DATA STRUCTURES FOR THE NONLINEAR SUBGRID STABILIZATION METHOD

Transcription

1 DATA STRUCTURES FOR TE NONLINEAR SUBGRID STABILIZATION METOD Isaac P. Santos Lucia Catabriga Departamento de Informática Universidade Federal do Espírito Santo - UFES Av. Fernando Ferrari, 54 - Vitória, ES, Brazil CEP: {ipsantos,luciac}@inf.ufes.br Regina C. Almeida Departamento de Mecânica Computacional Laboratório Nacional de Computação Científica - LNCC/MCT Av. Getúlio Vargas, Petrópolis, RJ, Brazil CEP: rcca@lncc.br Abstract. In this work we present a new implementation strateg for the Nonlinear Subgrid Stabilization (NSGS) method (Santos and Almeida, 27, 28). The NSGS method is a methodolog for approimating transport problems, based on a multiscale decomposition of the approimation space. The two-scale viscosit model is obtained b adding to the Galerkin formulation a nonlinear operator acting onl on the unresolved mesh scales. The resulting method is stable and does not require an tune-up parameter. owever, it requires the solution of a sstem twice as large as that related to the final resolved scale resolution. To avoid this drawback, we propose here a new strateg to approimate the effects of the unresolved scales on the resolved scales combined with an element-based and edge-based data structures. The resulting methodolog ields superior performance, which is demonstrated b a variet of numerical eperiments covering the regimes of dominant advection and dominant absorption. Kewords: Nonlinear subgrid viscosit, Edge-based data structure, Krlov iterative methods

2 . INTRODUCTION The numerical solution of transport equations using the Galerkin formulation ma ehibit global spurious oscillations when the diffusive term is smaller than the advective or reactive ones. More accurate and stable results can be obtained using stabilized or multiscale methods. In a general sense, each of these alternative methodologies are based on adding some tpe of linear stabilization to the standard Galerkin formulation and usuall depends on the definition of one or more tuning parameters. The multiscale approach consists of decomposing the approimation space into a resolved coarse scale and an unresolved subgrid scale spaces such that the weak form of the problem is split into two sub-problems: one for the coarse scales and one for the subgrid scales. From this point of view, the attempts to recover stabilit of the resolved scale solution ma be interpreted as a wa of improving the simulation b considering the effects of the smallest scales on the larger ones Brezzi and Marini (22). The Nonlinear Subgrid Stabilization (NSGS) method is a step towards the development of two-scale methods whose stabilit and convergence properties do not rel on tune-up parameters. Its first version was presented in (Santos and Almeida, 27) for advectiondiffusion problems and was based on adding a nonlinear artificial viscosit term onl to the subgrid scales. The amount of the subgrid viscosit depends on assuming an additive decomposition of the velocit field and solving the resolved scale equation at the element level, ielding a self adaptive method. A linear subgrid stabilization was kept when the resolved scale solution was accurate enough. The NSGS method was etended in (Santos and Almeida, 28) to advection-diffusionreaction problems and some improvements were introduced in smooth regions so as to recover the Galerkin formulation for regular resolved scale solutions. To improve the convergence of the linearization process, an alternative iterative algorithm to solve the nonlinear problem was also proposed. The resulting self adapted methodolog has ver good stabilit and convergence properties for a wide range of transport problems, but presents the drawback of requiring the solution of a sstem twice as large as that associated with the resolved scale resolution. To overcome this drawback, in this work we present a strateg for building the NSGS method using element-based (EBE) and edge-based (EDS) implementations (Catabriga et al., 998; Catabriga and Coutinho, 22). The present strateg locall approimates the effects of the subgrid scales on the resolved scales b performing a static condensation of the subgrid degrees of freedom of the associated macro element. Solution efficienc is improved b using an edge based data structure and the final algebraic sstem is solved b using the generalized minimal residual method (GMRES) (Saad, 995). The outline of this paper is as follows. We briefl describe the Nonlinear Subgrid Stabilization (NSGS) method in Section 2. Section 3 presents the proposed strateg of incorporating the subgrid effects into the resolved scale and Section 4 presents the related element-based (EBE) and edge-based (EDS) data structures. Numerical eamples are conducted in Section 5 and Section 6 concludes this paper b pointing out the main achievements, difficulties and perspectives. 2. TE NONLINEAR SUBGRID STABILIZATION METOD Consider the stead scalar advection-diffusion-reaction problem whose solution satisfies ǫ u + β u + σu = f in Ω; () u = on Ω, (2) where Ω R d (d = 2) is an open bounded domain with a Lipschitz boundar Ω and unit outward normal n, β is the velocit field, σ is the reaction coefficient, < ǫ is the

3 (constant) diffusion coefficient, and f is the source term. For simplicit, we onl consider homogeneous Dirichlet boundar conditions and it is assumed that β W, (Ω), σ L (Ω) and f L 2 (Ω). The weak formulation of the problem ()-(2) reads: find u (Ω) such that where B(u, v) = (f, v) v (Ω), (3) B(u, v) = ǫ( u, v) + (β u, v) + (σu, v) (4) and (, ) stands for the usual inner product in L 2 (Ω). It is also assumed that σ and that there eists a constant σ such that σ 2 β σ >. (5) The La-Milgram Lemma implies that the problem (3) has an unique solution. To define the discrete model, we consider a triangular partition T = {T } of the domain Ω where stands for the mesh parameter. From each triangle T T, four triangles are created b connecting the midpoints of the edges. We set h = /2 and denote b T h = {T h } the resulting finer triangulation. A two-level piecewise linear finite element approimation is defined b introducing the following two spaces: X = {u (Ω) u T P (T), T T }; (6) X h = {u h (Ω) u h Th P (T h ), T h T h }. (7) Moreover, we introduce an additional discrete space Xh decomposition holds: X h = X X h, X h, such that the following (8) where X is the resolved (coarse) scale space whereas Xh is the subgrid (fine) scale space. We define also the projection operator P : X h X so that Xh = (I P )X h, where I is the identit operator. Then, given u h X h and u X such that u h and u coincide in the coarse scale nodes, the space decomposition (8) implies that u h = (I P )u h = u h u, u h X h. The couple (X, X h ) will be referred to as the two-level P setting (see Fig. ) Guermond (999). T X nodes X h nodes Figure : Schematic representation of the two-level P setting

4 The Nonlinear Subgrid Stabilization Method (NSGS) (see Santos and Almeida (27, 28) for details) is given b Find u h X h such that B(u h, v h ) + T h T h D(u, u h, v h ) = (f, v h), v h X h, (9) where the non-linear operator D(u, u h, v h ) in each element T h T h is given b with D(u, u h, v h T ) = ξ(u ) u h v h dω, () h ξ(u ) = 2 µ(h) β h and µ(h) stands for the subgrid characteristic lehgth. For the two-level P setting, we use µ(h) = h/2. The subgrid velocit field β h is given b β h = { R(u ) u 2 u, if u ;, otherwise, () where R(u ) = ǫ u + β u + σu f. (2) Setting ξ(u ) = c b h, with c b > constant, we obtain the Linear Subgrid Stabilization (SGS) method (Guermond, 999, 2; Laton, 22). When u =, the formulation (9) recovers the standard Galerkin method. The NSGS method is solved using an iterative procedure for which the SGS solution is the initial guess (with c b = ) and ξ(u ) (or β h ) is delaed one iteration (see Algorithm ). The iterative NSGS process is defined b: Given u n h, we find u n h satisfing B(u n h, v h) + ) u;n h vh dω = (f, v h), v h X h, (3) T h T h T h where ) = c b,t h µ(h) is locall adjusted depending on the residual of the resolved scale solution. The iterative NSGS procedure is given b (Algorithm ), where u i and β;i h stand for the approimate resolved scale solution and the subgrid velocit field, respectivel, at iteration i; maiter is the maimum number of iterations and tol is the prescribed tolerance. Thus, the desired resolved solution u is obtained when the convergence of (Algorithm ) is attained. At each iteration, a linear sstem of the form A = b, with A nonsmmetric, has to be solved in order to determine u i (step 2). The sstem dimension is equal to the number of degrees freedom of the mesh T h. As mentioned before, this is the method major drawback and ma be overcome b appling the strateg proposed in the net section. 3. APPROXIMATING TE SUBGRID EFFECTS In this section we propose a new implementation strateg for the NSGS method, in which the non-linear formulation (9) is solved for the coarse mesh T. Let ũ be the approimation of the resolved scale solution u. ũ is obtained b locall approimating the effects of each

5 Algorithm Require: This algorithm receive as input u - calculated using SGS with c b = Ensure: The output is the approimate solution u : for (each element T h T h ) do 2: c T h c b 3: end for 4: i 5: repeat 6: i i + 7: for (each element T h T h ) do 8: determine β ;i 9: c i T h 2 [ h c i T h + 2 β ;i h : c b,th c i T h : end for 2: determine ( u i ) 3: until i maiter or ma u i ;j u i ;j tol j=,...,dof ] X h v 3 X v 3 T 4 h v 6 v 5 P v +v 3 2 T 4 h v 2 +v 3 2 T h T 3 h v v 2 v 4 T 2 h T h T 3 h v v 2 v +v 2 2 Figure 2: Macro element T T and its corresponding micro elements T h,t 2 h,t 3 h and T 4 h T h: relationship among the degrees of freedom of T and T h, due to the projection operator P. T 2 h

6 subgrid degree of freedom belonging to T T. To show how this approimation is performed, let us first notice that both linear and linearized operators in (3) can be built b summing up the contributions of each macro element T T. Each 6 6 local matri results from assembling the contributions of its corresponding micro elements Th i T, i =, 2, 3, 4 (see Fig. 2). Moreover, the degrees of freedom of the micro elements are connected to the resolved scale nodes of the related macro element due to the linearized operator D(u n ; u;n h, vh ), which prevents the use of an edge based structure. This can be clearl seen b rewriting it in the following wa: ) u;n h vh dω = ) (un h un ) (v h v ) dω T T T T T T = where I = III = T T T T ) un h v hdω; II = T ) un v hdω and IV = ( I II III + IV T ) un h v dω; ) un v dω. ), (4) The term I is similar to the diffusive term coming from the bilinear form B(u n h, v h). In the other three terms (II, III, IV ) the T resolved degrees of freedom are coupled with the T h degrees of freedom. For each Th i T, i =, 2, 3, 4, the former are obtained b projecting the latter into the macro element T using the projection operator P (Fig. 2). After assembling the contributions of each micro element T h T, the local problem associated with each T T ma be defined as K T U T = F T. (5) The local vector of unknowns U T = {u T h;, ut h;2, ut h;3, ut h;4, ut h;5, ut h;6 }t ma also be written as U T = {u T ;, ut ;2, ut ;3, ut h;4, ut h;5, ut h;6 }t, since u h and u coincide in the coarse scale nodes. The local sstem (5) can be partitioned in the following wa [ ][ ] [ ] A B U F = (6) C D U 2 F 2 where U = {u T ;, ut ;2, ut ;3 }t and U 2 = {u T h;4, ut h;5, ut h;6 }t. This local sstem shows quite well the coupling of the resolved scale and the subgrid nodes. Although we are interested in the resolved scale solution, such coupling should not be disregard. Besides, such local connection spreads to the macro element neighbors due to the support of the interpolation functions (twolevel P setting). This propert prevents the use of an EBE and EDS data structure. owever, we ma approimate the problem b disregarding the global subgrid connectivit and performing a static condensation of the unknowns U 2 at each macro element level. Thus, assuming that the matri D is nonsingular, the condensed local problem becomes K T U T = F T (7) where K T = (A BD C) is the 3 3 local matri which approimates the macro element matri of T T. The respective local vector is F T = (F BD F 2 ). After assembling the contributions of all T T, the following new global linear sstem is obtained KŨ = F, (8)

7 where K = nel A T= KT ; F = nel A T= FT ; Ũ = nel A T= UT. (9) In this epression, nel is the total number of macro elements of the coarse mesh T. The linear sstem (8) is then solved for the resolved scale unknown vector Ũ = {ũ ;,..., ũ ;Neq } t, where Neq is the total number of resolved scale nodes. Such approach ields a remarkable decrease of the computational cost, which can also be improved b using an EBE or EDS data structure. The resulting methodolog can be seen as an approimation of the original NSGS method. Although lacking a full mathematical eplanation, the numerical results are promising, as will be discussed in the following. 4. ELEMENT-BASED AND EDGE-BASED STRUCTURES Element-based and Edge-based structures have been etensivel used in finite element implementations, resulting in considerable improvements comparing to standard implementations. The success of this solution strateg requires an efficient implementation of matri-vector products and the choice of a suitable preconditioner. Generall, the edgebased data structure reduces processing time and requires around one half of the storage area to hold the coefficient matri when compared to an enhanced element-based implementation. We perform an implementation of the NSGS method using element-based and edge-based implementations. The conventional finite element data structure associates to each triangle e is its connectivit, that is, the mesh nodes I, J and K. In the edge-based data structure each edge s is associated to the adjacent elements e and f, thus to the nodes I, J, K and L, as shown in Figure 3. J K e s f L I Figure 3: Elements adjacent to edge s, formed b nodes I e J. Each element matri can be disassembled into its contributions to three edges, s, s + and s + 2, with connectivities IJ, JK and KI, that is, = + +, (2) } {{ } element e } {{ } edge s } {{ } edge s + } {{ } edge s + 2

8 where and are matri coefficients defined from (3). Thus, all the contributions belonging to edge s will be present in the adjacent elements e and f. The resulting edge matri is the sum of the corresponding sub-element matrices containing all the contributions to nodes I and J, that is, [ ] [ ] [ ] = +. (2) }{{} edge s }{{} element e }{{} element f Considering a conventional elementwise description of a given finite element mesh, the topological informations are manipulated, generating a new edge-based mesh description. Thus, the assembled global matri given in equation (9) ma be written now as, K = nedges A s= Ks where nedges is the total number of edges of the macro mesh T. The edge matri K s is obtained from the contributions of all the element matrices K T that share the edge s. In the element b element (EBE) implementation strateg, the coefficients of the global matri are stored in each macro element matri as defined b (9). The global matri K is stored in a compact form of size nel 9. On the other hand, in the edge-based (EDS) strateg the coefficients of the global matri, defined b (22), are also stored in a compact form of size nedge NUMERICAL RESULTS In this section we evaluate the numerical performance of the proposed Condensed NSGS on some academic test cases. In all of them the computational domain Ω = (, ) (, ) is discretized b using triangular meshes with 2 2, 4 4 and 8 8 cells, where each cell is subdivided in two triangles (macro elements). Besides comparing the resolved solution in the lights of the condensed and the original NSGS, we investigate the efficienc of both EBE and EDS implementations of the new methodolog. The tolerance of the GMRES algorithm is 7. It is used 5 vectors in the Krlov basis in eamples and 2 and vectors in the eample 3. The convergence of the NSGS method is attained for a prescribed tolerance (tol = 2 to the eamples and 2, and tol = 4 to the eample 3) or maiter = Eample : Advection-diffusion problem This eample simulates a two-dimensional advection dominated advection-diffusion problem with ǫ = 2, β = (, ) and f = σ =. The Dirichlet boundar conditions are given b {,.3 ; u(, ) = u(, ) = u(, ) = and u(, ) =, >.3. These conditions ield a solution with an internal laer in the direction of the velocit field starting at (.3,.) and an eponential eternal laer at =. Figure 4 shows the standard NSGS solution and the EBE/EDS-based condensed NSGS solution using the coarsest mesh (2 2). Thus, the mesh T is composed b 8 macro elements with 36 unknowns, while the mesh T h is formed b 32 elements with 52 unknowns. Note that the EBE/EDS-based condensed NSGS solution, obtained on T, is quite accurate. (22)

9 Table presents the computational performance of the element-based (EBE) and edgebased (EDS) data structures, respectivel, for the condensed NSGS method. In Tab., Neq is the number of unknowns related to coarse mesh and the CPU times are reported. The edgebased approach is about 6% faster than the element-based one. For T with 2 2 and 4 4, the CPU times for the standard NSGS (resolved for T h, direct solver) is 96.6 and seconds, respectivel. Thus, either using EBE or EDS, remarkable computational improvements are obtained with the condensed NSGS strateg (a) standard NSGS (b) EBE/EDS-based condensed NSGS Figure 4: Eample - NSGS method solutions. Table : Eample : Computational costs - EBE and EDS structures Mesh(T ) Neq Time (s) EBE EDS Eample 2: Advection-diffusion problem with source term This eample simulates a two-dimensional advection dominated advection-diffusion problem with ǫ = 9, β = (, ), σ = and a constant source term f =. We set u = on all boundar so that the eact solution is a 45 o slope, possessing parabolic laers at =, = and an eponential laer at =. The standard NSGS solution and the edge-based condensed NSGS solution using the mesh with 2 2 cells are shown in Fig. 5. The standard NSGS method presents some oscillations in the neighborhood of the parabolic laers ( = and = ). The condensed NSGS solution presents a diffusive behavior in the these laers and a small overshoot in =. owever, there is no oscillation at the eternals laers. Table 2 presents the computational performance for the condensed NSGS method considering both data structures. The edge-based approach is about 6% faster than the elementbased one, as in the eample. For T with 2 2 and 4 4, the CPU times for the standard NSGS (resolved for T h, direct solver) is and seconds, respectivel.

10 .2.2 (a) standard NSGS (b) EBE/EDS-based condensed NSGS Figure 5: Eample 2 - NSGS method solutions. Table 2: Eample 2: Computational costs: EBE and EDS structures Mesh(T ) Neq Time (s) EBE EDS Eample 3: Diffusion - reaction problem In this eample, we consider the singular perturbed case where ǫ = 2, σ = and f =.5, with the following boundar conditions: u(, ) = u(, ) = and u(, ) = u(, ) =. The Galerkin solution obtained on macro mesh 2 2 is shown in Fig. 6(a), which presents some locallized oscillations in the neighborhood of the eternal laers. Fig. 6(b) and 6(c) show the corresponding standard and condensed NSGS solution, respectivel. Although some oscillations still remain at eternal laers, the edge-based condensed NSGS solution is more accurate than the NSGS standard one. The computational performance for the condensed EBE and EDS based NSGS implementation are presented in Tab. 3. For this eample the element-based data structure is more effective than the edge-based, that is, the EBE CPU time for all meshes is smaller than the EDS CPU time. Usuall, EDS is more effective than EBE due to the matri-vector product float-point operations. owever, as the number of the GMRES iterations is ver small in the present eample, the solution required fewer number of matri-vector products floatpoint operations. Thus, the computational costs are mainl concentrated on the generation of the edge-based mesh. Table 3: Eample 3: Computational costs: EBE and EDS structures Mesh(T ) Neq() Time (s) EBE EDS

11 (a) Galerkin (b) standard NSGS (c) EBE/EDS-based condensed NSGS Figure 6: Eample 3 - Galerkin and NSGS methods solutions.

12 6. CONCLUSIONS We propose a new implementation strateg for the Nonlinear Subgrid Stabilization (NSGS) method. The NSGS method is a free parameter two-level subgrid method with good stabilit and convergence properties, owing to a subgrid viscosit that acts onl on the unresolved scales. It requires the solution of linear sstems associated with a mesh with characteristic length h = /2 to obtain a solution whose resolution is. The Condensed NSGS method proposed here introduces a procedure to overcome this drawback. This is performed b locall approimating the effects of the unresolved scale, ielded b a static condensation of the not null subgrid degrees of freedom at the macro element level. Then, the Condensed NSGS method can be seen as an approimation of the original NSGS method. Although lacking a full mathematical eplanation, such simple procedure results in a remarkable computational gain. Moreover, some numerical eperiments indicate that it ields etra regularization. Since the resolved scale solution is obtained directl for the macro finite element mesh, we incorporate the EBE and EDS data structures, ielding additional computational efficienc. Contrasting other techniques that use tune-up parameters to improve accurac, the present free Condensed EBE/EDS NSGS method combines simplicit with computational efficienc. Some important issues associated with the convergence of the nonlinear procedure, the mathematical and numerical analsis, application to nonlinear and transient problems remain to be investigated in forthcoming works. Acknowledgements The first author would like to thank the fellowship provided b the Brazilian Government, through the Agenc CNPq/Brazil Pos-Doc Junior (PDJ) 5649/27-3. The second and third authors acknowledge the financial support from the Agenc CNPq/Brazil, contract/grant numbers 6265/26-5 and 3573/27-, respectivel. REFERENCES Brezzi, F. & Marini, L. D., 22. Augmented spaces, two-level methods and stabilizing subgrids. International Journal for Numerical Methods in Fluids, vol. 4, pp Catabriga, L. & Coutinho, A., 22. Implicit SUPG solution of Euler equations using edgebased data structures. Computer Methods in Applied Mechanics and Engineering, vol. 9, pp Catabriga, L., Martins, M. A. D., Coutinho, A. L. G. A., & Alves, J. L. D., 998. Clustered edge-b-edge preconditioners for non-smmetric finite element equations. In Onate, E. & Idelsohn, S. R., eds, Computational Mechanics: New Trends and Applications, pp. 4, Buenos Aires, Argentina. 4th World Congress on Computational Mechanics, CD-ROM. Guermond, J.-L., 999. Stabilization of galerkin approimations of transport equation b subgrid modeling. Mathematical Modelling and Numerical Analsis, vol. 33, pp Guermond, J.-L., 2. Subgrid stabilization of galerkin approimations of linear monotone operators. IMA Journal of Numerical Analsis, vol. 2, pp Laton, W. J., 22. A connection between subgrid scale edd viscosit and mied methods. Appl. Math. and Comput., vol. 33, pp Saad, Y., 995. Iterative methods for sparse linear sstems. PWS Publishing Compan.

13 Santos, I. P. & Almeida, R. C., 27. A nonlinear subgrid method for advection-diffusion problems. Computer Methods in Applied Mechanics and Engineering, vol. 96, pp Santos, I. P. & Almeida, R. C., 28. A free parameter subgrid scale model for transport problems. Computer Methods in Applied Mechanics and Engineering.