Feel++

Learning Feel++

:leveloffset:+1 = Generic Partial Differential Equations

:leveloffset:+1

The Laplacian

Problem statement

We are interested in this section in the conforming finite element approximation of the following problem:

Laplacian problem

Look for \$u\$ such that

\$\left\{\begin{split} -\Delta u &= f \text{ in } \Omega\\ u &= g \text{ on } \partial \Omega_D\\ \frac{\partial u}{\partial n} &=h \text{ on } \partial \Omega_N\\ \frac{\partial u}{\partial n} + u &=l \text{ on } \partial \Omega_R \end{split}\right.\$
\$\partial \Omega_D\$, \$\partial \Omega_N\$ and \$\partial \Omega_R\$ can be empty sets. In the case \$\partial \Omega_D =\partial \Omega_R = \emptyset\$, then the solution is known up to a constant.

In the implementation presented later, \$\partial \Omega_D =\partial \Omega_N = \partial \Omega_R = \emptyset\$, then we set Dirichlet boundary conditions all over the boundary. The problem then reads like a standard laplacian with inhomogeneous Dirichlet boundary conditions:

Laplacian Problem with inhomogeneous Dirichlet conditions

Look for \$u\$ such that

Inhomogeneous Dirichlet Laplacian problem
\$-\Delta u = f\ \text{ in } \Omega,\quad u = g \text{ on } \partial \Omega\$

Variational formulation

We assume that \$f, h, l \in L^2(\Omega)\$. The weak formulation of the problem then reads:

Laplacian problem variational formulation

Look for \$u \in H^1_{g,\Gamma_D}(\Omega)\$ such that

Variational formulation
\$\displaystyle\int_\Omega \nabla u \cdot \nabla v +\int_{\Gamma_R} u v = \displaystyle \int_\Omega f\ v+ \int_{\Gamma_N} g\ v + \int_{\Gamma_R} l\ v,\quad \forall v \in H^1_{0,\Gamma_D}(\Omega)\$

Conforming Approximation

We now turn to the finite element approximation using Lagrange finite element. We assume \$\Omega\$ to be a segment in 1D, a polygon in 2D or a polyhedron in 3D. We denote \$V_\delta \subset H^1(\Omega)\$ an approximation space such that \$V_{g,\delta} \equiv P^k_{c,\delta}\cap H^1_{g,\Gamma_D}(\Omega)\$.

The weak formulation reads:

Laplacian problem weak formulation

Look for \$u_\delta \in V_\delta \$ such that

\$\displaystyle\int_{\Omega_\delta} \nabla u_{\delta} \cdot \nabla v_\delta +\int_{\Gamma_{R,\delta}} u_\delta\ v_\delta = \displaystyle \int_{\Omega_\delta} f\ v_\delta+ \int_{\Gamma_{N,\delta}} g\ v_\delta + \int_{\Gamma_{R,\delta}} l\ v_\delta,\quad \forall v_\delta \in V_{0,\delta}\$
from now on, we omit \$\delta\$ to lighten the notations. Be careful that it appears both the geometrical and approximation level.

Feel++ Implementation

In Feel++, \$V_{g,\delta}\$ is not built but rather \$P^k_{c,\delta}\$.

The Dirichlet boundary conditions can be treated using different techniques and we use from now on the elimination technique.

We start with the mesh

    auto mesh = loadMesh(_mesh=new Mesh<Simplex<FEELPP_DIM,1>>);
the keyword auto enables type inference, for more details see Wikipedia C++11 page.

Next the discretization setting by first defining Vh=Pch<k>(mesh) \$\equiv P^k_{c,h}\$, then elements of Vh and expressions f, n and g given by command line options or configuration file.

    auto Vh = Pch<2>( mesh );
    auto u = Vh->element("u");
    auto mu = doption(_name="mu");
    auto f = expr( soption(_name="functions.f"), "f" );
    auto r_1 = expr( soption(_name="functions.a"), "a" ); // Robin left hand side expression
    auto r_2 = expr( soption(_name="functions.b"), "b" ); // Robin right hand side expression
    auto n = expr( soption(_name="functions.c"), "c" ); // Neumann expression
    auto g = expr( soption(_name="functions.g"), "g" );
    auto v = Vh->element( g, "g" );

at the following line

    auto v = Vh->element( g, "g" );

v is set to the expression g, which means more precisely that v is the interpolant of g in Vh.

the variational formulation is implemented below, we define the bilinear form a and linear form l and we set strongly the Dirichlet boundary conditions with the keyword on using elimination. If we don’t find Dirichlet, Neumann or Robin in the list of physical markers in the mesh data structure then we impose Dirichlet boundary conditions all over the boundary.

    auto l = form1( _test=Vh );
    l = integrate(_range=elements(mesh),
                  _expr=f*id(v));
    l+=integrate(_range=markedfaces(mesh,"Robin"), _expr=r_2*id(v));
    l+=integrate(_range=markedfaces(mesh,"Neumann"), _expr=n*id(v));
    toc("l");

    tic();
    auto a = form2( _trial=Vh, _test=Vh);
    a = integrate(_range=elements(mesh),
                  _expr=mu*gradt(u)*trans(grad(v)) );
    a+=integrate(_range=markedfaces(mesh,"Robin"), _expr=r_1*idt(u)*id(v));
    a+=on(_range=markedfaces(mesh,"Dirichlet"), _rhs=l, _element=u, _expr=g );
    //! if no markers Robin Neumann or Dirichlet are present in the mesh then
    //! impose Dirichlet boundary conditions over the entire boundary
    if ( !mesh->hasAnyMarker({"Robin", "Neumann","Dirichlet"}) )
        a+=on(_range=boundaryfaces(mesh), _rhs=l, _element=u, _expr=g );
    toc("a");

    tic();
    //! solve the linear system, find u s.t. a(u,v)=l(v) for all v
    if ( !boption( "no-solve" ) )
        a.solve(_rhs=l,_solution=u);
    toc("a.solve");

    cout << "||u_h-g||_L2=" << normL2(_range=elements(mesh), _expr=idv(u)-g) << std::endl;

    tic();
    auto e = exporter( _mesh=mesh );
    e->addRegions();
    e->add( "u", u );
    e->add( "g", v );
    e->save();

    toc("Exporter");
    return 0;

}

We have the following correspondance:

Element sets Domain

elements(mesh)

\$\Omega\$

boundaryfaces(mesh)

\$\partial \Omega\$

markedfaces(mesh,"Dirichlet")

\$\Gamma_D\$

markedfaces(mesh,"Neumann")

\$\Gamma_R\$

markedfaces(mesh,"Robin")

\$\Gamma_R\$

next we solve the algebraic problem

Listing: solve algebraic system
    //! solve the linear system, find u s.t. a(u,v)=l(v) for all v
    if ( !boption( "no-solve" ) )
        a.solve(_rhs=l,_solution=u);

next we compute the \$L^2\$ norm of \$u_\delta-g\$, it could serve as an \$L^2\$ error if \$g\$ was manufactured to be the exact solution of the Laplacian problem.

    cout << "||u_h-g||_L2=" << normL2(_range=elements(mesh), _expr=idv(u)-g) << std::endl;

and finally we export the results, by default it is in the ensight gold format and the files can be read with Paraview and Ensight. We save both \$u\$ and \$g\$.

Listing: export Laplacian results
    auto e = exporter( _mesh=mesh );
    e->addRegions();
    e->add( "u", u );
    e->add( "g", v );
    e->save();

Testcases

The Feel++ Implementation comes with testcases in 2D and 3D.

circle

circle is a 2D testcase where \$\Omega\$ is a disk whose boundary has been split such that \$\partial \Omega=\partial \Omega_D \cup \partial \Omega_N \cup \partial \Omega_R\$.

Here are some results we can observe after use the following command

cd Testcases/quickstart/circle
mpirun -np 4 /usr/local/bin/feelpp_qs_laplacian_2d --config-file circle.cfg

This give us some data such as solution of our problem or the mesh used in the application.

ucircle

meshCircle

Solution \$u_\delta\$

Mesh

feelpp2d and feelpp3d

This testcase solves the Laplacian problem in \$\Omega\$ an quadrangle or hexadra containing the letters of Feel++

feelpp2d

After running the following command

cd Testcases/quickstart/feelpp2d
mpirun -np 4 /usr/local/bin/feelpp_qs_laplacian_2d --config-file feelpp2d.cfg

we obtain the result \$u_\delta\$ and also the mesh

ufeelpp2d

/images/Laplacian/TestCases/Feelpp2d/meshfeelpp2d.png[]

Solution \$u_\delta\$

Mesh

feelpp3d

We can launch this application with the current line

cd Testcases/quickstart/feelpp3d
mpirun -np 4 /usr/local/bin/feelpp_qs_laplacian_3d --config-file feelpp3d.cfg

When it’s finish, we can extract some informations

ufeelpp3d

meshfeelpp3d

Solution \$u_\delta\$

Mesh

Levelset

Having the possibility to determine where two regions meeting can be really useful in some scientific domains. That’s why the levelset method is born.

Levelset introduction

Levelset function

By using a scalar function \phi, define on all regions as the null value is obtained when it’s placed on an interface of two domains.

We denote \Omega_1 and \Omega_2 two domains with \Gamma the interface betwen them. Then \phi can be define as

\phi(\boldsymbol{x}) = \left\{ \begin{array}{cccc} \text{dist}(\boldsymbol{x}, \Gamma), & \boldsymbol{x}& \in &\Omega_1 \\ 0, & \boldsymbol{x}& \in &\Gamma\\ -\text{dist}(\boldsymbol{x}, \Gamma), & \boldsymbol{x}& \in & \Omega_2 \end{array} \right.

with \text{dist}(\boldsymbol{x}, \Gamma ) = \underset{\boldsymbol{y} \; \in \; \Gamma}{\min}( |\boldsymbol{x} - \boldsymbol{y}| ).

This function \phi had also the following property |\nabla\phi|=1.

Moreover, the unit normal vector \boldsymbol{n} outgoing from the interface and the curvature \mathcal{\kappa} can be obtained from the levelset function.

\boldsymbol{n}=\frac{\nabla\phi}{|\nabla\phi} \\ \mathcal{\kappa}=\nabla \cdot \boldsymbol{n}= \nabla \cdot \frac{\nabla\phi}{|\nabla\phi|}

Now we have exposed the levelset function, we need to define how the levelset will evolve and will spread into all the space. To do this, we use the following advection equation :

\partial_t\phi+\boldsymbol{u}\cdot\nabla\phi=0

where \boldsymbol{u} is an incompressible velocity field.

Heaviside and Dirac functions

We define also the regularized Heaviside function H_\varepsilon(\phi) = \left\{ \begin{array}{cc} 0, & \phi \leq - \varepsilon,\\ \dfrac{1}{2} \left(1+\dfrac{\phi}{\varepsilon}+\dfrac{\sin\left(\dfrac{\pi \phi}{\varepsilon}\right)}{\pi}\right), & -\varepsilon \leq \phi \leq \varepsilon, \\ 1, & \phi \geq \varepsilon \end{array} \right.

and the regularized Dirac function \delta_\varepsilon(\phi) = \left\{ \begin{array}{cc} 0, & \phi \leq - \varepsilon,\\ \displaystyle\dfrac{1}{2 \varepsilon} \left(1+\cos\left(\dfrac{\pi \phi}{\varepsilon}\right)\right), & -\varepsilon \leq \phi \leq \varepsilon, \\ 0, & \phi \geq \varepsilon. \end{array} \right.

The first one gives a different value to each side of the interface ( here 0 in and 1 out ). The second one allow us to define quantities, with value different from 0 at the interface. A typical value of \varepsilon in literature is 1.5h where h is the mesh step size.

It should be noted that these functions allow us to determine respectively the volume and the surface of the interface by V^+_{\varepsilon} = \int_{\Omega} H_\varepsilon \\ S^{\Gamma}_{\varepsilon} = \int_{\Omega} \delta_\varepsilon

Solid rotation of a slotted disk

We describe the benchmark proposed by Zalesak.

Computer codes, used for the acquisition of results, are from Vincent Doyeux.

Problem Description

In order to test our interface propagation method, i.e. the levelset method \phi, we will study the rotation of a slotted disk into a square domain. The geometry can be represented as

Slotted Disk Geometry
Figure 1 : Initial Geometry.

We denote \Omega, the square domain [0,1]\times[0,1]. The center of the slotted disk is placed at (0.5,0.75).
To model the rotation, we will apply an angular velocity, centered in (0.5,0.5), as the disk is back to its initial position after t_f=628.

During this test, we observe three different errors to measure the quality of our method. With these values, two kinds of convergence will be studied : the time convergence, with different time step on an imposed grid and the space one, where the space discretization and the time step are linked by a relation. Several stabilization methods are used such as CIP ( Continuous Interior Penalty ) or SUPG ( Streamline-Upwing/Petrov-Galerkin ).

Boundary conditions

We set a Neumann boundary condition on the boundary of the domain.

Initial conditions

The velocity is imposed as \boldsymbol{u}=\left( \frac{\pi}{314} (50-y),\frac{\pi}{314} (x-50) \right)

Here is the velocity look in the square domain

Velocity Geometry
Figure 2 : Imposed velocity in \Omega.

Inputs

The following table displays the various fixed and variables parameters of this test-case.

Table 1. Fixed and Variable Input Parameters

Name

Description

Nominal Value

Units

r

disk radius

0.15

m

l

slot base

0.05

m

h

slot height

0.25

m

t_f

slotted disk rotation period

628

s

Outputs

We observe during this benchmark three different errors.
First at all, the mass error, define by e_{\text{m}} = \frac{ \left| m_{\phi_f} - m_{\phi_0}\right| }{m_{\phi_0}} = \frac{ \left| \displaystyle \int_{\Omega} \chi( \phi_f < 0 ) - \displaystyle \int_{\Omega} \chi( \phi_0 < 0 ) \right| }{ \displaystyle \int_{\Omega} \chi( \phi_0 < 0 )}

where \chi is the characteristic function. \chi( f( \phi ) ) = \left\{ \begin{array}{rcl} 1 & \text{ if } & f( \phi ) \neq 0 \\ 0 & \text{ if } & f( \phi ) = 0 \end{array} \right.

However, mass is gain and loose at different emplacements on the mesh, and at the same time, with the level set method.

Secondly, the sign change error e_{\text{sc}} = \sqrt{ \int_\Omega \left( (1-H_0) - (1-H_f) \right)^2 }

with H_0=H_\epsilon(\phi_0) and H_f=H_\epsilon(\phi_f), H_\epsilon the smoothed Heaviside function of thickness 2ε.

This error is better to define the interface displacement. In fact, we can determine where \phi_0\phi_f<0, in other words where the interface has moved.

Finally, we define the classical L^2 error at the interface, as e_{L^2} = \sqrt{ \frac{1}{\displaystyle \int_\Omega \chi( \delta(\phi_0) > 0 ) } \int_\Omega (\phi_0 - \phi_f)^2 \chi( \delta(\phi_0) > 0 ) }.

Discretization

Time convergence

For this case, we set a fixed grid with mesh step size h=0.04, and so 72314 degree of freedom on a \mathbb{P}^1.

Then, after the disk made one round, we measure the errors obtained from two different discretizations ( BDF2 and Euler ) and compared them.

We repeat this with several time step dt\in \{2.14, 1, 0.5, 0.25, 0.20\}.

Only one stabilization method is used : SUPG

Space convergence

We define the following relation, between time step and mesh step size : dt=C\frac{h}{U_{max}}

where C<1 constant and U_{max} the maximum velocity of \Omega.

From the definition of our velocity, U_{max} is reached at the farthest point from the center of \Omega. In this case, we have U_{max}=0.007, and we set C=0.8.

We use the BDF2 method for time discretization. As in time convergence, we wait one round of the disk to measure the errors and we repeat this test for these values of h: 0.32, 0.16, 0.08, 0.04.

We compare the results from different stabilization methods : CIP, SUPG, GLS ( Galerkin-Least-Squares ) and SGS ( Sub-Grid Scale ).

Implementation

Results

Time convergence

Table 2. Time convergence with Euler scheme

dt

e_{L^2}

e_{sc}

e_m

2.14

0.0348851

0.202025

0.202025

1.00

0.0187567

0.147635

0.147635

0.5

0.0098661

0.10847

0.10847

0.25

0.008791

0.0782569

0.0782569

0.20

0.00803373

0.0670677

0.0670677

Table 3. Time convergence with BDF2 scheme

dt

e_{L^2}

e_{sc}

e_m

2.14

0.0118025

0.0906617

0.0492775

1.00

0.00436957

0.0445275

0.0163494

0.5

0.00173637

0.0216359

0.0100621

0.25

0.001003

0.0125971

0.00354644

0.20

0.000949343

0.0117449

0.00317368

Time mass error
Figure 2 : Mass error of Zalesac benchmark
Time sign change
Figure 3 : Sign change error of Zalesac benchmark
Time L2 error
Figure 4 : L^2 error of Zalesac benchmark
Time shape
Figure 5 : Slotted disk shape after a round

Space convergence

stab

h

e_{L^2}

e_{sc}

e_m

CIP

0.32

0.0074

0.072

0.00029

0.16

0.0046

0.055

0.00202

0.08

0.0025

0.033

0.00049

0.04

0.0023

0.020

0.00110

SUPG

0.32

0.012

0.065

0.01632

0.16

0.008

0.049

0.07052

0.08

0.004

0.030

0.00073

0.04

0.001

0.018

0.00831

GLS

0.32

0.013

0.066

0.02499

0.16

0.008

0.051

0.05180

0.08

0.004

0.031

0.00805

0.04

0.001

0.019

0.00672

SGS

0.32

0.012

0.065

0.01103

0.16

0.008

0.050

0.07570

0.08

0.004

0.030

0.00084

0.04

0.001

0.018

0.00850

Space sign change
Figure 6 : Sign change error of Zalesac benchmark
Space L2 error
Figure 7 : L^2 error of Zalesac benchmark
Space Shape
Figure 8 : Slotted disk shape after a round

Conclusion

Let’s begin with time convergence results. Tables shows us that sign change error is better to define the quality of the chosen scheme than the mass error. In fact, the loss of mass somewhere can be nullify by a gain of mass elsewhere. Sign change error shows half an order gain from Euler scheme to BDF2 one, as L^2 errors show us a gain of one order. For the slotted disk shape, BDF2 uses the two previous iterations to obtain the current result, while Euler only need the previous iteration. This explain why we can see an asymmetrical tendency in the first one.

As for space convergence, the different stabilization methods we used give us the same convergence rate equals to 0.6, with close error values, for the sign change error. For the L^2 error case, it’s not as evident as the previous one. Aside the CIP stabilization method with a 0.6 convergence rate, the others show us a convergence rate of 0.9.

Bibliography

References for this benchmark
  • [Zalesak] Steven T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids, Journal of Computational Physics, 1979.

  • [Doyeux] Vincent Doyeux, Modelisation et simulation de systemes multi-fluides. Application aux ecoulements sanguins., Physique Numérique [physics.comp-ph], Université de Grenoble, 2014

:leveloffset:-1

Computational Fluid Dynamics Benchmarks

Turek & Hron CFD Benchmark

Introduction

We implement the benchmark proposed by Turek and Hron, on the behavior of drag and lift forces of a flow around an object composed by a pole and a bar, see Figure Initial Geometry..

The software and the numerical results were initially obtained from Vincent Chabannes.

This benchmark is linked to the Turek Hron CSM and Turek Hron FSI benchmarks.

Problem Description

We consider a 2D model representative of a laminar incompressible flow around an obstacle. The flow domain, named \$\Omega_f\$, is contained into the rectangle \$ \lbrack 0,2.5 \rbrack \times \lbrack 0,0.41 \rbrack \$. It is characterised, in particular, by its dynamic viscosity \$\mu_f\$ and by its density \$\rho_f\$. In this case, the fluid material we used is glycerine.

TurekHron Geometry
Figure 1. Geometry of the Turek & Hron CFD Benchmark

In order to describe the flow, the incompressible Navier-Stokes model is chosen for this case, define by the conservation of momentum equation and the conservation of mass equation. At them, we add the material constitutive equation, that help us to define \$\boldsymbol{\sigma}_f\$

The goal of this benchmark is to study the behavior of lift forces \$F_L\$ and drag forces \$F_D\$, with three different fluid dynamics applied on the obstacle, i.e on \$\Gamma_{obst}\$, we made rigid by setting specific structure parameters. To simulate these cases, different mean inflow velocities, and thus different Reynolds numbers, will be used.

Boundary conditions

We set

  • on \$\Gamma_{in}\$, an inflow Dirichlet condition : \$ \boldsymbol{u}_f=(v_{in},0) \$

  • on \$\Gamma_{wall}\$ and \$\Gamma_{obst}\$, a homogeneous Dirichlet condition : \$ \boldsymbol{u}_f=\boldsymbol{0} \$

  • on \$\Gamma_{out}\$, a Neumann condition : \$ \boldsymbol{\sigma}_f\boldsymbol{ n }_f=\boldsymbol{0} \$

Initial conditions

We use a parabolic velocity profile, in order to describe the flow inlet by \$ \Gamma_{in} \$, which can be express by

v_{cst} = 1.5 \bar{U} \frac{4}{0.1681} y \left(0.41-y\right)

where \$\bar{U}\$ is the mean inflow velocity.

However, we want to impose a progressive increase of this velocity profile. That’s why we define

v_{in} =
\left\{
\begin{aligned}
 & v_{cst} \frac{1-\cos\left( \frac{\pi}{2} t \right) }{2}  \quad & \text{ if } t < 2 \\
 & v_{cst}  \quad & \text{ otherwise }
\end{aligned}
\right.

With t the time.

Moreover, in this case, there is no source term, so \$f_f\equiv 0\$.

Inputs

The following table displays the various fixed and variables parameters of this test-case.

Table 4. Fixed and Variable Input Parameters
Name Description Nominal Value Units

\$l\$

elastic structure length

\$0.35\$

\$m\$

\$h\$

elastic structure height

\$0.02\$

\$m\$

\$r\$

cylinder radius

\$0.05\$

\$m\$

\$C\$

cylinder center coordinates

\$(0.2,0.2)\$

\$m\$

\$\nu_f\$

kinematic viscosity

\$1\times 10^{-3}\$

\$m^2/s\$

\$\mu_f\$

dynamic viscosity

\$1\$

\$kg/(m \times s)\$

\$\rho_f\$

density

\$1000\$

\$kg/m^3\$

\$f_f\$

source term

0

\$kg/(m^3 \times s)\$

\$\bar{U}\$

characteristic inflow velocity

CFD1 CFD2 CFD3

\$0.2\$

\$1\$

\$2\$

\$m/s\$

Outputs

As defined above, the goal of this benchmark is to measure the drag and lift forces, \$F_D\$ and \$F_L\$, to control the fluid solver behavior. They can be obtain from

(F_D,F_L)=\int_{\Gamma_{obst}}\boldsymbol{\sigma}_f \boldsymbol{ n }_f

where \$\boldsymbol{n}_f\$ the outer unit normal vector from \$\partial \Omega_f\$.

Discretization

To realize these tests, we made the choice to used \$P_N\$-\$P_{N-1}\$ Taylor-Hood finite elements, described by Chabannes, to discretize space. With the time discretization, we use BDF, for Backward Differentation Formulation, schemes at different orders \$q\$.

Solvers

Here are the different solvers ( linear and non-linear ) used during results acquisition.

Table 5. KSP configuration

type

gmres

relative tolerance

1e-13

max iteration

1000

reuse preconditioner

false

Table 6. SNES configuration

relative tolerance

1e-8

steps tolerance

1e-8

max iteration

CFD1/CFD2 : 100 | CFD3 : 50

max iteration with reuse

CFD1/CFD2 : 100 | CFD3 : 50

reuse jacobian

false

reuse jacobian rebuild at first Newton step

true

Table 7. KSP in SNES configuration

relative tolerance

1e-5

max iteration

1000

max iteration with reuse

CFD1/CFD2 : 100 | CFD3 : 1000

reuse preconditioner

false

reuse preconditioner rebuild at first Newton step

false

Table 8. Preconditioner configuration

type

lu

package

mumps

Running the model

The configuration files are in toolboxes/fluid/TurekHron. The different cases are implemented in the corresponding .cfg files e.g. cfd1.cfg, cfd2.cfg and cfd3.cfg.

The command line in feelpp-toolboxes docker reads

Command line to execute CFD1 testcase
$ mpirun -np 4 /usr/local/bin/feelpp_toolbox_fluid_2d --config-file cfd1.cfg

The result files are then stored by default in

Results Directory
feel/applications/models/fluid/TurekHron/"case_name"/"velocity_space""pression_space""Geometric_order"/"processor_used"

For example, for CFD2 case executed on \$12\$ processors, with a \$P_2\$ velocity approximation space, a \$P_1\$ pressure approximation space and a geometric order of \$1\$, the path is

feel/toolboxes/fluid/TurekHron/cfd2/P2P1G1/np_12

Results

Here are results from the different cases studied in this benchmark.

CFD1

Table 9. Results for CFD1
\$\mathbf{N_{geo}}\$ \$\mathbf{N_{elt}}\$ \$\mathbf{N_{dof}}\$ Drag Lift

Reference Turek and Hron

14.29

1.119

1

9874

45533 (\$P_2/P_1\$)

14.217

1.116

1

38094

173608 (\$P_2/P_1\$)

14.253

1.120

1

59586

270867 (\$P_2/P_1\$)

14.262

1.119

2

7026

78758 (\$P_3/P_2\$)

14.263

1.121

2

59650

660518 (\$P_3/P_2\$)

14.278

1.119

3

7026

146057 (\$P_4/P_3\$)

14.270

1.120

3

59650

1228831 (\$P_4/P_3\$)

14.280

1.119

All the files used for this case can be found in this rep [geo file, config file, json file]

CFD2

Table 10. Results for CFD2
\$\mathbf{N_{geo}}\$ \$\mathbf{N_{elt}}\$ \$\mathbf{N_{dof}}\$ Drag Lift

Reference Turek and Hron

136.7

10.53

1

7020

32510 (\$P_2/P_1\$)

135.33

10.364

1

38094

173608 (\$P_2/P_1\$)

136.39

10.537

1

59586

270867 (\$P_2/P_1\$)

136.49

10.531

2

7026

78758 (\$P_3/P_2\$)

136.67

10.548

2

59650

660518 (\$P_3/P_2\$)

136.66

10.532

3

7026

146057 (\$P_4/P_3\$)

136.65

10.539

3

59650

1228831 (\$P_4/P_3\$)

136.66

10.533

All the files used for this case can be found in this rep [geo file, config file, json file]

CFD3

As CFD3 is time-dependent ( from BDF use ), results will be expressed as

mean ± amplitude [frequency]

where

  • mean is the average of the min and max values at the last period of oscillations.

mean=\frac{1}{2}(max+min)

  • amplitude is the difference of the max and the min at the last oscillation.

amplitude=\frac{1}{2}(max-min)

  • frequency can be obtain by Fourier analysis on periodic data and retrieve the lowest frequency or by the following formula, if we know the period time T.

frequency=\frac{1}{T}

Table 11. Results for CFD3
\$\mathbf{\Delta t}\$ \$\mathbf{N_{geo}}\$ \$\mathbf{N_{elt}}\$ \$\mathbf{N_{dof}}\$ \$\mathbf{N_{bdf}}\$ Drag Lift

0.005

Reference Turek and Hron

439.45 ± 5.6183[4.3956]

−11.893 ± 437.81[4.3956]

0.01

1

8042

37514 (\$P_2/P_1\$)

2

437.47 ± 5.3750[4.3457]

-9.786 ± 437.54[4.3457]

2

2334

26706 (\$P_3/P_2\$)

2

439.27 ± 5.1620[4.3457]

-8.887 ± 429.06[4.3457]

2

7970

89790 (\$P_2/P_2\$)

2

439.56 ± 5.2335[4.3457]

-11.719 ± 425.81[4.3457]

0.005

1

3509

39843\$(P_3/P_2)\$

2

438.24 ± 5.5375[4.3945]

-11.024 ± 433.90[4.3945]

1

8042

90582 (\$P_3/P_2\$)

2

439.25 ± 5.6130[4.3945]

-10.988 ± 437.70[4.3945]

2

2334

26706 (\$P_3/P_2\$)

2

439.49 ± 5.5985[4.3945]

-10.534 ± 441.02[4.3945]

2

7970

89790 (\$P_3/P_2\$)

2

439.71 ± 5.6410[4.3945]

-11.375 ± 438.37[4.3945]

3

3499

73440 (\$P_4/P_3\$)

3

439.93 ± 5.8072[4.3945]

-14.511 ± 440.96[4.3945]

4

2314

78168 (\$P_5/P_4\$)

2

439.66 ± 5.6412[4.3945]

-11.329 ± 438.93[4.3945]

0.002

2

7942

89482 (\$P_3/P_2)\$

2

439.81 ± 5.7370[4.3945]

-13.730 ± 439.30[4.3945]

3

2340

49389 (\$P_4/P_3\$)

2

440.03 ± 5.7321[4.3945]

-13.250 ± 439.64[4.3945]

3

2334

49266 (\$P_4/P_3\$)

3

440.06 ± 5.7773[4.3945]

-14.092 ± 440.07[4.3945]

All the files used for this case can be found in this rep [geo file, config file, json file].

TurekHron CFD3 results
Figure 2. Lift and drag forces

Geometrical Order

Add a section on geometrical order.

Conclusion

The reference results, Turek and Hron, have been obtained with a time step \$\Delta t=0.05\$. When we compare our results, with the same step and \$\mathrm{BDF}_2\$, we observe that they are in accordance with the reference results.

With a larger \$\Delta t\$, a discrepancy is observed, in particular for the drag force. It can also be seen at the same time step, with a higher order \$\mathrm{BDF}_n\$ ( e.g. \$\mathrm{BDF}_3\$ ). This suggests that the couple \$\Delta t=0.05\$ and \$\mathrm{BDF}_2\$ isn’t enough accurate.

Bibliography

References for this benchmark
  • [TurekHron] S. Turek and J. Hron, Proposal for numerical benchmarking of fluid-structure interaction between an elastic object and laminar incompressible flow, Lecture Notes in Computational Science and Engineering, 2006.

  • [Chabannes] Vincent Chabannes, Vers la simulation numérique des écoulements sanguins, Équations aux dérivées partielles [math.AP], Universitée de Grenoble, 2013.

Unresolved directive in CFD/README.adoc - include::MultiFluid/README.adoc[]

2D Drops Benchmark

This benchmark has been proposed and realised by Hysing. It allows us to verify our level set code, our Navier-Stokes solver and how they couple together.

Computer codes, used for the acquisition of results, are from Vincent Doyeux, with the use of Chabannes's Navier-Stokes code.

Problem Description

We want to simulate the rising of a 2D bubble in a Newtonian fluid. The bubble, made of a specific fluid, is placed into a second one, with a higher density. Like this, the bubble, due to its lowest density and by the action of gravity, rises.

The equations used to define fluid bubble rising in an other are the Navier-Stokes for the fluid and the advection one for the level set method. As for the bubble rising, two forces are defined :

  • The gravity force : \$\boldsymbol{f}_g=\rho_\phi\boldsymbol{g}\$

  • The surface tension force : \$\boldsymbol{f}_{st}=\int_\Gamma\sigma\kappa\boldsymbol{ n } \$

We denote \$ \Omega\times\rbrack0,3\rbrack \$ the interest domain with \$ \Omega=(0,1)\times(0,2) \$. \$\Omega\$ can be decompose into \$\Omega_1\$, the domain outside the bubble and \$\Omega_2\$ the domain inside the bubble and \$\Gamma\$ the interface between these two.

2D Bubble Geo
Figure 3. Geometry used in 2D Bubble Benchmark

Durig this benchmark, we will study two different cases : the first one with a ellipsoidal bubble and the second one with a squirted bubble.

Boundary conditions

  • On the lateral walls, we imposed slip conditions

\$\begin{eqnarray} \boldsymbol{u}\cdot\boldsymbol{n}&=&0 \\ t\cdot(\nabla\boldsymbol{u}+^t\nabla\boldsymbol{u})\cdot \boldsymbol{n}&=&0 \end{eqnarray}\$
  • On the horizontal walls, no slip conditions are imposed : \$\boldsymbol{u}=0 \$

Initial conditions

In order to let the bubble rise, its density must be inferior to the density of the exterior fluid, so \$\rho_1>\rho_2\$

Inputs

The following table displays the various fixed and variables parameters of this test-case.

Table 12. Fixed and Variable Input Parameters
Name Description Nominal Value Units

\$\boldsymbol{g}\$

gravity acceleration

\$(0,0.98)\$

\$m/s^2\$

\$l\$

length domain

\$1\$

\$m\$

\$h\$

height domain

\$2\$

\$m\$

\$r\$

bubble radius

\$0.25\$

\$m\$

\$B_c\$

bubble center

\$(0.5,0.5)\$

\$m\$

Outputs

In the first place, the quantities we want to measure are \$X_c\$ the position of the center of the mass of the bubble, the velocity of the center of the mass \$U_c\$ and the circularity \$c\$, define as the ratio between the perimeter of a circle and the perimeter of the bubble. They can be expressed by

\$\boldsymbol{X}_c = \dfrac{ \displaystyle \int_{\Omega_2} \boldsymbol{x}}{ \displaystyle \int_{\Omega_2} 1 } = \dfrac{ \displaystyle \int_\Omega \boldsymbol{x} (1-H_\varepsilon(\phi))}{ \displaystyle \int_\Omega (1-H_\varepsilon(\phi)) }\$
\$\boldsymbol{U}_c = \dfrac{\displaystyle \int_{\Omega_2} \boldsymbol{u}}{ \displaystyle \int_{\Omega_2} 1 } = \dfrac{\displaystyle \int_\Omega \boldsymbol{u} (1-H_\varepsilon(\phi))}{ \displaystyle \int_\Omega (1-H_\varepsilon(\phi)) }\$
\$c = \dfrac{\left(4 \pi \displaystyle \int_{\Omega_2} 1 \right)^{\frac{1}{2}}}{ \displaystyle \int_{\Gamma} 1} = \dfrac{ \left(4 \pi \displaystyle \int_{\Omega} (1 - H_\varepsilon(\phi)) \right) ^{\frac{1}{2}}}{ \displaystyle \int_{\Omega} \delta_\varepsilon(\phi)}\$

After that, we interest us to quantitative points for comparison as \$c_{min}\$, the minimum of the circularity and \$t_{c_{min}}\$, the time needed to obtain this minimum, \$u_{c_{max}}\$ and \$t_{u_{c_{max}}}\$ the maximum velocity and the time to attain it, or \$y_c(t=3)\$ the position of the bubble at the final time step. We add a second maximum velocity \$u_{max}\$ and \$u_{c_{max_2}}\$ and its time \$t_{u_{c_{max_2}}}\$ for the second test on the squirted bubble.

Discretization

This is the parameters associate to the two cases, which interest us here.

Case

\$\rho_1\$

\$\rho_2\$

\$\mu_1\$

\$\mu_2\$

\$\sigma\$

Re

\$E_0\$

ellipsoidal bubble (1)

1000

100

10

1

24.5

35

10

squirted bubble (2)

1000

1

10

0.1

1.96

35

125

Implementation

Results

Test 1

Test 2

We describe the different quantitative results for the two studied cases.

Table 13. Results comparison between benchmark values and our results for the ellipsoidal case

h

\$c_{min}\$

\$t_{c_{min}}\$

\$u_{c_{max}}\$

\$t_{u_{c_{max}}}\$

\$y_c(t=3)\$

lower bound

0.9011

1.8750

0.2417

0.9213

1.0799

upper bound

0.9013

1.9041

0.2421

0.9313

1.0817

0.02

0.8981

1.925

0.2400

0.9280

1.0787

0.01

0.8999

1.9

0.2410

0.9252

1.0812

0.00875

0.89998

1.9

0.2410

0.9259

1.0814

0.0075

0.9001

1.9

0.2412

0.9251

1.0812

0.00625

0.8981

1.9

0.2412

0.9248

1.0815

Table 14. Results comparison between benchmark values and our results for the squirted case

h

\$c_{min}\$

\$t_{c_{min}}\$

\$u_{c_{max_1}}\$

\$t_{u_{c_{max_1}}}\$

\$u_{c_{max_2}}\$

\$t_{u_{c_{max_2}}}\$

\$y_c(t=3)\$

lower bound

0.4647

2.4004

0.2502

0.7281

0.2393

1.9844

1.1249

upper bound

0.5869

3.0000

0.2524

0.7332

0.2440

2.0705

1.1380

0.02

0.4744

2.995

0.2464

0.7529

0.2207

1.8319

1.0810

0.01

0.4642

2.995

0.2493

0.7559

0.2315

1.8522

1.1012

0.00875

0.4629

2.995

0.2494

0.7565

0.2324

1.8622

1.1047

0.0075

0.4646

2.995

0.2495

0.7574

0.2333

1.8739

1.1111

0.00625

0.4616

2.995

0.2496

0.7574

0.2341

1.8828

1.1186

Conclusion

Bibliography

References for this benchmark
  • [Hysing] S. Hysing, S. Turek, D. Kuzmin, N. Parolini, E. Burman, S. Ganesan, and L. Tobiska, Quantitative benchmark computations of two-dimensional bubble dynamics, International Journal for Numerical Methods in Fluids, 2009.

  • [Chabannes] V. Chabannes, Vers la simulation numérique des écoulements sanguins, Équations aux dérivées partielles. PhD thesis, Université de Grenoble, 2013.

  • [Doyeux] V. Doyeux, Modélisation et simulation de systèmes multi-fluides, Application aux écoulements sanguins, PhD thesis, Université de Grenoble, 2014.

3D Drop benchmark

The previous section described the strategy we used to track the interface. We couple it now to the Navier Stokes equation solver described in \cite{chabannes11:_high}. In the current section, we present a 3D extension of the 2D benchmark introduced in \cite{Hysing2009} and realised using Feel++ in \cite{Doyeux2012}.

Benchmark problem

The benchmark objective is to simulate the rise of a 3D bubble in a Newtonian fluid. The equations solved are the incompressible Navier Stokes equations for the fluid and the advection for the level set:

\$\begin{array}[lll] \rho\rho(\phi(\mathbf{x}) ) \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) + \nabla p - \nabla \cdot \left( \nu(\phi(\mathbf{x})) (\nabla \mathbf{u} + (\nabla \mathbf{u})^T) \right) &=& \rho ( \phi(\mathbf{x}) ) \mathbf{g}, \\ \nabla \cdot \mathbf{u} &=& 0, \\ \frac{\partial \phi}{\partial t} + \mathbf{u} \cdot \nabla \phi &=& 0, \end{array}\$

where \$\rho\$ is the density of the fluid, \$\nu\$ its viscosity, and \$\mathbf{g} \approx (0, 0.98)^T\$ is the gravity acceleration.

The computational domain is \$\Omega \times \rbrack0, T\rbrack \$ where \$\Omega\$ is a cylinder which has a radius \$R\$ and a heigth \$H\$ so that \$R=0.5\$ and \$H=2\$ and \$T=3\$. We denote \$\Omega_1\$ the domain outside the bubble \$ \Omega_1= \{\mathbf{x} | \phi(\mathbf{x})>0 \} \$, \$\Omega_2\$ the domain inside the bubble \$ \Omega_2 = \{\mathbf{x} | \phi(\mathbf{x})<0 \} stem:[ and stem:[\Gamma\$ the interface \$ \Gamma = \{\mathbf{x} | \phi(\mathbf{x})=0 \} \$. On the lateral walls and on the bottom walls, no-slip boundary conditions are imposed, i.e. \$\mathbf{u} = 0\$ and \$\mathbf{t} \cdot (\nabla \mathbf{u} + (\nabla \mathbf{u})^T) \cdot \mathbf{n}=0\$ where \$\mathbf{n}\$ is the unit normal to the interface and \$\mathbf{t}\$ the unit tangent. Neumann condition is imposed on the top wall i.e. \$\dfrac{\partial \mathbf{u}}{\partial \mathbf{n}}=\mathbf{0}\$. The initial bubble is circular with a radius \$r_0 = 0.25\$ and centered on the point \$(0.5, 0.5, 0.)\$. A surface tension force \$\mathbf{f}_{st}\$ is applied on \$\Gamma\$, it reads : \$\mathbf{f}_{st} = \int_{\Gamma} \sigma \kappa \mathbf{n} \simeq \int_{\Omega} \sigma \kappa \mathbf{n} \delta_{\varepsilon}(\phi)\$ where \$\sigma\$ stands for the surface tension between the two-fluids and \$\kappa = \nabla \cdot (\frac{\nabla \mathbf{\phi}}{|\nabla \phi|})\$ is the curvature of the interface. Note that the normal vector \$\mathbf{n}\$ is defined here as \$\mathbf{n}=\frac{\nabla \phi}{|\nabla \phi|}\$.

We denote with indices \$1\$ and \$2\$ the quantities relative to the fluid in respectively in \$\Omega_1\$ and \$\Omega_2\$. The parameters of the benchmark are \$\rho_1\$, \$\rho_2\$, \$\nu_1\$, \$\nu_2\$ and \$\sigma\$ and we define two dimensionless numbers: first, the Reynolds number which is the ratio between inertial and viscous terms and is defined as : \$Re = \dfrac{\rho_1 \sqrt{|\mathbf{g}| (2r_0)^3}}{\nu_1}\$; second, the E\"otv\"os number which represents the ratio between the gravity force and the surface tension \$E_0 = \dfrac{4 \rho_1 |\mathbf{g}| r_0^2}{\sigma}\$. The table below reports the values of the parameters used for two different test cases proposed in~\cite{Hysing2009}.

Table 15. Numerical parameters taken for the benchmarks.

Tests

\$\rho_1\$

\$\rho_2\$

\$\nu_1\$

\$\nu_2\$

\$\sigma\$

Re

\$E_0\$

Test 1 (ellipsoidal bubble)

1000

100

10

1

24.5

35

10

Test 2 (skirted bubble)

1000

1

10

0.1

1.96

35

125

The quantities measured in \cite{Hysing2009} are \$\mathbf{X_c}\$ the center of mass of the bubble, \$\mathbf{U_c}\$ its velocity and the circularity. For the 3D case we extend the circularity to the sphericity defined as the ratio between the surface of a sphere which has the same volume and the surface of the bubble which reads \$\Psi(t) = \dfrac{4\pi\left(\dfrac{3}{4\pi} \int_{\Omega_2} 1 \right)^{\frac{2}{3}}}{\int_{\Gamma} 1}\$.

Simulations parameters

The simulations have been performed on the supercomputer SUPERMUC using 160 or 320 processors. The number of processors was chosen depending on the memory needed for the simulations. The table Numerical parameters used for the test 1 simulations: Mesh size, Number of processors, Time step, Average time per iteration, Total time of the simulation. summarize for the test 1 the different simulation properties and the table Mesh caracteristics: mesh size given, number of Tetrahedra, number of points, number of order 1 degrees of freedom, number of order 2 degrees of freedom give the carachteristics of each mesh.

Table 16. Numerical parameters used for the test 1 simulations: Mesh size, Number of processors, Time step, Average time per iteration, Total time of the simulation.

h

Number of processors

\$\Delta t\$

Time per iteration (s)

Total Time (h)

0.025

360

0.0125

18.7

1.25

0.02

360

0.01

36.1

3.0

0.0175

180

0.00875

93.5

8.9

0.015

180

0.0075

163.1

18.4

0.0125

180

0.00625

339.7

45.3

Table 17. Mesh caracteristics: mesh size given, number of Tetrahedra, number of points, number of order 1 degrees of freedom, number of order 2 degrees of freedom

h

Tetrahedra

Points

Order 1

Order 2

0.025

73010

14846

67770

1522578

0.02

121919

23291

128969

2928813

0.0175

154646

30338

187526

4468382

0.015

217344

41353

292548

6714918

0.0125

333527

59597

494484

11416557

The Navier-Stokes equations are linearized using the Newton’s method and we used a KSP method to solve the linear system. We use an Additive Schwarz Method for the preconditioning (GASM) and a LU method as a sub preconditionner. We run the simulations looking for solutions in finite element spaces spanned by Lagrange polynomials of order \$(2,1,1)\$ for respectively the velocity, the pressure and the level set.

Results Test 1: Ellipsoidal bubble

Accordind to the 2D results we expect that the drop would became ellipsoid. The figure~\ref{subfig:elli_sh} shows the shape of the drop at the final time step. The contour is quite similar to the one we obtained in the two dimensions simulations. The shapes are similar and seems to converge when the mesh size is decreasing. The drop reaches a stationary circularity and its topology does not change. The velocity increases until it attains a constant value. Figure~\ref{subfig:elli_uc} shows the results we obtained with different mesh sizes.

Bibliography

.

References for this benchmark
  • [cottet]

  • [Feelpp] C. Prud’homme et al.

  • [osher] Osher1988, book_Sethian, book_Osher

  • [Franca1992] Franca 1992

Computational Solid Mechanics

Computational solid mechanics ( or CSM ) is a part of mechanics that studies solid deformations or motions under applied forces or other parameters.

Second Newton’s law

The second Newton’s law allows us to define the fundamental equation of the solid mechanic, as follows

\$\rho^*_s \frac{\partial^2\boldsymbol{\eta}_s}{\partial t^2} - \nabla \cdot (\boldsymbol{F}_s\boldsymbol{\Sigma}_s) = \boldsymbol{f}^t_s\$

It’s define here into a Lagrangian frame.

Variables, symbols and units

Notation

Quantity

Unit

\$\rho_s^*\$

strucure density

\$kg/m^3\$

\$\boldsymbol{\eta}_s\$

displacement

\$m\$

\$\boldsymbol{F}_s\$

deformation gradient

\$\boldsymbol{\Sigma}_s\$

second Piola-Kirchhoff tensor

\$N/m^2\$

\$f_s^t\$

body force

\$N/m^2\$

Lamé coefficients

The Lamé coefficients are deducing from the Young’s modulus \$E_s\$ and the Poisson’s ratio \$\nu_s\$ of the material we work on and can be express

\$\lambda_s = \frac{E_s\nu_s}{(1+\nu_s)(1-2\nu_s)} \hspace{0.5 cm} , \hspace{0.5 cm} \mu_s = \frac{E_s}{2(1+\nu_s)}\$

Axisymmetric reduced model

We interest us here to a 1D reduced model, named generalized string.

The axisymmetric form, which will interest us here, is a tube of length \$L\$ and radius \$R_0\$. It is oriented following the axis \$z\$ and \$r\$ represent the radial axis. The reduced domain, named \$\Omega_s^*\$ is represented by the dotted line. So the domain, where radial displacement \$\eta_s\$ is calculated, is \$\Omega_s^*=\lbrack0,L\rbrack\$.

We introduce then \$\Omega_s^{'*}\$, where we also need to estimate a radial displacement as before. The unique variance is this displacement direction.

Reduced Model Geometry
Figure 1 : Geometry of the reduce model

The mathematical problem associated to this reduce model can be describe as

\$ \rho^*_s h \frac{\partial^2 \eta_s}{\partial t^2} - k G_s h \frac{\partial^2 \eta_s}{\partial x^2} + \frac{E_s h}{1-\nu_s^2} \frac{\eta_s}{R_0^2} - \gamma_v \frac{\partial^3 \eta}{\partial x^2 \partial t} = f_s.\$

where \$\eta_s\$, the radial displacement that satisfy this equation, \$k\$ is the Timoshenko’s correction factor, and \$\gamma_v\$ is a viscoelasticity parameter. The material is defined by its density \$\rho_s^*\$, its Young’s modulus \$E_s\$, its Poisson’s ratio \$\nu_s\$ and its shear modulus \$G_s\$

At the end, we take \$ \eta_s=0\text{ on }\partial\Omega_s^*\$ as a boundary condition, which will fixed the wall to its extremities.

inlude::FSI/README.adoc[]

Thermal Building Environment

ISO 10211:2007 Thermal bridges in building construction

Introduction

ISO 10211:2007 sets out the specifications for a three-dimensional and a two-dimensional geometrical model of a thermal bridge for the numerical calculation of:

  1. heat flows, in order to assess the overall heat loss from a building or part of it;

  2. minimum surface temperatures, in order to assess the risk of surface condensation.

These specifications include the geometrical boundaries and subdivisions of the model, the thermal boundary conditions, and the thermal values and relationships to be used.

ISO 10211:2007 is based upon the following assumptions:

  1. all physical properties are independent of temperature;

  2. there are no heat sources within the building element.

ISO 10211:2007 can also be used for the derivation of linear and point thermal transmittances and of surface temperature factors. More information here.

Implementation

Only the 2D specifications have been implemented.

Running the testcase

$ mpirun -np 4 /usr/local/bin/feelpp_thermodyn_2d --config-file thermo2dCase2.cfg

inlude::HeatFluid/README.adoc[]

Optimization problems

This section presents some mathematical optimization problems. The following examples source codes are located in "doc/manual/opt/".

:leveloffset:-1