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Shazil
Consider, that dating beneath your class site question interesting
I met a really nice guy a couple weeks ago and I want to go out with him but my parents don't allow it. The problem is that Im 17 and he is already 24 and a police men. Since my parents bough have high education, they say he is "lower" than me. The fact that you have to ask this and wonder about it should tell you that you don't like him enough anyway. Yes, 17 is almost a legal adult but when you are 24 you will realize how young 17 really is, and I'm betting you would not want to have anything to do with a 17 year old when you are Most 24 year olds would not even think about being serious with a 17 year old.
The whirlwind romance has been fun, but it hasn't been without roadblocks.
Dating outside your social class can be fraught with complications. People from different social classes may have trouble understanding the way other classes operate.
May 15, � Dating someone who is much lower in class status could lead to arguments and disagreements, but class is not just determined by profession in my book. The richest man could be horrible, while the middle class guy could be the most descent and end up making you the most happy. Choose a guy who makes you happy, consider both class and your feelings. Jul 11, � should you date someone out of your league? Here are five principles for creating a win-win relationship. The film "She's Out of My League" would hardly be confused with a Hollywood classic. Dec 09, � "An upper class person dating someone more blue collar might experience guilt or resentment." You might remember the Sex and the City .
The "New York Times" article "When Richer Weds Poorer, Money Isn't the Only Difference" describes a couple in which the lower-class husband did not fit in with people from his wealthy wife's social class - because he was a straight shooter and she and her friends talked around issues.
People from different social classes have different ways of acting - similar to a culture - that can take time to understand.
If your boyfriend has enough family money to buy designer clothing, drive his own sports car and apply to dozens of colleges, while you are flipping burgers at the local hamburger joint to scrape together enough money to attend the local community college, you may have trouble seeing eye to eye. You also might have issues when it comes to doing things together, since his tastes might outweigh what you can afford.
If your girlfriend is wealthy, and you come from a family with less money, you might feel as though there is a power imbalance in the relationship.
Often the person with more money ends up making most of the decisions - because she may be the one paying for things most of the time.
Although this is not a deal-breaker, it can take time to get comfortable with the idea that there is a natural imbalance of power in the relationship that will be hard to change. Sometimes the problem with dating outside your social class has nothing to do with the actual relationship. Instead, you might face harsh criticism from friends and family who believe that the two of you are a bad match.
10 Reasons We Keep Dating Guys That Are Beneath Us - It's Time To Quit It! What's the deal? it sucks to think about a future of spending Saturday nights alone with your cat and bottle of wine. But just because you're single right now doesn't mean you always will be. The best dating/relationships advice on the web. Oct 12, � But it usually doesn't work because when you're dating beneath you, you're not dating somebody who is your equally high vibrational partner. If you're constantly trying to bring someone up or get them to feel better about themselves, get them to learn about themselves, you're basically going into a teacher mode. "I married beneath me. All women do." -Lady Astor Some people become aware during their wedding ceremony that their partner is inferior to them. This kind of awareness can prove to be.
If you find yourself in this situation, it is important to remember that you are the one in the relationship - not the other people. You immediately connect with an awesome coach on text or over the phone in minutes.
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Oct 02, � The Truth About "Mixed-Collar" Dating - From the People Who Make These Relationships Work Understanding Cross-Class that means keeping your ego in check if you're dating someone who has Author: Kate Hakala. Marrying someone from a lower social class than yours hardly works. In the case of marrying someone with a lower social class, one argument about inequality can bring a relationship to an end. Personality and social psychology play a role when it is about getting married. Problems with Dating out of Your Social Class Dating outside your social class can be fraught with complications. Power Imbalance. If your girlfriend is wealthy, and you come from a family with less money, you might feel as though there is a power imbalance in the relationship. Often the person with more money ends up making most of the.
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Greece's proposal to the EU to restart tourism
by GTP
Greece is calling on the European Commission to adapt its proposal for a pre-travel coronavirus test to be instituted across the EU in order to allow travel to begin once the lockdown measures are lifted, GTP Headlines reported.
According to Politico and Kathimerini, Greece requested that the European Commission examine the possibility of lifting EU measures by June 15 and that its proposal for a test to be taken by travellers 72 hours ahead of travel be implemented by all EU countries.
Politico cites a non-paper issued by the Greek side requesting the "fair treatment" of all travellers, underlining that Greece stands against any "biased" health protocols that would be "citizenship-based" and stringent airline regulations such as leaving an empty seat on flights to avoid the potential spread of the virus.
It is reminded that last Friday, some of the rules included on Greece's proposed health protocols were made public.
It should also be noted that on Monday, the UK said it would gradually be "reopening" as the pandemic dies out requiring visitors, including UK citizens entering the country to self-isolate for 14 days at a specific facility, with the exception of the French.
"We cannot allow the single market to be fragmented when it comes to tourism services," the Greek document said, which is also requesting that quarantine measures be lifted as well as other "disproportionate or deterring requirements" to be implemented on the return of travellers to their country.
Greek Tourism Minister Harry Theoharis has reiterated Greece's proposal for travel in the coronavirus era, underlining that Covid-19 pre-flight screening 72 hours ahead of travel in the country of origin will ensure that travellers are healthy and "already feel safe when they board the plane" as well as the health of host countries.
In this direction, Greece has been putting pressure on the Commission to introduce a Union-wide health protocol that will be respected by all member states and thus allow travel to begin in efforts to save hard hit economies.
Theoharis has stressed however, that if the EU fails to do so, Greece is already exploring the possibility of creating a "safe travel zone" with Israel and Cyprus, as well as entering tourism pacts with specific partner countries.
According to reports, the Commission is expected to propose on Wednesday, easing EU lockdowns "in phases" before restoring free movement. At the same time, it may also suggest lifting bans in countries that have managed to contain the spread of Covid-19 and allowing travel between member states on condition that health checks are carried out.
*Source: GTP Headlines
Tags: break, European Commission, flight, flying, French, Greek economy, Greek Tourism Minister Harry Theoharis, holiday, lockdown measures, pre-travel coronavirus test, safe travel zone, Tourism, tourists, travel, travellers, UK, vacation
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Ross Street Roasting approached us initially for some Recharge AJAX cart integration issues, and we have since helped with various other improvements on the website too.
We have implemented a quick add to cart functionality on the collection page which allows customers to select the package size and ground level quickly.


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Daniel Kirkwood trial - barmaid describes fight that led to teenager's death
Barmaid gave evidence at Warwick Crown Court this morning
Ben Eccleston
Updated 13:52, 8 OCT 2018
Daniel Kirkwood
A barmaid has described how she ducked behind a bar as a fight, which ended with a teenager being killed, erupted inside a Coventry nightclub.
Daniel Kirkwood collapsed and died after a brawl at Society, in the city centre, in March last year.
He had suffered a fatal stab wound to his neck and also suffered a cardiac arrest while in an ambulance.
The barmaid told the trial of five men who stand accused of the 18-year-old's murder that she was among the bar staff working on the night of Saturday March 14, 2015.
She told the Warwick Crown Court jury that as the night entered the early hours the fight broke out and she saw somebody with "blood everywhere".
She was to later find out via social media that this was Daniel.
Story link:
Daniel Kirkwood trial: Friend tried to save life of Coventry teenager
Tobijah Thompson, Hasham Ali, Ramez Murtaza, Ezra Scott and Zaker Khan, who are all from Birmingham, are all accused of Daniel's murder.
They also face charges of wounding two other men with intent to cause them grievous bodily harm and violent disorder and assaulting a fourth alleged victim, causing him actual bodily harm.
Thompson, 25, of no fixed address, but previously of Blakesley Way, Bordesley Green, Birmingham, is also charged with possessing an offensive weapon - a knife - inside the nightclub.
The scene outside Society nightclub in Tower st, Coventry. Coventry teenager Daniel Kirkwood died in March 2015
The witness told the court that she first saw an argument between two men - one was Asian, about 20 and wearing a grey top, and the other "quarter-cast" with freckles.
She said: "I couldn't hear what they were arguing about due to the music. The Asian male then said something like 'I'm coming back' – I did hear that.
"He went off and the other just stayed where he was."
As she was worried something was "going to kick off", the barmaid alerted the club's bouncers before returning to the bar.
Daniel Kirkwood trial - bouncer saw stabbing inside nightclub, court hears
Soon after this she says a fight erupted inside the club.
"It was everywhere," said the barmaid. "I couldn't pick out anyone individually as it seemed everyone in the club was fighting or running or screaming.
"There were punches being thrown and I was ducking behind the bar.
"I remember seeing a lad running out with blood everywhere."
The barmaid described him as white, aged about 19, 5ft 5ins tall and wearing a grey zipped up hoodie.
After heading outside the club, the barmaid said she saw the same man lying in the road surrounded by ambulance staff .
Under cross-examination from Michael Borelli QC, representing Thompson, the barmaid told the jury that before she gave her statement to police she had seen comments on Facebook talking about somebody dying during a night out in Coventry.
After seeing Daniel Kirkwood's name mentioned, she searched for him on Facebook and says she recognised him as the male wearing the grey hoodie she had seen at the club that night.
Also giving evidence was Olamide Alabi, who helped organise the event at Society on March 14, 2015.
He was also the DJ for part of the evening and told the court he heard raised voices coming from behind the DJ box.
Mr Alabi said: "I saw a group of five or six males running towards another group of three or four males.
"There was fighting between the groups – the bigger group seemed to have the upper hand.
"I saw one of the Caucasian males go down to the floor and people were stamping."
Mr Alabi said that one person stood out from the crowd as he saw him aim a "flying kick" at somebody.
Daniel Kirkwood trial - close friend only told of teen's death in hospital, court hears
He says he was 6ft 1ins tall, with broad shoulders and "maybe of mixed race".
Mr Alabi then described seeing the same man on the dance floor screaming: "Who's next? Who wants it?"
"I was trying to keep an eye on him as he seemed the most dangerous guy," he added.
"It looked like he had something in his hand. He appeared to be grasping something."
Ali, 19, of Amberley Green, Great Barr, Birmingham, Khan, 25, of Belchers Lane, Small Heath, Birmingham, Murtaza, 21, of Bordesley Green East, Birmingham, Scott, 22, of Hawksyard Road, Erdington, and Thompson deny all of the charges.


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Synergies between the constitutive relation error concept and PGD model reduction for simplified V&V procedures
Ludovic Chamoin1,
Pierre-Eric Allier1 &
Basile Marchand1
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 so-called admissible fields which is a technical key-point of CRE. Numerical illustrations, addressing both model verification and model updating, are presented to assess the performances of the proposed approach.
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: so-called 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 goal-oriented 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 post-process 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 full-field 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 two-steps 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.
The objective of the paper is to present new numerical tools, based on model reduction techniques and offline–online strategy, that can be coupled to the CRE concept to make this latter fully implementable and exploitable for practical industrial applications. They particularly aim at decreasing the computational cost and technicality level which are required when computing admissible fields, leading to fast and inexpensive verification and validation (V&V) procedures. For that purpose, we decide to refer to the proper generalized decomposition (PGD) which is an a priori model reduction technique that has been extensively used over the last decade to solve multi-parametric problems (see [32–34]). Consider a general linear D-dimensional problem of the form:
$$\begin{aligned} \mathcal {L}u = g, \quad u \in \mathcal {X}= \mathcal {X}_1 \otimes \mathcal {X}_2 \otimes \cdots \otimes \mathcal {X}_D \end{aligned}$$
where \(\mathcal {L}\) is an operator defined on the tensor space \(\mathcal {X}\). PGD is a low-rank tensor method that consists in searching an approximation of u in a low-dimensional tensor subspace of \(\mathcal {X}_m \subset \mathcal {X}\) made of canonical format tensors of rank m:
$$\begin{aligned} u_m = \sum _{i=1}^m w_i^1 \otimes w_i^2 \cdots \otimes w_i^D, \quad w_i^{\mu } \in \mathcal {X}_{\mu } \end{aligned}$$
Among the various strategies to construct \(u_m\) [35], we focus on the one called progressive Galerkin. Introducing the global weak formulation of the problem:
$$\begin{aligned} \text {Find}\,\,u\in \mathcal {X}\,\,\mathrm{such~that} \; B(u,v)=F(v) \quad \forall v \in \mathcal {X}\end{aligned}$$
and assuming that the rank \(m-1\) decomposition \(u_{m-1}\) is known, the rank m decomposition \(u_m = u_{m-1} + w^1 \otimes w^2 \cdots \otimes w^D\) is searched such that:
$$\begin{aligned}&B(u_m,\delta v) = F(\delta v) \nonumber \\&\quad \forall \delta v= \delta w^1\otimes w^2 \cdots \otimes w^D + w^1\otimes \delta w^2 \cdots \otimes w^D + \dots + w^1\otimes w^2 \cdots \otimes \delta w^D \end{aligned}$$
with \(\delta w^{\mu } \in \mathcal {X}_{\mu }\). This formulation naturally leads to a nonlinear problem where a set of coupled low-dimensional problems has to be solved:
$$\begin{aligned} B(w^1 \otimes w^2 \cdots \otimes w^D,\delta w^1\otimes w^2 \cdots \otimes w^D)= & {} R_{m-1}(\delta w^1\otimes w^2 \cdots \otimes w^D) \quad \forall \delta w^1 \in \mathcal {X}_1 \nonumber \\ B(w^1 \otimes w^2 \cdots \otimes w^D,w^1\otimes \delta w^2 \cdots \otimes w^D)= & {} R_{m-1}(w^1\otimes \delta w^2 \cdots \otimes w^D) \quad \forall \delta w^2 \in \mathcal {X}_2 \nonumber \\ \vdots= & {} \vdots \nonumber \\ B(w^1 \otimes w^2 \cdots \otimes w^D,w^1\otimes w^2 \cdots \otimes \delta w^D)= & {} R_{m-1}(w^1\otimes w^2 \cdots \otimes \delta w^D) \quad \forall \delta w^D \in \mathcal {X}_D \end{aligned}$$
with \(R_{m-1}(v)=F(v)-B(u_{m-1},v)\). This problem is in practice solved with an iterative (fixed point) strategy.
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.
Reference problem and approximate FE solution
We consider a steady-state thermal problem that consists in finding the temperature/flux pair \((u,\varvec{q})\) such that:
$$\begin{aligned} \begin{array}{lll} &{}\quad u=0 \;\, \text {on}\,\; \Gamma _D &{}\quad \text {(kinematic constraints)}\\ -\varvec{\nabla }\cdot \varvec{q}= f \; \text {in}\; \Omega ; &{}\quad \varvec{q}\cdot \varvec{n}= g \;\, \text {on}\,\; \Gamma _N &{}\quad \text {(balance equations)}\\ &{}\quad \varvec{q}= \mathcal {K}\varvec{\nabla }u &{}\quad \text {(constitutive relation)} \end{array} \end{aligned}$$
\(\Omega \) is an open bounded subset of \(\mathbb {R}^d\) with Lipschitz boundary \(\partial \Omega \), and \(\Gamma _D\) and \(\Gamma _N\) are complementary parts of \(\partial \Omega \) such that \(\overline{\Gamma _D \cup \Gamma _N}=\partial \Omega \), \(\Gamma _D \cap \Gamma _N=\emptyset \), and \(|\Gamma _D| \ne 0\). We assume that \(f \in L^2(\Omega )\) and \(\mathcal {K}\in [L^{\infty }(\Omega )]^{d\times d}\) is a symmetric, uniformly bounded and positive matrix in the sense that there exists \(k_\mathrm{max} \ge k_\mathrm{min} > 0\) such that
$$\begin{aligned} \forall \varvec{\xi } \in \mathbb {R}^d, \quad k_\mathrm{min} |\varvec{\xi }|^2 \le \mathcal {K}\varvec{\xi } \cdot \varvec{\xi } \le k_\mathrm{max} |\varvec{\xi }|^2 \quad \text {a.e. in}\,\, \Omega . \end{aligned}$$
Considering the Hilbert space \(\mathcal {U}=\{v \!\in \! H^1(\Omega ), v=0 \; \text {on}\; \Gamma _D\}\) equipped with the \(H^1\)-norm \(\Vert v \Vert _1\), the weak formulation of (6) reads:
$$\begin{aligned} \text {Find}\,\,u \in \mathcal {U}\,\, \mathrm{such~that} \quad a(u,v)=l(v) \quad \forall v \in \mathcal {U}\end{aligned}$$
$$\begin{aligned} a(u,v) =\int _{\Omega }\mathcal {K}\varvec{\nabla }u \cdot \varvec{\nabla }v, \quad l(v)=\int _{\Omega }fv + \int _{\Gamma _N} gv. \end{aligned}$$
The bilinear form a is symmetric, continuous and coercive on \(\mathcal {U}\). It hence defines an inner product and induces the energy norm \(|||v|||_{\mathcal {U}}=\sqrt{a(v,v)}\) which is equivalent to \(\Vert v \Vert _1\) on \(\mathcal {U}\). We also denote \(\Vert v \Vert _0\) the \(L^2\)-norm on \(\mathcal {U}\). Existence and uniqueness of the solution u to (8) is provided by the Lax-Milgram theorem. We note that (8) is equivalent to the minimization of the potential energy \(J_1(v)=\frac{1}{2}a(v,v)-l(v)\) on \(\mathcal {U}\).
Let \(\mathcal {T}_h\) be a regular (non-degenerate) partition of \(\Omega \). Introducing the space \(\mathcal {U}_h\) of continuous and locally supported functions which are polynomials on each element \(K \in \mathcal {T}_h\), the conforming FE approximation of (8) reads:
$$\begin{aligned} \text {Find}\,\, u_h \in \mathcal {U}_h\,\, \mathrm{such~that} \quad a(u_h,v)=l(v) \quad \forall v \in \mathcal {U}_h \end{aligned}$$
We thus define the discretization error \(e=u_h-u \in \mathcal {U}\), for which a measure \(|||e|||_{\mathcal {U}}\) in terms of the energy norm can be introduced to express the global quality of the approximate solution \(u_h\). Introducing the residual functional and associated dual norm:
$$\begin{aligned} R(v) = l(v)-a(u_h,v); \quad \Vert R \Vert _{*} = \sup _{v \in \mathcal {U},v\ne 0}\frac{|R(v)|}{|||v|||_{\mathcal {U}}} \end{aligned}$$
leads to \(|||e|||_{\mathcal {U}} = \Vert R \Vert _{*}\). In the context of model verification, a main goal of a posteriori error estimators is to assess the value of \(|||e|||_{\mathcal {U}}\).
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.
Using the approximation approach with primal variational principle (10), that consists of minimizing the potential energy \(J_1(v)\) on \(\mathcal {U}_h\), leads to:
$$\begin{aligned} |||e|||_{\mathcal {U}}^2 = 2\left[ J_1(u_h)-J_1(u)\right] \ge 2\left[ J_1(u_h)-J_1(v) \right] \quad \forall v \in \mathcal {U}_h \end{aligned}$$
This shows that any \(v \in \mathcal {U}_h\) can only enable to compute a lower bound on the discretization error \(|||e|||_{\mathcal {U}}\); this lower bound is in practice usually poor unless v is chosen suitably.
Getting an upper bound on \(|||e|||_{\mathcal {U}}\) requires to use the complementary variational principle. Using the subspace \(\mathcal {S}\) of \(H(div,\Omega )=\{\varvec{\pi }\in [L^2(\Omega )]^d, \varvec{\nabla }\cdot \varvec{\pi }\in L^2(\Omega )\}\) defined as:
$$\begin{aligned} \mathcal {S}= & {} \left\{ \varvec{\pi }\in H(div,\Omega ),\varvec{\nabla }\cdot \varvec{\pi }+ f=0 \; \text {in}\;\Omega , \varvec{\pi }\cdot \varvec{n}= g\; \text {on}\;\Gamma _N \right\} \nonumber \\ \Longleftrightarrow \mathcal {S}= & {} \left\{ \varvec{\pi }\in H(div,\Omega ),\int _{\Omega }\varvec{\pi }\cdot \nabla v = \int _{\Omega }fv + \int _{\Gamma _N}gv \quad \forall v \in \mathcal {U}\right\} \end{aligned}$$
this complementary variational principle defines the solution flux field \(\varvec{q}=\mathcal {K}\varvec{\nabla }u\) as:
$$\begin{aligned} \varvec{q}= \arg \, \mathrm{{min}}_{\varvec{\pi }\in \mathcal {S}}J_2(\varvec{\pi }) \, ; \quad J_2(\varvec{\pi })=\frac{1}{2}\int _{\Omega }\mathcal {K}^{-1}\varvec{\pi }\cdot \varvec{\pi }= \frac{1}{2}|||\varvec{\pi }|||_{\mathcal {S}}^2 \end{aligned}$$
where \(|||\bullet |||_{\mathcal {S}}\) is the energy norm for flux fields. A direct consequence is:
$$\begin{aligned} |||\varvec{\pi }-\varvec{q}|||_{\mathcal {S}}^2 = 2\left[ J_2(\varvec{\pi })-J_2(\varvec{q}) \right] \end{aligned}$$
Noticing that \(J_1(u)=-J_2(\varvec{q})\) leads to the property (Prager-Synge equality):
$$\begin{aligned} |||e|||_{\mathcal {U}}^2= & {} 2[J_1(u_h)+J_2(\varvec{q})] = 2[J_1(u_h)+J_2(\varvec{\pi })] - |||\varvec{\pi }-\varvec{q}|||_{\mathcal {S}}^2 \nonumber \\= & {} 2E_{CRE}^2(u_h,\varvec{\pi })- |||\varvec{\pi }-\varvec{q}|||_{\mathcal {S}}^2 \end{aligned}$$
where we introduced the CRE functional \(E_{CRE}\) defined as:
$$\begin{aligned} E_{CRE}^2(v,\varvec{\pi }) = J_1(v)+J_2(\varvec{\pi }) = \frac{1}{2}|||\varvec{\pi }-\mathcal {K}\varvec{\nabla }v|||_{\mathcal {S}}^2 \quad \forall (v,\varvec{\pi }) \in \mathcal {U}\times \mathcal {S}\end{aligned}$$
that measures the non-verification of the constitutive relation for any pair \((v,\varvec{\pi }) \in \mathcal {U}\times \mathcal {S}\). In the following, such a pair is referred as admissible: a field \(v \in \mathcal {U}\) is said kinematically admissible (KA); a field \(\varvec{\pi }\in \mathcal {S}\) (i.e. verifying balance equations exactly) is said statically admissible (SA). Using the CRE concept, the reference problem (6) can be formulated as:
$$\begin{aligned} (u,\varvec{q})=\mathop {\text {arg min}}\limits _{(v,\varvec{\pi }) \in \mathcal {U}\times \mathcal {S}} E_{CRE}(v,\varvec{\pi }) \end{aligned}$$
For all material models described using internal variables and standard formulation, and introducing a suitable definition of admissibility spaces, a more general local (in space and time) expression of the CRE functional reads [2, 8]:
$$\begin{aligned} E_{CRE}^2(X,Y) = \phi (X)+\phi ^{*}(Y) - \langle X,Y \rangle \end{aligned}$$
where (X, Y) is a dual pair (with duality pairing \(\langle X,Y\rangle \)), and \(\phi \) and \(\phi ^{*}\) are dual (in the Legendre Fenchel sense) convex (pseudo-) potentials related to free energy or dissipation. For the present case, \(\phi (\varvec{\nabla }v)=\frac{1}{2} \mathcal {K}\varvec{\nabla }v \cdot \varvec{\nabla }v\) and \(\phi ^{*}(\varvec{\pi })=\frac{1}{2}\mathcal {K}^{-1}\varvec{\pi }\cdot \varvec{\pi }\).
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 non-conventional FE spaces. In "Coupling PGD with CRE in model verification" section, we present the basis of a technique (referred as hybrid-flux 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 post-processing of the FE field \(\varvec{q}_h\) at hand. The PGD strategy will be used within this technique in order to facilitate implementation issues.
It can be shown that using the hybrid-flux (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
We now consider that the material operator \(\mathcal {K}\) depends on a set \(\varvec{p}\in \mathcal {P}\) of parameters to be identified from experimental measurements. To solve the associated ill-posed inverse problem, we introduce the energy-based concept of modified constitutive relation error (mCRE) [28, 45], which can be seen as a direct extension of the CRE concept developed in the previous section. The mCRE functional, still based on duality between admissible primal and dual fields \((v,\varvec{\pi }) \in \mathcal {U}\times \mathcal {S}\), is defined as:
$$\begin{aligned} \mathcal {E}^2_{mCRE}(v,\varvec{\pi },\varvec{p},\varvec{s}) = \frac{1}{2}\left| \left| \left| \varvec{\pi }-\mathcal {K}(\varvec{p})\varvec{\nabla }v\right| \right| \right| ^2_{\mathcal {S}} + \frac{1}{2}\frac{r}{1-r}\sigma \Vert \varvec{\Pi }v-\varvec{s}\Vert ^2_0 \end{aligned}$$
where \(\varvec{s}\) is the set of experimental data, \(\varvec{\Pi }\) is an extraction operator, \(\sigma \) is a scaling coefficient, and \(r \in [0,1]\). The two terms that compose the mCRE functional are modeling error term (i.e. classical CRE term) and measurement error term, respectively; these terms are weighted depending on the value of r.
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 L-curve 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].
The solution of the inverse identification problem is then defined as the result of a double minimization:
$$\begin{aligned} \varvec{p}_0 = \mathop {\text {argmin}}\limits _{\varvec{p}\in \mathcal {P}}\left( \min _{(v,\varvec{\pi }) \in \mathcal {U}\times \mathcal {S}}\mathcal {E}^2_{mCRE}(v,\varvec{\pi },\varvec{p},\varvec{s})\right) \end{aligned}$$
In practice, this problem is solved using an iterative alternated minimization procedure with fixed point method as detailed in the following algorithm:
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 goal-oriented 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 trade-off 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 meta-model, 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 hybrid-flux technique
We consider the hybrid-flux (or EET) technique which enables to recover a flux field \(\widehat{\varvec{q}}_h \in \mathcal {S}\) from a post-processing of the FE field \(\varvec{q}_h\), with local independent computations [2, 6, 13]. It is a domain decomposition approach that consists of two steps:
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:
$$\begin{aligned} \int _K f + \int _{\partial K}\widehat{F} =0 \quad \forall K \in \mathcal {T}_h \end{aligned}$$
The construction of \(\widehat{F}\) is based on the following prolongation condition:
$$\begin{aligned} \int _K(\widehat{\varvec{q}}_h-\varvec{q}_h) \cdot \varvec{\nabla }\phi _i =0 \Longrightarrow \int _{\partial K} \widehat{F} \phi _i = \int _K (\varvec{q}_h \cdot \varvec{\nabla }\phi _i - f\phi _i) \end{aligned}$$
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 well-posed 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].
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:
$$\begin{aligned} -\varvec{\nabla }\cdot \widehat{\varvec{q}}_{h}= & {} f \quad \text {in}\; K \quad \Longleftrightarrow \; \int _K \widehat{\varvec{q}}_{h} \cdot \varvec{\nabla }v = \int _K fv + \int _{\partial K}\widehat{F} v \quad \forall v \in H^1(K) \nonumber \\ \widehat{\varvec{q}}_{h}\cdot \varvec{n}= & {} \widehat{F} \quad \text {on}\; \partial K \end{aligned}$$
The solution of (24) to get \(\widehat{\varvec{q}}_{h|K}\) 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 \(\widehat{\varvec{q}}_{h|K} =\mathcal {K}\varvec{\nabla }\rho \), with \(\rho \in H^1(K)\) verifying:
$$\begin{aligned} \int _K \mathcal {K}\varvec{\nabla }\rho \cdot \varvec{\nabla }v = \int _K f v + \int _{\partial K}\widehat{F} v \quad \forall v \in H^1(K) \end{aligned}$$
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.
Solving (25) is in practice the most costly part in the hybrid-flux method (in particular for 3D applications), as it involves high-order elements and has to be performed for each element K. We wish to use the PGD technique in order to find, in an offline phase, a parameterized solution to (25), valid for any configuration of the geometry and the loading. From equilibrated tractions computed in Step 1 and with respect to problem data (material parameters, mesh geometry, ...), this PGD solution would then be directly used in the online error estimation phase for each element K of the mesh.
Configuration at the element level. A 3-node triangle element is considered, and linear tractions \(\widehat{F}^{jl}\) are defined on element edges
Use of the PGD to solve problems at the element level
In the following:
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 \(\widehat{\varvec{q}}_{h|K}\); we thus set \(\mathcal {K}=\mathbb {I}\) when solving (25) and define \(\widehat{\varvec{q}}_{h|K} =\varvec{\nabla }\rho \). 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 3-node 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.
On each edge \(\Gamma ^{jl}\) between vertices j and l of any element K, tractions are linear combinations of FE shape functions and thus read, for the considered element type, \(\widehat{F}^{jl}(\varvec{x})=\widehat{F}_j^{jl}\phi _j(\varvec{x})+\widehat{F}_l^{jl}\phi _l(\varvec{x})\) with \(\left( \widehat{F}_j^{jl},\widehat{F}_l^{jl}\right) \in \mathbb {R}^2\). Consequently, the solution \(\rho \) to (25) can be written as a linear combination of elementary solutions:
$$\begin{aligned} \rho (\varvec{x}) = \sum _{(j,l)} \left[ \widehat{F}_j^{jl} \rho _j^{jl}(\varvec{x})+\widehat{F}_l^{jl} \rho _l^{jl}(\varvec{x})\right] \end{aligned}$$
where \(\rho _\ell ^{jl}\) (\(\ell =j,l\)) is the solution (up to a constant) to the elementary problem:
$$\begin{aligned} \int _K \varvec{\nabla }\rho _\ell ^{jl} \cdot \varvec{\nabla }v = \int _{\Gamma ^{jl}}\phi _\ell v - \int _K \frac{1}{|K|}\left( \int _{\Gamma ^{jl}}\phi _\ell \right) v \quad \forall v \in H^1(K) \end{aligned}$$
In the present case, there are 6 elementary problems.
Considering elasticity problems, (25) is changed in:
$$\begin{aligned} \int _K \mathcal {K}\underline{\underline{\nabla }}^s \varvec{\rho } : \underline{\underline{\nabla }}^s \varvec{v} = \int _K \varvec{f} \cdot \varvec{v} + \int _{\partial K}\widehat{\varvec{F}} \cdot \varvec{v} \quad \forall \varvec{v} \in [H^1(K)]^d \end{aligned}$$
where \(\mathcal {K}\) is the fourth-order symmetric elasticity tensor and \(\underline{\underline{\nabla }}^s\) is the symmetric part of the matrix gradient operator. For 3-node triangle elements, tractions on each edge \(\Gamma ^{jl}\) of K read \(\widehat{\varvec{F}}^{jl}(\varvec{x})=\widehat{\varvec{F}}_j^{jl}\phi _j(\varvec{x})+\widehat{\varvec{F}}_l^{jl}\phi _l(\varvec{x})\) or:
$$\begin{aligned} \widehat{\varvec{F}}^{jl}(\varvec{x})=\widehat{F}_{jx}^{jl}\varvec{\phi }^x_j(\varvec{x}) + \widehat{F}_{jy}^{jl}\varvec{\phi }^y_j(\varvec{x}) + \widehat{F}_{lx}^{jl}\varvec{\phi }^x_l(\varvec{x}) + \widehat{F}_{ly}^{jl}\varvec{\phi }^y_l(\varvec{x}) \end{aligned}$$
with \(\widehat{\varvec{F}}^{jl}_\ell = \left( \widehat{{F}}^{jl}_{\ell x},\widehat{{F}}^{jl}_{\ell y}\right) ^T\), \(\varvec{\phi }^x_\ell (\varvec{x})=(\phi _\ell (\varvec{x}),0)^T\) and \(\varvec{\phi }^y_\ell (\varvec{x})=(0,\phi _\ell (\varvec{x}))^T\) (\(\ell =j,l\)). We thus introduce solutions, defined up to a rigid body motion, to the following elementary problems [generalization of (27)]:
$$\begin{aligned} \int _K \mathcal {K}\underline{\underline{\nabla }}^s \varvec{\rho }_\ell ^{jl,x/y} : \underline{\underline{\nabla }}^s \varvec{v} = \int _{\Gamma ^{jl}}\varvec{\phi }^{x/y}_\ell \cdot \varvec{v} - \int _K (\varvec{a}_1 \wedge \varvec{X} + \varvec{a}_2) \cdot \varvec{v} \quad \forall \varvec{v} \in [H^1(K)]^2 \end{aligned}$$
where \(\varvec{X}\) are barycentric coordinates in element K, and \(\varvec{a}_1\) and \(\varvec{a}_2\) are defined as:
$$\begin{aligned} \varvec{a}_1 = \frac{\left( \int _{\Gamma ^{jl}}\varvec{X} \wedge \varvec{\phi }^{x/y}_\ell \right) \cdot \varvec{z}}{\int _K\varvec{X} \cdot \varvec{X}} \varvec{z}; \quad \varvec{a}_2 = \frac{1}{|K|}\int _{\Gamma ^{jl}}\varvec{\phi }^{x/y}_\ell \end{aligned}$$
with \(\varvec{z}\) the orthonormal vector to the 2D plane. The solution \(\varvec{\rho }\) to (28) is then recovered as:
$$\begin{aligned} \varvec{\rho }(\varvec{x}) = \sum _{(j,l)}\left[ \widehat{F}_{jx}^{jl} \varvec{\rho }_j^{jl,x}(\varvec{x})+\widehat{F}_{jy}^{jl} \varvec{\rho }_j^{jl,y}(\varvec{x})+\widehat{F}_{lx}^{jl} \varvec{\rho }_l^{jl,x}(\varvec{x})+\widehat{F}_{ly}^{jl} \varvec{\rho }_l^{jl,y}(\varvec{x})\right] \end{aligned}$$
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 parameter-dependent 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.
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.
Mapping between reference and parameterized elements. The mapping is defined as a combination of a scaling mapping \(\mathcal {M}_1\) between homothetic elements \(\overline{K}\) and K, and a linear mapping \(\mathcal {M}_2\), defined from an isoparametric formulation, between a reference element \(K_{ref}\) and \(\overline{K}\)
In the present case, the mapping is defined from three parameters (Fig. 2):
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:
$$\begin{aligned} \left( \begin{array}{c}x \\ y \end{array}\right) = \mathbb {T}_1\left( \begin{array}{c}\overline{x} \\ \overline{y} \end{array}\right) \, ; \quad \mathbb {T}_1 = \left[ \begin{array}{c@{\quad }c}\alpha &{} 0\\ 0 &{} \alpha \end{array}\right] = \alpha \mathbb {I}\end{aligned}$$
A second linear mapping \(\mathcal {M}_2:K_{ref} \rightarrow \overline{K}\) maps a reference element \(K_{ref}\) (right-angled isosceles triangle) to element \(\overline{K}\). This mapping reads, using an isoparametric formulation:
$$\begin{aligned} \left( \begin{array}{c} \overline{x} \\ \overline{y} \end{array}\right) = \left( \begin{array}{c} \phi _2(\eta ,\xi ) + \overline{x}_3 \phi _3(\eta ,\xi ) \\ \overline{y}_3 \phi _3(\eta ,\xi ) \end{array}\right) = \mathbb {T}_2\left( \begin{array}{c} \eta \\ \xi \end{array}\right) \, ; \quad \mathbb {T}_2 = \left[ \begin{array}{c@{\quad }c} 1 &{} \overline{x}_3\\ 0 &{} \overline{y}_3 \end{array}\right] \end{aligned}$$
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).
It thus involves 3 parameters and leads to the global mapping:
$$\begin{aligned} \mathcal {M}\left( \alpha , \overline{x}_3,\overline{y}_3 \right) = \mathcal {M}_1(\alpha ) \circ \mathcal {M}_2(\overline{x}_3,\overline{y}_3) \end{aligned}$$
with transformation matrix \(\mathbb {T}(\alpha , \overline{x}_3,\overline{y}_3)=\alpha \mathbb {T}_2(\overline{x}_3,\overline{y}_3)\), Jacobian matrix \(\mathbb {J}=\mathbb {T}\), and Jacobian \(J=det(\mathbb {J})=\alpha ^2 \overline{y}_3\).
Introducing \(\varvec{x}_{ref}=\left( \begin{array}{c}\eta \\ \xi \end{array}\right) \), approximations of solutions \(\rho ^{jl}_\ell \left( \varvec{x}_{ref},\alpha ,\overline{x}_3,\overline{y}_3 \right) \) are computed offline and once for all using the PGD technique with variable-separated modal decomposition. The parameter \(\alpha \) is included only for completeness of the description; it acts as a multiplicative constant in the solution \(\rho \) and disappears when computing \(\widehat{\varvec{q}}_{h|K} =\mathcal {K}\varvec{\nabla }\rho \). PGD solutions thus read:
$$\begin{aligned} \rho ^{jl}_{\ell ,m} \left( \varvec{x}_{ref},\alpha ,\overline{x}_3,\overline{y}_3 \right) = \alpha \sum _{i=1}^m\psi _i(\varvec{x}_{ref}) \delta ^x_i(\overline{x}_3) \delta ^y_i(\overline{y}_3) \end{aligned}$$
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, 6-node 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 4-node 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
The progressive Galerkin approach described in "Background" section is used with bilinear form B and linear form F constructed from the parameterized separated variable Jacobian transformation (all technical details can be found in [51, 52]). Introducing the interval \(I_{\alpha }\) (resp. \(I_{\overline{x}_3}\) and \(I_{\overline{y}_3}\)) in which \(\alpha \) (resp. \(\overline{x}_3\) and \(\overline{y}_3\)) evolves, these forms read:
$$\begin{aligned}&B(\rho _{\ell }^{jl},v) = \int _{I_{\alpha }}\int _{I_{\overline{x}_3}}\int _{I_{\overline{y}_3}}\int _{K_{ref}}J. \mathbb {J}^{-T}\varvec{\nabla }\rho _\ell ^{jl} \cdot \mathbb {J}^{-T}\varvec{\nabla }v \nonumber \\&F(v) = \int _{I_{\alpha }}\int _{I_{\overline{x}_3}}\int _{I_{\overline{y}_3}}\left[ \int _{\Gamma ^{jl}_{ref}}J_s \phi _\ell v - \int _{K_{ref}} J.\frac{1}{|K|}\left( \int _{\Gamma ^{jl}}\phi _\ell \right) v \right] \end{aligned}$$
Space functions \(\psi _i(\varvec{x}_{ref})\) in (36) are computed using the FEM with a single element of degree \(p+k\). Other functions are discretized using a fine grid over spaces \(I_{\alpha }\), \(I_{\overline{x}_3}\), and \(I_{\overline{y}_3}\).
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 EET-PGD" section.
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.
The PGD technique thus provides for a parameterized equilibrated flux field at the element level:
$$\begin{aligned}&\widehat{\varvec{q}}_{h,m|K}\left( \varvec{x}_{ref}, \left\{ \widehat{F}^{jl}_\ell \right\} ,\alpha ,\overline{x}_3,\overline{y}_3 \right) \nonumber \\&\quad = \mathbb {J}^{-T}(\alpha ,\overline{x}_3,\overline{y}_3)\varvec{\nabla }\left( \sum _{(j,l)}[\widehat{F}_j^{jl} \rho _{j,m}^{jl}(\varvec{x}_{ref},\alpha ,\overline{x}_3,\overline{y}_3)+\widehat{F}_l^{jl} \rho _{l,m}^{jl}(\varvec{x}_{ref},\alpha ,\overline{x}_3,\overline{y}_3)] \right) \end{aligned}$$
which can be directly used online in the a posteriori error estimation procedure.
Remark 10
Another study, which is not considered here, would benefit from the PGD representation \(\widehat{\varvec{q}}_{h,m|K} \left( \varvec{x}_{ref}, \left\{ \widehat{F}^{jl}_\ell \right\} ,\alpha ,\overline{x}_3,\overline{y}_3 \right) \). 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
The constrained minimization of the mCRE method (Step 2 in the algorithm given in "Extension of the CRE concept for model updating: modified CRE" section) is in practice performed finding the saddle-point of the following Lagrangian functional:
$$\begin{aligned} \mathcal {L}(v,\varvec{\pi },\varvec{p},\lambda ) = \mathcal {E}^2_{mCRE}(v,\varvec{\pi },\varvec{p},\varvec{s}) - \left[ \int _{\Omega }\varvec{\pi }\cdot \varvec{\nabla }\lambda - \int _{\Omega }f\lambda - \int _{\Gamma _N}g\lambda \right] \end{aligned}$$
for all \((v,\varvec{\pi },\lambda ) \in \mathcal {U}\times H(div,\Omega ) \times \mathcal {U}\). It leads to the solution \(\left( \widehat{u},\widehat{\varvec{q}},\widehat{\lambda } \right) \) of the coupled system:
$$\begin{aligned} \int _{\Omega }\left( \mathcal {K}(\varvec{p})\varvec{\nabla }\widehat{u} - \widehat{\varvec{q}}\right) \cdot \varvec{\nabla }v + \frac{r}{1-r}\sigma \left( \varvec{\Pi }\widehat{u} - \varvec{s}\right) \cdot \varvec{\Pi }v= & {} 0 \quad \forall v \in \mathcal {U}\nonumber \\ \int _{\Omega }\left( \widehat{\varvec{q}} - \mathcal {K}(\varvec{p})\varvec{\nabla }\widehat{u}\right) \cdot \mathcal {K}^{-1}(\varvec{p})\varvec{\pi }- \int _{\Omega }\varvec{\nabla }\widehat{\lambda } \cdot \varvec{\pi }= & {} 0 \quad \forall \varvec{\pi }\in H(div,\Omega ) \nonumber \\ \int _{\Omega }\widehat{\varvec{q}} \cdot \varvec{\nabla }\lambda - \int _{\Omega }f\lambda - \int _{\Gamma _N}g\lambda= & {} 0 \quad \forall \lambda \in \mathcal {U}\end{aligned}$$
The second relation yields \(\widehat{\varvec{q}}=\mathcal {K}(\varvec{p})\varvec{\nabla }\left( \widehat{u}+\widehat{\lambda }\right) \), and \((\widehat{u},\widehat{\lambda })\) is obtained solving:
$$\begin{aligned} -\int _{\Omega }\mathcal {K}(\varvec{p})\varvec{\nabla }\widehat{\lambda } \cdot \varvec{\nabla }v + \frac{r}{1-r}\sigma \varvec{\Pi }\widehat{u} \cdot \varvec{\Pi }v= & {} \frac{r}{1-r}\sigma \varvec{s}\cdot \varvec{\Pi }v \quad \forall v \in \mathcal {U}\nonumber \\ \int _{\Omega }\mathcal {K}(\varvec{p})\varvec{\nabla }\left( \widehat{u}+\widehat{\lambda }\right) \cdot \varvec{\nabla }\lambda= & {} \int _{\Omega }f\lambda + \int _{\Gamma _N}g\lambda \quad \forall \lambda \in \mathcal {U}\end{aligned}$$
The gradient of the cost function \(\mathcal {F}(\varvec{p})\), which is required when performing the second minimization with first order strategies, is then easily computed using the adjoint state method, as \(\varvec{\nabla }\mathcal {F}(\varvec{p})=\varvec{\nabla }_{\varvec{p}}\mathcal {L}\left( \widehat{u},\widehat{\varvec{q}},\varvec{p},\widehat{\lambda }\right) \).
Usually, the discretization error is assumed to be negligible in the mCRE formulation (it can be anyway controlled using classical verification procedures, see [58]) so that a discretized version of (21) can be written using FEM, and model updating is applied directly to the discretized representation. In particular, the strong equilibrium conditions involved in the admissibility space \(\mathcal {S}\) are replaced by weaker equilibrium conditions, in the FE sense only. We start from the following definitions:
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 \(\varvec{\mathcal {S}}_h\).
The discretized mCRE functional thus reads:
$$\begin{aligned} \mathcal {E}^2_h(\varvec{V},\varvec{W},\varvec{p},\varvec{s}) = \frac{1}{2}(\varvec{W}-\varvec{V})^T\mathbb {K}(\varvec{p})(\varvec{W}-\varvec{V}) + \frac{1}{2}\frac{r}{1-r}(\Pi \!{\Pi }\varvec{V}-\varvec{s})^T\mathbb {G}(\Pi \!{\Pi }\varvec{V}-\varvec{s}) \end{aligned}$$
where \(\mathbb {G}\) is a scaling diagonal matrix that integrates \(\sigma \). Defining the cost function \(\mathcal {F}_h(\varvec{p})\) as:
$$\begin{aligned} \mathcal {F}_h(\varvec{p})=\min _{(\varvec{V},\varvec{W})\in \varvec{\mathcal {U}}_h \times \varvec{\mathcal {S}}_h}\mathcal {E}^2_h(\varvec{V},\varvec{W},\varvec{p},\varvec{s}) \end{aligned}$$
the associated constrained minimization is performed introducing the Lagrangian:
$$\begin{aligned} \mathcal {L}_h(\varvec{V},\varvec{W},\varvec{p},\varvec{\Lambda }) = \mathcal {E}^2_h(\varvec{V},\varvec{W},\varvec{p},\varvec{s}) - \varvec{\Lambda }^T\left[ \mathbb {K}(\varvec{p}) \varvec{W}- \varvec{F}\right] \end{aligned}$$
and leads to \(\left( \widehat{\varvec{V}},\widehat{\varvec{W}},\widehat{\varvec{\Lambda }}\right) \) solution of the system:
$$\begin{aligned} \widetilde{\mathbb {K}}(\varvec{p})\left( \widehat{\varvec{U}}-\widehat{\varvec{W}}\right) + \frac{r}{1-r} \Pi \!{\Pi }^T\widetilde{\mathbb {G}}\left( \Pi \!{\Pi }\widehat{\varvec{U}}-\varvec{s}\right)= & {} \varvec{0} \nonumber \\ \mathbb {K}(\varvec{p})\left( \widehat{\varvec{W}}-\widehat{\varvec{U}}\right) - \mathbb {K}(\varvec{p})\widehat{\varvec{\Lambda }}= & {} \varvec{0} \nonumber \\ \widetilde{\mathbb {K}}(\varvec{p}) \widehat{\varvec{W}} - \widetilde{\varvec{F}}= & {} \varvec{0} \end{aligned}$$
where \(\widetilde{\mathbb {K}}\) (resp. \(\widetilde{\mathbb {G}}\) and \(\widetilde{\varvec{F}}\)) is the restriction of \(\mathbb {K}\) (resp. \(\mathbb {G}\) and \(\varvec{F}\)) in which lines corresponding to prescribed dofs in \(\varvec{\mathcal {U}}_h\) have been removed.
Use of the PGD for the first minimization
In this section, we implement a PGD meta-model to find, in an offline phase, parameterized solutions \(\left( \widehat{u},\widehat{\lambda }\right) \) to (41). Defining \(\sigma _r= \frac{r}{1-r}\sigma \in \Sigma _r\) (single parameter gathering scaling and weighting effects in mCRE) and assuming that \(\mathcal {P}=\otimes _{j=1}^P \mathcal {P}_j\), these are searched of the form:
$$\begin{aligned} \widehat{u}_m(\varvec{x},\sigma _r,\varvec{p})= & {} \sum _{i=1}^m \left[ \psi ^u_i(\varvec{x})\kappa _i^u(\sigma _r)\prod _{j=1}^P \chi ^u_{j,i}(p_i)\right] ; \nonumber \\&\widehat{\lambda }_m(\varvec{x},\sigma _r,\varvec{p}) = \sum _{i=1}^m \left[ \psi ^{\lambda }_i(\varvec{x})\kappa _i^{\lambda }(\sigma _r)\prod _{j=1}^P \chi ^{\lambda }_{j,i}(p_i)\right] \end{aligned}$$
Here again, the progressive Galerkin approach described in "Background" section is used with the following bilinear form B and linear form F:
$$\begin{aligned}&B((\widehat{u},\widehat{\lambda }),(v,\lambda )) = \int _{\Sigma _r}\int _{\mathcal {P}}\left[ \int _{\Omega }\mathcal {K}(\varvec{p})\varvec{\nabla }(\widehat{u}+\widehat{\lambda }) \cdot \varvec{\nabla }\lambda -\int _{\Omega }\mathcal {K}(\varvec{p})\varvec{\nabla }\widehat{\lambda } \cdot \varvec{\nabla }v + \sigma _r \varvec{\Pi }\widehat{u} \cdot \varvec{\Pi }v \right] \nonumber \\&F((v,\lambda )) = \int _{\Sigma _r}\int _{\mathcal {P}}\left[ \int _{\Omega }f\lambda + \int _{\Gamma _N}g\lambda + \sigma _r \varvec{s}\cdot \varvec{\Pi }v \right] \end{aligned}$$
Using then \((\widehat{u}_m,\widehat{\lambda }_m)\) in the online model updating phase with mCRE has several advantages:
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 L-curve method) straightforward.
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 time-dependent problems, they can be considered as extra-parameters 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 \(\mathcal {P}_j\) (\(j=1,\dots ,P\)).
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
We consider discretization error estimation on a 2D holed plate \(\Omega \), according to a given mesh composed of 3-nodes triangular elements (Fig. 3). We consider a steady-state thermal problem and homogeneous isotropic material properties with \(\mathcal {K}=\mathbb {I}\). A prescribed zero temperature is applied on the external boundary \(\Gamma _D\), while a flux \(g=1 \, W/m\) is imposed on the inner boundary \(\Gamma _N\). Owing to problem symmetries, only one quarter of the plate is studied.
Representation of the 2D domain, associated FE mesh, and applied loading. The holed plate is discretized with triangle elements, a zero temperature is prescribed on the external boundary, and a given flux is applied on the inner boundary
From the associated FE solution, equilibrated tractions are computed using the first step of the hybrid-flux (or EET) technique.
Details on the PGD solution
We compute a parametrized solution \(\rho ^{jl}_{\ell ,m}\left( \varvec{x}_{ref},\alpha ,\overline{x}_3,\overline{y}_3\right) = \alpha \sum _{i=1}^m\psi _i(\varvec{x}_{ref}) \delta ^x_i(\overline{x}_3) \delta ^y_i(\overline{y}_3)\) of (25) with a single 4th order FE element and 20 PGD modes (\(m=20\)). The domains \(I_{\overline{x}_3} = [0,1]\) and \(I_{\overline{y}_3} = [0.1,1]\) are discretized with 100 points each, after checking that this is sufficient to ensure an accurate description of the evolutions with respect to \(\overline{x}_3\) and \(\overline{y}_3\). The first three PGD modes of \(\rho ^{11}_{1,20}\) are shown in Fig. 4. In Fig. 5, we represent the PGD approximation of \(\rho ^{11}_1\) for different configurations of parameters \(\overline{x}_3\) and \(\overline{y}_3\). The computation of this PGD solution is done once for all, in an offline phase and stored for later use.
First PGD modes of \(\rho ^{11}_{1,20}\). Space functions \(\psi _i\) (top), and parameters functions \(\delta _i^x\) (center) and \(\delta _i^y\) (bottom) are represented with respect to the order i, for i = 1, 2, 3 (from left to right)
PGD solution \(\rho ^{11}_{1,20}\). The parameterized approximate solution is represented over the element \(\overline{K}\) for various parameter configurations
Identified PGD parameters in the mesh. The parameters values \(\alpha \), \(\overline{x}_3\) and \(\overline{y}_3\) are set for each element K of the domain
After identifying the PGD parameters \(\alpha \), \(\overline{x}_3\) and \(\overline{y}_3\) over each element of the mesh (see Fig. 6), an accurate PGD approximation of the admissible flux \(\widehat{\varvec{q}}_m\) can then be directly evaluated inside each element in an inexpensive online phase; this method is referred as EET-PGD method in the following.
Local contributions to the CRE estimate. We represent elementary contributions to the error estimate \(2E^2_{CRE}\) obtained using either the EET technique (left) or the EET-PGD technique (center), and elementary contributions to the exact error \(|||e|||^2_{\mathcal {U}}\) (right)
CRE estimate obtained from EET-PGD
From PGD solutions, we have all ingredients to estimate the discretization error using the CRE method. In Fig. 7, we compare local contributions to the CRE estimate \(2 E^2_{CRE}\), obtained from the EET-PGD technique when computing an admissible flux \(\widehat{\varvec{q}}_{h,m}\), with: (1) contributions to the CRE estimate obtained from the classical EET technique when computing an admissible flux \(\widehat{\varvec{q}}_h\); (2) contributions to the exact error \(|||e|||^2_{\mathcal {U}}\) evaluated using a highly refined mesh (overkill solution). One observes similarities between the two CRE estimations, showing up areas where the mesh needs to be refined. These areas are correctly predicted when comparing to the exact error distribution.
Influence on the effectivity index. We represent the value of the effectivity index with respect to the number of PGD modes used to evaluate the equilibrated flux in the CRE estimate
Error on the PGD construction. We represent the evolution of the relative error \(\left| \left| \left| \widehat{\varvec{q}}_{h,m}-\widehat{\varvec{q}}_{h}\right| \right| \right| _{\mathcal {S}}/\left| \left| \left| \widehat{\varvec{q}}_{h}\right| \right| \right| _{\mathcal {S}}\) with respect to the number of PGD modes used to compute \(\widehat{\varvec{q}}_{h,m}\)
Map of the error on the flux field computed with PGD. The spatial distribution of the error \(\left| \left| \left| \widehat{\varvec{q}}_{h,m}-\widehat{\varvec{q}}_{h}\right| \right| \right| _{\mathcal {S}}\) is represented for \(m=1\) (left), \(m=2\) (center), and \(m=3\) (right)
Choosing \(m=20\) to compute PGD solutions in the EET-PGD technique may be unnecessary. To analyze this point, we show in Fig. 8 values of the effectivity index \(\displaystyle {i_{eff} = \frac{\sqrt{2}E_{CRE}}{|||e|||_{\mathcal {U}}}}\) with respect to the number m of PGD modes used to evaluate the equilibrated flux \(\widehat{\varvec{q}}_{h,m}\). We also represent in Fig. 9 the evolution of the relative error \(\frac{|||\widehat{\varvec{q}}_{h,m}-\widehat{\varvec{q}}_{h}|||_{\mathcal {S}}}{|||\widehat{\varvec{q}}_{h}|||_{\mathcal {S}}}\) with respect to m, where \(\widehat{\varvec{q}}_{h}\) is the equilibrated flux field constructed with the EET technique. A map of \(|||\widehat{\varvec{q}}_{h,m}-\widehat{\varvec{q}}_{h}|||_{\mathcal {S}}\) for \(m=1\), \(m=2\), and \(m=3\) is given in Fig. 10. We observe that choosing \(m=7\) is enough to reconstruct an admissible flux solution which is equivalent to the one obtained with the classical EET technique. We also observe that \(m=3\) enables to capture the complexity of the local problems and to provide for a relevant error estimate, even though it is not guaranteed.
Comparison of CPU times. Values of CPU times required to compute the hybrid-flux steps are given depending on the equilibrated flux technique which is used
Speed-up 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 EET-PGD techniques share as much code as possible, and only the construction and solution of the matrix problem is replaced by a simple post-processing with PGD solutions in the EET-PGD technique. Naturally, the first step with construction of equilibrated tractions is similar for both techniques.
When using the EET-PGD technique, the offline CPU cost to compute the PGD solution is 312 s; this solution can then be used in a multi-query context. In the online step, computing the equilibrated flux from the classical EET technique (Cholesky factorization) takes 0.01509 s per element (0.0587 s for the whole mesh composed of 42 elements), whereas computing the equilibrated flux from a direct evaluation of the PGD solution takes 0.00426 s per element (0.0077 s for the whole mesh). We thus observe a speed-up of almost 10 in the second CRE step (construction of equilibrated fluxes in each element), and the global speed-up on the whole hybrid-flux technique (with associated CPU cost of 0.0960 s) is about a factor 2.
CPU gains. For meshes with increasing numbers of elements, the CPU gains on the equilibrated flux step are given. Gains on the local recovery alone or on the overall hybrid-flux technique are explicited
Sequence of refined meshes. This sequence, obtained from a refinement of the initial mesh, is used to assess gain on the CPU time with the EET-PGD technique
In Fig. 12, we represent this same speed-up for different levels of refinement of the initial mesh (corresponding meshes are given in Fig. 13). The speed-up increases as the mesh becomes finer, reaching a gain of 125 on a 2688 elements mesh for the local recovery, while the overall hybrid-flux technique shows a speed-up of magnitude 5 on this same mesh. An additional step would be to reduce the CPU time of the tractions reconstruction, by optimizing implementation, in order to fully benefit from the use of the PGD technique.
Problem geometry and sensors location. The structure is a two layers cylinder with a localized inclusion (represented in red), which is subjected to a given thermal flow on the inner boundary. The identification process is performed from sensors (red squares) located on the external boundary, and using a FE mesh made of tetrahedral elements
Example 2: model updating on a 3D structure
Identification problem
We consider a steady-state 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 \(p_{int}=20\).
To perform the identification process, the structure is discretized with a FE mesh made of 41,856 tetrahedra (13,164 nodes) as presented in Fig. 14. The noisy observation data are synthesized numerically by solving the direct problem with reference parameter values \((p^{ref}_{ext},p^{ref}_{inc})\), extracting the obtained nodal temperature values \(u_i\) at sensors positions, then adding a Gaussian white noise to get data \(s_i\):
$$\begin{aligned} s_i = \left( 1 + \mathcal {N}(0,\Upsilon ) \right) u_i \end{aligned}$$
with variance \(\Upsilon \). In the following, we choose \(\Upsilon =0.1\).
Spatial PGD modes for the identification with mCRE. The first five (normalized) modes for the primal field (left), and the adjoint field (right), are plotted
Extra-coordinates PGD modes for the identification with mCRE. The first five (normalized) modes for the primal field (left), and the adjoint field (right), are plotted
PGD model reduction
As detailed in "Coupling PGD with mCRE in model validation" section, PGD representations of the parameterized solutions \(\left( \widehat{u},\widehat{\lambda }\right) \) to (41) are computed in an offline phase. These read:
$$\begin{aligned} \widehat{u}_m(\varvec{x},\sigma _r,p_{ext},p_{inc})= & {} \sum _{i=1}^m\psi ^u_i(\varvec{x})\kappa _i^u(\sigma _r) \chi ^u_{1,i}(p_{ext})\chi ^u_{2,i}(p_{inc})\nonumber \\ \widehat{\lambda }_m(\varvec{x},\sigma _r,p_{ext},p_{inc})= & {} \sum _{i=1}^m \psi ^{\lambda }_i(\varvec{x})\kappa _i^{\lambda }(\sigma _r) \chi ^{\lambda }_{1,i}(p_{ext}) \chi ^{\lambda }_{2,i}(p_{inc}) \end{aligned}$$
The first five (normalized) PGD modes are represented in Figs. 15 and 16. In Fig. 17, we represent the energy norm of each PGD mode relative to the energy norm of the first PGD mode \(u_1\), which shows that their influence highly decreases with m. In practice, we choose \(m=15\).
Energy norm of each PGD mode. The graph on the right is a zoom for PGD modes with small values of the energy norm
Convex shape of the mCRE cost function. We represent the evolution of the mCRE functional with respect to \(p_{ext}\) and \(p_{inc}\)
Trade-off in the mCRE functional. We represent the evolution of the two terms which compose the mCRE functional, i.e. the model error term and the measurement error term, with respect to \(\sigma _r\)
Identification with PGD
Starting from the initial parameter values \(p_{ext}^0=20\) and \(p_{inc}^0=10\), we implement the model updating process using the mCRE method coupled with the previously computed PGD solutions. We represent in Fig. 18 the evolution of the cost function \(\mathcal {F}(p_{ext},p_{inc}) = \mathcal {E}^2_{mCRE}\left( \widehat{u}_m,\widehat{\varvec{q}}_m,p_{ext},p_{inc},\varvec{s}\right) \); it clearly shows the convex feature of this cost function, and therefore the uniqueness of the minimization solution. We also plot in Fig. 19 the evolution of the two terms of the cost function, i.e. the model error term and the measurement error term, with respect to the penalty coefficient \(\sigma _r\) for \((p_{ext},p_{inc})=\left( p^0_{ext},p^0_{inc} \right) \). The optimal value of \(\sigma _r\) is the one for which the two error terms are balanced (i.e. when the two curves intersect). Notice that these evolutions of the mCRE functional are easy to obtain as the PGD solutions (49) lead to explicit dependencies with respect to \(p_{ext}\), \(p_{inc}\), and \(\sigma _r\).
Identification results. Identified values of parameters \(p_{ext}\) and \(p_{inc}\) (left), and optimal value of the penalty coefficient \(\sigma _r\) (right), are given with respect to iterations
Influence of m on the identification results. We represent the relative error on the identified values of parameters \(p_{ext}\) (left) and \(p_{inc}\) (right), along the identification iterations, with respect to the value of m
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 \(p_{inc}\).
The use of PGD enables large computation gains. Using a direct solver with parallelization over 4 nodes, a classical identification process with mCRE would require about 4 h for this problem. Coupled with PGD, this same process takes only 5 min in the online phase (and additional 30 min to compute PGD solutions with 15 modes in the offline phase). All computations were performed with a Python FE code using the scipy.sparse module for matrix representation, and systems were solved with a dedicated direct solver based on the UMFPACK library. The speed-up thus comes from the difference between the original mCRE and the PGD-mCRE strategies. A crude complexity analysis of the two approaches can be conducted as follows:
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 sub-iterations, 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 PGD-mCRE, 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 sub-iterations (average), and a sub-iteration 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.
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 real-time 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 time-dependent problems. This will be the topic of forthcoming research works.
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Ammar A, Huerta A, Leygue A, Chinesta F, Cueto E. Parametric solutions involving geometry: a step towards efficient shape optimization. Comput Methods Appl Mech Eng. 2014;268(1):178–93.
Zlotnik S, Díez P, Modesto D, Huerta A. Proper generalized decomposition of a geometrically parametrized heat problem with geophysical applications. Int J Numer Methods Eng. 2015;103(10):737–58.
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All authors discussed the content of the article, and were involved in its writing. All authors read and approved the final manuscript.
LMT, ENS Cachan, CNRS, Université Paris Saclay, 61 Avenue du Président Wilson, 94235, Cachan, France
Ludovic Chamoin, Pierre-Eric Allier & Basile Marchand
Ludovic Chamoin
Pierre-Eric Allier
Basile Marchand
Correspondence to Ludovic Chamoin.
Chamoin, L., Allier, PE. & Marchand, B. Synergies between the constitutive relation error concept and PGD model reduction for simplified V&V procedures. Adv. Model. and Simul. in Eng. Sci. 3, 18 (2016). https://doi.org/10.1186/s40323-016-0073-9
Received: 07 February 2016
Accepted: 19 May 2016
Error estimation
Model updating
Constitutive relation error (CRE)
Proper generalized decomposition (PGD)


When Barack Obama took the oath of office in 2009 and addressed the nation during his first inauguration the air in Washington was bitter cold. An estimated 2 million people crammed the streets or watched on large-screen TVs in a throng that stretched from the Lincoln Memorial to the Washington Monument. The city was paralyzed.
They were there to witness the historic start for America�s first black president, one who promised hope and change through two tough campaigns.
Monday in a smaller but still festive ceremony, the 44th president, hair noticeably flecked with more gray, began his second term in more subdued but no less hopeful fashion. Officials estimated a crowd of 600,000 to 800,000 joined the festivities, still much higher than average for a second inaugural.
Speaking on a brisk sunny day, far more welcoming than the chill of 2009 (which perhaps was a harbinger of his future political battles), Obama, 51, spoke of an end to a decade of war, economic prosperity ahead, and the importance of freedom. When he took office four years ago, America was engaged in wars in both Iraq and Afghanistan while also facing the most daunting economic crisis in decades.
Inaugurals are historically more about broad themes and igniting supporters. Obama did both, while casting few specifics for immediate consumption. He spoke of rebuilding the middle class, evening the economic playing field for all Americans, climate change, immigration reform, and gun control. There�s also a nuclear threat in Iran and new terrorism in North Africa to contend with. It�s an ambitious agenda that ensures future political conflict, after four years of partisan gridlock.
What he didn�t say though is that the administration will also have to compromise as the economic recovery continues. Tax revisions and spending cuts must still be addressed. The fiscal cliff was averted, but the debt ceiling looms.
The window for flowery speeches has now officially closed on all sides. It�s time for Washington to stop talking. It�s time for action.


Perhaps out of all the lists we make at the end of the year, the EPs list best illustrates where our favorite music's been this year and where it's going.
Perhaps out of all the lists we make at the end of the year, the EPs list best illustrates where our favorite music's been this year and where it's going.
08 Beach Slang, Who Would Ever Want Anything So Broken?
Yeah, we know it's a cop out, but... shout out to Ben Khan's 1992 EP, Doss' Doss, Julianna Barwick's Rosabi EP, Kiesza's Hideaway, Kitty's Frostbite, Le1f's Hey, LIZ's Just Like You, Majid Jordan's A Place Like This, Róisín Murphy's Mi senti, Speedy Ortiz's Real Hair, SZA's Z, Tove Lo's Truth Serum, Yør's Lack of Being, and Yumi Zouma's Yumi Zouma. Phew.


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BOOKFEST TURNS OVER NEW LEAF - IN DOWNTOWN WEST PALM
SARAH LUNDY Staff WriterSUN-SENTINEL
Downtown West Palm Beach will soon be a bookworm's paradise.
Authors of hundreds of books _ biographies, mysteries, children's books, even cookbooks _ will be talking about their craft.
Bookfest '98 starts at noon on Fridayand goes to 5 p.m. Sundayat the International Pavilion on Flagler Drive.
"I love to hear authors talk about how they write and how they do research," said Angelica Carpenter, the Palm Springs Public Library director who started Bookfest eight years ago.
This year, the literary festival moved from the South Florida Fairgrounds, to the more sophisticated, air-conditioned pavilion.
Organizers hope the new location attracts more people.
They are also offering a room where visitors can get comfy in a big chair and read their favorite novel while sipping coffee, said Will Ray, president of the Cultural Council.
A smaller tent for children called Oz will provide storytelling and magic shows. "Our goal is 10,000 to 20,000 people," Ray said.
There is no official theme for Bookfest, but Ray said, "You'll see a continuing theme of books written about or in Florida."
The headliner for BookFest '98 is Dominick Dunne, the best-selling writer whose latest book, Another City, Not My Own, was published in November.
BookFest officials named him Palm Beach author laureate and lead guest author based on the popularity of his books, his career in film and television and his association with Palm Beach.
He will speak at 10 a.m. on Saturday and 11 a.m. on Sunday.
Some of the other featured writers:
* Kathryn Bel Monte, who wrote African-American Heroes and Heroines, will speak at 11 a.m. on Saturday.
* Stephen Birmingham, who wrote The Wrong Kind of Money, will speak at 7:30 p.m. on Saturday and 11 a.m. on Sunday.
* 3 p.m. _ Mystery writer Carol Higgins Clark will speak at 3 p.m. on Sunday.
Mystery writer Donald Westlake has been attending literary festivals for almost 20 years. He comes for two reasons: to meet the readers and to meet other authors he has admired.
"Being a freelance writer working at home, it means you never get to meet you co-workers," he said.
Bookfest runs from noon to 8 p.m. Friday, 10 a.m. to 10 p.m. Saturday and 10 a.m. to 5 p.m. Sunday. Tickets are $5 for adults and $3 for children 12 and younger. Three-day passes are $10 for adults and $5 for children. For more information, call Robin Spillias at 561-471-2901 or toll free from south county at 930-2787 or visit the Bookfest Web site at (www.pbccc.org.)


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How to Get to Roatan
Planning to visit our beautiful island? If you're going to be flying in, our Roatan airport, Juan Manuel Galvez International Airport, RTB as the letters to represent it, has domestic as well as international flights.
For Canadians (and Americans who might find it worthwhile to cross the border for their flight), in the winter months, there are direct flights offered from both Toronto and Montreal. Canadian airlines that offer flights to Roatan are the charter companies Air Transat and Sunwing (though they promote packages, it's also possible to just book the flight, Mon and Thurs flights only), and now West Jet is also starting to fly to Roatan for high season, on Sundays. You can sometimes find excellent deals. I find the Skyscanner site is a good way to search for best pricing- instead of plugging in specific dates, you can opt to search for through by month, which will come up with all the different pricing options. Travelers coming from the west side of Canada might find it more advantageous to fly with West Jet through Belize, catching a TropicAir flight from Belize straight to Roatan, or a red-eye with Avianca through El Salvador, then getting the direct flight from El Salvador to Roatan.
From the States, there are direct flights with Delta (from Atlanta) on Saturdays, United (through Houston) several days of the week, and American from both Dallas and Miami on select days.
If you're not finding good enough pricing, another option is to look for flights to San Pedro Sula. There are more airlines that fly to San Pedro Sula, including Spirit from Houston and from Fort Lauderdale, so sometimes it can work out cheaper going that route. You'll need to factor in the cost/time to get to Roatan from San Pedro Sula, to make sure it's worth your while. The domestic flights between these 2 spots takes about 40 minutes and costs around $115 each way (sometimes you can find one a bit cheaper), with CM usually being the most reliable, and SOSA and Avianca also offering options. If you'd rather take a bus to La Ceiba (3 1/2 hrs, $5-20 depending on the bus) and then catch the ferry from there (1 1/2 hrs, around $33), it does save you some money, but ties up quite a bit of time. Lanhsa also offers flights from La Ceiba that are almost the same cost as the ferry, if you wanted to bus and then fly from there.
Traveling through Central America and want to come over? Direct flights from Belize are daily with Tropic Air, and from El Salvador with Avianca. If you'd rather bus/boat your way, for ferries, there are 2 ferry companies. The Utila Dream has a daily ferry from the neighbouring island of Utila to Roatan (as well as ferries between Utila and La Ceiba), and the Galaxy Wave has twice-daily trips from la Ceiba to Roatan. There are no regularly scheduled boats or ferries between Belize and Roatan, though sometimes private boats make the trip.
Need some help? Send us a message and tell us what you're looking for, and we'll see if we can help you with finding the best route.