [ 1
√2,√2]. In he
Riemannian se ing, (5.7) ca ies o e o p o ide a measu e o size,
hM( ) = k k2
`M( ),(5.12)
wi h espec o ∈R2, which may be app oxima ed using quad a u e as in (5.10).
5.2 Gauging mesh quali y
Conside a ec o J∈RN, whe e Ndeno es he numbe o elemen s on he cu en mesh,
which will likely change alue a numbe o imes du ing a h-adap i e algo i hm. We ollow
[Piggo e al.,2005] in measu ing global mesh quali y using he ∞-no m,
F=kJk∞.(5.13)
The use o an ∞-no m means he mesh as a whole is held o ha e he quali y o he wo s
quali y elemen o he mesh, which we would like o imp o e, in a local sense.
The unc ional we conside coincides wi h he one used in PRAgMaTIc (Pa allel
aniso Ropic Adap i e Mesh ToolkI ).1, which p o ides aniso opic mesh adap i i y in Fi e-
d ake o meshes o simplexes and which unde pins he code w i en in his p ojec . In
1Quali y unc ional in o ma ion desc ibed in his sec ion was collec ed om he associa ed Gi Hub
page e e enced in Sec ion 1.3, in he in oduc ion. ex ile o he docs di ec o y.
21
we de ine he educed Riemannian space using me ic in e sec ion de ined in Sec ion 5.5,
M=n∩p∈H
Mp(x)∩
M(x)ox∈Ω.(5.37)
As such, we en o ce he s onges cons ain on elemen size ac oss me ics in (5.37).
Using linea in e pola ion o he me ic, we need only compu e (5.37) a each e ex.
Howe e , [Alauze ,2010] s a es ha he associa ed algo i hm is o quad a ic complexi y,
so is compu a ionally cumbe some. Ins ead, we ollow he app oach o only e alua ing
in e sec ions o e indi idual edges. Fo an edge pq ∈ H, we make he app oxima ion
M(p)≈ M(p)∩Mq(p),
M(q)≈ M(q)∩Mp(q).(5.38)
5.7 No es on he adap i e algo i hm code
Aniso opic mesh adap i i y unc ions used in his p ojec a e no cu en ly implemen ed
on he mas e b anch o Fi ed ake. Fo his eason, we use a di e en b anch3, which
will soon be me ged in o he mas e . As discussed in Sec ion 4.1, he e a e h ee cen al
pa s o he adap i e p ocedu e: e o es ima ion, mesh adap ion and in e pola ion. In
he ollowing we b ie ly elabo a e on he code equi ed o each o hese s eps.
As discussed in Sec ion 5.3, o compu a ional pu poses, e o es ima ion amoun s o
compu ing he Hessian o a scala ield. This is achie ed by he unc ion cons uc hessian.
The wo main modes o Hessian eco e y desc ibed in Sec ion 5.4 a e bo h implemen ed in
his unc ion, wi h selec ion be ween hem a ailable. Ha ing compu ed a Hessian, he co -
esponding me ic is compu ed using compu e s eady me ic, making modi ica ions de-
sc ibed in Sec ion 5.3. Con ol o he minimal and maximal elemen sizes ole a ed, as well
as he maximal aspec a io a, is allowed. By allowing la ge alues o a, we may inc ease
mesh aniso opy. In e sec ion o me ics by (5.30) is achie ed by me ic in e sec ion.
Func ions desc ibed he e a e ound in he u ili y sc ip adap i i y.py
The c ucial adap i i y cons uc used in his code is he Aniso opicAdap a ion
class. Membe s he eo ake as inpu a mesh and a me ic and ha e wo impo an
unc ions: adap ed mesh and ans e solu ion. The o me holds he adap ed mesh,
unde he me ic supplied, whils he la e in e pola es ields om he old mesh o he
new mesh. As such, he mesh adap ion p ocess is encapsula ed in adap ed mesh.
In e pola ion using ans e unc ion is only cu en ly suppo ed o Lag ange spaces,
and no mixed spaces. Since we conside a mixed Taylo -Hood p oblem, ou pu poses e-
qui e a modi ica ion. This is p o ided in he sc ip in e p.py, as he unc ion
in e p Taylo Hood. As well as ex ending o he mixed case, his unc ion in ol es a
p ocedu e accoun ing o when a me ic dic a es a node is mo ed ou side he domain. An
implemen a ion o me ic g ada ion is p o ided by he unc ion me ic g ada ion.
3Found a h ps://gi hub.com/ aupalosau us/ i ed ake/
28
5.8 Adap i i y code es s
Be o e conside ing ime-dependen PDE p oblems, which equi e se e al emeshing s eps
o e he solu ion p ocess, we es he adap i e algo i hm o some s eady p oblems. The
ollowing su aces conside ed we e in oduced in [Oli ie ,2011]:
u1(x, y) = x2+y2, u2(x, y) = an−10.1
sin(5y)−2x+ an−10.5
sin(3y)−7x.(5.39)
Using senso es s.py, we a e able o gene a e meshes adap ed o (5.39), as displayed
in Figu e 5.2. In each case double L2p ojec ion is used o Hessian econs uc ion. The
algo i hm appea s o adap well o he gi en senso unc ions, wi h he i s example yield-
ing a nea uni o m mesh in he domain in e io , as expec ed due o ∇∇Tu1= diag(2,2)
being cons an , al hough he e a e some issues on he bounda ies.
Le us now conside some empo ally a ying PDE p oblems. A simple such p oblem
o conside is he ad ec ion and di usion o a pollu an concen a ion φ, say, unde he
in luence o a cons an wind ield, u= (1,0) m s−1. Conside ing a di usion e m wi h a
non-negligible, bu s ill ela i ely small di usi i y pa ame e ν > 0, helps o smoo h ou
any small scale spu ious s uc u es in he p oblem. Bu ge s’ equa ion is gi en by
∂φ
∂ +u·∇φ−ν∇2φ= 0.(5.40)
(a) Senso u1(b) Mesh 1 (hmin = 10−3)
(c) Senso u2(d) Mesh 2 (hmin = 10−4)
Figu e 5.2: Tes senso unc ions (5.39) and hei associa ed adap ed meshes. The ini ial mesh
was uni o m wi h 200 e ices in each di ec ion. Maximal elemen size and aniso opy we e se
as hmax = 0.1 and amax = 103, wi h minimal elemen size as gi en.
29
Conside a ec angula domain Ω = [0,4] ×[0,1] m2and a Gaussian ini ial condi ion
φ(x, y)=0.001 exp(−25((x−α)2+ (y−β)2)),(5.41)
wi h α=β= 0.5. Code o his es case is p o ided by Bu ge s es .py.
Choosing hmin = 5 mm and hmax = 100 mm as de aul s, he algo i hm adap s he mesh
in such a way ha he as majo i y o he esolu ion is ocused a ound he bubble o high
concen a ion, as i is ad ec ed along he wind di ec ion, as illus a ed in Figu e 5.3 and
in acco dance wi h expec a ions. In his expe imen we ake he di usi i y pa ame e
ν= 10−3m2s−1and pe o m Hessian econs uc ion by double L2p ojec ion.
In he abo e, adap ion is pe o med wi h espec o a single scala dependen a iable,
φ, o a scala PDE (5.40). Inco po a ing adap i i y in o he SWEs is a li le mo e di icul ,
as hese comp ise a “2+1” pai o coupled PDEs as in (3.1), wi h ec o and scala
dependen a iables.4
Fo he nex es p oblem, we ake a s ep owa ds he ealis ic sunami case by con-
side ing a SW p oblem wi h a shel b eak discon inui y, as in Sec ion 4.4. Conside he
ex ension o ou ec angula domain o a squa e domain Ω = [0,4] ×[0,4] m2, wi h a shel
b eak om 1 cm dep h down o 10 cm dep h, occu ing 50 cm om he le -hand bounda y
and wi h he same ini ial condi ion as in (5.41), bu now wi h α=β= 2, so ha he
Gaussian bell is cen ed wi hin he domain. The sc ip ile o simple adap i e SW.py
adap i ely sol es he linea SWEs (3.2) o his se up. As is illus a ed in Figu e 5.4,
4The SWEs can also be in e p e ed as a sys em o h ee PDEs wi h h ee scala dependen a iables
o , in he linea case, as a single ec o PDE (3.3).
Figu e 5.3: Ad ec ion and di usion o he ini ial concen a ion (5.41) o α=β= 0.5, acco ding
o Bu ge ’s equa ion (5.40), wi h wind ield u= (1,0), pu ely in he x-di ec ion.
30
Figu e 5.4: Linea SW simula ion o ini ial condi ion (5.41) wi h α=β= 2 and a shel b eak
discon inui y.
he mesh esolu ion success ully ‘ ollows’ he ings p opaga ing ou wa ds om he ini-
ial sp eading o he ini ial condi ion. As we saw in conside ing he adjoin p oblem,
inc eased mesh esolu ion in he la e subplo indica es ha he mos signi ican egion
o he domain o his simula ion is he pa in he shallow wa e a ea ‘nea o he sho e’.
In he SW es case conside ed he e, we adap only o he luid speed, no he ee
su ace displacemen , o indi idual componen s o he luid eloci y. Using he heo y as
discussed in Sec ion 5.5, and me ic in e sec ion, we can adap o hese ields, oo.
The e is, howe e , an issue wi h he meshes gene a ed in Figu e 5.4. Examining hem
mo e closely, i becomes appa en ha he dense pa ches o mesh esolu ion a e o en
‘lagging behind’ he c es o he wa e i sel . This occu s mainly because he mesh is no
adap ed a e e y imes ep and, when i is, is adap ed based on a Hessian econs uc ed
om he cu en imes ep. In his way, he egions lagged as equi ing ine mesh es-
olu ion do no necessa ily ha e his equi emen o he coming imes eps un il he nex
mesh egene a ion. In o de o comba his issue, we nex conside goal-based adap i i y.
5.9 Goal-based mesh adap i i y
Thus a , adap ion has been based on a me ic calcula ed om solu ion da a a he
cu en imes ep.Goal-based mesh adap i i y seeks o imp o e his me ic, inco po a ing
app oxima ions o he dynamics wi hin egions o he solu ion ield which g ea ly a ec
hose in he spa ial (and empo al) egion o in e es . Fo he linea SW case, his
in ol es e alua ing an objec i e unc ional (3.13) and using he nume ical solu ion o he
adjoin equa ions (3.18). Al hough he me hod can be applied mo e gene ally, we es ic
a en ion o he linea SW case hence o h, o simplici y. Fo objec i e unc ional (3.13),
(x, ) =
1
A(x)η(x, ) =⇒∂
∂u=0,∂
∂η =
1
A(x),(5.42)
31
whe e Awe ha e made he ( easonable) assump ion ha he nea -coas al egion o im-
po ance Ais con ained wi hin he solu ion image Ωs. In his way, he adjoin equa ions
(3.18) ha e a cons an sou ce e m in he egion A.
Gi en ou nume ical solu ion uple o he SWEs (3.2), we i s es ima e he e o
made in he consequen compu a ion o he unc ional J, p o iding a measu e o wha
is deemed o be impo an . In his way, he e o es ima ion s age has he pu pose o
lagging ce ain a eas o he mesh o e inemen , coa sening o mo emen .
To quo e pp.1 o [Rannache ,2009], goal-based adap i i y “means he op imiza ion o
he mesh and possibly also he disc e e model i sel on he basis o an a pos e io i e o
es ima e”. Such an e o es ima e akes he gene al o m
J(q)−J(qh)≈e(qh) = X
K∈H
eK(qh),(5.43)
whe e qhdeno es ou FEM app oxima ion o he (exac ) solu ion uple qon a mesh
H, wi h local cell-o ien ed e o indica o s eKsumming o gi e a global e o indica o
e. Ha ing made an e ec i e choice o e o es ima o , we seek o minimise he e o
|J(q)−J(qh)|as in (5.43). To speci y he quali y o ou solu ion o his p oblem, we se
an uppe bound on he e o es ima e o accep ance.
Gi en an app oxima ion o he solu ion o he adjoin equa ions, we make a simila ,
ec o ised e sion o he a gumen ound in [Powe e al.,2006] ( o accoun o he ac
we ha e no jus one solu ion ield, bu h ee). In he s eady case (whe e ime de i a i e
e ms a e neglec ed) we ob ain he app oxima ion
J(q)−J(qh)≈ hR(qh),λhiΩs,(5.44)
which in ol es bo h he esidual o he o wa d p oblem and he app oxima ion o he
adjoin p oblem solu ion. Thus, in his app oach, (5.44) enables us o es ima e he as-
socia ed e o in ou app oxima ion o he objec i e unc ional. The e a e many o he
possible choices o e o es ima e, wo o which we discuss in he ollowing. The in eg al
igh hand side o (5.44) is app oxima ed using quad a u e, in he o m o (5.43).
We know he exac solu ion sa is ies R(q) = 0. Fo some linea ope a o Land wha
amoun s o a sou ce e m s, decomposing by R(qh) = Lqh−s, subs i u ing in (5.44)
and applying G een’s heo em (igno ing su ace e ms, as ecommended in [Powe e al.,
2006]) gi es an al e na i e e o es ima e
J(q)−J(qh)≈ hq−qh,L∗λhiΩs.(5.45)
Fo a mo e accu a e e o es ima e, o he app oaches a e a ailable. Some app oaches
conside he disc e e o m o he adjoin , as opposed o he con inuous one used abo e.
Ano he me hod, desc ibed in [Powe e al.,2006], is o make ‘coa se-g id’ app oxima ions
o he esidual. This in ol es sol ing he PDE on di e en mesh esolu ions, in o de o
32
gain some in o ma ion conce ning how much he esidual R(qh) s ays om ze o.
The SWEs (3.2) conside ed o m an uns eady PDE, so i is no so s aigh o wa d
as o apply s eady e o es ima es a each imes ep. An app oach o uns eady goal-
based aniso opic mesh adap i i y is desc ibed in [Belme e al.,2012], which de ails he
ex ensions equi ed o inne p oduc s. Fo simplici y, howe e , we ollow he app oach
o [Da is and LeVeque,2016] and conside he simpli ied e o es ima e o (5.44),
J(q)−J(qh)≈ hqh,λhiΩs.(5.46)
The es ima e in (5.46) is calcula ed using e ex-o ien ed e o indica o s
ep(qh; ) = max
τ∈[
e
, ]qh(p, )Tλh(p, τ),whe e e = max{ −Tend +Ts a ,0}.(5.47)
In his app oach, a ime , we conside he inne p oduc o o wa d and adjoin solu ion
app oxima ions o e a ime ange [e , ] wi hin which he objec i e (3.13) is non-ze o.
In he iso opic case, he disc e ised me ic ield M={Mp}p∈H is simply de ined by
Mp= diag(ep(qh; )−2, ep(qh; )−2). To cons uc he me ic in he aniso opic case, on
he o he hand, we begin by adap ing o a ield ela ed o he luid low, such as ee
su ace displacemen , as usual. We hen scale he me ic a each e ex by he e o
indica o . This may be in e p e ed as eplacing he e o equi emen o (5.17) wi h
e=
ep(qh; )(5.48)
Th ough (5.48), we insis ha he e o is smalle , and hence ha he solu ion is o
be e quali y in he ‘signi ican ’ egions lagged by he e o indica o . In egions o less
signi icance, e akes la ge alues, meaning ha a coa se mesh is allowed.
The gene al goal-based mesh adap i e p ocedu e is summa ised by Algo i hm 1.
Algo i hm 1: Goal-based mesh adap i e p ocess
/* Fixed mesh adjoin un */
Sol e he adjoin p oblem (backwa ds in ime, om =T) on a cons an , coa se
mesh and sa e he solu ion da a.;
/* Adap i e mesh o wa d un */
o each imes ep n( o wa d in ime, om = 0)do
Es ima e he e o a n, using an e o es ima e (5.43) and he sa ed da a.;
Feed he e o es ima e in o he adap i e algo i hm by eplacing he o (5.17)
by he upda ed e o equi emen (5.48).;
Regene a e he mesh, in e pola e a iables and sol e he o wa d p oblem.;
end
33
6. Tsunami applica ion
As discussed in Chap e 2, his p ojec aims o simula e he sunami which s uck he
coas o Fukushima, Japan, in 2011, bo h accu a ely and e icien ly, using aniso opic
mesh adap i i y. The p ocess by which his was achie ed is explained in his chap e ,
wi h esul s, analyses and conclusions p o ided in Sec ions 6.4 and 6.5.
6.1 Compu a ional se up o he T¯ohoku sunami
Figu e 6.1: Domain geome-
y o he T¯ohoku sunami.
Bounda y segmen s ∂ΩO,
∂ΩFand ∂ΩCa e shown in
blue, magen a and o ange,
espec i ely. Whi e num-
be s co espond o bounda y
iden i y ags. Ba hyme y
is shown using he QGIS
Google Maps plugin.
In o de o un compu a ions o he example conside ed,
a GIS domain bounda y ile was cons uc ed using sho e-
line da a om GSHHG, along wi h ba hyme y da a om
GEBCO. By sepa a ing he coas line in o pa s, we a e able
o indica e geog aphical egions whe e he ini ial mesh should
be ine , such as nea o Fukushima. Di e en GSHHG es-
olu ions a e used o he wo coas al bounda ies nea o
Fukushima and elsewhe e, so small, dis an island ea u es
a e no esol ed, o he wise incu ing compu a ional expense.
Figu e 6.1 illus a es he bounda y segmen s used, wi h
ba hyme y da a isible in he backg ound. Due o he use
o di e en da abases o ba hyme y and coas lines, wi h
disag eeing sea-le el con ou s, i is necessa y o cap he
ba hyme y so ha i is ne e shallowe han 30m, say.
The domain bounda y ∂Ω comp ises o he (disjoin )
union o segmen s ∂ΩO,∂ΩFand ∂ΩC, deno ing he open
ocean bounda y, egion o in e es su ounding Fukushima
and emaining coas al bounda y, espec i ely. A simpli i-
ca ion has been made along ∂ΩC, as indica ed in Figu e
6.1, whe e a ( ela i ely) small gap be ween land masses o
Hokkaido and Honshu, he Tsuga u S ai , has been closed, as i he e exis s a sea wall.
This is jus i ied as i is su icien ly a om bo h he ea hquake epicen e and he egion
o in e es ha any e ec he disc epancy may ha e on he inal solu ion will be negligible.
In sol ing luids p oblems on he en i e Ea h, o when conside ing imescales o he
o de o he ides, one should acknowledge bo h ha he Ea h is o a ing (as mani es ed in
34
he Co iolis o ce) and he ac he Ea h is (almos ) a sphe e. We ha e al eady discussed
he o me . Wi h espec o he la e , one app oach is o e-de ine he SWEs in sphe ical
pola co-o dina es. Howe e he Ea h is no qui e a sphe e and he e a e a numbe o
o he e ec i e me hods which a e egula ly used in he compu a ional geosciences.
One such app oach is ans o m o UTM coo dina es, p o iding no one single map
p ojec ion, bu a se ies o p ojec ions, aken ac oss he whole globe. The UTM app oach
di ides he Ea h in o 60 zones, whe e each band is 6◦in longi ude. In addi ion, zones a e
di ided in he la i udinal di ec ion, gi en zone le e s om he Roman alphabe . UTM
coo dina es ha e me ic uni s, aking he ho izon al alue 500km a zone midpoin s.
T ans o ma ion be ween la i ude-longi ude and UTM coo dina es is made possible in
Py hon using he unc ions con ained in con e sion.py1. Since he domain we conside
is con ained mainly in zone 54, bu also o e laps wi h zones 53 and 55, we use he
o ce zone numbe pa ame e op ion in ou con e sion, se o 54, o a oid discon inui ies.
Figu e 6.2: App oxima ion
o he ini ial su ace dis-
placemen o he T¯ohoku
sunami used in his p ojec .
Using he domain geome y gene a ed in QGIS, we can
cons uc meshes using QMESH online, ecen ly de eloped
a Impe ial College [A dis e al.,2017]. Each bounda y seg-
men o he GIS ile is gi en a ‘PhysID’ physical iden i ica-
ion ag, so a di e en le el o meshing can be applied wi hin
a g ada ion dis ance speci ied in he meshing p ocess. We
dis inguish be ween open ocean (PhysID=100) and coas al
bounda ies (PhysID=200), as displayed in Figu e 6.1 - neces-
sa y i we we e o en o ce Di ichle bounda y condi ions.
We also equi e a means o in e p e ing he sunami ini-
ial condi ions and ba hyme y in Fi ed ake. Ini ial condi-
ion da a was p o ided by he au ho o [Sai o e al.,2011]
as ini p o ile.xyz, con aining a se o disc e ised la i udinal and longi udinal coo di-
na es, along wi h ee su ace displacemen alues. This da a was p ocessed using GMT
o c ea e a Ne CDF su ace which could be in e pola ed in Fi ed ake, along wi h he
ba hyme y ield. The esul ing ini ial su ace p o ile is displayed in Figu e 6.2.
No ice om Figu e 6.2 ha ini ial p o ile o he sunami is e y close o he main
Japanese island o Honshu. Wi ness and measu emen epo s, such as hose used in
[Kazama and Noda,2012], [Okada,2011] and [Suzuki e al.,2012], men ion ha he
sunami wa e a i ed a many Japanese coas al loca ions wi hin 30 minu es and a some
in jus en minu es. Since we a e mainly in e es ed in modelling he wa e’s app oach o
and eaching o he coas , i is easonable o assume (and ai ly clea om expe imen s in
Figu e 6.3 o e lea ) ha he ocean-bound wa es will no each he open ocean bounda y
wi hin a 25 minu e ime ame. Hence, in ou applica ion o he SWEs o his case s udy,
a i s app oxima ion is o en o ce ze o lux o e he open ocean bounda ies. As dis-
cussed in Sec ion 3.3, his app oach is no pa icula ly physical when wa es in e ac wi h
1These ans o ma ion unc ions can be downloaded om h ps://pypi.py hon.o g/pypi/u m.
35
he sho eline and app oxima es wha should be open ocean bounda ies as impe meable.
None heless, i is su icien o ou pu poses, as we a e mos in e es ed in sligh ly o sho e
loca ions, a gauge loca ions P02 and P06 as displayed in Figu e 2.2.
Using code de eloped in he es p oblems, an adap i e algo i hm o sol ing he
T¯ohoku sunami SW p oblem is gi en by simple adap i e sunami.py. This sc ip al-
lows o expe imen a ion wi h di e en ini ial QMESH meshes and adap i i y pa ame e s.
Adjoin p oblem solu ion in o ma ion is inco po a ed in goal-based sunami.py.
6.2 Model e i ica ion
(a) Non- o ., linea (NL) (b) Non- o ., nonlinea (NN)
(c) Ro ., linea (RL) (d) Ro ., nonlinea (RN)
Figu e 6.3: F ee su ace displacemen app oxima ions in he
T¯ohoku p oblem a e 25 minu es o simula ion ime unde
ou SW equa ion se s, whe e ‘ o .’ deno es ‘ o a ional’.
A e es ablishing he go e ning
SWEs in Sec ion 3.1, we made
he simpli ica ion o linea i y, as
in (3.2). We expec he lin-
ea ised equa ions o be compu-
a ionally cheape o sol e han
he nonlinea case, hence p e-
e ing o use hem in ou cal-
cula ions. Be o e implemen ing
mesh adap i i y in he T¯ohoku
sunami case, le us b ie ly com-
pa e he wo app oaches o e -
i y he assump ion o linea i y
o ocean-scale SW dynamics.
A i s compa ison o hese
app oaches can be made using
an ‘eyeball no m’ conside a ion
o he solu ions p o ided ac oss he en i e domain, a a pa icula poin in ime. Hence-
o h an implici midpoin me hod is used, wi h imes eps o leng h 1 second. Sub igu es
6.3a and 6.3b display he shallow wa e solu ions gi en by he linea and nonlinea , non-
o a ional app oaches a e 25 minu es o simula ion ime, espec i ely. I is di icul o
dis inguish any signi ican di e ences in he ocean dynamics a his ime le el, suppo ing
he claim ha using he linea s andalone sol e migh be su icien . Th oughou his
sec ion, es s a e un on a ixed, ine mesh wi h 97,343 e ices. In he nonlinea case, we
ake di usi i y pa ame e ν= 10−3and bo om ic ion coe icien cb= 0.0025.
Addi ionally, in Sec ion 3.1 he Co iolis e ec was assumed o be negligible in sunami
modelling, due o he la ge di e ence in imescales and speeds on which he wo dynamics
ac . We can es his assump ion nume ically by sol ing he o a ional SWEs and com-
pa ing he esul s wi h he non- o a ional case used hus a . As men ioned in Sec ion
36
3.1, in he linea case, he momen um equa ion co esponding o (3.2) akes he o m
∂u
∂ + 2Ω×u+g∇η= 0.(6.1)
Equi alen code o he o a ional case o (6.1), along wi h he nonlinea equa ions, is
p o ided by model e i ica ion.py. E alua ing he ee su ace displacemen a e 25
minu es, as illus a ed in Sub igu es 6.3c and 6.3d, yields esul s indis inguishable by eye
o hose ob ained in he o he , non- o a ional cases o Figu e 6.3. The Co iolis pa ame e
is e alua ed e ex-wise using a ans o ma ion back in o la i ude-longi ude coo dina es.
The di icul y in making his analysis is ha we do no ha e he ue solu ion o he
luid dynamics which we seek o app oxima e. Fo a mo e igo ous analysis, we conside
a ime se ies o ee su ace alues a wo pa icula loca ions. A disad an age o his
app oach is i only gi es a localised, poin wise app oxima ion, a he han a global ones
gi en in Figu e 6.3. Howe e , i we choose a loca ion co esponding o an ocean gauge,
we can compa e model app oxima ions agains his o ical da a. Two such gauges a e he
o sho e bo om-p essu e gauges P02 and P06, illus a ed in Figu e 2.2. These p essu e
gauges, ope a ed by T¯ohoku Uni e si y, we e ac i e be ween June 2010 and May 2011.
We use ee su ace alues om P02 and P06 as calcula ed in he ini ial su ace in e -
sion analysis o [Sai o e al.,2011], as illus a ed by he blue lines o Sub igu es 6.4a and
6.4b. Gi en he gauge coo dina es (142.5016◦,38.5002◦) and (142.5838◦,38.6340◦), o he
nea es en housand h o a deg ee, model ou pu s may be compa ed agains hese da a.
Using model e i ica ion.py, we can un analyses o he a ious models, yielding he
cu es in Sub igu es 6.4a and 6.4b. The imese ies o he nonlinea sol e s a e unca ed
a 25 minu es simula ion ime, due o ex eme compu a ional expense a e his poin .
I is clea om Figu e 6.4 he e is e y li le di e ence be ween he model esul s, o
his ixed mesh and o hese ide gauges. Making use o Py hon’s clock unc ionali y, we
ob ain he pe o mance analyses displayed in he able o Sub igu e 6.8. He e he a e ages
we e aken o e wo p essu e gauge uns conside ed.2
We obse e om Figu e 6.8 ha he speedup in choosing he linea equa ions o e
he nonlinea ones is by a ac o o a leas 4. In addi ion, he e is a small speedup in
conside ing he linea , non- o a ional equa ions o e he linea , o a ional equa ions. I is
easonable o conclude we may exclusi ely use he linea , non- o a ional case hence o h.
Ou jus i ica ion is ha i p o ides an simila ly accu a e and a as ly compu a ionally
cheape app oach han he nonlinea cases, and yields a non-negligible sa ing in compu-
a ional e o o e he linea case whe ein he Co iolis e ec is conside ed.
I is also clea om Figu e 6.4 ha he ime se ies o ou solu ion ake he igh gene al
shape, mimicking he ini ial ise and all in ee su ace a he well, on his high esolu ion
mesh. Howe e , he la e 35 minu es o simula ion ime is no app oxima ed so well by
ou model, using any o he equa ions conside ed. This should be expec ed, because,
2Pe o mance es s we e un in se ies on a MacBook P o wi h OS X El Capi an.
37
Bibliog aphy
F Alauze . Size g ada ion con ol o aniso opic meshes. Fini e Elemen s in Analysis and
Design, 46(1):181–202, 2010.
F ´ed´e ic Alauze and Ad ien Loseille. A decade o p og ess on aniso opic mesh adap a ion
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