dataesr/scientific-paragraphs-categorization huggingface dataset:
Methods Computational procedure and force field As shown in figure 1, BuMe 3 N + and Tf 2 N -ions were described by using the united atom OPLS force field  [44]. [C 4 mim + ][Tf 2 N -] IL was modeled from a recent united atom force field  [45]. The CNT material was modeled as non-charged and flexible from the Tersoff potential  [46]. A perfect (without defaults) flexible armchair-typed CNT with a radius of 1. 2 nm and a length of 100 Å was considered corresponding to 3280 carbon atoms. A length of 10 nm was considered to avoid the size effect  [47]  that corresponds to 100 carbon atoms per nm length of CNT  [48]. All MD simulations were carried out with the DL_POLY package (version 4. 0)  [49]  using a combination of the velocity-Verlet and the SHAKE-RATTLE algorithms  [50]. The Nose-Hoover thermostat  [51, 52]  with a relaxation time of t = 0.
Microcapsules are liquid droplets enclosed by a thin elastic membrane. Being suspended in an external fluid, they undergo large deformations when flowing. Their deformation can be solved numerically, but the resolution of the fluidstructure interactions (FSI) requires extremely long computation times. This is a major constraint for instance when identifying the membrane mechanical properties from experimentations of microcapsules flowing in a microfluidic channel of similar size (Hu et al PFE 2013). We propose to apply Model Order Reduction (MOR) to predict in real time the steady-state capsule deformed shape, needed to determine the membrane elasticity. A database of the capsule deformed shapes was obtained numerically by solving the three-dimensional FSI through a finite element -boundary integral method coupling, and by varying systematically the two non-dimensionalized parameters of the problem: the capillary number, ratio of the viscous to the elastic forces, and the capsule-to-tube size ratio.
Among the MOR techniques, we chose to apply Proper Orthogonal Decomposition (POD) onto the database, which provides a vector basis of principal components, defining a multi-dimensional vector space. The advantage is that, when all the capsule shapes of the database are mapped into this new vector space, they form a manifold (smooth hypersurface) that represents all the admissible solutions of the problem. We show that POD with a manifold walking technique can be successfully applied to microcapsule data sets, whether they are Lagrangian (e. g. known position vector fields) or Eulerian (i. e. when data is acquired experimentally or numerically using methods like level sets). In both.
The latter have indeed made it possible to study the motion and deformation of microcapsules with high accuracy (e. g. Chu et al. (2011) ;  Koleva and Rehage (2012); De Loubens et al. (2014) ;  Gires et al. (2016) ). Numerical and experimental approaches have many points in common. In both cases, the generated data may either be Lagrangian or Eulerian, depending on the numerical schemes and experimental acquisition technique. But the large body of previous studies on capsules and cells has shown that a 3D Lagrangian follow-up of the points of the capsule membrane is necessary to determine the overall dynamics of the deformable particle (e. g. Fischer et al. (1978) ;  Abkarian et al. (2007) ;  Dupont et al. (2016) ). This explains why the classical numerical algorithm, since the pioneering work of  Barthès-Biesel and Rallison (1981) , consists of a Lagrangian tracking of the membrane nodes, whose time evolution is obtained from the integration of the capsule wall velocity provided by the fluid solver.
But novel strategies based on a pure Eulerian description are, however, emerging to solve FSI problems  (Cottet et al. , 2008). They rely on a numerical analysis of the particle shapes and surfaces on a fixed framework in order to determine the strain and deformation fields without any explicit parametrization of the moving particle. Recently, they have been successfully applied to capsules in flow  (Milcent and Maitre, 2016). Experimentally, the acquired data are by essence Eulerian, the only way to perform Lagrangian tracking being to place tracers on the capsule membrane, which is cumbersome and difficult to achieve on such small particles. Generally, experimental images thus consist of two-dimensional grayscale images, and their post-processing involves image processing techniques to extract relevant features such as the capsule deformed shape.
New techniques of characterization have been devised  (Chu et al. , 2011; Hu et al. , 2013; De Loubens et al. , 2014; Gubspun et al. , 2016)  as microfluidics matured over the last decade: the shear/Young moduli of the capsule wall have been determined in microcapillary/extensional flows by comparing the experimental steady-state capsule profiles to databases of profiles calculated numerically. Such experiments are much less cumbersome to perform on entire batches of microcapsules than atomic force microscopy indentation or micropipette aspiration, and can be automatized. But the inherent issue associated with them is the need to run the full theoretical fluid-structure interaction model corresponding to the experiment to identify the mechanical property, as it is computationally costly. The objective of the paper is to analyze whether MOR (and in particular Proper Orthogonal Decomposition) is suitable to reduce the dimensionality of the complex problem and sufficiently precise to determine the steady-state deformation of a capsule in flow without launching the full-simulation.
We aim at identifying if it is possible to accurately predict the capsule deformation in real time, and if it can be done with equal precision whether the data is Lagrangian or Eulerian. We focus on the identification technique based on the steady-state flow of a microcapsule in a microfluidic channel of similar characteristic size  (Chu et al. , 2011; Hu et al. , 2013). Its deformed profile (Figure  1 ) is known to be only a function of two non-dimensionalized parameters: the capsule-to-channel size ratio and the capillary number, ratio of the viscous to the elastic forces. The data sets used as input for the MOR procedure are the steady-state capsule shapes that we have computed off-line using a 3D FSI numerical scheme coupling the Finite Element (FE) method, used to solve for the capsule deformation, with the Boundary Integral (BI) Method, used to solve for the inner and outer fluid flows  (Hu et al.
, 2012). They are computed varying the two controlling parameters in a systematic way and compiled in a comprehensive database. The advantage of using numerical nodal data is that they are inherently Lagrangian, but can easily be transformed into a Eulerian field, by calculating, for instance, the signed distance of the points in space to the capsule membrane. We thus have the possibility to not only see whether reduced-order modeling approaches can be used to predict capsule shape, but also to compare the precision of POD when it is applied to Lagrangian and Eulerian similar-sized data sets. We will first describe, in §2, the physics of the problem at stake and the numerical method of the FE-BI code, and detail how POD is applied to the database containing the steady-state capsule shapes.
In §3, we will first analyze the precision of the capsule shapes reconstructed using POD by comparing them to the ones computed with the full FSI numerical code. We will try to predict the capsule deformation for parameter values that are not included within the database, by using a so-called manifold walking strategy. We will finally transform the Lagrangian numerical data into a Eulerian data field and compare the precision of the predictions for Lagrangian and Eulerian data sets. This will allow us to explore the potential of using POD to predict capsule shapes from experimental data (inherently Eulerian). In §4, we will eventually analyze the influence of the parameters of the POD method before providing concluding remarks in §5.
Database of capsule deformed shapes Simulations of capsules in flow have been run varying the two governing parameters in the range [0, 0. 2] for the capillary number and [0. 75, 1. 2] for the confinement ratio. Only the simulations for which a steady-state shape was reached were retained in the database. Let P be the set of values of capillary numbers Ca and confinement ratios a/ for the 95 simulations that have been selected (Fig. 5 ). No deformed capsule shape exists for simulation parameters above the dashed line. This phenomenon, described in  Barthès-Biesel (2011)  and clearly illustrated in the experimental study of  Chu et al. (2011) , is a direct consequence of the fact that many capsules (like the ones with a proteic membrane) exhibit a strain-softening mechanical behavior. The database is then constituted of the set of the three-dimensional deformed capsules shapes computed at steady-state for the 95 simulations.
Fig. 6  shows examples of capsule contours included in the database. At low values of Ca, the initially spherical capsule takes a bullet-like shape at steady state (Figs. 6a  and 6d ). When the capillary number is increased, a change in concavity occurs at the rear of the capsule: the capsule then has a parachuteshape (Figs. 6b-c  and 6e-f ). The capsule is limited in the x-and y-directions by the channel walls, but extends in the z-direction under the hydrodynamic forces: the capsule length depends on the confinement ratio and varies between 1. 3 and 2. 9 in the cases shown in Fig. 6. (a) (b) (c) (d) (e) (f).
Proper Orthogonal c i = (x c i1 , x c i2 , x c i3 ) be the position vector for the node (i) i=1:n of the c th capsule shape contained in the database. All the position vectors of the c th capsule shape can then be rearranged in a single vector x c = (x c 11 , x c 12 , x c 13 ,. , x c n1 , x c n2 , x c n3 ) of dimension N x = 3n. The proper orthogonal decomposition consists in finding an orthogonal basis of functions (ψ m ) m=1:M (i. e. the principal components) that allows to approximate the position vector x c by x c M m=1 α c m ψ m , (1) where M is the dimension of the POD basis (i. e. the number of modes) and α c m is the coefficient associated with the m th principal component. Eq. (  1 ) is written in the case of centered data (i. e. when the mean value of the vectors x c is equal to the null vector 0).
One of the main reasons is that these morphing techniques cannot guarantee the admissibility of the predicted shapes. To be admissible, the shape must indeed satisfy the constraints of the problem. In the case of the steady-state capsule shapes, the physical constraints are that a unique capsule deformed profile corresponds to each set of values of the parameters Ca and a/. It is generally accepted that all the admissible shapes of a given problem lie on a smooth manifold, which is a hypersurface in the α-space  (Meng et al. , 2016; Raghavan et al. , 2013a,b). The manifold is thus generated by projecting the database of shapes onto the POD basis: the capsule shape, originally represented as 2 562 nodes in the three-dimensional coordinate system, becomes a single point in the 94-dimensional α-space.
Interestingly, the number of dimensions of the manifold is equal to the intrinsic dimension of the problem. Since our problem is governed by two parameters, we know in advance that the manifold associated with our database is a 2D-surface. It is represented in Fig. 10  when POD is applied to the database of Lagrangian nodal position vectors: only the points lying on this manifold correspond to admissible capsule shapes. We now propose to use the manifold to determine the admissible capsule deformed shape that corresponds to a set of parameter values that are not present in the database. This allows to verify whether we have the capability of predicting capsule deformations using POD.
Manifold-based methods of prediction We use manifold walking schemes  (Raghavan et al. , 2013a)  to predict capsule shapes for parameter values θ not included in the database. The technique consists in remaining on the manifold (Fig. 10 ) when inferring a shape from a 1 Linear piecewise interpolation. 2 Cubic spline interpolation. 3 1 st -order polynomial regression. 4 2 nd -order polynomial regression. 5 3 rd -order polynomial regression. 6 4 th -order polynomial regression. 7 5 th -order polynomial regression. 8 2 nd -order diffuse approximation. 12-point adaptive region, A 1 /A 2 = 0. 4. set of known admissible shapes and finding the α -coefficients that best approximate it. We will test different manifold-based prediction approaches to find the α -coefficients: surface interpolation (piecewise linear and bicubic spline interpolation), polynomial regression (of first to fifth order), and diffuse approximation.
Evaluation of the accuracy of the manifold-based predictions In order to evaluate the capability of prediction of the manifold-based techniques and measure the precision of the predicted profiles, we propose to sequentially remove one data set from P and determine the capsule shape predicted for the parameter values corresponding to it (they become the components of θ ). The relative error ε rel is calculated by comparing the predicted contour to the removed capsule shape (originally computed with the full FSI numerical model). This procedure is repeated for all the points of the database and for the different manifold-based techniques using the conditions indicated in Table  1. The results are presented as heat maps, which allow to visualize how the relative error evolves with the location in the parameter space and thus provide a complete overview of the accuracy of each technique.
12b , in which the predicted profile has a small concavity whereas the actual data still has a convex rear shape. The largest inaccuracies are found in certain regions adjacent to the diagonal, where ε rel just exceeds 1% (and even 1. 5% at some points). This is linked to the fact that, close to the diagonal, fewer nearby points are available. The density in data points within the database has been adapted based on the observation of the error distribution. If the database has to cover the entire range of desired parameter values, the data point density also has to be increased in the regions where the capsule shape changes more abruptly with the parameters, as it is the case for the regions of low Ca and close to the dashed line. It is remarkable that, despite these border effects, the error never exceeds 2% and Fig.
POD of 3D Eulerian data fields Having established the capability of POD to predict capsule deformation when applied to Lagrangian data sets, we now investigate if it can equally be applied to Eulerian fields. In Lagrangian descriptions, fixed topologies of the shapes are assumed. The design variables can be any explicit parametrization of the surface boundaries, which can range from vectors of positions, lengths, angles or radii of the objects  (Meng et al. , 2016) , to boundary splines, basis shapes or free-form deformations  (Samareh, 1999; Sederberg and Parry, 1986). But the capsule datasets are not necessarily Lagrangian: they can also be in the form of a Eulerian field, whether they are obtained numerically (using purely Eulerian or meshless numerical schemes) or experimentally (the images being coded in greyscale).
In such Eulerian problems, design variables can still be determined, but they require the use of implicit strategies. Some examples of implicit parametrizations include the use of pixelized or voxelized representations  (Raghavan et al. , 2010) , phase fields  (Bourdin and Chambolle, 2006) , or level sets  (Burger, 2003; Burger and Osher, 2005). The issue is that these methods inherently add errors onto the datasets. In this section, we thus investigate whether it is possible to directly apply POD on the Eulerian dataset. Prior to applying POD, we first generate a scalar field from the Lagrangian datasets of the nodal position vectors obtained by the boundary element-finite element numerical method. We calculate the signed distance of all the points in space to the capsule membrane. After having developed a technique to reconstruct the 3D capsule contour from the level sets, we will compare the results to their Lagrangian counterparts and estimate the precision of POD when it is applied to 3D Eulerian data fields.
Generation of level sets We resort to level sets to emulate a Eulerian data set C E from the Lagrangian database C. To do so, we first create a three-dimensional regular grid capable of englobing all the capsule deformed contours for the parameters in P. We then generate a three-dimensional scalar field for each of the capsule deformed shapes, by measuring the shortest Euclidean distances between the grid points and the capsule surface centered at the middle of the grid. The distances are signed: they are positive for the grid points located outside the shape and negative for the grid points located inside. The scalar field is stored in the form of a high-dimensional vector. The level set grid has to be identical for all the shapes in C so that the new matrix X is correctly assembled. We choose a grid box that extends from -1.
Application of POD to C E and reconstruction of the capsule contours The Eulerian data set C E is composed of the N c level sets computed for each of the capsule deformed shape of C. In the Eulerian case, the vector x c of the c th capsule is x c = d c 1 , d c 2 ,. , d c p , where d c i is the shortest signed distance from the i th grid point to the surface of the c th capsule shape, and p is the total number of points in the level set. The POD technique is then applied to C E following exactly the same method as described in section 3. 1. It, similarly, returns M max = 94 components with non-zero variance. But the difference is that the computed POD reduced bases are now bases of level sets, and not bases of shapes. In order to compare the results with the Lagrangian results, there is thus the need to reconstruct the capsule shapes from the level sets.
One must note that applying POD to the Eulerian data set is computationally much costlier than in the Lagrangian case. Not only is the data set much bigger in size (p 3n), but also an additional step is needed to reconstruct the capsule shape. The capsule shapes are reconstructed from the level sets using the Marching Cubes algorithm  (Lorensen and Cline, 1987) , with which we generate a polygonal mesh of the isosurface d = 0, corresponding to the capsule surface. The size of the generated mesh is directly related to the density in grid points. The reconstructed capsule meshes are made of 969 nodes and 1 820 elements for the grid with p = 12 000 points, of 1 407 nodes and 2 656 elements for the grid with p = 23 750 points, and of 2 429 nodes and 4 628 elements for the grid with p = 50 176 points.
We have set the latter as the default grid, as it provides a mesh that has a similar amount of nodes and elements to the original capsule mesh, which has 2 562 nodes and 5 120 elements. This will guarantee the validity of the shape comparison. Fig. 16  shows the cumulative explained variance for the first five components. The first two components account for 97. 97% of the total variance of the data set, which, like in the POD on Lagrangian data, is consistent with the fact that the manifold is two-dimensional. It is interesting to note that the total variance of the first two modes is slightly higher than in the Lagrangian case. POD on Eulerian data appears as efficient, if not more, as POD on Lagrangian data.
17 ) are very similar to those obtained with Lagrangian data sets (Fig. 11 ). The predictions are the most accurate in the central region of the parametric domain, higher values of error being obtained for low values of Ca or parameter values close to the diagonal. The fact that the grey tones are globally more intense indicates that the predictions are generally less precise than in the Lagrangian case. But the error still hardly exceeds the value of 2%, which validates the concept of manifold-walking using Eulerian data to predict a capsule shape. The mean values of the relative error confirm that the results are slightly less precise than in the Lagrangian case. One can, for instance, note that εrel = 0. 49% for the 2 nd -order diffuse approximation, and 0. 51% for the cubic spline interpolation, which indicates that the difference is not major.
Discussion on the influence of the number of principal components In the previous sections, we have always used all the principal components provided by the POD to predict capsule shapes. We now propose to analyze the effects that the number of principal components has on the precision of the prediction and on the computational time. We will test different sum truncations, varying the value of M in Eq. 1. We have computed the relative error, when removing the capsule shapes of the database one by one, and reconstructing them using 1, 2, 3, 4, 5, 6, 10, 25, 50 or all the 93 principal components. The evolution of the mean relative error with the number of modes is shown in Fig. 18  for the cubic spline interpolation, which is representative of the results obtained in all the cases. We retrieve the  fact that the predictions errors for the level sets are slightly higher than for the nodal coordinate vectors.
The present algorithms have been written in Matlab, using the built-in function fit for interpolation and polynomial regression, and ad hoc functions for the diffuse approximation. Algorithms have been run on a desktop computer with an Intel Core i7 processor at 3. 40 GHz and 16 GB of RAM memory. Examples of the average values of the computational times are shown in Fig. 19 , along with the standard deviation, for the cubic spline interpolation and 2 nd -order diffuse approximation. The time required by the prediction algorithms to estimate a capsule shape is directly proportional to the number of principal components chosen to generate the estimation. For instance, using cubic spline interpolation, the capsule shapes are predicted in 2. 2 seconds using all the 93 principal components, and in approximately a third of that time us- ing 31 components.
that the capsule transits from a slug-shape to a parachute-shape. This increases the difficulty of shape prediction, but this inherent weakness can be mitigated by increasing the density of pre-calculated forms in this region of the database. We have also verified that POD-based prediction approaches can be applied to Lagrangian data (e. g. capsule nodal position vectors computed from numerical methods), as well as to Eulerian scalar fields (e. g. greyscale imaging data obtained from experiments). To emulate a Eulerian framework from our database, we have transformed the capsule shapes into scalar fields using level sets. The prediction results obtained on the level sets are approximately as accurate as the Lagrangian results. The slight difference in accuracy may be due to the fact that we had to go through a shape reconstruction step in order to estimate the accuracy.
Figure11: Heat maps of the relative error ε rel of the capsule shape prediction, obtained when the nodal coordinates are used as design variables. The maps are derived by sequentially excluding a parameter set from the database P. The error is calculated by comparing the predicted contour with the removed capsule shape (which was computed with the full FSI numerical model). (a) Piecewise linear interpolation, (b) cubic spline interpolation, (c) 5 thorder polynomial regression, (d) 2 nd -order diffuse approximation. The black dots correspond to the points, where capsule shape predictions could not be made by the method.
Figure17: Heat maps of the relative error ε rel of the capsule shape prediction, obtained when the level sets are used as design variables. The maps are derived by sequentially excluding a parameter set from the database P. The error is calculated by comparing the contour, reconstructed from the predicted level set, to the removed capsule shape (which was computed with the full FSI numerical model). (a) Piecewise linear interpolation, (b) cubic spline interpolation, (c) 5 th -order polynomial regression, (d) 2 nd -order diffuse approximation. The black dots correspond to the points, where capsule shape predictions could not be made by the method.