Embedded domain Reduced Basis Models for the shallow water hyperbolic equations with the Shifted Boundary Method
Introduction
The computational cost associated with the numerical solution of partial differential equations might be in some cases prohibitive. This is happening, for example, when the numerical solution is required in nearly real time or a when large number of system configurations need to be tested. Shape optimization problems are a typical example of the latter case, where a large number of different geometrical configurations need to be analyzed to converge to an optimal solution. Reduced order models demonstrated to be a viable approach to reduce the computational burden and have been developed for a large variety of different linear and nonlinear problems [1], [2].
In recent times, immersed/embedded/unfitted methods have seen a great development from the seminal ideas of Peskin [3]. The key ideas in embedded methods are the use of grids that are not body-fitted, in which the geometry of the shapes to be simulated is immersed by way of computational geometry techniques. In this work, we base the reduced order models on the Shifted Boundary Method (SBM), which is an embedded/unfitted finite element method originally proposed for the Poisson, Stokes and incompressible Navier–Stokes equations [4], [5] and recently extended to wave equations and shallow water equations (SWE) [6]. In the SBM, a surrogate boundary is introduced in proximity of the true immersed boundary, and the boundary conditions are imposed on the surrogate boundary, with appropriate corrections that rely on Taylor expansions [4], [5]. The SBM does not require complicated data structures and numerical quadratures to integrate the governing equations on cut element, typical of cutFEM/XFEM approaches. Compared to other embedded finite element methods such as XFEM or cutFEM [7], [8], [9], the SBM has also the advantage that the degrees of freedom (unknowns) stay the same for varying geometries, hence existing reduced order model methodologies are more easily adapted. Specifically, the total number of unknowns in SBM is determined by the background mesh and it is independent of the location of the embedded geometry. In contrast, XFEM and other enriched finite element methods alike introduce new degrees of freedom such as the Heaviside functions within cut elements, hence the total number of unknowns typically varies with the embedded geometry locations and/or depends on a computationally expensive cutting of elements procedure.
In this article we focus our attention on projection-based reduced order models specifically tailored to geometrically parametrized problems [10], [11]. The idea is to combine the recently proposed Shifted Boundary Method [4], [5], [12] with Reduced Order Models based on the Proper Orthogonal Decomposition (POD) with Galerkin projection (SBM-ROM). This combination, that has been recently proposed in previous works in a different setting [13], [14], [15], allows to avoid the map of all the parametrized solutions to a common reference geometry, see also [16], [17]. In this paper, the embedded methodology introduced in the mentioned research works is extended to shallow water equations with explicit time marching schemes. The idea of merging embedded approaches with reduced order models has been proposed also in [18] where a fictitious domain method was coupled with a Proper Generalized Decomposition approach to study uncertain geometries. In [19] the authors proposed a projection based reduced order model starting from an embedded full order simulation applied to evolving interfaces.
With embedded simulations it is in fact easy to work with a common background mesh also in the case of large geometrical changes. By comparison, body-fitted meshes often require sophisticated re-meshing techniques, when complex geometrical deformations are present, and maintaining the topology of the underlying mesh is a difficult task.
In addition, we introduce a new approach to handle degrees of freedom located in the “out of interest/ghost” region which is based on a radial basis function interpolation. We denote this new approach as SBM-iROM. As shown in the numerical examples, this approach allows to partially reduce the drawback associated with the slow decay of the Kolmogorov N-width when dealing with embedded full order models.
Remark 1.1 Embedded computations with parametrized geometries are characterized by a slow decay of the Kolmogorov N-width which is caused by two different aspects. The first one is related to the embedded geometry that can arbitrarily move within the background mesh. Therefore, the boundary condition is applied on different elements, depending on the position of the true boundary with respect to the background mesh. The second aspect, that we believe is of minor entity, is related to the discontinuity of the solution in the proximity of the jump between the active and inactive nodes. The methodology that we have developed contributes to partially deal only with the second aspect.
The article is organized as follows: in Section 2 we introduce the mathematical formulation of the full order problem, the associated weak formulation and the details concerning the specific discretization strategy. Section 3 describes in detail the approach used for the construction of the reduced order model with a focus on the relevant changes required for the specific full order model formulation and introduces the employed interpolation preprocessing. In Section 5 we introduce three numerical examples to show the properties and accuracy of the proposed methodology. Finally, in Section 6 we report some conclusions and outlooks for future developments.
Section snippets
The model problem and the full order approximation
Before introducing the shallow water model, we briefly review the relevant literature: for linearized shallow water equations arising from the equations of acoustics we refer to the work [20], where only the generation of the second harmonic wave is considered (the higher order harmonics being neglected) under the assumption of weak non-linearity, while a set of uncoupled equations for the primary and secondary wave is discretized in space by a finite element method, and then solved by using
Reduced order model with a POD-Galerkin method
The reduced order model proposed here is based on a POD-Galerkin approach. It means that the underlying system of equations is projected onto a linear subspace of smaller dimension spanned by a reduced number of global basis functions (POD modes). There are different techniques to generate this linear subspace and here we rely on the POD [42]. The overall methodology is based on the classic offline–online splitting approach [2], which is briefly recalled in what follows.
A possible implementation of hyper-reduction using the discrete empirical interpolation method
The most natural approach to have an efficient implementation of the methodology would be to perform hyper-reduction directly on the residual term . In what follows we present a possible implementation using the discrete empirical interpolation method as originally introduced in [47]. During the offline stage there is the need also to store snapshots of the residual vector and to assemble a snapshots matrix for the residual vector: Then the
Numerical experiments
We consider three test cases featuring symmetric SWE flow past a stationary cylinder, based on the configuration in Fig. 5.1. The straight channel is represented by the computational domain . Denoting the unit outer normal vector to by , the upper and lower boundary conditions are given by slippery walls, for which , the left boundary is set to a constant inflow flux , and the right boundary condition is a constant outflow with flux . The flow
Concluding remarks and future developments
In this article we have proposed and analyzed the coupling between the Shifted Boundary Method and POD-Galerkin methods for reduced order modeling in presence of geometrical parameters considering a case of hyperbolic systems. The methodology has been applied to shallow water equations discretized using an explicit time integration scheme and tested on three numerical benchmarks of increasing complexity.
In order to tackle one of the issues arising with the coupling of immersed methods and
Declaration of Competing Interest
The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Gianluigi Rozza reports financial support was provided by International School for Advanced Studies. Gianluigi Rozza reports a relationship with European Research Council that includes: funding grants.
Acknowledgments
This research has been supported by the Army Research Office (ARO) under Grant W911NF-18-1-0308 (GS), the U.S. National Science Foundation under Grant DMS-2137934, European Union Funding for Research and Innovation -Horizon 2020 Program- in the framework of European Research Council Executive Agency: Consolidator Grant H2020 ERC CoG 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics” (PI Prof. Gianluigi Rozza). We also acknowledge the
References (50)
Numerical analysis of blood flow in the heart
J. Comput. Phys.
(1977)- et al.
The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems
J. Comput. Phys.
(2018) - et al.
The shifted boundary method for embedded domain computations. part II: Linear advection-diffusion and incompressible Navier–Stokes equations
J. Comput. Phys.
(2018) - et al.
The shifted boundary method for hyperbolic systems: Embedded domain computations of linear waves and shallow water flows
J. Comput. Phys.
(2018) - et al.
An eXtended finite element method/Lagrange multiplier based approach for fluid-structure interaction
Comput. Methods Appl. Mech. Engrg.
(2008) - et al.
A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow
Comput. Methods Appl. Mech. Engrg.
(2019) - et al.
Fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains
Comput. Methods Appl. Mech. Engrg.
(2011) - et al.
Reduction of nonlinear embedded boundary models for problems with evolving interfaces
J. Comput. Phys.
(2014) - et al.
Finite element simulation of non-linear sound wave propagation
J. Sound Vib.
(1992) - et al.
Weak boundary conditions for wave propagation problems in confined domains: Formulation and implementation using a variational multiscale method
Comput. Methods Appl. Mech. Engrg.
(2012)