Examples

Examples#

This section documents reference examples distributed with Z3ST. Each example corresponds to a fully reproducible simulation located in the z3st/examples directory of the repository.

Thin slab#

This example considers a thin slab subjected to a prescribed temperature on a region, and an adiabatic condition on another region. The case is used as a thermo-mechanical verification benchmark, comparing Z3ST results against a analytical reference solutions.

Direct links to the documented examples:

Geometry and loading#

  • Slab thickness: L_x

  • In-plane dimensions: L_y \times L_z

The material is linear elastic and isotropic. Small-strain thermo-elasticity is assumed.

Mesh#

The domain is discretized with a finite element mesh.

_images/mesh.png

Fig. 1 Finite element mesh of the thin slab.#

_images/convergence.png

Fig. 2 Convergence of the solution during the simulation.#

Results#

Temperature field:

_images/temperature_with_mesh.png

Fig. 3 Temperature distribution with mesh overlay.#

Displacement field:

_images/displacement_norm_with_mesh.png

Fig. 4 Displacement norm with mesh overlay.#

Cylindrical shell under pressure#

This example shows a thick-walled cylindrical shell subjected to internal pressure. The case is used as a verification benchmark, comparing Z3ST results against the analytical Lamé solution.

Direct links to the documented examples:

Geometry and loading#

  • Inner radius: R_i

  • Outer radius: R_o

  • Length: L_z

  • Internal pressure: P_i

The material is assumed to be linear elastic and isotropic. Plane-strain conditions are enforced along the axial direction (:math: varepsilon_z=0)

Mesh#

The mesh is axisymmetric and extruded along the axial direction.

_images/mesh1.png

Fig. 5 Finite element mesh of the cylindrical shell.#

_images/convergence1.png

Fig. 6 Convergence of the solution during the simulation.#

Results#

The following figures compare numerical and analytical results along the radial direction at a fixed axial coordinate.

Displacement field:

_images/displacement_norm_with_mesh1.png

Fig. 7 Radial displacement norm with mesh overlay.#

Stress components:

_images/stress_comparison.png

Fig. 8 Radial, hoop and axial stresses compared with the Lamé solution.#

Strain components:

_images/strain_comparison.png

Fig. 9 Radial, hoop and axial strains compared with the analytical reference.#

Temperature field:

For thermo-mechanical runs, a temperature field can be prescribed and coupled to the mechanical problem.

_images/temperature_with_mesh1.png

Fig. 10 Temperature distribution with mesh overlay.#

Cluster dynamics in solids#

This example demonstrates the simulation of cluster evolution using the Cluster dynamics module. It solves the advection-diffusion equation in the cluster size space, ensuring mass conservation and modeling the evolution of clusters.

Direct links to the example:

Phase-field fracture in solids#

This example shows the simulation of crack propagation using the Phase-field fracture (Damage) module. It enables the study of complex fracture patterns, including crack initiation, propagation, branching, and merging, by solving an additional evolution equation for a scalar damage field.

Direct links to the examples:

The formulation supports both AT1 and AT2 models, with spectral or volumetric-deviatoric splits for the crack driving force.

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