Boundary Element Analysis of Viscous Flow

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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.

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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 add-on, high-fidelity methods such as finite element, finite volume or discontinuous Galerkin methods. Indeed, the choice of the high-fidelity solver can be made depending on the particular problem at hand and on pre-existing expertise and software availability. Current literature has explored a broad variety of options, including reduced order models based on a finite element high-fidelity discretization e.

More recently, investigations towards the coupling with discontinuous Galerkin methods for multiscale problems [ 23 ] or domain-decomposition approaches [ 24 — 26 ], spectral element methods [ 27 , 28 ], and extended finite element methods [ 29 , 30 ] have been carried out. 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 design-to-analysis simulations by replacing classical mesh generation, and employing the same class of functions used for geometry parameterization in CAD packages during the analysis process.

A robust and reliable solution for such passage is still lacking, making this step an open question. Once the three-dimensional 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 IGA-ROMs have been applied to steady potential flows [ 39 , 40 ], parabolic problems [ 41 ] or shell structural models [ 42 ]. In this work offline—online IGA-ROM is applied for the development of stable computational reduction strategies for viscous flows problems in parametrized shapes by FFD means.

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. 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.

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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 IGA-ROM, 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 free-form deformation to efficiently deform the computational domain by means of few geometrical parameters [ 43 ], and iii proper orthogonal decomposition -based 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 Isogeometric Boundary Element Method |

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 problem of interest throughout this work is a parametrized incompressible steady Stokes problem, obtained as a simplification of Navier—Stokes equations when inertial forces can be neglected, compared to viscous forces.

Isogeometric formulations of Stokes flows have been extensively studied in the literature. Different parameter values will produce different IGA control points and, thus, different computational domains. An example of 1-D and 2-D B-splines basis functions. The two dimensional basis functions are obtained by tensor product of the one dimensional ones.

For simplicity of exposition, we will use the same notation we used in Eq.

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To differentiate w. In this work we used a Taylor-Hood 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. In this work we seek for an offline—online decomposition of the computational stages, as required in the reduced order modelling context for an efficient evaluation of the ROM [ 1 ]. In order to achieve this, we require that matrices and vectors in 11 fullfil the following affine parametric dependence assumption:.

We employ the empirical interpolation method EIM [ 49 ] to approximate this assumption up to a desired tolerance.

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See also [ 50 — 53 ] for the application of EIM to viscous flows in parametrized domains. Unfortunately, choosing the IGA control points position as geometrical parameters i. The aim of this section is to introduce an efficient representation of the deformation of parametrized domains described by the IGA transformation 4. Free-form 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 high-fidelity discretization.

One of the drawbacks of FFD from practical point of view is the lack of immediate interpretation of its parameters. Indeed, FFD is not interpolatory, so the magnitude of the displacement of a control point is not exactly equal to the actual deformation obtained at that spatial location. Recent works have improved the versatility from the user point of view of complex shape parametrization techniques thanks to the automatic prescription of control points position based on more intuitive geometrical parameters [ 57 , 58 ].

For the case of two rotations, a summary of the meanline FFD is shown in Fig. For the four rotation case, the extension is straightforward. The reference meanline of D is divided in four intervals Fig.

In a similar way, FFD can also be employed to perform local variations to the section area. In particular, FFD is applied in Fig. This requires one geometrical parameter in 2D related to the height of the outlet section and two geometrical parameters in 3D related to the width and height of the outlet section. Change of the outflow section for problem 4: reference geometry red , morphed geometry blue and free form control points movement dashed line. The following snapshot matrices are then considered. A POD basis for the velocity and pressure reduced spaces are then obtained by a thin singular value decomposition SVD of the snapshot matrices, i.

Moreover, the so-called supremizer enrichment is employed in this work in order to satisfy the inf-sup stability also at the reduced order level [ 59 — 61 ]. Thus, for each training sample the following elliptic problem is solved.

Therefore, the following problem has to be solved:. Moreover, thanks to the affine dependence assumption 12 , during the online stage each block the ROM linear system 14 can be assembled as. We refer to Fig. In order to validate our framework, we first perform some tests on the high-fidelity method for problem with known, exact solution, both for the two dimensional and three dimensional case. The first test is to recover the divergence-free solution:.

The corresponding forcing term is. In Fig. 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 Poiseuille-like flows, such as the one depicted in Figs.

A summary of the computational details is given in Table 1. The resulting singular values are depicted in decreasing order in Figs. The time required for this offline stage ranges from about seconds for problems 1 and 3 to more than seconds for the three-dimensional problem 4. For the sake of visualization, we also report the reconstructed velocity and pressure fields in Fig. We can compare it with the visualization of the high-fidelity solution of Fig. Similar considerations apply for the other problems: see Fig. Table 1 also highlights several factors that slightly affect the online performance in terms of CPU time.

A first point to take into account is related to the number of terms resulting from the EIM approximation of parametrized tensors: comparing problems 1 to 2 and 3 we can see that both an increased high-fidelity discretization order and an higher number of parameters result in a larger number of EIM terms.

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A second factor to take into account is related to the reduced space dimension. This can be observed comparing problems 1 and 4, where the latter requires a larger reduced space due to a slower decay of POD singular values. In any case, computational speedups are of at least an order of magnitude. We now present the results of the shape optimization routine for the deformable pipe.

The aim is to find the parameter values that minimize the pressure drop in the pipe, for prescribed inflow section and parametrized meanline variation and outlet section, in case of problem 4. For prescribed outlet section, the exact result of the optimization procedure is the straight pipe, obtained for null value of the angles; for parametrized outlet section, the exact solution is characterized by null angles and maximum outlet area.

The error on the angles and on the pressure drop is negligible in the case of the high fidelity solver. 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 aero-hydrodynamic components.

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.