diff --git a/test/freeflow/shallowwater/bowl/CMakeLists.txt b/test/freeflow/shallowwater/bowl/CMakeLists.txt
index a456bb95c9060e5a9b0954bdd311769cb51fe22c..af740595755adf0f2b47277cca982d5836bb7e7e 100644
--- a/test/freeflow/shallowwater/bowl/CMakeLists.txt
+++ b/test/freeflow/shallowwater/bowl/CMakeLists.txt
@@ -1,11 +1,20 @@
dune_symlink_to_source_files(FILES "params.input")
dumux_add_test(NAME test_shallowwater_bowl
- SOURCES main.cc
- COMMAND ${CMAKE_SOURCE_DIR}/bin/testing/runtest.py
- CMD_ARGS --script fuzzy
- --files ${CMAKE_SOURCE_DIR}/test/references/test_ff_shallowwater_bowl-reference.vtu
- ${CMAKE_CURRENT_BINARY_DIR}/bowl-00001.vtu
- --zeroThreshold {"velocityY":1e-14}
- --command "${CMAKE_CURRENT_BINARY_DIR}/test_shallowwater_bowl params.input
- -Problem.Name bowl")
+ SOURCES main.cc
+ COMMAND ${CMAKE_SOURCE_DIR}/bin/testing/runtest.py
+ CMD_ARGS --script fuzzy
+ --files ${CMAKE_SOURCE_DIR}/test/references/test_ff_shallowwater_bowl-reference.vtu
+ ${CMAKE_CURRENT_BINARY_DIR}/bowl-00013.vtu
+ --zeroThreshold {"velocityY":1e-14}
+ --command "${CMAKE_CURRENT_BINARY_DIR}/test_shallowwater_bowl")
+
+dumux_add_test(NAME test_shallowwater_bowl_parallel
+ TARGET test_shallowwater_bowl
+ CMAKE_GUARD MPI_FOUND
+ COMMAND ${CMAKE_SOURCE_DIR}/bin/testing/runtest.py
+ CMD_ARGS --script fuzzy
+ --files ${CMAKE_SOURCE_DIR}/test/references/test_ff_shallowwater_bowl-reference.vtu
+ ${CMAKE_CURRENT_BINARY_DIR}/s0002-bowl-parallel-00013.pvtu
+ --zeroThreshold {"velocityY":1e-14,"process rank":100}
+ --command "${MPIEXEC} -np 2 ${CMAKE_CURRENT_BINARY_DIR}/test_shallowwater_bowl -Problem.Name bowl-parallel")
diff --git a/test/freeflow/shallowwater/bowl/main.cc b/test/freeflow/shallowwater/bowl/main.cc
index 606c60028622e613fe2317f4fd1e8a893c085e54..b5559bea76f7a2d5987ff3de02b8893dacca05bb 100644
--- a/test/freeflow/shallowwater/bowl/main.cc
+++ b/test/freeflow/shallowwater/bowl/main.cc
@@ -42,7 +42,83 @@
#include <dumux/assembly/fvassembler.hh>
-#include "problem.hh"
+#include "properties.hh"
+
+template<class SolutionVector, int i>
+class SolutionComponent
+{
+ const SolutionVector& sol_;
+public:
+ SolutionComponent(const SolutionVector& sol) : sol_(sol) {}
+ auto operator[] (std::size_t idx) const { return sol_[idx][i]; }
+ auto size() const { return sol_.size(); }
+};
+
+//! Compute the analytical solution at time t
+template<class SolutionVector, class Problem>
+SolutionVector computeAnalyticalSolution(const double t, const Problem& problem)
+{
+ const auto& gg = problem.gridGeometry();
+ SolutionVector exactSolution(gg.numDofs());
+ for (const auto& element : elements(gg.gridView()))
+ {
+ const auto eIdx = gg.elementMapper().index(element);
+ const auto& globalPos = element.geometry().center();
+ exactSolution[eIdx] = problem.analyticalSolution(t, globalPos);
+ }
+ return exactSolution;
+}
+
+//! Compute L2 error wrt the analytical solution at time t
+template<class SolutionVector, class GridGeometry>
+typename SolutionVector::block_type
+computeL2Error(const double t,
+ const SolutionVector& exactSolution,
+ const SolutionVector& curSol,
+ const GridGeometry& gridGeometry)
+{
+ typename SolutionVector::block_type l2Error(0.0);
+ for (const auto& element : elements(gridGeometry.gridView(), Dune::Partitions::interior))
+ {
+ auto fvGeometry = localView(gridGeometry);
+ fvGeometry.bindElement(element);
+
+ using std::pow;
+ for (auto&& scv : scvs(fvGeometry))
+ {
+ auto localDiffSq = exactSolution[scv.dofIndex()] - curSol[scv.dofIndex()];
+ for (int i = 0; i < localDiffSq.size(); ++i)
+ localDiffSq[i] *= localDiffSq[i];
+ l2Error += localDiffSq;
+ }
+ }
+ // sum over processes if we are running with multiple processes in parallel
+ if (gridGeometry.gridView().comm().size() > 0)
+ l2Error = gridGeometry.gridView().comm().sum(l2Error);
+
+ // square root of sum of squared errors is our absolute discrete l2 error
+ for (int i = 0; i < l2Error.size(); ++i)
+ l2Error[i] = std::sqrt(l2Error[i]);
+ return l2Error;
+}
+
+//! Compute and print L2 error wrt the analytical solution at time t
+template<class SolutionVector, class GridGeometry>
+void computeAndPrintL2Error(const double t,
+ const SolutionVector& exactSolution,
+ const SolutionVector& curSol,
+ const GridGeometry& gridGeometry)
+{
+ const auto l2Error = computeL2Error(t, exactSolution, curSol, gridGeometry);
+ auto numElements = gridGeometry.gridView().size(0);
+ // sum over processes if we are running with multiple processes in parallel
+ if (gridGeometry.gridView().comm().size() > 0)
+ numElements = gridGeometry.gridView().comm().sum(numElements);
+
+ if (gridGeometry.gridView().comm().rank() == 0)
+ std::cout << "L2 error in (h, u, v) at t = " << t << " seconds for " << numElements << " elements: "
+ << std::scientific << l2Error << std::endl;
+}
////////////////////////
// the main function
@@ -86,7 +162,7 @@ int main(int argc, char** argv) try
// the solution vector
using SolutionVector = GetPropType<TypeTag, Properties::SolutionVector>;
- SolutionVector x(gridGeometry->numDofs());
+ SolutionVector x;
problem->applyInitialSolution(x);
auto xOld = x;
@@ -97,16 +173,20 @@ int main(int argc, char** argv) try
// get some time loop parameters
using Scalar = GetPropType<TypeTag, Properties::Scalar>;
- const auto tEnd = getParam<Scalar>("TimeLoop.TEnd");
+ const auto tEnd = 2.0*problem->oscillationPeriodInSeconds();
const auto maxDt = getParam<Scalar>("TimeLoop.MaxTimeStepSize");
auto dt = getParam<Scalar>("TimeLoop.DtInitial");
- // intialize the vtk output module
- using IOFields = GetPropType<TypeTag, Properties::IOFields>;
-
+ // intialize the vtk output module and analytical solution
VtkOutputModule<GridVariables, SolutionVector> vtkWriter(*gridVariables,x, problem->name());
- vtkWriter.addField(problem->getExactWaterDepth(), "exactWaterDepth");
- problem->updateAnalyticalSolution(0.0);
+ auto exactSolution = computeAnalyticalSolution<SolutionVector>(0.0, *problem);
+ const auto exactWaterDepth = SolutionComponent<SolutionVector, 0>(exactSolution);
+ const auto exactVelocityX = SolutionComponent<SolutionVector, 1>(exactSolution);
+ const auto exactVelocityY = SolutionComponent<SolutionVector, 2>(exactSolution);
+ vtkWriter.addField(exactWaterDepth, "exactWaterDepth");
+ vtkWriter.addField(exactVelocityX, "exactVelocityX");
+ vtkWriter.addField(exactVelocityY, "exactVelocityY");
+ using IOFields = GetPropType<TypeTag, Properties::IOFields>;
IOFields::initOutputModule(vtkWriter);
vtkWriter.write(0.0);
@@ -126,11 +206,8 @@ int main(int argc, char** argv) try
using NewtonSolver = Dumux::NewtonSolver<Assembler, LinearSolver>;
NewtonSolver nonLinearSolver(assembler, linearSolver);
- //! set some check point at the end of the time loop
- timeLoop->setCheckPoint(tEnd);
-
- //! Compute L2 error for the initial solution (should be zero)
- problem->computeL2error(timeLoop->time(), x, *gridVariables);
+ //! Compute L2 error for the initial solution (should be zero because initial solution is exact)
+ computeAndPrintL2Error(0.0, exactSolution, x, *gridGeometry);
// time loop
timeLoop->start(); do
@@ -141,15 +218,15 @@ int main(int argc, char** argv) try
xOld = x;
gridVariables->advanceTimeStep();
- // update the analytical solution
- problem->updateAnalyticalSolution(timeLoop->time());
- problem->computeL2error(timeLoop->time(), x, *gridVariables);
+ // update the analytical solution and print current l2 error
+ exactSolution = computeAnalyticalSolution<SolutionVector>(timeLoop->time(), *problem);
+ computeAndPrintL2Error(timeLoop->time(), exactSolution, x, *gridGeometry);
// advance to the time loop to the next step
timeLoop->advanceTimeStep();
- // write vtk output
- if (timeLoop->isCheckPoint())
+ // write vtk output every 10 time steps
+ if (!(timeLoop->timeStepIndex() % 10))
vtkWriter.write(timeLoop->time());
// report statistics of this time step
@@ -176,28 +253,14 @@ int main(int argc, char** argv) try
return 0;
}
-
+// Error handling
catch (const Dumux::ParameterException &e)
{
std::cerr << std::endl << e << " ---> Abort!" << std::endl;
return 1;
}
-catch (const Dune::DGFException & e)
-{
- std::cerr << "DGF exception thrown (" << e <<
- "). Most likely, the DGF file name is wrong "
- "or the DGF file is corrupted, "
- "e.g. missing hash at end of file or wrong number (dimensions) of entries."
- << " ---> Abort!" << std::endl;
- return 2;
-}
catch (const Dune::Exception &e)
{
std::cerr << "Dune reported error: " << e << " ---> Abort!" << std::endl;
- return 3;
-}
-catch (...)
-{
- std::cerr << "Unknown exception thrown! ---> Abort!" << std::endl;
- return 4;
+ return 2;
}
diff --git a/test/freeflow/shallowwater/bowl/params.input b/test/freeflow/shallowwater/bowl/params.input
index 6c91f7db0eaf8223372baaa16384c73d0cc74e07..c5fa78af94dc8da2e3940dbf96752845f474b106 100644
--- a/test/freeflow/shallowwater/bowl/params.input
+++ b/test/freeflow/shallowwater/bowl/params.input
@@ -11,29 +11,14 @@ LowerWaterDepth = 1.0E-7
UpwindFluxLimiting = true
[TimeLoop]
-TEnd = 13.104 # [s] 6.552 is one period
MaxTimeStepSize = 0.1 # [s]
-DtInitial = 0.01 # [s]
+DtInitial = 0.1 # [s]
[Grid]
Positions0 = -2.0 2.0
Positions1 = -2.0 2.0
-Cells0 = 50
-Cells1 = 50
+Cells0 = 40
+Cells1 = 40
[Newton]
-EnableAbsoluteResidualCriterion = true
-MaxAbsoluteResidual = 1e-6
-RetryTimeStepReductionFactor = 0.5
-SatisfyResidualAndShiftCriterion = "false"
EnableDynamicOutput = false
-EnablePartialReassembly = false
-TargetSteps = 6
-MaxSteps = 10
-UseLineSearch = true
-
-[LinearSolver]
-MaxIterations = 200
-ResidualReduction = 1e-5
-
-
diff --git a/test/freeflow/shallowwater/bowl/problem.hh b/test/freeflow/shallowwater/bowl/problem.hh
index 89a09b7200e55b6cb15605304074545a13b86551..3cc77a886a982e8740a8ddd715eeb20aa1546ce6 100644
--- a/test/freeflow/shallowwater/bowl/problem.hh
+++ b/test/freeflow/shallowwater/bowl/problem.hh
@@ -24,59 +24,11 @@
#ifndef DUMUX_BOWL_TEST_PROBLEM_HH
#define DUMUX_BOWL_TEST_PROBLEM_HH
-#include <dune/grid/yaspgrid.hh>
-#include <dumux/discretization/cctpfa.hh>
-#include "spatialparams.hh"
-#include <dumux/common/parameters.hh>
-
-#include <dumux/freeflow/shallowwater/model.hh>
#include <dumux/freeflow/shallowwater/problem.hh>
#include <dumux/freeflow/shallowwater/boundaryfluxes.hh>
namespace Dumux {
-template <class TypeTag>
-class BowlProblem;
-
-// Specify the properties for the problem
-namespace Properties {
-
-// Create new type tags
-namespace TTag {
-struct Bowl { using InheritsFrom = std::tuple<ShallowWater, CCTpfaModel>; };
-} // end namespace TTag
-
-template<class TypeTag>
-struct Grid<TypeTag, TTag::Bowl>
-{ using type = Dune::YaspGrid<2, Dune::TensorProductCoordinates<GetPropType<TypeTag, Properties::Scalar>, 2> >; };
-
-// Set the problem property
-template<class TypeTag>
-struct Problem<TypeTag, TTag::Bowl>
-{ using type = Dumux::BowlProblem<TypeTag>; };
-
-// Set the spatial parameters
-template<class TypeTag>
-struct SpatialParams<TypeTag, TTag::Bowl>
-{
-private:
- using GridGeometry = GetPropType<TypeTag, Properties::GridGeometry>;
- using Scalar = GetPropType<TypeTag, Properties::Scalar>;
- using ElementVolumeVariables = typename GetPropType<TypeTag, Properties::GridVolumeVariables>::LocalView;
- using VolumeVariables = typename ElementVolumeVariables::VolumeVariables;
-public:
- using type = BowlSpatialParams<GridGeometry, Scalar, VolumeVariables>;
-};
-
-template<class TypeTag>
-struct EnableGridGeometryCache<TypeTag, TTag::Bowl>
-{ static constexpr bool value = true; };
-
-template<class TypeTag>
-struct EnableGridVolumeVariablesCache<TypeTag, TTag::Bowl>
-{ static constexpr bool value = false; };
-} // end namespace Properties
-
/*!
* \ingroup ShallowWaterTests
* \brief A wetting and drying test with sloshing water in a bowl.
@@ -89,11 +41,14 @@ struct EnableGridVolumeVariablesCache<TypeTag, TTag::Bowl>
* (UpwindFluxLimiting = true) will help to improve the convergence for such cases.
*
* This test uses a low mesh resoultion and only ensures that UpwindFluxLimiting for the mobility
- * works.
+ * works. For low mesh resolution the solution is very diffusive so that the oscillation is dampened.
+ * This gets better with grid refinement (not tested here).
*
* The results are checked against a analytical solution which is based on the "Thacker-Solution"
* (William Thacker, "Some exact solutions to the nonlinear shallow-water wave equations", Journal
- * of Fluid Mechanics, 107:499–508, 1981). Further examples and details on the solution are given
+ * of Fluid Mechanics, 107:499–508, 1981, doi: https://doi.org/10.1017/S0022112081001882).
+ * This implements the oscillating solution in a circular bowl (Section 4 in the paper).
+ * Further examples and details on the solution are given
* in SWASHES (Shallow Water Analytic Solutions for Hydraulic and Environmental Studies,
* https://www.idpoisson.fr/swashes/).
*
@@ -103,125 +58,93 @@ template <class TypeTag>
class BowlProblem : public ShallowWaterProblem<TypeTag>
{
using ParentType = ShallowWaterProblem<TypeTag>;
- using PrimaryVariables = GetPropType<TypeTag, Properties::PrimaryVariables>;
- using BoundaryTypes = GetPropType<TypeTag, Properties::BoundaryTypes>;
- using Scalar = GetPropType<TypeTag, Properties::Scalar>;
- using Indices = typename GetPropType<TypeTag, Properties::ModelTraits>::Indices;
+
using GridGeometry = GetPropType<TypeTag, Properties::GridGeometry>;
- using NeumannFluxes = GetPropType<TypeTag, Properties::NumEqVector>;
- using ElementVolumeVariables = typename GetPropType<TypeTag, Properties::GridVolumeVariables>::LocalView;
- using GridVariables = GetPropType<TypeTag, Properties::GridVariables>;
- using ElementFluxVariablesCache = typename GridVariables::GridFluxVariablesCache::LocalView;
- using VolumeVariables = typename ElementVolumeVariables::VolumeVariables;
using FVElementGeometry = typename GetPropType<TypeTag, Properties::GridGeometry>::LocalView;
+ using SubControlVolume = typename FVElementGeometry::SubControlVolume;
using SubControlVolumeFace = typename FVElementGeometry::SubControlVolumeFace;
using GridView = GetPropType<TypeTag, Properties::GridView>;
using Element = typename GridView::template Codim<0>::Entity;
using GlobalPosition = typename Element::Geometry::GlobalCoordinate;
+
+ using Scalar = GetPropType<TypeTag, Properties::Scalar>;
using NumEqVector = GetPropType<TypeTag, Properties::NumEqVector>;
- using SubControlVolume = typename FVElementGeometry::SubControlVolume;
- using SolutionVector = GetPropType<TypeTag, Properties::SolutionVector>;
+ using PrimaryVariables = GetPropType<TypeTag, Properties::PrimaryVariables>;
+
+ using NeumannFluxes = GetPropType<TypeTag, Properties::NumEqVector>;
+ using ElementVolumeVariables = typename GetPropType<TypeTag, Properties::GridVolumeVariables>::LocalView;
+ using Indices = typename GetPropType<TypeTag, Properties::ModelTraits>::Indices;
+ using BoundaryTypes = GetPropType<TypeTag, Properties::BoundaryTypes>;
+
public:
BowlProblem(std::shared_ptr<const GridGeometry> gridGeometry)
: ParentType(gridGeometry)
{
- using std::sqrt;
- using std::pow;
name_ = getParam<std::string>("Problem.Name");
- gravity_ = getParam<Scalar>("Problem.Gravity");
bowlDepthAtCenter_ = getParam<Scalar>("Problem.BowlDepthAtCenter");
bowlParaboloidRadius_ = getParam<Scalar>("Problem.BowlParaboloidRadius");
- bowlInitialWaterElevationAtCenter_ = getParam<Scalar>("Problem.BowlInitialWaterElevationAtCenter");
- bowlAnalyticParameterOmega_ = sqrt(8.0 * gravity_ * bowlDepthAtCenter_) / bowlParaboloidRadius_;
- bowlAnalyticParameterA_ = (pow(bowlDepthAtCenter_ + bowlInitialWaterElevationAtCenter_, 2.0) -
- pow(bowlDepthAtCenter_, 2.0)) /
- (pow(bowlDepthAtCenter_ + bowlInitialWaterElevationAtCenter_, 2.0)+
- pow(bowlDepthAtCenter_, 2.0));
- exactWaterDepth_.resize(gridGeometry->numDofs(), 0.0);
- updateAnalyticalSolution(0.0);
- }
- //! Get the analytical water depth
- const std::vector<Scalar>& getExactWaterDepth()
- {
- return exactWaterDepth_;
- }
-
- //! Get the analctic water depth at
- Scalar calculateAnalyticWaterDepth(Scalar time, Scalar x, Scalar y)
- {
- using std::max;
+ // Thacker (1981) Eq. (43)
using std::sqrt;
- using std::pow;
- using std::cos;
-
- auto waterDepth = max(bowlDepthAtCenter_* ((sqrt(1.0 - pow(bowlAnalyticParameterA_, 2.0)))/
- (1.0 - bowlAnalyticParameterA_ * cos(bowlAnalyticParameterOmega_ * time))-
- 1.0 - (pow(x, 2.0) + pow(y, 2.0)) / pow(bowlParaboloidRadius_, 2.0) *
- ((1.0- pow(bowlAnalyticParameterA_, 2.0)) /
- (pow(1.0 - bowlAnalyticParameterA_ *
- cos(bowlAnalyticParameterOmega_ * time), 2.0)) - 1.0)) -
- bowlDepthAtCenter_ *((pow(x, 2.0) + pow(y, 2.0))/
- pow(bowlParaboloidRadius_, 2.0) - 1.0), 1.0E-5);
-
- return waterDepth;
+ bowlAnalyticParameterOmega_ = sqrt(8.0 * getParam<Scalar>("Problem.Gravity") * bowlDepthAtCenter_) / bowlParaboloidRadius_;
+ std::cout << "One full oscillation period of the water table is: "
+ << oscillationPeriodInSeconds() << " seconds." << std::endl;
+
+ // Thacker (1981) Eq. (50)
+ const auto D0PlusEta = bowlDepthAtCenter_ + getParam<Scalar>("Problem.BowlInitialWaterElevationAtCenter");
+ const auto D0PlusEtaSquared = D0PlusEta*D0PlusEta;
+ const auto D0Squared = bowlDepthAtCenter_*bowlDepthAtCenter_;
+ bowlAnalyticParameterA_ = (D0PlusEtaSquared - D0Squared)/(D0PlusEtaSquared + D0Squared);
+
+ // check constraint Thacker (1981) text after Eq. (49)
+ if (bowlAnalyticParameterA_ >= 1.0)
+ DUNE_THROW(Dune::InvalidStateException, "Parameter A has to be smaller than unity!");
}
- //! Compute L2 error
- void computeL2error(const Scalar time,
- const SolutionVector& curSol,
- const GridVariables& gridVariables)
- {
- Scalar l2error = 0.0;
-
- //first ensure that the Analytical solution is up-to-date
- updateAnalyticalSolution(time);
-
- for (const auto& element : elements(this->gridGeometry().gridView()))
- {
- const auto eIdx = this->gridGeometry().elementMapper().index(element);
- const auto& globalPos = element.geometry().center();
- auto fvGeometry = localView(this->gridGeometry());
- fvGeometry.bindElement(element);
- auto elemVolVars = localView(gridVariables.curGridVolVars());
- elemVolVars.bindElement(element, fvGeometry, curSol);
- exactWaterDepth_[eIdx] = calculateAnalyticWaterDepth(time, globalPos[0], globalPos[1]);
-
- for (auto&& scv : scvs(fvGeometry))
- {
- using std::pow;
- l2error += pow(exactWaterDepth_[eIdx] - elemVolVars[scv].waterDepth(), 2.0);
- }
- }
- using std::sqrt;
- l2error = sqrt(l2error);
- l2error = this->gridGeometry().gridView().comm().sum(l2error);
-
- if (this->gridGeometry().gridView().comm().rank() == 0)
- {
- std::cout << "L2 error at t = "
- << time << " seconds "
- << " for "
- << std::setw(6) << this->gridGeometry().gridView().size(0)
- << " elements: "
- << std::scientific
- << l2error
- << std::endl;
- }
- }
+ //! One oscillation period of the water table (analytically this goes on forever)
+ Scalar oscillationPeriodInSeconds() const
+ { return 2*M_PI/bowlAnalyticParameterOmega_; }
- //! Udpate the analytical solution
- void updateAnalyticalSolution(const Scalar time)
+ //! Get the analytical water depth at time t and position pos
+ PrimaryVariables analyticalSolution(const Scalar t, const GlobalPosition& pos) const
{
- for (const auto& element : elements(this->gridGeometry().gridView()))
- {
- const auto eIdx = this->gridGeometry().elementMapper().index(element);
- const auto& globalPos = element.geometry().center();
-
- exactWaterDepth_[eIdx] = calculateAnalyticWaterDepth(time, globalPos[0], globalPos[1]);
- }
+ using std::sqrt;
+ using std::cos;
+ using std::sin;
+
+ // see Thacker (1981) Eq. (51) for formula
+ const auto radiusSquared = pos[0]*pos[0] + pos[1]*pos[1];
+ const auto LSquared = bowlParaboloidRadius_*bowlParaboloidRadius_;
+ const auto A = bowlAnalyticParameterA_;
+ const auto omega = bowlAnalyticParameterOmega_;
+ const auto D0 = bowlDepthAtCenter_;
+
+ const auto oneMinusASq = 1.0 - A*A;
+ const auto oneMinusACosOmegaT = 1.0 - A*cos(omega*t);
+ const auto ratioSq = oneMinusASq / (oneMinusACosOmegaT*oneMinusACosOmegaT);
+ const auto localRadiusSq = radiusSquared / LSquared;
+ // bowl depth function (cf. D in Thacker (1981))
+ const auto D = D0*(1.0 - localRadiusSq);
+ // height above equilibrium water level (cf. h in Thacker (1981))
+ const auto h = D0*(sqrt(ratioSq) - 1.0 - localRadiusSq*(ratioSq - 1.0));
+ // see remark about total water depth in Thacker (1981) beginning section 2
+ const auto analyticalWaterDepth = h + D;
+
+ const auto halfOmegaASinOmegaT = 0.5*omega*A*sin(omega*t);
+ const auto analyticalVelocityX = pos[0]*halfOmegaASinOmegaT/oneMinusACosOmegaT;
+ const auto analyticalVelocityY = pos[1]*halfOmegaASinOmegaT/oneMinusACosOmegaT;
+
+ // The radius of the shoreline (where h + D = 0), Eq. (48)
+ const auto h0 = D0*(sqrt(ratioSq) - 1.0); // h in the middle of the bowl (r=0)
+ const auto shoreLineRadiusSquared = LSquared*(D0/(D0 + h0));
+
+ // outside shoreline the water height and velocity is zero
+ if (radiusSquared > shoreLineRadiusSquared)
+ return { 0.0, 0.0, 0.0 };
+ else
+ return { analyticalWaterDepth, analyticalVelocityX, analyticalVelocityY };
}
/*!
@@ -231,41 +154,10 @@ public:
/*!
* \brief The problem name
- *
* This is used as a prefix for files generated by the simulation.
*/
const std::string& name() const
- {
- return name_;
- }
-
- /*!
- * \brief Evaluate the source term for all balance equations within a given
- * sub-control-volume.
- *
- * This is the method for the case where the source term is
- * potentially solution dependent and requires some quantities that
- * are specific to the fully-implicit method.
- *
- * \param element The finite element
- * \param fvGeometry The finite-volume geometry
- * \param elemVolVars All volume variables for the element
- * \param scv The sub control volume
- *
- * For this method, the \a values parameter stores the conserved quantity rate
- * generated or annihilate per volume unit. Positive values mean
- * that the conserved quantity is created, negative ones mean that it vanishes.
- * E.g. for the mass balance that would be a mass rate in \f$ [ kg / (m^3 \cdot s)] \f$.
- */
- NumEqVector source(const Element& element,
- const FVElementGeometry& fvGeometry,
- const ElementVolumeVariables& elemVolVars,
- const SubControlVolume &scv) const
- {
- NumEqVector source (0.0);
-
- return source;
- }
+ { return name_; }
// \}
@@ -277,8 +169,6 @@ public:
/*!
* \brief Specifies which kind of boundary condition should be
* used for which equation on a given boundary segment.
- *
- * \param globalPos The position for which the boundary type is set
*/
BoundaryTypes boundaryTypesAtPos(const GlobalPosition &globalPos) const
{
@@ -293,12 +183,8 @@ public:
* We need the Riemann invariants to compute the values depending of the boundary type.
* Since we use a weak imposition we do not have a dirichlet value. We impose fluxes
* based on q, h, etc. computed with the Riemann invariants
- *
- * \param element
- * \param fvGeometry
- * \param elemVolVars
- * \param scvf
*/
+ template<class ElementFluxVariablesCache>
NeumannFluxes neumann(const Element& element,
const FVElementGeometry& fvGeometry,
const ElementVolumeVariables& elemVolVars,
@@ -321,16 +207,12 @@ public:
boundaryStateVariables[1] = nxy[0] * vNormalGhost - nxy[1] * vTangentialGhost;
boundaryStateVariables[2] = nxy[1] * vNormalGhost + nxy[0] * vTangentialGhost;
- auto riemannFlux = ShallowWater::riemannProblem(insideVolVars.waterDepth(),
- boundaryStateVariables[0],
- insideVolVars.velocity(0),
- boundaryStateVariables[1],
- insideVolVars.velocity(1),
- boundaryStateVariables[2],
- insideVolVars.bedSurface(),
- insideVolVars.bedSurface(),
- gravity,
- nxy);
+ const auto riemannFlux =
+ ShallowWater::riemannProblem(insideVolVars.waterDepth(), boundaryStateVariables[0],
+ insideVolVars.velocity(0), boundaryStateVariables[1],
+ insideVolVars.velocity(1), boundaryStateVariables[2],
+ insideVolVars.bedSurface(), insideVolVars.bedSurface(),
+ gravity, nxy);
values[Indices::massBalanceIdx] = riemannFlux[0];
values[Indices::velocityXIdx] = riemannFlux[1];
@@ -347,33 +229,19 @@ public:
// \{
/*!
- * \brief Evaluate the initial values for a control volume.
- *
- * For this method, the \a values parameter stores primary
- * variables.
- *
- * \param globalPos The position for which the boundary type is set
+ * \brief Evaluate the initial values at position globalPos
*/
- PrimaryVariables initial(const Element& element) const
+ PrimaryVariables initialAtPos(const GlobalPosition& globalPos) const
{
- PrimaryVariables values(0.0);
-
- auto elemId = this->gridGeometry().elementMapper().index(element);
- values[0] = exactWaterDepth_[elemId];
- values[1] = 0.0;
- values[2] = 0.0;
-
- return values;
+ using std::max; // regularize so that we are virtually dry but not completely dry
+ return { max(analyticalSolution(0, globalPos)[0], 1e-5), 0.0, 0.0 };
};
// \}
private:
- std::vector<Scalar> exactWaterDepth_;
- Scalar gravity_;
Scalar bowlDepthAtCenter_;
Scalar bowlParaboloidRadius_;
- Scalar bowlInitialWaterElevationAtCenter_;
Scalar bowlAnalyticParameterOmega_;
Scalar bowlAnalyticParameterA_;
static constexpr Scalar eps_ = 1.0e-6;
diff --git a/test/freeflow/shallowwater/bowl/properties.hh b/test/freeflow/shallowwater/bowl/properties.hh
new file mode 100644
index 0000000000000000000000000000000000000000..68a91568151edabc0ed4fe24bafff13e680eb1c3
--- /dev/null
+++ b/test/freeflow/shallowwater/bowl/properties.hh
@@ -0,0 +1,73 @@
+// -*- mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
+// vi: set et ts=4 sw=4 sts=4:
+/*****************************************************************************
+ * See the file COPYING for full copying permissions. *
+ * *
+ * This program is free software: you can redistribute it and/or modify *
+ * it under the terms of the GNU General Public License as published by *
+ * the Free Software Foundation, either version 2 of the License, or *
+ * (at your option) any later version. *
+ * *
+ * This program is distributed in the hope that it will be useful, *
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of *
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *
+ * GNU General Public License for more details. *
+ * *
+ * You should have received a copy of the GNU General Public License *
+ * along with this program. If not, see <http://www.gnu.org/licenses/>. *
+ *****************************************************************************/
+/*!
+ * \file
+ * \brief The properties for the shallow water bowl test
+ */
+#ifndef DUMUX_BOWL_TEST_PROPERTIES_HH
+#define DUMUX_BOWL_TEST_PROPERTIES_HH
+
+#include <dune/grid/yaspgrid.hh>
+#include <dumux/discretization/cctpfa.hh>
+#include <dumux/common/properties.hh>
+#include <dumux/freeflow/shallowwater/model.hh>
+
+#include "spatialparams.hh"
+#include "problem.hh"
+
+namespace Dumux::Properties {
+
+// Create new type tags
+namespace TTag {
+struct Bowl { using InheritsFrom = std::tuple<ShallowWater, CCTpfaModel>; };
+} // end namespace TTag
+
+template<class TypeTag>
+struct Grid<TypeTag, TTag::Bowl>
+{ using type = Dune::YaspGrid<2, Dune::TensorProductCoordinates<GetPropType<TypeTag, Properties::Scalar>, 2> >; };
+
+// Set the problem property
+template<class TypeTag>
+struct Problem<TypeTag, TTag::Bowl>
+{ using type = Dumux::BowlProblem<TypeTag>; };
+
+// Set the spatial parameters
+template<class TypeTag>
+struct SpatialParams<TypeTag, TTag::Bowl>
+{
+private:
+ using GridGeometry = GetPropType<TypeTag, Properties::GridGeometry>;
+ using Scalar = GetPropType<TypeTag, Properties::Scalar>;
+ using ElementVolumeVariables = typename GetPropType<TypeTag, Properties::GridVolumeVariables>::LocalView;
+ using VolumeVariables = typename ElementVolumeVariables::VolumeVariables;
+public:
+ using type = BowlSpatialParams<GridGeometry, Scalar, VolumeVariables>;
+};
+
+template<class TypeTag>
+struct EnableGridGeometryCache<TypeTag, TTag::Bowl>
+{ static constexpr bool value = true; };
+
+template<class TypeTag>
+struct EnableGridVolumeVariablesCache<TypeTag, TTag::Bowl>
+{ static constexpr bool value = false; };
+
+} // end namespace Dumux::Properties
+
+#endif
diff --git a/test/freeflow/shallowwater/bowl/spatialparams.hh b/test/freeflow/shallowwater/bowl/spatialparams.hh
index b69824c5c61126aff5806bd81f7ce909b511f859..916ce70ab0983c65a152fb5b088554e64e40a28a 100644
--- a/test/freeflow/shallowwater/bowl/spatialparams.hh
+++ b/test/freeflow/shallowwater/bowl/spatialparams.hh
@@ -24,8 +24,8 @@
#ifndef DUMUX_BOWL_SPATIAL_PARAMETERS_HH
#define DUMUX_BOWL_SPATIAL_PARAMETERS_HH
-#include <dumux/material/spatialparams/fv.hh>
#include <dumux/common/parameters.hh>
+#include <dumux/material/spatialparams/fv.hh>
#include <dumux/material/fluidmatrixinteractions/frictionlaws/frictionlaw.hh>
#include <dumux/material/fluidmatrixinteractions/frictionlaws/nofriction.hh>
@@ -34,7 +34,9 @@ namespace Dumux {
/*!
* \ingroup ShallowWaterTests
* \brief The spatial parameters class for the bowl test.
- *
+ * \note Analytical solution from Thacker (1981) section 4
+ * William Thacker, "Some exact solutions to the nonlinear shallow-water wave equations", Journal
+ * of Fluid Mechanics, 107:499–508, 1981
*/
template<class GridGeometry, class Scalar, class VolumeVariables>
class BowlSpatialParams
@@ -54,29 +56,22 @@ public:
: ParentType(gridGeometry)
{
gravity_ = getParam<Scalar>("Problem.Gravity");
- const Scalar bowlDepthAtCenter = getParam<Scalar>("Problem.BowlDepthAtCenter");
- const Scalar bowlParaboloidRadius = getParam<Scalar>("Problem.BowlParaboloidRadius");
- initFrictionLaw();
+ frictionLaw_ = std::make_unique<FrictionLawNoFriction<VolumeVariables>>();
bedSurface_.assign(this->gridGeometry().gridView().size(0), 0.0);
- //compute the bedSurface for all elements
+ // compute the bedSurface for all elements
+ const Scalar D0 = getParam<Scalar>("Problem.BowlDepthAtCenter");
+ const Scalar L = getParam<Scalar>("Problem.BowlParaboloidRadius");
for (const auto& element : elements(this->gridGeometry().gridView()))
{
const auto eIdx = this->gridGeometry().elementMapper().index(element);
const auto& globalPos = element.geometry().center();
- bedSurface_[eIdx] = bowlDepthAtCenter *(((globalPos[0] * globalPos[0]) +
- (globalPos[1] * globalPos[1])) /
- (bowlParaboloidRadius * bowlParaboloidRadius) - 1.0) + 1.0;
+ // see Thacker (1981) Eq. (13) + shift bowl so that midpoint is a z=0
+ // the shift is arbitrary and doesn't affect the analytical solution
+ const auto radiusSquared = globalPos[0]*globalPos[0] + globalPos[1]*globalPos[1];
+ const auto D = D0*(1.0 - radiusSquared/(L*L));
+ bedSurface_[eIdx] = D0 - D;
}
-
- }
-
- /*!
- * \brief Initialize the FrictionLaw
- */
- void initFrictionLaw()
- {
- frictionLaw_ = std::make_unique<FrictionLawNoFriction<VolumeVariables>>();
}
/*! \brief Define the gravitation.
@@ -88,15 +83,6 @@ public:
return gravity_;
}
- /*! \brief Define the gravitation.
- *
- * \return gravity constant
- */
- Scalar gravity() const
- {
- return gravity_;
- }
-
/*! \brief Get the frictionLaw.
*
* Get the frictionLaw, which already includes the friction value.
@@ -106,9 +92,7 @@ public:
const FrictionLaw<VolumeVariables>& frictionLaw(const Element& element,
const SubControlVolume& scv) const
- {
- return *frictionLaw_;
- }
+ { return *frictionLaw_; }
/*! \brief Define the bed surface
*
@@ -119,10 +103,7 @@ public:
*/
Scalar bedSurface(const Element& element,
const SubControlVolume& scv) const
- {
- auto eIdx = scv.elementIndex();
- return bedSurface_[eIdx];
- }
+ { return bedSurface_[scv.elementIndex()]; }
private:
std::vector<Scalar> bedSurface_;
diff --git a/test/references/test_ff_shallowwater_bowl-reference.vtu b/test/references/test_ff_shallowwater_bowl-reference.vtu
index 51a73f2ea78ecae93e17f84131bcc5faac8efb73..5a89864c231e8135518eedbb700ae210f8583a26 100644
--- a/test/references/test_ff_shallowwater_bowl-reference.vtu
+++ b/test/references/test_ff_shallowwater_bowl-reference.vtu
@@ -1,1062 +1,687 @@
<?xml version="1.0"?>
<VTKFile type="UnstructuredGrid" version="0.1" byte_order="LittleEndian">
<UnstructuredGrid>
- <Piece NumberOfCells="2500" NumberOfPoints="2601">
+ <Piece NumberOfCells="1600" NumberOfPoints="1681">
<CellData Scalars="waterDepth">
<DataArray type="Float32" Name="waterDepth" NumberOfComponents="1" format="ascii">
- 4.97967e-06 5.45262e-06 5.85379e-06 6.12838e-06 6.28307e-06 6.35455e-06 6.38208e-06 6.39111e-06 6.39368e-06 6.39434e-06 6.39449e-06 6.39452e-06
- 6.39452e-06 6.39452e-06 6.39452e-06 6.39452e-06 6.39452e-06 6.39452e-06 6.39452e-06 6.39452e-06 6.39452e-06 6.39452e-06 6.39452e-06 6.39452e-06
- 9.75747e-06 9.75747e-06 6.39452e-06 6.39452e-06 6.39452e-06 6.39452e-06 6.39452e-06 6.39452e-06 6.39452e-06 6.39452e-06 6.39452e-06 6.39452e-06
- 6.39452e-06 6.39452e-06 6.39452e-06 6.39449e-06 6.39434e-06 6.39368e-06 6.39111e-06 6.38208e-06 6.35455e-06 6.28307e-06 6.12838e-06 5.85379e-06
- 5.45262e-06 4.97967e-06 5.45262e-06 6.05905e-06 6.63775e-06 7.06219e-06 7.31107e-06 7.42984e-06 7.47732e-06 7.4936e-06 7.49848e-06 7.49978e-06
- 7.50009e-06 7.50016e-06 7.50017e-06 7.50017e-06 7.50017e-06 7.50017e-06 7.50017e-06 7.50017e-06 7.50017e-06 7.50017e-06 7.50017e-06 7.50017e-06
- 7.50017e-06 7.50017e-06 1.29211e-05 1.29211e-05 7.50017e-06 7.50017e-06 7.50017e-06 7.50017e-06 7.50017e-06 7.50017e-06 7.50017e-06 7.50017e-06
- 7.50017e-06 7.50017e-06 7.50017e-06 7.50017e-06 7.50016e-06 7.50009e-06 7.49978e-06 7.49848e-06 7.4936e-06 7.47732e-06 7.42984e-06 7.31107e-06
- 7.06219e-06 6.63775e-06 6.05905e-06 5.45262e-06 5.85379e-06 6.63775e-06 7.42978e-06 8.0105e-06 8.34165e-06 8.49638e-06 8.5579e-06 8.57912e-06
- 8.58556e-06 8.5873e-06 8.58772e-06 8.58781e-06 8.58783e-06 8.58784e-06 8.58784e-06 8.58784e-06 8.58784e-06 8.58784e-06 8.58784e-06 8.58784e-06
- 8.58784e-06 8.58784e-06 8.58784e-06 8.58784e-06 1.5404e-05 1.5404e-05 8.58784e-06 8.58784e-06 8.58784e-06 8.58784e-06 8.58784e-06 8.58784e-06
- 8.58784e-06 8.58784e-06 8.58784e-06 8.58784e-06 8.58784e-06 8.58783e-06 8.58781e-06 8.58772e-06 8.5873e-06 8.58556e-06 8.57912e-06 8.5579e-06
- 8.49638e-06 8.34165e-06 8.0105e-06 7.42978e-06 6.63775e-06 5.85379e-06 6.12838e-06 7.06219e-06 8.0105e-06 8.68279e-06 9.05252e-06 9.22233e-06
- 9.28962e-06 9.3129e-06 9.32001e-06 9.32194e-06 9.32242e-06 9.32253e-06 9.32255e-06 9.32255e-06 9.32255e-06 9.32255e-06 9.32255e-06 9.32255e-06
- 9.32255e-06 9.32255e-06 9.32255e-06 9.32255e-06 9.32255e-06 9.32255e-06 1.69614e-05 1.69614e-05 9.32255e-06 9.32255e-06 9.32255e-06 9.32255e-06
- 9.32255e-06 9.32255e-06 9.32255e-06 9.32255e-06 9.32255e-06 9.32255e-06 9.32255e-06 9.32255e-06 9.32253e-06 9.32242e-06 9.32194e-06 9.32001e-06
- 9.3129e-06 9.28962e-06 9.22233e-06 9.05252e-06 8.68279e-06 8.0105e-06 7.06219e-06 6.12838e-06 6.28307e-06 7.31107e-06 8.34165e-06 9.05252e-06
- 9.43617e-06 9.61139e-06 9.68087e-06 9.70497e-06 9.71235e-06 9.71437e-06 9.71486e-06 9.71498e-06 9.715e-06 9.715e-06 9.71501e-06 9.71501e-06
- 9.71501e-06 9.71501e-06 9.71501e-06 9.71501e-06 9.71501e-06 5.61796e-05 6.16977e-05 8.81145e-05 0.000201285 0.000201285 8.81145e-05 6.16977e-05
- 5.61796e-05 9.71501e-06 9.71501e-06 9.71501e-06 9.71501e-06 9.71501e-06 9.71501e-06 9.71501e-06 9.715e-06 9.715e-06 9.71498e-06 9.71486e-06
- 9.71437e-06 9.71235e-06 9.70497e-06 9.68087e-06 9.61139e-06 9.43617e-06 9.05252e-06 8.34165e-06 7.31107e-06 6.28307e-06 6.35455e-06 7.42984e-06
- 8.49638e-06 9.22233e-06 9.61139e-06 9.78886e-06 9.85928e-06 9.88374e-06 9.89125e-06 9.8933e-06 9.89381e-06 9.89392e-06 9.89395e-06 9.89395e-06
- 9.89395e-06 9.89395e-06 9.89395e-06 5.61841e-05 3.51334e-05 3.55726e-05 0.000418909 0.00089611 0.00125089 0.00148811 0.00160839 0.00160839
- 0.00148811 0.00125089 0.00089611 0.000418909 3.55726e-05 3.51334e-05 5.61841e-05 9.89395e-06 9.89395e-06 9.89395e-06 9.89395e-06 9.89395e-06
- 9.89392e-06 9.89381e-06 9.8933e-06 9.89125e-06 9.88374e-06 9.85928e-06 9.78886e-06 9.61139e-06 9.22233e-06 8.49638e-06 7.42984e-06 6.35455e-06
- 6.38208e-06 7.47732e-06 8.5579e-06 9.28962e-06 9.68087e-06 9.85928e-06 9.9301e-06 9.95471e-06 9.96226e-06 9.96433e-06 9.96485e-06 9.96496e-06
- 9.96499e-06 9.96499e-06 9.96499e-06 5.10079e-05 3.09133e-05 0.000514883 0.00115083 0.00194582 0.00270055 0.00333359 0.00379681 0.00409798
- 0.00424502 0.00424502 0.00409798 0.00379681 0.00333359 0.00270055 0.00194582 0.00115083 0.000514883 3.09133e-05 5.10079e-05 9.96499e-06
- 9.96499e-06 9.96499e-06 9.96496e-06 9.96485e-06 9.96433e-06 9.96226e-06 9.95471e-06 9.9301e-06 9.85928e-06 9.68087e-06 9.28962e-06 8.5579e-06
- 7.47732e-06 6.38208e-06 6.39111e-06 7.4936e-06 8.57912e-06 9.3129e-06 9.70497e-06 9.88374e-06 9.95471e-06 9.97938e-06 9.98695e-06 9.98902e-06
- 9.98954e-06 9.98965e-06 9.98968e-06 6.13224e-05 2.38563e-05 0.000516693 0.00140891 0.00255584 0.00362139 0.00455437 0.00534618 0.00598455
- 0.00645932 0.00677164 0.0069265 0.0069265 0.00677164 0.00645932 0.00598455 0.00534618 0.00455437 0.00362139 0.00255584 0.00140891
- 0.000516693 2.38563e-05 6.13224e-05 9.98968e-06 9.98965e-06 9.98954e-06 9.98902e-06 9.98695e-06 9.97938e-06 9.95471e-06 9.88374e-06 9.70497e-06
- 9.3129e-06 8.57912e-06 7.4936e-06 6.39111e-06 6.39368e-06 7.49848e-06 8.58556e-06 9.32001e-06 9.71235e-06 9.89125e-06 9.96226e-06 9.98695e-06
- 9.99452e-06 9.9966e-06 9.99712e-06 9.99723e-06 3.20343e-05 0.000346156 0.00109015 0.00246955 0.00383475 0.00508197 0.00617885 0.00712069
- 0.00790999 0.00854383 0.00901822 0.00933268 0.00948981 0.00948981 0.00933268 0.00901822 0.00854383 0.00790999 0.00712069 0.00617885
- 0.00508197 0.00383475 0.00246955 0.00109015 0.000346156 3.20343e-05 9.99723e-06 9.99712e-06 9.9966e-06 9.99452e-06 9.98695e-06 9.96226e-06
- 9.89125e-06 9.71235e-06 9.32001e-06 8.58556e-06 7.49848e-06 6.39368e-06 6.39434e-06 7.49978e-06 8.5873e-06 9.32194e-06 9.71437e-06 9.8933e-06
- 9.96433e-06 9.98902e-06 9.9966e-06 9.99868e-06 9.99919e-06 1.9095e-05 0.000472582 0.00169702 0.00332648 0.00487046 0.00626909 0.00751952
- 0.00861504 0.00955467 0.0103398 0.0109695 0.0114421 0.0117568 0.0119146 0.0119146 0.0117568 0.0114421 0.0109695 0.0103398
- 0.00955467 0.00861504 0.00751952 0.00626909 0.00487046 0.00332648 0.00169702 0.000472582 1.9095e-05 9.99919e-06 9.99868e-06 9.9966e-06
- 9.98902e-06 9.96433e-06 9.8933e-06 9.71437e-06 9.32194e-06 8.5873e-06 7.49978e-06 6.39434e-06 6.39449e-06 7.50009e-06 8.58772e-06 9.32242e-06
- 9.71486e-06 9.89381e-06 9.96485e-06 9.98954e-06 9.99712e-06 9.99919e-06 1.60408e-05 0.000552876 0.00208705 0.00390344 0.00560987 0.00716621
- 0.00856661 0.00981424 0.0109068 0.0118438 0.0126257 0.0132523 0.0137229 0.0140371 0.014195 0.014195 0.0140371 0.0137229
- 0.0132523 0.0126257 0.0118438 0.0109068 0.00981424 0.00856661 0.00716621 0.00560987 0.00390344 0.00208705 0.000552876 1.60408e-05
- 9.99919e-06 9.99712e-06 9.98954e-06 9.96485e-06 9.89381e-06 9.71486e-06 9.32242e-06 8.58772e-06 7.50009e-06 6.39449e-06 6.39452e-06 7.50016e-06
- 8.58781e-06 9.32253e-06 9.71498e-06 9.89392e-06 9.96496e-06 9.98965e-06 9.99723e-06 1.9095e-05 0.000552876 0.00222464 0.00419761 0.00605485
- 0.00776286 0.00931701 0.0107155 0.01196 0.0130497 0.0139844 0.0147638 0.0153881 0.015857 0.0161704 0.0163281 0.0163281
- 0.0161704 0.015857 0.0153881 0.0147638 0.0139844 0.0130497 0.01196 0.0107155 0.00931701 0.00776286 0.00605485 0.00419761
- 0.00222464 0.000552876 1.9095e-05 9.99723e-06 9.98965e-06 9.96496e-06 9.89392e-06 9.71498e-06 9.32253e-06 8.58781e-06 7.50016e-06 6.39452e-06
- 6.39452e-06 7.50017e-06 8.58783e-06 9.32255e-06 9.715e-06 9.89395e-06 9.96499e-06 9.98968e-06 3.20343e-05 0.000472582 0.00208705 0.00419761
- 0.00620501 0.00806181 0.00976652 0.0113173 0.0127133 0.013955 0.0150422 0.0159748 0.0167522 0.0173747 0.0178422 0.0181548
- 0.0183121 0.0183121 0.0181548 0.0178422 0.0173747 0.0167522 0.0159748 0.0150422 0.013955 0.0127133 0.0113173 0.00976652
- 0.00806181 0.00620501 0.00419761 0.00208705 0.000472582 3.20343e-05 9.98968e-06 9.96499e-06 9.89395e-06 9.715e-06 9.32255e-06 8.58783e-06
- 7.50017e-06 6.39452e-06 6.39452e-06 7.50017e-06 8.58784e-06 9.32255e-06 9.715e-06 9.89395e-06 9.96499e-06 6.13224e-05 0.000346156 0.00169702
- 0.00390344 0.00605485 0.00806181 0.00991585 0.0116175 0.0131653 0.014559 0.0157983 0.0168834 0.0178141 0.0185899 0.0192109
- 0.0196772 0.019989 0.0201458 0.0201458 0.019989 0.0196772 0.0192109 0.0185899 0.0178141 0.0168834 0.0157983 0.014559
- 0.0131653 0.0116175 0.00991585 0.00806181 0.00605485 0.00390344 0.00169702 0.000346156 6.13224e-05 9.96499e-06 9.89395e-06 9.715e-06
- 9.32255e-06 8.58784e-06 7.50017e-06 6.39452e-06 6.39452e-06 7.50017e-06 8.58784e-06 9.32255e-06 9.71501e-06 9.89395e-06 9.96499e-06 2.38563e-05
- 0.00109015 0.00332648 0.00560987 0.00776286 0.00976652 0.0116175 0.0133161 0.0148611 0.0162524 0.0174896 0.0185728 0.0195017
- 0.0202761 0.0208959 0.0213612 0.0216723 0.0218286 0.0218286 0.0216723 0.0213612 0.0208959 0.0202761 0.0195017 0.0185728
- 0.0174896 0.0162524 0.0148611 0.0133161 0.0116175 0.00976652 0.00776286 0.00560987 0.00332648 0.00109015 2.38563e-05 9.96499e-06
- 9.89395e-06 9.71501e-06 9.32255e-06 8.58784e-06 7.50017e-06 6.39452e-06 6.39452e-06 7.50017e-06 8.58784e-06 9.32255e-06 9.71501e-06 9.89395e-06
- 5.10079e-05 0.000516693 0.00246955 0.00487046 0.00716621 0.00931701 0.0113173 0.0131653 0.0148611 0.0164037 0.0177929 0.0190281
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