doc: Add open-source alternative to Ansys-based shapeopt to showcase folder (#408)

<!--
Please use a PR title that conforms to *conventional commits*:
"<commit_type>: Describe your change"; for example: "fix: prevent race
condition". Some other commit types are: fix, feat, ci, doc, refactor...
For a full list of commit types visit
https://www.conventionalcommits.org/en/v1.0.0/
-->

#### Relevant issue or PR
<!-- If the changes resolve an issue or follow some other PR, link to
them here. Only link something if it is directly relevant. -->

#### Description of changes
<!-- Add a high-level description of changes, focusing on the *what* and
*why*. -->

#### Testing done
<!-- Describe how the changes were tested; e.g., "CI passes", "Tested
manually in stagingrepo#123", screenshots of a terminal session that
verify the changes, or any other evidence of testing the changes. -->

---------

Co-authored-by: Dion Häfner <dion.haefner@simulation.science>

Author: andrinr

demo/_showcase/ansys-shapeopt/README.md

@@ -13,6 +13,17 @@ The main entry point of this demo is `optimization.ipynb`. The evolution of the
 where the loss decay is plotted here:
 
 ![Workflow](imgs/loss.png)
+
+> [!TIP]
+> In `optimization_os.ipynb` we provide an alternative workflow that is relying on the open-source FEM solver [JAX-FEM](https://github.com/deepmodeling/jax-fem), and does not require an active Ansys license.
+
+
+
+| Mesh | Loss |
+|----------|----------|
+| ![Workflow](imgs/mesh_optim.gif)    | ![Workflow](imgs/loss_os.png)     |
+
+
 ## Get Started
 
 ### Prerequisites

demo/_showcase/ansys-shapeopt/bars_3d/tesseract_api.py

@@ -0,0 +1,148 @@
+import numpy as np
+import pyvista as pv
+import trimesh
+from pydantic import BaseModel, Field
+
+from tesseract_core.runtime import Array, Float32
+
+#
+# Schemata
+#
+
+
+class InputSchema(BaseModel):
+    """Input schema for bar geometry design and SDF generation."""
+
+    differentiable_parameters: list[
+        Array[
+            (None,),
+            Float32,
+        ]
+    ] = Field(
+        description=(
+            "Vertex positions of the bar geometry. "
+            "The shape is (num_bars, num_vertices, 3), where num_bars is the number of bars "
+            "and num_vertices is the number of vertices per bar. The last dimension represents "
+            "the x, y, z coordinates of each vertex."
+        )
+    )
+
+    non_differentiable_parameters: list[
+        Array[
+            (None,),
+            Float32,
+        ]
+    ] = Field(description="Flattened array of non-differentiable geometry parameters.")
+
+    static_parameters: list[list[int]] = Field(
+        description=(
+            "List of integers used to construct the geometry."
+            " The first integer is the number of bars, and the second integer is the number of vertices per bar."
+        )
+    )
+
+    string_parameters: list[str] = Field(
+        description="List of string parameters for geometry construction."
+    )
+
+
+class TriangularMesh(BaseModel):
+    """Triangular mesh representation with fixed-size arrays."""
+
+    points: Array[(None, 3), Float32] = Field(description="Array of vertex positions.")
+    faces: Array[(None, 3), Float32] = Field(
+        description="Array of triangular faces defined by indices into the points array."
+    )
+
+
+class OutputSchema(BaseModel):
+    """Output schema for generated geometry."""
+
+    meshes: list[TriangularMesh] = Field(
+        description="Triangular meshes representing the geometries"
+    )
+
+
+def pyvista_to_trimesh(mesh: pv.PolyData) -> trimesh.Trimesh:
+    """Convert a pyvista mesh to a trimesh style polygon mesh."""
+    points = mesh.points
+    points_per_face = mesh.faces[0]
+    n_faces = mesh.faces.shape[0] // (points_per_face + 1)
+
+    faces = mesh.faces.reshape(n_faces, (points_per_face + 1))[:, 1:]
+
+    return trimesh.Trimesh(vertices=points, faces=faces)
+
+
+def build_geometries(
+    differentiable_parameters: list[np.ndarray],
+    non_differentiable_parameters: list[np.ndarray],
+    static_parameters: list[list[int]],
+    string_parameters: list[str],
+) -> list[trimesh.Trimesh]:
+    """Build a pyvista geometry from the parameters.
+
+    The parameters are expected to be of shape (n_chains, n_edges_per_chain + 1, 3),
+    """
+    n_geometrics = len(differentiable_parameters)
+    geometries = []
+    for i in range(n_geometrics):
+        n_chains = static_parameters[i][0]
+        n_vertices_per_chain = static_parameters[i][1]
+        geometry = []
+
+        params = differentiable_parameters[i].reshape(
+            (n_chains, n_vertices_per_chain, 3)
+        )
+
+        radius = non_differentiable_parameters[i][0]
+
+        for chain in range(n_chains):
+            tube = pv.Spline(points=params[chain]).tube(
+                radius=radius, capping=True, n_sides=30
+            )
+            tube = tube.triangulate()
+            tube = pyvista_to_trimesh(tube)
+            geometry.append(tube)
+
+        # convert each geometry in a trimesh style mesh and combine them
+        mesh = geometry[0]
+
+        for geom in geometry[1:]:
+            mesh = mesh.union(geom)
+
+        geometries.append(mesh)
+
+    return geometries
+
+
+#
+# Tesseract endpoints
+#
+
+
+def apply(inputs: InputSchema) -> OutputSchema:
+    """Generate mesh and SDF from bar geometry parameters.
+
+    Args:
+        inputs: Input schema containing bar geometry parameters.
+
+    Returns:
+        Output schema with generated mesh and SDF field.
+    """
+    meshes = build_geometries(
+        differentiable_parameters=inputs.differentiable_parameters,
+        non_differentiable_parameters=inputs.non_differentiable_parameters,
+        static_parameters=inputs.static_parameters,
+        string_parameters=inputs.string_parameters,
+    )
+
+    return OutputSchema(
+        meshes=[
+            TriangularMesh(
+                points=mesh.vertices.astype(np.float32),
+                faces=mesh.faces.astype(np.int32),
+            )
+            for mesh in meshes
+        ],
+    )

demo/_showcase/ansys-shapeopt/bars_3d/tesseract_config.yaml

@@ -0,0 +1,9 @@
+name: bars_3d
+version: "0.1.0"
+description: |
+  Tesseract that generates 3D bar geometry.
+
+  Parameters are expected to define the control points and radii of piecewise linear tubes in 3D space.
+
+build_config:
+  target_platform: "linux/x86_64"

demo/_showcase/ansys-shapeopt/bars_3d/tesseract_requirements.txt

@@ -0,0 +1,4 @@
+numpy==1.26.4
+pyvista==0.45.2
+trimesh==4.9.0
+manifold3d==3.3.2