Link-wise Artificial Compressibility Method: a simple way to deal with complex geometries

Link-wise Artiﬁcial Compressibility Method:
a simple way to deal with complex geometries

Antonio F. Di Rienzo, Pietro Asinari, Eliodoro Chiavazzo

Dipartimento di Energetica, Politecnico di Torino, Torino, Italy,
e-mail: antonio.dirienzo@polito.it

XXIX National Heat Transfer Conference, Torino 20 - 22 June, 2011

Antonio F. Di Rienzo (Politecnico di Torino) Link-wise ACM Torino, 21st June 2011 1 / 22

Outline of this talk

1 Impact on the engineering community

2 The Lattice Boltzmann Method (LBM)

3 Link-wise Artiﬁcial Compressibility Method (LW-ACM)

4 LW-ACM at work


Impact on the engineering community

Outline Compass




4 LW-ACM at work



Unstructured meshing

In engineering computational fluid dynamics (CFD), accurate
meshing is most of the times the main concern. In case of
complex objects, meshing can require up to 90% of the efforts
invested into computations, affecting the whole computer aided
engineering (CAE) process.
Classical finite-volume (FVM) and finite-element (FEM) methods
use unstructured mesh in the attempt to adapt the computational
grid to the real object.
Generating unstructured meshes with high quality is a challenging
computational task by itself, which requires advanced algorithms
(Ruppert’s algorithm, Chew’s second algorithm, Delaunay
triangulation, etc.).
Unfortunately these algorithms imply an additional computational
overhead and, more seriously, may not converge to an acceptable
solution.


Structured meshing: reviving the past!

The old approach based on structured meshing can offer an
alternative, as far as new computational algorithms, like Lattice
Boltzmann Method (LBM) and Immersed Boundary Method (IBM),
and new data structures, like hierarchical block grids and recursive
reﬁnement, can overcome old limitations.
Many examples have been already turned into commercialization:
e.g. PowerFlow by Exa Corporation and XFlow by XFlow-CFD
both based on LBM; Karalit-CFD by Karalit based on IBM
(sponsor of this Congress!).
In this presentation, we focus on the Lattice Boltzmann Method
and we try to simplify it by preserving only its essential features,
namely (a) the artiﬁcial compressibility and (b) the link-wise
formulation taking advantage of the theory of characteristics.


The Lattice Boltzmann Method (LBM)

Outline Compass




4 LW-ACM at work



What is the Lattice Boltzmann Method?
“The lattice Boltzmann method (LBM) is used for the numerical
simulation of physical phenomena and serves as an alternative to
classical solvers of partial differential equations (PDEs)”
[www.lbmethod.org/]. The main unknown is the discrete
distribution function, from which all relevant macroscopic
quantities (satisfying some target PDEs) can be derived.
The operative formula consists of the (a) relaxation process and
the (b) advection process

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
f (x + v, t + 1) − f (x, t) = −ω f (x, t) − fEQ (x, t) , (1)

while the computer implementation consists of the collision and
the streaming step, deﬁned as follows:

f ∗ (x, t + 1) = f (x, t) − ω f (x, t) − fEQ (x, t) ,
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ (2)

f (x + v, t + 1) = f ∗ (x, t + 1).
ˆ ˆ ˆ ˆ (3)


Dealing with complex boundaries (Bouzidi et al., 2001)



Hierarchical block grids for recursive reﬁnement

Courtesy of prof. Manfred Krafczyk, Institute for Computational
Modeling in Civil Engineering, TU Braunschweig (Germany).



Structured mesh at work

Tölke J., Freudiger S.,Krafczyk M., Comp. Fluids, 2006
Courtesy of prof. Manfred Krafczyk, Institute for Computational
Modeling in Civil Engineering, TU Braunschweig (Germany)


LBM limitations and beyond LBM

LBM works with an enlarged set of unknowns, including the
so-called ghost variables (due to the pseudo-kinetic origin of the
method) beyond the ﬂuid-dynamic variables.
Ohwada & Asinari (JCP, 2010) proposed to revive the Artiﬁcial
Compressibility Method (ACM) as a high-order accurate numerical
method for incompressible Navier-Stokes equations, even for
transient simulations (linking the compressibility with the mesh
spacing). This method borrows the main idea from LBM but it
eliminates the ghost variables.
Unfortunately standard ACM relies on standard meshing
techniques for dealing with complex boundaries.


Link-wise Artiﬁcial Compressibility Method (LW-ACM)

Outline Compass




4 LW-ACM at work


Link-wise Artiﬁcial Compressibility Method (LW-ACM)

Link-wise Artiﬁcial Compressibility Method (ACM)
Let us consider the following link-wise formula
2 (e)
fi (x, t + ) = fi (x − vi , t)
ω−1 (e,o) (e,o)
+2 fi (x, t) − fi (x − vi , t) ,
ω
(4)
(e) (e,o)
where fi and fi are local functions of ρ = i fi and
ρu = i vi fi , but they do not depend on the ghost quantities.
The asymptotic analysis of the previous formula, in the limit 1,
proves that the quantities p = p/ 2 and u = u/ satisfy the
¯ ¯
incompressible Navier-Stokes equations, with the viscosity ν
1 1 1
ν= − . (5)
3 ω 2
Accurately solving Navier-Stokes equations requires that
2 /ν 1, as usual in Lattice Boltzmann Method (LBM) as well.

LW-ACM at work

Outline Compass




4 LW-ACM at work


LW-ACM at work

Couette Flow and Couette Flow with Wall Injection


LW-ACM at work

Couette Flow and Couette Flow with Wall Injection

M a ∼ Kn ∼ O(δx) M a ∼ O(1), Kn ∼ O(δx)
δx Ma νLB δx Ma νLB
1e-1 3e-2 3e-2 1e-1 3e-1 3e-3
5e-2 1.5e-2 3e-2 5e-2 3e-1 1.5e-3
2.5e-2 7.5e-3 3e-2 2.5e-2 3e-1 7.5e-4


LW-ACM at work

Complex boundaries: stealing from LBM... :)

Circular Couette Flow

LBM
M. Bouzidi, M. Firdaouss, P. Lallemand, Momentum transfer of
Boltzmann-lattice ﬂuid with boundaries, Phys. Fluid, 2001

LW-ACM at work



LW-ACM at work

Step 1: pre-combining

(e) ω−1 (e,o)
fi∗ (x, t + 2
) = fi (x, t) − 2 fi (x, t). (6)
ω
Step 2: streaming (depending on the distance q from the wall
along that link)

∗∗ 2
2qfi∗ (x, t + 2) + (1 − 2q)fi∗ (x − vi , t + 2 ),
fBB(i) (x, t + )= 1 ∗ 2) 1 ∗ 2 ),
2q fi (x, t + + 1− 2q fBB(i) (x, t +

where BB(i) is the bounce-back operator giving the lattice link
opposite to i-th.
Step 3: post-combining

2 ∗∗ 2 ω−1 (e,o)
fBB(i) (x, t + ) = fBB(i) (x, t + )+2 fBB(i) (x, t). (7)
ω


LW-ACM at work

Circular Couette Flow
M a ∼ Kn ∼ O(δx)
max max
δx Ma νLB νACM = 7e − 6 ⇒ ωACM = 1.99991
5e-2 3e-1 4e-2
2.55e-2 1.5e-1 4e-2 max max
νLBM = 3.5e − 3 ⇒ ωLBM = 1.95
1.25e-2 7.5e-2 4e-2
Extended pratical stability range!


Conclusions

LW-ACM in a nutshell
LW-ACM is purely formulated in terms of hydrodynamic variables:
no longer ghost quantities.
For simple geometries, boundary conditions are imposed as simply
as in CFD (Dirichlet, Neumann BCs).

LW-ACM nothing more than CFD disguised as kinetic model: LBM
link-wise formulation is preserved:
easiness in dealing with complex boundaries: no body-ﬁtting is
required and meshing time demand drastically reduced (!)

LW-ACM is an appealing alternative to classical FVMs and FEMs
as well as LBMs.


Thank you for your attention!


Link-wise Artificial Compressibility Method: a simple way to deal with complex geometries

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