On the Extension of Goal-Oriented Error Estimation and Hierarchical Modeling to Discrete Lattice Models

Transcription

1 On the Etension of Goal-Oriented Error Estimation and Hierarchical Modeling to Discrete Lattice Models J. T. Oden, S. Prudhomme, and P. Bauman Institute for Computational Engineering and Sciences The Universit of Teas at Austin August 4, 24 Abstract The Goals algorithm for adaptive modeling is etended to the case of discrete models such as those characterized b lattice structures (or the static behavior molecular sstems) b using surrogate models obtained from continuum approimations of the lattice. The result is a technique that could provide for scale-bridging between continuum models and atomistic models, although the present development concerns onl simple algebraic sstems. An eample is provided in which quantities of interest in a large number of degrees of freedom sstem are computed to a preset tolerance in relativel few low-order approimations. Kewords: multiscale models, adaptive modeling, error estimates, continuum/lattice models. Introduction In earlier work [6, 8, 4], we developed techniques for assessing model and approimation error in solid and continuum mechanics, and methods for adapting the mathematical models of certain events to control modeling error. These techniques fall under the general heading of hierarchical modeling and Goal-oriented a posteriori error estimation and model adaptivit. A general theor for estimating modeling error was given in [4]. In the present paper, we etend this approach to the equilibrium analsis of atomic lattices, where the base model is defined b a regular periodic lattice and the surrogate models are obtained from continuum models characterized b PDE s; precisel the opposite situation is encountered in modeling micro-scale effects in multi-phase heterogeneous materials [6, 8]. An important bproduct of our analsis is a new approach to modeling the transition from continuum models to molecular models. In particular, we present techniques in which the notion of convergence of finite element models of the continuum is unambiguousl preserved, and in which convergence of the base molecular or lattice model is trivial. Thus, the notion of a continuum atomistic interface and coupling is precisel determined in a rigorous wa. The paper is organized as follows: In Section 2, we provide a brief summar of the procedure proposed in [4] to estimate the modeling errors with respect to some quantit of interest. We present in Section 3 a model problem for static equilibrium of a material lattice and introduce in Section 4 a simple problem which will be later referred to as the continuum model. The

2 continuum model is given in terms of a partial differential equation and will be considered here as a surrogate problem of the lattice model problem. We appl in Section 5 the technique described in Section 2 to the continuum/lattice problems for estimating modeling errors, i.e. the difference between the solutions of the continuum and lattice models, and describe an adaptive algorithm for controlling modeling errors. We demonstrate in Section 6 the efficienc of the adaptive algorithm on two numerical eamples. Conclusions are given in Section 7. 2 Modeling Errors in Quantities of Interest The abstract setting for the theor of Goal-oriented estimation of modeling error is this: we wish to find a vector u in a topological vector space V such that B(u, v) = F (v) v V () where B(, ) is a bilinear (or semilinear) form on V and F is a linear functional on V. Of interest is a quantit of interest Q(u), Q being a functional on V. To characterize Q, we seek an influence vector p which is a solution of the dual problem, B(v, p) = Q(v) v V (2) Problem () is called the primal base problem; problem (2) the dual base problem. If B(, ) is nonlinear in u, we replace (2) b B (u; v, p) = Q (u; v), where B ( ;, ) and Q ( ; ) are functional (Gâteau) derivatives of B(, ) and Q( ) respectivel. The basic idea is that () and (2) are intractable: too comple or too large to be solved b an practical means. Therefore, we seek simpler surrogate problems that are tractable and of the form B (u, v) = F (v) B (v, p ) = Q(v) v V v V (3) B (, ) being a new bilinear form. In [4] (see also [2], it is shown that where R(u, v) is the residual functional Q(u) Q(u ) = R(u, p) (4) R(u, v) = F (v) B(u, v) (5) If the problem is nonlinear, the right-hand-side of (4) is replaced b R(u, p)+r, where r is a functional involving quadratic and higher terms in the errors e = u u and ε = p p. In applications, we often neglect such higher order terms, and use R(u, p) as an approimation of Q(u) Q(u ). The use of such relations to develop adaptive modeling schemes for estimating and controlling errors in modeling heterogeneous materials is described in [6, 8]. Our aim is to use (4) as a basis for estimating and controlling errors in models involving a combination of discrete and continuum attributes. Here, the base model is one provided b the discrete lattice structure, representing, for instance, a molecular dnamics sstem in equilibrium, and in which the surrogate model is to be constructed from a continuum model of the material. 2

3 H H Figure : A regular periodic lattice in R 2. 3 A Model Lattice Problem To fi ideas, we consider a simple case of static equilibrium of a material lattice with regular periodic micro-structure. The regular lattice L has lattice uniform width H and etends indefinitel in R d, d = 2 or 3. For the sake of simplicit, we onl consider two-dimensional domains here, i.e. d = 2, as indicated in Fig.. All results presented in this report are easil etended to three-dimensional problems. The goal is to determine a discrete scalar field u which takes on values u i,j at each lattice point i,j, (i, j) Z 2, and which satisfies the equilibrium equation (i,j) Z 2 a i,j u i,j f i,j = (6) where f i,j = f( i,j ), f being an (i, j)-periodic prescribed force field, and a i,j are appropriate constants. To further restrict the problem while also establishing conditions sufficient to guarantee the eistence of solutions to (), we shall make the following simplifications:. A subdomain Ω = (, a) (, b) R 2 is identified with origin (, ) at a lattice point, as shown in Fig. 2, and the lengths a and b are naturall chosen as multiples of H. Note that in more general settings, Ω can be a more comple domain than just a rectangle and ma depend on the structure of the lattice as well. 2. The boundar of Ω is denoted Ω and u i,j = at points i,j Ω. 3. f i,j is given at all points i,j Ω. 4. The coefficients representing the interactions of adjacent atoms at sites in the lattice are given. For simplicit, one eample is the difference stencil, H 2 u i,j H 2 u i,j + 4 H 2 u i,j H 2 u i+,j H 2 u i,j+ f i,j = (7) which we write in matri form as Au = f (8) 3

4 b k4 k7 k3 ω * k8 k9 k6 Ω (,) a k k5 k2 Figure 2: A subdomain Ω of R 2 and an interior patch ω of four cells surrounding lattice site = (, ). where u and f are N-vectors and A is an N N matri, assuming there are N lattice points in the closure of Ω. In a more abstract setting, we introduce the finite dimensional vector space V L = {v R N, v k = component of v corresponding to and the bilinear and linear forms, lattice point k Ω, v k = on Ω} (9) B : V L V L R; F : V L R; Then, the lattice equilibrium problem is B(u, v) = v T Au F (v) = v T f () Find u V L such that B(u, v) = F (v) v V L () Our goal is not to calculate u, but some functional of u defined b a quantit of interest Q(u). For eample, if ω is a patch of four cells surrounding a particular lattice point (see Fig. 2), we ma be interested in the weighted average value of u over the cell: Q(u) = ] [(u k + u k2 + u k3 + u k4 ) + 2(u k5 + u k6 + u k7 + u k8 ) + 4u k9 6 9 (2) = w kl u kl l= w kl being the indicated weights. This corresponds to the average of the interpolant of u over the patch ω. The problem of finding a vector p V L such that B(v, p) = Q(v) v V L (3) is then the base dual problem for the functional Q, and p is the dual solution or influence vector for the lattice problem. 4

5 4 A Continuum Surrogate Problem We propose here a simple continuum model deduced from the lattice model. Our principal motivation in this paper is to focus on the adaptive algorithm rather than modeling issues. Various techniques have been suggested to derive continuum models, see for eample [3, 7]. For the lattice model (8), in the limit as H, we consider here the following continuum model of the problem: Find u V such that B (u, v) = F (v) v V (4) and where now Find p V such that B (v, p ) = Q (v) v V (5) B (u, v) = u v dd (6) Ω F (v) = fv dd (7) Ω Q (v) = v dd (8) ω ω V = {v H (Ω) : v = on Ω} = H (Ω) (9) Clearl, in the limit as H, the lattice model (8) converges to the model of the Poisson problem u = f in Ω, u = on Ω. Of course, the spaces V L and V are now incompatible, V L being of dimension N and V infinite dimensional. We must net map the continuum description onto the lattice model to be able to compare errors in the quantities of interest. This is done as follows. Let Π : V V L denote a (collocation) mapping from V onto V L defined b Πw = v = {v k } N k= v k = w ρ ( k ) = k ρ (, k )w() d B ρ( k ) ρ H k = lattice site in L (2) Here B ρ ( k ) is a ball of radius ρ centered at the lattice node k, k ρ (, k ) is a smooth kernel vanishing outside B ρ ( k ) and normalized so that k ρ (, k ) d =. B ρ( k ) Thus, w ρ is the mollifier (or mollification) of w. The operator Π thus produces an N-vector in V L defined on the lattice whose components are simpl the values of the functions w in V, smoothed so that their pointwise values are well-defined (the functions w V, being in H (Ω), are not necessaril continuous for 2D and 3D domains). In practice, we replace w ρ b finite element approimations w h for mesh size h sufficientl small. This is discussed in Section 5, to follow. Appling the ideas of Section 2, we now derive an estimate for the modeling error. Let u V be the unique solution of (4) and p V L be the influence vector for Q satisfing 5

6 (3). Then, according to (4), the modeling error e = u Πu = u u in the quantit of interest Q is Q(e ) = Q(u) Q(u ) = R(u, p). (2) We can rewrite the residual as: R(u, p) = R(u, p ) + R(u, ε ) (22) = R(u, p ) + B(e, ε ) (23) where p = Πp, p being the influence function for the surrogate problem (5), and ε = p p. Note that (23) follows from (22) b making use of the error equation B(e, v) = R(u, v), v V and b taking v = ε. An estimate of the modeling error is then obtained b neglecting the higher order term B(e, ε ) so that Q(e ) R(u, p ), or b computing bounds on B(e, ε ) as in [6], or computing the eact dual solution vector p. 5 Approimations of the Continuum Model 5. Discrete approimations In general, the continuum models (4) and (5) cannot be solved eactl, and numerical approimations, such as finite element approimations, of u and p, must be obtained. Then, instead of (2), a more direct construction of a mapping from V onto V L can be defined. Let {P h } denote a famil of partitions of Ω into finite elements K each being the image under invertible maps F K of a master element ˆK over which polnomial test functions are defined (see, e.g. []). We shall assume that the partitions are regular and that the usual interpolation properties of finite element interpolation of functions in Sobolev spaces are in force [5]. In this wa, we generate a famil of finite-dimensional subspaces {V h } of V, with h> V h everwhere dense in, sa, H (Ω). For a given subspace V h, the finite element approimations of (4) and (5) are thus B (u h, vh ) = F (v h ) v h V h (24) B (v h, p h ) = Q (v h ) v h V h (25) Instead of (2), we introduce the collocation operator Π h : V h V L such that Π h v h = v h = {(v h ) k } N k= (v h ) k = v h ( k ) k = lattice site in Ω (26) 5.2 Estimate of modeling error To estimate the error in Q(u h ), we proceed as before. Using u h = Π h u h and p h = Π h p h instead of u and p, respectivel, the estimate (2) becomes Q(u) Q(u h ) = R(u h, p) = R(u h, ph ) + B(u uh, p ph ) (27) 6

7 Neglecting the higher order term in u u h, we then obtain the error estimate: E = R(u h, ph ) Q(u) Q(uh ) (28) It is important to note that there is no connection between the lattice dimension H and the mesh size h. Convergence of the finite element approimation is unambiguous: as h, u h u in H (Ω), independentl of H (for quasi-uniform refinements, with h = ma h K, h K = dia(k)). 5.3 Adaptive modeling strateg We now describe an adaptive modeling strateg aimed at reducing computational costs. For instance, let us suppose that we cannot solve the N-order sstem (), N being a large number, but that we wish to compute for the vector u the quantit of interest Q(u) to within a tolerance γ tol, i.e. Q(u) Q(u h ) γ tol. Q(u) We propose a Goal-oriented hierarchical modeling algorithm that emplos concepts similar to those developed in [6, 8] for computing local features in a lattice. Again, let L denote the lattice of N sites covering the domain Ω with lattice width H. In addition, let Ω be partitioned into K cells of size larger than H (see Fig. 3). The error estimate E can then be decomposed into contributions over each cell as E = R(u h, p h ) = K R n (u h, p h ) (29) where R n (u h, ph ) denotes the contribution in the nth cell to the total error E. More eplicitl, recall that n= R(u h, p h ) = F (p h ) B(u h, p h ) = p ht f pht Auh = p ht ( ) f Au h ( ) N N = p h k f k A kl u h l Then the contribution E n from the n th cell Θ n is computed according to [ ( )] N N E n = R n (u h, p h ) = f k A kl u h l k= k= β n k where the coefficients βk n depend on the location of the sites k with respect to the cell Θ n. Let θ k be the number of cells that contain point k. This number can take on the values 4, 2, or, depending on whether k is an interior site of L, or lies on one of the boundar edges of L, or is one of the four corners of L, respectivel. Then { βk n = if k Θ n if k Θ n θ k The algorithm for adaptivel modeling the lattice model with a continuum model is as follows: p h k l= l= (3) 7

8 . Set s = 2. Replace the lattice model with the continuum model (4) and solve for a (ver accurate) finite element approimation u h of u. Compute also an approimation p h of p. 3. Evaluate u h = Π h u h, p h = Π h p h, and the quantit of interest Q(u h ). 4. Estimate the error in Q(u h ) using (28), i.e. compute E () = R(u h, p h ). 5. Check whether E () / Q(u h ) + E() γ tol? If es, stop. If not, start the adaptive process. 6. Set s = s + (a) Compute the cell contributions E n = R n (u h s, ph s ), n =,..., K as in (3). Construct the patch L s made of the cells that have not been alread refined and that satisf E n ma n E n α where α is a user-defined parameter. (b) Solve the reduced lattice problem on L s for lattice vectors u h s and p h s, with u h s = uh and ph s = ph at the points outside of L s. (c) Estimate the error in Q(u h s ), i.e. compute E (s) = R(u h s, ph s ). (d) Check whether E (s) / Q(u h s )+E(s) γ tol? If the answer is negative, go to step 6. If the answer is affirmative, stop the adaptive process. In this manner, we create a sequence of surrogate problems P s, whose solutions are given b (u h s, p h s ), s =,,.... The initial surrogate problem P is obtained using the continuum model onl. The subsequent problems P s, s =, 2,..., combine the solutions from the continuum model and from the lattice model used onl in the reduced lattice L s. Obviousl, this is onl one eample of man possible variants of the adaptive process. Note that the overall approach is also reminiscent of the multigrid method in which the lattice model stands for the the fine-scale grid, and the surrogate model for the coarse-scale grid model. 6 Numerical Eamples To demonstrate the Goals algorithm for adaptive modeling, we consider eamples of regular lattices L on the square domain Ω = (, ) 2 with lattice width H = (m ), m =, 2,..., 6. The lattice L is then made of N = m 2 sites such that: L = {( i, j ): i, j N, i, j m, i = ih, j = jh} and will be referred to as a m m lattice. We emphasize that the lattice solution is our solution of reference, although in these eamples, the scales provided b the finite element solutions are finer. Again our onl concern here is to demonstrate the feasibilit of the adaptive strateg. In all eamples, we partition the domain Ω into 25 cells of dimension.2.2 (see Fig. 3) and the quantit of interest is the weighted average over the square domain ω of dimension 2H 2H located at the center of the lattice, as defined in (2). In other words, the 8

9 (,) Cells (,) Figure 3: A lattice in R 2. ω (,) quantit of interest is the weighted average of u at the lattice points i,j where i, j {(m )/2, (m )/2, (m )/2 + }. Note that the quantit of interest depends indirectl on the lattice width H. The eact solution to the continuum model problem is given b u = f, in Ω u = on Ω where the datum f is determined such that u (, ) is known analticall. We will consider in the following two different functions u. The solution of the Poisson problem is approimated using a mesh of 2 2 bilinear elements. In all numerical eperiments, the parameter α introduced in step 6(a) of the adaptive algorithm will be chosen to be Eample one In this eample, the continuum and lattice problems are set up in such a wa that the eact continuum solution is given b u (, ) = ( ) 2 2 ( ) 2 e ((.25)2 +(.25) 2 ) The function u is essentiall a differenced Gaussian centered at the point (.25,.25) as shown in Fig. 4. The term 2 ( ) 2 2 ( ) 2 ensures that u and the derivatives of u are zero on the boundar of Ω. The corresponding lattice solutions are shown in Fig. 5 for the and 3 3 lattices. We show in Figs. 6 and 7 the continuum influence function p and lattice influence influence p associated with the weighted average defined above, for the and 3 3 lattices, respectivel. As epected, we observe that the influence function becomes more localized as the lattice width decreases since, in all cases, the quantit of interest is defined with respect to the nine lattice points at the center of the lattice. 9

10 Figure 4: Accurate approimation of the continuum solution u for eample. u u Figure 5: Lattice solutions of eample computed on the and 3 3 lattices. In the net set of numerical eperiments, we test the adaptive modeling strateg proposed in Section 5.3. We set the tolerance on the error γ tol to 5 percent. Figures 8 and 9 show the sequence of refinements for the 2 2 lattice. In these plots, and in all similar plots that follow, the gre area represents the subregion in which the solution has been computed using the continuum model, that is, u. The complementar subregion displaing the lattice corresponds to the reduced lattice L s, where s is the number of iterations alread performed, as eplained in Section 5.3. The number l s in the various cells indicates the iteration at which the cell has been added to the lattice L s. In this case, eight refinements were necessar to achieve a tolerance of 5 percent which resulted in all but three cells being refined. Note that all plots are smmetric with respect to the diagonal =, as epected. We show in Fig. the final configurations obtained b adaptive modeling for all lattices. We observe that eight iterations are needed for the and 2 2 lattices to achieve a tolerance of 5 percent on the error in the quantit of interest. For all the other lattices, the algorithm ields the same final configuration after 5 iterations, but note that the cells are

11 p Figure 6: Influence functions associated with the lattice: (left) continuum solution p and (right) lattice solution p p Figure 7: Influence functions associated with the 3 3 lattice: (left) continuum solution p and (right) lattice solution p. not necessaril refined in the same order in all cases. In Fig., we show the effectivit indices and the evolution of the relative errors with respect to the number of refinements obtained b adaptive modeling. The effectivit indices provide a measure of the qualit of the error estimates and are calculated as the ratio of the error estimate E (s) at iteration s and the eact error, i.e. λ = E (s) Q(u) Q(u h s ) Thus, an effectivit inde of eactl one indicates that the error estimator perfectl estimates the eact error. From Fig., we observe that the effectivit indices oscillates between.95 and. ecept for one value. The rates of convergence are mostl monotonic apart from the case of the lattice. In this case, we believe that some internal boundar conditions obtained from the continuum solution are in error and greatl increase the relative error in the quantit of interest until these boundaries become part of the lattice b including the adjacent cells. Finall, we show in Fig. 2, for illustration purposes, the solutions obtained on the reduced lattices L 3 (after the third iteration) for the and 3 3 lattices. These plots can be qualitativel compared to the solutions that were computed on either the full continuum

12 Figure 8: Sequence of surrogate models obtained b adaptive refinement for the 2 2 lattice (iterations s = to 4). In these plots, the gre area represents the subregion in which the solution is obtained using the continuum model. The complementar subregion corresponds to the reduced lattice L s. The number l s in the cells indicates the iteration at which the cell has been added to the lattice L s. model or the full lattice model; see Figs. 4 and Eample two We repeat in this section the same tpes of numerical eperiments as before but for a different eact solution of the continuum model. Here the solution u (see Fig. 3) is obtained b superposition of two Gaussian functions centered at the points (.25,.25) and (.5,.8) such that: u (, ) =28 2 ( ) 2 2 ( ) 2 e ((.25)2 +(.25) 2 ) ( ) 2 2 ( ) 2 e ((.5)2 +(.8) 2 ) In this case, the weighted average error, still computed over the nine points at the center of the lattices, is essentiall influenced b the sources of error generated at the location of the 2

13 Figure 9: Sequence of surrogate models obtained b adaptive refinement for the 2 2 lattice (iterations s = 5 to 8). In these plots, the gre area represents the subregion in which the solution is obtained using the continuum model. The complementar subregion corresponds to the reduced lattice L s. The number l s in the cells indicates the iteration at which the cell has been added to the lattice L s. two Gaussian functions. The corresponding lattice solutions are shown in Fig. 4 for the and 3 3 lattices. Since the quantities of interest are the same as in eample, so are the associated influence functions. For testing the adaptive modeling strateg proposed in Section 5.3, we select two tolerances, i.e. γ tol = percent and γ tol = 5 percent. Fig. 5 shows the sequence of refinements for the 2 2 lattice. The algorithm automaticall performed five refinements in order to achieve a tolerance of percent. We observe that the procedure essentiall refines the cells that cover the peaks of the Gaussian functions and the features of the quantit of interest. The final configurations for all lattices are shown in Fig. 6 for γ tol = percent and in Fig. 7 for γ tol = 5 percent. We observe that, apart from the lattice, two to three etra iterations are necessar to reduce the error from percent to 5 percent. The evolution of the effectivit indices and relative errors with respect to the number of refinements are displaed in Figs. 8 and 9. The convergence results suggest that the adaptive algorithm could be improved for our particular choice of quantit of interest. The erratic behavior prior 3

14 to a dramatic reduction in error ma be attributed to the fact that the sequence of model enrichments emplos local error measures rather than global error measures. Regardless of the rate of convergence, quite acceptable, indeed ecellent, estimates of the modeling error are obtained. A more detailed stud of various adaptive strategies and convergence will be undertaken for more comple sstems in future studies. Finall the solutions obtained on the reduced lattices L 2 for the and 3 3 lattices are given in Fig. 2 and should be compared with the solutions of Figs. 3 and 4. These pictures clearl demonstrate that the proposed adaptive procedure automaticall refines the regions in which the large sources of modeling error contribute the most to the error in the quantit of interest. 7 Concluding Remarks We have described in this paper a method to etend the Goals algorithm for adaptive modeling to the case of discrete models b using a continuum model for the surrogate problem. The method involves the derivation of a Goal-oriented error estimator to obtain computable error measures of local quantities of interest. The proposed adaptive algorithm is an iterative procedure which allows one to determine the regions of the domain in which it is necessar to use the lattice base model to control the error in the quantit of interest to within some preset tolerance. The solution of the continuum model is merel used to prescribe the internal boundar conditions for the reduced lattice problems. The methodolog was tested here on two numerical eamples and the results clearl demonstrate the great potential of this approach. In these eamples, the continuum model is defined as a Poisson problem and the lattice model was derived b using a five-point central difference stencil for the Poisson equation. The main conclusion from this stud is that it is actuall feasible to automaticall select the sites of the lattice that need be included in the reduced lattice problem to obtain accurate quantities of interest. Whenever the lattice problem is more epensive to solve than the continuum problem, the use of this approach would allow for substantial cost reductions. Our objective in the near future is to etend this methodolog to molecular statics or molecular dnamics for problems in nanomechanics. The main issue will be to construct adequate surrogate problems from the lattice problems for multiscale modeling and to deal with the loss of scale information due to the transition from the molecular model to the continuum model. Acknowledgment The support of the work b ONR under contract N is gratefull acknowledged. The work of Paul Bauman was supported b a DOE Computational Science Graduate Fellowship. References [] M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analsis, John Wile & Sons, 2. [2] M. Braack and A. Ern, A posteriori control of modeling errors and discretization errors, Multiscale Model. Simul, (23), pp [3] P.-J. Martinsson and G. J. Rodin, Asmptotic epansions of lattice Green s functions, Proc. R. Soc. Lond. A, 458 (22), pp

15 [4] J. T. Oden and S. Prudhomme, Estimation of modeling error in computational mechanics, Journal of Computational Phsics, 82 (22), pp [5] J. T. Oden and J. N. Redd, An Introduction to the Mathematical Theor of Finite Elements, John Wile & Sons, New-York, 976. [6] J. T. Oden and K. Vemaganti, Estimation of local modeling error and goal-oriented modeling of heterogeneous materials; Part I: error estimates and adaptive algorithms, Journal of Computational Phsics, 64 (2), pp [7] V. B. Sheno, R. Miller, E. B. Tadmor, D. Rodne, R. Phillips, and M. Ortiz, An adpative finite element approach to atomic-scale mechanics the quasicontinuum approach, Journal of the Mechanics and Phsics of Solids, 47 (999), pp [8] K. Vemaganti and J. T. Oden, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials: II. a computational environment for adaptive modeling of heterogeneous elastic solids, Comput. Meth. Appl. Mech. Engrg., 9 (2), pp

16 Figure : Adaptive modeling for the lattices, 2 2,..., 6 6 for eample. For each case the final configuration which achieves a tolerance of 5 percent in the error in the quantit of interest is shown. 6

17 Effectivit Inde.95.9 Relative Error Number of Refinements Number of Refinements Figure : Effectivit indices (left) and relative errors (right) versus the number of refinements for eample. Figure 2: Solutions obtained on the reduced lattices L 3 (after iteration 3) for the lattice (top) and the 3 3 lattice (bottom). 7

18 Figure 3: Accurate approimation of the continuum solution u for eample u.3.2 u Figure 4: Lattice solutions of eample 2 computed on the and 3 3 lattices. 8

19 Figure 5: Sequence of surrogate models obtained b adaptive refinement for the 2 2 lattice (iterations s = to 5). Again, the gre area represents the subregion in which the solution is obtained using the continuum model. The complementar subregion corresponds to the reduced lattice L s. The number l s in the cells indicates the iteration at which the cell has been added to the lattice L s. 9

20 Figure 6: Adaptive modeling for the lattices, 2 2,..., 6 6 for eample 2. For each case the final configuration which achieves a tolerance of percent in the error in the quantit of interest is shown. 2

21 Figure 7: Adaptive modeling for the lattices, 2 2,..., 6 6 for eample 2. For each case the final configuration which achieves a tolerance of 5 percent in the error in the quantit of interest is shown. 2

22 Effectivit Inde 2.5 Relative Error Number of Refinements Number of Refinements Figure 8: Effectivit indices (left) and relative errors (right) versus the number of refinements for eample 2 and γ tol = percent Effectivit Inde 2.5 Relative Error Number of Refinements Number of Refinements Figure 9: Effectivit indices (left) and relative errors (right) versus the number of refinements for eample 2 and γ tol = 5 percent. 22

23 Figure 2: Solutions obtained on the reduced lattices L 2 (after iteration 2) for the lattice (top) and the 3 3 lattice (bottom). 23