Estimation of the validity domain of hyperreduction approximations in generalized standard elastoviscoplasticity
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Abstract
Background
We propose an a posteriori estimator of the error of hyperreduced predictions for elastoviscoplastic problems. For a given fixed mesh, this error estimator aims to forecast the validity domain in the parameter space, of hyperreduction approximations. This error estimator evaluates if the simulation outputs generated by the hyperreduced model represent a convenient approximation of the outputs that the finite element simulation would have predicted. We do not account for the approximation error related to the finite element approximation upon which the hyperreduction approximation is introduced.
Methods
We restrict our attention to generalized standard materials. Upon use of incremental variational principles, we propose an error in constitutive relation. This error is split into three terms including a tailored norm of the hyperreduction approximation error. This error norm is defined by using the convexity of an incremental potential introduced to state the constitutive equations. The second term of the a posteriori error is related to the stress recovery technique that generates stresses fulfilling the finite element equilibrium equations. The last term is a coupling term between the hyperreduction approximation error at each time step and the errors committed before this time step. Unfortunately, this last term prevents error certification. In this paper, we restrict our attention to outputs extracted by a Lipschitz function of the displacements.
Results
In the proposed numerical examples, we show very good preliminary results in predicting the validity domain of hyperreduction approximations. The average computational time of the predictions obtained by hyper reduction, is accelerated by a factor of 6 compared to that of finite element simulations. This speedup incorporates the computational time devoted to the error estimation.
Conclusions
The numerical implementation of the proposed error estimator is straightforward. It does not require the computation of the incremental potential. In the numerical results, the estimated validity domain of hyperreduced approximations is inside the reference validity domain. This paper is a first attempt for a posteriori error estimation of hyperreduction approximations.
Keywords
Reduced integration domain Hyperreduction POD Incremental variational principleAbbreviations
 PDE
Partial differential equation
 RID
Reduced integration domain
 HR
Hyperreduced
 FE
Finite element
 HRAE
Hyperreduction approximation error
 CRE
Constitutive relation error
Background
ReducedOrder models aim at reducing the computational time required to obtain solutions of Partial Differential Equations (PDE) which are physicallybased and parameter dependent. They reduce the computational complexity of optimization procedures or parametric analyses [13].
ReducedOrder Models are very useful to model complex mechanical phenomena, when parametric studies are mandatory to setup convenient nonlinear constitutive equations or boundary conditions. In such situations, many open questions about physical assumptions arise and the scientist’s intuition alone cannot guarantee that convenient simplified numerical approximations are chosen. Here, we consider hyperreduced (HR) simulations, involving both a reducedbasis approximation and reducedcoordinate estimation by using a mesh restricted to Reduced Integration Domain (RID) [4,5]. The introduction of the RID is crucial for elastoviscoplastic models, because in many practical cases, no speedup is achieved if the mesh is not restricted to the RID. In such a framework, error estimation provides a valuable algorithmic approach to check if hyperreduced simulations have been performed in the validity domain of the hyperreduced model, and if they allow a physical understanding of the simulation outputs.
HR simulations may not be performed in faster time than predictions using metamodeling, response surface methods, or virtual charts [6]. But, unlike these methods, ReducedOrder Models involve equations related to physical principles (i.e. the balance of momentum), physical constitutive equations and data, validated by experiments. Such an approach enables certification of outputs of interest [7], or error estimation of predictions obtained by using ReducedOrder Models [711]. Moreover, error estimation is mandatory in many model reduction of parametric Partial Differential Equations, when one needs to evaluate, in the parameter space, a trust region or a validity domain related to ReducedBasis accuracy [1214]. Error estimation is also mandatory to perform a priori model reduction by using greedy algorithms [15,16] or the A Priori Hyperreduction method [4].
Unfortunately, in nonlinear mechanics of materials, parametric PDEs are rarely affine in parameter, and it is quite difficult to reduce equations and data to an approximate affine form as proposed in [15] by using the Empirical Interpolation Method [17,18]. The novelty of this paper is the use of an incremental variational principle [1921] to develop an a posteriori estimator of errors for HR predictions in case of generalized standard elastoviscoplastic models. As usual, the finite element (FE) solution fulfills incremental equations in time, but no differential equations related to the continuous time variable. The incremental variational principle enables to preserve formulations discrete in time when comparing the HR solution to the FE solution. This approach enables to state a relationship between a tailored norm of the hyperreduction approximation error and the a posteriori estimator of this error. Here, we term hyperreduction approximation error (HRAE) the difference between the FE solution of the PDE and its estimation by the hyperreduced model. The reference solution is the FE solution, not the exact solution of the PDE. In the sequel, the proposed tailored norm of the displacement is denoted by ∥u∥_{ u }, where u is the predicted displacement over the domain Ω. The error estimator is denoted by η. We follow the theory of constitutive relation error (CRE) proposed by P. Ladevèze [22]. This kind of a posteriori error has been developed for a large set of constitutive equations [9,2326]. Although constitutive relation error (CRE) can handle all approximation errors due to time discretization, FE approximation, and eventually a separated representation of the displacements [27], we restrict our attention to approximation errors due to the additional assumptions incorporated in the numerical model by the hyperreduced formulation. Here, the CRE η requires the construction of stresses fulfilling FE equilibrium equations.
where c _{ η } is a substitute for the Lipschitz constant c _{ o }, η is a substitute for the tailored norm of the HRAE, \((\textbf {u}^{n}_{\textit {HR}})_{n=1}^{N_{t}}\) is the sequence of hyperreduced predictions over the full time interval and \((\widehat {{\boldsymbol \sigma }}^{n})_{n=1}^{N_{t}}\) is a sequence of stresses fulfilling the FE equilibrium equation at each discrete time t ^{ n }, where \(\widehat {{\boldsymbol \sigma }}^{n}\) is an affine function of the stress estimated from \(\textbf {u}^{n}_{\textit {HR}}\). In the sequel, both sequences \((\textbf {u}^{n}_{\textit {HR}})_{n=1}^{N_{t}}\) and \((\widehat {{\boldsymbol \sigma }}^{n})_{n=1}^{N_{t}}\) are denoted by u _{ HR } and \(\widehat {{\boldsymbol \sigma }}\) respectively.
In this paper μ _{1} is arbitrary chosen. Nevertheless, the knowledge of an approximate validity domain aims to validate this choice or to adapt it. In the sequel, for clarity, we do not mention the vector of parameters μ in the notations.
Methods
This section is organized as follows. The generalized standard constitutive equations are presented first. They are based on an incremental variational principle proposed in [33]. Then, we introduce the hyperreduced incremental problem to be solved. The following section is devoted to the construction of the FEequilibrated stress within the framework of the hyperreduced modeling. The next sections introduce the error estimator, the tailored norm related to the CRE and the partition of the CRE. We finish by remarks on the numerical implementation of the proposed approach.
Incremental potential for the constitutive equations
The constitutive laws are described by using an incremental potential in the framework of the irreversible thermodynamic processes. A priori error estimators and incremental variational formulations were introduced in [33] for mechanical problems of bodies undergoing large dynamic deformations. Extensions of this approach were proposed in [20,21] for effective response predictions of heterogeneous materials. The strain history is taken into account by using internal variables denoted by α. These variables are the lump sum of the history of material changes. This approach has its roots in the works by Biot [34], Ziegler [35], Germain [36] or Halpen and Nguyen [37] and has proven its ability to cover a broad spectrum of models in viscoelasticity, viscoplasticity, plasticity and also continuum damage mechanics. The FE solution is approximated by an hyperreduced predictions denoted by \(\textbf {u}_{\textit {HR}}^{n}\), \(\boldsymbol {\alpha }_{\textit {HR}}^{n}\), \(\boldsymbol {\sigma }_{\textit {HR}}^{n}\) respectively for the displacements, the internal variables and the Cauchy stress, at discrete time t ^{ n }. For the sake of simplicity, we denote by \(\boldsymbol {\varepsilon }_{\textit {HR}}^{n}\) the strain tensor \(\boldsymbol {\varepsilon }(\textbf {u}_{\textit {HR}}^{n})\), in the framework of the infinitesimal strain theory.
where w(ε,α) is the free energy of the material, and \(\varphi (\dot {\boldsymbol {\alpha }})\) is its dissipation potential [36]. The two potentials w and φ are convex functions of their arguments (ε,α) and \(\dot {\boldsymbol {\alpha }}\) respectively, according to the theory of generalized standard materials [36,37]. Examples of constitutive laws can be found in [38]. A detailed example is given in the last section of this paper.
The convexity of w _{ Δ } is proved in [21] under the assumption that w and φ are convex functions. In the sequel, the explicit knowledge of the condensed incremental potential is not required for the mathematical formulation of the error estimator.
Hyperreduced setting
where : is the contracted product for secondorder tensors. We must emphasize the fact that Equation (24) gives access to the displacement in all the domain Ω although the equilibrium is set only on Ω _{ Z }. Hence, when the hyperreduced solution is known, the stress \(\boldsymbol {\sigma }^{n}_{\textit {HR}}\) can be estimated on the full domain Ω by using Equation (10).

the shape functions \((\boldsymbol {\xi }_{i})_{i=1}^{N_{\xi }}\) for \((\boldsymbol {\psi }_{k})_{k=1}^{N_{\psi }}\),

the test functions \((\boldsymbol {\xi }_{i})_{i=1}^{N_{\xi }}\) for \(({\boldsymbol {\psi }_{k}^{Z}})_{k=1}^{N_{\psi }}\),

the full domain Ω for Ω _{ Z },

the variables \(((q_{i})_{i=1}^{N_{\xi }},\textbf {u}^{n}_{\textit {FE}}, \boldsymbol {\sigma }^{n}_{\textit {FE}}, \boldsymbol {\alpha }^{n}_{\textit {FE}})\) for \(\left ((\gamma _{k})_{k=1}^{N_{\psi }},\textbf {u}^{n}_{\textit {HR}}, \boldsymbol {\sigma }^{n}_{\textit {HR}}, \boldsymbol {\alpha }^{n}_{\textit {HR}}\right)\).
where \(\boldsymbol {\varepsilon }_{\textit {FE}}^{n} = \boldsymbol {\varepsilon }(\textbf {u}_{\textit {FE}}^{n})\) and δ σ ^{ n } account for the variation of the internal variables due to the difference between \(\left (\boldsymbol {\varepsilon }_{\textit {FE}}^{i}\right)_{i=1}^{n1}\) and \(\left (\boldsymbol {\varepsilon }_{\textit {HR}}^{i}\right)_{i=1}^{n1}\) at time instants before t ^{ n }. The convexity of w and ϕ insures that the FE solution is unique, if no rigid mode is available.
Dual reducedsubspace
where \(E_{o} \in \mathbb {R}^{+}\) is an abritary constant.
Similarly to the approach proposed in [10] for elasticity, we introduce a dual reducedsubspace denoted by \(\mathcal {S}_{\textit {ROM}} \subset \mathcal {S}_{h}\). \(\mathcal {S}_{\textit {ROM}}\) is generated by the usual POD applied to \(\left (\boldsymbol {\sigma }^{n}_{\textit {FE}}(\cdot ;{\boldsymbol \mu }^{1})  \boldsymbol {\sigma }^{n}_{N}\right)_{n=1}^{N_{t}}\). Its dimension is denoted by \(N^{\sigma }_{\psi }\), and it is such that \(N^{\sigma }_{\psi } \le N_{t}\).
The above minimization problem is a global L ^{2} projection of the stress \(\boldsymbol {\sigma }^{n}_{\textit {HR}}\) onto the reduced basis that span \(\mathcal {S}_{\textit {ROM}}\). The dual basis being a POD basis, it is orthonormal with respect to the L ^{2} scalar product. So the computational complexity of the stresses projection scales linearly with the number of Gauss points of the mesh and \(N^{\sigma }_{\psi }\). In practice, this computational complexity is negligible compared to the complexity of the evaluation of \(\boldsymbol {\sigma }^{n}_{\textit {HR}}\), by using the constitutive equations.
The constitutive relation error and its partition
The reference solution being incremental in time, no continuous formulation in time is considered for the error estimator. The error estimation proposed in [33] for incremental variational formulation, is related to an asymptotic convergence assumption in order to get an upper bound of the approximation error related to the FE discretization. This upper bound depends on a constant related of the weak form of the partial differential equations. Here, we propose to apply the Constitutive Relation Error method proposed in [22] to estimate the constant c _{ η } without assuming an asymptotic convergence of the HRAE.
The first property (36) comes from the Legendre transformation (14). The property (37) comes from Equation (19). The proof of Property (38) is the following. If \(\eta (\textbf {u}^{n}_{\textit {HR}}, \: \widehat {\boldsymbol {\sigma }}^{n}) = 0\forall \: n \in \{1,\ldots,N_{t}\}\) then \((\textbf {u}^{n}_{\textit {HR}}, \widehat {\boldsymbol {\sigma }}^{n}, \boldsymbol {\alpha }^{n}_{\textit {HR}})\) fulfills the constitutive equations, the Dirichlet conditions and the FE equilibrium equation. As \((\textbf {u}^{n}_{\textit {HR}}, \widehat {\boldsymbol {\sigma }}^{n}, \boldsymbol {\alpha }^{n}_{\textit {HR}})\) fulfills all the equations of the FE problem, it is a solution of the FE problem and e ^{ n }=0.
where ε(·) is the symmetric part of the gradient and G a symmetric positivedefinite fourthorder tensor. If the identity tensor is substituted for G, we obtain a usual H ^{1}(Ω) norm. We assume that there is no rigid mode, neither in the FE solution nor in the reduced basis.
Property.
The last term of the sum is a coupling term between the error e ^{ n } and the HRAE committed before the discrete time t ^{ n }. Since we can’t certify that this term is positive, η is not an upper bound of the HRAE.
This ends the proof.
Property.
In the general case, the closer \(\widehat {\boldsymbol {\sigma }}^{n}\) to \(\boldsymbol {\sigma }_{\textit {FE}}^{n}  \delta \boldsymbol {\sigma }^{n}\) the better the error estimation. The proposed error estimator incorporates errors related to the projection of the stress onto the dual reduced basis.
Numerical implementation of the CRE
Here, \(\widehat {\boldsymbol {\varepsilon }}^{n}(\lambda)\) is the strain tensor given by the constitutive equation upon the stress τ(λ) at time t ^{ n } and the internal variables \(\boldsymbol {\alpha }^{n1}_{\textit {HR}}\) at time t ^{ n−1}. Hence, the numerical estimation of \(\widehat {\boldsymbol {\varepsilon }}^{n}(\lambda)\) can be performed by the usual implementation of the constitutive equations of generalized standard materials [4042].
In the following numerical simulations, we have estimated the integral on λ by the value of first derivative of g for λ=1.
Numerical estimation of the constant c _{ η }
Hence the constraint (9) is fulfilled. Let us denote by \(N^{c}_{\psi }\) the number of displacement modes for which the maximum in Equation (66) is reached. In our opinion, if we expect that the error estimator behaves like an upper bound, we should not take \(N^{c}_{\psi }\) modes to generate the final HR model. In the following example, we are setting \(N_{\psi } = N^{c}_{\psi }+1\).
Results and discussion
The estimation of the validity domain is shown on the right of Figure 6. It is quite restrictive. This means that most of the points in \(\widetilde {\mathcal {D}}_{V}\) are in the reference validity domain \(\mathcal {D}_{V}\). Here, for small values of the parameter h, the prediction of the validity domain is too conservative. Such a situation can appear when the average influence of a parameter on the solution is larger than its influence on the output, because the error estimator is linked to the HRAE in an average sense.
We can notice that the higher h and C, the smaller the plastic strains in the beam. Since the hyperreduced model is accurate for low plastic strains, the validity domain covers the high values of h and C.
Conclusions
We propose an a posteriori estimator of hyperreduction errors that aims to evaluate if the simulation outputs predicted by hyperreduced models are convenient approximations of the outputs that the finite element simulation would have predicted. By choosing a tailored norm of the approximation error on displacement, we show how the proposed error estimator is related to the HR approximation error. This error estimator receives the contribution of a coupling term between the error at time step n and the error committed before. We show that this term prevents rigorous error certification. But in practice, numerical examples show a restrictive estimation of the validity domain of the HR model. By restrictive, we mean that this domain is inside the reference validity domain computed by using both the FE solution and the HR solution. The speedup achieved by the HR model including the estimation of the constitutive relation error is 6 on the proposed numerical elastoviscoplastic examples. The estimation of the validity domain requires the computation of a constant similar to a Lipschitz constant. This constant is estimated without any additional simulation of FE solution, but the FE solution at the sampling point μ=μ ^{1}. As shown in is this paper, the numerical implementation of the proposed error estimator is very simple. It does not require the computation of the incremental potential or of its dual. This paper is a first attempt for a posteriori error estimation of hyperreduction approximations. More numerical experiments are in preparation for the assessment of the proposed numerical approach.
Notes
Acknowledgements
We would like to thank Pierre Ladevèze, Gérard Coffignal and Yvon Maday for the discussions we had about this work.
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