Hybrid Discontinuous Galerkin User Manual

In many application context, not only do we need the primal variable (e.g. the potential or the displacement) but also the dual variable (e.g the flux or the stress).

Feel++ supports HdG methods and enables primal/dual variables solves.

1. Mixed Poisson

1.1. Models

Mixed Poisson equations are

\[\begin{align} \boldsymbol{u} + k\nabla p &= \boldsymbol{f}\\ \nabla\cdot \boldsymbol{u} &= g \end{align}\]

completed by boundary conditions and where \(k\) represents the conductivity.

Those equations can represent several problem types, such as electrostatics problems or heat transfer problems. One can choose to use field names and material properties in the model file for a particular physic by using a MixedPoissonPhyics in the constructor of the toolbox.

By default, no physic is used (MixedPoissonPhysics::None). Electric (MixedPoissonPhysics::Electric) and heat (MixedPoissonPhysics::Heat) are also available.

1.2. Materials

Table 1. Materials properties defined in the MixedPoisson toolbox with MixedPoissonPhysics::None
Name	Symbol	Shape	Description
conductivity	cond	scalar	conductivity

Table 2. Materials properties defined in the MixedPoisson toolbox with MixedPoissonPhysics::Electric
Name	Symbol	Shape	Description
electric-conductivity	sigma	scalar	electric conductivity

Table 3. Materials properties defined in the MixedPoisson toolbox with MixedPoissonPhysics::Heat
Name	Symbol	Shape	Description
thermal-conductivity	k	scalar	thermal conductivity

1.3. Fields and symbols expressions

Table 4. Fields available with MixedPoissonPhysics::None
Name	Description	Shape
potential	the potential field	scalar
flux	the flux field	vectorial

Table 5. Symbols expressions with MixedPoissonPhysics::None
Symbol	Expression	Description
poisson_P	\(P\)	evaluate the potential
poisson_grad_P_0	\(\frac{\partial p}{\partial x}\)	evaluate the first component of gradient of potential
poisson_grad_P_1	\(\frac{\partial p}{\partial y}\)	evaluate the second component of gradient of potential
poisson_grad_P_2	\(\frac{\partial p}{\partial z}\)	evaluate the third component of gradient of potential
poisson_dn_P	\(\nabla p \cdot \boldsymbol{n}\)	evaluate the normal derivative of potential
poisson_F_0	\(\boldsymbol{u}_x\)	evaluate the x component of the flux
poisson_F_1	\(\boldsymbol{u}_y\)	evaluate the y component of the flux
poisson_F_2	\(\boldsymbol{u}_z\)	evaluate the z component of the flux

Table 6. Fields available with MixedPoissonPhysics::Electric
Name	Description	Shape
electric-potential	the electric potential field	scalar
current-density	the current density field	vectorial

Table 7. Symbols expressions with MixedPoissonPhysics::Electric
Symbol	Expression	Description
electric_V	\(V\)	evaluate the potential
electric_grad_V_0	\(\frac{\partial V}{\partial x}\)	evaluate the first component of gradient of electric potential
electric_grad_V_1	\(\frac{\partial V}{\partial y}\)	evaluate the second component of gradient of electric potential
electric_grad_V_2	\(\frac{\partial V}{\partial z}\)	evaluate the third component of gradient of electric potential
electric_dn_V	\(\nabla V \cdot \boldsymbol{n}\)	evaluate the normal derivative of electric potential
electric_C_0	\(\boldsymbol{j}_x\)	evaluate the x component of the current density
electric_C_1	\(\boldsymbol{j}_y\)	evaluate the y component of the current density
electric_C_2	\(\boldsymbol{j}_z\)	evaluate the z component of the current density

Table 8. Fields available with MixedPoissonPhysics::Heat
Name	Description	Shape
temperature	the temperature field	scalar
heat-flux	the heat flux field	vectorial

Table 9. Symbols expressions with MixedPoissonPhysics::Heat
Symbol	Expression	Description
heat_T	\(T\)	evaluate the temperature
heat_grad_T_0	\(\frac{\partial T}{\partial x}\)	evaluate the first component of gradient of temperature
heat_grad_T_1	\(\frac{\partial T}{\partial y}\)	evaluate the second component of gradient of temperature
heat_grad_T_2	\(\frac{\partial T}{\partial z}\)	evaluate the third component of gradient of temperature
heat_dn_T	\(\nabla T \cdot \boldsymbol{n}\)	evaluate the normal derivative of temperature
heat_F_0	\(\boldsymbol{u}_x\)	evaluate the x component of the heat flux
heat_F_1	\(\boldsymbol{u}_y\)	evaluate the y component of the heat flux
heat_F_2	\(\boldsymbol{u}_z\)	evaluate the z component of the heat flux

1.4. Boundary Conditions

1.4.1. Dirichlet

On the potential/electric-potential/temperature:

\[P = g \quad \text{ on } \Gamma\]

1.4.2. Neumann

On the potential/electric-potential/temperature:

\[-k \nabla P \cdot \boldsymbol{n} = g \quad \text{ on } \Gamma\]

1.4.3. Robin

On the potential/electric-potential/temperature:

\[-k \nabla P \cdot \boldsymbol{n} = h \left( T - g \right) \quad \text{ on } \Gamma\]

1.4.4. Integral

On the flux/current-density/heat-flux:

\[\int_\Gamma \boldsymbol{u}\cdot\boldsymbol{n} = g \quad \text{ on } \Gamma\\ |\Gamma|p - \int_\Gamma p = 0 \quad \text{ on } \Gamma\]

1.5. Initial Conditions

1.6. PostProcessing

1.6.1. Exports

Table 10. Fields allowed to be exported in the `fields` section with MixedPoissonPhysics::None are:
Name	Description
potential	the potential field
flux	the flux field
pid	the mesh partitioning
all	all fields available

Table 11. Fields allowed to be exported in the `fields` section with MixedPoissonPhysics::Electric are:
Name	Description
electric-potential	the electric potential field
current-density	the current density flux
pid	the mesh partitioning
all	all fields available

Table 12. Fields allowed to be exported in the `fields` section with MixedPoissonPhysics::Heat are:
Name	Description
temperature	the temperature field
heat-flux	the heat flux field
pid	the mesh partitioning
all	all fields available

All materials properties given in the section Materials can be also exported by specifying the name in the fields entry.

2. Coupled Mixed Poisson

2.1. Models

The equations of the model are:

\[\begin{align} \boldsymbol{u} + k\nabla p &= \boldsymbol{f}\\ \frac{1}{M}\frac{\partial p}{\partial t} + \nabla\cdot \boldsymbol{u} &= g \end{align}\]

coupled with a system of ODE:

\[\begin{align} \frac{\partial \boldsymbol{y}}{\partial t} + \underline{\underline{A}}(\boldsymbol{y},t)\boldsymbol{y} = s(\boldsymbol{y},t) + b(\boldsymbol{y},t) \end{align}\]

where \(\Pi_1\in \boldsymbol{y}\) and the boundary condition:

\[\int_\Gamma \boldsymbol{u}\cdot\boldsymbol{n} = \frac{P_I - \Pi_1}{R_b} \quad \text{ on } \Gamma\\ |\Gamma|p - \int_\Gamma p = 0 \quad \text{ on } \Gamma\]

with \(P_I = p_{|\Gamma}\).

Most of the configuration is the same as for the Mixed Poisson toolbox, except for the boundary condition where the coupling happens.

2.2. Boundary Conditions

2.2.1. Coupling

On the flux/current-density/heat-flux:

Example of a Coupling boundary condition

"Coupling":
{
    "buffer":
    {
        "markers":"top", (1)
        "capacitor": "Cbuffer.C", (2)
        "resistor": "Rbuffer.R", (3)
        "circuit": "$cfgdir/test3d0d_linear/test3d0d_linear.fmu", (4)
        "buffer": "Pi_1.phi" (5)
    }
}

1	marker for \(\Gamma\)
2	name of the variable in the FMU for \(C_b\)
3	name of the variable in the FMU for \(R_b\)
4	path of the FMU
5	name of the variable in the FMU for \(\Pi_1\)

3. Mixed Elasticity

3.1. Notations and units

Notation

Quantity

\(\mathcal A\)

compliance operator

\(\lambda\)

first Lamé parameter

\(\mu\)

second Lamé parameter

\(\rho\)

mass density

3.2. Equations

Mixed Elasticity equations are

\[\newcommand{\uv}{\underline{\underline{\mathbf v}}} \newcommand{\usigma}{\underline{\underline{\boldsymbol\sigma}}} \newcommand{\ueps}{\underline{\underline{\boldsymbol\varepsilon}}} \newcommand{\uw}{\underline{\mathbf w}} \newcommand{\un}{\underline{\mathbf n}} \newcommand{\uu}{\underline{\mathbf u}} \newcommand{\uzero}{\underline{\mathbf 0}} \newcommand{\ug}{\underline{\mathbf g}} \newcommand{\umu}{\underline{\boldsymbol\mu}} \newcommand{\ux}{\underline{\mathbf x}} \begin{align} \mathcal A\usigma - \ueps(\uu) &= \uzero & &\text{in }\Omega\times (0,T)\subset\mathbb R^n\times\mathbb{R} \\ \rho \dfrac{\partial^2 \uu}{\partial t^2} &= \nabla\cdot\usigma + \underline{\mathbf F}_{ext} & &\text{in }\Omega\times (0,T) \\ \usigma\;\un &= \underline{\mathbf g}_N & &\text{on }\Gamma_N \\ \uu &= \underline{\mathbf g}_D & &\text{on }\Gamma_D \\ \int_{\Gamma_I} \usigma \; \un &= \underline{\mathbf{F}}_{target} & & \text{on } \Gamma_I \\ \uu(x,t) &= \uu(t) & & \text{on } \Gamma_I \end{align}\]

The compliance operator is defined as follow:

\begin{equation} \mathcal A\uv = c_1 \uv + c_2 \text{trace}(\uv) \underline{\underline{\mathbf{I}}} \end{equation}

where

\begin{align} & c_1 = \dfrac{1}{2\mu} & c_2 = \dfrac{-\lambda}{2\mu \left( 3\lambda + 2\mu \right)} \end{align}

3.3. MixedElasticity Toolbox

The model is described in a json file which path is given by the option mixedelasticity.model_json. The construction of this file is detailed in the following sections.

3.3.1. Models

The models are not considered for now.

Model section

"Model": "HDG"

3.3.2. Materials

In this part we define the two Lamé parameters \(\lambda\) and \(\mu\) and the mass density \(\rho\). It is always necessary to define also the material we work on.

Material section

"Materials":
{
    "<marker>":
    {
        "name": "copper",
	"rho":"1",
        "lambda":"1",
	"mu":"1"
    }
}

3.3.3. Boundary Conditions

All boundary conditions are described in the same way

Listing : boundary conditions in json

"BoundaryConditions":
{
    "<field>":
    {
        "<bc_type>":
        {
            "<marker>":
            {
                "<option1>":"<value1>",
                "<option2>":"<value2>",
                // ...
            }
        }
    }
}

Different types of boundary condition are available.

Dirichlet condition

\[\uu = \ug_D\]

Field

Type

Option

Value

displacement

Dirichlet

expr

\(\ug_D\)

Neumann condition

\[\usigma\;\un = \ug_N\]

Field

Type

Option

Value

stress

Neumann

expr

\(\ug_DN\)

Integral boundary condition

\[\int_{\Gamma_I} \usigma \; \un = \underline{\mathbf{F}}_{target}\]

Field

Type

Option

Value

stress

Integral

expr

\(\underline{\mathbf{F}}_{target}\)

3.3.4. Source Term

The source term \(\underline{\mathbf{F}}_{ext}\) is treated as a boundary condition.

Field

Type

Option

Value

stress

SourceTerm

expr

\(\underline{\mathbf{F}}_{ext}\)

3.3.5. Post Process

Two fields can be exported, the displacement \(\uu\) and the stress \(\usigma\).

Post Process section

"PostProcess":
{
    "Fields":["displacement","stress"]
}

Moreover it is possible to apply a scaling after the computation and then export the scaled field, in particular we define in the material section the scale factor for the elements with a specific marker.

Material section

"Materials":
{
    "<marker>":
    {
	"scale_displacement":"1",
	"scale_stress":"1"
    }
}

Post Process section

"PostProcess":
{
    "Fields":["displacement","stress","scaled_displacement","scaled_stress"]
}

3.4. Create applications

In order to solve linear elasticity problem, an application should contain at least

Minimal Elasticity case

typedef FeelModels::MixedElasticity<FEELPP_DIM,FEELPP_ORDER> me_type;
auto ME = me_type::New("mixedelasticity");
ME->init();
ME->solve();
ME->exportResults();

The assembling for the constant part is inside the initialization, while the assembling of the non-constant part (e.g. the right hand side) is in the solve method.

3.5. Run simulations

Programme available to run simulations:

mpirun -np 4 feelpp_toolbox_mixed-elasticity-model_3DP{<polynomial_order>}_G{<geometric_order>}`

with =0,1,2,3,4 and =1,2 .