1Solution of Singular ProblemsUsing h-p CloudsJ. Tinsley Oden1 and C. Armando Duarte2Texas Institute for Computational and Applied MathematicsThe University of Texas at AustinTaylor Hall 2.400Austin, Texas, 78712, U.S.A.1Professor, Director, TICAM. e-mail: oden@ticam.utexas.edu2Research Assistant, TICAM. e-mail: armando@ticam.utexas.edu1.1 INTRODUCTIONThe h-p cloud technique [DO, DO95a] is a generalization of a family of so-calledmeshless methods (see, e.g., [BKOF96, LCJ+, Dua95] for an overview) proposed inrecent months that provide both h and p (spectral) type approximations of boundary-value problems while freeing the analyst from traditional diculties due to meshconnectivities. In these methods, the bounded domain of the solution of an ellipticboundary-value problem is covered by the union of a collection of open sets (theclouds) over which spectral-type approximation can be constructed. The mathematicalfoundation of these techniques applied to linear elliptic problems is discussed in [DO].A-posteriori estimates and adaptive h-p clouds are developed in [DO95a].In the present investigation, the generalization of h-p clouds to problems withsingularities is presented. The theory of h-p clouds is reviewed following thisintroduction and adaptive h-p cloud techniques are outlined as well. Then theapplication of h-p clouds to problems with singularities is developed, particularattention being given to the calculation of stress intensity factors in linear fracturemechanics. A new scheme is presented for computing very accurate approximationsof stress intensity factors. Several numerical examples are presented to support thetheoretical developments.

SOLUTION OF SINGULAR PROBLEMS USING H-P CLOUDS 21.2 H-P CLOUD APPROXIMATIONSThe fundamental idea in the h-p cloud method is to use a partition of unity to constructa hierarchy of functions that can represent polynomials of any degree. This familyof functions, called FpN , has many interesting properties like compact support andthe ability to reproduce, through linear combinations, polynomials of any degree.In addition, the functions FpN can be built with any degree of regularity. Anotherremarkable feature of the functions FpN is that there is no need to partition the domaininto smaller subdomains, e.g., nite elements, to construct these functions|all thatis required is an arbitrarily placed set of nodes in the domain . The construction ofthe functions FpN , also known as h-p cloud functions, is described in the next section.1.2.1 CONSTRUCTION OF A PARTITION OF UNITYLet be an open bounded domain in IRn, n = 1; 2 or 3 and QN denote an arbitrarilychosen set of N points x 2 referred to as nodes:QN = fx1;x2; : : : ;xNg; x 2 Let TN := f!gN=1 denote a nite open covering of consisting of N clouds !with centers at x; = 1; : : : ; N and having the following property N[=1 ! (1.1)(a) Open covering build usingcircles. (b) Open covering build usingcircles, ellipses and rectangles.Figure 1 Examples of open coverings.A cloud ! can be almost anything. In two dimensions, for example, it can be arectangle, an ellipse or a circle. Figure 1 shows examples of valid open coverings and

3associated clouds. In this paper we shall restrict ourselves to the case where a cloud! is dened by ! := fy 2 IRnj kx ykIRn < hg (1.2)This corresponds to the case shown in Fig. 1(a).A class of functions SN := f'gN=1 is called a partition of unity subordinated tothe open covering TN if it possesses the following properties [OR76]:1) ' 2 C10 (!); 1 N2) PN=1'(x) = 1; 8 x 2 Note that '(x) may be negative.There is no unique way to build functions ' satisfying the above requirements.Each approach has its own merits and embedded costs. The choice of a particularpartition of unity should be based on the class of problems to be solved, e.g. linear or non-linear problems, the complexity of the geometry of the domain, the regularity required from the approximation, e.g. C0, C1, or higher, the importance of the meshless character of the approximation, etc.The following algorithm is currently used the h-p cloud method to build partitions ofunity.LetW : IRn ! IR denote a weighting function with compact support ! that belongsto the space Cs0(!); s 0 and suppose thatW(x) 0 8 x 2 In the case of the clouds dened in (1.2), the weighting functions W can beimplemented with any degree of regularity using \ridge" functions. More specically,the weighting functions W can be implemented through the compositionW(x) := g(r)where g is, e.g., a B-spline with compact support [1; 1] and r is the functionalr := kx xkIRnhIn the computations presented in Section 1.3, g is a quartic C3() B-spline. Detailson the construction of the B-splines can be found in, e.g., [deB78].The partition of unity functions ' can then be dened by'(x) = W(x)PW(x) (1.3)which are known as Shepard functions [She68]. The main advantages of this particularpartition of unity are low computational cost and simplicity of computation, it is meshless|there is no need to partition the domain to build this partition ofunity, it can easily be implemented in any dimension, it can be constructed with any degree of regularity it allows easy implementation of h adaptivity, as is demonstrated in Section 1.3.1.

SOLUTION OF SINGULAR PROBLEMS USING H-P CLOUDS 41.2.2 THE FAMILY FpNThe construction of the h-p cloud functions is very straightforward after a partitionof unity is derived, such as the one described above, for the domain .Let bLp := nbLioi2I , denote a set of basis functions bLi dened on the unit cloudb! := f 2 IRnj kkIRn < 1gsatisfying bL1() 1 (1.4)Pp(b!) span( bLp)In the above, I denotes an index set and Pp denotes the space of polynomials of degreeless or equal to p.The family of functions FpN is dened byFpN := n'(x)bLi F1 (x) j = 1; : : : ; N ; i 2 Io (1.5)where N is the number of nodes in the domain andF1 : ! ! b! (1.6)F1 (x) := x xhAccording to the above denition, FpN is constructed by multiplying each partitionof unity function ' 2 SN by the elements from the set bLp. One element from thespace of h-p cloud functions can therefore be written asuhp(x) = NX=1Xi2I hai'(x)(bLi F1 (x))iThe following theorems are proved in [DO].Theorem 1 Let Lp := fLigi2I and Li be the same functions bLi but dened on .Then Lp spanfFpNg,Therefore the elements from the set Lp can be recovered through linear combinationsof the h-p clouds functions. This is one of the most fundamental properties of the h-pcloud functions.Theorem 2 Let bLi; i 2 I, be the basis functions from the set bLp and W; =1; : : : ; N be the weighting functions used to construct the partition of unity functions' dened in (1.3). Suppose that bLi; i 2 I 2 C l(b!) and W; = 1; : : : ; N 2 Cq0(!).Then the h-p cloud functions FpN dened in (1.5) belong to the space Cmin(l;q)0 ().Figure 2(a) shows the partition of unity function ' associated with a node x atthe origin. A uniform 5 5 node arrangement and quartic splines are used to build it.Figure 2(b) shows the function xy' from the families Fp2N=25.

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Z(b) 2-D h-p cloud function xy' from thefamilies Fp2N=25.Figure 2 2-D h-p cloud functions.Remark 1 There are many cases in which there is some knowledge about the functionbeing approximated and the requirement that Pp(b!) span( bLp) can be weakenedwithout deteriorating the approximating properties of the set bLp [BM95]. There arealso situations in which the inclusion in the set bLp of specially tailored functions tomodel, e.g., boundary layers, shocks, singularities, etc., can be very advantageous. Thisis the case in the stress analysis of cracks where the quantities of interest are the stressintensity factors. This case is discussed in detail in Section 1.3.2.An a-priori estimateIn this section we summarize some results presented in [DO].Let Xhp be the restriction to (\ !) of the elements from the two parameters (pand h(N )) family of spacesfXhp (!) := spannbLi F1 oi2Iwhere the mapping F1 is dened in (1.6) and bLi 2 bLp.Theorem 3 Let u 2 Hr+1(); r 0. Suppose that the following hold for all localspaces Xhp Pr( \ !) Xhp H1( \ !)Then for xed h and p there is uhp 2Xhp := spanfFpNg such thatku uhpkL2() C1hr+1 kukHr+1() ; (1.7)ku uhpkH1() C2hr kukHr+1() (1.8)

SOLUTION OF SINGULAR PROBLEMS USING H-P CLOUDS 6where the constants C1 and C2 do not depend on h or u but depend on p, andh = max=1;:::;N(h) h.An a-posteriori error estimateLet IR2 be a bounded domain with Lipschitz boundary @. Consider the modelelliptic boundary-value problem of nding the solution u ofu+ cu = f in subject to the boundary conditions@u@n = g on Nu = 0 on DThe variational form of this problem is to nd u 2 VD such thatB(u; v) = L(v) 8 v 2 VDwhere VD is the space VD = fv 2 H1() : v = 0 on Dgand where B(u; v) = Z (ru rv + cuv)dxL(v) = Z fvdx + ZN gvdxSuppose that Xhp VD is a subspace built using h-p clouds. Then the h-p cloud-Galerkin approximation of this problem is to nd uhp 2Xhp such thatB(uhp; vhp) = L(vhp) 8 vhp 2XhpThe following is proved in [DO95a].Theorem 4 Suppose that Xhp (C2() \ VD). Let r denote the interior residualr = f +uhp cuhp in and R denote the boundary residualR = g @uhp@n on NThen the energy norm of the discretization error e = u uhp satiseskekE; 1=2 C X 2!1=2

7where the contributions from the clouds ! are error indicators and are given by2 = h2p2 krk2L2(!\) + hp kRk2L2(@(!\)\N ) (1.9)and 2 IN satises cardfjx 2 !g 8 x 2 The error indicators are used in Section 1.3.1 to implement h, p and h-padaptivity in the h-p cloud method.1.3 NUMERICAL EXAMPLESIn this section, the techniques described in Section 1.2 are used to constructappropriate nite dimensional subspaces of functions used in the Galerkin method.The resulting approach is denoted as the h-p cloud method. Two problems are solvedin this section using the h-p cloud method. The rst problem demonstrates how h, pand h-p adaptivity can be implemented in the h-p cloud method and how the methodcan handle the presence of a singularity. The second problem deals with the stressanalysis of cracks in linear elasticity and the computation of stress intensity factors.We explore the fact that the h-p cloud spaces are able include almost any kind offunction and develop a novel and very ecient approach to extract stress intensityfactors. It is demonstrated numerically that the computed stress intensity factorsconverge at the same rate as the error in energy and, therefore, at twice the rate ofthe error in the energy norm.In all problems solved the following is adopted: The essential boundary conditions are imposed using the method of Lagrangemultipliers. The domain integrations are performed using a background cell structure thatexactly covers the domains. Nonetheless, there is no relationship between thebackground cell structures and the nodes x used in the discretizations. It shouldbe noted that the generation of a background cell structure is much easier than thegeneration of a traditional nite element mesh. The algorithm presented in [DO95a] is used to handle re-entrant corners. In all problems analyzed, the size of the supports of the h-p cloud functions areautomatically set using the algorithm presented in [DO95b, DO].1.3.1 POTENTIAL FLOW IN A L-SHAPED DOMAINIn this section, a potential ow problem in a L-shaped domain is solved using theh-p cloud method. The problem statement is given below. The domain and theboundary segments 1 and 2 are depicted in Fig. 3. The value of u is set to zeroat (1; 1) in order to make the solution unique. This simple boundary-value problemis used to demonstrate how h, p and h-p adaptivity can be implemented in the h-pcloud method without using the concept of a mesh and how adaptivity can be usedin the h-p cloud method to handle the presence of singularities.

SOLUTION OF SINGULAR PROBLEMS USING H-P CLOUDS 8Find u such thatu = 0 in @u@n = 5 on 1 (1.10)@u@n = 5 on 2@u@n = 0 on @n(1 [ 2) Ω

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2Figure 3 Boundary conditions, initialnodal arrangement and background cellstructure used for quadrature.For this problem, the set of functions bLp(b!) dened in (1.4) is composed of thefollowing monomials: bLp(b!) = fij j0 i+ j pgThat is, the family of h-p cloud functions, FpN , is constructed by multiplying thepartition of unity functions '; = 1; : : : ; N , by the smallest set of completepolynomials of degree less or equal to p.The background cell structure used to perform the numerical quadrature isrepresented in Fig. 3. The nodal arrangement used in the rst step of h, p and h-padaptation is also shown in the gure.h adaptivityTheorem 3 demonstrates that the discretization error of the h-p cloud solution canbe controlled by decreasing the radius h of the clouds. This, of course, has to befollowed by the addition of more nodes to the discretization in order to guarantee thata valid covering for the domain still exists. It should be noted, however, that there is noconstraint on how these nodes are placed in the domain|they can simply be insertedinto the regions of interest. This strategy is denoted the h version of the h-p cloudmethod. This approach is demonstrated in the solution of problem (1.10). Details ofthe algorithm used in the h version of the h-p cloud method can be found [DO95a].The nodal arrangement represented in Fig. 3 and clouds with polynomial degree p = 1are used to obtain the rst approximate solution. The error indicators given by (1.9)are then computed for each cloud !. Clouds with errors above a preset value areselected to be rened while the polynomial order is kept xed. The renement ofclouds involves the addition of news nodes around each of the rened clouds. The sizeof the supports ! are then automatically reset to take into account the new nodesadded to the discretization (details of the algorithm can be found in [DO]). A newsolution is then computed using the new discretization and the process is repeateduntil the quality of the solution is deemed acceptable.Figure 4 shows the nodal arrangement and the covering obtained after seven stepsof h adaptation. A high concentration of nodes near the corner at (0; 0) is observed.

9Figure 5 depicts the uxes computed using the discretization of Fig. 4. The colorassociated with each arrow represents the computed potential uh. The ux distributionindicates the presence of a singularity at the re-entrant corner at (0; 0).0

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0.806425Figure 5 Flux distribution.p adaptivityIn the p version of the h-p cloud method, the solution space is enriched by keeping xedthe size h of the clouds and increasing the polynomial order p associated with eachcloud. It is noted that each cloud ! can have a dierent polynomial order associatedwith it, independently of the polynomial order associated with neighboring clouds.Figure 6(a) shows the polynomial orders associated with each cloud after six stepsof p adaptation. The polynomial orders assigned to each cloud is automatically chosenusing the error indicators given by (1.9). The polynomial orders range from p = 1 to

SOLUTION OF SINGULAR PROBLEMS USING H-P CLOUDS 10p = 7. Figure 6(b) represents the potential up computed using the discretization ofFig. 6(a).0

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0.790367(b) Solution obtained using p adaptiv-ity.Figure 6 Results from the p version of the h-p cloud methodh-p adaptivityThe h-p version of the h-p cloud method is implemented by combining the h andp algorithms described previously. Two or more steps of h adaptation are initiallyperformed in order to isolate any singularity present in the solution. This is followedby a number of p steps until the estimated error is below a preset value. In thecase of problem (1.10), two h steps are initially performed, starting from the nodalarrangement of Fig. 3. After that, the error is controlled through p enrichment of theclouds. The nal discretization is presented in Fig. 7. Figure 8(a) shows the computed uxes and potential. The three dimensional plot of Fig. 8(b) presents the computed ux in the y direction using the discretization of Fig. 7. The presence of a singularityat (0; 0) is evident.1.3.2 MODELING OF CORNER SINGULARITIES AND COMPUTATION OFSTRESS INTENSITY FACTORS IN LINEAR FRACTURE MECHANICSIn this section, we demonstrate how corner singularities in plane elasticity problemscan be eciently modeled using the framework of h-p clouds. These singularities occurwhen the solution domain has corners, abrupt changes in boundary data or consistsof two or more materials [OB95]. The h-p cloud results are compared with those ofthe p version of the nite element method. We also present a new approach for theextraction of the amplitude of the singular terms associated with corner singularities.The accuracy and simplicity of this novel approach is demonstrated numerically.

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SOLUTION OF SINGULAR PROBLEMS USING H-P CLOUDS 12Near crack tip expansionIn the neighborhood of a crack, assuming traction-free crack surfaces and in theabsence of body forces, each component of the displacement vector u = fux; uygTcan be written as [SB91, SB88]ux(r; ) = MXj=1 A(1)j u(1)xj (r; ) +A(2)j u(2)xj (r; )+ ~ux(r; ) (1.11)uy(r; ) = MXj=1 A(1)j u(1)yj (r; ) +A(2)j u(2)yj (r; )+ ~uy(r; ) (1.12)where (r; ) are the polar coordinates associated with the crack tip (cf. Fig. 9), ~ux(r; )and ~uy(r; ) are functions smoother than any term in the sum and the eigenfunctionsare given byu(1)xj (r; ) = rj2G n[Q(1)j (j + 1)] cosj j cos (j 2)ou(2)xj (r; ) = rj2G n[Q(2)j (j + 1)] sinj j sin (j 2)ou(1)yj (r; ) = rj2G n[+Q(1)j (j + 1)] sinj + j sin (j 2)ou(2)yj (r; ) = rj2G n[+ Q(2)j (j + 1)] cosj + j cos (j 2)owhere the eigenvalues j are1 = 12 ; j = j + 12 j 2and Q(1)j = 1 j = 3; 5; 7; : : :j j = 1; 2; 4; 6; : : : Q(2)j = 1 j = 1; 2; 4; 6; : : :j j = 3; 5; 7; : : :where j = j 1j + 1and the material constant and G are = 3 4 plane strain31+ plane stress G = E2(1 + )where E is the Young's modulus and is the Poisson's ratio.The coecients A(1)1 and A(2)1 in (1.11) and (1.12) are related to the Mode I andMode II stress intensity factors of linear elasticity fracture mechanics, usually denotedby KI and KII , as follows:A(1)1 = KIp2 A(2)1 = KIIp2

13Enrichment of the h-p cloud spaces and extraction of stress intensityfactorsSuppose that the eigenfunctions u(1)xj ; u(2)xj ; u(1)yj ; u(2)yj ; j = 1; : : : ;M are added to theset bLp dened in Section 1.2.2 givingbLp := nbLioi2I [ nu(1)xj ; u(2)xj ; u(1)yj ; u(2)yj oj=1;:::;M (1.13)Then, the enriched h-p cloud functions are dened as followsFpN := n '(x) hbLi F1 (x)i [ '(x) hu(1)xj (x); u(2)xj (x); u(1)yj (x); u(2)yj (x)ij = 1; : : : ; N ; i 2 I; j = 1; : : : ;M g (1.14)where u(1)xj (x) := u(1)xj T1(x)and T1 is the transformation from rectangular to polar coordinates. The other barquantities are computed similarly and the transformation F1 is dened in (1.6). Notethat all elements from FpN have compact support.The h-p cloud approximation to the displacement eld (1.11),(1.12) can be writtenas uhpx (x) = NX=1Xi2I haix '(x)(bLi F1 (x))i+ NX=1 MXj=1 hb(1)j '(x)u(1)xj (x) + b(2)j'(x)u(2)xj (x)iuhpy (x) = NX=1Xi2I haiy '(x)(bLi F1 (x))i+ NX=1 MXj=1 hb(1)j '(x)u(1)yj (x) + b(2)j'(x)u(2)yj (x)iIn the above expansion, for a xed x 2 , only a few terms are nonzero since allthe functions have compact support. The expansion for, e.g., uhpx can be rewritten asuhpx (x) = NX=1Xi2I haix '(x)(bLi F1 (x))i| z polynomial reproducing part + MXj=1[u(1)xj (x) A(1)jz | NX=1 b(1)j'(x)]| z singular part

SOLUTION OF SINGULAR PROBLEMS USING H-P CLOUDS 14+ MXj=1[u(2)xj (x) A(2)jz | NX=1 b(2)j '(x)]| z singular part (1.15)Comparing (1.15) with (1.11) leads to the following conclusion: the term ~ux can beapproximated very accurately by the polynomial part of uhpx since it is a smoothfunction. Therefore the dierence between ~ux and the polynomial part of uhpx can bemade as small as we want (possibly exponentially) by increasing the polynomial orderassociated with the clouds; consequently we must expect thatNX=1 b(1)j ' p!1! A(1)j and NX=1 b(2)j' p!1! A(2)jTherefore the amplitude of any number of generalized stress intensity factors can beobtained directly from the coecient of the h-p cloud functions without any additionalwork. The rst and second stress intensity factors from linear fracture mechanics aretherefore given by KI = p2A(1)1 = p2 NX=1 b(1)1'KII = p2A(2)1 = p2 NX=1 b(2)1'Remark 2 The case of more complicated boundary conditions at the crack surfacescan be treated exactly as above [OB95]. In the case of anisotropic or nonhomogenousmaterials, the numerical approach proposed by [PB95] can be used to compute theeigenvalues and eigenfunctions of the asymptotic expansion near the crack tip. Theapproach described above can then be used without any modications.Cracked panelIn this section, the edge-cracked panel shown in Fig. 9 is analyzed using the enrichedh-p cloud spaces FpN dened in (1.14). The h-p cloud results are compared with thosepresented by Szabo [Sza86] and Duarte and Barcellos [DdB91] for the p version ofthe nite element method using strongly graded meshes.A state of plane-strain, Poisson's ratio of 0:3 and unity thickness are assumed.The tractions corresponding to the rst symmetric term of the asymptotic expansion(1.11), (1.12) are applied on @. The two components of the displacement at (0; 0)and the vertical component at (d; 0) are set to zero to make the solution unique.The components of the stress tensor associated with the rst term of the asymptoticexpansion are given by [SB91](1)ij = KIp2rf (1)ij () (1.16)

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0.0225dFigure 9 Cracked panel and geometric mesh with three layers of nite elements.Figure 10 h-p cloud discretization for the cracked panel.f (1)11 () = cos21 sin2 sin32 (1.17)f (1)22 () = cos21 + sin2 sin32 (1.18)f (1)12 () = cos2 sin2 cos32 (1.19)where (r; ) is the polar coordinate system shown in Fig. 9.Due to symmetry, only half of the panel is modeled. The nodal arrangement andassociated covering used in the h-p cloud discretization are depicted in Fig. 10. Nospecial nodal arrangement is used near the singularity at (0; 0). Figure 9 shows thenite element mesh used by Szabo [Sza86]. The dimensions of the nite elementsdecreases in geometric progression towards the singularity. Duarte and Barcellos[DdB91] used a mesh with the same geometric progression but with ve layers ofelements instead of the three layers as shown in Fig. 9.The exact strain energy for half of the panel is [Sza86]U (u(1)x1 ; u(1)y1 ) = 0:23706469K2I dEThe values E = KI = d = 1 are adopted in the calculations.In the following, an h-p cloud node is said to be enriched if the set bLp dened in(1.13) is used to build the h-p cloud functions associated with that node. For this

SOLUTION OF SINGULAR PROBLEMS USING H-P CLOUDS 16problem the set bLp has always the following elementsbLp := f1; ; g[ nu(1)xj ; u(2)xj ; u(1)yj ; u(2)yj oj=1;:::;p1That is, the set bLp has linear polynomials and p 1 symmetric and antisymmetricmodes from the near crack tip expansion.Figure 11 shows the convergence in the energy norm of the h-p cloud and niteelement solutions. There are two curves for the h-p cloud method. The solid curvecorresponds to the case in which all six nodes shown in Fig. 10 are enriched nodes.The error associated with this discretization is zero for p 2 because bLp2 containthe exact solution (cf. Theorem 1). The error (of O(106) in strain energy) shown inFig. 11 for p = 2 is due to integration errors.0.001

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Figure 11 Convergence in the energy norm of the h-p cloud and p nite elementsolutions.This problem was constructed so that the exact solution is equal to the rstsymmetric mode all over the domain. In practical problems no such restriction appliesand the expansion (1.11) (1.12) is valid only in a neighborhood r < r0, r0 > 0 [SB88].However, far from the crack tip the solution is smooth and can be approximated wellby polynomials. Therefore in the general case enriched nodes should be used only inthe vicinity of the crack tip. The second curve for the h-p cloud method shown in Fig.11 corresponds to the use of this strategy. More precisely, the curve corresponds tothe use of the discretization shown in Fig. 10 with only one enriched nodethe one atthe crack tip. It can be observed that the h-p cloud solution converges at a very highrate, as if the solution was smooth. The convergence of the p nite element solutionusing graded meshes with three (cf. Fig. 9) and ve layers of elements is also shownin Fig. 11. It can be observed that the h-p cloud discretization requires much fewer

17degree of freedoms than the nite element counterpart. An additional advantage ofusing special functions, made possible by the partition of unity methods framework, isthat the stress intensity factors can be obtained, without using elaborate extractionsprocesses, via the approach described in Section 1.3.2.-0.2

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(b) Convergence of the strain energyand the Mode 1 stress-intensity factorextracted from h-p cloud solutions.Figure 12 Convergence of stress intensity factors.Figure 12(a) shows the values of KI and KII obtained using the h-p clouddiscretization with only one enriched node and the technique described in Section1.3.2. The relative errors in strain energy (not energy norm) and the absolute value ofthe relative error in KI , computed from the coecients of the h-p cloud solution, areplotted against the number of degrees of freedom in Fig. 12(b). The stress intensityfactor converges at about the same rate as the strain energy and therefore much fasterthan the error in the energy norm.Figure 13 shows the computed stress component 22. The h-p cloud solutioncorresponds to the discretization with only one enriched node (at (0; 0)) and p = 5. Theplot of Fig. 13(a) is in the (; r); 0 ; 0:00001 r 1, plane; correspondsto the x direction and r corresponds to the y direction shown in the picture. Theeectiveness of the h-p cloud approximation to capture the singularity at (0; 0) isevident. Indeed Maxj22 hp22 jMaxj22j = 0:000144where the maximumis taken over all points used to plot Fig. 13(a). Figure 13(b) showsthe same results of Fig. 13(a) but this time the plot is done in the (x; y) plane.

SOLUTION OF SINGULAR PROBLEMS USING H-P CLOUDS 180

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0.0413418(b) Plot of 22 in the (x; y) plane.Figure 13 Stress component 22 computed using h-p clouds.1.4 CONCLUSIONSThe h-p cloud method is shown to be an extremely eective tool for handlingsingularities and is particularly eective in computing stress intensity factors inlinear elastic fracture mechanics. Experiments suggests the method is exponentiallyconvergent and superior in performance to comparable h-p nite element methods.Acknowledgment: Preliminary work on this meshless technology was performedunder the support of the Army Research Oce (J. T. Oden) under contract DAAH04-96-0062. The support of the present work by the National Science Foundation undercontract DMI-9561365 and of the CNPq Graduate Fellowship Program (C. A. Duarte)grant # 200498/92-4 is gratefully acknowledged. The authors thank Professor IvoBabuska from the University of Texas at Austin for fruitful discussion during thecourse of this investigation.

References[BKOF96] Belytschko T., Krongauz Y., Organ D., and Fleming M. (1996) Meshlessmethods: An overview and recent developments. To appear in a SpecialIssue of Computer Methods in Applied Mechanics and Engineering onmeshless methods.[BM95] Babuska I. and Melenk J. M. (July 1995) The partition of unity niteelement method. Technical Report BN-1185, Inst. for Phys. Sc. and Tech.,University of Maryland.[DdB91] Duarte C. A. M. and de Barcellos C. S. (December 1991) The elementresidual method for elasticity and potential problems. In de Las CasasE. B. and de Paula F. A. (eds) Symposium on Computational Mechanics,pages 328335. Belo Horizonte, MG, Brazil. In Portuguese.[deB78] deBoor C. (1978) A Practical Guide to Splines. Springer-Verlag, New York.[DO] Duarte C. A. M. and Oden J. T.Hp clouds|an hp meshless method.Numerical Methods for Partial Dierential Equations (to appear).[DO95a] Duarte C. A. M. and Oden J. T. (1995) An hp adaptive method usingclouds. Technical report, TICAM, The University of Texas at Austin. Toappear in Computer Methods in Applied Mechanics and Engineering.[DO95b] Duarte C. A. M. and Oden J. T. (1995) Hp cloudsa meshless methodto solve boundary-value problems. Technical Report 95-05, TICAM, TheUniversity of Texas at Austin.[Dua95] Duarte C. A. M. (1995) A review of some meshless methods to solve partialdierential equations. Technical Report 95-06, TICAM, The University ofTexas at Austin.[LCJ+] Liu W. K., Chen Y., Jun S., Belytschko T., Pan C., Uras R. A., and ChangC. T.Overview and applications of the reproducing kernel particle methods.Archives of Computational Methods in Engineering. State of Art Reviews(to appear).[OB95] Oh H.-S. and Babuska I. (1995) The method of auxiliary mapping forthe nite element solutions of elasticity problems containing singularities.Journal of Computational Physics 121: 193212.

SOLUTION OF SINGULAR PROBLEMS USING H-P CLOUDS 20[OR76] Oden J. T. and Reddy J. N. (1976) An Introduction to the MathematicalTheory of Finite Elements. John Wiley and Sons, New York.[PB95] Papadakis P. J. and Babuska I. (1995) A numerical procedure for thedetermination of certain quantities related to the stress intensity factors intwo-dimensional elasticity. Computer Methods in Applied Mechanics andEngineering 122: 6992.[SB88] Szabo B. A. and Babuska I. (1988) Computation of the amplitude ofstress singular terms for cracks and reentrant corners. In Cruse T. A.(ed) Fracture Mechanics: Nineteenth Symposium, ASTM STP 969, pages101124.[SB91] Szabo B. and Babuska I. (1991) Finite Element Analysis. John Wiley andSons, New York.[She68] Shepard D. (1968) A two-dimensional function for irregularly spaced data.In ACM National Conference, pages 517524.[Sza86] Szabo B. A. (1986) Estimation and control of error based on p convergence.In Babuska I., Zienkiewicz O. C., Gago J., and Oliveira E. R. d. A.(eds) Accuracy Estimates and Adaptive Renements in Finite ElementsComputations. John Wiley and Sons, New York.