# Fields

Fields are **global** variables provided by Taichi. **Global** indicates that fields can be read/written from both the Python scope and the Taichi scope. A field can be considered as a multi-dimensional array of elements, and it can be either **dense** or **sparse**. Similar to a NumPy `ndarray`

object, a field has a data type and a shape. Moreover, an element of a field can be a scalar, a **vector**, a **matrix**, or a **struct**.

The term **field** is borrowed from mathematics and physics. If you
have already known scalar field (e.g., heat field) or vector field (e.g., gravitational field) in mathematics and physics, it will be straightforward to understand the fields in Taichi.

To be noticed:

- Fields are always accessed by indices.
- Field values are initially zero.
- Sparse fields are initially inactive.

##### tip

In earlier versions of Taichi, you could not allocate new fields after executing the first kernel. Since Taichi v0.8.0, you can use a new class `FieldsBuilder`

for dynamic field allocation and destruction. For more details, please see Field (advanced).

## #

Scalar fieldsA simple example might help you understand scalar fields. Assume you have a rectangular wok on the top of a fire. At each point of the wok, there would be a temperature. The surface of the wok forms a heat field. The width and height of the wok are similar to the `shape`

of the Taichi scalar field. The temperature (0-D scalar) is like the element of the Taichi scalar field. We could use the following field to represent the
heat field on the wok:

`heat_field = ti.field(dtype=ti.f32, shape=(width_wok, height_wok))`

### #

Access elements of scalar fields- If
`x`

is a 3D scalar field (`ti.field(dtype=ti.f32, shape=(10, 20, 30)`

), access its element with`x[i, j, k]`

(`0 <= i < 10, 0 <= j < 20, 0 <= k < 30`

). - When accessing 0-D field
`x`

, use`x[None] = 0`

instead of`x = 0`

. A 0-D field looks like`energy = ti.field(dtype=ti.f32, shape=())`

.

##### caution

Please **always** use indexing to access entries in fields.

## #

Vector fieldsWe are all live in a gravitational field which is a vector field. At each position of the 3D space, there is a gravity force vector. The gravitational field could be represented with:

`gravitational_field = ti.Vector.field(n=3, dtype=ti.f32, shape=(x, y, z))`

`x, y, z`

are the sizes of each dimension of the 3D space respectively. `n`

is the number of elements of the gravity force vector.

### #

Access elements of vector fields- The gravity force vector could be accessed by
`gravitational_field[i, j, k]`

(`0 <= i < x, 0 <= j < y, 0 <= k < z`

). - The
`p`

-th member of the gravity force vector could be accessed by`gravitational_field[i, j, k][p]`

(`0 <= p < n`

). - The 0-D vector field
`x = ti.Vector.field(n=3, dtype=ti.f32, shape=())`

should be accessed by`x[None][p]`

(`0 <= p < n`

).

##### note

As you may have noticed, there are **two** indexing operators `[]`

when you access a member of a vector from a vector field: the first is for field indexing, and the second is for vector indexing.

## #

Matrix fieldsField elements can also be matrices. In continuum mechanics, each infinitesimal point in a material exists a strain and a stress tensor. The strain and stress tensor is a 3 by 3 matrix in the 3D space. To represent this tensor field we could use:

`strain_tensor_field = ti.Matrix.field(n=3, m=3, dtype=ti.f32, shape=(x, y, z))`

`x, y, z`

are the sizes of each dimension of the 3D material respectively. `n, m`

are the dimensions of the strain tensor.

In a general case, suppose you have a `128 x 64`

field called `A`

, and each element is
a `3 x 2`

matrix, you can define it with `A = ti.Matrix.field(3, 2, dtype=ti.f32, shape=(128, 64))`

.

### #

Access elements of matrix fields- If you want to get the matrix of grid node
`i, j`

, please use`mat = A[i, j]`

.`mat`

is simply a`3 x 2`

matrix. - To get the element on the first row and second column of that
matrix, use
`mat[0, 1]`

or`A[i, j][0, 1]`

. - The 0-D matrix field
`x = ti.Matrix.field(n=3, m=4, dtype=ti.f32, shape=())`

should be accessed by`x[None][p, q]`

(`0 <= p < n, 0 <= q < m`

).

##### note

- As you may have noticed, there are
**two**indexing operators`[]`

when you access a member of a matrix from a matrix field: the first is for field indexing, and the second is for matrix indexing. `ti.Vector`

is simply an alias of`ti.Matrix`

.

### #

Matrix sizeFor performance reasons matrix operations will be unrolled during the compile stage, therefore we
suggest using only small matrices. For example, `2x1`

, `3x3`

, `4x4`

matrices are fine, yet `32x6`

is probably too big as a matrix size.

##### caution

Due to the unrolling mechanisms, operating on large matrices (e.g.
`32x128`

) can lead to a very long compilation time and low performance.

If you have a dimension that is too large (e.g. `64`

), it's better to
declare a field of size `64`

. E.g., instead of declaring
`ti.Matrix.field(64, 32, dtype=ti.f32, shape=(3, 2))`

, declare
`ti.Matrix.field(3, 2, dtype=ti.f32, shape=(64, 32))`

. Try to put large
dimensions to fields instead of matrices.

## #

Struct fieldsIn addition to vectors and matrices, field elements can be user-defined structs. A struct variable may contain scalars, vectors/matrices, or other structs as its members. A struct field is created by providing a dictionary of the name and data type of each member. For example, a 1D field of particles with position, velocity, acceleration, and mass for each particle can be represented as:

`particle_field = ti.Struct.field({ "pos": ti.types.vector(3, ti.f32), "vel": ti.types.vector(3, ti.f32), "acc": ti.types.vector(3, ti.f32), "mass": ti.f32, }, shape=(n,))`

Compound types (`ti.types.vector`

, `ti.types.matrix`

, and `ti.types.struct`

) need to be used to create vectors, matrices, or structs as field members. Apart from using `ti.Struct.field`

, the above particle field can be alternatively created using field creation from compound types as:

`vec3f = ti.types.vector(3, ti.f32)particle = ti.types.struct( pos=vec3f, vel=vec3f, acc=vec3f, mass=ti.f32,)particle_field = particle.field(shape=(n,))`

Members of a struct field can be accessed either locally (i.e., member of a struct field element) or globally (i.e., member field of a struct field):

```
# set the position of the first particle to originparticle_field[0] # local ti.Structparticle_field[0].pos = ti.Vector([0.0, 0.0, 0.0])
# set the first member of the second position to 1.0particle_field[1].pos[0] = 1.0
# make the mass of all particles be 1particle_field.mass # global ti.Vector.fieldparticle_field.mass.fill(1.0)
```