Convergent meshfree approximation schemes of arbitrary order and ...

Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103Contents lists available at SciVerse ScienceDirectComput. Methods Appl. Mech. Engrg.journal homepage: www.elsevier.com/locate/cmaConvergent meshfree approximation schemes of arbitrary order and smoothnessA. Bompadre a , L.E. Perotti a , C.J. Cyron b , M. Ortiz a,⇑a Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125, United Statesb Institute for Computational Mechanics, Technische Universität München, 85748 Garching, GermanyarticleinfoabstractArticle history:Received 3 August 2011Received in revised form 25 November 2011Accepted 31 January 2012Available online 15 February 2012Keywords:Meshfree interpolationConvergence analysisHigh-order interpolationSmooth interpolationLocal Maximum-Entropy (LME) approximation schemes are meshfree approximation schemes that satisfyconsistency conditions of order one, i.e., they approximate affine functions exactly. In addition,LME approximation schemes converge in the Sobolev space W 1;p , i.e., they are C 0 -continuous in the conventionalterminology of finite-element interpolation. Here we present a generalization of the Local Max-Ent approximation schemes that are consistent to arbitrary order, i.e., interpolate polynomials of arbitrarydegree exactly, and which converge in W k;p , i.e., they are C k -continuous to arbitrary order k. Werefer to these approximation schemes as High Order Local Maximum-Entropy Approximation Schemes(HOLMES). We prove uniform error bounds for the HOLMES approximates and their derivatives up toorder k. Moreover, we show that the HOLMES of order k is dense in the Sobolev space W k;p , for any1 6 p < 1. The good performance of HOLMES relative to other meshfree schemes in selected test casesis also critically appraised.Ó 2012 Elsevier B.V. All rights reserved.1. IntroductionThe Local Maximum-Entropy (LME) approximation scheme is ameshfree approximation scheme introduced in [2] (see also [3,29]).The LME approximation scheme combines, by means of a singleparameter, shape functions of least width and maximum-entropy,in the sense of statistical inference. The LME approximation schemeis convex, in the sense that the shape functions are non-negative,and satisfy consistency constrains up to order one. In particular,they approximate affine functions exactly and satisfy a weak Kroneckerdelta condition, which simplifies the imposition of boundaryconditions. Maximum-entropy approximates defined using adifferent entropy measure were also developed in [27,28].The LME approximation scheme can be used to solve PDEsnumerically, in the style of meshfree Galerkin methods, see, e.g.,[5,14]. This was done, in particular, in [2,22], where the LMEapproximation scheme was shown to give better results than finiteelement schemes. An optimal transportation meshfree (OTM)method for simulating general solid and fluid flows that builds onthe LME approximation scheme was developed in [17]. An extensionof LME to thin shells was presented in [20]. Related approximationmethods include the Moving Least Squares (MLS) methods[21,6,18], and radial basis functions [31]. A study of the convergenceproperties of the LME approximation scheme was carried⇑ Corresponding author.E-mail addresses: abompadr@alum.mit.edu (A. Bompadre), luigiemp@caltech.edu (L.E. Perotti), cyron@lnm.mw.tum.de (C.J. Cyron), ortiz@aero.caltech.edu (M.Ortiz).out in [7]. In that paper, it was shown that the error in the LMEapproximates is quadratic in h, a measure of the density of the pointset on the domain. Moreover, the error in the derivatives of the LMEapproximates is linear in h. As a result, the LME approximates aredense in W 1;q , for any 1 6 q < 1. In the conventional terminologyof finite-element interpolation, LME approximation schemes are‘C 0 -continuous’ and, as such, can only be expected to convergewhen applied to the solution of second-order problems.A natural generalization of the LME approximation schemes isto append ever higher-order consistency constraints. However, asalready pointed out in [2], simply adding second-order LME constraintsto the current set of constraints may lead to an infeasiblemethod. Recently, this difficulty was overcome in [10,13] by relaxingsome of the LME constraints.In this paper, we formulate a generalization of the LME approximationscheme to high-order consistency, which we refer to as the High-Order Local Maximum-Entropy Approximation Scheme (HOLMES).The shape functions of the HOLMES of order n satisfy consistency constraintsup to order n, i.e., they interpolate exactly all polynomials ofdegree less or equal to n. In order to guarantee feasibility, the nonnegativityconstraint on the shape functions is eliminated. A drawback ofnot enforcing non-negativity is that, unlike LME, HOLMES does notsatisfy a weak Kronecker delta property on the boundary of the domain.As a result, the imposition of boundary conditions for HOLMESis somewhat more involved than in the LME case. However, imposingessential boundary conditions can be accomplished by recourse tostandard Lagrangian multipliers, as in [14,9].We analyze the theoretical convergence of the HOLMES approximatesof order n. We prove that, for every k 6 n, the error in the0045-7825/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.cma.2012.01.020

A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103 85For the HOLMES defined in Section 4, we assume the correspondingpoint set P to be an h-covering of order n with constantj, and to have h-density bounded by s. In particular, we areinterested on sequences P k with the following properties.Definition 3 (Strongly regular sequence of point sets). Let X R d bea domain. Let P k be a sequence of point sets of X. We say that P k is astrongly regular sequence of point sets of order n if there is a sequenceof positive real numbers h k ! 0 and parameters j; s > 0 such that(i) For every k; P k is an h k -covering of X of order n and constantj.(ii) For every k, the node set P k has h k -density bounded by s.3. Approximation schemesLet X R d be a domain. By an approximation scheme fI; W; Pgwe mean a collection W ¼fw a ; a 2 Ig of shape functions and apoint set P, both indexed by a finite set I. Given an approximationscheme fI; W; Pg, we approximate functions u : X ! R byfunctions in the span X of W of the formu I ðxÞ ¼ X a2Iuðx a Þw a ðxÞ;provided that this operation is well defined. More generally, weshall consider sequences of approximation schemes fI k ; W k ; P k gand letu k ðxÞ ¼ X a2I kuðx a Þw a ðxÞ;be the corresponding sequence of approximations to u in the sequenceX k of finite-dimensional spaces of functions spanned by W k .A theoretical study of the convergence of general approximationschemes is carried out in [7]. In that paper, it is shown thatapproximation schemes fI k ; W k ; P k g that1. exactly approximate polynomials of degree less or equal to n,and2. have sufficiently smooth shape functions of sufficiently highpolynomial decay,are guaranteed to be dense in a suitable Sobolev space. Forcompleteness, we next state the relevant definitions and results of[7] that we use in this paper.Definition 4. We say that W is consistent of order n P 0 inXrelative to a point set P if it exactly interpolates polynomials ofdegree less or equal to n, i.e., ifx a ¼ X a2Ix a a w aðxÞeverywhere in X and for all multiindices a of order jaj 6 n.A simple binomial expansion shows that (9) can be equivalentlyreplaced byXw a ðxÞ ¼1;ð10aÞa2IXw a ðxÞðx x a Þ a ¼ 0; 8a 2 N d ; 0 < jaj 6 n; ð10bÞa2Iin the definition of consistency. We call the system of Eqs. (9) or(10) the Consistency Conditions of order n.Definition 5 (Approximation scheme with polynomial decay). Wesay that an approximation scheme fI; W; Pg has a polynomial decayof order ðr; sÞ for constants c > 0 and h > 0 if the basis W is in C r ðXÞ,andð7Þð8Þð9Þ sup sup sup 1 þ x x 2 s a h jaj jD a w a ðxÞj < c: ð11Þjaj6r x2X a2I hA sequence of approximation schemes fI k ; W k ; P k g has a uniformpolynomial decay of order ðr; sÞ if there exists a constant c > 0 anda sequence h k ! 0 such that, for each k, fI k ; W k ; P k g has a polynomialdecay of order ðr; sÞ for constants c and h k .The following error bound holds for nth-order consistentapproximation schemes of sufficiently high polynomial decay.Theorem 1 [7], Uniform interpolation error bound. LetfI k ; W k ; P k g be an approximation scheme in X. Suppose that:(i) The approximation scheme is n-consistent, n P 0.(ii) There exists s > 0 such that P k has h k -density bounded by s forall k.(iii) The approximation scheme has polynomial decay of order ðr; sÞwith 2s > d þ m þ 1, where m ¼ minfn; ‘g and ‘ P 0 is aninteger.Let m 0 ¼ minfn; ‘;rg. Then, there exists a constant C < 1 such thatD a u k ðxÞ D a uðxÞ 6 C D mþ1 u h mþ1 jajk; ð12Þ1for every k; u 2 C ‘þ1 ðconvðXÞÞ, jaj 6 m 0 , and x 2 X.Convergence in Sobolev spaces follows from standard theory ofapproximation by continuous functions (cf. e. g., [1]). By way ofexample, we recall that a domain X satisfies the segment conditionif, for all x in the boundary of X, there exists a neighborhood U x anda direction y x – 0 such that, for any point z 2 X \ U x , the pointz þ ty x belongs to X, for all 0 < t < 1([1], Section 3.21). Convex domainssatisfy the segment condition without additional restrictionson their boundaries. We additionally recall ([1], thm. 3.22)that, if X satisfies the segment condition, then the functions ofC 1 c ðRd Þ restricted to X are dense in W j;q ðXÞ for 1 6 q < 1.Theorem 2 [7]. Let X be a domain satisfying the segment condition.Suppose that the assumptions of Theorem 1 hold and that h k ! 0. Let1 6 q < 1 and j ¼ minfn; rg. Then, for every u 2 W j;q ðXÞ there existsa sequence u k 2 X k such that u k ! u.3.1. Feasibility of the consistency conditions of order nIn this subsection, we analyze conditions on the node set P thatguarantee feasibility of the conditions (10). In particular, we showthat a sufficient condition for feasibility is that P is an h-covering oforder n of X (cf. Def. 1).Proposition 2. Let P be an h-covering of order n of X, for someh; j > 0. Then, for any point x 2 X, there exists a vector wðxÞ 2R jIjthat satisfies the system of consistency conditions (10) of order n.Proof. Let x 2 X be fixed. Let P x P be a set of D n node points sothat the inf–sup condition (1) holds on P x . The inf–sup conditionimplies that the linear system of equationsXh jaj ðx x a Þ a u a ¼ 0; 8x a 2 P x ; ð13Þa2N d :jaj6nhas u ¼ 0 as its unique solution. In particular, since this system hasD n unknowns and D n constraints, the adjoint systemXw a ðxÞ ¼1;x a2P xXw a ðxÞðx x a Þ a ð14Þ¼ 0; 8a 2 N d ; 0 < jaj 6 n;x a2P x

86 A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103has a unique solution fw a ðxÞ : x a 2 P x g. By zeroing out the remainingvalues, the solution fw a ðxÞ : x a 2 P x g can be extended to a solutionof the consistency conditions (10) on P. h4. High-Order Local Maximum-Entropy ApproximationSchemesIn this section we define the High-Order Local Maximum-Entropy Approximation Schemes (HOLMES), and establish someof their properties. We start by reviewing the LME approximationscheme.4.1. Local maximum-entropy approximation schemesThe local maximum entropy (LME) approximation scheme wasintroduced in [2] as a method for approximating functions over aconvex domain. The LME shape functions are nonnegative andsatisfy the consistency conditions (10) of order one. The LMEapproximation scheme is a convex approximation scheme thatendeavors to satisfy two objectives simultaneously:1. Unbiased statistical inference based on the nodal data.2. Shape functions of least width.Since, for each point x, the shape functions of a convex approximationscheme are nonnegative and add up to 1, they can be thoughtof as the probability distribution of a random variable. The statisticalinference of the shape functions is then measured by the entropyof the associated probability distribution, as defined in informationtheory [25,15,16]. Thus, the entropy of a probability distribution pover I is:HðpÞ ¼X a2Ip a log p a ;ð15Þwhere 0 log 0 ¼ 0. The least biased probability distribution p is thatwhich maximizes the entropy. In addition, the width of a non-negativefunction w about a point n is identified with the secondmomentZU n ðwÞ ¼ wðxÞjx nj 2 dx: ð16ÞXThus, the width U n ðwÞ measures how concentrated w is about n.According to this measure, the most local shape functions are theones that minimize the total width [23]UðWÞ ¼ X ZXU a ðw a Þ¼ w a ðxÞjx x a j 2 dx: ð17Þa2IX a2IThe local maximum-entropy approximation schemes combine thefunctionals (15) and (17) into a single objective functional. Moreprecisely, for a parameter b > 0, the LME approximation schemeis the minimizer of the functionalF b ðWÞ ¼bUðWÞ HðWÞ ð18Þunder the restrictions of first-order consistency and nonnegativity ofthe shape functions. We note that the minimizer of (18) can be computedpointwise by means of a convex optimization problem.4.2. Definition of high-order local maximum entropy approximationschemesAs already pointed out in [2], simply adding second-order consistencyconditions to the set of constrains of LME leads to aninfeasible problem. Therefore, in order to extend the LME schemesto higher-order consistency, some of the constrains of LME mustnecessarily to be relaxed. In [10], a second order maximum entropy(SME) approximation scheme was introduced by introducing anonnegative gap function into the second-order consistency constraint.Alternative higher-order extensions of LME have been presentedin [13,29].In this section, we formulate a high-order generalization of theLME approximation scheme, which we name High-Order LocalMaximum-Entropy Approximation Scheme (HOLMES). For an integern P 0, the HOLMES of order n enforces the consistency conditionsup to order n, and jettisons the nonnegativity constraint onthe shape functions. The shape functions w a ðxÞ; a 2 I, can then bewritten as w a ðxÞ ¼w þ a ðxÞ w a ðxÞ, where wþ a ðxÞ and w aðxÞ are nonnegative.We define the following modified entropy measureHðw þ ðxÞ; w ðxÞÞ ¼ w þ a ðxÞ log wþ a ðxÞXw aðxÞ log w aðxÞ:Xa2Ia2Ið19ÞThe total width of w þ ; w with respect to a p-norm is furtherdefined asZU p ðw þ ; w Þ¼Xw þ a ðxÞþw a ðxÞ jx x a j p pdx: ð20ÞXa2INote that U p ðw þ ; w Þ can be minimized pointwise subject to theconsistency constraints. More precisely, the minimizers of U p alsominimize the functionU p ðw þ ðxÞ; w ðxÞÞ ¼ X a2Iw þ a ðxÞþw a ðxÞ jx x a j p p ; ð21Þalmost everywhere. Let c; h > 0 be two parameters. As in LME, theshape functions of HOLMES are chosen to minimize a combinationof the objective functions (19) and (21), namely, for each point x,min f c ðx; w þ ðxÞ; w ðxÞÞ ¼ ch p U p ðw þ ðxÞ; w ðxÞÞHðw þ ðxÞ; w ðxÞÞ;ð22Þsubject to the consistency constrains of order n. An equivalentmathematical formulation of the HOLMES of order n at a point x ismin f c ðx; w þ ðxÞ; w ðxÞÞ ¼ ch p Pw þ a ðxÞþw a ðxÞ jx x a j p 9pþ P w þ a ðxÞ log wþ a ðxÞþP w aðxÞ log w aðxÞ;a2Ia2Isubject to :w þ a ðxÞ; w aðxÞ P 0; a 2 I;Pw þ a ðxÞ w a ðxÞ ¼ 1;a2IPw þ a ðxÞ w a ðxÞ h jaj ðx x a Þ a ¼ 0; 8a 2 N d ; 1 6 jaj 6 n:a2Ia2I>=>;ðHOLMESÞNote that, in this formulation, h is used as a scaling factor in theobjective function as well as in the consistency constrains. Clearly,the problem can be equivalently formulated without h. However,the scaling h conveniently simplifies the analysis of the Lagrangianmultipliers that follows. The HOLMES shape functions are, then,w a ¼ w þ aw a; ð23Þand the corresponding shape function basis is W ¼fw a : w a ¼w þ aw a; a 2 Ig.We note that, when n ¼ 1 and p ¼ 2, the HOLMES approximationscheme is similar to (but not equal to) the LME approximationscheme of [2] of parameter b ¼ c=h 2 . One difference in the formulationof both optimization problems is that HOLMES allows for negativeshape functions. Consequently, the shape functions do notdefine a probability distribution on each point x and Hðw þ ; w Þ isno longer the entropy of a probability distribution. In HOLMES,the term Hðw þ ; w Þ in the objective function penalizes values ofw þ and of w greater than one. Moreover, as in LME, the resultingshape functions decrease exponentially, as shown in the sequel.

A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103 87Next, we discuss the feasibility of (HOLMES). Since the constraintsof (HOLMES) correspond to the consistency constrains(10), (HOLMES) is feasible as long as the node set is an h-coveringof order n of X (see Proposition 2). The following Proposition showsa stronger result, namely, the existence of a solution of (HOLMES)with all of its coordinates being strictly positive. We use this resultsubsequently for proving uniqueness of the optimal dual solution.Proposition 3. Let P ¼fx a : a 2 Ig be a nodal set that is anh-covering of X of order n and constant j > 0. Then, for every pointx 2 X, the problem (HOLMES) is feasible. Moreover, there exist shapefunctions w þ þþ ; w þþ 2 RjIj þþ that satisfy the constraints of (HOLMES)for x.Proof. By Proposition 2, there exists a vector w 2 R jIj that satisfieseq. (10) for point x. The vector w can be decomposed as w ¼w þ w for some nonnegative vectors w þ ; w 2 R jIjP0, and thereforeðw þ ; w Þ is a solution of (HOLMES). Next, we modify ðw þ ; w Þ toobtain a solution ðw þ þþ ; w þþÞ of (HOLMES) with all of its coordinatesbeing strictly positive. Let M :¼ maxf1; jw þ j 1; jw j 1g, andlet M 2 R jIj be the vector with all its coordinates being M. Then,the vectorsw þ þþ ¼ wþ þ 2M;ð24aÞw þþ¼ w þ 2M; ð24bÞhave all their coordinates strictly positive and satisfy (HOLMES).4.3. Dual formulation of HOLMESWe make use of duality theory in convex optimization (cf. [24])to characterize the optimal solution of HOLMES of order n. Forevery multiindex a 2 N d ; jaj 6 n, let k a 2 R be a variable. Inaddition, let k 2 R Dn be the vector containing all the variables k a ,namely, k ¼ðk a Þ a2N d ; jaj6n . For each node point x a 2 P, we definethe functions fn;a þ ; f n;a : Rd R Dn! R as follows.01f þ n;a ðx; kÞ ¼ 1 h Xpcjx x a j p @ph jaj k a ðx x a Þ a A; ð25aÞa2N d ;jaj6n0f n;aðx; kÞ ¼ 1 h p cjx x a j p p þ X@a2N d ;jaj6nh jaj k a ðx1hx a Þ a A: ð25bÞMoreover, we define the functions w þ a ; w a : Rd R Dn ! R ash iw þ a ðx; kÞ ¼exp f þ n;a ðx;kÞ; ð26aÞh iw aðx; kÞ ¼exp f n;aðx; kÞ: ð26bÞIn addition, we define the partition function Z n : R d R Dn ! R,Z n ðx; kÞ ¼k 0 þ X h iexp f þ n;aðx; kÞ þ X h iexp f n;aðx; kÞa2Ia2I¼ k 0 þ X a2Iw þ a ðx; kÞþw a ðx; kÞ :ð27ÞThe optimal solution of HOLMES can then be described in terms ofthe unique minimizer of the partition function, as the nextproposition shows. We observe that this result is similar to thecharacterization of the optimal shape functions of LME given in [2].Proposition 4. Let P be a node set that is an h-covering of order n withconstant j of X. Then, the optimal solution of (HOLMES) for x 2 X isw þ aðxÞ ¼w þ að x; k ðxÞÞ; a 2 I; ð28aÞw aðxÞ ¼w aðx; k ðxÞÞ; a 2 I; ð28bÞwherek ðxÞ ¼argmin Z n ðx; kÞ: ð29Þk2R DnMoreover, the minimizer k ðxÞ is unique.Proof. Let x 2 X be fixed. Let L : R jIjþ RjIj þ ! R be theRDnLagrangian of (HOLMES)!XLðw þ ; w ; kÞ ¼f c ðx; w þ ; w Þþk 0 ðw þ aw aÞ 1þX16jaj6nh jaj k aLet g : R Dn! R be the Lagrangian dual functiongðkÞ ¼infw þ ; w 2R jIjþLðw þ ; w ; kÞ:a2IXðw þ aw aÞðx x a Þ!: a ð30ÞThe Karush–Kuhn–Tucker (KKT) conditions of (HOLMES) are0 ¼ h p cjx x a j p p þ log wþ a þ 1 þ Xh jaj k a ðx x a Þ!; a a 2 I;a2Ijaj6nð31Þð32aÞ0 ¼ h p cjx x a j p p þ log w a þ 1 Xh jaj k a ðx x a Þ!; a a 2 Ijaj6nð32bÞThe KKT conditions can be written ash iw þ a¼ exp f þ n;aðx; kÞ ; a 2 I; ð33aÞh i¼ exp f n;aðx; kÞ ; a 2 I; ð33bÞw awhich are Eqs. (26a) and (26b). hProblem (HOLMES) is a convex optimization problem since it isdefined by a convex objective function and linear constrains.Moreover, by Proposition 3, there exists a solution w þ ðxÞ; w ðxÞ 2R jIjþþ that satisfies the constraints, and thus (HOLMES) satisfiesSlater’s condition [24]. As a result, strong duality holds (cf.Theorem 6.9 in [24]). Therefore, the KKT conditions of (HOLMES)are necessary and sufficient conditions for optimality. In particular,ðw þ ; w Þ and k, feasible solutions of the (HOLMES) problem and itsdual (31), respectively, are optimal solutions of their respectiveproblems if and only if they satisfy (33). Since any vector in R Dnisfeasible for the dual problem, to find an optimal solution of(HOLMES) entails finding k 2 R Dnsuch that the correspondingvectors w þ , w defined by (26) are a feasible solution of (HOLMES).We claim that the partition function (27) has a uniqueminimizer k ðxÞ, and that k ðxÞ is the desired dual vector. Weshow first that Z n is a strictly convex function of k. To this end, letrðx; kÞ @ k Z n ðx; kÞ, namely,Xr 0 ðx; kÞ @ k0 Z n ðx; kÞ ¼1 w þ a ðx; kÞ w a ðx; kÞ ; ð34aÞa2Ir a ðx; kÞ @ ka Z n ðx; kÞ ¼ h X jaj w þ aðx; kÞ w aðx; kÞa2Iðx x a Þ a ; 1 6 jaj 6 n: ð34bÞIn addition, the second derivatives of the partition function withrespect to k are@ ka @ kb Z n ðx; kÞ ¼h X ðjajþjbjÞ ðw þ a ðx; kÞþw aðx; kÞÞðx x aÞ aþb ; ea2I0 6 jaj; jbj 6 n: ð35ÞLet Jðx; kÞ @ k @ k Z n ðx; kÞ be the Hessian matrix of Z n ðx; kÞ. Considernow a vector u 2 R Dn . Then, the following identities hold,

88 A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103u T Jðx; kÞu ¼Xjaj;jbj6n¼ X a2I0@Xh ðjajþjbjÞ u a u b ðw þ a ðx; kÞþw aðx; kÞÞðx xa2Iw þ a ðx; kÞþw Xaðx; kÞjaj6naÞ aþb! 21h jaj u a ðx x a Þ a A:ð36ÞFrom this equation it follows that u T Jðx; kÞu P 0, and therefore thepartition function is convex in k. We next show that Z n is strictlyconvex on k. Note that w þ a ðx; kÞ and w aðx; kÞ are strictly greater thanzero (see Eqs. (26a) and (26b)). In particular, in order for u T Jðx; kÞuto be zero, the following must hold,Xh jaj u a ðx x a Þ a ¼ 0; 8a 2 I: ð37Þjaj6nBecause of the inf–sup condition (1), the system of Eq. (37) has zeroas its unique solution, and therefore Z n is strictly convex in k. Inparticular, Z n has a unique minimizer k ðxÞ.It remains to show that ðw þ ; w Þ, defined in (28), is a feasiblesolution of (HOLMES). We note that k ðxÞ satisfies rðx; k ðxÞÞ 0.Moreover, rðx; k ðxÞÞ 0 corresponds to the consistency conditionsof order n in the associated solution w þ ; w . Therefore, the uniqueminimizer of Z n ðx; kÞ defines an optimal solution of (HOLMES). hThe smoothness of the shape functions depends on the exponentp chosen, as the next proposition shows.Proposition 5. Assume that the assumptions of Proposition 4 hold.Then, the HOLMES shape functions, w þ a and w a ; a 2 I, are C 1 ðXÞ if pis even, and C p 1 ðXÞ if p is odd.Proof. Let r ¼ @ k Z n ðx; kÞ, where Z n is the partition function (27).Since the function jxj p p is C1 if p is even, and C p 1 if p is odd, thefunction r is C 1 if p is even, and C p 1 if p is odd.Let k : X ! R Dnbe the dual variables given by eq. (29). Notethat rðx; k ðxÞÞ 0, and that detð@ k rðx; k ðxÞÞÞ > 0, as shown in theproof of Proposition 4. Therefore, by the implicit function theorem,k is C 1 if p is even, and C p 1 if p is odd. In addition, because of thecharacterization of w þ a and w a by Eq. (28), w þ a and w a have thesame smoothness as k h.In particular, it follows from the preceding proposition that ifp P n þ 1, then the HOLMES shape functions are C n functions.Lemma 1. Let X R d be a bounded set. Let h; s > 0. Let P be apoint set of X that has h-density bounded by s. Let Z n ðx; kÞ be thepartition function (27) defined using parameters h; c > 0, and p > n.Then, there exists a constant c Z that depends on c; s; p, and d, suchthatZ n ðx; 0Þ 6 c Z ;for every x 2 X.ð38ÞProof. For every nonnegative integer t P 1, let U t ðxÞ be the subsetof node points U t ðxÞ ¼fx a 2 P : ðt 1Þh 6 jx a xj p< thg. Note that,by Proposition 1, the number of points in U t ðxÞ is at most ct d , for aconstant c < 1 that depends on s and d. Therefore, Z n ðx; 0Þ satisfiesthe following inequalities,Z n ðx; 0Þ ¼ X exp½f þ n;a ðx; 0ÞŠ þ exp½f n;aðx; 0ÞŠx a2P¼ 2 X e 1 ch p jx x aj p p ¼ 2e1 X1x a2Pt¼1t¼1Xx a2U tðxÞt¼1e ch p jx x aj p p6 2e 1 X1 cðt 1Þp#U t ðxÞe 6 2e 1 X1ct d e cðt 1Þp ¼ c Z :!ð39ÞThe series in (39) is finite. Moreover, because this series is definedin terms of c; s; p, and d, its limit c Z also depends on these parametersonly. hThe following upper bound holds uniformly for the minimizerof the partition function (27).Lemma 2. Let X R d be a bounded set. Let n P 0 be an integer. Leth; j; s > 0. Let P be a point set of X that is an h-covering of X of ordern with constant j, and has h-density bounded by s. Let c > 0. Forevery point x in X, let k ðxÞ 2R Dn be the minimizer of thecorresponding partition function (27), i.e.,k ðxÞ ¼argmin Z n ðx; kÞ: ð40Þk2R DnThen, there exists a constant c k > 0 (that depends on c; s; n; p; d, andj) such that5. Convergence Analysis of High-Order LMEjk ðxÞj 6 c k ;ð41ÞIn Section 5.3, we prove that the HOLMES approximates of ordern are dense in a suitable Sobolev space. To this end, we rely on thesufficient conditions for density of meshfree methods establishedin [7] and summarized in Section 3.Note that, by definition, the HOLMES of order n satisfies theconsistency conditions of order n. Moreover, the shape functionsare C n (Proposition 5). In order for the HOLMES of order n to satisfythe assumptions of Theorems 1 and 2, it remains to show that theshape functions have sufficiently high polynomial decay. Since theshape functions are defined implicitly through a convex optimizationproblem, showing that they have the requisite polynomial decayis not immediate. The next two subsections are devoted tobounding the dual variable k ðxÞ and its derivatives. In particular,the polynomial decay of the HOLMES shape functions is provedin Proposition 13 of Section 5.2.5.1. Bound on the optimal dual variablesIn Lemma 2 we bound the norm of the optimal dual variables of(HOLMES). We start by showing that the partition function (27),evaluated at k ¼ 0, can be bounded uniformly.for every x 2 X.Proof. By Lemma 1, Z n ðx; 0Þ 6 c Z for a constant c Z . Then, anecessary condition for a vector k 2 R Dnto be optimal for x 2 X isthatZ n ðx; kÞ 6 Z n ðx; 0Þ 6 c Z :ð42ÞNote that, for any x, the following inequality holds,h i h ijkjþexp f þ n;aðx; kÞ þ exp f n;aðx; kÞ 6 Z n ðx; kÞ; 8x a 2 P:ð43ÞBy the definition of fn;a þ and of f n;a, the following inequality holds,23exp 1 h p cjx x a j p p þ Xh jaj k a ðx x a Þ a45a2N d ;jaj6nh i h i6 exp f þ n;aðx; kÞ þ exp f n;aðx; kÞ :

A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103 89Combining the last two inequalities, it follows that a necessary conditionfor k to be optimal is that23jkjþexp 1 h p cjx x a j p p þ Xh jaj k a ðx x a Þ a45a2N d ;jaj6n6 c Z ; 8x a 2 P: ð44ÞSince P satisfies the inf–sup condition (1) for constant j, for everypoint x 2 X and every vector k, there exists a node point x k , at distanceat most h of x, such thatXh jaj k a ðx x k Þ a P jjkj:ð45Þa2N d ;jaj6nTherefore, a necessary condition for (44) to hold is thathijkjþexp 1 h p cjx x k j p p þ jjkj c Z 6 0:Since h p jxx k j p p6 1, it follows that jkj must satisfyð46Þjkjþexp ½ 1 c þ jjkjŠ c Z 6 0: ð47ÞSince j > 0, it is readily seen that lim jkj!1 jkjþexp ½ 1 cþjjkjŠ c Z ¼þ1. In particular, there exists a constant c k (that dependson c; s; n; p; d, and j) such that every k satisfying (47) alsoverifies that jkj 6 c k . h5.2. Bounds on the derivatives of the shape functionsIn this subsection, we bound the derivatives of degree less orequal to n of the HOLMES shape functions. This bound issubsequently used to show that the shape functions have polynomialdecay of order ðn; sÞ, where n is the consistency order, and sis arbitrarily large.First, we state a multidimensional chain rule formula (see [19]and references therein). Let F : R m ! R l , and G : R l ! R be twosmooth functions. A multiindex a 2 N m is said to be decomposedinto N parts d 1 ; ...; d N in N m with multiplicities ‘ 1 ; ...;‘ N in N l ,respectively, ifa ¼ XNt¼1j‘ t jd t : ð48ÞLet d ¼ðd 1 ; ...; d N Þ and let ‘ ¼ð‘ 1 ; ...;‘ N Þ. We call ðN; d; ‘Þ a decompositionof a, and we denote by D the set of all decompositions of a.Let ‘ ¼ P Nt¼1 ‘ t in N l be the total multiplicity of ðN; d; ‘Þ. Then, the followingmultidimensional Faà di Bruno’s formula holds [19],D a ðG FÞ ¼XC ðN; d; ‘Þ D ‘ G YN Y l ‘tiD dt F i ; ð49aÞwhereC ðN; d; ‘Þ ¼ a! YNðN; d; ‘Þ2Dt¼11 1‘ t ! d t ! :t¼1 i¼1ð49bÞIn what follows, we use the following notational convention. Givena function Gðx; kÞ : R d R Dn! R m , we denote by G ðxÞ the compositefunction G ðxÞ ¼Gðx; k ðxÞÞ, where k ðxÞ is the optimal dualvector given by (29). A partial derivative of Gðx; kÞ is denoted byD ð‘1 ;‘ 2Þ G i ðx; kÞ, where 1 6 i 6 m, and ‘ 1 2 N d , ‘ 2 2 N Dnare multiindices.We additionally denote by id : R d ! R d the identity function.The following two corollaries of the multidimensional chainrule formula (49) then hold.Corollary 1. Assume that the assumptions of Proposition 4 hold. LetZ n ðx; kÞ be the partition function (27), and let k ðxÞ be its minimizer.Let w þ a ; w a be the shape functions defined in (28). Leta 2 N d ; jaj 6 n be a multiindex. Then, the a-derivative of a shapefunction w a can be expressed asD a w a¼ XðN; d; ‘Þ2DC ðN; d; ‘Þ D ð‘1 ;‘ 2Þ Y Nw aY dt¼1 i¼1 ‘1Y DnD dt tiid ii¼1 ‘2D dt k tii:ð50ÞProof. The proof is an immediate consequence of (49) applied tothe composite function w a K, where w aðx; kÞ is the functiondefined in (26), and K : R d ! R d R Dnis the function KðxÞ ¼ðx; k ðxÞÞ. hCorollary 2. Assume that the assumptions of Proposition 4 hold. LetZ n ðx; kÞ be the partition function (27), and let k ðxÞ be its minimizer.Let r ¼ @Zn . Let @k J ¼@k @r . Let a 2 N d , jaj 6 n be a multiindex. Then,the a-derivative of k can be expressed as01D a Xk ¼ J 1 B@ðN; d; ‘Þ2D;ðj‘ 1 j; j‘ 2 jÞ–ð0; 1ÞC ðN; d; ‘Þ D ð‘1 ;‘ 2Þ Y NrY dt¼1 i¼1 ‘1Y DnD dt tiid ii¼1 ‘2D dt k tii CA :ð51ÞProof. Let r ðxÞ ¼rðx; k ðxÞÞ. Since k ðxÞ is the minimizer of Z n ðx; kÞ,it follows that r ðxÞ 0. By (49) the following identities hold0 D a r ¼XðN; d; ‘Þ2D¼ J D a k þC ðN; d; ‘Þ D ð‘1 ;‘ 2Þ Y NrXðN; d; ‘Þ2D;ðj‘ 1 j;j‘ 2 jÞ–ð0; 1ÞC ðN; d; ‘ÞY dt¼1 i¼1 ‘D dt 1 Y Dntiid i D ð‘1 ;‘ 2Þ Y Nri¼1Y dt¼1 i¼1D dt k i ‘2ti ‘D dt 1 Y Dntiid ii¼1D dt k i ‘ 2ti:ð52ÞAs shown in Proposition 4, the matrix J ¼ @ k r, and in particular J ,isinvertible, whence Eq. (51) follows. hLet fn;a ðx; kÞ : Rd R Dn! R be the functions defined in (25). Werecall that the functions fn;a are Cn if p P n þ 1. Next, we wish tobound D ð‘1 ;‘ 2Þ fn;a ðx; kÞ for multiindices ‘1 in N d ;‘ 2 in N Dn ,j‘ 1 jþj‘ 2 j 6 n. We note that the partial derivative with respect tothe multiindices ‘ 1 2 N d and ‘ 2 2 N Dn of the term ch p jx x a j p p iszero when j‘ 2 j > 0, or when there exist two different indices1 6 i; j 6 d such that ‘ 1 i ;‘1 j> 0. Otherwise,D ð‘1 ;‘ 2 Þch p jx x a j p p¼ Ch ‘1 i jh 1 ðx i x ai Þj p ‘1 i ; ð53Þwhere 1 6 i 6 d is the unique index such that ‘ 1 i> 0, andC ¼ c Q‘1 iðp kÞk¼0ðsgnðx i x ai ÞÞ ‘1 i . Therefore, the following boundson the partial derivatives of fn;a and of w ahold.Proposition 6. Let fn;a ðx; kÞ : Rd R Dn! R be the functions definedin (25). Then, there exists a constant C such thatjD ð‘1 ;‘ 2Þ f n;a ðx; kÞj 6 Cð1 þjkjÞh j‘1 jX pi¼0h i jx x a j i !; ð54Þfor every x 2 R d ; k 2 R Dn , every node point x a , and every multiindex ‘ 1in N d , ‘ 2 in N Dn ; j‘ 1 jþj‘ 2 j 6 n.Proposition 7. Let w a ðx; kÞ : Rd R Dn! R be the functions definedin (26). Then, there exist constants C and M such thatD ð‘1 ;‘ 2Þ w aðx; kÞ 6 Cw a ðx; kÞð1 þjkjÞM h j‘1 jX Mi¼0h i jx x a j i !; ð55Þ

A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103 91every x a 2 P x ; w þ a ðxÞþw a ðxÞ P c w > 0 for some constant c w > 0that depends on c; s; n; p; d, and j. Then,ju T J ðxÞuj P X a2P xP c wXw þaðxÞþw a ðxÞ Xa2P xjaj6nh jaj u a ðx x a Þ a ! 2! 2Xh jaj u a ðx x a Þ a : ð67Þjaj6nBecause of the inf–sup condition (1), it follows thatXa2P x! 2Xh jaj u a ðx x a Þ a P j 2 juj 2 ; ð68Þjaj6nand therefore ju T J ðxÞuj P c J juj 2 for a constant c J ¼ c w j 2 > 0 thatdepends on c; s, n; p; d, and j. hTherefore, the following bound on kJ 1 k holds.Corollary 4. Suppose that the assumptions of Lemma 2 hold. Forevery x 2 X, let J ðxÞ 1 be the inverse of the Hessian of the partitionfunction (36). Then,J ðxÞ 1 6 c 1J; ð69Þfor all x 2 X, where c J > 0 is the constant of Proposition 10.Proposition 11. Let fI k ; W k g be a HOLMES approximation schemeand P k a corresponding family of node sets. Suppose there exists asequence h k ! 0 such that P k is a strongly regular sequence of pointsets of order n. Let k ðxÞ be the optimal dual variables. Leta 2 N d ; jaj 6 n be a multiindex. Then, there exists a constant C suchthatD a k jaj6 Chk; ð70Þfor every x 2 X.Proof. We proceed by induction on jaj. Note that, when jaj ¼0, theresult holds because of the bound of jk j of Lemma 2.Let a 2 N d be a multiindex, jaj > 0, and suppose that theinductive hypothesis holds for every multiindex d 2 N d ; jdj < jaj.By Corollary 2,01jD a X k j 6 kJ 1 kC ðN; d; ‘Þ D ð‘1 ;‘ 2Þ r YN Y dYD dt id i ‘1 Dnti D dt k ‘2 tiBi C@A :ðN; d; ‘Þ2D;ðj‘ 1 j;j‘ 2 jÞ–ð0; 1Þt¼1 i¼1ð71Þ By Corollary 4, kJ 1 k 6 cJ 1 .ByProposition 9, D ð‘1 ;‘ 2Þ r 6 Ch j‘1 jk.Then,01jD a XYk j 6 C 0C ðN; d; ‘Þ h j‘1 jN Y dYkD dt id i ‘1 Dnti D dt k ‘2 tiBi C@t¼1 i¼1i¼1ðN; d; ‘Þ2D;ðj‘ 1 j;j‘ 2 jÞ–ð0; 1Þi¼1A ; ð72Þfor some constant C 0 . Note that, for a decomposition ðN; d; ‘Þ of (72),either jd t j < jaj, or ‘ 2 t¼ 0. Combining this observation with theinductive hypothesis, it follows that, for all the terms of the righthand side of (72),Y Dni¼1D dt k ‘2 tii6 Ch j‘2 t jjdtjk: ð73ÞNote that, a term D dt id i ‘1 tiin (72) is nonzero only if either ‘ 1 ti ¼ 0orjd t j¼1. In either case, j‘ 1 t j¼j‘1 t jjd tj for nonzero terms in (72).Combining this observation with the identityP Nt¼1 ðj‘1 t jþj‘2 t jÞjd tj¼jaj, it follows thatX Nt¼1j‘ 1 t jþj‘2 t jjd tj¼jaj ð74Þholds for every nonzero term of (72). Therefore, a satisfies theinductive hypothesis as well. hProposition 12. Let fI k ; W k g be a HOLMES approximation schemeand P k a corresponding family of node sets. Suppose that there existsa sequence h k ! 0 such that P k is a strongly regular sequence of pointsets of order n. Let a 2 N d ; jaj 6 n be a multiindex. Then, there existconstants C and M such thatD a w jaj6 Chakw aX Mi¼0for every x 2 X, and every x a 2 P k .Proof. By Corollary 1,jD a w a j 6 X C ðN; d; ‘Þ D ð‘1 ;‘ 2Þ w aðN; d; ‘Þ2Dh ik jx x aj i !; ð75Þ YNY dk¼1 i¼1D d kid iY ‘1 Dnkii¼1D d kk ‘2 kii:ð76ÞThen, (75) follows by applying to this equation the bounds onjD ð‘ 1;‘ 2 Þ w a j and of jDa k j of Propositions 7 and 11, respectively. hWe are now in a position to prove the polynomial decay of theHOLMES shape functions.Proposition 13. Let fI k ; W k g be a HOLMES approximation schemeof order n, and P k a corresponding family of node sets. Supposethere exists a sequence h k ! 0 such that P k is a stronglyregular sequence of point sets of order n. Then, for any positiveinteger s, the shape functions fI k ; W k g have polynomial decay of orderðn; sÞ in X.Proof. We have to show that there exists a constant c < 1 suchthat, for any k, shsup sup sup 1 þ h 2kjx x a j 2 jajkD a w þ aðxÞ w aðxÞ jaj6n x2X a2I kBy Proposition 12, there exist constants C and M such thatD a w þ aðxÞ þ Da wa ðxÞ jaj6 Ch ðw þ aðxÞThen, it follows thatsup supjaj6n x2X6 sup supjaj6n x2Xþ w a ðxÞÞ6 sup supjaj6n x2Xkþ w a ðxÞÞX Mi¼0 shsup 1 þ h 2kjx x a j 2 jajkD a w þ aðxÞa2I k shsup 1 þ h 2kjx x a j 2 jajka2I k!X Mh ik jx x aj ii¼0sup C 0 ðw þ aðxÞþw aðxÞÞa2I k 6 c:ð77Þh ik jx x aj i !: ð78Þw aðxÞ jajChkðw þ aðxÞXMþ2si¼0h ik jx x aj i !; ð79Þfor some constant C 0 . Combining this bound with that of Proposition8, the claim of the proposition follows. h

92 A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–1035.3. Convergence of HOLMESCombining the sufficient conditions for convergence of a generalapproximation scheme given in Section 3 with the precedinganalysis of HOLMES, we have the following two theorems.Theorem 3 (Uniform interpolation error bound for HOLMES). LetfI k ; W k g be a HOLMES approximation scheme of order n P 0, and P k acorresponding family of point sets. Suppose that P k is a stronglyregular sequence of order n for a sequence h k . Let ‘ P 0 be an integer.Let m ¼ minfn; ‘g. Then, there exists a constant C < 1 such thatD a u k ðxÞ D a uðxÞ 6 C D mþ1 u h mþ1 jajk; ð80Þ1for every k; u 2 C ‘þ1 ðconvðXÞÞ, jaj 6 m, and x 2 X.Proof. Since HOLMES has polynomial decay ðn; sÞ for s sufficientlyhigh (Proposition 13), the bound of Theorem 1 holds. hTheorem 4. Let X be a domain satisfying the segment condition. Supposethat the assumptions of Theorem 3 hold and that h k ! 0. Let1 6 q < 1. Then, for every u 2 W n;q ðXÞ there exists a sequenceu k 2 X k such that u k ! u.Proof. The proof is a direct consequence of Theorem 3 and of thedensity of approximation schemes of Theorem 2. hWe note that other meshfree methods satisfy similar errorbounds, notably Moving Least Squares (MLS) [12,4], and MovingLeast-Square Reproducing Kernel (MLSRK) methods [18].6. ImplementationIn this section, we focus on the numerical implementation ofHOLMES of order two. However, similar considerations apply toHOLMES of general order. The convexity of the nonlinear optimizationproblem (HOLMES) opens the way for its numerical solutionby means of Newton–Raphson iterations or similar methods. Themain difficulty encountered in practice concerns poor convergencefor bad initial guesses of k a . This poor convergence is especiallyvexing for large values of c and close to the boundary. However,limiting the step size Dk a of the Newton–Raphson iterations tojDk a jK c supplies a simple and reliable remedy.The number of Newton–Raphson iterations required to solvethe nonlinear convex optimization problem depends on differentfactors such as the nodal distribution, the location at which theHOLMES functions need to be evaluated, the value of c, and thenumerical tolerance for the convergence of the Newton–Raphsonmethod. We have investigated the number of required Newton–Raphson iterations for the example shown in Fig. 2 andsubsequently in the examples presented in Section 7.6.Fig. 2 depicts a non-uniform distribution of 40 points inside aunit circle. The convex domain is divided in an inner and outerregion defined in terms of the radius R measured from the centerof the domain. The inner region contains the evaluationpoints for which R < 0:9 and the outer region contains the evaluationpoints for which R P 0:9. An auxiliary Delaunay triangulationis used to determine the points at which to evaluate theHOLMES approximation functions, which we choose as theGauss quadrature points of the auxiliary triangulation. A 9th orderGauss quadrature rule is used in each triangle correspondingto 1091 and 163 evaluation points in the inner and outer regions,respectively. The number of Newton–Raphson iterationsto convergence is observed to increase with an increasing valueof c, with a decreasing convergence tolerance and with proximityto the boundary of the domain. In all cases considered, anaverage of 4 to 7 Newton–Raphson iterations is required forconvergence.Table 1 lists the average number of Newton–Raphson iterationsto convergence encountered in the analyses presented in Section7.6, where the HOLMES shape functions have been evaluatedon a uniform nodal distribution using p ¼ 3 and c ¼ 0:3 orc ¼ 0:8. An average of 5 or 6 iterations is sufficient in all cases. Inaddition, the number of iterations to convergence decreases withincreasing node density since a smaller fraction of the evaluationpoints is close to the domain boundary.In Figs. 3 and 4 second order HOLMES functions as well as theirfirst derivatives are plotted in one and two dimensions. Fig. 5shows the shape functions of boundary nodes with uniform andnon-uniform node set. The strict support of the HOLMES basisfunctions in d dimensions spans the entire space R d . However, inpractice the locality term in (HOLMES) renders their amplituderapidly decreasing with increasing distance from their respectivenodes. By virtue of this decay, the basis functions may be truncatedwithin numerical precision at an l p -distance r p;HOLMES . This rangecan be approximated considering (25) and (28) and assumingk a 0asFig. 2. Left: Newton–Raphson iterations to convergence required for the computation of k a as function of c, location at which HOLMES functions are evaluated and numericaltolerance. Right: non-uniform nodal distribution and auxiliary Delaunay triangulation for the location of the sampling points. The inner and the outer regions of the convexdomain are divided by the dashed line corresponding to R ¼ 0:9.

A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103 93Table 1Newton–Raphson iterations required in the plate examples reported in Section 7.6 as function of c and of the number of nodes in the analysis domain. The Newton–Raphsonconvergence tolerance adopted in calculations is 10 12 .Number of nodes9 9 11 11 15 15 19 19 25 25 31 31 41 41 55 55c ¼ 0:3 5.09 5.06 5.04 5.03 5.02 5.02 5.01 5.01c ¼ 0:8 6.62 6.50 6.35 6.27 6.20 6.16 6.12 6.09Basis functions10.80.60.40.20First Derivatives3020100−10−20−0.20 0.2 0.4 0.6 0.8 1x1−300 0.2 0.4 0.6 0.8 1x20Basis functions0.80.60.40.20First Derivatives100−10−0.20 0.2 0.4 0.6 0.8 1x1−200 0.2 0.4 0.6 0.8 1x20Basis functions0.80.60.40.20First Derivatives100−10Basis functions−0.20 0.2 0.4 0.6 0.8 1x10.80.60.40.20−0.20 0.2 0.4 0.6 0.8 1xFirst Derivatives−200 0.2 0.4 0.6 0.8 1x151050−5−10−150 0.2 0.4 0.6 0.8 1xFig. 3. HOLMES basis functions (left) and their first derivatives (right) for p ¼ 3 and c ¼ 2:0; c ¼ 1:0; c ¼ 0:5; c ¼ 0:1 (top down).r p;HOLMES 1ln þ 1 p;ð81Þcwhere represents the truncation tolerance. In the computation ofthe basis functions at a point x in the domain, the optimizationproblem (HOLMES) may then be solved neglecting the influenceof all nodes at an l p -distance larger than r p;HOLMES .7. ExamplesIn this section we study the convergence and accuracy of secondorder HOLMES approximations. To this end we compare the performanceof HOLMES with second order B-splines, moving leastsquares (MLS) functions and Lagrange polynomials as used in thefinite element method (FEM). For the MLS functions, third order

94 A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103Fig. 4. HOLMES basis functions (left) in x-y-plane and their partial derivatives with respect to x (right) with p ¼ 3; c ¼ 2:0; c ¼ 1:0; c ¼ 0:5; c ¼ 0:1 (top down) and adiscretization of 7 7 nodes (black dots) on a unit square domain.Fig. 5. 2nd order HOLMES boundary functions for uniform (left) and non-uniform (right) discretization. For the non-uniform discretization case, the discretization length inthe x-direction has been increased by 25% for x < 0:5, which makes the basis function spread out farther in the negative than in the positive x-direction.

A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103 95B-splines are used as window functions. All examples presented inthis section make use of a uniform discretization with characteristicdiscretization length h along each coordinate axis, and Gaussquadrature subordinate to an auxiliary triangulation is used fornumerical integration. In the case of B-splines, both the nodes inthe physical space and the knots in the parameter space are uniformlydistributed. Numerical quadrature is performed in d dimensionsfor FEM basis functions with n d GPGauss points per finiteelement. For B-splines Gauss integration with n d GPGauss points isperformed in the parameter space on the intervals between adjacentknots and for HOLMES and MLS functions with n d GPGausspoints on the intervals between adjacent nodes in the physicalspace. Boundary conditions on derivatives are imposed by meansof Lagrange multipliers, as pointed out in Section 5.2.2 of [9], whereverB-splines were used. For HOLMES and MLS functions, Lagrangemultipliers are used to impose boundary conditions bothon function values and derivatives following the collocation pointmethod as described in Subsection 3.3.2, Chapter 10 of [14].According to the collocation point method, the interpolation functionsfor Lagrange multipliers are Dirac-delta functions centered ata set of collocation points where the boundary conditions are imposedexactly. In all our examples, the collocation points correspondto the discretization nodes on the Dirichlet boundary ofthe domain.Before applying HOLMES to solve PDE’s, in the following numericalexamples we assess the ability of HOLMES to reproduce asmooth function, and to pass a ‘‘patch test’’ in the spirit of Example5.1 of [2]. We test HOLMES to reproduce the functions uðxÞ ¼sinðxÞin the interval ½0; 2pŠ, and uðxÞ ¼expðxÞ in the interval ½0; 1Š usingp ¼ 3 and c ¼ 0:3 orc ¼ 0:8. As shown in Fig. 6 and in Table 2, theL 2 ; H 1 , and H 2 error seminorms converge at a rate equal to, or higherthan, the theoretical rate of convergence given by Theorem 3, forboth functions and values of c. For the patch test, we consider thedisplacement uðxÞ of an elastic beam in the interval ½0; 1Š, subjectedto the boundary conditions uð0Þ ¼uð1Þ ¼0; u 0 ð0Þ ¼1, andu 0 ð1Þ ¼ 1, so that the deformed configuration is represented by aquadratic polynomial. Second-order HOLMES can reproduce exactlyquadratic functions and, as discussed in [2], the patch-test isintended to assess the consistency of HOLMES and the accuracy ofthe numerical quadrature used. We use seven nodes to discretizethe domain [0,1], using both uniform and non uniform nodal spacing.The patch test is performed using p ¼ 3 and c ¼ 0:3. Table 3tabulates the error seminorms with respect to the exact solutionfor different numbers of quadrature points between adjacent nodes.When a uniform nodal distribution is used, the exact solution isrecovered within machine precision with 32 quadrature points.Table 2Convergence rates for HOLMES approximation to sinðxÞ in the interval ½0; 2pŠ, and toexpðxÞ in the interval ½0; 1Š, for c ¼ 0:3 and c ¼ 0:8.uðxÞ Domain c L 2 H 1 H 2sinðxÞ ½0; 2pŠ 0.3 3.46 2.23 1.030.8 3.39 2.13 1.03expðxÞ ½0; 1Š 0.3 3.45 2.18 1.020.8 3.37 2.09 1.02For the non-uniform nodal distribution test, the convergence towardthe exact solution is slower and requires more quadraturepoints, as already observed in [2].In all subsequent examples, the number of quadrature points ischosen so as to ensure accuracy in the evaluation of HOLMES.7.1. Elastic rod under sine loadThe static displacements uðxÞ of a linearly elastic rod with unitmaterial and geometry parameters and fixed ends under sine loadare governed by the partial differential equationu xx ðxÞ ¼sinð2pxÞ;x 2½0; 1Š; uð0Þ ¼uð1Þ ¼0:ð82ÞApproximate solutions to (82) can be obtained by a Bubnov–Galerkin method using different types of basis functions. In Fig. 7we compare the convergence resulting from second-order HOLMES,B-spline, MLS and FEM, respectively. To this end, we use HOLMESand MLS shape functions with the same range r 3;HOLMES ¼r MLS ¼ 3h. The HOLMES shape functions are constructed and evaluatedwith p ¼ 3; c ¼ 0:8, and a numerical tolerance ¼ 1e 10with respect to (81). HOLMES is expected to attain optimal ratesof convergence for both the L 2 - and H 1 -error norms. The accuracyof B-splines, MLS functions, and HOLMES is comparable and markedlybetter than the one achieved by finite element basis functions.We choose n GP ¼ 2 for the FEM and B-spline basis functions, andn GP ¼ 6 for the HOLMES and MLS functions, respectively. Thus,HOLMES and MLS require a slightly larger range and more integrationpoints than B-splines for a comparable performance. However,in contrast to B-splines, HOLMES and MLS can be applied withunstructured nodal sets, which greatly increases their range ofapplicability. We mention in passing that the computational costof the evaluation of HOLMES and MLS basis functions in our MATLABimplementation is similar. Figs. 8 and 9 show how the accuracyobtained from HOLMES depends on c and p. In all cases, optimalconvergence is achieved for p P 3, as expected from the analysis10 −210 −2Error seminorms10 0 h10 −410 −610 −810 −2 10 −1L 2 η = 3.46H 1 η = 2.23H 2 η = 1.03Error seminorms10 −410 −610 −810 −1010 0 hL 2 η = 3.45H 1 η = 2.18H 2 η = 1.0210 −2 10 −1Fig. 6. Convergence of the L 2 ; H 1 , and H 2 error seminorms with respect to uniform nodal spacing h for HOLMES approximation to sinðxÞ in the interval ½0; 2pŠ (left), and toexpðxÞ in the interval ½0; 1Š (right) for c ¼ 0:3. The rates of convergence g for the error seminorms are reported in each plot.

96 A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103Table 3Seminorms of the errors between the exact quadratic polynomial solution and the solution computed in the patch test as function of the number of quadrature points n GP used ineach nodal interval. Uniform and non uniform nodal distributions are considered.n GP Uniform spacing Non uniform spacingL 2 H 1 H 2 L 2 H 1 H 24 5:3 10 4 1:9 10 3 1:7 10 2 1:4 10 3 5:9 10 3 6:8 10 28 1:3 10 5 4:6 10 5 4:2 10 4 2:2 10 4 9:0 10 4 1:0 10 216 2:8 10 9 1:0 10 8 9:1 10 8 2:2 10 6 8:9 10 6 9:9 10 532 1:3 10 16 6:7 10 16 1:8 10 14 1:8 10 11 7:4 10 11 8:3 10 10ln(L 2 −error)−4−6−8−10−12−143 : 1HOLMESB−splineMLSFEM−16−6 −5 −4 −3 −2ln(h)ln(H 1 −error)0−2−4−6−82 : 1HOLMESB−splineMLSFEM−10−6 −5 −4 −3 −2ln(h)Fig. 7. L 2 - and H 1 -error convergence of the displacement field of an elastic rod under sine load approximated with different types of basis functions: higher order max-entfunctions (HOLMES), B-splines, moving least squares functions (MLS) and Lagrange polynomials (FEM).ln(L 2 −error)−4−6−8−10 3 : 1−12γ = 2.4γ = 1.6γ = 0.8−14γ = 0.4γ = 0.2−16−6 −5 −4 −3 −2ln(h)ln(H 1 −error)0−2−4−6−82 : 1γ = 2.4γ = 1.6γ = 0.8γ = 0.4γ = 0.2−10−6 −5 −4 −3 −2ln(h)Fig. 8. Comparison of L 2 - and H 1 -error convergence with HOLMES basis functions for p ¼ 3 and varying values of c.−50−2ln(L 2 −error)−103 : 1p=4p=3p=2−15−6 −5 −4 −3 −2ln(h)ln(H 1 −error)−4−6−82 : 1p=4p=3p=2−10−6 −5 −4 −3 −2ln(h)Fig. 9. Comparison of L 2 - and H 1 -error convergence with HOLMES basis functions for c ¼ 0:8 and varying values of p.of Section 5, and even for p ¼ 2. Evidently, the error increases andthe range decreases with increasing c and p, which suggests thatlarger supports generally result in increased accuracy. In Table 4,range and Gauss-point number requirements are presented forp ¼ 3 with varying c, and for c ¼ 0:8 with varying p. The numberof Gauss quadrature points is chosen as the minimal number forwhich no ostensible loss of optimal convergence is observed forany of the discretization lengths employed. Likewise, the tolerance is set to the minimal value for which no ostensible loss of convergenceis observed. The numerical tests reveal that formula (81) is a

A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103 97Table 4Effective range of HOLMES basis functions and number of employed Gauss quadrature points n GP for different values of c with p ¼ 3 (left) and different values of p with c ¼ 0:8(right).c 2.4 1.6 0.8 0.4 0.2 p 2 3 4r 3;HOLMES 2.0 2.4 3.0 3.8 4.8 r p;HOLMES 6.7 3.0 2.0n GP 8 6 6 10 >12 n GP 10 6 10 1e 10 1e 10 1e 10 1e 10 1e 10 n GP 1e 16 1e 10 1e 6good approximation for p ¼ 3, but that it slightly underestimatesthe range for p < 3 and slightly overestimates it for p > 3. These errorscan be compensated for by employing a smaller or larger in(81), respectively.From Table 4 and Fig. 9 it may be concluded that p ¼ 3 is a goodtrade-off between computational cost and accuracy: for p ¼ 2 theresults are only slightly better, but the support and the numberof required Gauss quadrature points is considerably larger, resultingin a marked increase in computational cost both as regardsshape function evaluation and solution of the system of equations.Likewise, for p ¼ 4 the accuracy is worse than for p ¼ 3 and thesmaller support is offset by a substantially larger number of Gausspoints, which effectively wipes out any computational gains. Inaddition, Table 4 and Fig. 8 suggest that c 0:8 is a good tradeoff: for larger c, the accuracy steadily decreases and the smallerrange is offset by an increase in the number of Gauss points; forsmaller c, both range and number of required Gauss points increasesteadily without compensating gains in accuracy. In summary,p ¼ 3 and c ¼ 0:8 are optimal choices of parameters in thisexample.7.2. Eigenfrequencies of an elastic rodThe eigenfrequencies x n ¼ np; n ¼ 1; 2; 3; ... of a linearlyelastic rod with unit material and geometry parameters and fixedends can be computed analytically as the solutions of the standardlinear eigenvalue problemu xx ðxÞþx 2 uðxÞ ¼0;x 2½0; 1Š; uð0Þ ¼uð1Þ ¼0ð83ÞIn addition, an application of Galerkin’s method and a discretizationinto N degrees of freedom gives the approximate eigenfrequenciesx h n; n 2f1; 2; ...; Ng. The error in the n-th approximate eigenfrequencymay be defined ase n ¼ xh nx n1: ð84ÞIn Fig. 10 this error is compared for HOLMES, B-spline, MLS and FEMfor a problem size N ¼ 999. The range of HOLMES and MLS is chosenr 3;HOLMES ¼ r MLS ¼ 3h and the HOLMES parameters are set to p ¼ 3and c ¼ 0:8. Numerical solutions using FEM and B-splines are obtainedwith n GP ¼ 3, and for MLS functions with n GP ¼ 6. Parametersand settings for HOLMES are taken from Table 4, as the requirementsfor the eigenfrequency analysis are found to be similar tothose pertaining to the static example in Section 7.1. As demonstratedin Fig. 10, HOLMES and MLS basis functions with a supportof 3h considerably outperform finite elements and B-splines. This isespecially remarkable, as positivity of the shape functions is suggestedin [9] to be the reason for the superior performance of higher-orderB-splines over FEM basis functions in the present example.However, neither HOLMES nor MLS basis functions satisfy a positivitycondition. Figs. 11 and 12 show that, as a general rule of thumb,larger supports result in increased accuracy of HOLMES approximations,as already found for the static example in Section 7.1. Thisgeneral trend notwithstanding, accuracy eventually deterioratesfor very small values of c, as observed for MLS by [2]. As in the staticexample of Section 7.1, the choice of parameters p ¼ 3 and c ¼ 0:8again appears to represent a good trade-off between accuracy andcomputational cost.7.3. Elastic beam under sine loadA first simple test of the performance of HOLMES in fourth-orderproblems is furnished by the problem of computing the staticdeflections of an elastic beam. The static displacements uðxÞ of alinearly elastic Euler–Bernoulli cantilever beam with unit materialand geometry parameters under sine load are governed by the partialdifferential equationu xxxx ðxÞ ¼sinð2pxÞ;x 2½0; 1Š; uð0Þ ¼u x ð0Þ ¼u xx ð1Þ ¼u xxx ð1Þ ¼0:ð85ÞIn Table 5 the different parameter values and settings of theHOLMES basis functions applied to this example are presented,and in Figs. 13–15 the L 2 - and H 1 -error norms of numerical solutionof (85) with HOLMES, B-spline and MLS basis functions are compared.The general trends are similar to those observed for the elasticrod in Section 7.1. Again p ¼ 3 and c ¼ 0:8 seems to be a goodparameter choice for HOLMES. Altogether, the performance ofHOLMES is slightly better than that of both B-spline and MLS.ω nh / ωn − 10.40.30.2HOLMESB−splineMLSFEMω nh / ωn − 10.40.30.2HOLMESB−SplineMLS0.10.100 0.2 0.4 0.6 0.8 1n/N00 0.2 0.4 0.6 0.8 1n/NFig. 10. Error e n of approximated eigenfrequencies of a linearly elastic rod (left) and beam (right) with different basis functions.

98 A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–1030.40.3p=4p=3p=20.80.6p=4p=3p=2ω nh / ωn − 10.2ω nh / ωn − 10.40.10.200 0.2 0.4 0.6 0.8 1n/N00 0.2 0.4 0.6 0.8 1n/NFig. 11. Error e n of approximated eigenfrequencies of a linearly elastic rod (left) and beam (right) with HOLMES basis functions for c ¼ 0:8 and varying values of p.ω nh / ωn − 10.40.30.2γ = 2.4γ = 1.6γ = 0.8γ = 0.4γ = 0.2ω nh / ωn − 121.51γ = 2.4γ = 1.6γ = 0.8γ = 0.4γ = 0.20.10.500 0.2 0.4 0.6 0.8 1n/N00 0.2 0.4 0.6 0.8 1n/NFig. 12. Error e n of approximated eigenfrequencies of a linearly elastic rod (left) and beam (right) with HOLMES basis functions for p ¼ 3 and varying values of c.Table 5Effective range of HOLMES basis functions and number of employed Gauss quadrature points n GP for different values of c with p ¼ 3 (left) and different values of p with c ¼ 0:8(right).c 2.4 1.6 0.8 0.4 0.2 p 2 3 4r 3;HOLMES 2.0 2.4 3.0 3.8 5.1 r p;HOLMES 6.5 3.0 2.0n GP 16 8 6 5 12 n GP 10 6 10 1e 10 1e 10 1e 10 1e 10 1e 12 1e 15 1e 10 1e 6−4−4ln(L 2 −error)−6−8−102 : 1HOLMESB−SplineMLS−12−6 −5 −4 −3 −2ln(h)ln(H 1 −error)−6−8−102 : 1HOLMESB−SplineMLS−12−6 −5 −4 −3 −2ln(h)Fig. 13. L 2 - and H 1 -error convergence of the displacement field of an Euler–Bernoulli cantilever under sine load approximated with different types of basis functions: higherorder max-ent functions (HOLMES), B-splines, moving least squares functions (MLS).7.4. Eigenfrequencies of an elastic beamA first simple test of the performance of HOLMES in dynamicfourth-order problems is furnished by the modal analysis of anelastic beam. The eigenfrequencies x of a linearly elastic simplysupported Euler–Bernoulli beam with unit material and geometryparameters can be computed analytically as the solutions of thestandard linear eigenvalue problemu xxxx ðxÞ x 2 uðxÞ ¼0;x 2½0; 1Š; uð0Þ ¼uð1Þ ¼u xx ð0Þ ¼u xx ð1Þ ¼0ð86ÞThe errors of the corresponding numerical approximations obtainedfrom B-splines, HOLMES and MLS are depicted in Fig. 10. TheHOLMES solutions are obtained using the same parameters as inthe dynamic rod problem of Section 7.2, but the number of nodesis now reduced to N ¼ 199. An analysis of different HOLMES param-

A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103 9900−52 : 1−52 : 1ln(L 2 −error)−10−15γ=2.4γ=1.6γ=0.8γ=0.4γ=0.2−20−6 −5 −4 −3 −2ln(h)ln(H 1 −error)−10−15γ=2.4γ=1.6γ=0.8γ=0.4γ=0.2−20−6 −5 −4 −3 −2ln(h)Fig. 14. Comparison of L 2 - and H 1 -error convergence with HOLMES basis functions for p ¼ 3 and varying values of c.ln(L 2 −error)−2−4−6−8−10−122 : 1p=4p=3p=2−14−6 −5 −4 −3 −2ln(h)ln(H 1 −error)−2−4−6−82 : 1p=4−10p=3p=2−12−6 −5 −4 −3 −2ln(h)Fig. 15. Comparison of L 2 - and H 1 -error convergence with HOLMES basis functions for c ¼ 0:8 and varying values of p.eters is presented in Figs. 11 and 12. The requirements on the numberof Gauss quadrature points and the numerical tolerance arefound to be similar to those of the static example in Section 7.3and, accordingly, the HOLMES parameters and settings are chosenfrom Table 5. The main findings in Figs. 10–12 are very similar forthe rod and the beam example. Overall, the choice of parametersp ¼ 3 and c 0:8 is suggested by the analysis as the best tradeoffbetween accuracy and computational cost.7.5. Membrane under sine loadWe proceed to extend the analysis of rods presented in Section7.1 to two dimensions by considering the static deformationsof a membrane. The example is intended to test the performance ofHOLMES in applications involving second-order problems in multipledimensions. The static displacements uðx; yÞ of a linearly elasticmembrane with unit material and geometry parameters and fixedboundary under sine load in the x and y directions are governed bythe partial differential equationu xx ðx; yÞþu yy ðx; yÞ ¼sinð2pxÞ sinð2pyÞ;ðx; yÞ 2½0; 1Š½0; 1Š; uð0; yÞ ¼uð1; yÞ ¼uðx; 0Þ ¼uðx; 1Þ ¼0:ð87ÞFig. 16 shows convergence plots for HOLMES, B-spline, MLS andFEM obtained with the same parameters and settings as in Section7.1. The general trends follow closely those of the rod problem,which suggests that the observations and conclusions put forth inSection 7.1 carry over mutatis mutandi to higher dimensions.7.6. Kirchhoff plateWe proceed to extend the analysis of beams presented inSection 7.3 to two dimensions by considering the static deformationsof a plate. This example effectively tests the performance ofHOLMES in applications involving fourth order problems in multipledimensions. The static displacements uðx; yÞ of a linearly elasticKirchhoff plate with unit material and geometry parameters undera load qðx; yÞ obey the partial differential equationu xxxx ðx; yÞþu yyyy ðx; yÞþ2u xxyy ðx; yÞ ¼qðx; yÞðx; yÞ 2½0; 1Š½0; 1ŠþBCð88Þwhere the term BC indicates the boundary conditions on plate displacementsand rotations.The problem of the deflections of a Kirchhoff plate provides auseful test case for investigating issues pertaining to the impositionof boundary conditions. To this end, we solve several exampleswith the following boundary and loading conditions: clampedboundary with concentrated and uniform load; simply supportedboundary on two opposite edges with uniform load; and simplysupported boundary on all four edges with concentrated, uniformand sine load. In all calculations, the HOLMES shape functionsare computed using p ¼ 3; c ¼ 0:3 orc ¼ 0:8, and a search radiusequal to 5:6h, which corresponds to a numerical tolerance ofthe order of machine precision, Eq. (81). Eight different uniformnodal distributions varying from 9 9 points to 55 55 pointsare used in the convergence analyses. Gauss quadrature is performedwith 121 points per element of an auxiliary quadrilateralmesh of the node set. Whereas all examples presented in the precedingsections have been solved using a MATLAB implementation,the Kirchhoff plate examples presented in this section require significantlylarger computational power and memory and thereforethey have been solved using a different implementation withinour in-house C++ code Eureka, which in turn uses the linear solverSuperLU [11].The convergence of HOLMES applied to the Kirchhoff plateproblem is assessed in terms of the convergence of the L 2 - and

100 A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103ln(L 2 −error)−8−10−12−14−16HOLMESB−SplineMLSFEM3 : 1ln(H 1 −error)−4−6−8−10HOLMESB−SplineMLSFEM2 : 1−18−12−20−5 −4.5 −4 −3.5 −3 −2.5 −2ln(h)−14−5 −4.5 −4 −3.5 −3 −2.5 −2ln(h)Fig. 16. L 2 - and H 1 -error convergence of the displacement field of an elastic membrane under sine load approximated with different types of basis functions: higher ordermax-ent functions (HOLMES), B-splines, moving least squares functions (MLS) and Lagrange polynomials (FEM).H 1 -error norms of the displacement field. The L 2 - and H 1 -errornorms are calculated with respect to the exact analytical solutionsfor plate bending problems found in [30]. We also compute the H 1 -error norm restricted to the boundary of the plate domain wherethe boundary conditions are imposed. The convergence of theH 1 -error norm restricted to the constrained boundaries assuresthat boundary conditions are imposed correctly using Lagrangemultipliers. The L 2 - and H 1 -error norms are computed approximatelyusing 2:5 10 5 uniformly distributed sampling points inthe plate domain.The order of convergence of the error norms computed in theplate examples is reported in Table 6 for two different values ofc. It is noteworthy that the order of convergence is consistentlyclose to, or higher than, the theoretical convergence order of 2for both values of c and throughout the numerical examples, whichdiffer strongly with respect to loading and boundary conditions. Asin the case of second-order problems, a comparison of Table 6 andFig. 13 reveals that the general trends in one dimension carry overto higher dimensions. For reasons of brevity, it is not possible to reportthe full convergence plots and deformed configurations for allTable 6Convergence rates for different displacement error norms computed in several numerical examples of a Kirchhoff plate subjected to different loads with different boundaryconditions (the abbreviation SS refers to simply supported boundary conditions). Two different values of c are used to compute the HOLMES approximation functions.L 2 H 1 H 1 boundaryc ¼ 0:3 c ¼ 0:8 c ¼ 0:3 c ¼ 0:8 c ¼ 0:3 c ¼ 0:8Clamped – concentrated load 2.17 2.41 2.03 2.16 3.44 2.35Clamped – uniform load 2.18 2.45 2.12 2.33 4.10 2.17SS on 2 edges – uniform load 2.26 2.39 2.23 2.36 2.38 2.34SS on 4 edges – concentrated load 2.15 2.32 1.99 1.85 2.27 1.94SS on 4 edges – uniform load 2.13 2.65 2.04 2.31 2.17 2.60SS on 4 edges – sine load 2.14 2.79 2.11 2.49 2.22 2.9710 −2 h10 −3L 2H 1H 1 boundary η = 2.1610 −4η = 2.41Error norms10 −510 −6η = 2.3510 −710 −810 −910 −2 10 −1Fig. 17. Elastic Kirchhoff plate clamped on all four edges and subjected to a unit centered concentrated load. Convergence of the L 2 - and H 1 -error norms of the displacementfield (left) and deformed configuration for the finest nodal distribution (55 55 nodes) used in the convergence analysis (right). The order of convergence g for each errornorm is also reported.

A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103 10110 −2 h10 −3L 2H 1H 1 boundaryη=2.3610 −4η=2.39Error norms10 −510 −6η=2.3410 −710 −810 −2 10 −1Fig. 18. Elastic Kirchhoff plate simply supported on two opposite edges and subjected to a unit uniform load. Convergence of the L 2 - and H 1 -error norms of the displacementfield (left) and deformed configuration for the finest nodal distribution (55 55 nodes) used in the convergence analysis (right). The order of convergence g for each errornorm is also shown.10 −3 h10 −4L 2H 1H 1 boundaryη=2.49Error norms10 −510 −6η=2.79η=2.9710 −710 −810 −910 −2 10 −1Fig. 19. Elastic Kirchhoff plate simply supported on all four edges and subjected to a sine load. Convergence of the L 2 - and H 1 -error norms of the displacement field (left) anddeformed configuration for the finest nodal distribution (55 55 nodes) used in the convergence analysis (right). The order of convergence g for each error norm is alsoreported.the examples considered, and we focus on three representativeexamples instead. Figs. 17–19 show the convergence in the L 2 -and H 1 -error norms together with the deformed configurationsfor a clamped plate subjected to a concentrated load, a plate simplysupported on two opposite edges subjected to a uniform load, anda plate simply supported on all four edges subjected to a sine load.In all these cases, HOLMES shape functions are computed usingc ¼ 0:8. The convergence plots show that the L 2 and H 1 error normsdecrease monotonically as a function of the nodal spacing h. Moreover,both error bounds exhibit convergence rates close to, or higherthan, h 2 , which is the theoretical convergence rate in both errornorms when quadratic interpolation is applied to fourth orderPDEs [26]. The smoothness and regularity of the deformed configurationsis also noteworthy.8. Summary and conclusionsWe have developed a family of approximation schemes, whichwe have termed High-Order Local Maximum-Entropy Schemes(HOLMES), that deliver approximations of arbitrary order andsmoothness, in the sense of converge in general Sobolev spacesW k;p . Thus, the HOLMES shape functions span polynomials of arbitrarydegree and result in C k -interpolation of arbitrary degree k ofsmoothness. HOLMES approximations are constructed in the spiritof the Local Maximum-Entropy (LME) approximation schemes ofArroyo and Ortiz [2] but differ from those schemes in an importantrespect, namely, the shape functions are not required to be positive.This relaxation effectively bypasses difficulties with lack offeasibility of convex interpolation schemes of high order, albeit

102 A. Bompadre et al. / Comput. Methods Appl. Mech. Engrg. 221–222 (2012) 83–103at the expense of losing the Kronecker-delta property of LMEapproximation schemes. We have proved uniform error boundsfor the HOLMES approximates and their derivatives up to orderk. Moreover, we have shown that the HOLMES of order k is densein the Sobolev Space W k;p , for any 1 6 p < 1. The good performanceof HOLMES relative to other meshfree schemes has alsobeen critically appraised in selected test cases.On the basis of these results, HOLMES appears attractive for usein a number of areas of application. Thus, because of its meshfreecharacter, HOLMES provides an avenue for highly-accurate simulationof unconstrained flows, be them solid or fluid [17]. In addition,the high-order and degree of differentiability of HOLMES can beexploited in constrained problems, e. g., to account for incompressibilityexactly by the introduction of flow potentials, or in higherorderproblems such as plates and shells. These potential applicationsof HOLMES suggest themselves as worthwhile directions forfurther study.ACKNOWLEDGEMENTSThe support of the Department of Energy National Nuclear SecurityAdministration under Award Number DE-FC52-08NA28613through Caltech’s ASC/PSAAP Center for the Predictive Modelingand Simulation of High Energy Density Dynamic Response of Materialsis gratefully acknowledged. The third author (C.J.C.) gratefullyacknowledges the support by the International Graduate School ofScience and Engineering of the Technische Universitat Munchen.Appendix A. First and second order derivatives of the shapefunctionsWe present in more detail the formulas of the first- and secondorderderivatives of the shape functions of HOLMES of order n. Westart by summarizing a number of formulas defined throughoutthe paper.Let P ¼fx a : a 2 Ig be a node set of a domain X, indexed by anindex set I. For each index a 2 I, we define the functionsfn;a þ ; f n;a : Rd R Dn! R as follows.01f þ n;a ðx; kÞ ¼ 1 h Xpcjx x a j p @ph jaj k a ðx x a Þ a A; ð89aÞa2N d ;jaj6n0f n;aðx; kÞ ¼ 1 h p cjx x a j p p þ X@a2N d ;jaj6nLet w þ a ; w a : Rd R Dn ! R be the followingh jaj k a ðx1x a Þ a A: ð89bÞw þ a ðx; kÞ ¼exp½f þ n;að x; k ÞŠ; a 2 I; ð90aÞw aðx; kÞ ¼exp½f n;aðx; kÞŠ; a 2 I: ð90bÞLet Z n : R d R Dn! R be the partition functionZ n ðx; kÞ ¼k 0 þ X h iexp f þ n;aðx; kÞ þ X h iexp f n;aðx; kÞa2Ia2I¼ k 0 þ X a2IFor every x 2 X, letw þ a ðx; kÞþw a ðx; kÞ :ð91Þk ðxÞ ¼argmin Z n ðx; kÞ: ð92Þk2R DnThe shape functions of HOLMES of order n arew þ aðxÞ ¼w þ a ðx; k ðxÞÞ; a 2 I; ð93aÞw aðxÞ ¼w aðx; k ðxÞÞ; a 2 I: ð93bÞIn what follows, we use the following notational conventions. Givena function Gðx; kÞ : R d R Dn! R m , we denote by G ðxÞ the compositefunction G ðxÞ ¼Gðx; k ðxÞÞ, where k ðxÞ is the optimal dual vector.By the chain rule, we can write the first-order partial derivativevector of a shape function w aas rw a ðxÞ ¼ @w aþ @w aDk : ð94Þ@x @kApplying the chain rule again, for every pair of indices 1 6 i; j 6 d,the second-order partial derivative of w awith respect to x i and x jfollows as.@ 2 w a@x i @x j¼þ@2 w a@x i @x j! þXb2N d ;jbj6n @k b@x j@k m@x iþXb2N d ;jbj6n! @ 2 w a @kbþ@k b @x i @x jXb2N d ;jbj6n! @ 2 w a @kb@k b @x j @x iXb;m2N d ;jbj;jmj6n w @ 2 k ab:@k b @x i @x j! @ 2 w a@k b @k mð95ÞNote that, in (94) as well as in (95), the terms not readily availableare the first and second order derivatives of k . Next, we outline theformulas for computing them.Let r : R d R Dn ! R Dn and J : R d R Dn ! R DnDn be the functionsrðx; kÞ r k Z n ðx; kÞ;Jðx; kÞ @ k rðx; kÞ ¼@ k @ k Z n ðx; kÞ:ð96Þð97ÞLet r be the function r ðxÞ ¼rðx; k ðxÞÞ, and let J be the functionJ ðxÞ ¼Jðx; k ðxÞÞ. The function r is identically zero. Then, by theimplicit function theorem,Dr ðxÞ 0 ¼@r þ @r Dk ; ð98Þ@x @kand in particular, Dk ðxÞ ¼ ðJ Þ 1 @r: ð99Þ@xFor any a 2 N d ; jaj 6 n, and any two indices 1 6 i; j 6 d, the followingholds,@ 2 r a ðxÞ@x i @x j 0 ¼ @2 r a@x i @x j! þþþXb2N d ;jbj6nXb2N d ;jbj6nXb2N d ; jbj6n! @ 2 r a @kbþ@k b @x j @x i @r a @ 2 k b:@k b @x i @x j! @ 2 r a @kb@k b @x i @x jXb;m2N d ;jbj;jmj6n! @ 2 r a @kb @k m@k b @k m @x i @x jð100ÞWe rewrite this equation in a more convenient form. For everya 2 N d ; jaj 6 n, and every two indices 1 6 i; j 6 d, let F a;ði;jÞ ðxÞ bethe functionF a;ði;jÞ ðxÞ ¼þþ@2 r a@x i @x j! þXb2N d ;jbj6nXb;m2N d ;jbj;jmj6nXb2N d ;jbj6n! @ 2 r a @kb@k b @x j @x i! @ 2 r a @kb@k b @x i @x j! @ 2 r a @kb @k m:@k b @k m @x i @x jð101ÞThen, we can rewrite (100) asX @r a @ 2 k b¼@k b @x i @x jF a; ði; jÞ ðxÞ; 8a 2 N d ; jaj 6 n; 1 6 i; j 6 n:b2N d ;jbj6nð102Þ

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