Example problem: Spatially adaptive solution of the 2D unsteady heat equation with flux boundary conditions in a moving domain: ALE methods.

In this example we consider the solution of the unsteady heat equation in a domain with moving boundaries. We demonstrate that the presence of moving boundaries only requires trivial changes to driver codes for corresponding fixed-mesh computations.

The two-dimensional unsteady heat equation with flux boundary conditions in a moving domain. in the domain , bounded by the coordinate axes and the time-dependent ellipse subject to Neumann boundary conditions, along the horizontal domain boundary , and to Dirichlet boundary conditions, elsewhere. where the functions and are given. |

We choose the functions and so that

is the exact solution.

The solution represents the "usual" tanh profile, whose steepness is controlled by the parameter so that for the solution approaches a step. The step is oriented at an angle against the axis and its position varies periodically. The parameter controls the amplitude of the step's lateral displacement, while determines the rate at which its position changes. For , the step remains stationary for most of the period and then translates rapidly parallel to the axis, making this a very challenging problem.

The figure below shows a snapshot of the animated solution, obtained from the spatially adaptive simulation discussed below, for the parameter values

The mesh adaptation in response to the translation of the step can be seen more clearly in this contour plot, taken from another animation of the solution.

`oomph-lib's`

`UnsteadyHeatEquations`

, the equation class that forms the basis for the isoparametric `QUnsteadyHeatElements`

, is based on the Arbitrary Lagrangian Eulerian (ALE) formulation of the weak form of the governing PDE, (1). Within each element, the solution is represented by interpolation between the element's nodal values , i.e.

where and are the element's two local coordinates. The mapping between the local and global (Eulerian) coordinates is based on the interpolation between the nodal coordinates,

where is the i-th global (Eulerian) coordinate of node j in the element. In moving-domain problems, where the nodal positions vary as function of time, the time-derivative of the nodal value, , represents the rate-of-change of at the moving node, rather than the time-derivative of at a fixed Eulerian position, the quantity required in the PDE, (1).

The rate of change of at a moving node, , may also be expressed by the material derivative,

where

is the i-th velocity component of node j, often referred to as the "mesh velocity". The rate of change of , experienced at the (fixed) spatial position that coincides with the current position of node j is therefore given by

This is the form in which the time-derivative in (1) is implemented in `oomph-lib's`

unsteady heat elements. The mesh velocity is determined automatically, using the `Node's`

positional `TimeStepper`

and the history values of the nodal positions. [By default, the positional `TimeStepper`

is the same as the `TimeStepper`

used for the evaluation of the time-derivatives of the nodal values; a different one may be assigned with the access function `Node::position_time_stepper_pt()`

.] This is why it is important to initialise the "previous nodal positions" in computations on fixed meshes, as demonstrated in the exercise in our earlier example: The previous nodal positions are initialised to zero when a `Node`

is first created. Therefore, if the previous positions are not overwritten with the actual nodal positions, the positional `TimeStepper`

would compute non-zero mesh velocities, even though the `Nodes`

are stationary.

`oomph-lib's`

unsteady Newton solvers automatically advance the time-history of the nodal positions when computing a new timestep. Performing computations on moving meshes is therefore extremely straightforward: The only task to be performed by the "user" is to update the nodal positions before a new timestep is taken. This is best done in the function `Problem::actions_before_implicit_timestep()`

.

The update of the nodal positions may be performed "manually", by assigning new nodal positions directly, using the function `Node::x`

(...). However, in most cases, the deformation of the domain will be driven by the motion of its boundaries. We discussed in an earlier example, that in `oomph-lib`

curvilinear, moving boundaries are typically represented by (time-dependent) `GeomObjects`

which define the `MacroElement`

boundaries of the `Domain`

object associated with the `Mesh`

. In this case, the update of the nodal positions may be performed by updating the parameters that control the shape of the `GeomObject`

and calling the `Mesh's`

node-update function `Mesh::node_update()`

, as illustrated in the earlier example.

It is also possible (and, in fact, much easier) to include the time-dependence of the domain boundaries into the definition of the `GeomObject`

, by making its shape, described by `GeomObject::position`

(...), a function of time. In that case, the update of the nodal positions in response to the boundary motion merely requires a call to `Mesh::node_update()`

in `Problem::actions_before_implicit_timestep()`

. This is the approach we take in the present problem.

As usual, we store the problem parameters in a namespace, `TanhSolnForUnsteadyHeat`

, in which we also specify the source function, the prescribed flux along the Neumann boundary and the exact solution. The namespace is identical to that used in the fixed domain example.

As discussed above, we will incorporate the time-dependence of the moving curvilinear boundary into the specification of the `GeomObject`

that defines that boundary. For this purpose we represent the elliptical boundary by the `GeomObject`

`MyEllipse`

. Its constructor stores the geometric parameters (the mean values of the ellipse's half axes, and , the amplitude of their temporal variations and , and the period of the oscillation, , in its private member data. We also store a pointer to a `Time`

object to give the `GeomObject`

access to the "current" and "previous" values of the "continuous time". The destructor can remain empty.

//============start_of_MyEllipse===========================================

/// \short Oscillating ellipse

/// \f[ x = (a + \widehat{a} \sin(2\Pi t/T)) \cos(\xi) \f]

/// \f[ y = (b + \widehat{b} \sin(2\Pi t/T)) \sin(\xi) \f]

//=========================================================================

{

public:

/// \short Constructor: Pass half axes, amplitudes of their variation, period

/// of oscillation and pointer to time object.

const double& a_hat, const double& b_hat,

const double& period, Time* time_pt) :

/// Destructor: Empty

virtual ~MyEllipse() {}

The "steady" version of the `position`

(...) function must return the position vector to the point on the `GeomObject`

, identified by its intrinsic coordinate `xi`

, and evaluated at the current value of the continuous time, which we extract from `Time::time()`

:

/// \short Current position vector to material point at

/// Lagrangian coordinate xi

void position(const Vector<double>& xi, Vector<double>& r) const

{

// Get current time:

double time=Time_pt->time();

// Position vector

r[0] = (A+A_hat*sin(2.0*MathematicalConstants::Pi*time/T))*cos(xi[0]);

r[1] = (B+B_hat*sin(2.0*MathematicalConstants::Pi*time/T))*sin(xi[0]);

} // end of position(...)

The "time-dependent" version of the `position`

(...) function must return the position vector to the `GeomObject`

, evaluated at the `t`

- th previous timestep. The value of the continuous time at that timestep is available from from `Time::time(t)`

:

/// \short Parametrised position on object: r(xi). Evaluated at

/// previous time level. t=0: current time; t>0: previous

/// time level.

void position(const unsigned& t, const Vector<double>& xi,

Vector<double>& r) const

{

// Get current time:

double time=Time_pt->time(t);

// Position vector

r[0] = (A+A_hat*sin(2.0*MathematicalConstants::Pi*time/T))*cos(xi[0]);

r[1] = (B+B_hat*sin(2.0*MathematicalConstants::Pi*time/T))*sin(xi[0]);

} // end of position(...)

We omit the code that defines the private member data.

Since the deformation of the domain and the update of the nodal positions will be handled automatically by adding a call to `Mesh::node_update()`

to `Problem::actions_before_implicit_timestep()`

, the driver code for this problem is exactly the same as that in the previous example in a fixed domain.

The `Problem`

class and most of its member functions are exactly the same as in the previous example.

The `Problem`

constructor is identical to the fixed-mesh version in the previous example, apart from the fact that we use the `MyEllipse`

`GeomObject`

to define the curvilinear `Mesh`

boundary. Here is the relevant code fragment:

// Setup mesh

//-----------

// Build geometric object that forms the curvilinear domain boundary:

// an oscillating ellipse

// Half axes

double a=1.0;

double b=1.0;

// Variations of half axes

double a_hat= 0.1;

double b_hat=-0.1;

// Period of the oscillation

double period=1.0;

// Create GeomObject

Boundary_pt=new MyEllipse(a,b,a_hat,b_hat,period,Problem::time_pt());

// Start and end coordinates of curvilinear domain boundary on ellipse

double xi_lo=0.0;

double xi_hi=MathematicalConstants::Pi/2.0;

// Now create the bulk mesh. Separating line between the two

// elements next to the curvilinear boundary is located half-way

// along the boundary.

double fract_mid=0.5;

Bulk_mesh_pt = new RefineableQuarterCircleSectorMesh<ELEMENT>(

Boundary_pt,xi_lo,fract_mid,xi_hi,time_stepper_pt());

As discussed above, the addition of a single line to `Problem::actions_before_implicit_timestep()`

suffices to update the nodal positions in response to the changes in the domain boundary.

//=========start of actions_before_implicit_timestep===============================

/// \short Actions before timestep: Update the domain shape, then set the

/// boundary conditions for the current time.

//========================================================================

template<class ELEMENT>

{

// Update the domain shape

Bulk_mesh_pt->node_update();

The rest of this function is identical to the that in the fixed-domain version and updates the nodal values on the Dirichlet boundaries according to the values given by the exact solution.

The only other change to the code occurs in the assignment of the initial conditions. The `Nodes'`

positional history values are given by the positions at which the `Nodes`

would have been at previous timesteps. Similarly, the history values themselves must be computed by evaluating the exact solution at the position at which the `Nodes`

would have been at those timesteps.

This is achieved with a few minor changes to the previous version of this function. We loop over the previous timesteps, reconstruct the value of the continuous time at that timestep, and temporarily over-write the value of the continuous time stored in `Time::time()`

. This ensures that the call to `MyEllipse::position`

(...) during the node update operation returns the position vector to the domain boundary at that timestep. Following the update of the nodal positions (which moves them into the position they would have occupied at the previous timestep) we copy their positions and the value of the exact solution into the appropriate history values. Here is the relevant code fragment from the `set_initial_condition()`

function:

// Loop over current & previous timesteps (in outer loop because

// the mesh also moves!)

for (int itime=nprev_steps;itime>=0;itime--)

{

double time=prev_time[itime];

// Set global time (because this is how the geometric object refers

// to continous time

time_pt()->time()=time;

cout << "setting IC at time =" << time << std::endl;

// Update the mesh for this value of the continuous time

// (The wall object reads the continous time from the same

// global time object)

Bulk_mesh_pt->node_update();

// Loop over the nodes to set initial guess everywhere

for (unsigned jnod=0;jnod<num_nod;jnod++)

{

// Get nodal coordinates

x[0]=Bulk_mesh_pt->node_pt(jnod)->x(0);

x[1]=Bulk_mesh_pt->node_pt(jnod)->x(1);

// Get intial solution

TanhSolnForUnsteadyHeat::get_exact_u(time,x,soln);

// Assign solution

Bulk_mesh_pt->node_pt(jnod)->set_value(itime,0,soln[0]);

// Loop over coordinate directions

for (unsigned i=0;i<2;i++)

{

Bulk_mesh_pt->node_pt(jnod)->x(itime,i)=x[i];

}

}

} // end of loop over previous timesteps

While the spatial adaptivity resolves the rapid spatial variations in the solution, the time-integration with a fixed timestep introduces errors during the phases when the solution undergoes rapid temporal variations. The animations of the exact and computed solutions show clearly that the computed solution lags behind the exact one during these phases. In the next example we will therefore demonstrate how to combine temporal and spatial adaptivity.

- The source files for this tutorial are located in the directory:
demo_drivers/unsteady_heat/two_d_unsteady_heat_ALE/ - The driver code is:
demo_drivers/unsteady_heat/two_d_unsteady_heat_ALE/two_d_unsteady_heat_ALE.cc

A pdf version of this document is available.