Second-Order Linear Solver (solvcon.parcel.linear)

This package implements a basic second-order, three-dimensional CESE solver that uses the CFL-insensitive \(c\mbox{-}\tau\) scheme. The basic solver can be extended to solver for any linear system of first-order hyperbolic PDEs. An example is the velocity-stress equation solver in .velstress.

Numerical Implementation (._algorithm)

Let

• \(u_m\) be the \(m\)-th solution variable and \(m = 1, \ldots, M\). \(M\) is the total number of variables.
• \(u_{mx_{\mu}}\) be the component of the gradient of \(u_m\) along the \(x_{\mu}\) axis in a Cartesian coordinate system. \(\mu = 1, 2\) in two-dimensional space and \(\mu = 1, 2, 3\) in three-dimensional space.

Common Data Structure

sc_linear_algorithm_t

Basic Information of the Solver

int neq

double time

double time_increment

sc_linear_algorithm_t.neq is the number of equations or the number of variables in a mesh cell. sc_linear_algorithm_t.time is the current time of the solver. sc_linear_algorithm_t.time_increment is \(\Delta t\).

Parameters to the \(c\mbox{-}\tau\) Scheme

int alpha

double sigma0

double taylor

double cnbfac

double sftfac

double taumin

double tauscale

Metric Arrays for the CESE Method

double *cecnd

double *cevol

double *sfmrc

Group Data

int ngroup

int gdlen

double *grpda

Scalar Parameters

int nsca

double *amsca

Vector Parameters

int nvec

double *amvec

Solution Arrays

double *sol

double *soln

double *solt

double *dsol

double *dsoln

double *stm

double *cfl

double *ocfl

Metric for CEs & SEs

void sc_linear_prepare_ce_3d(sc_mesh_t *msd, sc_linear_algorithm_t *alg)

void sc_linear_prepare_ce_2d(sc_mesh_t *msd, sc_linear_algorithm_t *alg)

Calculate the volume and centroid of conservation elements.

void sc_linear_prepare_sf_3d(sc_mesh_t *msd, sc_linear_algorithm_t *alg)

void sc_linear_prepare_sf_2d(sc_mesh_t *msd, sc_linear_algorithm_t *alg)

Calculate the centroid and normal of hyperplanes of conservation elements and solution elements.

Time Marching

void sc_linear_calc_soln_3d(sc_mesh_t *msd, sc_linear_algorithm_t *alg)

void sc_linear_calc_soln_2d(sc_mesh_t *msd, sc_linear_algorithm_t *alg)

Calculate the solutions of the next half time step (\((u_m)_j^{n+1/2})\) based on the informaiton at the current time step (\(n\)).

void sc_linear_calc_solt_3d(sc_mesh_t *msd, sc_linear_algorithm_t *alg)

void sc_linear_calc_solt_2d(sc_mesh_t *msd, sc_linear_algorithm_t *alg)

Calculate the changing rate of solutions (\((u_mt)_j^n\)).

void sc_linear_calc_jaco_3d(sc_mesh_t *msd, sc_linear_algorithm_t *alg, int icl, double fcn[NEQ][NDIM], double jacos[NEQ][NEQ][NDIM])

void sc_linear_calc_jaco_2d(sc_mesh_t *msd, sc_linear_algorithm_t *alg, int icl, double fcn[NEQ][NDIM], double jacos[NEQ][NEQ][NDIM])

Calculate the Jacobian matrices.

void sc_linear_calc_dsoln_3d(sc_mesh_t *msd, sc_linear_algorithm_t *alg)

void sc_linear_calc_dsoln_2d(sc_mesh_t *msd, sc_linear_algorithm_t *alg)

Calculate the gradient of solutions of the next half time step (\((u_{mx_{\mu}})_j^{n+1/2}\)) based on the information at the current time step (\(n\)).

void sc_linear_calc_cfl_3d(sc_mesh_t *msd, sc_linear_algorithm_t *alg)

void sc_linear_calc_cfl_2d(sc_mesh_t *msd, sc_linear_algorithm_t *alg)

Calculate the CFL number based on the eigenvalues of the linear Jacobian matrices.

Plane Wave Solution

void sc_linear_calc_planewave_3d(sc_mesh_t *msd, sc_linear_algorithm_t *alg, double *asol, double *adsol, double *amp, double *ctr, double *wvec, double afreq)

void sc_linear_calc_planewave_2d(sc_mesh_t *msd, sc_linear_algorithm_t *alg, double *asol, double *adsol, double *amp, double *ctr, double *wvec, double afreq)

Calculate the plane-wave solutions.

Wrapper for Numerical Code

class solvcon.parcel.linear.LinearAlgorithm

This class wraps around the C functions for the second-order CESE method.

Numerical Controller (.solver)

class solvcon.parcel.linear.solver.LinearSolver(blk, **kw)

This class controls the underneath algorithm LinearAlgorithm.

class solvcon.parcel.linear.solver.LinearPeriodic(**kw)

General periodic boundary condition for sequential runs.

Simulation Controller (.case)

class solvcon.parcel.linear.case.LinearCase(**kw)

Basic case with linear CESE method.

Helpers for Plane Wave (.planewave)

class solvcon.parcel.linear.planewave.PlaneWaveSolution(**kw)

class solvcon.parcel.linear.planewave.PlaneWaveAnchor(svr, planewaves=None, **kw)

Use PlaneWaveSolution to calculate plane-wave solutions for LinearSolver.

planewaves = None

Sequence of PlaneWaveSolution objects.

class solvcon.parcel.linear.planewave.PlaneWaveHook(svr, planewaves=None, **kw)

planewaves = None

Sequence of PlaneWaveSolution objects.

norm = None

A dict holding the calculated norm.

Helpers for I/O (.inout)

class solvcon.parcel.linear.inout.MeshInfoHook(cse, show_bclist=False, perffn=None, **kw)

Print mesh information.

show_bclist = None

Flag to show the list of boundary conditions. Default is False.

perffn = None

Performance file name.

class solvcon.parcel.linear.inout.ProgressHook(cse, linewidth=50, **kw)

Print simulation progess.

linewidth = None

The maximum width for progress mark.

class solvcon.parcel.linear.inout.FillAnchor(svr, mappers=None, **kw)

Fill the specified arrays of a LinearSolver with corresponding value.

mappers = None

A dict maps the names of attributes of the MeshAnchor.svr to the filling value.

class solvcon.parcel.linear.inout.CflAnchor(svr, rsteps=None, **kw)

Counting CFL numbers. Use MeshSolver.marchret to return results. Implements postmarch() method.

rsteps = None

Steps to run (int).

class solvcon.parcel.linear.inout.CflHook(cse, name='cfl', cflmin=0.0, cflmax=1.0, fullstop=True, rsteps=None, **kw)

Makes sure CFL number is bounded and print averaged CFL number over time. Reports CFL information per time step and on finishing. Implements (i) postmarch() and (ii) postloop() methods.

rsteps = None

Steps to run.

name = None

Name of the CFL tool.

cflmin = None

Miminum CFL value.

cflmax = None

Maximum CFL value.

fullstop = None

Flag to stop when CFL is out of bound. Default is True.

aCFL = None

Accumulated CFL.

mCFL = None

Mean CFL.

hnCFL = None

Hereditary minimum CFL.

hxCFL = None

Hereditary maximum CFL.

aadj = None

Number of adjusted CFL accumulated since last report.

haadj = None

Total number of adjusted CFL since simulation started.

class solvcon.parcel.linear.inout.MarchSaveAnchor(svr, anames=None, compressor=None, fpdtype=None, psteps=None, vtkfn_tmpl=None, **kw)

Save solution data into VTK XML format for a solver.

anames = None

The arrays in LinearSolver or MeshSolver.der to be saved.

compressor = None

Compressor for binary data. Can be either 'gz' or ''.

fpdtype = None

String for floating point data type (NumPy convention).

psteps = None

The interval in step to save data.

vtkfn_tmpl = None

The template string for the VTK file.

class solvcon.parcel.linear.inout.PMarchSave(cse, anames=None, compressor='gz', fpdtype=None, altdir='', altsym='', vtkfn_tmpl=None, **kw)

Save the geometry and variables in a case when time marching in parallel VTK XML format.

anames = None

The arrays in LinearSolver or MeshSolver.der to be saved. Format is (name, inder, ndim), (name, inder, ndim) … For ndim > 0 the array is a spatial vector, for ndim == 0 a simple scalar, and ndim < 0 a list of scalar.

compressor = None

Compressor for binary data. Can be either 'gz' or ''.

fpdtype = None

String for floating point data type (NumPy convention).

altdir = None

The alternate directory to save the VTK files.

altsym = None

The symbolic link in basedir pointing to the alternate directory to save the VTK files.

vtkfn_tmpl = None

The template string for the VTK file.

pextmpl = None

Velocity-Stress Equation Solver (.velstress)

See [1] for the basic formulation.

Solver Logic (.velstress.logic)

class solvcon.parcel.linear.velstress.logic.VslinPWSolution(**kw)

Plane-wave solutions for the velocity-stress equations.

class solvcon.parcel.linear.velstress.logic.VslinSolver(blk, mtrldict=None, **kw)

Basic elastic solver.

mtrldict = None

A dict that maps names to Material object.

mtrllist = None

A list of all Material objects.

class solvcon.parcel.linear.velstress.logic.VslinCase(**kw)

Case for anisotropic elastic solids.

Material Definition (.velstress.material)

solvcon.parcel.linear.velstress.material.mltregy = {'Acmite': <class 'solvcon.parcel.linear.velstress.material.Acmite'>, 'Albite': <class 'solvcon.parcel.linear.velstress.material.Albite'>, 'AlphaQuartz': <class 'solvcon.parcel.linear.velstress.material.AlphaQuartz'>, 'AlphaUranium': <class 'solvcon.parcel.linear.velstress.material.AlphaUranium'>, 'BariumTitanate': <class 'solvcon.parcel.linear.velstress.material.BariumTitanate'>, 'Beryl': <class 'solvcon.parcel.linear.velstress.material.Beryl'>, 'CdS': <class 'solvcon.parcel.linear.velstress.material.CdS'>, 'Cubic': <class 'solvcon.parcel.linear.velstress.material.Cubic'>, 'GaAs': <class 'solvcon.parcel.linear.velstress.material.GaAs'>, 'Hexagonal': <class 'solvcon.parcel.linear.velstress.material.Hexagonal'>, 'Isotropic': <class 'solvcon.parcel.linear.velstress.material.Isotropic'>, 'Material': <class 'solvcon.parcel.linear.velstress.material.Material'>, 'Monoclinic': <class 'solvcon.parcel.linear.velstress.material.Monoclinic'>, 'Orthorhombic': <class 'solvcon.parcel.linear.velstress.material.Orthorhombic'>, 'RickerSample': <class 'solvcon.parcel.linear.velstress.material.RickerSample'>, 'RickerSampleLight': <class 'solvcon.parcel.linear.velstress.material.RickerSampleLight'>, 'Tetragonal': <class 'solvcon.parcel.linear.velstress.material.Tetragonal'>, 'Triclinic': <class 'solvcon.parcel.linear.velstress.material.Triclinic'>, 'Trigonal': <class 'solvcon.parcel.linear.velstress.material.Trigonal'>, 'Zinc': <class 'solvcon.parcel.linear.velstress.material.Zinc'>, 'ZnO': <class 'solvcon.parcel.linear.velstress.material.ZnO'>}

Registry class for the name of types.

class solvcon.parcel.linear.velstress.material.Material(rho=None, al=None, be=None, ga=None, **kw)

Material properties. The constitutive relation needs not be symmetric.

rho = None

Density.

al = None

Alpha angle.

be = None

Beta angle.

ga = None

Gamma angle.

origstiff = None

Stiffness matrix in the crystal coordinate.

stiff = None

Stiffness matrix in the transformed global coordinate.

Bibliography

[1]	Yung-Yu Chen, Lixiang Yang, and Sheng-Tao John Yu, “Hyperbolicity of Velocity-Stress Equations for Waves in Anisotropic Elastic Solids”, Journal of Elasticity, Volume 106, Number 2, Feb. 2012, Page 149-164. doi: 10.1007/s10659-011-9315-8 <http://dx.doi.org/10.1007/s10659-011-9315-8>.