A reduced-order variational multiscale interpolating element free Galerkin technique based on proper orthogonal decomposition for solving Navier–Stokes equations coupled with a heat transfer equation: Nonstationary incompressible Boussinesq equations

https://doi.org/10.1016/j.jcp.2020.109875Get rights and content

Highlights

  • A new meshless method is developed for nonstationary Boussinesq (NB) equations.

  • The proposed idea is based on the variational multiscale interpolating EFG method.

  • The proper orthogonal decomposition method is used to generate a reduced order model.

  • The POD-VMIEFG method gives acceptable results for simulating the NB equations.

Abstract

In the recent decade, meshless methods have been handled for solving some PDEs due to their easiness. One of the most efficient meshless methods is the element free Galerkin (EFG) method. The test and trial functions of the EFG are based upon the special basis. Recently, some modifications have been developed to improve the EFG method. One of these improvements is the variational multiscale EFG (VMEFG) procedure. In the current article, the shape functions of interpolating moving least squares (IMLS) approximation are applied to the variational multiscale EFG technique to numerical study the Navier–Stokes equations coupled with a heat transfer equation such that this model is well-known as two-dimensional nonstationary Boussinesq equations. In order to reduce the computational time of simulation, we employ a reduced order model (ROM) based on the proper orthogonal decomposition (POD) technique. In the current paper, we developed a new reduced order model based on the meshless numerical procedure for solving an important model in fluid mechanics. To illustrate the reduction in CPU time as well as the efficiency of the proposed method, we investigate two-dimensional cases.

Introduction

This work devoted to coupling of the incompressible Navier–Stokes equations with a heat conduction problem. The resulting system is the so-called nonstationary Boussinesq approximation [1], [2]. The Boussinesq equations can be used for modeling large scale atmospheric and oceanic flows that are responsible for cold fronts and the jet stream. Furthermore, the Boussinesq equations have important role in the study of Rayleigh-Benard convection [3]. Thus, we considerut(x,t)εΔu(x,t)+(u(x,t))u(x,t)+p(x,t)=wj,(x,t)Ω×I,u(x,t)=0,(x,t)Ω×I,wt(x,t)γΔw(x,t)+(u(x,t))w(x,t)=0,(x,t)Ω×I,u(x,t)=f(x,t),w(x,t)=h(x,t),(x,t)Ω×I,u(x,0)=u0(x),w(x,0)=w0(x),xΩ, where j=(0,1) is unit vector, Ω is the computational domain and I=(0,Tf) such that Tf is the final time. The mathematical model (1.1)-(1.4) has been studied by some numerical techniques for example mixed finite element formulation [4], POD mixed finite volume element procedure [5], POD Galerkin type with error estimation [2] and Crank–Nicolson mixed finite volume-element procedure [1]. Also, the existence and uniqueness of the solutions of model (1.1)-(1.5) are studied in [8]. The extended version of Eqs. (1.1)-(1.5) is [4], [5]utε(2ux2+2uy2)+uux+vuy+px=0,(x,t)Ω×I,vtε(2vx2+2vy2)+uvx+vvy+py=w,(x,t)Ω×I,ux+vy=0,(x,t)Ω×I,wtγ(2wx2+2wy2)+uwx+vwy=0,(x,t)Ω×I,u(x,t)=f(x,t),v(x,t)=g(x,t),(x,t)Ω×I,w(x,t)=h(x,t),(x,t)Ω×I,u(x,0)=u0(x),v(x,t)=v0(x),xΩ,w(x,0)=w0(x),xΩ, where

  • x is (x,y),

  • u and v are the velocity components of the fluid in the x- and y-directions, respectively,

  • w is the temperature of the fluid,

  • p presents the pressure of the fluid,

  • ε=Pr(Re)1,

  • Re denotes the Reynolds number,

  • Pr interprets the Prandtl number,

  • γ=(Re)(Pr),

  • f, g and h are the boundary conditions for the velocity in the x- and y-directions and the temperature of the fluid, respectively,

  • Furthermore, u0, v0 and w0 are the initial conditions for the velocity in the x- and y-directions and the temperature of the fluid, respectively.

The proper orthogonal decomposition (POD) idea is a method used to construct reduced order models (ROMs) [9], [10]. The POD technique can be found in several research papers for solving different physical models. The POD technique is considered by many scholars. The main aim of [11], [12] is to evaluate and compare the efficiencies of techniques for constructing reduced-order models for finite difference (FD) and finite element (FE) algorithms obtained via discretizing the systems of unsteady nonlinear partial differential equations (PDEs). A new approach to enhance the accuracy of a novel Proper Orthogonal Decomposition (POD) model applied to moderate Reynolds number flows (of the type typically encountered in ocean models) is developed in [13]. The authors of [14] proposed a non-intrusive reduced order model for general, dynamic partial differential equations based on the proper orthogonal decomposition (POD) and Smolyak sparse grid collocation. Reduced-order models are derived in [15], [16] from low-order bases computed by applying proper orthogonal decomposition (POD) on an a priori ensemble of data of the Navier–Stokes model. A non-intrusive model reduction computational method is developed in [17] using hypersurfaces representation for reservoir simulation and further it was applied to 3D fluvial channel problems. Recently, authors of [18] presented a non-intrusive reduced order model based on machine learning.

A non-intrusive reduced order method is employed in [19] to model a solid interacting with compressible fluid flows to simulate crack initiation and propagation. A new reduced order model is proposed in [20] based upon the POD for solving the Navier–Stokes equations as the novelty of the method lies in its treatment of the equation's non-linear operator. The main aim of [21] is to develop a new nonlinear POD Petrov–Galerkin approach for the Navier–Stokes equations. A new non-intrusive model reduction method is proposed in [22] for the Navier–Stokes equations based on the radial basis function (RBF) multi-dimensional interpolation instead of the traditional approach of projecting the equations onto the reduced space. A fast and stabilized meshless method that combines a variational multi-scale element free Galerkin (VMEFG) method and the POD method is developed in [23] to solve convection-diffusion problems. The POD technique is applied for the meshless method in [24] for transient heat conduction problems. A combination of POD method with finite difference technique has been proposed in [25], [26] to solve the parabolized Navier–Stokes (PNS) equations. A POD technique is used in [5] for model reduction of mixed finite element (MFE) for the nonstationary Navier–Stokes equations and error estimates between a reference solution and the POD solution of reduced MFE formulation are studied. Authors of [6] proposed a framework for orthogonal decomposition of swirling flows applied to problems originating from turbomachines. A combination of proper orthogonal decomposition with radial basis functions is developed in [7] for solving fluid flow problems.

The interpolating moving least-squares (IMLS) method based on a nonsingular weight function is used in [27] to construct the approximation function, the weak form of the problem of inhomogeneous swelling of polymer gels is used to obtain the final discretized equations, and penalty method is applied to impose the displacement boundary condition, then an improved element-free Galerkin (IEFG) method for the problem of the inhomogeneous swelling of polymer gels is presented. The improved element-free Galerkin (IEFG) method based on the improved MLS approximation and a nonsingular weight function is proposed in [28] for solving elastoplastic large deformation problems. Improved complex variable moving least-squares (ICVMLS) approximation is applied in [29] to construct the shape function and then modified Galerkin weak form of wave propagation problems is employed for obtaining the final system equations. The improved element-free Galerkin (IEFG) method is presented in [30] based on the improved MLS approximation to solve three-dimensional elastoplasticity. The authors of [31] developed an interpolating element-free Galerkin (IEFG) method for solving three-dimensional potential problems based on the improved interpolating moving least-squares (IIMLS) method. By combining the dimension splitting method and the improved complex variable element–free Galerkin method, the dimension splitting and improved complex variable element-free Galerkin (DS–ICVEFG) method is developed in [32] for solving 3D transient heat conduction problems. Furthermore, authors of [33] combined the dimension splitting method with the improved complex variable element-free Galerkin method to get a hybrid improved complex variable element-free Galerkin (H–ICVEFG) method for solving three-dimensional advection-diffusion problems. The main aim of [34] is to develop a fast and efficient local meshless method based on the POD method and RBF-generated FD technique for solving shallow water equations in one- and two-dimensional cases. The authors of [35] employed the shape functions of the reproducing kernel particle method in the meshless local Petrov–Galerkin procedure for solving two-dimensional nonstationary incompressible Boussinesq equations. The main propose of [36] is to introduce a numerical procedure based on the POD method and local RBF-generated FD formulation to simulate the time dependent incompressible Navier–Stokes equation with variable density. The Oldroyd model as a generalized incompressible Navier–Stokes equation is investigated in [37] via the interpolating stabilized element free Galerkin technique. An upwind local radial basis functions-differential quadrature (RBFs-DQ) technique is developed in [38] to simulate some models arising in water sciences. The authors of [39] developed a meshless numerical procedure based on the interpolating element free Galerkin (IEFG) method to simulate the groundwater equation (GWE). The main aim of [40] is to propose a POD reduced-order discontinuous Galerkin method for solving the generalized Swift–Hohenberg equation with application in biological science and mechanical engineering.

In the current research work, we replace the MLS shape functions with the interpolating MLS shape functions to directly apply the essential conditions. Also, we employ a variational multiscale (VM) approach based on increasing the order of approximation to improve the numerical results. Furthermore, to decrease the computational cost of the new scheme, the POD technique is utilized.

The structure of this paper is: the shape functions of interpolating moving least squares approximation are explained in Section 2, the proper orthogonal decomposition is described in Section 3, the discretization of the temporal variable is developed in Section 4, the variational multiscale element free Galerkin method is explained in Section 5, some numerical experiments are investigated in Section 6 to show the efficiency and accuracy of the new numerical formulation and conclusion of paper has been noted in Section 7.

Section snippets

Shape functions of interpolating MLS approximation

Here, we explain constructing the shape functions of the interpolating MLS (IMLS) approximation. The shape functions of MLS approximation do not have δ-Kronecker property thus the Dirichlet boundary condition cannot be applied, directly. However, the shape functions of IMLS approximation are built based on a singular weight function as according to this alteration, the new shape functions have δ-Kronecker property.

Let X={ςi}i=1N be a set of distributed nodes in ΩRn. The fill distance parameter

Construction of the POD basis

Let prni be known for 1n1<n2<<nLN and 1iL, we defineV=span{prn1,prn2,,prnL}, and also {Φ}j=1l is an orthogonal basis of V such thatprni=j=1l(prni,Φj)ωΦj,i=1,2,,L, in which(prni,Φj)ω=(prni,Φj).

Definition 3.1

[4], [5] In the POD idea, we want to find an orthogonal basis Φj such that for every 1dl{min{Φ}j=1d1Li=1Lprnij=1d(prni,Φj)ωΦjω2subjectto:(Φi,Φj)ω=δij,1id,1ji, whereprniω2=prni2.

On other hand, problem (3.4) is equivalent to [4], [5]max(Φ,Φ)ω=ΦL2(Ω)21Li=1L|(prni,Φ)ω|2. We use

Discretization of the temporal variable

We define tk=kdt for k=0,1,,N, where dt=T/N. To approximate the time-derivative the backward finite difference method is used, thus we haveu(x,y,tk)t=uk+1(x,y)uk(x,y)dt,v(x,y,tk)t=vk+1(x,y)vk(x,y)dt. According to the main problem (1.6)-(1.9), we can write the following time discretizationuk+1ukdtε(2ukx2+2uky2)+ukukx+vkuky+pkx=0,vk+1vkdtε(2vkx2+2vky2)+ukvkx+vvky+pky=wk,uk+1x+vk+1y=0,wk+1wkdtγ(2wkx2+2wky2)+ukwkx+vkwky=0. Simplifying relations (4.3)-

Variational multiscale interpolating EFG procedure

In the mid 90's Hughes [48], [49] reviewed the stabilization schemes for the two-scale problems which are commonly known as the variational multiscale (VM) method. There are several research papers that the VM idea is combined with finite element method such as multiscale/stabilized (FEM) formulations for solving the incompressible Navier–Stokes equations [50], the advection-diffusion equation [51], the heat transfer problem [52], the Darcy flow model [53], the Fokker–Planck equation [54].

Numerical argument

The numerical results are carried out using MATLAB 2018b software on an Intel Core i7 machine with 16 GB of memory.

Conclusion

In this paper, we developed a new reduced order model based on the meshless variational multiscale interpolating element free Galerkin (IEFG) method for solving the two-dimensional nonstationary Boussinesq equations. The interpolating moving least squares approximation is employed in the IEFG technique to derive an improved meshless weak form formulation. First, the time variable is discretized by a finite difference scheme. The time-discrete plane is based on a two-step formulation such that

CRediT authorship contribution statement

Author's contributions: Six authors contributed equally and significantly in writing this article. Authors wrote, read and approved the final manuscript.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors are very grateful to the reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper. A. Khodadadian and C. Heitzinger acknowledge financial support by FWF Austrian Science Fund START Project no. Y660 PDE Models for Nanotechnology.

References (66)

  • X. Zhang et al.

    A fast meshless method based on proper orthogonal decomposition for the transient heat conduction problems

    Int. J. Heat Mass Transf.

    (2015)
  • J. Du et al.

    Reduced order modeling based on POD of a parabolized Navier–Stokes equations model II: trust region POD 4D VAR data assimilation

    Comput. Math. Appl.

    (2013)
  • H. Cheng et al.

    Analyzing wave propagation problems with the improved complex variable element-free Galerkin method

    Eng. Anal. Bound. Elem.

    (2019)
  • S. Yu et al.

    The improved element-free Galerkin method for three-dimensional elastoplasticity problems

    Eng. Anal. Bound. Elem.

    (2019)
  • D. Liu et al.

    The interpolating element-free Galerkin (IEFG) method for three-dimensional potential problems

    Eng. Anal. Bound. Elem.

    (2019)
  • H. Cheng et al.

    A hybrid improved complex variable element-free Galerkin method for three-dimensional advection-diffusion problems

    Eng. Anal. Bound. Elem.

    (2018)
  • M. Dehghan et al.

    The use of proper orthogonal decomposition (POD) meshless RBF-FD technique to simulate the shallow water equations

    J. Comput. Phys.

    (2017)
  • M. Abbaszadeh et al.

    The reproducing kernel particle Petrov–Galerkin method for solving two-dimensional nonstationary incompressible Boussinesq equations

    Eng. Anal. Bound. Elem.

    (2019)
  • M. Abbaszadeh et al.

    Reduced order modeling of time-dependent incompressible Navier–Stokes equation with variable density based on a local radial basis functions-finite difference (LRBF-FD) technique and the POD/DEIM method

    Comput. Methods Appl. Mech. Eng.

    (2020)
  • M. Abbaszadeh et al.

    Investigation of the Oldroyd model as a generalized incompressible Navier–Stokes equation via the interpolating stabilized element free Galerkin technique

    Appl. Numer. Math.

    (2020)
  • M. Abbaszadeh et al.

    An upwind local radial basis functions-differential quadrature (RBFs-DQ) technique to simulate some models arising in water sciences

    Ocean Eng.

    (2020)
  • M. Abbaszadeh et al.

    Analysis and application of the interpolating element free Galerkin (IEFG) method to simulate the prevention of groundwater contamination with application in fluid flow

    J. Comput. Appl. Math.

    (2020)
  • X. Li et al.

    Analysis of the inherent instability of the interpolating moving least squares method when using improper polynomial bases

    Eng. Anal. Bound. Elem.

    (2016)
  • X. Li

    Error estimates for the moving least-square approximation and the element-free Galerkin method in n-dimensional spaces

    Appl. Numer. Math.

    (2016)
  • L. Chen et al.

    The boundary element-free method for 2D interior and exterior Helmholtz problems

    Comput. Math. Appl.

    (2019)
  • L. Chen et al.

    Boundary element-free methods for exterior acoustic problems with arbitrary and high wavenumbers

    Appl. Math. Model.

    (2019)
  • X. Li et al.

    Analysis of the element-free Galerkin method for Signorini problems

    Appl. Math. Comput.

    (2019)
  • X. Li

    A meshless interpolating Galerkin boundary node method for Stokes flows

    Eng. Anal. Bound. Elem.

    (2015)
  • T.J. Hughes

    Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods

    Comput. Methods Appl. Mech. Eng.

    (1995)
  • T.J. Hughes et al.

    The variational multiscale method—a paradigm for computational mechanics

    Comput. Methods Appl. Mech. Eng.

    (1998)
  • A. Masud et al.

    A multiscale/stabilized finite element method for the advection–diffusion equation

    Comput. Methods Appl. Mech. Eng.

    (2004)
  • A. Masud et al.

    A stabilized mixed finite element method for Darcy flow

    Comput. Methods Appl. Mech. Eng.

    (2002)
  • A. Masud et al.

    Application of multi-scale finite element methods to the solution of the Fokker–Planck equation

    Comput. Methods Appl. Mech. Eng.

    (2005)
  • Cited by (28)

    • Probabilistic failure mechanisms via Monte Carlo simulations of complex microstructures

      2022, Computer Methods in Applied Mechanics and Engineering
      Citation Excerpt :

      In numerical optimization using adjoint methods (the adjoint problem is linear, but is running backward in time) resulting in a high computational cost. Consequently, the general natural idea is to use dimension reduction techniques, as proposed in [98]. For reducing the computational costs of the phase-field failure analysis in a probabilistic framework (mainly Bayesian inversion), a non-intrusive global–local approach is recently introduced, rather than using fine-scale high-fidelity finite elements [99].

    View all citing articles on Scopus
    View full text