PROGRESS ON PRACTICAL METHODS OF ERROR ESTIMATION FOR ENGINEERING CALCULATIONS

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1 ECCM-2001 European Conference on Computational Mechanics June 26-29, 2001 PROGRESS ON PRACTICAL METHODS OF ERROR ESTIMATION FOR ENGINEERING CALCULATIONS J. Tinsley Oden and Serge Prudhomme, The Texas Institute for Computational and Applied Mathamatics The University of Texas at Austin, Austin, TX 78712, U.S.A. Tim Westermann and Jon Bass Altair Engineering Inc. Austin, Texas, U.S.A. Key words: Goal-oriented error estimation, Displacement error, Error in stress, Upper and lower bounds, Dual problem, Residual method. Abstract. In the present paper, the goal-oriented error estimation method is applied to several complex engineering analysis of three-dimensional elastic solids. In the examples considered, the errors are estimated with respect to displacement or stress components at a given point in the solid. The numerical results indicate that the method can be used effectively for complex engineering applications.

2 J.T. Oden, S. Prudhomme, T. Westermann, J. Bass 1 Introduction The use of a posteriori error estimation as a means for verification of accuracies of computer simulations has been a topic of research for nearly two decades. Until recently, such error estimates have been global in nature, giving bounds on global approximation errors in energy-type norms. Extensions of global estimation methods to procedures for estimating errors in local quantities of interest has been described in recent papers [I, 2]. However, applications of these new techniques have been confined to very simple model problems involving scalar-valued functions on one- and two-dimensional domains. In the present investigation, we review the theory of goal-oriented error estimation and discuss applications to several complex engineering analysis of three-dimensional elastic solids. These examples indicate that the goal-oriented methods can be used effectively in quite complex engineering applications. 2 Model problem and approximation error Let 0 be a bounded open domain in lr d, d = 2,3, with Lipschitz continuous boundary ao. The region D is occupied by a linearly elastic material body in static equilibrium under the action of body forces f E (L 2 (D))d and surface tractions g E (L 2 (r N ))d, where r N C ao. The portion of the boundary on which displacements are prescribed is denoted r D, such that rdurn = ao, and rdnrn = 0. The stress 0', related to the displacement u by the constitutive equation (T = ~E : (Vu + Vu T ), satisfies inside the linearly elastic the momentum equations: -V u = f, in D. (1) Here E is the elasticity tensor, which is assumed to satisfy the standard symmetry and uniform ellipticity conditions. Moreover, the displacement and stress are subjected to the boundary conditions: u = 0, { (2) u n=g, Let s be a positive integer and 8 an open region in jrd (8 may stand for the whole domain 0, for a subdomain of 0 or for a portion of the boundary). Denote by HS(8) the Sobolev space of functions in L2(8) with distributional derivatives of order ~ s in L2(8) as well. The space of admissible displacements V is therefore defined as: (3) the boundary values being understood in the sense of traces of Hl-functions. The classical variational form associated with the elasticity problem consists in finding the displacement u E V such that: B(u, v) = F(v), VVEV, (4) 2

3 ECCM-2001, where B(u, v) F(v) = InVv : EVu dx, 1 f.v d+1g' v ds. n fn The bilinear form B(,.) is symmetric positive-definite on Y x Y and induces a norm, the so-called energy norm, Ilvlle =.jb(v, v). (6) Let Ph denote a partition of n, Le. Ph is a finite collection of Ne open elements K i, i = 1,2,..., Ne. The partition is assumed to satisfy 0 = UKiE'Ph K i, and Ki n Kj = 0 whenever i is different from j. The size h K of an element K is measured in terms of its diameter. We associate with the partition the parameter h = maxke'ph hk. In order to approximate shape functions such as the solution u, one constructs a finite element space of hierarchical v h = {v E H1(O); V = 0 on 80 and VIK = v 0 Fj(l, v E PPK(K), VK E Ph} (7) where FK is the affine mapping from the master element K to the element K in the partition, and PpK(K) is the space of polynomial functions defined on K of degree at most PK. We employ here hierarchical shape functions based on the integrated Legendre polynomials [3, 4]. These shape functions can be classified into nodal, edge,face and interior bubble functions in three dimensions. We associate with the partition Ph the parameter P = minke'phpk. Then, the finite element space yh,p, or, for the sake of simplicity in the notation, yh, is defined as: Using the classical Galerkin method, a finite element approximation Uh E yh of U is given as the solution of B(Uh, v) = F(v), (9) The numerical error in the approximation shall be defined as e = U - Uh, e E Y. (5) (8) 3 Goal-oriented error estimates Rather than estimating the numerical error in the energy norm, we aim here at evaluating the error with respect to a quantity of interest. The quantity of interest represents the goal of the computational analysis, and is supposed to be defined mathematically as a bounded linear functional on the space of trial displacements Y. Many quantities of interest actually fall into that category. Let L denote such a functional. Therefore, our goal is to predict the quantity L(u). The numerical error in which we are now interested in is given by the quantity e L E lr, which, by linearity of L, reduces to: (10) 3

4 J.T. Oden, S. Prudhomme, T. Westermann, J. Bass We briefly explain how to estimate L(e). The reader is referred to [2,5] for more details. The starting point is to find a function w which relates the quantity L(e) to the residual 'R obtained from the momentum equation. This function, the so-called influence function, is shown to be the solution of the dual problem: B(v, w) = L(v), VVEV, (11) and is approximated in Vh by the dinite element solution wh E V h of the discrete dual problem: B(v, Wh) = L(v), (12) As for the error in the solution, we introduce the error in the approximate influence function as g = W - Who Then follows the remarkable result which establishes the relation between the error quantity L(e) and the energy norms of linear combinations of e and g. In Equation (13), s defines a scaling factor in R Several methods to obtain bounds on errors in the energy norm are now available in the literature. In the following numerical experiments, we shall use a simple method described in [6]. In order to derive an estimate of and bounds on L( e), we suppose that the following global error estimates are available: 1J~w ~ lise + s-lglle ~ 1J~p' 1J1~w ~ lise - s-lglle ~ 1J;;µp' Then, it is straightforward to show that the quantities 1Jtw and 1J!;µp, defined as: (14) L _ 1( +)2 1( - )2 1J/ow - 4 1J/ow - 4 1Jupp, L _ 1( +)2 1( - )2 1JIIPP - 4 1JIIPP - 4 1J/ow. (15) produce respectively a lower and an upper bounds on L( e), i.e. 1Jtw ~ L( e) ~ 1J~P' (16) An estimate of L(e) can be derived by taking the average of 1Jtw and 1J/~P' that is: L 1( L L 1Jest = "2 1J/ow + 1Jupp)' (17) Finally, the value of s is chosen so that the quantities Jlselleand Jls-lglle have same amplitudes, i.e. IIseile = Iis-lglle, which implies that: 8= (18) Such a choice of s is justified because it minimizes the quantities Jlse+ g/sll~ and lise - g/sll~. 4

5 ECCM-2001, Figure 1: Mesh topology and boundary conditions for cantelever beam. 4 Numerical experiments The main purpose of the present paper is to provide an account on the performance of goaloriented error estimation for three-dimensional elasticity problems. Our estimates and error bounds are verified by comparing the results with a refined and/or enriched mesh which is believed to provide a nearly "exact solution". We consider here as quantities of interest pointwise displacements and stresses. By poinwise quantities, we actually mean to take local averages using mollifying kernels defined in the form: ifr < c otherwise (19) The constant C corresponds to a normalizing factor and is chosen such that In ke;(x) dx = 1. It follows that the quantity of interest corresponding to the displacement at a point Xo E 0 will be represented by the linear functional: L(u) = Inke(lx - xol)u(x)dx ~ u(xo). (20) Likewise, for the stress a xx at a given point Xo, we shall consider the quantity of interest: L(u) = Inke;(lx - xol)axxdx ~ axx(xo). (21) The parameter c can be prescribed by the user and varies according to the problem and quantity of interest. In most cases, c is chosen of the order of the mesh size h. 4.1 The cantilever beam In the first example, we consider a cantilever beam with dimension L = (10,1,2) subjected to a uniform load. The beam is fixed at x = O. The mesh topology and boundary conditions are shown in Fig. 1. The material properties are set to E = 200 OPa and v = 0.3 where E is the Young's modulus and v the Poisson's ratio. 5

6 J.T. Oden, S. Prudhomme, T. Westermann, J. Bass Mesh Estimated error "Exact error" Elements p Estimate Lower Upper L(u - Uh) 4x1x x1x x 2 x x 4 x Table 1: Error estimates and bounds for the quantity of interest (displacement Uz at Xo (10.0,0.5, 1.0) in the cantilever beam). InIIueru:e fwlction for W iu x direction ~~~ InOueuco function for W in \' direction Ih1)(rSolve l.ot4s......,!"......,..,... Figure 2: Influence function corresponding to the displacement Uz at Xo = (10.0,0.5,1.0) (left) in x-direction (right) in y-direction Displacement at a point We estimate here the error in the displacement Uz (in direction z) at the free end-point Xo = (10.0,0.5,1.0). We select a sequence of meshes which are uniformly refined by dividing each element into two new elements in one or several directions. Estimates and bounds on the error in the displacement are shown in Table 1. The "exact errors" are computed here by substracting the discrete displacement from the displacement obtained on the refined mesh 32 x 4 x 8 with spectral order 2. The displacement at (10.0,0.5,1.0) on the mesh of reference reads We observe in Table 1 that the bounds and estimates, although very accurate, all slightly underestimate the exact errors. We attribute this behavior to the fact that the global estimates of the solution and influence function in the energy norm use edge bubble functions only and hence are rather coarse. Finally, we show in Fig. 2 the x- and y-component of the corresponding influence function Stress at a point In the next example, we estimate the error in the stress component a xx at Xo = (5.0,0.5,2.0). This point is located exactly at the center of the upper face of the beam, as it can be observed from the shape of the influence function shown in Fig. 3. As before we collect in Table 2 the estimates of the errors in the quantity of interest. This time, the estimates are rather accurate, but 6

7 ECCM-2001, Mesh Estimated error "Exact error" h p Estimate Lower Upper L(u - Uh) 4x1x x1x2 1 -] x 2 x x 4 x Table 2: Error estimates and bounds for the quantity of interest (Stress O'xx at Xo = (5.0,0.5,2.0) in the cantilever beam). Hu)erSoh"C' HvperSolv~ 02 Figure 3: Influence function corresponding to the stress O'xx at Xo = (5.0,0.5,2.0) (left) in x-direction (right) in z-direction. the bounds have the tendency to diverge from the exact error. This behavior was also observed in [2] for two-dimensional problems. We finally show in Fig. 4 the distribution of the elementwise contributions to the error in the displacement and stress component to be used as refinement indicators. We clearly observe the error in the displacement at the free-end is sensitive to the inaccuracy of the solution at the fixed-end, as this can be expected. On the other hand, we notice that the error in the stress has a more local origin. 4.2 The "hoop" problem The "hoop" problem is concerned with the simulation of a portion of a tire under a specific loading, as shown in Fig. 5. The blue arrows show the direction in which the hoop is held fixed while the orange arrows indicate a uniformly distributed load. The elements of the mesh are of order 2 along the surface of the tire and linear in the transversal direction and the provided mesh consists of 200 elements. Here, we analyze the stress component O'xx at Xo = (16,61,37) and show the errors for a sequence of meshes. In this case, as shown in Table 3, the bounds for the quantity of interest 7

8 J.T. Oden, S. Prudhomme, T. Westermann, J. Bass Contributions to the error ingol for W ~'i'ig,i~~e h.o.,... 1.&. 0" eo.dbuljom 10lhe enoc!nqoi for SIOl ,0.+00 rf' t.o.o.t I~ & Figure 4: Distribution of the contributions to the error in the quantity of interest (left) for the displacement at Xo = (10.0,0.5, 1.0) (right) for the stress at Xo = (5.0,0.5,2.0). Mesh and Loads lit_solve Figure 5: Mesh topology and boundary conditions for the "hoop" problem. are more accurate as compared to the beam problem. We also show in Fig. 6 the corresponding influence function and the distribution of the contributions to the error in the stress component. 4.3 The control arm In the final example, we consider a control arm whose geometry and loading are shown in Fig. 7. Once again, we are interested here in the stress component a xx whose distribution we show in Fig. 7 as well. The goal is to estimate the error in the computed stress at location Xo = (150,25,0) on the given mesh. We found the following estimate and bounds: rl:s~'" = CT",,,, 2 T/lolV = T/I~;; = to be compared with the "exact" error in the stress e(a xx ) = obtained by using a refined mesh with higher spectral order. We also show a plot of the influence function and the 8

9 ECCM-2001, Mesh Estimated error "Exact error" Elements p Estimate Lower Upper Initial mesh (1,2) Refine in (zeta) (1,2) Refine in (xi, zeta) (1,2) Table 3: Error estimates and bounds for the quantity of interest (Stress O'xx at Xo = (16,61,37) for the hoop problem). IIVJ'l(rSolve ~lothcem:cinq..~r[(lhellfui ~~I,. ",. I.' I ID )p I " "D Figure 6: Distribution of the contributions to the error in the quantity of interest (left) for the displacement at Xo = (57,23,37) (right) for the stress at Xo = (16,61,37). distribution of the error in the stress component in Fig. 8. On the latter, we observe that the contributions are not just local and that remote locations do contribute as well. An adapted mesh based on these elementwise contributions would then control the error in the quantity of interest by taking care of all contributing sources, near and remote. 5 Conclusions In this paper, we have applied the goal-oriented error estimation technique to the case of threedimensional linear elasticity problems. We have concentrated on estimating pointwise errors in displacement and stress components. The results obtained on the various examples shown here are very promising. In all cases, the error was predicted with very good accuracy. We would like to point out that relatively simple estimates of the error in the energy norm were used in those numerical experiments, which, to our own surprise, provided us with very reasonable estimates and bounds on the error. Acknowledgements This research was supported by a contract with the General Motors Research and Development Center. This support and the encouragement of Dr. Mark Botkin of GM are greatfully acknowledged. 9

10 J.T. Oden, S. Prudhomme, T. Westermann, J. Bass Control ami JbP<'rSolve Stress Sxx M~W,'~e ,;;;- "'",0> 100,... '''''' '00,(. - '00..., Figure 7: The control arm problem: (left) Geometry and loading (right) Distribution of the stress component axx. Influence function " 11 -~. u (.". Figure 8: The control arm problem: (left) Influence function in x-direction corresponding to the pointwise stress axx atxo = (150,25,0) (right) Distribution of the error in the stress component axx at Xo = (150,25,0). References [1] S. Prudhomme and J.T. aden. On goal-oriented error estimation for elliptic problems: Application to the control of pointwise errors. Compo Meth. in Appl. Mech. and Eng., 176, , (1999). [2] J.T. aden and S. Prudhomme. Goal-oriented error estimation and adaptivity for the finite element method. Computers Math. Applic., 41(5-6), , (2001). [3] B. Szab6 and J. Babuska. Finite Element Analysis. John Wiley & Sons, (1991). [4] J.T. aden. Optimal h-p finite element methods. Compo Meth. in Appl. Mech. and Eng., 112, , (1994). 10

11 ECCM-2001, [5] M. Ainsworth and J.T. Oden. A Posteriori Error Estimation in Finite Element Analysis. John Wiley & Sons, (2000). [6] S. Prudhomme and J.T. Oden. Simple techniques to improve the reliability of a posteriori error estimates for finite element approximations. ECCN 2001, (2001). 11