Extruded Meshes in Firedrake

Introduction

Firedrake provides several utility functions for the creation of semi-structured meshes from an unstructured base mesh. Firedrake also provides a wide range of finite element spaces, both simple and sophisticated, for use with such meshes.

These meshes may be particularly appropriate when carrying out simulations on high aspect ratio domains. More mundanely, they allow a two-dimensional mesh to be built from square or rectangular cells.

The partial structure can be exploited to give performance advantages when iterating over the mesh, relative to a fully unstructured traversal of the same mesh. Firedrake exploits these benefits when extruded meshes are used.

Structured, Unstructured and Semi-Structured Meshes

Structured and unstructured meshes differ in the way the topology of the mesh is specified.

In a fully structured mesh, the array indices of mesh entities can be computed directly. For example, given the index of the current cell, the indices of the cell’s vertices can be computed using a simple mathematical expression. This means that data can be directly addressed, using expressions of the form A[i].

In a fully unstructured mesh, there is no simple relation between the indices of different mesh entities. Instead, the relationships have to be explicitly stored. For example, given the index of the current cell, the indices of the cell’s vertices can only be found by looking up the information in a separate array. It follows that data must be indirectly addressed, using expressions of the form A[B[i]].

Memory access latency makes indirect addressing more expensive than direct addressing: it is usually more efficient to compute the array index directly than to look it up from memory.

The characteristics of a semi-structured or extruded mesh lie somewhere between the two extremes above. An extruded mesh has an unstructured base mesh. Each cell of the base mesh corresponds to a column of cells in the extruded mesh. Visiting the first cell in each column requires indirect addressing. However, visiting subsequent cells in the column can be done using direct addressing. As the number of cells in the column increases, the performance should approach that of a fully structured mesh.

Generating Extruded Meshes in Firedrake

Extruded meshes are built using ExtrudedMesh(). There are several built-in extrusion types that generate commonly-used extruded meshes. To create a more complicated extruded mesh, one can either pass a hand-written kernel to ExtrudedMesh(), or one can use a built-in extrusion type and modify the coordinate field afterwards.

The following information may be passed in to the constructor:

  • a Mesh object, which will be used as the base mesh.
  • the desired number of cell layers in the extruded mesh.
  • the extrusion_type, which can be one of the built-in “uniform”, “radial” or “radial_hedgehog” – these are described below – or “custom”. If this argument is omitted, the “uniform” extrusion type will be used.
  • the layer_height, which is needed for the built-in extrusion types.
  • a kernel, only if the custom extrusion type is used
  • the appropriate gdim, describing the geometric dimension of the mesh, only if the custom extrusion type is used.

Uniform Extrusion

Uniform extrusion adds another spatial dimension to the mesh. For example, a 2D base mesh becomes a 3D extruded mesh. The coordinates of the extruded mesh are computed on the assumption that the layers are evenly spaced (hence the word ‘uniform’).

Let m be a standard UnitSquareMesh(). The following code produces the extruded mesh, whose base mesh is m, with 5 mesh layers and a layer thickness of 0.2:

m = UnitSquareMesh(4, 4)
mesh = ExtrudedMesh(m, 5, layer_height=0.2, extrusion_type='uniform')

This can be simplified slightly. The extrusion_type defaults to ‘uniform’, so this can be omitted. Furthermore, the layer_height, if omitted, defaults to the reciprocal of the number of layers. The following code therefore has the same effect:

m = UnitSquareMesh(4, 4)
mesh = ExtrudedMesh(m, 5)

The base mesh and extruded mesh are shown below.

_images/UnitSquare44.png
_images/UnifExt.png

Radial Extrusion

Radial extrusion extrudes cells radially outwards from the origin, without increasing the number of spatial dimensions. An example in 2 dimensions, in which a circle is extruded into an annulus, is:

m = CircleManifoldMesh(20, radius=2)
mesh = ExtrudedMesh(m, 5, extrusion_type='radial')

The base mesh and extruded mesh are shown below.

_images/CircleMM20.png
_images/RadExt2D.png

An example in 3 dimensions, in which a sphere is extruded into a spherical annulus, is:

m = IcosahedralSphereMesh(radius=3, refinement_level=3)
mesh = ExtrudedMesh(m, 5, layer_height=0.1, extrusion_type='radial')

The base mesh and part of the extruded mesh are shown below.

_images/Icos3.png
_images/RadExt3D.png

Hedgehog Extrusion

Hedgehog extrusion is similar to radial extrusion, but the cells are extruded outwards in a direction normal to the base cell. This produces a discontinuous coordinate field.

m = CircleManifoldMesh(20, radius=2)
mesh = ExtrudedMesh(m, 5, extrusion_type='radial_hedgehog')

An example in 3 dimensions, in which a sphere is extruded into a spherical annulus, is:

m = UnitIcosahedralSphereMesh(refinement_level=2)
mesh = ExtrudedMesh(m, 5, layer_height=0.1, extrusion_type='radial_hedgehog')

The 2D and 3D hedgehog-extruded meshes are shown below.

_images/HedgeExt2D.png
_images/HedgeExt3D.png

Custom Extrusion

For a more complicated extruded mesh, a custom kernel can be given by the user. Since this is a mesh-wide operation, a PyOP2 parallel loop is constructed by Firedrake.

m = UnitSquareMesh(5, 5)
kernel = op2.Kernel("""
    void extrusion_kernel(double **base_coords, double **ext_coords,
                          int **layer, double *layer_height) {
        for (int i=0; i<6; i++) {
            ext_coords[i][0] = base_coords[i / 2][0]; // X
            ext_coords[i][1] = base_coords[i / 2][1]; // Y
            ext_coords[i][2] = 0.1 * (layer[0][0] + (i % 2)) + 0.5 * base_coords[i / 2][1]; // Z
        }
    }
""", "extrusion_kernel")
mesh = ExtrudedMesh(m, 5, extrusion_type='custom', kernel=kernel, gdim=3)

Function Spaces on Extruded Meshes

The syntax for building a FunctionSpace on an extruded mesh is an extension of the existing syntax used with normal meshes. On a non-extruded mesh, the following syntax is used:

mesh = UnitSquareMesh(4, 4)
V = FunctionSpace(mesh, "RT", 1)

To allow maximal flexibility in constructing function spaces, Firedrake supports a more general syntax:

V = FunctionSpace(mesh, element)

where element is a UFL FiniteElement object. This requires generation and manipulation of FiniteElement objects.

Geometrically, an extruded mesh cell is the product of a base, “horizontal”, cell with a “vertical” interval. The construction of function spaces on extruded meshes makes use of this. Firedrake supports all function spaces whose local element can be expressed as the product of an element defined on the base cell with an element defined on an interval.

We will now introduce the new operators which act on FiniteElement objects.

The TensorProductElement operator

To create an element compatible with an extruded mesh, one should use the TensorProductElement operator. For example,

horiz_elt = FiniteElement("CG", triangle, 1)
vert_elt = FiniteElement("CG", interval, 1)
elt = TensorProductElement(horiz_elt, vert_elt)
V = FunctionSpace(mesh, elt)

will give a continuous, scalar-valued function space. The resulting space contains functions which vary linearly in the horizontal direction and linearly in the vertical direction.

_images/cg1cg1_prism.svg

The product of a CG1 triangle element with a CG1 interval element

The degree and continuity may differ; for example

horiz_elt = FiniteElement("DG", triangle, 0)
vert_elt = FiniteElement("CG", interval, 2)
elt = TensorProductElement(horiz_elt, vert_elt)
V = FunctionSpace(mesh, elt)

will give a function space which is continuous between cells in a column, but discontinuous between horizontally-neighbouring cells. In addition, the function may vary piecewise-quadratically in the vertical direction, but is piecewise constant horizontally.

_images/dg0cg2_prism.svg

The product of a DG0 triangle element with a CG2 interval element

A more complicated element, like a Mini horizontal element with linear variation in the vertical direction, may be built using the EnrichedElement functionality in either of the following ways:

mini_horiz_1 = FiniteElement("CG", triangle, 1)
mini_horiz_2 = FiniteElement("B", triangle, 3)
mini_horiz = mini_horiz_1 + mini_horiz_2  # Enriched element
mini_vert = FiniteElement("CG", interval, 1)
mini_elt = TensorProductElement(mini_horiz, mini_vert)
V = FunctionSpace(mesh, mini_elt)

or

mini_horiz_1 = FiniteElement("CG", triangle, 1)
mini_horiz_2 = FiniteElement("B", triangle, 3)
mini_vert = FiniteElement("CG", interval, 1)
mini_elt_1 = TensorProductElement(mini_horiz_1, mini_vert)
mini_elt_2 = TensorProductElement(mini_horiz_2, mini_vert)
mini_elt = mini_elt_1 + mini_elt_2  # Enriched element
V = FunctionSpace(mesh, mini_elt)
_images/mini_prism.svg

The product of a Mini triangle element with a CG1 interval element

The HDivElement and HCurlElement operators

For moderately complicated vector-valued elements, TensorProductElement does not give enough information to unambiguously produce the desired space. As an example, consider the lowest-order Raviart-Thomas element on a quadrilateral. The degrees of freedom live on the facets, and consist of a single evaluation of the component of the vector field normal to each facet.

The following element is closely related to the desired Raviart-Thomas element:

CG_1 = FiniteElement("CG", interval, 1)
DG_0 = FiniteElement("DG", interval, 0)
P1P0 = TensorProductElement(CG_1, DG_0)
P0P1 = TensorProductElement(DG_0, CG_1)
elt = P1P0 + P0P1
_images/rt_quad_pre.svg

The element created above

However, this is only scalar-valued. There are two natural vector-valued elements that can be generated from this: one of them preserves tangential continuity between elements, and the other preserves normal continuity between elements. To obtain the Raviart-Thomas element, we must use the HDivElement operator:

CG_1 = FiniteElement("CG", interval, 1)
DG_0 = FiniteElement("DG", interval, 0)
P1P0 = TensorProductElement(CG_1, DG_0)
RT_horiz = HDivElement(P1P0)
P0P1 = TensorProductElement(DG_0, CG_1)
RT_vert = HDivElement(P0P1)
elt = RT_horiz + RT_vert
_images/rt_quad_post.svg

The RT quadrilateral element, requiring the use of HDivElement

Another reason to use these operators is when expanding a vector into a higher dimensional space. Consider the lowest-order Nedelec element of the 2nd kind on a triangle:

N2_1 = FiniteElement("N2curl", triangle, 1)

This is naturally vector-valued, and has two components. Suppose we form the product of this with a continuous element on an interval:

CG_2 = FiniteElement("CG", interval, 2)
N2CG = TensorProductElement(N2_1, CG_2)

This element still only has two components. To expand this into a three-dimensional curl-conforming element, we must use the HCurlElement operator; the syntax is:

Ned_horiz = HCurlElement(N2CG)
_images/ned_prism.svg

This gives the horizontal part of a Nedelec edge element on a triangular prism. The full element can be built as follows:

N2_1 = FiniteElement("N2curl", triangle, 1)
CG_2 = FiniteElement("CG", interval, 2)
N2CG = TensorProductElement(N2_1, CG_2)
Ned_horiz = HCurlElement(N2CG)
P2tr = FiniteElement("CG", triangle, 2)
P1dg = FiniteElement("DG", interval, 1)
P2P1 = TensorProductElement(P2tr, P1dg)
Ned_vert = HCurlElement(P2P1)
Ned_wedge = Ned_horiz + Ned_vert
V = FunctionSpace(mesh, Ned_wedge)

Shortcuts for simple spaces

Simple scalar-valued spaces can be created using a variation on the existing syntax, if the HDivElement, HCurlElement and enrichment operations are not required. To create a function space of degree 2 in the horizontal direction, degree 1 in the vertical direction and possibly discontinuous between layers, the short syntax is

fspace = FunctionSpace(mesh, "CG", 2, vfamily="DG", vdegree=1)

If the horizontal and vertical parts have the same family and degree, the vfamily and vdegree arguments may be omitted. If mesh is an ExtrudedMesh then the following are equivalent:

fspace = FunctionSpace(mesh, "Lagrange", 1)

and

fspace = FunctionSpace(mesh, "Lagrange", 1, vfamily="Lagrange", vdegree=1)

Solving Equations on Extruded Meshes

Once the mesh and function spaces have been declared, extruded meshes behave almost identically to normal meshes. However, there are some small differences, which are listed below.

  1. Surface integrals are no longer denoted by ds. Since extruded meshes have multiple types of surfaces, the following notation is used:
    • ds_v is used to denote an integral over side facets of the mesh. This can be combined with boundary markers from the base mesh, such as ds_v(1).
    • ds_t is used to denote an integral over the top surface of the mesh.
    • ds_b is used to denote an integral over the bottom surface of the mesh.
    • ds_tb is used to denote an integral over both the top and bottom surfaces of the mesh.
  2. Interior facet integrals are no longer denoted by dS. The horizontal and vertical interior facets may require different numerical treatment. To facilitate this, the following notation is used:
    • dS_h is used to denote an integral over horizontal interior facets (between cells that are vertically-adjacent).
    • dS_v is used to denote an integral over vertical interior facets (between cells that are horizontally-adjacent).
  3. When setting strong boundary conditions, the boundary markers from the base mesh can be used to set boundary conditions on the relevant side of the extruded mesh. To set boundary conditions on the top or bottom, the label is replaced by:
    • top, to set a boundary condition on the top surface.
    • bottom, to set a boundary condition on the bottom surface.