Isogeometric analysisbased reduced order modelling for incompressible linear viscous flows in parametrized shapes
 Filippo Salmoiraghi^{1},
 Francesco Ballarin^{1},
 Luca Heltai^{1} and
 Gianluigi Rozza^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s4032301600766
© The Author(s) 2016
Received: 2 February 2016
Accepted: 23 June 2016
Published: 20 July 2016
Abstract
In this work we provide a combination of isogeometric analysis with reduced order modelling techniques, based on proper orthogonal decomposition, to guarantee computational reduction for the numerical model, and with freeform deformation, for versatile geometrical parametrization. We apply it to computational fluid dynamics problems considering a Stokes flow model. The proposed reduced order model combines efficient shape deformation and accurate and stable velocity and pressure approximation for incompressible viscous flows, computed with a reduced order method. Efficient offline–online computational decomposition is guaranteed in view of repetitive calculations for parametric design and optimization problems. Numerical test cases show the efficiency and accuracy of the proposed reduced order model.
Keywords
Isogeometric analysis (IGA) Reduced order models (ROM) Proper orthogonal decomposition (POD) Stokes flows Free form deformation (FFD) Computational fluid dynamics (CFD)Focus and motivation
The capability to perform fast simulations is becoming increasingly relevant for several applications in engineering sciences, related for instance to naval and aeronautical engineering, as well as biomedicine. To this end, reduced basis methods [1, 2], proper orthogonal decomposition [3–5], proper generalized decomposition [6, 7], hierarchical model reduction [8–10], or more in general reduced order modelling (ROM) techniques [11], have received considerable attention in the last decades. ROMs do not replace, but rather build upon as an addon, highfidelity methods such as finite element, finite volume or discontinuous Galerkin methods. Indeed, the choice of the highfidelity solver can be made depending on the particular problem at hand and on preexisting expertise and software availability. Current literature has explored a broad variety of options, including reduced order models based on a finite element highfidelity discretization (e.g. [2, 12–15]), finite volume (e.g. [16–19]) and finite difference methods (e.g. [20–22]). More recently, investigations towards the coupling with discontinuous Galerkin methods for multiscale problems [23] or domaindecomposition approaches [24–26], spectral element methods [27, 28], and extended finite element methods [29, 30] have been carried out.
The aim of this work is to embed isogeometric analysis (IGA) [31, 32] as a highfidelity discretization option in a ROM setting, for the simulation of incompressible linear viscous flows [33–36] and to propose a complete workflow (pipeline) integrated with free from deformation (FFD) as efficient geometrical parametrisation. The latter is enhanced into an IGA context ready to be used within reduced order method (POD). A considerable advantage of IGA with respect to classical finite element analysis is the possibility to avoid any geometrical approximation error and to perform direct designtoanalysis simulations by replacing classical mesh generation, and employing the same class of functions used for geometry parameterization in CAD packages during the analysis process. Even though most modern CAD tools are based on boundary representation (BRep) objects, it is still possible to use them in threedimensional isogeometric analysis, by extending the computational domain inside (or outside) the enclosing (or enclosed) CAD surface (see, for example, [37]). A robust and reliable solution for such passage is still lacking, making this step an open question. However, the superior approximation properties of IGA methods make their adoption appealing also in biomedical and bioengineering applications [38], notwithstanding the fact that in this case the geometry is normally obtained through an approximate NURBS reconstruction of medical images.
Once the threedimensional tensor product representation of the geometry is available, there is no distinction in computational cost or implementation complexity, with respect to simulations done on elementary geometries.
Preliminary related IGAROMs have been applied to steady potential flows [39, 40], parabolic problems [41] or shell structural models [42]. In this work offline–online IGAROM is applied for the development of stable computational reduction strategies for viscous flows problems in parametrized shapes by FFD means. We investigate IGAROMs in a different context with respect to earlier works [39, 40]. In [40] the authors neglect viscous terms and formulate the highfidelity discretization in terms of boundary integral equations and boundary element methods (BEM) to study external flows. The main novelty of the present work, besides the investigation of the other side of the spectrum of incompressible regimes (that is, when the Reynolds number tends to zero), is the coupling of FFD techniques applied to IGA geometries, for internal flows, and using finite element based IGA, in view of studies dealing with nonlinear viscous flows, for which BEM is not suited.
We would like to remark here that, although the background idea is the same as the one presented in [40], several technical issues are fundamentally different. One of the most obvious one is that the discrete systems obtained through boundary integral formulations are in general full, which implies that higher order and higher continuity finite element spaces do not influence the bandwidth of the resulting matrix. In finite element formulations of IGA methods, however, this is an important issue, and it may result in reduced performances also of the final reduced order model. In this work we show how the increased bandwidth of the high fidelity solver does not influence negatively on the combination IGAROM, provided that stable approximations are used for the high fidelity solver.
The proposed integrated approach is composed of the following numerical techniques: (i) isogeometric analysis, that integrates the geometrical representation of the domain and the finite dimensional approximation of the fluid dynamics problem [32], (ii) freeform deformation to efficiently deform the computational domain by means of few geometrical parameters [43], and (iii) proper orthogonal decompositionbased reduced order modelling to generate a stable reduced basis to be queried to cut down the computational cost of numerical simulations [44]. This integration has been introduced in a preliminary version in [45].
The approach we present is completely integrated and automatic from CAD to simulation, taking advantage of IGA and FFD perspectives for the accurate and efficient management of parametrized domains and shapes. The split between offline and online computational steps is crucial and it allows the versatility of bringing this proposed computational approach on very different devices, scenarios and situations in design and optimization, for instance.
The structure of the work is as follows. The parametrized formulation and the IGA method are introduced in “Problem formulation and isogeometric analysisbased highfidelity approximation” section; necessary assumptions related to the offline–online decomposition are also summarized. “Shape parametrization by freeform deformation” section summarizes the freeform deformation map which is employed to prescribe geometrical variations. The proposed stable POD–Galerkin ROM is introduced in “A PODGalerkin ROM for parametrized Stokes equations” section, and 2D and 3D numerical tests are performed in “Numerical results” section into an optimisation framework. Finally, conclusions and perspectives follow in “Conclusions and future work” section.
Problem formulation and isogeometric analysisbased highfidelity approximation
Parametrized formulation
Isogeometric formulations of Stokes flows have been extensively studied in the literature. We refer to [46] for a comprehensive analysis of stable choices of isogeometric finite element spaces, and to [47] for an alternative formulation based on boundary integral equations.
Isogeometric description of the parametrized domain
A CAD representation of the domain is usually obtained through a set of control points \(\left\{ {\varvec{P}}_i\right\} _{i=1}^{\mathcal N_g}\), where in general \({\varvec{P}}_i \in \mathbb {R}^d\) is a ddimensional IGA control point,^{1} whose position depends on the geometrical parameters \({\varvec{\mu }}\).
Weak formulation on the reference domain and discrete problem
If one chooses to use the same basis functions for the geometry and the velocity (for example), then \(\phi _i\) are vector versions of \(B_i\), and \(\mathcal N_u = d\mathcal N_g\), where \(\mathcal N_g\) is the number of the geometry basis functions. For an extensive discussion on the choices of stable pairs of isogeometric finite element approximations of Stokes flows, we refer the reader to [46] and the references therein. In this work we used a TaylorHood approximation (as presented, for example, in [36]), in which the pressure space is taken to be one degree less of the velocity space, maintaining the same knot vectors of the geometry and velocity spaces, i.e., we consider pairs of spaces given by \(({\varvec{\mathcal {S}}}^{p\phantom {1}, \dots , p\phantom {1}}_{p1, \dots , p1} {\mathcal {S}}^{p1, \dots , p1}_{p2, \dots , p2})\) which satisfy the infsup condition and represent a good balance between attainable accuracy and computational efficiency.
Affine parametric dependence assumption
Shape parametrization by freeform deformation
In this section we show how to relate geometrical parameters \({\varvec{\mu }}\) to the IGA control points position \({\varvec{P}}_i({\varvec{\mu }})\). Unfortunately, choosing the IGA control points position as geometrical parameters (i.e. \(G = d \mathcal {N}_g\) and \([{\varvec{P}}_i({\varvec{\mu }})]_j = {\varvec{\mu }}_{(i  1) d + j}\), \(i = 1, \ldots , \mathcal {N}_g\), \(j = 1, \ldots , d\)) results in an extremely high parameter space dimension \(G \gg 1\) which, in turn, may lead to poor performance of the reduced order model [e.g. due to an intractable number of terms in the affine expansions (12)]. The aim of this section is to introduce an efficient representation of the deformation of parametrized domains described by the IGA transformation (4).
Freeform deformation map
Freeform deformation (FFD) techniques, introduced in [43] in the late 80s, are a powerful tool for the deformation of a computational domain by means of a small number of displacements. FFD maps have been employed in the reduced order modelling framework for the first time in [54], as well as applied to shape optimization problems in [55], in both cases considering an underlying finite element highfidelity discretization. FFD has been exploited in [54, 55] to handle the deformation of \(\overline{\Omega }\) into \(\Omega ({\varvec{\mu }})\) as the result of the application of the FFD map to each node of the finite element mesh. In contrast, in this work, we apply FFD to IGA control points to obtain their deformed position \(\left\{ {\varvec{P}}_i({\varvec{\mu }})\right\} _{i=1}^{\mathcal N_g}\), and then rely on the map \({\varvec{c}}(\mathbf {s}; {\varvec{\cdot }})\) in (4) to describe the deformed domain \(\Omega ({\varvec{\mu }})\). To further highlight the sequential nature between the highfidelity IGA spatial description and the application of FFD map to its control points we will follow the original derivation in [43], that uses a different set of basis functions (Bernstein polynomials) than the more general ones employed in “Problem formulation and isogeometric analysisbased highfidelity approximation” section. In any case, further extensions to Bsplines or NURBS can also be pursued [56].
Denote by \(D \subset \mathbb {R}^d\) a box that contains all IGA control points \(\left\{ {\varvec{P}}_i({\varvec{0}})\right\} _{i=1}^{\mathcal N_g}\) obtained (e.g.) for \({\varvec{\mu }} = {\varvec{0}}\). Moreover, in order to apply Bernstein polynomials defined on the reference hypercube^{2} \(\overline{D} = [0,1]^d\), let \({\varvec{\psi }}(\varvec{p})\) be the affine function that maps D to \(\overline{D}\). A (second) set of equispaced control points \(\{\varvec{Q}_{j}\}_{j = 1}^{N_g}\), namely the FFD control points is introduced, where \(N_g := \prod _{k=1}^d N_{g,k}\) being \(N_{g,k}\) the number of FFD control points in the coordinate direction k. The deformed position of the jth control point is then obtained as \(\varvec{Q}_{j} + {\varvec{\mu }}_j\). Since it is possible for some FFD control points to be fixed or to be allowed to move only in some prescribed coordinate direction, the parameter vector \({\varvec{\mu }} \in \mathbb {R}^G\) will contain only the nonzero displacement components, so that \(G \le d N_g\). Effective computational reduction is obtained if \(N_g \ll \mathcal {N}_g\); numerical tests will show that only a small number of FFD control points will be necessary to obtain a large range of admissible shapes.
More practical geometrical parameters in channel configurations
A PODGalerkin ROM for parametrized Stokes equations
In this section we summarize a reduced order model (ROM) for parametrized Stokes equations based on a POD method and a Galerkin projection (see [59] for a deeper insight in the subject).
Reduced basis construction through Proper Orthogonal Decomposition

\(\underline{\mathbf {X}}_{\mathbf {u}} \in \mathbb {R}^{\mathcal {N}_{\mathbf {u}} \times \mathcal {N}_{\mathbf {u}}}\) (\(\underline{\mathbf {X}}_{p} \in \mathbb {R}^{\mathcal {N}_{p} \times \mathcal {N}_{p}}\), respectively) is the matrix representing the velocity (pressure, respectively) inner product;

\(\underline{\mathbf {U}}_{\mathbf {u}} \in \mathbb {R}^{\mathcal {N}_{\mathbf {u}} \times N_{\text {train}}}\) (\(\underline{\mathbf {U}}_p \in \mathbb {R}^{\mathcal {N}_{p} \times N_{\text {train}}}\), respectively) contains the velocity (pressure, respectively) left singular vectors of \(\underline{\mathbf {S}}_{\mathbf {u}}\) (\(\underline{\mathbf {S}}_{p}\), respectively);

\(\underline{\mathbf {W}}_{\mathbf {u}} \in \mathbb {R}^{N_{\text {train}} \times N_{\text {train}}}\) (\(\underline{\mathbf {W}}_p \in \mathbb {R}^{N_{\text {train}} \times N_{\text {train}}}\), respectively) is an orthogonal matrices of the velocity (pressure, respectively) right singular vectors of \(\underline{\mathbf {S}}_{\mathbf {u}}\) (\(\underline{\mathbf {S}}_{p}\), respectively);

\(\underline{\mathbf {\Sigma }}_{\mathbf {u}} \in \mathbb {R}^{N_{\text {train}} \times N_{\text {train}}}\) (\(\underline{\mathbf {\Sigma }}_{p} \in \mathbb {R}^{N_{\text {train}} \times N_{\text {train}}}\), respectively) is a diagonal matrix, containing the singular values of \(\underline{\mathbf {S}}_{\mathbf {u}}\) (\(\underline{\mathbf {S}}_{p}\), respectively) sorted in descending order.
Finally, the reduced spaces dimensions \(N_{\mathbf {u}}\) are chosen such that the retained energy \(I_{\mathbf {u}}\), given by the sum of the squares of the singular values up to \(N_{\mathbf {u}}\) normalized by the sum up to \(N_{\text {train}}\), is larger than a prescribed treshold. A similar procedure is applied to choose \(N_{\mathbf {s}}\) and \(N_{p}\). The basis functions of the reduced velocity space \({\varvec{V}}_N\) are then obtained as the union of the first \(N_{\mathbf {u}}\) left singular vectors of \(\underline{\mathbf {X}}_{\mathbf {u}}^{1/2} \underline{\mathbf {S}}_{\mathbf {u}}\) to the first \(N_{\mathbf {s}}\) left singular vectors of \(\underline{\mathbf {X}}_{\mathbf {u}}^{1/2} \underline{\mathbf {S}}_{\mathbf {s}}\). Similarly, the basis functions of the reduced pressure space \(Q_N\) are given by the first \(N_{p}\) left singular vectors of \(\underline{\mathbf {X}}_{p}^{1/2} \underline{\mathbf {S}}_{p}\). The corresponding basis function matrices, that hold the basis functions as column vectors, are denoted by \(\mathcal {Z}_{\mathbf {u},\mathbf {s}}\) and \(\mathcal {Z}_{p}\), respectively.
Reduced order approximation through Galerkin projection on the reduced spaces
Numerical results
Highfidelity IGA solver validation
In Fig. 5 we plot the convergence test for the solution over several refinement cycles on a uniform grid. The rate of convergence is the one predicted by an a priori analysis, as shown in [46]. In Fig. 6 the numerical solution for the last iteration is shown.
Reduced order approximation of Poiseuillelike flows with meanline FFD
Once the code for the Poiseuille flow has been validated, we keep the same model and boundary conditions and deform the original rectangle (for the two dimensional problem) or parallelepiped (for the three dimensional case) domain through FFD, obtaining a family of possible different configuration of Poiseuillelike flows, such as the one depicted in Figs. 9, 10, 21, and provide main results regarding the ROM framework explained in “A PODGalerkin ROM for parametrized Stokes equations” section.
Computational details about the highfidelity model and the model order reduction
Problem number  1  2  3  4 

Space dimension  2D  2D  2D  3D 
IGA space dimension \((\mathcal {N}_v, \mathcal {N}_p)\)  (2178, 1024)  (2592, 1225)  (2178, 1024)  (6591, 343) 
Number of geometrical parameters  2 rotations  2 rotations  4 rotations  4 = 2 rotations+ outflow variation (length and width) 
Geometrical parameters range  \([75 ^\circ , 75 ^\circ ]^2\)  \([75 ^\circ , 75 ^\circ ]^2\)  \([45 ^\circ , 45 ^\circ ]^4\)  \([75 ^\circ , 75 ^\circ ]^2 \times [0,2]^2\) 
Number of IGA control points  1089  1296  1089  2197 
Number of FFD control points  10  10  20  40 
EIM tolerance  \(10^{3}\)  \(10^{3}\)  \(10^{3}\)  \(10^{3}\) 
EIM terms \(Q_K + Q_B + Q_f\)  27 + 14 + 0  89 + 22 + 0  50 + 22 + 0  104 + 44 + 0 
Number of snapshots  500  500  500  500 
POD tolerance I(N)  \(10^{3}\)  \(10^{2}\)  \(10^{2}\)  \(2*10^{2}\) 
POD space dimension \((N_{\mathbf {u},\mathbf {s}}, N_p)\)  (20, 10)  (20, 10)  (20, 10)  (40, 20) 
HF evaluation time  1.5 s  6.1 s  1.5 s  27 s 
POD offline construction time  250 s  2344 s  250 s  12325 s 
POD evaluation time  0.07 s  0.08 s  0.08 s  0.11 s 
Computational speedup POD  20  76  18  245 
Shape optimization of Poiseuillelike flows with ROM and meanline FFD
Details about the optimization algorithm
Geometrical parameters range  Problem 1  Problem 2  Problem 3  Problem 4 

[−75 deg, 75 deg]  [−45 deg, 45 deg]  [−75 deg, 75 deg]  
Optimization algorithm  MATLAB fmincon  
Cost functional J  \(\int _{\Gamma _{in}} p \ d\Gamma  \int _{\Gamma _{out}} p \ d\Gamma \) 
Main results for the optimization process
Problem 1  Problem 2  Problem 3  Problem 4  

IGA  POD  IGA  POD  IGA  POD  IGA  POD  
Opt. CPU time (s)  90  2.5  280  2.5  151  7  1994  5 
Opt. speedup  –  36  –  112  –  21  –  400 
\(\Vert \varvec{\mu }  \varvec{\mu }^* \Vert \)  \(10^{7}\)  \(10^{4}\)  \(10^{5}\)  \(10^{2}\)  \(10^{5}\)  \(10^{3}\)  \(10^{6}\)  \(10^{2}\) 
Pressure drop (J)  80  79.997  80  79.997  80  80.0003  126.43  126.43 
Relative error on J  0  \(O\,(10^{5})\)  0  \(O\,(10^{5})\)  0  \(O\,(10^{6})\)  0  \(O\,(10^{6})\) 
Details about the optimization algorithm are summarized in Table 2. In Table 3 we summarize the main results for the optimization process, both for the high fidelity solver and for the reduced order model. The error on the angles and on the pressure drop is negligible in the case of the high fidelity solver. The error for the ROM is of the order of \(10^{4}\) (\(10^{4}\), respectively), and we obtain a computational speedup of about 36, for the two rotation case. Interestingly, such speedup is considerably higher than the speedup for a single simulation (which is around 20), most likely because it is generally easier for optimization software to explore a smaller state space, and some smarter procedure may be used internally to save computational effort. This behaviour is less evident for the four rotation case (problem 3). We expect that also in the nonlinear case the computational speedup would increase more considerably.
This simple shape optimization test case highlights the capability of the proposed reduced order model (in terms of reducing the computational cost). In future more complex applications will deal with the optimal design process of aerohydrodynamic components.
Conclusions and future work
We have presented a complete parametric design pipeline from CAD to accurate and efficient numerical simulation, by introducing geometrical parametrization based on FFD, high order simulations based on IGA and efficient and stable computational reduction strategies based on proper orthogonal decomposition, after the enrichment of the velocity space with suited supremizers. This setting is motivated and developed by industrial applications in mechanical, nautical and naval engineering at low Reynolds number (e.g. microfluidics devices characterized by low velocity flows and in small geometrical configurations). Results look promising to continue with the implementation of a viscous nonlinear model and more complex physical and geometrical problems in order to deal with more advanced fluid mechanics indexes (vorticity, viscous stresses, viscous energy dissipation), derived from the state equations. For example, we mention the project UBE (Underwater Blue Efficiency) whose goal is the shape optimization of immersed parts of motor yachts, including exhaust flow devices, for the reduction of emissions and vibrations , in order to increase onboard comfort. This parametric design automatic embedded pipeline is motivating also the investigation and improvement of some computational aspects related with FFD and the already mentioned EIM.
In the next section we will introduce another set of control points, related to the freeform deformation, which will be denoted FFD control points.
Even though actually \(\overline{D} = \overline{\Omega }\), we use different symbols to stress the fact that the two reference domains can be, in principle, different depending on the choice of IGA and FFD basis functions.
Since in this case \(\Gamma _N = \emptyset \) we take \(Q = L^2_0(\Omega ) := \{ q \in L^2(\Omega ) s.t. \int _{\Omega } q\ \text {d}{\varvec{s}} = 0 \}\)
Declarations
Author's contributions
All authors have prepared the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This work has been supported by the projects “Underwater Blue Efficiency” and “OpenViewSHIP”, both funded by Regione Friuli Venezia Giulia (FVG)— PAR FSC 20072013, Programma Attuativo Regionale, Fondo per lo Sviluppo e la Coesione, coordinated by the technological cluster MARE TC FVG, and by the project INDAMGNCS 2015, “Computational Reduction Strategies for CFD and FluidStructure Interaction 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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