Feel++

Table of Contents

Feel++ Book Contributors https://github.com/feelpp/book.feelpp.org/graphs/contributors

Hybrid Discontinuous Galerkin

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1. Mixed Poisson

1.1. Notations and units

Notation

Quantity

\(k\)

conductivity

\(\Lambda\)

resistivity

1.2. Equations

Mixed Poisson equations are

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

completed by boundary conditions.

The conductivity \(k\) can be non-linear. If it depends only on the primal variable \(p\), it is already handle by the model, in other cases, one needs to provide the corresponding expression.

1.3. MixedPoisson Toolbox

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

1.3.1. Models

The models are not considered for now.

Model section
"Model": "HDG"

1.3.2. Materials

The definition of the conductivity \(k\) depends on the material, it can be linear or non-linear. In the linear case, it is given in the material we work on by the keyword cond and in the non-linear case, by condNL.

Material section
"Materials":
{
    "<marker>":
    {
        "name": "copper",
        "cond": "1",
	"condNL": "2*p:p"
    }
}

The keywords cond and condNL can be changed respectively by the options mixedpoisson.conductivity_json and mixedpoisson.conductivityNL_json.

1.3.3. 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
\[p = g_D\]

Field

Type

Option

Value

potential

Dirichlet

expr

\(g_D\)
Neumann condition
\[-k\nabla p \cdot\boldsymbol{n} = g_N\]

Field

Type

Option

Value

potential

Neumann

expr

\(g_DN\) or \(-k\nabla p\)

The choice between \(g_DN\) or \(-k\nabla p\) is base on the dimension of the expression.

Robin condition
\[-k\nabla p \cdot\boldsymbol{n} + g_R^1 p = g_R^2\]

Field

Type

Option

Value

potential

Robin

expr1

\(g_R^1\)

potential

Robin

expr2

\(g_R^2\)
Integral boundary condition
\[\int_{\Gamma_I} \boldsymbol{u}\cdot\boldsymbol{n} = g_I\]

Field

Type

Option

Value

flux

Integral

expr

\(g_I\)

1.3.4. Source Terms

The source terms \(f\) and \(\boldsymbol{g}\) are treated as boundary condtions.

Field

Type

Option

Value

potential

SourceTerm

expr

\(f\)

flux

SourceTerm

expr

\(\boldsymbol{g}\)

1.3.5. Post Process

Two fields can be exported, the potential \(p\) and the flux \(\boldsymbol{u}\).

Post Process section
"PostProcess":
{
    "Fields":["potential","flux"]
}

1.4. Create applications

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

Minimal Linear case
    typedef FeelModels::MixedPoisson<FEELPP_DIM,FEELPP_ORDER> mp_type;
    auto MP = mp_type::New("mixedpoisson");
    MP->init();
    MP->assembleAll();
    MP->solve();
    MP->exportResults();

2. Mixed Elasticity

2.1. Notations and units

Notation

Quantity

\(\mathcal A\)

compliance operator

\(\lambda\)

first Lamé parameter

\(\mu\)

second Lamé parameter

\(\rho\)

mass density

2.2. 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}

2.3. 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.

2.3.1. Models

The models are not considered for now.

Model section
"Model": "HDG"

2.3.2. 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"
    }
}

2.3.3. 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

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

2.3.4. Source Term

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

Field

Type

Option

Value

stress

SourceTerm

expr

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

2.3.5. 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"]
}

2.4. 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.

2.5. Run simulations

Programme available to run simulations:

feelpp_toolbox_mixed-elasticity-model_3DP{<polynomial_order>}_G{<geometric_order>}

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

3. Benchmarks

3.1. BM-1: HDG Benchmark