Nonlinear heat conduction in a sphere | PDDO-MultiPhysics Examples

1The Problem

Curved boundaries are where grid-based methods lose accuracy. Here the same nonlinear equation as in the box test is solved in a unit sphere: Dirichlet data on the upper half of the surface, a convective flux through the true surface normal on the lower half. The manufactured solution again makes the error measurable.

$$\frac{\partial T}{\partial t} = \nabla\cdot\big(k(T)\nabla T\big) + s \quad\text{in } |\mathbf{x}| < 1$$

$$T = T_{\text{exact}} \text{ on } z > 0, \qquad -k\,\nabla T\cdot\mathbf{n} = h_c\,T + g \text{ on } z \le 0$$

2The Problem File

The domain is a primitive, {primitive: sphere, radius: 1.0}, with a node spacing of 0.1. Its surface is a single boundary named surface; the first condition restricts itself to the upper half with where: "z > 0", and the second covers the rest, because earlier conditions take precedence where they apply.

The complete problem file, examples/geometry/heat_sphere_mms.yaml

# Verification on a curved geometry: nonlinear heat conduction in a sphere with a
# manufactured solution. Dirichlet on the upper half, convective flux (true normals) on the
# lower half. Run with --set domain.spacing=0.1 / 0.05 to measure the convergence order.
"k">name: heat_sphere_mms
"k">domain:
  "k">type: geometry
  "k">geometry: {primitive: sphere, radius: 1.0}
  "k">spacing: 0.1
"k">parameters:
  "k">k0: 1.0
  "k">beta: 0.5
  "k">hc: 2.0
"k">relations:
  "k">k: "k0*(1 + beta*T**2)"
  "k">q: "-k*grad(T)"
"k">fields:
  "k">T: scalar
"k">equations:
  "k">T: "dt(T) = -div(q)"
"k">boundary_conditions:
  - "k">faces: [surface]
    "k">where: "z > 0"
    "k">T: {dirichlet: 0}
  - "k">faces: [surface]
    "k">T: {equation: "dot(q, n) = hc*T"}
"k">manufactured:
  "k">T: "1 + 0.5*sin(1.3*x + 0.4)*cos(0.9*y - 0.2)*exp(0.4*z)*exp(-t)"
"k">time:
  "k">end: 0.02
  "k">dt: 2.5e-4
  "k">scheme: bdf2
"k">output:
  "k">every: 20

**Figure 1.** The model in the GUI: the Model Builder tree on the left, the settings of the selected node in the middle, the graphics window on the right.

A boundary condition with an Only where expression: Dirichlet data on the upper half of the sphere. — **Figure 2.** A boundary condition with an `Only where` expression: Dirichlet data on the upper half of the sphere.

3Discretization

The point cloud fills the sphere with a regular grid and adds nodes on the surface, so that the boundary conditions act exactly on the surface with its exact normal. Near the surface the PDDO families become one-sided and are enlarged automatically.

**Figure 3.** The point cloud: interior grid nodes and surface nodes, coloured by node type.

4Results

Nodes: 5,048 (point cloud)
PDDO order: 2, horizon 2.015 Δx
Time scheme: bdf2
Steps: 80 × Δt = 0.00025
Newton iterations: 81
Factorizations: 1
Run time: 28 s

At the final time the relative L2 error is 0.0017 % and the largest error 6.2×10^-5. Refining the spacing from 0.1 to 0.05 shows second-order convergence on the curved boundary as well.

**Figure 4.** Temperature on the sphere's surface at the final time.

**Figure 5.** Error monitors of the manufactured solution against time.

5Run It Yourself

The problem file ships with PDDO-MultiPhysics. Open it in the graphical interface and press Compute, or run it from the command line:

python -m pddo_mp gui examples/geometry/heat_sphere_mms.yaml
python -m pddo_mp run examples/geometry/heat_sphere_mms.yaml

Results go to results/heat_sphere_mms/ next to the problem file: a ParaView time series (.pvd), the monitors as CSV, and the final state. Any entry can be changed on the command line, for example --set time.end=0.01 for a run to half the end time. The user manual describes every part of the problem file and of the interface.

1The Problem

2The Problem File

3Discretization

4Results

5Run It Yourself

Have a coupled problem of your own?