General treatment of essential boundary conditions in reduced order models for nonlinear problems
 Alejandro Cosimo^{1},
 Alberto Cardona^{1} and
 Sergio Idelsohn^{1, 2}Email author
https://doi.org/10.1186/s4032301600588
© The Author(s) 2016
Received: 27 October 2015
Accepted: 30 January 2016
Published: 8 March 2016
Abstract
Inhomogeneous essential boundary conditions must be carefully treated in the formulation of Reduced Order Models (ROMs) for nonlinear problems. In order to investigate this issue, two methods are analysed: one in which the boundary conditions are imposed in an strong way, and a second one in which a weak imposition of boundary conditions is made. The ideas presented in this work apply to the big realm of a posteriori ROMs. Nevertheless, an a posteriori hyperreduction method is specifically considered in order to deal with the cost associated to the nonlinearity of the problems. Applications to nonlinear transient heat conduction problems with temperature dependent thermophysical properties and time dependent essential boundary conditions are studied. However, the strategies introduced in this work are of general application.
Keywords
Background
Currently, many engineering problems of practical importance are suffering from the socalled “curse of dimensionality” [1]. In this context, the need of optimising nonlinear multiphysics problems makes necessary to develop numerical techniques which can efficiently deal with the high computational cost characterising such applications. A widespread strategy is to consider the formulation of Reduced Order Models, which can be implemented by adopting either the Proper Orthogonal Decomposition (POD) method [2, 3], or the proper generalised decomposition (PGD) technique [4, 5]. The discussion in this paper only considers PODbased ROMs, from now on referred to as ROMs. The ideas presented here apply to the big realm of a posteriori ROMs, despite the fact that an a posteriori hyperreduction method, referred to as Hyper Reduced Order Model (HROM), is specifically considered in order to deal with the cost associated to the nonlinearity of the problems.
In what follows, let \(\mathcal {S}^h \subset \mathcal {S}\) and \(\mathcal {V}^h \subset \mathcal {V}\) be the trial and test finite dimension subspaces of the functional spaces \(\mathcal {S}\) and \(\mathcal {V}\) used in the definition of a variational problem. Generally, in the formulation of ROMs, an approximate solution \(\widehat{T}^h\) to \(T^h \in \mathcal {S}^h\) is sought in a subspace of \(\mathcal {S}^h\) by defining a new basis \({\varvec{X}} \in \mathbb {R}^{N \times k}\), where N is the number of degrees of freedom (DOFs) of the High Fidelity (HF) model and k is the dimension of the basis spanning the subspace of \(\mathcal {S}^h\). If a BubnovGalerkin projection is used, approximate versions \(\widehat{w}^h \in {span}\{ {\varvec{X}} \}\) of the test functions \(w^h\) are built, and functions \(T^h \in \mathcal {S}^h\) are approximated by affine translations of the test functions \(\widehat{w}^h\). In PODbased ROMs, the new basis \({\varvec{X}}\) is built by computing the singular value decomposition (SVD) [6] of a set of snapshots that are given by time instances of the spatial distribution of the solution of a training problem [7]. It is wellknown that the vectors comprising this basis inherit the behaviour of the snapshots [8], hindering the possibility of reproducing nonadmissible test functions. That is why careful attention must be paid on how the snapshots for building \({\varvec{X}}\) are collected. This issue is studied in detail in this work.
The concept of consistent snapshots collection procedures for nonlinear problems was first introduced by Carlberg et al. [9, 10]. As they pointed out in [10] “most nonlinear model reduction techniques reported in the literature employ a POD basis computed using as snapshots \(\{{\varvec{T}}_n  n=1, \cdots , n_t\}\) ^{1}, which do not lead to a consistent projection”. In the last expression \(n_t\) is given by the number of time steps comprising the training problem and \({\varvec{T}}_n\) are the parameters such that \(T^h_n={\varvec{N}}^T{\varvec{T}}_n\) with \({\varvec{N}}\) given by the shape functions used for interpolation. The lack of consistency of these formulations is produced by the fact that when computing the POD basis with time instances of \(T^h\), that is by \(\{{\varvec{T}}_n  n=1, \ldots , n_t\}\), if nonzero essential boundary conditions are present, \({span}\{ {\varvec{X}} \} \not \subset \mathcal {V}^h \) because some elements \({\varvec{v}} \in {span}\{ {\varvec{X}} \}\) are not identically zero at the portion of the boundary with nonhomogeneous essential conditions.

Snapshots of the form \(\{{\varvec{T}}_n  {\varvec{T}}_{n1}  n=1, \ldots , n_t\}\). The problem with this strategy is that the set of snapshots is characterised by a high frequency content, giving a less compressible SVD spectrum [11] than when using a collection procedure based on the snapshots of the solution. Another problem of this strategy is the handling of time dependent essential boundary conditions. In this case, it cannot be guaranteed that the snapshots will be identically zero at the boundary with essential boundary conditions.

Snapshots of the form \(\{{\varvec{T}}_n  {\varvec{T}}_0  n=1, \ldots , n_t \}\), where \({\varvec{T}}_0\) is the initial condition. With this strategy it cannot be guaranteed that functions in \({span}\{ {\varvec{X}} \}\) will be test functions, for instance, when essential boundary conditions are different from \(T^h_0\). Amsallem et al. [12] have observed that this strategy leads to more accurate ROMs than the previous strategy. As it is discussed in that work, using a different initial condition \({\varvec{T}}^*_0\) in the online stage requires in principle recomputing the snapshots for reconstructing the POD modes for projection. Several fast alternatives to solve this problem are proposed in [12].
In the work of González et al. [1], the problem of imposing nonhomogeneous essential boundary conditions in the context of a priori model order reduction methodologies (PGD) is tackled. They imposed the Dirichlet conditions by constructing a global function that verifies the essential boundary conditions, using the technique of transfinite interpolation [14]. A good example of interpolation functions is given by the inverse distance function, and as exposed by Rvachev et al. [14], different interpolation functions can be built based on the theory of Rfunctions. Although the methodology presented by González et al. is really appealing and show particular advantages in the PGD context, its use requires large symbolic algebra computations, leading to very complex algebraic expressions even in the case of quite simple academic problems that could hinder its application to practical problems. In the current study, we are looking to develop physicallybased techniques that can be easily applied to domains coming from threedimensional industrial problems.
In this work we analyse the treatment of time dependent inhomogeneous essential boundary conditions from a general point of view, taking into consideration the costs associated to nonlinear problems and to the strong imposition of the essential boundary conditions. Alternatives based on the weak imposition of the boundary conditions are evaluated, combined with a reduction of the number of degrees of freedom at the boundary. The presented ideas are applied to nonlinear transient heat conduction problems with temperature dependent thermophysical properties; however, the introduced strategies are of widely general application.
Methods
This section describes first the problem statement and two variational formulations, one weakly imposing Dirichlet boundary conditions and the other strongly imposing those conditions. Then, an HROM that considers strong enforcement of boundary conditions is presented. Finally, the formulation of two alternative HROMs that weakly impose Dirichlet conditions is introduced.
Problem statement, variational formulation and finite element discretisation
HROM formulation by strongly enforcing boundary conditions
The HROM associated to the formulation given by Eq. (3) is here introduced. Each nonlinear contribution to \(\varvec{\varPi }_n\) is hyperreduced separately as done by Cosimo et al. [11]. Therefore, each of these terms has associated a particular POD basis \(\varvec{\varPhi }_i\) for its gappy data reconstruction [15–17]. In what follows, suffices \(i \in \{ c,k,f,q\}\) are used to identify each term. We emphasise that the sampling is performed independently for each term, but the number of sampling points \(n_s\) and the number of gappy modes \(n_g\) are always the same for all of them. In what follows, \(\widehat{\cdot }\) denotes the vector of \(n_s\) components sampled from the associated complete term. To compute the POD modes \(\varvec{\varPhi }_i\), snapshots are taken for each individual contribution at each time step, after convergence of the NewtonRaphson scheme.
In order to get admissible test functions \(\widehat{w}^h\), the restriction \(\widehat{w}^h_{\Gamma _d}=0\) must be satisfied. That is why, for the design of a consistent snapshot collection strategy, the snapshots must be of the form \({\varvec{T}}  {\varvec{T}}_d\). Then, the problem resides in the correct description of \(T_d^h\). A possible solution is to describe it as in standard FEM, i.e., \(T_d^h = {\varvec{N}}^{B,T}{\varvec{T}}^B_n\), but this could lead to a snapshots set with a very high frequency content, decreasing the compressibility of the signal [11].
Then, the approximation \(\widehat{T}^h\) is given by \(\widehat{T}^h = {\varvec{N}}^T {\varvec{\varPhi }}_B {\varvec{T}}^B_n + {\varvec{N}}^T {\varvec{X}} {\varvec{a}}_n \simeq T^h\). Note that the static modes \({\varvec{\varPhi }}_B\) have the property to be interpolatory at the boundary \(\Gamma _d\), thus \({\varvec{T}}^B_n\) has the physical interpretation to be the value of the field at the nodes lying on \(\Gamma _d\).

The snapshots given by \(S_p\) tend to preserve the compressibility posed by the field \(T^h\).

General essential boundary conditions can be represented by \({\varvec{\varPhi }}_B\), while keeping simple the process of imposing essential boundary conditions because of the interpolatory property of \({\varvec{\varPhi }}_B\) at \(\Gamma _d\).

Using different initial conditions in the online stage does not require recomputing the snapshots for \({\varvec{X}}\) or considering another alternative.
Remark 1
In the examples section, all the static modes associated to the portion of the boundary with essential boundary conditions are retained, and no other approximation is applied to the boundary DOFs.
HROM formulation by weakly enforcing boundary conditions

Test functions for the temperature field are not required to meet the constraint \(T_{\Gamma _d} = 0\).

As previously introduced, the cost of computing the product \({\varvec{\varPhi }}_B {\varvec{T}}^B_n\) can be a penalising factor when a large number of static modes is required. By using Lagrange multipliers, this problem can be avoided.
Remark 2
The techniques presented here apply to higher order problems as well. For instance, let us consider a fourth order onedimensional problem in which Hermite polynomials are used in the FEM discretisation. In the case of imposing the essential boundary conditions strongly, the procedure for computing static modes is applied exactly in the same way as described before, with static modes obtained by imposing unit displacements and unit rotations at the boundary. In the case of imposing weakly the essential boundary conditions, the only difference with the thermal case is that we will have independent Lagrange multipliers fields for each degree of freedom. We note finally that an extension to fourth order problems of the techniques presented in [1] in the context of the a priori reduced order method PGD, was proposed by Quesada et al. [24].
Results and discussion
We will show the application of the proposed snapshot collection strategies to two nonlinear transient heat conduction problems with time dependent essential boundary conditions. To assess the performance and robustness of the proposed methods, we study the relative error introduced by the HROM. The relative error \(\epsilon \) characterising the HROM as a function of time is measured as \(\frac{\Vert T_R  T_H\Vert }{\max \limits _{t}\Vert T_H\Vert }\), where \(T_R\) is the solution obtained with the HROM, \(T_H\) is the High Fidelity solution for same problem and \(\Vert \cdot \Vert \) denotes the \(L_2\) norm. Trilinear hexahedral elements are used in the examples to interpolate the temperature field. The Lagrange multipliers field is interpolated with bilinear quadrilateral elements. In what follows, \(n_p\) is used to denote the number of POD modes for \({\varvec{T}}_n\), and \(n_{\lambda }\) is used to denote the number of POD modes for \(\varvec{\lambda }_n\).
Example 1
The error obtained using static modes to represent the essential boundary condition can be observed in Fig. 2, where different numbers of projection modes were considered. As it can be seen, good results are obtained. Additionally, the results are comparable to the ones obtained by Gunzburger et al.
When comparing the three different alternatives, it is observed that the lowest error option is when using static modes. Nevertheless, the cost is higher than in the strategies that impose the Dirichlet boundary conditions weakly. Concerning the two latter alternatives, it is observed that the strategy of reducing \({\varvec{T}}_n\) and \(\varvec{\lambda }_n\) as a unit leads to the lowest errors, for the same number of reduced DOFs. For example, when using that alternative with \(n_p=12\) the error for the temperature field is \(O(10^{4})\), but when reducing \({\varvec{T}}_n\) and \(\varvec{\lambda }_n\) separately with \(n_p=8\) and \(n_\lambda =4\), the error is \(O(10^{3})\). The approximation error to \(\varvec{\lambda }_n\) is always lower when reducing \({\varvec{T}}_n\) and \(\varvec{\lambda }_n\) as a unit. However, as seen in the numerical experiments, the number of POD modes needed to describe \({\varvec{T}}_n\) and \(\varvec{\lambda }_n\) as a unit must be fairly large in order to provide enough freedom to the temperature field to satisfy the restrictions imposed by the Lagrange multipliers.
Example 2
Conclusion
Several alternatives for building HyperReduced Order Models to solve nonlinear thermal problems with time dependent inhomogeneous essential boundary conditions were analysed and compared.
One strategy considers the use of static modes for strongly imposing the boundary conditions. This approach is similar to the method presented by Gunzburger et al. [13] who proposed to use particular solutions instead of static modes. A good behaviour was obtained by using static modes and the results were comparable to the ones obtained by Gunzburger et al. Even though this method proved to be a robust technique for describing essential boundary conditions, the associated computational cost is high for models that require a large number of static modes.
In order to work out the disadvantages of the static modes approach, two other alternatives that are based on a weak imposition of the essential boundary conditions were studied. One alternative consists in reducing the primal and the secondary fields as a unit, while the other consists in reducing them separately. It was observed that, for the same number of reduced DOFs, the former approach led to the lowest errors for the primal (temperature) field. The performed numerical experiments also made evident that the number of POD modes used for describing the primal and the secondary fields as a unit, must be large enough in order to provide enough freedom to the primal (temperature) field to satisfy the restrictions imposed by the Lagrange multipliers.
In a future work, the case with time dependent variation of the support of the essential boundary conditions will be studied.
For the sake of conciseness, in this work we do not consider the objective function \(T^h\) to depend on a set of analysis parameters \({\varvec{\mu }}\). If this were the case, the snapshots collection strategies introduced herein apply directly just by applying them to each of the training parameters \({\varvec{\mu }}_i\).
Declarations
Acknowledgments
All authors contributed to the development of the theory. The computer code for the numerical simulations was developed by AC. All authors read and approved the final manuscript.
Acknowledgements
This work received financial support from CONICET Consejo Nacional de Investigaciones Científicas y Técnicas (PIP 1105), Agencia Nacional de Promoción Científica y Tecnológica (PICT 20132894), and Universidad Nacional del Litoral (CAI+D2011) from Argentina, and from the European Research Council under the Advanced Grant: ERC2009AdG “Real Time Computational Mechanics Techniques for MultiFluid Problems”.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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