Cube with Spherical Cavity

First, create a geometry consisting of a cube with a spherical cavity. This geometry has one cell.

Create a 3-D rectangular mesh grid.

[xg, yg, zg] = meshgrid(-2:0.25:2);
Pcube = [xg(:) yg(:), zg(:)];

Extract the grid points located outside of the unit spherical region.

Pcavitycube = Pcube(vecnorm(Pcube') > 1,:);

Create points on the unit sphere.

[x1,y1,z1] = sphere(24);
Psphere = [x1(:) y1(:) z1(:)];
Psphere = unique(Psphere,'rows');

Combine the coordinates of the rectangular grid (without the points inside the sphere) and the surface coordinates of the unit sphere.

Pcombined = [Pcavitycube;Psphere];

Create an alphaShape object representing the cube with the spherical cavity.

shpCubeWithSphericalCavity = alphaShape(Pcombined(:,1), ...
                                        Pcombined(:,2), ...
                                        Pcombined(:,3));

figure
plot(shpCubeWithSphericalCavity,'FaceAlpha',0.4)
title('alphaShape: Cube with Spherical Cavity')

Recover the triangulation that defines the domain of the alphaShape object.

[tri,loc] = alphaTriangulation(shpCubeWithSphericalCavity);

Create a PDE model.

modelCube = createpde;

Create a geometry from the mesh and import the geometry and the mesh into the model.

[gCube,mshCube] = geometryFromMesh(modelCube,loc',tri');

Plot the resulting geometry.

figure
pdegplot(modelCube,'FaceAlpha',0.5,'CellLabels','on')
title('PDEModel: Cube with Spherical Cavity')

Solid Sphere Nested in Cube

Create tetrahedral elements to form a solid sphere by using the spherical shell and adding a new node at the center. First, obtain the spherical shell by extracting facets of the spherical boundary.

sphereFacets = boundaryFacets(mshCube,'Face',3);
sphereNodes = findNodes(mshCube,'region','Face',3);

Add a new node at the center.

newNodeID = size(mshCube.Nodes,2) + 1;

Construct the tetrahedral elements by using each of the three nodes on the spherical boundary facets and the new node at the origin.

sphereTets =  [sphereFacets; newNodeID*ones(1,size(sphereFacets,2))];

Create a model that combines the cube with the spherical cavity and a sphere.

model = createpde;

Create a vector that maps all mshCube elements to cell 1, and all elements of the solid sphere to cell 2.

e2c = [ones(1,size(mshCube.Elements,2)), 2*ones(1,size(sphereTets,2))];

Add a new node at the center [0;0;0] to the nodes of the cube with the cavity.

combinedNodes = [mshCube.Nodes,[0;0;0]];

Combine the element connectivity matrices.

combinedElements = [mshCube.Elements,sphereTets];

Create a two-cell geometry from the mesh.

[g,msh] = geometryFromMesh(model,combinedNodes,combinedElements,e2c);
 
figure
pdegplot(model,'FaceAlpha',0.5,'CellLabels','on')
title('Solid Sphere in Cube')