General finite-volume framework for saddle-point problems of various physics

Kirill M. Terekhov

doi:10.1515/rnam-2021-0029

Article

General finite-volume framework for saddle-point problems of various physics

Kirill M. Terekhov

Published/Copyright: December 3, 2021

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From the journal Russian Journal of Numerical Analysis and Mathematical Modelling Volume 36 Issue 6

Abstract

This article is dedicated to the general finite-volume framework used to discretize and solve saddle-point problems of various physics. The framework applies the Ostrogradsky–Gauss theorem to transform a divergent part of the partial differential equation into a surface integral, approximated by the summation of vector fluxes over interfaces. The interface vector fluxes are reconstructed using the harmonic averaging point concept resulting in the unique vector flux even in a heterogeneous anisotropic medium. The vector flux is modified with the consideration of eigenvalues in matrix coefficients at vector unknowns to address both the hyperbolic and saddle-point problems, causing nonphysical oscillations and an inf-sup stability issue. We apply the framework to several problems of various physics, namely incompressible elasticity problem, incompressible Navier–Stokes, Brinkman–Hazen–Dupuit–Darcy, Biot, and Maxwell equations and explain several nuances of the application. Finally, we test the framework on simple analytical solutions.

Keywords: Finite volume method; inf-sup stability; harmonic averaging point; multi-physics; Brinkman–Darcy equations; Navier–Stokes equations; Maxwell equations

MSC 2010: 74S10; 74F10; 76M12; 76D05; 76S05; 78M25

Acknowledgment

The author would like to thank Yuri Vassilevski for valuable comments and corrections to the manuscript.

Funding: The work was supported by the Russian Science Foundation through the grant No. 21-71-20024.

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A Analytical solutions

A.1 Beam under shear

The analytical solution for the displacement is given by [3, 10]:

ux=−34FνExyz,uy=F8E3νzx2−y2−z3uz=F8E3yz2+νyy2−3x2+3F(1+ν)4Ey−y33+Fν4E3x2y−y−3Fνπ3E∑n=1∞(−1)nn3cosh⁡(nπ)cos⁡(nπx)sinh⁡(nπy). (A.1)

The analytical stress is given by

σxx=σyy=σxy=0,σzz=3F4yzσxz=3Fν2π2(1+ν)∑n=1∞(−1)nn2cosh⁡(nπ)sin⁡(nπx)sinh⁡(nπy)σyz=3F81−y2+Fν8(1+ν)3x2−1−3Fν2π2(1+ν)∑n=1∞(−1)nn2cosh⁡(nπ)cos⁡(nπx)cosh⁡(nπy). (A.2)

A.2 Beam under torsion

The solution for the displacement is given by [3, 10]:

ux=−βyz,uy=βxzuz=βxy+32βπ3∑n=1∞(−1)n(2n−1)3cosh(2n−1)π2sin(2n−1)πx2sinh(2n−1)πy2. (A.3)

The stress is given by

σxx=σyy=σzz=σxy=0σxz=8Eβπ2(1+ν)∑n=1∞(−1)n(2n−1)2cosh(2n−1)π2cos(2n−1)πx2sinh(2n−1)πy2σyz=Eβ1+νx+8Eβπ2(1+ν)∑n=1∞(−1)n(2n−1)2cosh(2n−1)π2sin(2n−1)πx2cosh(2n−1)πy2, (A.4)

A.3 Barry–Mercer problem with pulsating source

The analytical solution is given in terms of double summation [4]:

gij=βsin⁡(λix0)sin⁡(λjy0)θλij2+ω2λijsin⁡(ωθt)−ωcos⁡(ωθt)+ω\e−λijθtux=−4∑i=1∞∑j=1∞λiλijgij(t)cos⁡(λix)sin⁡(λjy)uy=−4∑i=1∞∑j=1∞λjλijgij(t)sin⁡(λix)cos⁡(λjy)pf=4E(1−ν)(1+ν)(1−2ν)∑i=1∞∑j=1∞gij(t)sin⁡(λix)sin⁡(λjy) (A.5)

where λ_i := πi, λ_j := πj, λ_ij := λi2+λj2 . The source term is given by

q=βsin⁡(ωθt)δ(x−x0)δ(y−y0) (A.6)

where δ(⋅) is the Dirac function, θ=kE(1−ν)(1+ν)(1−2ν).

A.4 Two-layered Terzaghi column

For ξ := h₁ – x, ξ ∈ [–h₂, h₁] the analytical solution reads as [39]:

gm=2BFωmexp−ωm2c2t/h22(1+βθ)cos⁡(θωm)sin⁡(ωm)+(β+θ)sin⁡(θωm)cos⁡(ωm)pf=∑m=0∞gmcos⁡(ωm)cosθωmξ/h1−βsin⁡(ωm)sinθωmξ/h1,ξ>0cos⁡(ωm)cosωmξ/h2−sin⁡(ωm)sinωmξ/h2,ξ<0ux=Fm1ξ+m2h2,ξ>0m2(ξ+h2),ξ<0−∑m=0∞gmθωmα1m1h1cos⁡(ωm)sinθωmξ/h1+α1m1h1βsin⁡(ωm)cosθωmξ/h1−α1m1h1βsin⁡(ωm)+α2m2h2θsin⁡(ωm),ξ>0α2m2h2θcos⁡(ωm)sinωmξ/h2+α2m2h2θsin⁡(ωm)cosωmξ/h2,ξ<0. (A.7)

The Skempton’s coefficients B_i, the consolidation coefficients c_i, and the confined compressibility coefficients m_i are given by

Bi=αimiMi1+αi2miMi,ci=kiμMi1+αi2miMi,mi=(1+ν)(1−2ν)E(1−ν) (A.8)

where k_i is the permeability, E_i is the Young modulus, ν_i is the Poisson’s ratio, and μ is the fluid viscosity.

Parameter M₂ is chosen so that B₁ = B₂ = B. The expressions for M₂ and coefficients β and θ are

M2=α1m1α2m2M11+α1m1(α1−α2)M1,β=k2k1c1c2,θ=h1h2c2c1. (A.9)

Coefficients ω_m = ω̄_m / (1 + θ) are the positive roots of the equation

cosω¯m−β−1β+1cosθ−1θ+1ω¯m=0. (A.10)

The initial pressure and displacement are given by

pf=BF,ux=Fm1(1−α1B)ξ+m2(1−α2B)h2,ξ>0m2(1−α2B)(ξ+h2),ξ<0. (A.11)

A.5 Ethier–Steinman

The analytical solution for Navier–Stokes equations reads as [7]:

u˙x=−π4eπx/4sinπ4(y+2z)+eπz/4cosπ4(x+2y)e−μπ2t/4u˙y=−π4eπy/4sinπ4(z+2x)+eπx/4cosπ4(y+2z)e−μπ2t/4u˙z=−π4eπz/4sinπ4(x+2y)+eπy/4cosπ4(z+2x)e−μπ2t/4 (A.12)

pf=−π232eπx/2+eπy/2+eπz/2+2sinπ4(x+2y)cosπ4(z+2x)eπ(y+z)/4+2sinπ4(y+2z)cosπ4(x+2y)eπ(z+x)/4+2sinπ4(z+2x)cosπ4(y+2z)eπ(x+y)/4e−μπ2t/2. (A.13)

A.6 Poiseuille flow

The flow in a cylindrical pipe with radius R and length L, base centers x₀ = (x₀, y₀, z₀)^T and x₁ = (x₀, y₀, z₀ + L)^T, and shear rate γ is described by the steady-state Poiseuille solution:

u˙x=u˙y=0,u˙z=γR2−r22R,pf=p0+2μγL−(z−z0)R (A.14)

where r=(x−x0)2+(y−y0)2.

A.7 Wang’s solution for flow over a sphere

The steady-state solution for the incompressible fluid flow in a porous medium over impermeable sphere of radius L, described by the Darcy–Brinkman equations is given in terms of the streamfunction in the spherical polar coordinates [40]:

ψ=sin⁡(θ)22ϰ2r3(ϰr+1)eϰ(1−r)+ϰ2r3−ϰ2−3ϰ−3 (A.15)

where ϰ=Lφ/k with porosity φ and scalar permeability k, and

p=−μcosθ2r2Lϰ2+2ϰ2r3+3ϰ+3,vr=∂θψr2sin⁡(θ),vθ=−∂rψrsin⁡(θ). (A.16)

The spherical polar coordinates r, θ ∈ [0, π], φ ∈ [0, 2π] are obtained from Cartesian ones with

r=x2+y2+z2,tan⁡(θ)=x2+y2z,tan⁡(φ)=yx (A.17)

and the backward conversion is

x=rsin⁡(θ)cos⁡(φ),y=rsin⁡(θ)cos⁡(φ),z=rcos⁡(θ). (A.18)

The velocity in Cartesian coordinates is given by

vx=cos⁡(φ)cos⁡(θ)vθ+cos⁡(φ)sin⁡(θ)vrvy=sin⁡(φ)cos⁡(θ)vθ+sin⁡(φ)sin⁡(θ)vrvz=cos⁡(θ)vr−sin⁡(θ)vθ. (A.19)

A.8 Bounded square cavity

The analytical solution for (1, 1) mode of electric and magnetic fields in a square cavity with side L, filled with an anisotropic material, and bounded by a perfect electric conductor, reads as [15]:

Ex(x,y,t)=−πωLεxxcosπLxsinπLysin⁡(ωt)Ey(x,y,t)=πωLεyysinπLxcosπLysin⁡(ωt)Hz(x,y,t)=cosπLxcosπLycos⁡(ωt)Ez=Hx=Hy=0 (A.20)

with the permittivity, permeability, conductivity, and ω given by

ε=εxxεyyεzz,μ=1,σ=0,ω=πL1εxx+1εyy. (A.21)

Received: 2021-10-29

Accepted: 2021-11-08

Published Online: 2021-12-03

Published in Print: 2021-12-20

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Articles in the same Issue

https://doi.org/10.1515/rnam-2021-0029

Keywords for this article

Finite volume method; inf-sup stability; harmonic averaging point; multi-physics; Brinkman–Darcy equations; Navier–Stokes equations; Maxwell equations