First Edition: Theory of Global Estimation of Modeling Error J. Tinsley Oden, 1 Serge Prudhomme, 1 Daniel C. Hammerand, 2 and Mieczyslaw S. Kuczma 3 1

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1 First Edition: Theory of Global Estimation of Modeling Error J. Tinsley Oden, 1 Serge Prudhomme, 1 Daniel C. Hammerand, 2 and Mieczyslaw S. Kuczma 3 1 Texas Institute for Computational and Applied Mathematics The University of Texas at Austin 2 TICAM Visiting Scientist, Sandia National Laboratories 3 TICAM Visiting Scientist, Poznan University of Technology Abstract. In this report, computable global bounds on errors due to the use of various mathematical models of physical phenomena are derived. The procedure involves identifying a so-called fine model among a class of models of certain events and then using that model as a datum with respect to which coarser models can be compared. The error inherent in a coarse model, compared to the fine datum, can be bounded by residual functionals unambiguously defined by solutions of the coarse model. Whenever there exist hierarchical classes of models in which levels of sophistication of various coarse models can be defined, an adaptive modeling strategy can be implemented to control modeling error. In the present work, the class of models is within those embodied in nonlinear continuum mechanics. Keywords: mathematical model, modeling error, continuum mechanics, model adaptivity 1 Introduction Computational Engineering and Science (CES) is concerned with the use of mathematical and computational models to simulate physical events and engi- 1

2 neering systems. In recent decades, the remarkable predictivepower of computer models has led to advances in virtually every area of science and technology. Still, the most basic and fundamental step in developing a computer simulation of natural events is the selection of the model itself, a step left largely to heuristic arguments, judgment of the analyst, or based on incomplete empirical data. This most important step in computer simulation, if not done correctly, can be, and generally is, the source of the largest error. In recognition of this fact, code validation" (defined tersely by Roache [9, p.13] as solving the right equations"), has become a major issue in CES. According to Roache, Ultimately, code validation will come down to comparison (directly or indirectly) of code predictions with physical experiments." While we certainly agree with this pronouncement, we offer a supplementary approach to raw experimental validation that we refer to as hierarchical modeling, by which we mean the selection of various models from a large class of mathematical models of varying complexity and sophistication. The issue of validation, then, is first set in a mathematical framework in which modeling error can be defined with some precision, it being understood that various properties of the datum with respect to which such models are assessed must, itself, ultimately be compared with physical experiments. The notion of hierarchical modeling and modeling error estimation was first advanced in connection with errors due to homogenization of heterogeneous materials [4, 5, 7, 8, 10] and of errors inherent in dimensional reduction to produce plate and shell models of three-dimensional bodies [6, 1, 2]. In these works, the notion of model adaptivity is introduced, in which modeling error estimates are used to adaptively select models capable of producing acceptable simulations (compared to a datum model). In the present report, we present a general theory of error estimation and adaptivity within the broad framework of nonlinear continuum mechanics. We derive global error bounds for models of general classes of materials, and we present an algorithm for model adaptivity. The organization of the work is outlined as follows: 1. Introduction 2. Preliminaries: Problem Setting and Notation 2.1 Fine and Coarse Models 2.2 Problem Setting: The Fine Model 2.3 Functional Setting 2

3 2.4 The Coarse Model 3. A Global Estimate of Modeling Error 3.1 Residual Modeling Error 3.2 Remarks and Extensions The Space V Fine Model Density The Error Measure E Other Model Errors 4. Model Adaptivity Based on Global Bounds References 2 Preliminaries: Problem Setting and Notation 2.1 Fine and Coarse Models Our general goal is to estimate and control error due to the choice of a mathematical model within a class of possible models of certain physical phenomena. The modeling error is measured relative to what we shall call the fine model, which itself is also a mathematical model, but generally one of such detail, sophistication, and complexity, that all phenomena of interest are confidently captured, predicted, and simulated by it with adequate accuracy. All other models within the class of models are coarser or simplified models. The fine model, in general, maybeintractable; while general and inclusive, it is often too complex to be used to obtain quantitative results. The solution" of the fine model are, thus, never actually computed (except for possibly very special cases). The fine model is used only for a datum with respect to which modeling error in coarser models is measured. The fine model, for example, may characterize phenomena occurring at many spatial and temporal scales, and may include many interacting physical effects, while the coarser model may be characterized by averaged mechanical properties and simplified physical laws. Ultimately, the suitability of the fine model itself must be estimated by determining its predictive limits within the context of still larger classes of models or, more specifically, through physical experiments. 3

4 In this investigation, the fine model is provided by the nonlinear theory of continuous media: the fundamental laws and equations of nonlinear continuum mechanics. 2.2 Problem Setting: The Fine Model We begin the characterization of the fine model by considering a material body in motion and under the action of prescribed external forces. We adopt a material ( Lagrangian") description of the motion in which the material particles (points) of the body in a fixed reference configuration are identified with positions X of points in the cosure of an open bounded region ρ R N ; N =1; 2, or 3. The motion is characterized by the function χ: R +! R N ; Ω t = χ( ;t) with x = χ(x;t) the place of material particle X at time t. The displacement of X at time t is the vector u(x;t)=χ(x;t) X; X 2 : In classical continuum mechanics, the motion (and deformation) of the body is governed by the linear and angular momentum equations, ρ 0 2 (X;t) Div (F (u; X;t)P (X;t)) = ρ 0(X)b(X;t) P (X;t)=P (X;t) T 9 = ; (2.1) for X 2 ; t > 0, where Div is the divergence operator in the reference configuration and ρ 0 = the mass density in the reference configuration F = rχ = I + ru = the deformation gradient (I being the unit tensor) P = the second Piola-Kirchhoff stress tensor b = the body force per unit mass in the reference configuration The displacement field (equivalently, the motion) satisfies the boundary conditions, u(x;t)=u(x;t); X 2 0 D ; t 0 F (u; X;t)P (X;t) n 0 (X) =g(x;t); X 2 0 D ; t 0 ) (2.2) 4

5 where u is a prescribed displacement on a portion 0 D of the and g is a prescribed surface traction per unit area in the reference configuration on a portion 0 = 0 [ D 0 N. At t =0,u also satisfies initial conditions, u(x; 0) = U (X; 0) = V 0(X) (2.3) U 0 and V 0 being prescribed initial displacements and velocities, respectively. Equations (2.1)-(2.2) define a classical model of the behavior of continuous media without regard to regularity of the data, the constitutive equations for stress or other physical quantities, and without thermodynamic considerations. To put the fine model in a more general and useful setting, we shall now introduce a weak or variational form of the model. 2.3 Functional Setting Let us introduce the space V (= V ( )) of kinematically admissible displacements with domain. Typically, we can take V = n v 2 H 1 ( ) N ; fl D v = 0 on 0 D o (2.4) where fl D is the trace operator. In many cases, depending on the structure and definition of P, more general spaces may be identified for V (e.g. (H 1 (Ω)) N may be replaced by (W 1;p (Ω)) N ; p 6= 2). The Piola-Kirchhoff stress P is assumed to be determined by histories of the gradient of the motion (displacement), the temperature, and other internal variables, entropy, etc.), by constitutive equations of the form P = P ru t (s); A t (s) where we suppress the dependence on X and t for simplicity, and where ru t (s) = the displacement gradient history = fru( ;t s); u( ; t s) 2 V; t» s» 0g and A t (s) denotes the histories of other constitutive variables. Hereafter, we shall suppress the histories A t (s) and write, for simplicity, P = P(ru t (s);t). 5

6 Summing up, a weak or variational form of the momentum equation is embodied in the initial-value problem: Given ( ; 0 D; 0 N;ρ 0 ; b; g; U 0 ; V 0 ), find, for each t 0, the displacement field u( ;t) 2 V + fug and the stress P = P(ru t (s);t) such that ρ 0 ü(t) v dx + F (u; t)p ru t (s);t : rv dx = F(u(t); v;t) 8 v 2 V; t>0 (2.5) wherein u; _u satisfy (2.3) and u 2 V is such that fl 0 U = u on 0 D: Here ü 2 u=@t 2, _u dx = dx 1 dx 2 dx 3, and the dependence of the various integrand functions on X is understood. The functional on the right side of (2.5) is the loading functional, F(u(t); v;t)= ρ 0 b(t) v dx + 0 N g(t) v ds 0 (2.6) ds 0 being an element of surface area on 0 N We note that, in terms of the tractions eg per unit surface area in the current configuration χ ( 0 N)= N, g(x;t) = det F (u; t) fi fi F (u; t) T n 0 (X) fi fi eg (χ(x;t);t) : (2.7) For example, if eg is a pressure load of intensity ß on N, (see, e.g., Ciarlet [3]), g(x;t)= ß det F (u; t) F (u; t) T n 0 (X): Thus F(u(t); v;t)may be a highly nonlinear function of u(t) (precisely, ru(t)); hence the notation used on the left of (2.6). 2.4 The Coarse Model We now consider the motion of a material body occupying the same reference configuration at t = 0 as that appearing in the fine model and subjected to the same external forces (b; g), boundary conditions, and initial conditions, but whose mechanical response is characterized by a simplified" collection of 6

7 constitutive equations. The displacement field in this case is denoted u 0. Instead of (2.7), the weak form of the linear momentum equation for the coarse model is ρ 0 ü 0 (t) v dx + F (u 0 ; t)p 0 ru t 0(s);t : rv dx = F(u 0 ; v;t) 8 v 2 V; t>0 (2.8) where u 0 ( ;t) 2 V + fug and P = P 0 (ru t 0(s);t) satisfy the same boundary and initial conditions as u and P in (2.5). It may also be possible that the density ρ for the coarse model differs from that of the fine model. We address this possibility later. 3 A Global Estimate of Modeling Error 3.1 Residual Modeling Error Let us assume that the motion (particularly, the displacement field) u 0 (t) 2 V of the coarse model is known for all t 2 [0;T]. Then the modeling error at time t is defined by the functions eu(t) =u(t) u 0 (t) E P (t) =P (ru t (s);t) P (ru t 0(s);t) e F (v;t)=f(u(t); v;t) F(u 0 (t); v;t) 9 >= >; (3.1) Introducing (3.1) into (2.7) (by eliminating u(t), P(u t (s);t) and F(u(t); v;t)), we arrive at the equation ρ 0 ëu(t) v dx + + reu(t)p(ru t 0(s);t): rv dx F (u 0 + eu; t)e P (t): rv dx e F (v;t)=r(u t 0(s); v;t); 8v 2 V; t>0 (3.2) where the functional R( ; v;t) is the global modeling residual, R(u t 0(s); v;t)= F (u 0 ; t) P ru t 0(s);t : rv dx (3.3) 7

8 with P the stress error, P ru0(s);t t := P 0 ru t 0(s);t P ru0(s);t t : (3.4) Thus, once the coarse model is used to obtain u 0 (t) for all t 2 [0;T] of interest, the residual functional R( ; v;t) is determined from known data (modulo other mechanical and thermodynamical effects characterized by other constitutive equations, constraints, and the balance of energy equations.) Next, let E denote the linear functional on V, defined by the nonlinear evolutions of error components on the left side of (3.2): E(v;t):= ρ 0 ëu(t) v dx + + reu(t)p(ru t 0(s);t): rv dx F (u 0 + eu; t)e P (t): rv dx e F (v;t); With these notations, (3.2) can be rewritten compactly as 8v 2 V; t>0: (3.5) E(v;t)=R (u t 0(s); v;t) ; 8 v 2 V; t>0 : (3.6) From (3.3) and Schwarz's inequality, R u t 0(s); v;t» (t)kvk 1;Ω0 (3.7) where (t) istheglobal, a posteriori, modeling error indicator, ρ ff 1=2 (t) = (3.8) and kvk 1;Ω0 = fi fif (u0 (t); t) P(ru t 0(s);t) fi fi 2 dx ρ ff 1=2 (v v + rv : rv) dx : (3.9) Introducing (3.7) into (3.6), dividing by kvk 1;Ω0, and taking the supremum of the result over all V, we have the global error bound: ke(t)k V 0» (t) : (3.10) Here k k V 0 denotes the norm of the dual space V 0 of V. 8

9 3.2 Remarks and Extensions Several remarks are in order concerning the error bound (3.10) and the steps leading to it The Space V The spaces of admissible displacement for the fine and coarse models may differ. Thus, (2.8) maybevalid for all v in a smaller space V 0, the ensuing error analysis remains valid whenever V 0 ρ V Fine Model Density The mass density ρ of the fine and coarse models may differ. For example, the fine model may involve characterizations of heterogeneous material with ρ 0 = ρ 0 (X) varying rapidly over, while the coarse model may involve the use of effective properties of materials with ρ 0 a constant. Let ρ 00 denote the mass R density field in the coarse model. The first term on the left of (2.8) becomes ρ 00 ü 0 v dx and, instead of (3.6), we have, for v 2 V, where be(v;t)=e(v;t) e ρ ü 0 (t) v dx (3.11) e ρ = ρ 0 ρ 00 : (3.12) The remainder of the analysis is the same with E replaced by b E The Error Measure E It is not necessary to calculate the quantity kek V 0. Inequality (3.10) merely indicates what measure of error is bounded by the residual indicator at time t. Nevertheless, an interpretation of this error can be given by introducing the H 1 -inner product, (u; v) 1;Ω0 : = (ru: rv + u v)dx; u; v 2 V: (3.13) 9

10 Then Next, let G(t) be the solution of the boundary-value problem, (G(t); v) 1;Ω0 = E(v;t); 8 v 2 V: (3.14) kg(t)k 1;Ω0 = ke(t)k V 0» (t): (3.15) The error measure G(t) can thus be thought of as an H 1 -projection of the various error functionals defined by E. Indeed, if h ; i denotes duality paring on V 0 V, and v = ' 2 D( ), h( + I)G(t) E(t); 'i =0; 8 ' 2 D( ) (3.16) where is the vector-laplacian. Then G(t) represents the global smoothing, G(t) =( + I) 1 E(t) Other Modeling Errors The bound (3.10) is obtained by comparing only models of the linear momentum equation; (angular momentum is presumed to be conserved in both fine and coarse models). Other physical processes may be part of the fine and coarse models which are not adequately captured by this analysis. For a more complete picture, thermodynamic models must be included and we must augment the momentum equation with the energy equations and possibly with other constraints not shared by the coarse model. We shall take up these issues in a later work. 4 Model Adaptivity Based on Global Bounds is partitioned into a collection of non- Let us suppose that the domain overlapping subdomains K : P : = N (P) [ K=1 K ; K J =?; K 6= J: (4.17) The global model error indicator (t) of (3.8) can be written as the sum, (t) = 8 < N (P) X : K=1 2 K(t) 9 = ; 1=2 (4.18) 10

11 where 2 K(t) = with P defined by (3.4). K fi fif (u0 (t); t) P ru t 0(s);t fi fi 2 dx (4.19) The K (t) are thus the contributions of the error over K to the global modeling error (t). They do not represent local modeling errors, as the actual local error is generally polluted by errors in remote subdomains. Nevertheless, the K (t) are used as an indication of the relative error in various subdomains and to identify subdomains where the error in the predictions of the coarse model may belarge. For example, if K (t) <ffmax K2P where ff 2 (0; 1) is a preselected parameter, we may declare the coarse model acceptable for subdomain K ; otherwise, the modeling error must be reduced by adding more sophistication to the coarse model. Exactly how this can be done will vary depending on characteristics of the fine model and, since the fine model is often intractable, the process can fail if an error tolerance cannot be met by any coarser model in the class. K ; References [1] J. R. Cho and J. T. Oden. A priori modeling error estimates of hierarchical models for elasticity problems for plate- and shell-like structures. Mathematics of Computer Modeling, 23(10): , [2] J. R. Cho and J. T. Oden. Local a posteriori error estimation of hierarchical models for plate- and shell-like structures. Computers in Mathematics with Applications, 149:33 48, [3] P. G. Ciarlet. Mathematical Elasticity, Volume 1: Three Dimensional Elasticity, volume 20 of Stud. Math. and Appl. North-Holland, Amsterdam, [4] N. Moës, J. T. Oden, and K. Vemaganti. Atwo-scale strategy and a posteriori error estimation for modeling heterogeneous structures. In P. Ladev eze and J. T. Oden, editors, Advances in Adaptive Computational Methods in Mechanics, volume 47 of Stud. Appl. Mech., pages Elsevier, Amsterdam,

12 [5] N. Moës, J. T. Oden, K. Vemaganti, and J.-F. Remacle. Simplified methods and a posteriori error estimation for the homogenization of representative volume elements (RVE). Comput. Methods Appl. Mech. Engrg., 176: , [6] J. T. Oden and J. R. Cho. Adaptive hpq-finite element methods of hierarchical models for plate-and shell-like structures. Comput. Methods Appl. Mech. Engrg., 136(3-4): , [7] J. T. Oden and K. Vemaganti. Adaptive hierarchical modeling of heterogeneous structures. predictability: quantifying uncertainty in models of complex phenomena. Phys. D, 133: , [8] J. T. Oden and K. Vemaganti. Estimation of local modeling error and goaloriented modeling of heterogeneous materials; Part I: error estimates and adaptive algorithms. Journal of Computational Physics, To appear. [9] P. J. Roache. Verification and Validation in Computational Science and Engineering. Hermosa, Albuquerque, NM, [10] K. Vemaganti and J. T. Oden. Estimation of local modeling error and goal-oriented modeling of heterogeneous materials; Part II: A computational environment for adaptive modeling of heterogeneous elastic solids. In preparation. 12