Geometric Error Estimation

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1 Geometrc Error Estmaton Houman Borouchak Project-team GAMMA 3 UTT Troyes, France Emal: houman.borouchak@utt.fr Patrck Laug Project-team GAMMA 3 INRIA Pars - Rocquencourt, France Emal: patrck.laug@nra.fr Abstract An essental prerequste for the numercal fnte element smulaton of physcal problems expressed n terms of PDEs s the constructon of an adequate mesh of the doman. Ths frst stage, whch usually nvolves a fully automatc mesh generaton method, s then followed by a computatonal step. One can show that the qualty of the soluton strongly depends on the shape qualty of the mesh of the doman. At the second stage, the numercal soluton obtaned wth the ntal mesh s generally analyzed usng an approprate a posteror error estmator whch, based on the qualty of the soluton, ndcates whether or not the soluton s accurate. The qualty of the soluton s closely related to how well the mesh corresponds to the underlyng physcal phenomenon, whch can be quantfed by the element szes of the mesh. An a posteror error estmaton based on the nterpolaton error dependng on the Hessan of the soluton seems to be well adapted to the purpose of adaptve meshng. In ths paper, we propose a new nterpolaton error estmaton based on the local deformaton of the Cartesan surface representng the soluton. Ths methodology s generally used n the context of surface meshng. In our example, the proposed methodology s appled to mnmze the nterpolaton error on an mage whose grey level s consdered as beng the soluton. eywords-a posteror error estmaton; nterpolaton error; mesh adaptaton; surface curvature. I. INTRODUCTION Dfferent knds of estmators are avalable to a posteror control the error made on a fnte element soluton []. Usng such an estmator, t s possble to control the mesh by h- adaptaton so that the correspondng soluton of the PDE problem has a gven accuracy. Some of these estmators are based on the nterpolaton error and, n ths sense, are purely geometrc snce they gnore the nature of the operator consdered. Ths knd of estmators has been studed by many authors [2] [5]. However, most of these studes le on the fact that a parameter h, representng the sze of the elements, s small or tends to zero, and thereby they are asymptotc studes. The estmator s thus based on approprate Taylor expansons, and gves n ths manner some ndcatons on the admssble sze h. Nevertheless, as ths sze s not necessarly small, we propose a novel approach whch, although closely related to the prevous ones, does not assume any partcular hypothess on ths parameter, and therefore s probably more justfed. Our approach s besdes rather smlar, n ts sprt, to certan solutons used n a dfferent doman, namely the mesh generaton of parametrc patches [6] [8]. Secton 2 gves the mathematcal formulaton of the problem and revews the related works. Secton 3 ntroduces a new class of measures to quantfy the nterpolaton error dependng on the local deformaton or curvature of the Cartesan surface correspondng to the soluton. A numercal example s llustrated n Secton 4 and fnally, the last secton provdes a bref concluson. II. DEFINITION OF THE PROBLEM AND STATE OF THE ART Let Ω be a doman of R d wth d =, 2 or 3 and let T be a smplcal mesh of Ω composed of lnear smplces P or quadratc smplces P 2. We suppose that, n order to solve a problem gven n terms of PDEs on Ω, we have made a fnte element computaton on Ω usng T, and we have obtaned the scalar soluton u T. Denotng by u the exact soluton, the problem frstly conssts n evaluatng the gap e T = u u T between u and u T representng the error nvolved by the fnte element soluton, and secondly deducng n general by boundng ths gap another mesh T such that the estmated gap between u and the soluton u T usng mesh T s bounded by a gven threshold. Several ponts must be more precsely explaned: how to quantfy the gap e T between u and u T? how to use the latter nformaton for buldng a new mesh on whch the gap between the correspondng fnte element soluton and the exact soluton s bounded by a gven threshold? The soluton u T obtaned by the fnte element method s not nterpolatng.e. the soluton obtaned at the nodes of T does not concde wth the exact value of u at these nodes. Moreover, for each element of the mesh, t cannot be guaranteed that the soluton u T concde wth the exact value of u at one pont at least of the element. Then, t seems dffcult to explctly quantfy the gap e T. However, the drect study of ths gap has been dealt n several works [9]. But, n the general case, ts quantfcaton remans an open problem. Consequently, other ndrect approaches have been proposed to quantfy or rather bound ths gap. Let us denote by ũ T the functon nterpolatng u on the mesh T

2 whch s a pecewse lnear or quadratc functon, dependng on the degree of the elements of T and by ẽ T the gap u ũ T between u and ũ T, called the nterpolaton error on u along mesh T. To be able to quantfy the gap e T, we suppose the followng relaton holds Céa s lemma: e T C ẽ T where. denotes a norm and C s a constant not dependng on T. In other words, we suppose that the fnte element error s bounded by the nterpolaton error. The orgnal problem s then smplfed by consderng the followng problem: gven an nterpolaton ũ T of u along a mesh T, how to buld another mesh T for whch the nterpolaton error s bounded by a gven threshold? As ũ T can be seen as a dscrete representaton of u, the problem now reduces to a characterzaton of meshes for whch the nterpolaton error s bounded by ths threshold. Ths problem has been the subject of several studes see for nstance [5] and, n most of them, the examnaton of a measure of the nterpolaton error provdes some constrants assocated wth the mesh elements. In the context of mesh adaptaton methods, h- methods or sze adaptaton are partcularly relevant, and the constrants are specfed n terms of element szes. In the followng, some classcal measures of ths error are recalled, as well as resultng constrants on the mesh elements. To quantfy the nterpolaton error, two knds of measures can be consdered: contnuous or dscrete. A classcal contnuous measure of ths error s the square of the L 2 norm of ẽ T : ẽ T 2 L = ẽ 2 2 T dω = ẽ 2 L 2 T T wth ẽ 2 L = ẽ 2 2 dω, where ẽ s the nterpolaton error on each element of T, and dω s an elementary volume of R d. In two dmensons, consderng lnear elements and assumng that the Hessan H u of u restrcted to the elements s constant, Nadler [0] gves an analytcal expresson of the measure of the nterpolaton error ẽ 2 L on as a functon of 2 the area A of and the quanttes d = 2 at H u a second drectonal dervatves along the edges where a s the vector jonng vertces and + of : ẽ 2 dx dy = A 80 2 d + d 2. Berzns [] extends ths result n three dmensons for lnear elements and shows stll assumng that the Hessan H u of u s constant n element that: 2 e 2 T dx dy dz = V d + d d d 4 d 2 d 5 d 3 d 6 where V s the volume of and quanttes d are smlar to the 2D case. Berzns deduces from ths expresson a measure of the qualty of the elements, and thus characterzes the mesh. However, t s unclear to nterpret ths nformaton n terms of element sze. The extenson of these results to the case of an arbtrary Hessan H u remans open. An alternatve measure, well suted to problem solvng by the fnte element method, conssts n consderng Sobolev norms of ẽ, n partcular the H norm whose square s defned by: ẽ 2 H = ẽ2 + ẽ 2 dω, where represents the gradent and. s the usual Eucldan norm. In two dmensons and consderng lnear elements, Zlamal [2], as also Babuska and Azz [2], ndependently propose an upper bound of ẽ 2 H by the semnorm u 2 of the Sobolev space H 2 whose square s defned by: u 2 2 = 2 2 u x u L x y u 2 L y 2. 2 L 2 Indeed, they show that: ẽ 2 H Γθ u 2, where Γθ s a functon dependng on the dameter of and ts nternal angles. An extenson n three dmensons of ths relaton has been proposed by rzek [3]. Agan, t seems dffcult to establsh a constrant n terms of element sze for ths norm. Another measure, whch s smpler, conssts n consderng the L 2 norm of the gradent of ẽ. It s gven by: ẽ 2 L = ẽ 2 2 dω. An explct expresson of ths error measure related to lnear elements has been proposed by Bank and Smth [4] n two dmensons n the case where the Hessan H u s constant n. An approxmaton of ths expresson s gven by: a 2 d 2 ẽ 2 L A They use ths measure for relocatng the nodes of the mesh n order to mnmze the error. Among the dscrete measures, one can menton the L norm of the nterpolaton error, whch s defned by: ẽ L = max x ẽ x,,

3 where pont x sweeps element. Smlarly, assumng that Hessan H u s constant on each element, Manz et al. [5] propose an approxmaton of the measure ẽ L from an expresson of error e gven by D Azevedo and Smpson [3] for lnear elements n two dmensons: ẽ L δ 6 deth u A 2, where δ = a T H u a, H u beng the absolute value of the Hessan of u. Usng ths approxmaton, they show that f the sze h of along all drectons verfes h T H u h 3 ε then ẽ L ε. Ths sze constrant proves well-suted to h-methods and the results obtaned by the authors show the smplcty and the effcency of ths method. In the context of surface trangulaton by lnear elements, Anglada et al. [7] propose, n the general case where the Hessan of u s arbtrary, an upper bound of ẽ L gven by: ẽ L 2 9 sup pq T H u x pq, x where pont x sweeps element, p s the vertex of such that the barycentrc coordnate of x n wth respect to p s maxmal, and q the ntersecton pont of the straght lne p x wth the edge of opposte to p. They nfer that the nterpolaton error s bounded by a threshold f element les n regons defned and centered at the vertces of. Therefore, these regons can be defned at every ponts of the doman and then consttute constrants for the element szes. Accordng to the above descrpton of dfferent works on the subject although ths lst s far from beng exhaustve, a dscrete measure lnkng error bound and mesh element sze seems more approprate n the scope of error estmaton for mesh adaptaton. The followng secton detals ths ssue. III. A NOVEL APPROACH BASED ON SURFACE GEOMETRY In ths secton, we recall the approach proposed by [6] whch consders soluton u as a Cartesan surface, and we gve a new error estmaton n the case of ansotropc geometrc surface meshng. Let Ω be the computatonal doman, T Ω a mesh of Ω, and uω the physcal soluton obtaned on Ω usng the mesh T Ω. The couple T Ω, uω defnes a Cartesan surface Σ u T. Gven Σ u T, the problem of mnmzng the nterpolaton error conssts n defnng an optmal mesh T opt Ω of Ω for whch surface Σ u T opt would be as smooth as possble. For ths purpose, we propose to locally characterze the surface n the neghborhood of a vertex. Two methods are ntroduced: the frst one, based on local deformaton, can be appled for an sotropc adaptaton whle the second one, based on local curvature, s sutable to an ansotropc adaptaton. A. Local deformaton of a surface The man dea conssts n locally characterzng the devaton of order 0 of a surface mesh Σ u T n the neghborhood of a vertex wth respect to a reference plane, n partcular the tangent plane to the surface at ths vertex. Ths devaton can be quantfed by consderng the Hessan along the normal to the surface.e. the second fundamental form of the surface. Let P be a vertex of the soluton surface Σ u T. Locally, n the neghborhood of P, ths surface admts a parametrc representaton σu, v, u, v beng the parameters, wth P = σ0, 0. The Taylor expanson at order 2 to σ n the neghborhood of P gves: σu, v = σ0, 0 + σ u u + σ v v + 2 σ uu u σ uv u v + σ vv v 2 + ou 2 + v 2 e, where e =,,. If νp denotes the normal to the surface at P, then the quantty νp, σu, v σ0, 0.,. denotng the dot product representng the gap between pont σu, v and the tangent plane at P, expressed by: νp, σ 2 uu u νp, σ uv u v + νp, σ vv v 2 + ou 2 + v 2, s therefore proportonal to the second fundamental form of the surface for u 2 + v 2 small enough. The local deformaton of the surface at P s defned as the maxmum gap between vertces adjacent to P and the tangent plane to the surface at P. If P denotes these vertces, then the local deformaton εp of the surface at P s gven by: εp = max νp, P P. Consequently, the optmal mesh of Ω for Σ u T s a mesh whose sze at each node p s nversely proportonal to εp where P = p, up. More formally, the optmal sze h opt p assocated wth a node p reads: h opt = hp ε εp, where ε denotes the mposed devaton threshold and hp the element sze n the neghborhood of p n mesh T Ω. It can be notced that the local deformaton s a very smple characterzaton of the local devaton of the surface, whch does not requre the explct computaton of the Hessan of the soluton. The only dsadvantage of ths measure s that the resultng adaptve meshes can only be sotropc. In the same context local devaton mnmzaton, the noton of curvature provdes a more precse and ansotropc analyss of ths devaton.

4 B. Local curvature of a surface The analyss of the local geometrc curvature of the surface representng the soluton can be used to mnmze also the devaton of order between the tangent planes of the nterpolatng soluton and those of the exact soluton. Indeed, n the context of sotropc surface mesh generaton, we show [8] that the two devatons of order 0 and of the surface are bounded by a gven threshold f, at any pont of the surface, the sze of the surface elements s proportonal to the mnmal radus of curvature. Let P = p, up be a vertex of Σ u T, let ρ P and ρ 2 P wth ρ P ρ 2 P be the two prncpal rad of curvature at P, and let e P, e 2 P be the two unt vectors n the correspondng prncpal drectons. The deal sze for a surface mesh element at P s [8]: h Σ optp = γ ρ P, where γ s a coeffcent dependng on the mposed devaton threshold. Ths sze s defned n the tangent plane to the surface at P. In the reference system P, e P, e 2 P of ths plane, the deal sze n a gven drecton s a vector v P = h Σ e P + h Σ 2 e 2 P whose components h Σ and satsfy the followng relaton: h Σ 2 h Σ h Σ 2 I 2 γ 2 ρ 2 P h Σ h Σ 2 =. Ths expresson, where I 2 denotes the 2 2 dentty matrx, represents the equaton of a crcle wth center P and radus γ ρ P n the tangent plane to the surface at P. By an orthogonal projecton of ths crcle n the plane of Ω, the sze constrant at p s obtaned. If v p and v 2 p are the respectve orthogonal projectons of e P and e 2 P n the plane of Ω, then ths sze constrant n the reference system p,, j =, 0 and j = 0, s gven by: h h 2 P T I 2 γ 2 ρ 2 P where P = v p v2 p P h h 2 =, and h, h 2 are the coordnates n the reference system p,, j of the projecton of the deal sze vector v P n the plane of Ω. Ths relaton defnes, among others, a metrc generally ansotropc at p. Ths metrc may produce an mportant number of elements owng to the sotropc feature of surface elements. To mnmze ths number of elements, and n the context of ansotropc geometrc surface meshng, we have establshed [7] a relaton whch s smlar to the sotropc case and depends on both prncpal rad of curvature. Now, the deal sze of the surface elements s gven by a metrc, called geometrc, whch can be expressed at a vertex P of Σ u T : h Σ h Σ γ 2 2 ρ 2 P 0 h Σ =, 0 η 2 ρ 2 2 P h Σ 2 where γ = 2 ε 2 ε, η = 2 ε ρ P ρ 2 P 2 ε ρ P ρ 2 P, n whch ε s the prescrbed gap n drecton e P. Ths relaton generally represents an ellpse n the tangent plane to the surface at P whch contaned a crcle n the sotropc case. Agan, by projectng ths ellpse n the plane of Ω, the correspondng metrc at p n ths plane s obtaned. Ths measures also provde a means to control the nterpolaton error n H norm boundng the error on the soluton but also on ts dervatves, and thereby seams more adequate compared to an sotropc measure. In practce, to compute the local curvature, several steps are necessary. Frst, at each vertex of the surface mesh, the normal hence the gradent s determned by a weghted average of unt normals to the adjacent elements. Then, n the local reference system composed of the tangent plane and the normal assocated wth each vertex, a quadrc centered at ths vertex and approachng at best the adjacent vertces s bult. Afterwards, the Hessan s locally approxmated by the Hessan to ths quadrc. nowng the gradent and the Hessan of the soluton at the nodes of T Ω, the curvatures and prncpal drectons at each vertex of surface Σ u T are obtaned. IV. NUMERICAL EXAMPLE To llustrate the proposed method, we consder an mage of pxels and the feld of ts grey levels. Fgure shows the orgnal color mage, a reproducton of The Adoraton of the Mag crca 500. Its author was the North Italan Renassance panter Andrea Mantegna, whose early career was shaped by mpressons of Florentne works. The mage s frstly represented by a regular grd of quadrlaterals, defnng ts ntal mesh. The analyss of the local geometrc curvature of the Cartesan surface representng the feld leads to the determnaton of an ansotropc geometrc sze map assocated wth the ntal mesh, n order to bound the nterpolaton error here ε = 0.. Fgure 2 general vew and 3 close-up show the adapted ansotropc mesh. Ths mesh contans 227,557 vertces and 453,265 trangles. It has been realzed usng the ansotropc adaptve mesh generator BL2D [8]. The resultng nterpolaton error s n average. V. CONCLUSION A novel approach connectng the problem of a posteror error estmaton and some technques of surface meshng has been ntroduced. It consttutes an alternatve method to classcal approaches usng the Hessan of the soluton.

5 Fgure. Orgnal color mage pantng by Andrea Mantegna, crca 500. To llustrate our methodology, a numercal example has been presented. The proposed a posteror error estmaton can be used n any computatonal problem where a statc feld must be calculated. In the case of dynamc felds, the adaptve computaton s consttuted by a calculaton loop: at each teraton, begnnng at the same global ntal tme and endng at a dfferent tme, a combnaton of the current metrc and the prevous metrcs s appled. REFERENCES [] M. Fortn, Estmaton a posteror et adaptaton de mallages, Rev. Europ. des Éléments Fns, vol. 9, n o 4, [2] I. Babuska and. Azz, On the angle condton n the fnte element method, Sam J. Numer. Anal., vol. 3, n o 2, 976, pp [3] E.F. D Azevedo and B. Smpson, On optmal trangular meshes for mnmzng the gradent error, Numer. Math., vol. 59, 99, pp [4] S. Rppa, Long and thn trangles can be good for lnear nterpolaton, Sam J. Numer. Anal., vol. 29, n o, 992, pp [5] M. Berzns, Mesh Qualty : a Functon of Geometry, Error Estmates or Both?, Eng. wth Comp., vol. 5, 999, pp [6] X. Sheng and B.E. Hrsch, Trangulaton of trmmed surfaces n parametrc space, Comp. Ad. Des., vol. 24, n o 8, 992, pp [7] M.V. Anglada, N.P. Garca, and P.B. Crosa, Drectonal adaptve surface trangulaton, Comp. Ad. Des., vol. 6, 999, pp [8] H. Borouchak, P. Laug, and P.L. George, Parametrc Surface Meshng Usng a Combned Advancng-Front Generalzed-Delaunay Approach, Int. Journal for Numercal Methods n Engneerng, vol. 49, 2000, pp [9] R. Verfürth, A Revew of A Posteror Error Estmaton and Adaptve Mesh-Refnement Technques, Wley & Teubner, 996. [0] E.J. Nadler, Pecewse lnear best l 2 approxmaton on trangles, Approxmaton Theory V : Proc. Ffth Inter. Symposum on Approx. Theory, Academc Press, New York, 986, pp [] M. Berzns, Soluton-based Mesh Qualty for Trangular and Tetrahedral Meshes, Proc. 6th Internatonal Meshng Roundtable, Sanda Lab., 997, pp [2] M. Zlamal, On the fnte element method, Numer. Math., vol. 2, pp , 968. [3] M. rzek, On the maxmum angle condton for lnear tetrahedral elements, Sam J. Numer. Anal., vol. 29, n o 2, 992, pp [4] R.E. Bank and R.. Smth, Mesh smoothng usng a posteror error estmates, Sam J. Numer. Anal., vol. 34, n o 3, 997, pp [5] C. Manz, F. Rapett, and L. Formagga, Functon approxmaton on trangular grds : some numercal results usng adaptve technques, Appl. Numer. Math., vol. 32, n o 4, 2000, pp [6] P.J. Frey and H. Borouchak, Surface meshng usng a geometrc error estmate, Int. Journal for Numercal Methods n Engneerng, vol. 58, 2003, pp [7] P. Laug and H. Borouchak, Interpolatng and Meshng 3-D Surface Grds, Int. Journal for Numercal Methods n Engneerng, vol. 58, 2003, pp [8] P. Laug and H. Borouchak, The BL2D Mesh Generator Begnner s Gude, User s and Programmer s Manual, INRIA Techncal Report RT-094, July 996.

6 Fgure 2. Adapted mesh correspondng to the nterpolaton error ε = 0.. Fgure 3. Enlargement of a selected regon.

2x x l. Module 3: Element Properties Lecture 4: Lagrange and Serendipity Elements

2x x l. Module 3: Element Properties Lecture 4: Lagrange and Serendipity Elements Module 3: Element Propertes Lecture : Lagrange and Serendpty Elements 5 In last lecture note, the nterpolaton functons are derved on the bass of assumed polynomal from Pascal s trangle for the fled varable.