Mondaic - Full Waveform Solutions

This documentation is not for the latest stable Salvus version.

This tutorial is presented as Python code running inside a Jupyter Notebook, the recommended way to use Salvus. To run it yourself you can copy/type each individual cell or directly download the full notebook, including all required files.

Download Jupyter Notebook [120.55 KB]

Reference solution: analytic
Physis: Diffusion equation

In this notebook we benchmark the numerical solution of the diffusion equation in a simple test case for which an analytic solution exists. Specifically, we are considering the fundamental solution of the diffusion equation in homogeneous media, which is often called diffusion kernel.

To this end, we consider the diffusion equation with a Dirac measure as initial values and a constant diffusion tensor $\mathcal{D}(\mathbf{x}) = \left(\begin{array}{cc}k & 0\\0 & k \end{array}\right)$ with $k > 0$ :

\begin{aligned} \partial_t u(\mathbf{x},t) - \nabla \cdot \mathcal{D}(\mathbf{x})\, \nabla u(\mathbf{x},t) &= 0,\\ u(\mathbf{x},0) &= \delta(\mathbf{x}-\mathbf{\bar{x}}). \end{aligned}

On $\mathbf{R}^d$ , ( $d=2,3$ ), this equation has a unique solution given by:

u_{\text{exact}} = \frac{1}{(k\pi\,t)^{d/2}} \exp\left( - \frac{(\mathbf{x}-\mathbf{\bar{x}})^T (\mathbf{x}-\mathbf{\bar{x}})}{4\,k\,t}\right),

which can also be interpreted as a Wiener process.

Remarks:

Representing the delta source on the finite element mesh requires a proper projection onto the finite element basis. Instead, we simulate the time interval from 0.01 s to 0.02 s.
While the analytic solution is defined on the unbounded domain $\mathbf{R}^d$ , we restrict the computational to a box or sphere. To avoid artifacts from the artificial boundaries, we use a fairly short simulation time.

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%matplotlib inline
%config Completer.use_jedi = False

import os
import numpy as np

import salvus.namespace as sn

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# Number of processes SalvusCompute will run with.
# Get it from the environment or default to 4.
MPI_RANKS = int(os.environ.get("NUM_MPI_RANKS", 4))
# Choose on which site to run this.
SALVUS_FLOW_SITE_NAME = os.environ.get("SITE_NAME", "local")

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domain_size = 2.0
k = 0.5
FINAL_TIME = 0.01

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def wiener_process(time, dim, mean, points, k):

    x_bar = points - mean
    u_exact = (
        1
        / (np.sqrt((k * np.pi * time) ** dim))
        * np.exp(-np.einsum("ijk,ijk->ij", x_bar, x_bar) / (4 * k * time))
    )
    return u_exact

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def create_mesh(dim, mode, tensor_order, domain_size, k):

    if dim == 2 and mode == "Cartesian":
        m = sn.simple_mesh.CartesianHomogeneousAcoustic2D(
            vp=1.0,
            rho=1.0,
            x_max=domain_size,
            y_max=domain_size,
            max_frequency=30.0,
        )
        m.advanced.tensor_order = tensor_order
        mesh = m.create_mesh()

    elif dim == 2 and mode == "spherical":
        m = sn.simple_mesh.basic_mesh.SphericalHomogeneousAcoustic2D(
            radius=domain_size / 2, vp=2.0, rho=1.0, max_frequency=20.0
        )
        m.advanced.tensor_order = tensor_order
        mesh = m.create_mesh()
        mesh.points[:, 0] += 1.0
        mesh.points[:, 1] += 1.0

    elif dim == 3 and mode == "Cartesian":

        m = sn.simple_mesh.CartesianHomogeneousAcoustic3D(
            vp=1.0,
            rho=1.0,
            x_max=domain_size,
            y_max=domain_size,
            z_max=domain_size,
            max_frequency=10.0,
        )
        m.advanced.tensor_order = tensor_order
        mesh = m.create_mesh()

    f = np.ones_like(mesh.elemental_fields["VP"])
    del mesh.elemental_fields["VP"]
    del mesh.elemental_fields["RHO"]

    mesh.attach_field("M0", 1.0 * f)
    mesh.attach_field("M1", k * f)

    mesh.attach_field(
        "uinit",
        wiener_process(
            0.01,
            mesh.ndim,
            [1.0] * mesh.ndim,
            mesh.points[mesh.connectivity],
            k,
        ),
    )

    return mesh

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def simulate(mesh, final_time):

    mesh.write_h5("init.h5")

    w = sn.simple_config.simulation.Diffusion(mesh=mesh)

    w.domain.polynomial_order = mesh.shape_order

    w.physics.diffusion_equation.initial_values.filename = "init.h5"
    w.physics.diffusion_equation.initial_values.format = "hdf5"
    w.physics.diffusion_equation.initial_values.field = "uinit"

    w.physics.diffusion_equation.final_values.filename = "final.h5"
    w.physics.diffusion_equation.end_time_in_seconds = final_time

    w.validate()

    sn.api.run(
        input_file=w,
        site_name=SALVUS_FLOW_SITE_NAME,
        output_folder="diffusion",
        overwrite=True,
        ranks=MPI_RANKS,
    )

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def analysis(mesh, final_time):

    tensor_order = mesh.shape_order

    if tensor_order == 1:
        TOL = 1e-2
    elif tensor_order == 2:
        TOL = 1e-3
    else:
        TOL = 1e-4

    solution = sn.UnstructuredMesh.from_h5("diffusion/final.h5")
    solution.attach_field("init", mesh.elemental_fields["uinit"])

    u_analytic = wiener_process(
        0.01 + final_time,
        mesh.ndim,
        [1.0] * mesh.ndim,
        solution.points[solution.connectivity],
        k,
    )
    u_salvus = solution.elemental_fields["uinit"]
    residuals = u_salvus - u_analytic

    u_max = u_salvus.max()
    ref_max = u_analytic.max()

    solution.attach_field("analytic_solution", u_analytic)
    solution.attach_field("solution", u_salvus)
    del solution.elemental_fields["uinit"]
    solution.attach_field("residuals", residuals)
    print(
        f"simulation time: {final_time}\n",
        f"|| u ||_inf = {u_max}\n",
        f"|| u_exact ||_inf = {ref_max}\n",
        f"|| u - u_exact ||_inf = {np.abs(residuals).max()}\n",
        f"|| u - u_exact ||_inf / || u_exact ||_inf = {np.abs(residuals).max() / ref_max}\n",
    )
    np.testing.assert_allclose(
        u_salvus, u_analytic, rtol=1e-6, atol=ref_max * TOL
    )

    return solution

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mesh = create_mesh(
    dim=2, mode="Cartesian", tensor_order=1, domain_size=2.0, k=0.5
)
simulate(mesh, 0.01)
solution = analysis(mesh, 0.01)
solution

Job `job_2011031823548467_f775e90c88` running on `local_f64` with 4 rank(s).
Site information:
  * Salvus version: 0.11.20
  * Floating point size: 64
-> Current Task: Time loop complete* Downloaded 1.8 MB of results to `diffusion`.
* Total run time: 1.43 seconds.
* Pure simulation time: 0.63 seconds.
simulation time: 0.01
 || u ||_inf = 31.959188667283925
 || u_exact ||_inf = 31.830988618379067
 || u - u_exact ||_inf = 0.12820004890485848
 || u - u_exact ||_inf / || u_exact ||_inf = 0.004027523318293556