Synergies between the constitutive relation error concept and PGD model reduction for simplified V&V procedures
 Ludovic Chamoin^{1}Email author,
 PierreEric Allier^{1} and
 Basile Marchand^{1}
DOI: 10.1186/s4032301600739
© The Author(s) 2016
Received: 7 February 2016
Accepted: 19 May 2016
Published: 3 June 2016
Abstract
The paper deals with the constitutive relation error (CRE) concept which has been widely used over the last 40 years for verification and validation of computational mechanics models. It more specifically focuses on the beneficial use of model reduction based on proper generalized decomposition (PGD) into this CRE concept. Indeed, it is shown that a PGD formulation can facilitate the construction of socalled admissible fields which is a technical keypoint of CRE. Numerical illustrations, addressing both model verification and model updating, are presented to assess the performances of the proposed approach.
Keywords
Error estimation Model updating Model reduction Constitutive relation error (CRE) Proper generalized decomposition (PGD)Background
Mathematical models and their solutions, either analytical or numerical, are fundamental in science and engineering activities as they constitute the basic ingredient of simulations that enable to predict the behavior of physical phenomena. Consequently, a permanent issue is the verification and validation of these models, which nowadays can attain very high levels of complexity, in order to certify the quality of numerical simulations. On the one hand, verification deals with the assessment of the numerical (FE) model with respect to initial mathematical model, and implies the estimation of discretization error in order to control the quality of the approximate numerical solution. In this context, a large set of a posteriori error estimates has appeared over the last thirty years (see [1–3] for an overview). On the other hand, validation addresses the capability of mathematical models to represent a faithful abstraction of the real (physical) world. It aims at identifying or updating model parameters in order to minimize the discrepancy between numerical predictions and experimental measurements, and leads to the solution of inverse problems [4].
In the context of model verification and validation, and particularly for computational mechanics models in which the constitutive relation is a major component, the constitutive relation error (CRE) concept is a convenient and powerful tool. The idea of CRE is rather simple: socalled admissible fields verifying all equations of the model except the constitutive relation are constructed, then the residual associated with the constitutive relation is measured. The CRE concept was first introduced as a robust a posteriori error estimator in FE computations [5], enabling to compute both strict and effective discretization error bounds for linear and more generally convex structural mechanics problems, and to lead mesh adaptivity processes. It was primarily used for linear thermal and elasticity problems [6, 7] before being extended to nonlinear time dependent problems [8, 9] and to goaloriented error estimation [10–12]. The use of CRE for model verification, for which a general overview can be found in [2], requires in particular the computation of admissible dual fields which are fully equilibrated. This requirement, which is the main practical issue both in terms of computational cost and implementation technicality, was addressed by means several techniques that postprocess the FE solution at hand [2, 6, 13–20]. During the 90s, the CRE concept was extended to model identification/updating. First introduced for dynamics models [21–24], this method was latter successfully used in many calibration applications including defects [25], uncertain measurements and behaviors [26, 27], or corrupted measurements [28, 29]. It was also used in the context of fullfield measurements [30, 31]. After initial studies in which measurements were included as additional admissibility constraints, a more flexible and effective strategy was developed. Denoted as modified CRE (mCRE), this strategy consists in relaxing constraints on measurements and other uncertain data, proposing a general framework in which reliable theoretical and experimental information (equilibrium, sensor position,...) is favored to define admissibility spaces, and residual on complementary information (material behavior, sensor measurements,...) is measured. It acts in an iterative twosteps algorithm, in which optimal admissible fields are first computed, before minimizing the obtained mCRE functional with respect to model parameters. The use of mCRE presents interesting advantages; it has excellent capacities to localize structural defects spatially, it is very robust with respect to noisy measurements, and it has good convexity properties.
On the one hand, in the context of model verification, the CRE concept was already used to control PGD approximations (see a posteriori error estimates developed in [36, 37]) or to directly drive the PGD process with CRE minimization [38]. Nevertheless, the use of PGD in CRE implementation has never been investigated and we wish to show here that there are major advantages to do so, in particular for the construction of equilibrated fields. On the other hand, in the context of model validation, PGD was used for model updating within classical procedures with least square minimization [39]. It was also recently used in particular applications involving robust model updating with the CRE concept [40, 41]. Here, the goal is to give a general framework on the effective use of PGD for model updating with CRE. For the sake of simplicity and clarity, we consider scalar linear elliptic (stationary thermal) problems even though extensions to elasticity or more complex problems (nonlinear or transient analyses), briefly addressed in this paper, are possible with regards to existing literature [2, 8].
The paper outline is as follows: after presenting the mathematical model of interest in “Reference problem and approximate FE solution” section, the CRE concept is reviewed in “Basics on the CRE concept” section; its extension to model validation with the mCRE concept is addressed in “Extension of the CRE concept for model updating: modified CRE” section; the use of PGD in addition to CRE for the construction of admissible fields is shown in details in “Coupling PGD with CRE in model verification” section for model verification, and in “Coupling PGD with mCRE in model validation” section for model validation; illustrative numerical results are reported in “Results and discussion” section; conclusions are drawn in “Conclusions” section.
Methods
Reference problem and approximate FE solution
Basics on the CRE concept
We present here the foundations and implementation of the CRE concept, built from a dual approach and measuring the residual on the constitutive relation \(\varvec{q}=\mathcal {K}\varvec{\nabla }u\), in the context of model verification.
Remark 1
Consequently, and provided that a flux field \(\varvec{\pi }\in \mathcal {S}\) is available, we observe from (16) that the term \(\sqrt{2}E_{CRE}(u_h,\varvec{\pi })\) is a computable upper bound on \(e_{\mathcal {U}}\). The quality of this bound depends on that of \(\varvec{\pi }\).
The constraints in space \(\mathcal {S}\) make the construction of SA solutions awkward. A first possibility, which is the most effective, would consist in using a FE discretization with equilibrium elements on the complementary problem (14) (dual approach, see [42–44]). However, this is in practice unrealistic as it would require the solution of an additional global problem, with large computational efforts and nonconventional FE spaces. In “Coupling PGD with CRE in model verification” section, we present the basis of a technique (referred as hybridflux or EET in the literature) that enables to compute a flux field \(\widehat{\varvec{q}}_h \in \mathcal {S}\) [and therefore the a posteriori error estimate \(\sqrt{2}E_{CRE}(u_h,\widehat{\varvec{q}}_h)\)] from a postprocessing of the FE field \(\varvec{q}_h\) at hand. The PGD strategy will be used within this technique in order to facilitate implementation issues.
Remark 2
It can be shown that using the hybridflux (or EET) technique to construct an admissible flux field \(\widehat{\varvec{q}}_h\) enables to obtain a lower error bound from the CRE functional [2, 6]; it is of the form \(E_{CRE}(u_h,\widehat{\varvec{q}}_h) \le C e_{\mathcal {U}}\), where C is a constant independent of the mesh size, proving that the constructed error estimate has the same convergence rate as the true discretization error.
Extension of the CRE concept for model updating: modified CRE
Remark 3
The value of r should generally be set in regards to the a priori reliability on both model and measurements. For instance, the Morozov principle or Lcurve method [46] may be used to define r with respect to data noise. The influence of r on the sensitivity with respect to measurement uncertainties, and therefore on the quality of the updating performed using mCRE, was illustrated in [47].
Remark 4
When some parameters in \(\varvec{p}\) describe a field (material parameter field for instance), a localization step after spatial splitting of the cost function \(\mathcal {F}(\varvec{p})\) can be added at the end of the first minimization (Step 2). It consists in selecting the highest local contributions to \(\mathcal {F}(\varvec{p})\) and updating first the associated parameters. Moreover, a goaloriented version of the model updating with mCRE, in which only parameters which have influence for the prediction of an output of interest are updated, can be constructed [48].
The mCRE formulation is thus based on a tradeoff between modeling and measurement errors, which enables it to be less sensitive to noise. It inherits all the convenient properties of the CRE concept; it can be in particular extended to complex constitutive models [involving e.g. (visco)plasticity or damage] and leads to a natural regularization.
Notice to conclude that the mCRE strategy, without adding particular techniques, is costly. In particular, the iterative strategy requires to compute optimal admissible fields \((\widehat{u},\widehat{\varvec{q}})\) at each iteration, i.e. each time \(\varvec{p}_0\) is updated. This can be highly facilitated using a PGD metamodel, as shown in “Coupling PGD with mCRE in model validation” section.
Coupling PGD with CRE in model verification
In this section, we explain how PGD can be advantageously used when implementing CRE for model verification (see “Basics on the CRE concept” section).
Constructing admissible flux fields with the hybridflux technique
 1.Step 1 construction of equilibrated tractions \(\widehat{F}\) on the boundary \(\partial K\) of each element \(K \in \mathcal {T}_h\), with \(\widehat{F}=g\) if \(\partial K \subset \Gamma _N\), so that equilibration at the element level is verified:The construction of \(\widehat{F}\) is based on the following prolongation condition:$$\begin{aligned} \int _K f + \int _{\partial K}\widehat{F} =0 \quad \forall K \in \mathcal {T}_h \end{aligned}$$(22)applied to each element \(K \in \mathcal {T}_h\) and each FE node i connected to K; \(\phi _i\) is the FE shape function associated to node i. This condition automatically yields equilibrated tractions \(\widehat{F}\) and leads to the solution of local wellposed systems over patches of elements connected to each node i. In practice, tractions \(\widehat{F}\) are found as linear combinations of functions \(\phi _i\). All technical details on the construction of \(\widehat{F}\) can be found in [2, 20].$$\begin{aligned} \int _K(\widehat{\varvec{q}}_h\varvec{q}_h) \cdot \varvec{\nabla }\phi _i =0 \Longrightarrow \int _{\partial K} \widehat{F} \phi _i = \int _K (\varvec{q}_h \cdot \varvec{\nabla }\phi _i  f\phi _i) \end{aligned}$$(23)
 2.Step 2 local construction, for given tractions \(\widehat{F}\) and over each element \(K \in \mathcal {T}_h\), of \(\widehat{\varvec{q}}_{h}\) solving the following Neumann problem:The solution of (24) to get \(\widehat{\varvec{q}}_{hK}\) may be performed analytically, using polynomial functions with sufficiently high degree, provided the source term f is polynomial as well [49]. In practice, an alternative approach with numerical solution is preferred. For fixed tractions \(\widehat{F}\), the optimal admissible flux \(\widehat{\varvec{q}}_h\) inside each element K is the one that minimizes the local error estimate on K \(\widehat{\varvec{q}}\varvec{q}_h_{\mathcal {S},K}\) (or equivalently \(\widehat{\varvec{q}}_{\mathcal {S},K}\)) among all fluxes \(\widehat{\varvec{q}}\) verifying (24). Duality arguments show that this is equivalent to taking \(\widehat{\varvec{q}}_{hK} =\mathcal {K}\varvec{\nabla }\rho \), with \(\rho \in H^1(K)\) verifying:$$\begin{aligned} \varvec{\nabla }\cdot \widehat{\varvec{q}}_{h}= & {} f \quad \text {in}\; K \quad \Longleftrightarrow \; \int _K \widehat{\varvec{q}}_{h} \cdot \varvec{\nabla }v = \int _K fv + \int _{\partial K}\widehat{F} v \quad \forall v \in H^1(K) \nonumber \\ \widehat{\varvec{q}}_{h}\cdot \varvec{n}= & {} \widehat{F} \quad \text {on}\; \partial K \end{aligned}$$(24)A numerical approximation of the solution of (25) (defined up to an additive constant) can be obtained using the FEM with a single finite element of high degree \(p+k\), where p denotes the polynomial degree used to compute \(u_h \in \mathcal {U}_h\) and k denotes the extra degree. Numerical studies performed in [50] showed that analytical and numerical approaches give similar CRE error estimates choosing \(k\ge 3\), even though the flux field is not rigorously equilibrated in each element K with the latter approach. We consider the numerical approach in the following.$$\begin{aligned} \int _K \mathcal {K}\varvec{\nabla }\rho \cdot \varvec{\nabla }v = \int _K f v + \int _{\partial K}\widehat{F} v \quad \forall v \in H^1(K) \end{aligned}$$(25)
Use of the PGD to solve problems at the element level

We consider that the material behavior is isotropic and that material parameters are constant over each element K, so that their values have no influence on \(\widehat{\varvec{q}}_{hK}\); we thus set \(\mathcal {K}=\mathbb {I}\) when solving (25) and define \(\widehat{\varvec{q}}_{hK} =\varvec{\nabla }\rho \). In cases where \(\mathcal {K}\) is not constant over each element, its evolution could be parameterized and additional material parameters would be introduced in the PGD decomposition;

We consider, as an illustrative example, the case of 3node triangle elements (Fig. 1). Nevertheless, the proposed strategy is generic (based on element shape functions and nodes coordinates alone) and can be straightforwardly applied to other elements.
Remark 5
The solution \(\rho _\ell ^{jl}\) to each problem (27) can be computed with the PGD technique, for any element K, parameterizing the geometry of K with a set of parameters \(\varvec{p}_{geo} \in \mathcal {P}\). Following the approach described in [51–53], we reformulate the weak problem (27) by introducing a parameterdependent mapping \(\mathcal {M}(\varvec{p}_{geo}):K_{ref} \rightarrow K(\varvec{p}_{geo})\) from a reference fixed element \(K_{ref}\) to the geometrically parameterized element \(K(\varvec{p}_{geo})\). Such a geometrical transformation then allows defining the weak problem in a tensor product space and applying the PGD method, in order to compute generic parameterized solutions \(\rho _\ell ^{jl}(\varvec{p}_{geo})\) which can be used for any element geometry.
Remark 6
In the presence of geometrical variabilities, an alternative approach described in [54, 55] could also be used. It consists in embedding the parameterized domain into a fixed fictitious domain.

A first scaling mapping \(\mathcal {M}_1:\overline{K} \rightarrow K\) maps a homothetic element \(\overline{K}\) with diameter 1 to the actual element K with diameter \(\alpha \). This mapping reads:$$\begin{aligned} \left( \begin{array}{c}x \\ y \end{array}\right) = \mathbb {T}_1\left( \begin{array}{c}\overline{x} \\ \overline{y} \end{array}\right) \, ; \quad \mathbb {T}_1 = \left[ \begin{array}{c@{\quad }c}\alpha &{} 0\\ 0 &{} \alpha \end{array}\right] = \alpha \mathbb {I}\end{aligned}$$(33)

A second linear mapping \(\mathcal {M}_2:K_{ref} \rightarrow \overline{K}\) maps a reference element \(K_{ref}\) (rightangled isosceles triangle) to element \(\overline{K}\). This mapping reads, using an isoparametric formulation:where \((\overline{x}_3,\overline{y}_3)\) are local coordinates of node 3 in the coordinates system associated with element \(\overline{K}\), and \((\eta ,\xi )\) are local coordinates in the coordinates system associated with element \(K_{ref}\) (see Fig. 2).$$\begin{aligned} \left( \begin{array}{c} \overline{x} \\ \overline{y} \end{array}\right) = \left( \begin{array}{c} \phi _2(\eta ,\xi ) + \overline{x}_3 \phi _3(\eta ,\xi ) \\ \overline{y}_3 \phi _3(\eta ,\xi ) \end{array}\right) = \mathbb {T}_2\left( \begin{array}{c} \eta \\ \xi \end{array}\right) \, ; \quad \mathbb {T}_2 = \left[ \begin{array}{c@{\quad }c} 1 &{} \overline{x}_3\\ 0 &{} \overline{y}_3 \end{array}\right] \end{aligned}$$(34)
Remark 7
The number of elementary problems (27) and the number of geometrical parameters involved in the mapping \(\mathcal {M}\) depend on the FE element type; for instance, 6node triangle elements would involve 9 elementary problems (3 for each of the three edges) and 9 geometrical parameters (12 degrees of freedom with three rigid body motions), whereas 4node tetrahedron elements would involve 12 elementary problems (3 for each of the four edges) and 6 geometrical parameters (12 degrees of freedom with six rigid body motions).
Implementation of the PGD
Remark 8
The number m of PGD modes which is required to get accurate solutions \(\rho ^{jl}_{\ell ,m}\) can be rigorously defined using classical a posteriori error estimation tools devoted to PGD [36, 37, 56, 57]. A numerical assessment of the value m that yields sufficient accuracy is provided in “CRE estimate obtained from EETPGD” section.
Remark 9
In order to save computational time and storage needs, symmetries in the local parameterized solutions \(\rho ^{jl}_\ell \) can be used. For instance, the relation \(\rho ^{12}_1 \left( \eta ,\xi ,\alpha ,\overline{x}_3,\overline{y}_3 \right) =\rho ^{12}_2 \left( 1\eta ,\xi ,\alpha ,1\overline{x}_3,\overline{y}_3 \right) \) holds.
Remark 10
Another study, which is not considered here, would benefit from the PGD representation \(\widehat{\varvec{q}}_{h,mK} \left( \varvec{x}_{ref}, \left\{ \widehat{F}^{jl}_\ell \right\} ,\alpha ,\overline{x}_3,\overline{y}_3 \right) \). It addresses the optimization of equilibrated tractions \(\left\{ \widehat{F}^{jl}_\ell \right\} \) considering a global problem in which the complementary energy is minimized. This procedure, first developed in [49], is very costly in the general case but can be highly facilitated by the explicit dependency on \(\left\{ \widehat{F}^{jl}_\ell \right\} \) provided by the PGD.
Coupling PGD with mCRE in model validation
In this section, we explain how PGD can be advantageously used when implementing mCRE for model updating (see “Extension of the CRE concept for model updating: modified CRE” section).
Performing minimizations in the mCRE method
Remark 11

The discretized field \(\varvec{V}\) is KA if it verifies the (discretized) kinematic constraints of (6), so that it contains prescribed dofs. The associated admissibility space is denoted \(\varvec{\mathcal {U}}_h\);

The discretized field \(\varvec{W}\) is SA if it verifies the FE equilibrium equations \(\varvec{V}^T(\mathbb {K}\varvec{W} \varvec{F})=0\) for all \(\varvec{V}\in \varvec{\mathcal {U}}_h\), where \(\mathbb {K}\) and \(\varvec{F}\) are the global stiffness matrix and load vector, respectively, of the FE system. The associated admissibility space is denoted \(\varvec{\mathcal {S}}_h\).
Use of the PGD for the first minimization

The explicit dependency on parameters \(\varvec{p}\) enables: (1) to evaluate very fast and for any values of \(\varvec{p}\) the optimal admissible fields arising from the first constrained minimization; (2) to compute gradients of the cost function \(\mathcal {F}(\varvec{p})\) analytically and thus perform the second minimization step very easily;

The explicit dependency on parameter \(\sigma _r\) makes the definition of the optimal value of \(\sigma _r\) (primarily with respect to measurement noise using the Lcurve method) straightforward.
Remark 12
In the present work, we assume that measurement values in \(\varvec{s}\) are known upstream to the updating procedure, and that this procedure is conducted for a single set of measurement values. In other cases such as data assimilation on timedependent problems, they can be considered as extraparameters in the PGD decomposition as performed in [40, 41].
In practice, space functions \(\psi ^u_i(\varvec{x})\) and \(\psi ^{\lambda }_i(\varvec{x})\) are computed using the FEM, and other functions appearing in PGD modes are discretized using a fine grid over spaces \(\Sigma _r\) and \(\mathcal {P}_j\) (\(j=1,\dots ,P\)).
Results and discussion
In this section, we illustrate and analyze performances of the approach proposed in “Coupling PGD with CRE in model verification” and “Coupling PGD with mCRE in model validation” sections. “Example 1: a posteriori error estimation on a 2D structure” section deals with model verification using a CRE error estimate coupled with PGD, whereas “Example 2: model updating on a 3D structure” section addresses model updating using a mCRE formulation coupled with PGD.
Example 1: a posteriori error estimation on a 2D structure
Problem geometry and data
From the associated FE solution, equilibrated tractions are computed using the first step of the hybridflux (or EET) technique.
Details on the PGD solution
CRE estimate obtained from EETPGD
Speedup obtained using the PGD solution
Eventually, we compare the CPU time required to compute the equilibrated flux field depending on which method is used (Fig. 11). All the computations were performed on an Intel Core i5 2.4 GHz with 8 GB of RAM, without parallelization. Classical EET and EETPGD techniques share as much code as possible, and only the construction and solution of the matrix problem is replaced by a simple postprocessing with PGD solutions in the EETPGD technique. Naturally, the first step with construction of equilibrated tractions is similar for both techniques.
Example 2: model updating on a 3D structure
Identification problem
We consider a steadystate thermal problem on the 3D geometry shown in Fig. 14. It is a two layers cylinder (length \(L=100\), internal radius \(R_{int}=10\), external radius \(R_{ext}=14\)) with a localized inclusion (length \(L_{inc}=10\)) in the middle of the cylinder. The internal layer (resp. external layer, and inclusion) is represented in green (resp. blue, and red) color in Fig. 14. In each of the layers and in the inclusion, the material is supposed to be isotropic and homogeneous with respective material operators \(\mathcal {K}_{int}=p_{int}\mathbb {I}\), \(\mathcal {K}_{ext}=p_{ext}\mathbb {I}\), and \(\mathcal {K}_{inc}=p_{inc}\mathbb {I}\). The applied boundary conditions are: (1) homogeneous Dirichlet boundary conditions on one end of the cylinder; (2) given thermal flow \(q^d=1\) on the inner boundary; (3) zero thermal flow (free surface) on all other boundaries.
We wish to identify thermal conductivity parameters \((p_{ext},p_{inc})\) from noisy measurements given by a set of 12 sensors. These sensors are placed on four horizontal rows with \(\pi /6\) angle spacing (see Fig. 14). The reference values for parameters \((p_{ext},p_{inc})\) to be identified are \(p_{ext}^{ref}=10\) and \(p_{inc}^{ref}=1\). Furthermore, we fix \(p_{int}=20\).
PGD model reduction
Identification with PGD
We now perform the iterative process using a first order (gradient) minimization method. For each iteration, we show in Fig. 20 the identified values of \((p_{ext}, p_{inc})\), as well as the optimal value \(\sigma _r\) used for this iteration and defined as previously. We observe that the method converges to identified values of \((p_{ext}, p_{inc})\) which are very close to the reference values \(\left( p_{ext}^{ref}, p_{inc}^{ref} \right) \). In addition, we study the incidence of the number m of considered PGD modes on the identification results. The convergence of the identification process is represented in Fig. 21 for several values of m. We clearly observe that the accuracy of the identification results is highly impacted by the value chosen for m, and that the process leads to a relative error lower than \(10\, \%\) for both parameters \(p_{ext}\) and \(p_{inc}\) when using \(m=15\). It is also interesting to notice that the PGD representation with \(m=10\) is suitable for the identification of \(p_{ext}\), which is the parameter with greater weight on the overall solution, but still fails for the identification of \(p_{inc}\).

Considering original mCRE, each iteration with update of the value of \(\sigma _r\) requires to solve a Pareto problem to find the optimal value of \(\sigma _r\). This involves about 40 subiterations, each of them corresponding to the solution of a linear system with the size of the problem in space [resulting from (41)]. Considering 10 iterations in the mCRE identification process thus leads to the solution of about 400 linear systems of the space problem size;

Considering PGDmCRE, the offline computational cost is due to the use of a greedy algorithm to compute 15 PGD modes. At each iteration of this algorithm, we implement a fixed point procedure which converges in 3 subiterations (average), and a subiteration requires the solution of the space problem. Consequently, the computation of the parametric PGD decomposition requires to solve about 45 linear systems with the size of the problem in space. Then, no more solutions of linear systems are required in the online step, merely some inexpensive evaluations of parametric functions.
Conclusions
We presented a general framework that highlights the beneficial use of PGD in V&V procedures performed by means of the CRE concept. Based on an offline/online strategy, it drastically decreases the computational cost and technicalities which are essentially associated with the computation of admissible fields. We believe this work paves the way to both robust, practical, and realtime methods for controlling computational mechanics models. Furthermore, as the proposed technique is focused on balance equations alone, it should be possible to extend it to nonlinear timedependent problems. This will be the topic of forthcoming research works.
Declarations
Authors' contributions
All authors discussed the content of the article, and were involved in its writing. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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