Towards a combined use of Kalman filtering and Error in Constitutive Relation 

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Energy-based functionals. Introduction to the Error in Constitutive Relation

In the case of the least-squares or the auxiliary fields approaches, the quality of a model is measured by either the distance between the measured data to the solution of the direct problem or the reciprocity gap. On the contrary, the energy-based functionals propose to measure the model error by evaluating the difference between kinematically and statically admissible fields using an energy norm. Finding the best admissible fields itself leads to the resolution of a secondary problem, leading to a two-step inverse solution procedure for the general identification process.
This technique has first been introduced by P. Ladeveze` in 1975 as a method to evaluate the quality of the solution of a FE model [55], where the concept of Error in constitutive Relation (ECR) first appeared. Since then, several versions and many applications have been proposed, see for example [57, 60, 70] for model quality assessment or [19, 37, 42, 2, 67, 26] for model updating and identification problems.
The background idea of Error in Constitutive Relation can be introduced from two different points of view:
• A first approach valid for Generalized Standard Materials (GSM) [43]. For such materials, the constitutive equations are described from the expressions of the energy potential and the dissi-pation potential. The difference between two admissible states associated with both force and displacement boundary conditions are characterized by the residual:
e(ε, σ) = φ(σ) + φ∗ (ε) − σ : ε (1.4)
where φ∗ is the potential defined by the Legendre-Fenchel transformation of φ. In the case of elasticity, those potentials are defined by:
1 T −1 ∗ 1 T
φ(σ) = σ K σ, φ (ε) = ε K ε (1.5)
where K is the Hooke tensor. This error is positive-definite and vanishes when the admissible states are compatible with the constitutive relation. Thus, it can be used as a measure to build ECR functionals.
• The second point of view comes from the principle of stability in the sense of Drucker which, for a given structure, stipulates that for any couple of evolution states (considering the same initial state) the following inequality stands:
(σ2 − σ1 ) : (ε2 − ε1 )dzdt ≥ 0∀t ∈ [0, T ], ∀z ∈ Ω (1.6)
It can be shown that the Drucker error vanishes for the case where the two couples (σ1 , ǫ1 ) and (σ2 , ǫ2 ) are compatible with both the history of boundary conditions and the constitutive relation, the Drucker error being positive otherwise. Within the small perturbations principle, the response of a given structure is unique for all evolution coming from the same boundary condition history. This uniqueness property is verified in most of the constitutive relations such as elasticity, plastic-ity, viscoelasticity, etc. except from some singular cases (damage laws). Indeed, the integration of the virtual work equations for a given structure gives:
T T 1 − u˙2 )2 dzdt = 0
Ω (σ2 − σ1 ) : (ε2 − ε1 )dzdt + ρ(u˙1 (1.7)
0 0Ω 2
and considering the Drucker inequality (1.6) one can easily obtain:
1 ρ(u˙1 − u˙2 )2 dzdt ≤ 0 t [0, T ], z ∈ Ω (1.8)
Ω 2 ∀ ∈ ∀
Hence, it is clear that for a given structure u˙1 and u˙2 are necessarily the same for all t ∈ [0, T ] as the two evolutions have the same initial condition. Thus, deriving this expression, the history of deformations must be identical and so has to be the history of constrains from the constitutive relation law. This hypothesis has therefore been exploited to define a residual indicator based in the Drucker inequality as a measure of the compatibility of the constitutive relation of a structure with respect to the boundary conditions.
Since the introduction of such a concept, several studies have successfully applied this principle to different applications such as model verification [58] and model updating problems [56, 59], under linear or nonlinear conditions [10, 19], either in the frequency domain [27] or the time domain [36, 37]. Consequently, the constitutive relation appears to be an appropriate indicator of the quality of a model with respect to measured data history and some particularly good properties deserve to be highlighted:
• Excellent ability to locate erroneously modeled regions in space. Indeed, in [10] it is demonstrated that regions where the ECR density is high correspond to those which contain the most erroneous constitutive relations.
• Strong robustness in presence of noisy data [37].
• Good convexity properties of cost functions [42].
In the following, classical formulations of the ECR are reviewed for the static and the dynamic cases. Moreover, a frequency domain formulation is presented with a special interest for the case of a FE framework.

Classic formulation. The static case.

In the field of static elasticity, the ECR can be formulated in one of its most classical forms. Consider the case where ∂0 Ω = ∅ and ∂uf Ω = ∅. Thus, the elasticity tensor C that better represents the available data u¯ and f is sought by solving the following minimization problem:
Find the kinematic admissible field uK A ∈ U (u¯), the static admissible field σSA ∈ S(f ), and the constitutive relation C that minimize:
J (uK A, σSA, C) = Ω (σSA − C : ε(uK A)) : C−1 : (σSA − C : ε(uK A))dΩ (1.9)
where the admissible spaces are defined by:
U (u¯) = {u(z) s.r. |u(z) = u¯ for z ∈ ∂uΩ} S(f ) = {σ(z) s.r. |σ(z) • n = f for z ∈ ∂f Ω, div(σ(z))
In this static case, it can be shown that the solution fields uK A and σSA are uncoupled. In practice, this means that both fields can be computed separately as a Neumann and a Dirichlet problem respec-tively.

Modified formulation. The dynamic case.

In the case of elastodynamics, nevertheless, it has been proved in [36, 2] that the resolution of an ECR introduces an additional issue since the kinematic and dynamically admissible fields are coupled not only by the constitutive relation (1.2) but also by the equilibrium equation (1.1). In addition, a regularization technique is introduced to deal with the presence of noisy measurements. In this context, a modified approach has been proposed by Feissel and Allix in [36]. In their investigations, they considered the case with ∂0 Ω = ∅ and ∂f Ω = ∂uΩ = ∂f uΩ. Their formulation of the ECR problem is summarized below:
Find the kinematic admissible field uK A ∈ U (ub), the dynamic admissible field σDA ∈ S(fb, u), and the constitutive relation C that minimize: J (uK A, σDA, C) = 0Ω (σDA − C : ε(uK A)) : C−1 : (σDA − C : ε(uK A))dΩdt T (1.10) + du(ub, u¯)dS + dt df (fb, f )dS 0 ∂u Ω ∂f Ω
where the admissible spaces are defined by:
U (ub) = {u(z, t) s.r. |u(z, t) = ub for z ∈ ∂uΩ, u(z, 0) = u0, u˙(z, 0) = u˙0 }
S(fb, u) = {σ(z, t) s.r. |σ(z, t) • n = fb for z ∈ ∂f Ω, −ρu¨(z, t) + div(σ(z, t)) = 0 for z ∈ Ω}
where the terms du and df represent a discrepancy measure to be defined (usually based on the L2 norm, e.g. du(v, w) = v − w 2 )).
In all the cases, the ECR approach relies upon distinguishing between two sets of relations: reliable and unreliable. The latter will therefore be relaxed by simply introducing them in the ECR functional and finding a solution that best fulfills them. In the case of elastodynamics defined in (1.10) the following sets of relations can be introduced:
• Reliable relations:
−ρu¨ + div(σ) = 0
u˙(z, 0) = u˙0
u(z, 0) = u0 (1.11)
• Unreliable relations and quantities:
σ=Cε
u¯ (1.12)
An interesting work by Feissel, Allix and Nguyen in [37, 67] has recently shown that in the identifi-cation process by means of ECR functionals, the obtainment of admissible fields uK A and σDA and the correction of the constitutive relation C are two well-separated steps and therefore one can use different functionals to solve each problem. In particular in one of the examples the identification problem is defined as follows:
Given a constitutive relation C, find the fields σDA ∈ S(fb, u), uK A ∈ U (ub) minimizing
J (uK A, σDA, C) = 0Ω (σDA − C : ε(uK A)) : C−1 : (σDA − C : ε(uK A))dΩdt
T ¯
+ du(ub, u¯)dS +
df (fb, f )dS dt
0 ∂u Ω ∂f Ω (1.13)
where S(fb, u) and U (ub) are defined in (1.10) and the functional used to measure the discrepancy on the constitutive relation is G(uK A, σDA, C) = T (σDA − C : ε(uK A)) : C−1 : (σDA − C : ε(uK A))dΩdt (1.14) 0 Ω
This approach has been tested with satisfying results in a 1-D case where the Young’s modulus is sought and the measurement noise reached 60%. Experiments were realized for both homogeneous and heterogeneous moduli. In all the cases, the ECR approach presented excellent properties of robustness against noisy data as well as good convexity properties of cost functions. However, the research per-formed in [36] and [67] clearly pinpointed one of the major limitations of this method: its computational cost when solving the minimization problem. As a matter of fact, the resolution of the ECR problem in elastodynamics leads to a large system of space-time equations where the admissible fields are coupled to the solution of a time-backwards adjoint problem. Hence the application of such a formulation for an industrial size problem is still an area of open research. In the following we will use the frequency domain formulation which is presented below.

Frequency-domain formulation. Application to a FE formulation.

In the scope of the present work and with the aim to avoid prohibitive computational costs that would prevent the use of the ECR in industrial cases, a frequency-domain formulation will be adopted from now on. This derivation of the ECR was studied by [19] in a FE framework and further adopted by [26, 27] to a high DOF case. Thus, this version of the ECR is the most suitable when dealing with linear FE models of industrial size.
To fix the ideas, consider the above equations (1.1) and (1.2) and assume that the sought-after solu-tions are of the form: ℜ(uω (z)eiωt), ℜ(σω (z)eiωt) (1.15)
where ℜ(•) represents the real part of a complex number and ω is the angular frequency. Then, equation (1.1) can be rewritten as: − ρω2 uω (z) + div(σω (z)) = 0 (1.16)
When it comes to the constitutive relation, the following expressions are considered:
σω (z) = (K + iωC)ǫ(uω (z)) (1.17)
ω (z) = −ρω2 uω (z)
where ω represents the inertial forces, and K From the above equations (1.17), the following and C are respectively the Hooke and damping tensors.
spaces are defined:
U (u¯) = {uω (z)
S ¯ {
(f ) = σω (z)
D(σ) = { (z)
ω
s.r. |uω (z) = u¯ for z ∈ ∂uΩ} ¯
s.r. |σω (z) = (K + iωC)ǫ(vω (z)), σ(z) • n = f for
s.r. | ω (z) = −ρω2 wω (z), ω (z) + div(σω (z)) = 0
z ∈ ∂f Ω} (1.18)
for z ∈ Ω}
where uω (z), vω (z) and wω (z) are displacement fields and in the sequel will be denoted u, v and w respectively for the sake of clarity. Thus, the Drucker inequality (1.6) can be rewritten, for a given angular frequency ω, by considering a triple of displacement fields only, as:
ξω2 (u, v, w) = { Trace[(K + T ω2 C)(ǫ(v) − ǫ(u))∗ (ǫ(v) − ǫ(u))]
Ω 2 1 − γ (1.19) + ρω2 (u − w)∗ (u − w) dΩ 2 }
where γ ∈ [0, 1] is a weighting scalar indicating the relative quality of the constitutive relations (1.17) and the superscript “∗ ” represents the complex conjugate. Hence, from the above definition (1.18) of admissible spaces we will refer to u as a kinematically admissible field, v as a dynamically admis-sible field related to the K and C tensors and w as a dynamically admissible field related to inertial forces.
Since the expression of the Drucker error has been defined for the frequency domain, the following expressions can be developed in order to be applied for the study of real structures.
A relative structural error can be defined as:
ξ2
ξ2 = ω (1.20)
ωr Dω2
where Dω2 represents the reference structural energy defined by:
Dω2=( γ Trace[(K + T ω2 C)ǫ(u)∗ ǫ(u)] + 1 − γ ρω2 u∗ u)dΩ (1.21)
Besides, if we are interested in studying the influence of different regions of Ω to the global error, nE we might consider a subdivision of sub-domains E ∈ E of Ω in a way that Ω = Ei. Thus, the global i=1 error can be interpreted as the contribution of all the local errors and we obtain:
ξ2 = ξ2 (1.22)
ωr ωEr
E∈Ω
Moreover, when studying the behavior of a structure in the frequency domain, it is natural to be interested in its behavior in a bandwidth of interest [ωmin, ωmax] of angular frequencies. Thus, we can define the bandwidth relative error as:
ξ2 ωmax
= η(ω)ξ2 dω (1.23)
T r ωmin ωr
where η(ω) is a weighting function defined over [ωmin, ωmax] satisfying the following condition:
ωmax η(ω) ≥ 0
ωmin η(ω)dω = 1 (1.24)
In most of the industrial and application cases, and in the particular scope of interest of this work, the study of structural dynamic behavior is performed by means of Finite Element models. In order to adopt the above error expressions in a FE framework, the discretization of equation (1.16) leads to the following matrix equation:
[−ω2 [M ] + jω[C] + [K]]{q} = {F } (1.25)
where [M ], [C] and [K] are the so-called mass, damping and stiffness matrices respectively. Moreover, {F } and {q} are the vectors of nodal forces and displacements. Within this framework, the following considerations will be made with regard to the inverse problem we aim to solve:
• ∂0Ω=∅
The prescribed loading f over ∂f Ω is considered as a reliable information (e.g. external loadings, free surfaces, etc.) and directly embedded in {F }.
• A set of unreliable displacement data u˜ (e.g. sensor measurement on a free surface) is available over ∂f uΩ.
• Displacement data u¯ are restricted to the boundary ∂u f Ω = ∂uΩ − ∂f uΩ and considered as a reliable information (e.g. clampings). This reliable kinematic information is generally enforced in the construction of model matrices by either introducing Lagrange multipliers or by considering matrices with active DOFs only, which is the solution adopted in the sequel.
Hence, the above FE matrix equation (1.25) and the Drucker inequality (1.19) leads to the following expression of the modified Error in Constitutive Relation in a FE framework:
e2ω (Tω ,θ)
Find the kinematic admissible field u ∈ U (Π, u¯), and the dynamic admissible fields (v, w) ∈
D({F }, u¯) minimizing:
e2 ( u , v , { w ) = γ u v ∗[K + T ω2C] u v + 1 − γ u w ∗ ω2 [M ] u w
2 { − − } 2 { − − }
ω { } { } } } { } {
+ r {Πu − u˜}∗ [GR]{Πu − u˜}
1 − r (1.26)
where the admissible spaces are defined by:
U ( F , u¯) = w , | N N v ϕ (z) = N w ϕ (z ) = u¯ for z ∂ Ω,
v s.r.
(Π, u¯) = { {u} s.r. i=1 uiϕi(z) = u¯ for z ∈ ∂u f Ω, u(z) = Π{u} for z ∈ ∂f uΩ}
D { } {{ }{} | i=1 i i i=1 i i ∈ u f
2 [M ]{w} = {F} }
[K + iωC]{v} − ω
where (ϕ1 , , ϕ N ) are the basis functions, r is a weighting scalar, Π a projection operator from the space of structural nodal displacements to the observation space and [GR] represents a symmetric positive-definite matrix. Notice that, in this formulation, unreliable displacements u˜ are introduced as an additional term in the ECR functional (1.26). As described for the dynamic formulation in time domain, this consists of the modified formulation and corresponds to the regularization term stabilizing the solu-tions in case of noisy data (e.g. experimental measurements). Moreover, although the choice of matrix [GR] is not a priori defined it is usually chosen to be dimensionally consistent with the induced energy norm as proposed in [26, 27]. In our case the following choice is made: [GR] = γ [[KR] + T ω2[CR]] + 1 − γ ω2[MR] (1.27)
where the index “R” represents the Guyan reduction on the observation space.
Hence, in order to evaluate the discrepancy of a FE model with respect to a set of measurements, a two step method is adopted:
1. Given a set of model parameters θ that parametrize [M ] = [M (θ)], [C] = [C(θ)], [K] = [K(θ)], obtain the triple of admissible fields Tω = (u,ˆ v,ˆ wˆ) that minimizes (1.26). This minimization
problem leads to the resolution of linear equations as developed in Appendix B.
2. Evaluate the model error by computing e2 (Tω , θ), or its relative form ω Dω2 (u,θˆ)
In the present work, we aim not only at studying model errors in a bandwidth of frequencies, but also at monitoring their spatial distribution over Ω. For these reasons, from now on, the following expression is adopted to evaluate the total model error: η(ω) T dω
ξT r = 2 (1.28)
2 ωmax E∈Ω eEω2 ( ω, θ)
ωmin Dω (u,ˆ θ)
Thus, since the triple of admissible fields solution of (1.26) depends on θ (Tω = Tω (θ)), the problem of finding the best set of model parameters θ that better represents the noisy data u¯ can be written as:
ˆ 2 (1.29)
θ = arg min ξT r (θ)
θ∈Θ
where Θ is the space of admissible parameters.
The well-behaved nature of functional (1.28) with respect to θ is one of the main features that we aim to take advantage of in this work. A visual example is presented in Figure 1.3, where a ξT2 r functional is evaluated in a 4-DOF dynamic system for different values of stiffness (k) and mass (m) as described in Figure 1.3(a), where all displacements are supposed to be observed and the external load is supposed to contain a single frequency. This surface is compared to the one obtained with a least square functional of the form u˜ − Πqdirect 2 , where qdirect represents the solution of the associated direct problem. As it can be seen in Figure 1.3(b) and Figure 1.3(c), the ECR functional exhibits a clear minimum while the least square surface presents several local minima and peaks.

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Table of contents :

Introduction and general overview 
EDF’s industrial need
Considered methods
Overview of the thesis
I Introduction to Error in Constitutive Relation and Data Assimilation methods 
1 Identification methods and Error in Constitutive Relation 
1.1 Reference Problem
1.2 Energy-based functionals. Introduction to the Error in Constitutive Relation
1.3 Conclusions
2 Data Assimilation 
2.1 Introduction
2.2 Concepts and classic notation in data assimilation
2.3 Sequential and variational formalisms: Kalman filter and 4D-Var
2.3.1 Variational formalism: 4D-Var
2.3.2 Sequential formalism: The Kalman filter
2.4 Example of nonlinear identification by means of the Unscented KF
2.5 Conclusions
II Towards a combined use of Kalman filtering and Error in Constitutive Relation 
3 A Kalman filter and ECR strategy for structural dynamics model identification 
3.1 Purpose
3.2 Improving a priori knowledge with the ECR
3.3 Introducing the ECR functionals into Kalman Filtering
3.4 Solving the identification problem by using ECR – UKF coupled method
3.5 Numerical example of structural parameter identification
3.6 Conclusions
4 ECR and UKF for model enhancement in problems of industrial relevance 
4.1 Damage identification through the ECR-UKF strategy for high DOF models
4.1.1 Case of evolving parameters
4.2 Identifying incorrect modelling of boundary conditions
4.2.1 A time-domain approach for the identification of mis-modeled boundaries
4.3 Comparison of ECR and BLUE methods for structural field reconstruction
4.4 Conclusions
5 Improvements of the ECR-UKF algorithm 
5.1 Introducing algebraic constraints in the Unscented Kalman Filter
5.2 Parametric study of ECR-UKF parameter error covariance matrix
5.3 Conclusions
III Applications 
6 ECR in civil structures assessment: application to the SMART benchmark 
6.1 Introduction
6.2 Main results
6.3 Conclusions and further work on the SMART benchmark
7 Study of a reinforced concrete beam with strong boundary coupling 
7.1 Experimental setup and problem description
7.2 Boundary impedances identification
7.2.1 A new approach to identify boundary conditions based in ECR functionals
7.3 Study of the evolving structural damage
7.4 Conclusions
Conclusions and future research 
Appendices
A Stochastic interpolation: the BLUE formalism
B Minimization of the ECR functional and first order derivatives in a FE framework.
C The Unscented Kalman filter
D Implementation within Code Aster FE software
E Application of the ECR to the SMART benchmark
Bibliography

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