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.

Inheritance diagram of LinearSolver

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

General periodic boundary condition for sequential runs.

Inheritance diagram of LinearPeriodic

Simulation Controller (`.case`)¶

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

Basic case with linear CESE method.

Inheritance diagram of LinearCase

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.

Inheritance diagram of PlaneWaveAnchor

planewaves = None¶: Sequence of PlaneWaveSolution objects.

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

Inheritance diagram of PlaneWaveHook

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.

Inheritance diagram of MeshInfoHook

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.

Inheritance diagram of ProgressHook

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.

Inheritance diagram of FillAnchor

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.

Inheritance diagram of CflAnchor

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.

Inheritance diagram of CflHook

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.

Inheritance diagram of MarchSaveAnchor

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.

Inheritance diagram of PMarchSave

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

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