Boundary and Initial data

First, we initialize the new worksheet; Note:

1. This tutorial can be found in the source code repository as as BoundaryAndInitialData.ipynb. One can directly load this into Jupyter to interactively work with the following code examples.
2. In the following line, the reference to BoSSSpad.dll is required. You must either set #r "BoSSSpad.dll" to something which is appropirate for your computer (e.g. C:\Program Files (x86)\FDY\BoSSS\bin\Release\net5.0\BoSSSpad.dll if you installed the binary distribution), or, if you are working with the source code, you must compile BoSSSpad and put it side-by-side to this worksheet file (from the original location in the repository, you can use the scripts getbossspad.sh, resp. getbossspad.bat).

#r "BoSSSpad.dll"
using System;
using System.Collections.Generic;
using System.Linq;
using ilPSP;
using ilPSP.Utils;
using BoSSS.Platform;
using BoSSS.Foundation;
using BoSSS.Foundation.Grid;
using BoSSS.Foundation.Grid.Classic;
using BoSSS.Foundation.IO;
using BoSSS.Solution;
using BoSSS.Solution.Control;
using BoSSS.Solution.GridImport;
using BoSSS.Solution.Statistic;
using BoSSS.Solution.Utils;
using BoSSS.Solution.Gnuplot;
using BoSSS.Application.BoSSSpad;
using BoSSS.Application.XNSE_Solver;
using static BoSSS.Application.BoSSSpad.BoSSSshell;
Init();

The below script needs to be able to find the current output cell; this is an easy method to get it.

This tutorial demostrates the definition, resp. the import of data for boundary and initial values.

In order to demonstrate the usage, we employ the exemplaric Poisson solver.

using BoSSS.Application.SipPoisson;

We use the following helper function to create a template for the multiple solver runs.

Func<SipControl> PreDefinedControl = delegate() {
    SipControl c = new SipControl();
 
    c.SetDGdegree(2);
 
    c.GridFunc = delegate() {
        // define a grid of 10x10 cells
        double[] nodes = GenericBlas.Linspace(-1, 1, 11);
        var grd = Grid2D.Cartesian2DGrid(nodes, nodes);
 
        // set the entire boundary to Dirichlet b.c.
        grd.DefineEdgeTags(delegate (double[] X) {
            return BoundaryType.Dirichlet.ToString();
        });
 
        return grd;
    };
 
    c.SetDatabase(wmg.DefaultDatabase);
    c.savetodb = true; 
 
    return c;    
};

Again, we are using the workflow management

BoSSSshell.WorkflowMgm.Init("Demo_BoundaryAndInitialData");

Project name is set to 'Demo_BoundaryAndInitialData'.

Default Execution queue is chosen for the database.

Creating database '\\fdygitrunner\BoSSStests\Demo_BoundaryAndInitialData'.

Textual and Embedded formulas

SipControl c1 = PreDefinedControl();

Provide initial data as a text:

c1.AddInitialValue("RHS","X => Math.Sin(X[0])*Math.Cos(X[1])",
                        TimeDependent:false);

Finally, all initial data is stored in the AppControl.InitialValues dictionary and all boundary data is stored in the AppControl.BoundaryValues dictionary.

The common interface for all varinats to specify boundary and initial data is IBoundaryAndInitialData

The snippet above is only a shortcut to add a Formula object, which implements the IBoundaryAndInitialData interface.

c1.InitialValues

key
RHS

c1.InitialValues["RHS"]

X => Math.Sin(X[0])*Math.Cos(X[1])

In BoSSSpad, such objects can also be extracted from static methods of classes; note that these should not depend on any other object in the worksheet.

Formula BndyFormula = new Formula(
    "BndyValue.BndyFunction",
    false,
"static class BndyValue {"+
"    public static double BndyFunction(double[] X) {"+
"        return 1.0;"+
"    }"+
"}");

c1.AddBoundaryValue(BoundaryType.Dirichlet.ToString(),
                    "T",
                    BndyFormula);

Creates a Job named J1 and runs it

var J1 = c1.RunBatch();

Deployments so far (0): ;

Success: 0

job submit count: 0

unable to determine job status - unknown

Deploying job UnnamedJob_1 ... 

Control object contains grid function. Trying to Serialize the grid...

Grid Edge Tags changed.

Control object modified.

Deploying executables and additional files ...

Deployment directory: \\fdygitrunner\BoSSStests\Demo_BoundaryAndInitialData-ipPoisson2023Dec05_001823.724709

copied 37 files.

   written file: control.obj

   copied 'win\amd64' runtime.

deployment finished.

The next line prints the Status of the Job J1.

BoSSSshell.WorkflowMgm.BlockUntilAllJobsTerminate(3600*4);

All jobs finished.

We can print the Status of the Job J1.

J1.Status

FinishedSuccessful

We can also check via a method if the Job J1 is truly finished

NUnit.Framework.Assert.IsTrue(J1.Status == JobStatus.FinishedSuccessful);

1D Splines

Splines can be used to interpolate nodal data onto a DG field; currently, only 1D is supported.

SipControl c2 = PreDefinedControl();

// create test data for the spline
double[] xNodes = GenericBlas.Linspace(-2,2,13);
double[] yNodes = xNodes.Select(x => x*0.4).ToArray();

var rhsSpline = new Spline1D(xNodes, yNodes,
                             0,
                             Spline1D.OutOfBoundsBehave.Extrapolate);

/// BoSSScmdSilent
double err = 0;
// test the spline: a line must be interpolated exactly.
foreach(double xtst in GenericBlas.Linspace(-3,3,77)) { 
   double sVal = rhsSpline.Evaluate(new double[] {xtst , 0, 0 }, 0.0);
   double rVal = xtst*0.4;
   err += Math.Abs(sVal - rVal);
}
NUnit.Framework.Assert.Less(err, 1.0e-10, "Slpine implementation fail.");

c2.AddInitialValue("RHS", rhsSpline);

var J2 = c2.RunBatch();

Deployments so far (0): ;

Success: 0

job submit count: 0

unable to determine job status - unknown

Deploying job UnnamedJob_2 ... 

Control object contains grid function. Trying to Serialize the grid...

Grid Edge Tags changed.

An equivalent grid (5dd19fe9-7251-4dbf-bf6e-0b141f6a9b93) is already present in the database -- the grid will not be saved.

Control object modified.

Deploying executables and additional files ...

Deployment directory: \\fdygitrunner\BoSSStests\Demo_BoundaryAndInitialData-ipPoisson2023Dec05_001922.498703

copied 37 files.

   written file: control.obj

   copied 'win\amd64' runtime.

deployment finished.

BoSSSshell.WorkflowMgm.BlockUntilAllJobsTerminate(3600*4);

All jobs finished.

J2.Status

FinishedSuccessful

/// BoSSScmdSilent
NUnit.Framework.Assert.IsTrue(J2.Status == JobStatus.FinishedSuccessful);

Interpolating values from other Calculations

For demonstrational purposes, we use the result (i.e. the last time-step) of a previous calculation as a right-hand-side for the next calculation.

var j2Sess = J2.LatestSession;

j2Sess

Demo_BoundaryAndInitialData	UnnamedJob_2	12/05/2023 00:19:41	bd21f59e...

j2Sess.Timesteps

#0:  { Time-step: 0; Physical time: 0s; Fields: T, ResidualKP1, Error, MgLevel_0, MgLevel_1, MgLevel_2, MgLevel_3, MgLevel_4, MPIrank; Name:  }

#1:  { Time-step: 1; Physical time: 0s; Fields: T, ResidualKP1, Error, MgLevel_0, MgLevel_1, MgLevel_2, MgLevel_3, MgLevel_4, MPIrank; Name:  }

var lastTimeStep = j2Sess.Timesteps.Last();

We encapsulate the value T in the ForeignGridValue object, which allows interpolation between different meshes:

var newForeignMesh = new ForeignGridValue(lastTimeStep,"T");

/// Use different mesh in the control file:
SipControl c3 = PreDefinedControl();

c3.GridFunc = delegate() {
   // define a grid of *triangle* cells
   double[] nodes = GenericBlas.Linspace(-1, 1, 11);
   var grd = Grid2D.UnstructuredTriangleGrid(nodes, nodes);
 
   // set the entire boundary to Dirichlet b.c.
   grd.DefineEdgeTags(delegate (double[] X) {
       return BoundaryType.Dirichlet.ToString();
   });
 
   return grd;
};
// we also save the RHS in the database
c3.AddFieldOption("RHS", SaveOpt: FieldOpts.SaveToDBOpt.TRUE);

/// finally, we define the RHS:
c3.AddInitialValue("RHS", newForeignMesh);

/// BoSSScmdSilent
double orgProbe = newForeignMesh.Evaluate(new double[] {0.5,0.5}, 0.0);
double newProbe = lastTimeStep.GetField("T").ProbeAt(new double[] {0.5,0.5});
NUnit.Framework.Assert.Less(Math.Abs(orgProbe - newProbe), 1.0e-10, "Check (1) on ForeignGridValue failed");

var J3 = c3.RunBatch();

Deployments so far (0): ;

Success: 0

job submit count: 0

unable to determine job status - unknown

Deploying job UnnamedJob_3 ... 

Control object contains grid function. Trying to Serialize the grid...

Grid Edge Tags changed.

Control object modified.

Deploying executables and additional files ...

Deployment directory: \\fdygitrunner\BoSSStests\Demo_BoundaryAndInitialData-ipPoisson2023Dec05_002000.712680

copied 37 files.

   written file: control.obj

   copied 'win\amd64' runtime.

deployment finished.

BoSSSshell.WorkflowMgm.BlockUntilAllJobsTerminate(3600*4);

All jobs finished.

J3.Status

FinishedSuccessful

/// BoSSScmdSilent
NUnit.Framework.Assert.IsTrue(J3.Status == JobStatus.FinishedSuccessful);

Since the quadrilateral mesh used for the original right-hand-side is geometrically embedded in the triangular mesh the interpolation error should be zero (up to machine precision).

var firstTimeStep = J3.LatestSession.Timesteps.First();

DGField RhsOnTriangles = firstTimeStep.GetField("rhs"); // case-insensitive!
DGField RhsOriginal    = lastTimeStep.GetField("T");

// note: we have to cast DGField to ConventionalDGField in order to use
// the 'L2Distance' function:
((ConventionalDGField)RhsOnTriangles).L2Distance((ConventionalDGField)RhsOriginal)

1.2356612792540816E-17

/// BoSSScmdSilent
var H1err = ((ConventionalDGField)RhsOnTriangles).H1Distance((ConventionalDGField)RhsOriginal);
NUnit.Framework.Assert.Less(H1err, 1.0e-10, "Check (2) on ForeignGridValue failed.");

Restart from Dummy-Sessions

Dummy sessions are kind of fake siolver runs, with the only purpose of using them for a restart.

DGField RHSforRestart = firstTimeStep.GetField("RHS");

We save the DG field RHSforRestart in the database; This automatically creates a timestep and a session which host the DG field:

var RestartTimestep = wmg.DefaultDatabase.SaveTimestep(RHSforRestart);

RestartTimestep

 { Time-step: 0; Physical time: 0s; Fields: RHS; Name:  }

RestartTimestep.Session

Demo_BoundaryAndInitialData	InitialValueSession	12/05/2023 00:20:39	2da702e0...

This time step can be used as a restart value.:

var c4 = PreDefinedControl();

c4.InitialValues.Clear();
c4.SetRestart(RestartTimestep);

var J4 = c4.RunBatch();

Deployments so far (0): ;

Success: 0

job submit count: 0

unable to determine job status - unknown

Deploying job UnnamedJob_4 ... 

Deploying executables and additional files ...

Deployment directory: \\fdygitrunner\BoSSStests\Demo_BoundaryAndInitialData-ipPoisson2023Dec05_002043.151058

copied 37 files.

   written file: control.obj

   copied 'win\amd64' runtime.

deployment finished.

BoSSSshell.WorkflowMgm.BlockUntilAllJobsTerminate(3600*4);

All jobs finished.

J4.Status

FinishedSuccessful

/// BoSSScmdSilent
NUnit.Framework.Assert.IsTrue(J4.Status == JobStatus.FinishedSuccessful);

Note

Since no mesh interpolation is performed for the restart, it is much faster than ForeignGridValue but less flexible (a restart is always performed on the same mesh).

To avoid multiple mesh interpolations (e.g. when multiple runs are required) one coudl therefore speed up the process by doing the mesh interpolation once use ProjectFromForeignGrid in BoSSSpad and save the interpolation in a dummy session.