An automatic mesh coarsening technique for three dimensional anisotropic meshes

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HAL Id: inria-00186037

https://hal.inria.fr/inria-00186037v2

Submitted on 8 Nov 2007

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An automatic mesh coarsening technique for three dimensional anisotropic meshes

Youssef Mesri, Hervé Guillard

To cite this version:

Youssef Mesri, Hervé Guillard. An automatic mesh coarsening technique for three dimensional anisotropic meshes. [Research Report] RR-6344, INRIA. 2007, pp.52. �inria-00186037v2�

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a p p o r t

d e r e c h e r c h e

Thème NUM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

An automatic mesh coarsening technique for three dimensional anisotropic meshes

Youssef Mesri — Hervé Guillard

N° 6344

November 7, 2007

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M^−1/2 gj^#pF§dcx¦`bgm8cmoU`hMPR.Qqp&`bfhg»,pv^hR6wmvc`bMPR.w_gpvvm8cpog-.4p&`bgm8c,mo

M = V.D.V^T

x6pojxdPjp&`bR6w!]\`bpv]gcP

M^−1/2=V.D^−1/2.V^T g©`bMFpq^rgQ#gjpoftwPR¬cPg`hgmvc.§m8f

M⁻²ªLNMPRu2m&R6fb^

−1/2^pocw −2M¥p¸vR2R*R6cµxyMPm8^hR*c.`bm(8g8R`hmq`bMPRR*»_uPfhR4^h^hgmvc±8³`hMPRQ#R6pocgcP(mo p¸vR6fbpvvgcP`hMPR xyM¥pofypvx¦`bR*fbg^r`hgjxtQ#R4^rM¤^hg-.6Rm&8R*f`hMRR*R*Q#R*c;`b^

T ∈ T(i)pvw&nrpvxR6c;``bm

i^ª

- Á

¸

½

v

½ Â ½

Á

"H_

HÁ Á,+t½

"½ ½ ½

6

v

Á Á

Â ½

LNMRcm]wpolQ#R`bfhgjx§fhm8QY^hR6x¦`bgm8cPª}ª Mp8^¥R6R*c,xm8c^r`hfbdx¦`bR6wmvc`hMPRqR6R6QR6c;`QR*`hfbgx6^*pocwpv^

^hdxyM^hMPmvdw`ypovRUgc;`hmpvx6xmvdc8`@`bMPRUm_x*pv¥^hMpou2RWpvcwqw_gfhR4x¦`hgmvcqmvpocPgj^hmo`hfbmvugR4^mopvw&nrpvxR6c;` Q#R6^hM

R6R6Q#R*c;`b^6ª &Um&R6vR6f6_c]dPQ#R*fbgjx*poR»_u¥R6fhgQ#R*c;`bÛ^rMPm& `hM¥p&Ùg`Ww_m]R6^tcPmoÙ8g8RvPgcF^hmvQ#Rx*p8^rR4^*i;dPg`hR

R*»Ppvx¦`Ngc_§mvfbQqp&`hgmvc(mvc`bMPRp¸vR6fbpvvRtpv^hu2R6x¦`Nfyp&`hgmmv`bMPRp8w¸nrpvx*R*c;`NR*R*Q#R*c;`b^Npvcwq`bMp&`g`m&8R*fy^rg/.*R4^

`bMPRF^hQ#pvR6^r`xyMpofypvx`hR6fhgj^a`bgx(Q#R6^hM ^hg-.6R,±§R*¥ª `hMPR.2mvdPc¥wPpofb\jp¸\vR*f`bMPgxy]cPR4^h^y³¦ª°LNMPgj^q^hg©`bdp&`bgm8c

(12)

M¥pouPu2R*c^6¹§m8fR*»PpoQ#uPRv«m8cEpxdPfbvR4wµ2mvdcwPpofb\¤jp¸\vR*fg`hM,p(Mg8MEp8^ru2R6x`m8fbwPR*fq±§m8fbw_R6f

10⁴ ^mvf

Q#m8fhR¸³¦ª

Mi

D ^g`hM

R6g8R*c]&podPR6^mvfyw_R*fbR6wEgcpocp8^hx*R*cw_gcPFm8fbwPR*fm8c`hMPRw_gpvvm8cpo pvcw,pocmvfh`hMPm8cPmvfbQqpo@Qqp&`bfhg»Emv

x*mvfbfhR4^ru2mvc¥w_gcR*gvR6c]vR6x`hm8fb^]\

V

M =V.D.V^T

D¯i

p8^§mvm&t^

D¯i= ( 1 card(T(i))

X

T∈T(i)

D⁻_T^1/2)⁻² ^±+8³

LNMPR¬¥cpo¹cPm_wPpv«Q#R`bfhgjxg^N`bM]d^NvgvR6c!]\

Mi=Vi.D¯i.V_i^T ^± ^8³

MPR6fhR

Vi

Mi

d^hR6wgc§m8fhQdPjp

±8³ªLNMPgj^uPfbm_xR6wPdPfhR`bM;d¥^tfbR*uPjpvx*R6^`hMRR*gvR*c]&podPR4^Nmo

Mi

]\`hMRxm8fhfbR6x`hR6w!R*gvR6c]¸pvdR6^NvgvR6c

]\µ±+8³`bMp&``bfhgR6^`bmM¥p¸vRp¥R*`r`hR6ftR6^r`hgQqp&`bRmv`bMPRpv^hu2R6x¦`fyp&`bgm#mo`hMPRR6R6Q#R*c;`b^po`tcPm_w_R

i^ª

½ Á "²_ ²Á Pq Á

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LNMR¤uPfhm8PR*Q mvgc;`hR6fhu2mvjp&`bgcPE`bR*c^hmvfq¬¥R*jwP^qMp8^qfhR4xR6c8`b\fhR4xR*gvR4wpo`r`bR*c;`hgmvc gc `hMPRFpvfhR4pmv

gQqpo8RuPfbm]x*R6^b^rgcP* ,Iª R.w_R6^bxfbg¥Rgc`bMPR§m8m&gcP¤`am^hR6x¦`bgm8c^6`amµw_g-'CR*fbR*c;`^r`hfyp&`bR*vgR6^

`bmFw_R¬¥cPRp.cPm_wPpo@Q#R*`hfbgx¬R6w,x*mvQ#gcP§fhm8Q¨`hMg^fbR6^hR6pvfbxyMEpofbR6pª R`d¥^po;pogcxm8c^hgw_R6f

T(i)

`bMPR¤^rR*`qmoWpvR6R6QR6c;`b^

T ^MPgjxyM xmvc;`ypogcpcPm_w_R

i i ∈ T^ª WcR6pvxyMR*R*Q#R*c;`

T gj^qw_R*¬cPR6w

MT

ª R

pvfhRm]mv]gc#§mvfUpqdcPgi;dPRQR*`hfbgx

Mi

`hMpoÙfbR*uPfbR6^hR*c;`b^Ngc.`hMR2R6^rÙu¥m;^h^hgPRNp¸\q`bMPg^W^rR*Ùmo`bR*c_

^hmvfy^6ªLNMPgj^NuPfbmvPR*Q x*poc!¥R^hmvvR4w(p8^pQ#gcPgQ#g/.6po`hgmvcuPfhm8PR*Q uPfbm&]gw_R4wRpofbRpvPR`bm#wPR¬cPR

pWxm8cxR6u_`mvw_gj^a`ypocx*R2R`aR*R6c`hR6c^hmvfy^*ª®mvf`hMPgj^RMp¸8R@`hmfhR4x*pv]^rm8Q#Rw_R*¬cPg`hgmvc^poc¥wfhR4^rd©`y^*ª

®gfb^r`6WR*`!d^xm8c^rgjw_R6f(`bMPRw_R*¬cPg`hgmvc mo`hMR8R*m_w_R6^hgjxFw_gjâ`ypocx*Rµ2RàR*R*c àm^h\]Q#QR*`hfbgx

u2m8^hg`hgvRw_R*¬cPg`hR`bR*c^hmvfy^6ª·¢c`bMPR!^hR6i;dPR6HRgwPR*cPmv`hR

Sym(n) `bMPR!^hR`mvt^r\]Q#Q#R`hfbgjxqfhR4po

nQqp&`bfhgjxR4^pvcw

Sym⁺∗(n)_`hMR^rR*`Nmv@Z_k ¡¯fbR6pv

n Qqp&`bfhgjxR4^0

½ ²Á

LNMRgc]&pofbgpvc8`tw_gj^r`bpoc¥xR¥R*`aR6R*c

M, N ∈Sym⁺∗(n)gj^tw_R*¬cPR4w(]\0

d(M, N) =dist(Id, M⁻^1/2N M⁻^1/2) =N(M⁻^1/2N M⁻^1/2)

pvcw

N(M)) =tr(log(D)²).

(13)

MR*fbR¡¯g^`bMPRR*gvR6c;&pvdPRQqp&`bfhg»!p8^h^hm_xgjp&`hR4wq`bm

M^ª

ÄUcPmo`bMPR*fNp¸\(`bmw_R*`hR6fhQ#gcPR`bMPRgc]&pofbgpvc8`w_gj^a`ypocx*Rgj^U`bMPfbmvdP8M.`hMPR#sgR*QqpocPcgpvcµQ#R`bfhgjxm8c

`bMPR^hupvx*RmvZ]k¡z`hR6c^hmvfy^*ª

½ ²Á

g W1

pvcw

W2

pvfhRW`am`ypocP8R*c;`vR6x`hm8fb^Np&`

M_Rw_R¬¥cPR`hMPR^bx*pvpvfNuPfhm_w_d¥x¦`tp&`

M ^]\0 (W1, W2)M = (M^−1/2W1M^−1/2, M^−1/2W2M^−1/2) =T r(M^−1/2W1M⁻¹W2M^−1/2)

½ ²Á

LNMRstgR6Q#pvcPcPgjpoc.R»_u2mvcPR6c;`hgjpo¹Qqpou.pv^b^rm_x*gpo`hR6w`hmpvc!gc]¸pvfhgjpoc;`NQ#R`hfbgjxg^w_R*¬cPR4w!]\0

expM(W) =M^1/2exp(M^−1/2W M^−1/2)M^1/2

`bMPgj^tw_g-'2R6mvQ#mvfbuPMPgj^rQ'g^N8m8poIPpocw!Rx6pocdPcPgji;dPR*\!w_R*¬cPR`bMPRgc]8R*fy^rRQqpouPuPgcP#]\0

logM(W) =M^1/2log(M⁻^1/2W M⁻^1/2)M^1/2

* ½ Á ¸ ½ P _ HÁ " Á °

°g`hMµ`hMPRuPfhR6]gm8d^r\.w_R¬cR6wFx*mvcx*R*u_`mo@vR6m]wPR6^hgxwPg^r`bpvcxR¥R*`aR6R*cZ]k¡ `bR*c^hmvfy^6Rx*pvc

wPR¬cPR`bMPRcPm_wPpo¹Q#R`bfhgjx

Mi

pv^`hMRgc;`hfbgc¥^rgjxUQ#R4pocmo`bMPR^rR*`moQ#R`bfhgjx*^

{MT;T∈ T(i)} ^* ^|,Iª

ERNfhR4x*pv;`hM¥p&``hMPRgc;`hfbgc¥^rgjx Q#R4pocmo2pWfypocw_m8Qz&pofbgpvPRgcpvcpvfhPg`hfypofb\Q#R`hfbgjx^hup8xRg^@w_R*¬cPR4w

gc#`hMRU^rR6c^hRtmo«®Pfbe6xyMR` pv^l`hMPRUu¥m8gc;`l`bMp&` Q#gcPgQg/.*R4^@`bMPRUR*»]u2R6x`hR4w#¸pvdRtmoC`hMRU^rdQ mo´ ¢^bi8d¥pofbR6w

wPg^r`bpvcxR(§dPcx¦`bgm8c 0!Ãm8c^rgjw_R6fp^rR*`moWu¥m8gc;`

A = {x1,· · ·, xn} mvcpµsgR*QqpoccPgpvc^hupvx*R

E^ª

LNMR*c`bMPRgc;`bfhgc^hgxWQ#R6poc(mou2mvgc;`b^gc

Ag^Nw_R*¬cPR4w(pv^NpQ#gcPgQdQ mo«`bMPRW§dPcx¦`bgm8c

jA

]`hM¥p&`Ng^6

µ=pofbvQ#gc

x∈EjA(x) ^± ^³

MR*fbR

jA(x) = 1 2n

n

X

i=1

d(µ, xi)²

pvcw

dg^`hMPRvR6m_w_R6^hgxw_gj^r`bpoc¥xRmvc

E^ª

LNMPRuPfbmvu2R*fh`hgR6^mv`hMRgc;`hfbgc¥^rgjxUQ#R4pocMp¸vRW¥R6R*c.^a`bdw_gR6w!;\ pofyxyMPR*f* ,Iª Z_gcx*R`hMPRQ#R6pvc

gj^NvgvR6c(]\q`bMPRQ#gcPgQ#g-.4p&`bgm8c!uPfbmvR6QY± ³¦PRQd^r`vR6fhg§\q`hM¥p&`U^hdxyM!pQ#gcgQdPQ'R*»]gj^r`b^poc¥w

gj^dPcPgji;dPRvªpvfbxyMPR6f^hMPm&t^`bMp&`@§m8flpWQqpvcPg©§m8wg`hM#cPm8c_ Iu2m8^hg`hgvR^rR4x¦`bgm8cpo_xdfh&p&`bdPfbR`bMPRQ#R6pvc

gj^dPcPgji;dPR*\qw_R¬cR6w¹ª@·¢c(²pvx¦`4;`hMPRZ]k ¡z^hup8xRWwPm;R4^ M¥p¸vRUcmvc_ u¥m;^rg`hgvR^rR4x¦`bgm8cpoCxdPfb&p&`bdPfhR8;pvcw¹

(14)

`bM]d^*`hMRQ#R6poc,gj^dPcgi;dPR6\w_R¬¥cPR6w¹ª.·¢c * 6,Np.vfypvw_gR*c;`w_R4^hx*R*c;`povm8fhg`hMQ¶`hmµx*mvQ#uPd_`bRq`hMg^

Q#R4pocg^8g8R*c«ª@LNMPR8fbp8w_gR6c;`Nmv

jA

gj^NvgvR6c!]\

∇jA(x) =−1 n

X

i=1,n

Logx(xi)

pvcw.`bM;d¥^U`bMPR#gc;`bfhgc^hgxQR4pocmop!^rR*`mvl`hR6c^rm8fb^x*pvcF2Rx*mvQ#uPd_`bR6w;\`bMPR§m8m&gcP(8fbp8w_gR6c;`

wPR6^bxR*c;`tpovm8fhg`hMQ 0

"

Á « 6 ! ½ Á,+½ Á &

·¢cPuPd_` 0

M1,· · ·, Mn∈SP D ^{^hupvx*R}

Wd_`huPdP`0

R∈SP D ^hupvx*Ro_`bMPRgc;`bfhgc^hgxQ#R6pvc

R0=M1

¡Wm

Xi=_n¹P

k=1,nLogR_i(Mk) Ri+1=ExpRi(Xi)

°MgR

(Xi, Xi)Ri>

½ Á "²_ ²Á Á

¿ #"Â ½ P ½

WcRWmv«`hMPRw_gºxdP`hgR6^Ngc!w_R4pogcg`hM!QR*`hfbgx6^g^`hM¥p&``hMPgj^^ru¥pvxRWg^cPmv`NpvR4x¦`bmvfN^rup8xRU§mvf`hMPR

d¥^rdpv_Qqp&`hfbg»pocw#^bx*pvpvfQdP`hguPgjx*po`hgmvc#mvu2R*fyp&`bgm8c^6ªLNMPRuPdfhu2m8^hRNmo2`hMPgj^l^hR6x¦`bgm8cgj^`bMPRd^hRmo2p

Sym⁺∗(n)`hMpo`Ng28g8RU`bm`bMPgj^^rup8xRp8R6x¦`bmvf

^hup8xR ^r`hfbdx¦`bdPfbR * 6, ªLNMPR4^rRmvu2R*fyp&`bgm8c^¹`bMp&`vR*cR*fypog-.6R@`hMPRd^hdpo;Qqp&`hfbg»pvcw^bx*pvpvf«QdP`hguPgx6p&

`bgm8c^@pofbRNgc;`bR*fbR6^r`hgcPgc`hR6fhQ mo2`hMPR6gf@Qqpo`hMPR6Q#po`hgjx*^@ufhm8u¥R6fr`bgR4^@^rdxyM#pv^`bMPRU^hgQ#gpvfhg`a\; Igc]&pofbgpvc8`4

½ ²Á «R*`

M1, M2

g^@w_R*¬cPR4w

p8^§mvm&0

M1M2:=exp(log(M1) +log(M2)) ^± ^v³

⊗gj^tw_R*¬cPR6w]\0

λ⊗M1:=exp(λ.log(M1)) =M₁^λ ^± ^}8³

* Á Á ²Á

`bMPR8fhm8dPu.gc]vR6fb^hRmolpvcµZ]k ¡ Q#po`hfbg©»g^Ug©`y^Ugc]vR6fb^hRgc.`bMPRQqp&`bfhg»^hR*c¥^rR8ªNVFmvfbR*m&vR6f6_MR*cPR6vR*f

`amZ]k ¡:Q#po`hfbgx*R6^lx*mvQ#QdP`hRNgc#`bMPRQqp&`bfhg»^hR*c^hRvo`hMPR6c#`hMPRtmv;pofbg©`bMPQ#gxQd©`bguPgjx*p&`bgm8c#g^@R4i;dpo