- Research Article
- Open Access

# An error estimator for real-time simulators based on model order reduction

- Icíar Alfaro†
^{1}, - David González†
^{1}, - Sergio Zlotnik†
^{2}, - Pedro Díez†
^{2}, - Elías Cueto†
^{1}Email authorView ORCID ID profile and - Francisco Chinesta†
^{3}

**2**:30

https://doi.org/10.1186/s40323-015-0050-8

© Alfaro et al. 2015

**Received: **4 June 2015

**Accepted: **16 October 2015

**Published: **26 November 2015

## Abstract

Model order reduction is one of the most appealing choices for real-time simulation of non-linear solids. In this work a method is presented in which real time performance is achieved by means of the off-line solution of a (high dimensional) parametric problem that provides a sort of response surface or *computational vademecum*. This solution is then evaluated in real-time at feedback rates compatible with haptic devices, for instance (i.e., more than 1 kHz). This high dimensional problem can be solved without the limitations imposed by the curse of dimensionality by employing proper generalized decomposition (PGD) methods. Essentially, PGD assumes a separated representation for the essential field of the problem. Here, an error estimator is proposed for this type of solutions that takes into account the non-linear character of the studied problems. This error estimator allows to compute the necessary number of modes employed to obtain an approximation to the solution within a prescribed error tolerance in a given quantity of interest.

## Keywords

- Error estimation
- Real time
- Model order reduction
- Proper generalized decomposition

## Background

Real-time simulation of non-linear solids is always a delicate task due to the heavy computational cost associated with the linearization of the equations. Applications are ubiquitous, ranging from industrial uses [1] to surgery planning and training [2, 3] or movies [4].

Probably, the field in which more effort has been paid to the development of real-time simulation techniques is that of computational surgery [5–10]. This is because surgery training systems are equipped with haptic peripherals, those that provide the user with realistic touch sensations (force feedback). Just like some 25 pictures per second are necessary for a realistic perception of movement in films, haptic feedback needs for some 500 Hz to 1 kHz in order to achieve the necessary realism. The difficulty of the task is thus readily understood: to perform 500–1000 simulations of highly non-linear solids (soft tissues are frequently assumed to be hyperelastic), possibly suffering contact, cutting, etc.

Equation (1) thus represents a sort of *response surface* in the sense that it provides with the solution for any physical coordinate, time instant and value of the *p* parameters. Instead of *response surface*, and to highlight the fact that no set of *a priori* experiments will be necessary to obtain such a response in our method, we prefer to call Eq. (1) a *computational vademecum* [17], inspired by the work of ancient engineers (such as Bernoulli, for instance [18]), who compiled sets of known solutions to problems of interest.

The problem with an approach such as that introduced in Eq. (1) is that such an expression is inherently high dimensional. If we try to discretize the governing equations of the problem so as to obtain an approximation to Eq. (1), and do it by a mesh-based method such as finite differences, volumes or elements, we will soon realize that the complexity of the problem will make us run out of computer memory very soon. This is due to the well-known exponential growth of the number of degrees of freedom (nodes of the mesh) with the number of dimensions of the problem. In other words, the well-known *curse of dimensionality* [19].

*a priori*. Such a technique has been coined as proper generalized decomposition (PGD) [20–24] and its main characteristic is to assume that the essential field (1) can be approximated in a separated form, i.e.,

One crucial problem related to such an approximation, see Eq. (2), is the choice of the number of terms *n* employed in the approximation. Being the main objetive of a simulator to provide the user with a realistic force feedback, the aim of the work presented herein is to develop a suitable error estimator that allows us to fix the number of functional products *n* necessary for a given tolerance in the error of the transmitted force. The literature on error estimation for model order reduction is vast, see [25–31], to name but a few. In “Formulation of the problem in a PGD setting” we recall the basics of the PGD approach to the problem at hand. In “One possible explicit linearization of the formulation” we revisit one of the possible linearization of the problem and, finally, in “An error estimator based on the dual formulation” we develop the sought error estimator for the force feedback. The paper is completed with two different numerical examples in “Numerical examples” that show the performance of the method.

## Formulation of the problem in a PGD setting

As a model non-linear problem hyperelasticity has been chosen. This constitutes a sufficiently general theory, with important implications in the simulation of soft living tissues [32, 33], for instance, and therefore in surgical simulators as an ubiquitous example of the restrictions placed by real-time constraints.

*find the displacement*\(\varvec{u}\in \mathcal H^1\)

*such that for all*\(\varvec{u}^*\in \mathcal H^1_0\):

*z*-coordinate axis.

*m*represents the order of truncation and \(f^i_j, g^i_j\) represent the

*j*th component of vectorial functions in space and boundary position, respectively. Following Eq. (2), the high dimensional solution of the problem will be sought as

*j*-th component of the displacement vector, \(j=1,2,3\) and functions \(\varvec{X}^k\) and \(\varvec{Y}^k\) represent the separated functions used to approximate the unknown field.

### Computation of \(S(\varvec{s})\) assuming \(R(\varvec{x})\) is known

### Computation of \(R(\varvec{x})\) assuming \(S(\varvec{s})\) is known

## One possible explicit linearization of the formulation

However, a critical issue remains in this case (or, in general, when dealing with PGD approximations of non-linear problems), which is that of selecting the number of terms *n* composing the approximation, see Eq. (4). This must be done on the basis of predictions given by a suitable error estimator, which is the main objective of this work and will be detailed in the following section.

## An error estimator based on the dual formulation

*p*the weak form of the problem looks like

*n*of terms. Secondly, the sought functions \(\varvec{F}_i\), \(\varvec{G}_i\), ..., are actually expressed by projecting them onto a finite element mesh of size

*h*. In brief, the following diagram depicts the situation:

where we have denoted \(e_{\text {PGD}}=\Vert \varvec{u}_h^{n=\infty }-\varvec{u}_h^{n}\Vert = \Vert \varvec{u}-\varvec{u}_{h=0}^n\Vert \) and \(e_{\text {FEM}} = \Vert \varvec{u}_{h=0}^n-\varvec{u}_h^n\Vert =\Vert \varvec{u} -\varvec{u}_h\Vert \). Finally, the sought, total committed error would be \(e= \Vert \varvec{u} -\varvec{u}_h^n\Vert \).

It is noteworthy to mention that, if the FE mesh size, *h* is not chosen judiciously, the total error in the simulation, composed by the sum of the FEM error plus the PGD error, will never get below a prescribed tolerance despite the number of modes added to the PGD approximation. Therefore, care must be paid not only to the number of terms *n* in the PGD approximation, but to the mesh size, *h*.

Following [34] (although other approaches are equally feasible for PGD, see [35–37]), the error in the quantity of interest is obtained through an auxiliary problem, often referred to as dual or adjoint problem. In [34], the exact solution of the auxiliary problem is replaced by a more accurate solution, which in a PGD context can naturally be obtained by performing some extra enrichment increments (i.e., letting *n* grow sufficiently to a value *N*).

*N*. For instance, results taking \(N=n+5\), \(N=2n\) or, simply, \(N=n\) were analyzed. In general some extra terms, say 5, are enough to determine a good dual solution.

## Numerical examples

### Cantilever beam

We consider the example of a cantilevered Kirchhoff-Saint Venant beam whose geometry is shown in Fig. 2. Beam nodes are assumed fixed at one of the ends, while the rest of the degrees of freedom are assumed to be free. The mesh is composed by tetrahedral elements, with \(3\times 3\) nodes in the \(40\times 40\) mm\(^2\) cross-section and 21 nodes in the longitudinal direction, 400 mm long. Material parameters were Young’s modulus \(E=2\times 10^{11}\) Pa and Poisson’s coefficient \(\nu =0.3\). The applied force is assumed to be always vertical and its value taken as \(10^8\) N.

*n*employed in the computation of the primal variable is shown in Fig. 6.

###
*Remark 1*

It is important to note that, despite the fact that we have considered 24 possible positions for the load vector, the fact of finding up to 60 modes to express the solution is not an inconsistency. One could think that obtaining a singular value decomposition of the 24 displacement vectors corresponding to the distinct 24 possible load positions would give up to 24 possible modes to express the high-dimensional solution \(\varvec{u} (\varvec{x}, \varvec{s})\). However, in this case we have performed an explicit, incremental solution of the non-linear problem, by dividing it into \(p=8\) pseudo-time steps. Therefore, in none of the examples shown the limit number of 24 modes has been reached. The highest number of modes for a particular load increment was 11, thus very far from 24. This is consistent with our previous experience in the development of computational vademecums by PGD techniques.

###
*Remark 2*

In addition, modes for the different load steps are not mutually orthogonal. An additional compression of the modes with the so-called PGD-projection, see [38], provides with a very restricted number of modes. In this case, the modes could be compressed so as to consider less than 12 modes for the whole loading process without further increase in the error in the approximation.

### Palpation of the liver

In this example we apply the dust developed error estimator to the simulation of liver palpation. The liver model, already shown in Fig. 1, is essentially the same developed in previous references by the authors, see details in [13, 14].

The dual problem was solved by applying a stopping criterion such that \(\Vert \varvec{\varphi }^{n+1}-\varvec{\varphi }^{n}\Vert \le 10^{-8}\). The evolution of the predicted error with the number of modes *n* employed in the computation of the primal variable is shown in Fig. 10.

## Conclusions

In applications with haptic response, the development of a suitable error indicator of the force being transmitted to the user is of utmost importance, as can be readily understood. In this paper, we have developed a method for the error estimation in such a quantity of interest for a real-time simulator based on the use of reduced order models. In particular, proper generalized decomposition techniques have been employed.

Based on previous developments of the authors, an explicit linearization of the originally non-linear constitutive equations in the framework of PGD has been employed. This renders the problem in the form of a sequence of linear problems, for which an error estimator in the spirit of [34] has been employed. It is based on the employ of the so-called dual problem as a stopping criterion for the original (or primal) one.

The result is the first example (up to our knowledge) of an error estimator for non-linear problems in the framework of PGD methods in general, and haptic simulators in particular.

## Notes

## Declarations

### Authors' contributions

IA, DG and SZ participated in the development of the proposed technique and implemented it in Matlab. PD, EC and FCh developed the technique, checked the results and wrote the manuscript. All authors read and approved the final manuscript.

### Acknowledgements

This work has been supported by the Spanish Ministry of Economy and Competitiveness through Grant Numbers CICYT DPI2014-51844-C2-1-R and 2-R and by the Generalitat de Catalunya, Grant Number 2014-SGR-1471. This support is gratefully acknowledged.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This 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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