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%% Cell type:code id: tags:
``` python
from pystencils.session import *
```
%% Cell type:markdown id: tags:
# Tutorial 02: Basic Kernel generation with *pystencils*
Now that you have an [overview of pystencils](01_tutorial_getting_started.ipynb),
this tutorial shows in more detail how to formulate, optimize and run stencil kernels.
## 1) Kernel Definition
### a) Defining kernels with assignment lists and the `kernel` decorator
*pystencils* gets a symbolic formulation of the kernel. This can be either an `Assignment` or a sequence of `Assignment`s that follow a set of restrictions.
Lets first create a kernel that consists of multiple assignments:
%% Cell type:code id: tags:
``` python
src_arr = np.zeros([20, 30])
dst_arr = np.zeros_like(src_arr)
dst, src = ps.fields(dst=dst_arr, src=src_arr)
```
%% Cell type:code id: tags:
``` python
grad_x, grad_y = sp.symbols("grad_x, grad_y")
symbolic_description = [
ps.Assignment(grad_x, (src[1, 0] - src[-1, 0]) / 2),
ps.Assignment(grad_y, (src[0, 1] - src[0, -1]) / 2),
ps.Assignment(dst[0, 0], grad_x + grad_y),
]
kernel = ps.create_kernel(symbolic_description)
symbolic_description
```
%% Output
$\displaystyle \left[ grad_{x} \leftarrow_{} \frac{{src}_{(1,0)}}{2} - \frac{{src}_{(-1,0)}}{2}, \ grad_{y} \leftarrow_{} \frac{{src}_{(0,1)}}{2} - \frac{{src}_{(0,-1)}}{2}, \ {dst}_{(0,0)} \leftarrow_{} grad_{x} + grad_{y}\right]$
⎡ src_E src_W src_N src_S ⎤
⎢gradₓ := ───── - ─────, grad_y := ───── - ─────, dst_C := gradₓ + grad_y⎥
⎣ 2 2 2 2 ⎦
%% Cell type:markdown id: tags:
We created subexpressions, using standard sympy symbols on the left hand side, to split the kernel into multiple assignments. Defining a kernel using a list of `Assignment`s is quite tedious and hard to read.
To simplify the formulation of a kernel, *pystencils* offers the `kernel` decorator, that transforms a normal Python function with `@=` assignments into an assignment list that can be passed to `create_kernel`.
%% Cell type:code id: tags:
``` python
@ps.kernel
def symbolic_description_using_function():
grad_x @= (src[1, 0] - src[-1, 0]) / 2
grad_y @= (src[0, 1] - src[0, -1]) / 2
dst[0, 0] @= grad_x + grad_y
symbolic_description_using_function
```
%% Output
$\displaystyle \left[ grad_{x} \leftarrow_{} \frac{{src}_{(1,0)}}{2} - \frac{{src}_{(-1,0)}}{2}, \ grad_{y} \leftarrow_{} \frac{{src}_{(0,1)}}{2} - \frac{{src}_{(0,-1)}}{2}, \ {dst}_{(0,0)} \leftarrow_{} grad_{x} + grad_{y}\right]$
⎡ src_E src_W src_N src_S ⎤
⎢gradₓ := ───── - ─────, grad_y := ───── - ─────, dst_C := gradₓ + grad_y⎥
⎣ 2 2 2 2 ⎦
%% Cell type:markdown id: tags:
The decorated function can contain any Python code, only the `@=` operator, and the ternary inline `if-else` operator have different meaning.
### b) Ternary 'if' with `Piecewise`
The ternary operator maps to `sympy.Piecewise` functions, that can be used to introduce branching into the kernel. Piecewise defined functions must give a value for every input, i.e. there must be a 'otherwise' clause in the end that is indicated by the condition `True`. Piecewise objects are standard sympy terms that can be integrated into bigger expressions:
%% Cell type:code id: tags:
``` python
sp.Piecewise((1.0, src[0,1] > 0), (0.0, True)) + src[1, 0]
```
%% Output
$\displaystyle {src}_{(1,0)} + \begin{cases} 1.0 & \text{for}\: {src}_{(0,1)} > 0 \\0.0 & \text{otherwise} \end{cases}$
⎛⎧1.0 for src_N > 0⎞
src_E + ⎜⎨ ⎟
⎝⎩0.0 otherwise ⎠
%% Cell type:markdown id: tags:
Piecewise objects are created by the `kernel` decorator for ternary if-else statements.
%% Cell type:code id: tags:
``` python
@ps.kernel
def kernel_with_piecewise():
grad_x @= (src[1, 0] - src[-1, 0]) / 2 if src[-1, 0] > 0 else 0.0
kernel_with_piecewise
```
%% Output
$\displaystyle \left[ grad_{x} \leftarrow_{} \begin{cases} \frac{{src}_{(1,0)}}{2} - \frac{{src}_{(-1,0)}}{2} & \text{for}\: {src}_{(-1,0)} > 0 \\0.0 & \text{otherwise} \end{cases}\right]$
⎡ ⎧src_E src_W ⎤
⎢ ⎪───── - ───── for src_W > 0⎥
⎢gradₓ := ⎨ 2 2 ⎥
⎢ ⎪ ⎥
⎣ ⎩ 0.0 otherwise ⎦
%% Cell type:markdown id: tags:
### c) Assignment level optimizations using `AssignmentCollection`
When the kernels get larger and more complex, it is helpful to organize the list of assignment into a more structured way. The `AssignmentCollection` offers optimizating transformation on a list of assignments. It holds two assignment lists, one for subexpressions and one for the main assignments. Main assignments are typically those that write to an array.
%% Cell type:code id: tags:
``` python
@ps.kernel
def somewhat_longer_dummy_kernel(s):
s.a @= src[0, 1] + src[-1, 0]
s.b @= 2 * src[1, 0] + src[0, -1]
s.c @= src[0, 1] + 2 * src[1, 0] + src[-1, 0] + src[0, -1] - src[0,0]
dst[0, 0] @= s.a + s.b + s.c
ac = ps.AssignmentCollection(main_assignments=somewhat_longer_dummy_kernel[-1:],
subexpressions=somewhat_longer_dummy_kernel[:-1])
ac
```
%% Output
AssignmentCollection: dst_C, <- f(src_C, src_W, src_S, src_N, src_E)
%% Cell type:code id: tags:
``` python
ac.operation_count
```
%% Output
{'adds': 8,
'muls': 2,
'divs': 0,
'sqrts': 0,
'fast_sqrts': 0,
'fast_inv_sqrts': 0,
'fast_div': 0}
%% Cell type:markdown id: tags:
The `pystencils.simp` submodule offers several functions to optimize a collection of assignments.
It also offers functionality to group optimization into strategies and evaluate them.
In this example we reduce the number of operations by reusing existing subexpressions to get rid of two unnecessary floating point additions. For more information about assignment collections and simplifications see the [demo notebook](demo_assignment_collection.ipynb).
%% Cell type:code id: tags:
``` python
opt_ac = ps.simp.subexpression_substitution_in_existing_subexpressions(ac)
opt_ac
```
%% Output
AssignmentCollection: dst_C, <- f(src_C, src_W, src_S, src_N, src_E)
%% Cell type:code id: tags:
``` python
opt_ac.operation_count
```
%% Output
{'adds': 6,
'muls': 1,
'divs': 0,
'sqrts': 0,
'fast_sqrts': 0,
'fast_inv_sqrts': 0,
'fast_div': 0}
%% Cell type:markdown id: tags:
### d) Ghost layers and iteration region
When creating a kernel with neighbor accesses, *pystencils* automatically restricts the iteration region, such that all accesses are safe.
%% Cell type:code id: tags:
``` python
kernel = ps.create_kernel(ps.Assignment(dst[0,0], src[2, 0] + src[-1, 0]))
ps.show_code(kernel)
```
%% Output
%% Cell type:markdown id: tags:
When no additional ghost layer information is given, *pystencils* looks at all neighboring field accesses and introduces the required number of ghost layers **for all directions**. In the example above the largest neighbor accesses was ``src[2, 0]``, so theoretically we would need 2 ghost layers only the the end of the x coordinate.
By default *pystencils* introduces 2 ghost layers at all borders of the domain. The next cell shows how to change this behavior. Be careful with manual ghost layer specification, wrong values may lead to SEGFAULTs.
%% Cell type:code id: tags:
``` python
gl_spec = [(0, 2), # 0 ghost layers at the left, 2 at the right border
(1, 0)] # 1 ghost layer at the lower y, one at the upper y coordinate
kernel = ps.create_kernel(ps.Assignment(dst[0,0], src[2, 0] + src[-1, 0]), ghost_layers=gl_spec)
ps.show_code(kernel)
```
%% Output
%% Cell type:markdown id: tags:
## 2 ) Restrictions
### a) Independence Restriction
*pystencils* only works for kernels where each array element can be updated independently from all other elements. This restriction ensures that the kernels can be easily parallelized and also be run on the GPU. Trying to define kernels where the results depends on the iteration order, leads to a ValueError.
%% Cell type:code id: tags:
``` python
invalid_description = [
ps.Assignment(dst[1, 0], src[1, 0] + src[-1, 0]),
ps.Assignment(dst[0, 0], src[1, 0] - src[-1, 0]),
]
try:
invalid_kernel = ps.create_kernel(invalid_description)
assert False, "Should never be executed"
except ValueError as e:
print(e)
```
%% Output
Field dst is written at two different locations
%% Cell type:markdown id: tags:
The independence restriction makes sure that the kernel can be safely parallelized by checking the following conditions: If a field is modified inside the kernel, it may only be modified at a single spatial position. In that case the field may also only be read at this position. Fields that are not modified may be read at multiple neighboring positions.
Specifically, this rule allows for in-place updates that don't access neighbors.
%% Cell type:code id: tags:
``` python
valid_kernel = ps.create_kernel(ps.Assignment(src[0,0], 2*src[0,0] + 42))
```
%% Cell type:markdown id: tags:
If a field stores multiple values per cell, as in the next example, this restriction only applies for accesses with the same index.
%% Cell type:code id: tags:
``` python
v = ps.fields("v(2): double[2D]")
valid_kernel = ps.create_kernel([ps.Assignment(v[0,0](1), 2*v[0,0](1) + 42),
ps.Assignment(v[0,1](0), 2*v[0,1](0) + 42)])
```
%% Cell type:markdown id: tags:
### b) Static Single Assignment Form
All assignments that don't write to a field must be in SSA form
1. Each sympy symbol may only occur once as a left-hand-side (fields can be written multiple times)
2. A symbol has to be defined before it is used. If it is never defined it is introduced as function parameter
The next cell demonstrates the first SSA restriction:
%% Cell type:code id: tags:
``` python
@ps.kernel
def not_allowed():
a, b = sp.symbols("a b")
a @= src[0, 0]
b @= a + 3
a @= src[-1, 0]
dst[0, 0] @= a + b
try:
ps.create_kernel(not_allowed)
assert False
except ValueError as e:
print(e)
```
%% Output
Assignments not in SSA form, multiple assignments to a
%% Cell type:markdown id: tags:
Also it is not allowed to write a field at the same location
%% Cell type:code id: tags:
``` python
@ps.kernel
def not_allowed():
dst[0, 0] @= src[0, 1] + src[1, 0]
dst[0, 0] @= 2 * dst[0, 0]
try:
ps.create_kernel(not_allowed)
assert False
except ValueError as e:
print(e)
```
%% Output
Field dst is written twice at the same location
%% Cell type:markdown id: tags:
This situation should be resolved by introducing temporary variables
%% Cell type:code id: tags:
``` python
tmp_var = sp.Symbol("a")
@ps.kernel
def allowed():
tmp_var @= src[0, 1] + src[1, 0]
dst[0, 0] @= 2 * tmp_var
ast = ps.create_kernel(allowed)
ps.show_code(ast)
```
%% Output
doc/notebooks/03_tutorial_datahandling.ipynb 0 → 100644
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%% Cell type:code id: tags:
``` python
import psutil
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import pyplot, cm
from pystencils.session import *
from pystencils.boundaries import add_neumann_boundary, Neumann, Dirichlet, BoundaryHandling
from pystencils.slicing import slice_from_direction
import math
import time
%matplotlib inline
```
%% Cell type:markdown id: tags:
Test to see if cupy is installed which is needed to run calculations on the GPU
%% Cell type:code id: tags:
``` python
try:
import cupy
gpu = True
except ImportError:
gpu = False
cupy = None
print('No cupy installed')
```
%% Cell type:markdown id: tags:
# Tutorial 03: Datahandling
%% Cell type:markdown id: tags:
This is a tutorial about the `DataHandling` class of pystencils. This class is an abstraction layer to
- link numpy arrays to pystencils fields
- handle CPU-GPU array transfer, such that one can write code that works on CPU and GPU
- makes it possible to write MPI parallel simulations to run on distributed-memory clusters using the waLBerla library
We will look at a small and easy example to demonstrate the usage of `DataHandling` objects. We will define an averaging kernel to every cell of an array, that writes the average of the neighbor cell values to the center.
%% Cell type:markdown id: tags:
## 1. Manual
### 1.1. CPU kernels
In this first part, we set up a scenario manually without a `DataHandling`. In the next sections we then repeat the same setup with the help of the data handling.
One concept of *pystencils* that may be confusing at first, is the differences between pystencils fields and numpy arrays. Fields are used to describe the computation *symbolically* with sympy, while numpy arrays hold the actual values where the computation is executed on.
One option to create and execute a *pystencils* kernel is listed below. For reasons that become clear later we call this the **variable-field-size workflow**:
1. define pystencils fields
2. use sympy and the pystencils fields to define an update rule, that describes what should be done on *every cell*
3. compile the update rule to a real function, that can be called from Python. For each field that was referenced in the symbolic description the function expects a numpy array, passed as named parameter
4. create some numpy arrays with actual data
5. call the kernel - usually many times
Now, lets see how this actually looks in Python code:
%% Cell type:code id: tags:
``` python
# 1. field definitions
src_field, dst_field = ps.fields("src, dst:[2D]")
# 2. define update rule
update_rule = [ps.Assignment(lhs=dst_field[0, 0],
rhs=(src_field[1, 0] + src_field[-1, 0] +
src_field[0, 1] + src_field[0, -1]) / 4)]
# 3. compile update rule to function
kernel_function = ps.create_kernel(update_rule).compile()
# 4. create numpy arrays and call kernel
src_arr, dst_arr = np.random.rand(30, 30), np.zeros([30, 30])
# 5. call kernel
kernel_function(src=src_arr, dst=dst_arr) # names of arguments have to match names passed to ps.fields()
```
%% Cell type:markdown id: tags:
This workflow separates the symbolic and the numeric stages very cleanly. The separation also makes it possible to stop after step 3, write the C-code to a file and call the kernel from a C program. Speaking of the C-Code - lets have a look at the generated sources:
%% Cell type:code id: tags:
``` python
ps.show_code(kernel_function.ast)
```
%% Output
%% Cell type:markdown id: tags:
Even if it looks very ugly and low-level :) lets look at this code in a bit more detail. The code is generated in a way that it works for different array sizes. The size of the array is passed in the `_size_dst_` variables that specifiy the shape of the array for each dimension. Also, the memory layout (linearization) of the array can be different. That means the array could be stored in row-major or column-major order - if we pass in the array strides correctly the kernel does the right thing. If you're not familiar with the concept of strides check out [this stackoverflow post](https://stackoverflow.com/questions/53097952/how-to-understand-numpy-strides-for-layman) or search in the numpy documentation for strides - C vs Fortran order.
The goal of *pystencils* is to produce the fastest possible code. One technique to do this is to use all available information already on compile time and generate code that is highly adapted to the specific problem. In our case we already know the shape and strides of the arrays we want to apply the kernel to, so we can make use of this information. This idea leads to the **fixed-field-size workflow**. The main difference there is that we define the arrays first and therefore let *pystencils* know about the array shapes and strides, so that it can generate more specific code:
1. create numpy arrays that hold your data
2. define pystencils fields, this time telling pystencils already which arrays they correspond to, so that it knows about the size and strides
in the other steps nothing has changed
3. define the update rule
4. compile update rule to kernel
5. run the kernel
%% Cell type:code id: tags:
``` python
# 1. create arrays first
src_arr, dst_arr = np.random.rand(30, 30), np.zeros([30, 30])
# 2. define symbolic fields - note the additional parameter that link an array to each field
src_field, dst_field = ps.fields("src, dst:[2D]", src=src_arr, dst=dst_arr)
# 3. define update rule
update_rule = [ps.Assignment(lhs=dst_field[0, 0],
rhs=(src_field[1, 0] + src_field[-1, 0] +
src_field[0, 1] + src_field[0, -1]) / 4)]
# 4. compile it
kernel_function = ps.create_kernel(update_rule).compile()
# 5. call kernel
kernel_function(src=src_arr, dst=dst_arr) # names of arguments have to match names passed to ps.fields()
```
%% Cell type:markdown id: tags:
Functionally, both variants are equivalent. We see the difference only when we look at the generated code
%% Cell type:code id: tags:
``` python
ps.show_code(kernel_function.ast)
```
%% Output
%% Cell type:markdown id: tags:
Compare this to the code above! It looks much simpler. The reason is that all index computations are already simplified since the exact field sizes and strides are known. This kernel now only works on arrays of the previously specified size.
Lets try what happens if we use a different array:
%% Cell type:code id: tags:
``` python
src_arr2, dst_arr2 = np.random.rand(40, 40), np.zeros([40, 40])
try:
kernel_function(src=src_arr2, dst=dst_arr2)
except ValueError as e:
print(e)
```
%% Output
Wrong shape of array dst. Expected (30, 30)
%% Cell type:markdown id: tags:
### 1.2. GPU simulations
Let's now jump to a seemingly unrelated topic: running kernels on the GPU.
When creating the kernel, an additional parameter `target=ps.Target.GPU` has to be passed. Also, the compiled kernel cannot be called with numpy arrays directly, but has to be called with `cupy.array`s instead. That means, we have to transfer our numpy array to GPU first. From this step we obtain a gpuarray, then we can run the kernel, hopefully multiple times so that the data transfer was worth the time. Finally we transfer the finished result back to CPU:
%% Cell type:code id: tags:
``` python
if cupy:
config = ps.CreateKernelConfig(target=ps.Target.GPU)
kernel_function_gpu = ps.create_kernel(update_rule, config=config).compile()
# transfer to GPU
src_arr_gpu = cupy.asarray(src_arr)
dst_arr_gpu = cupy.asarray(dst_arr)
# run kernel on GPU, this is done many times in real setups
kernel_function_gpu(src=src_arr_gpu, dst=dst_arr_gpu)
# transfer result back to CPU
dst_arr = dst_arr_gpu.get()
```
%% Cell type:markdown id: tags:
### 1.3. Summary: manual way
- Don't confuse *pystencils* fields and *numpy* arrays
- fields are symbolic
- arrays are numeric
- Use the fixed-field-size workflow whenever possible, since code might be faster. Create arrays first, then create fields from arrays
- if we run GPU kernels, arrays have to transferred to the GPU first
As demonstrated in the examples above we have to define 2 or 3 corresponding objects for each grid:
- symbolic pystencils field
- numpy array on CPU
- for GPU use also cupy to mirror the data on the GPU
Managing these three objects manually is tedious and error-prone. We'll see in the next section how the data handling object takes care of this problem.
%% Cell type:markdown id: tags:
## 2. Introducing the data handling - serial version
### 2.1. Example for CPU simulations
The data handling internally keeps a mapping between symbolic fields and numpy arrays. When we create a field, automatically a corresponding array is allocated as well. Optionally we can also allocate memory on the GPU for the array as well. Lets dive right in and see how our example looks like, when implemented with a data handling.
%% Cell type:code id: tags:
``` python
dh = ps.create_data_handling(domain_size=(30, 30))
# fields are now created using the data handling
src_field = dh.add_array('src', values_per_cell=1)
dst_field = dh.add_array('dst', values_per_cell=1)
# kernel is created just like before
update_rule = [ps.Assignment(lhs=dst_field[0, 0],
rhs=(src_field[1, 0] + src_field[-1, 0] + src_field[0, 1] + src_field[0, -1]) / 4)]
kernel_function = ps.create_kernel(update_rule).compile()
# have a look at the generated code - it uses
# the fast version where array sizes are compiled-in
# ps.show_code(kernel_function.ast)
```
%% Cell type:markdown id: tags:
The data handling has methods to create fields - but where are the corresponding arrays?
In the serial case you can access them as a member of the data handling, for example to initialize our 'src' array we can write
%% Cell type:code id: tags:
``` python
src_arr = dh.cpu_arrays['src']
src_arr.fill(0.0)
```
%% Cell type:markdown id: tags:
This method is nice and easy, but you should not use it if you want your simulation to run on distributed-memory clusters. We'll see why in the last section. So it is good habit to not access the arrays directly but use the data handling to do so. We can, for example, initialize the array also with the following code:
%% Cell type:code id: tags:
``` python
dh.fill('src', 0.0)
```
%% Cell type:markdown id: tags:
To run the kernels with the same code as before, we would also need the arrays. We could do that accessing the `cpu_arrays`:
%% Cell type:code id: tags:
``` python
kernel_function(src=dh.cpu_arrays['src'],
dst=dh.cpu_arrays['dst'])
```
%% Cell type:markdown id: tags:
but to be prepared for MPI parallel simulations, again a method of the data handling should be used for this.
Besides, this method is also simpler to use - since it automatically detects which arrays a kernel uses and passes them in.
%% Cell type:code id: tags:
``` python
dh.run_kernel(kernel_function)
```
%% Cell type:markdown id: tags:
To access the data read-only instead of using `cpu_arrays` the gather function should be used.
This function gives you a read-only copy of the domain or part of the domain.
We will discuss this function later in more detail when we look at MPI parallel simulations.
For serial simulations keep in mind that modifying the resulting array does not change your original data!
%% Cell type:code id: tags:
``` python
read_only_copy = dh.gather_array('src', ps.make_slice[:, :], ghost_layers=False)
```
%% Cell type:markdown id: tags:
### 2.2. Example for GPU simulations
In this section we have a look at GPU simulations using the data handling. Only minimal changes are required.
When creating the data handling we can pass a 'default_target'. This means for every added field an array on the CPU and the GPU is allocated. This is a useful default, for more fine-grained control the `add_array` method also takes additional parameters controlling where the array should be allocated.
Additionally we also need to compile a GPU version of the kernel.
%% Cell type:code id: tags:
``` python
if gpu is False:
dh = ps.create_data_handling(domain_size=(30, 30), default_target=ps.Target.CPU)
else:
dh = ps.create_data_handling(domain_size=(30, 30), default_target=ps.Target.GPU)
# fields are now created using the data handling
src_field = dh.add_array('src', values_per_cell=1)
dst_field = dh.add_array('dst', values_per_cell=1)
# kernel is created just like before
update_rule = [ps.Assignment(lhs=dst_field[0, 0],
rhs=(src_field[1, 0] + src_field[-1, 0] + src_field[0, 1] + src_field[0, -1]) / 4)]
config = ps.CreateKernelConfig(target=dh.default_target)
kernel_function = ps.create_kernel(update_rule, config=config).compile()
dh.fill('src', 0.0)
```
%% Cell type:markdown id: tags:
The data handling provides function to transfer data between CPU and GPU
%% Cell type:code id: tags:
``` python
if gpu:
dh.to_gpu('src')
dh.run_kernel(kernel_function)
if gpu:
dh.to_cpu('dst')
```
%% Cell type:markdown id: tags:
usually one wants to transfer all fields that have been allocated on CPU and GPU at the same time:
%% Cell type:code id: tags:
``` python
dh.all_to_gpu()
dh.run_kernel(kernel_function)
dh.all_to_cpu()
```
%% Cell type:markdown id: tags:
We can always include the `all_to_*` functions in our code, since they do nothing if there are no arrays allocated on the GPU. Thus there is only a single point in the code where we can switch between CPU and GPU version: the `default_target` of the data handling.
%% Cell type:markdown id: tags:
### 2.2 Ghost Layers and periodicity
The data handling can also provide periodic boundary conditions. Therefor the domain is extended by one layer of cells, the so-called ghost layer or halo layer.
%% Cell type:code id: tags:
``` python
print("Shape of domain ", dh.shape)
print("Direct access to arrays ", dh.cpu_arrays['src'].shape)
print("Gather ", dh.gather_array('src', ghost_layers=True).shape)
```
%% Output
Shape of domain (30, 30)
Direct access to arrays (32, 32)
Gather (32, 32)
%% Cell type:markdown id: tags:
So the actual arrays are 2 cells larger than what you asked for. This additional layer is used to copy over the data from the other end of the array, such that for the stencil algorithm effectively the domain is periodic. This copying operation has to be started manually though:
%% Cell type:code id: tags:
``` python
dh = ps.create_data_handling((4, 4), periodicity=(True, False))
dh.add_array("src")
# get copy function
copy_fct = dh.synchronization_function(['src'])
dh.fill('src', 0.0, ghost_layers=True)
dh.fill('src', 3.0, ghost_layers=False)
before_sync = dh.gather_array('src', ghost_layers=True).copy()
copy_fct() # copy over to get periodicity in x direction
after_sync = dh.gather_array('src', ghost_layers=True).copy()
plt.subplot(1,2,1)
plt.scalar_field(before_sync);
plt.title("Before")
plt.subplot(1,2,2)
plt.scalar_field(after_sync);
plt.title("After");
```
%% Output
%% Cell type:markdown id: tags:
## 3. Going (MPI) parallel - the parallel data handling
### 3.1. Conceptual overview
To run MPI parallel simulations the waLBerla framework is used. waLBerla has to be compiled against your local MPI library and thus is a bit hard to install. We suggest to use the version shipped with conda for testing. For production, when you want to run on a cluster the best option is to build waLBerla yourself against the MPI library that is installed on your cluster.
Now lets have a look on how to write code that runs MPI parallel.
Since the data is distributed, we don't have access to the full array any more but only to the part that is stored locally. The domain is split into so called blocks, where one process might get one (or sometimes multiple) blocks. To do anything with the local part of the data we iterate over the **local** blocks to get the contents as numpy arrays. The blocks returned in the loop differ from process to process.
Copy the following snippet to a python file and run with multiple processes e.g.:
``mpirun -np 4 myscript.py`` you will see that there are as many blocks as processes.
%% Cell type:code id: tags:
``` python
dh = ps.create_data_handling(domain_size=(30, 30), parallel=True)
field = dh.add_array('field')
for block in dh.iterate():
# offset is in global coordinates, where first inner cell has coordiante (0,0)
# and ghost layers have negative coordinates
print(block.offset, block['field'].shape)
# use offset to create a local array 'my_data' for the part of the domain
#np.copyto(block[field.name], my_data)
```
%% Output
(-1, -1) (32, 32)
[0][PROGRESS]------(0.000 sec) Initializing SetupBlockForest:
[0] - AABB: [ <0,0,0>, <30,30,1> ]
[0] - forest size (root blocks / blocks on the initial grid): 1 x 1 x 1
[0] - periodicity: false x false x false
[0][PROGRESS]------(0.000 sec) Initializing SetupBlockForest: Allocating root blocks ...
[0][PROGRESS]------(0.000 sec) Initializing SetupBlockForest: Setting up neighborhood information for each block (with respect to periodicity) ...
[0][PROGRESS]------(0.000 sec) Initializing SetupBlockForest: Assigning workload, memory requirements, and SUIDs to blocks ...
[0][PROGRESS]------(0.000 sec) Initializing SetupBlockForest: finished!
[0] The following block structure has been created:
[0] - AABB: [ <0,0,0>, <30,30,1> ]
[0] - initial decomposition: 1 x 1 x 1 (= forest size)
[0] - periodicity: false x false x false
[0] - number of blocks discarded from the initial grid: 0 (= 0 %)
[0] - number of levels: 1
[0] - tree ID digits: 1 (-> block ID bytes = 1)
[0] - total number of blocks: 1
[0] - blocks have not yet been distributed to processes
[0][PROGRESS]------(0.000 sec) Balancing SetupBlockForest: Creating a process distribution for 1 process(es) ...
[0][PROGRESS]------(0.000 sec) Balancing SetupBlockForest: process distribution to 1 process(es) finished!
[0] - number of worker processes: 1
[0] - number of empty buffer processes: 0
[0] - buffer processes are inserted into the process network: no
[0] The resulting block structure looks like as follows:
[0] - AABB: [ <0,0,0>, <30,30,1> ]
[0] - initial decomposition: 1 x 1 x 1 (= forest size)
[0] - periodicity: false x false x false
[0] - number of blocks discarded from the initial grid: 0 (= 0 %)
[0] - number of levels: 1
[0] - tree ID digits: 1 (-> block ID bytes = 1)
[0] - total number of blocks: 1
[0] - number of processes: 1 (1 worker process(es) / 0 empty buffer process(es))
[0] - buffer processes are inserted into the process network: no
[0] - process ID bytes: 0
[0] - blocks/memory/workload per process:
[0] + blocks:
[0] - min = 1
[0] - max = 1
[0] - avg = 1
[0] - stdDev = 0
[0] - relStdDev = 0
[0] + memory:
[0] - min = 1
[0] - max = 1
[0] - avg = 1
[0] - stdDev = 0
[0] - relStdDev = 0
[0] + workload:
[0] - min = 1
[0] - max = 1
[0] - avg = 1
[0] - stdDev = 0
[0] - relStdDev = 0
[0][PROGRESS]------(0.000 sec) Adding block data ("cell bounding box")
[0][PROGRESS]------(0.000 sec) Adding block data ("field")
%% Cell type:markdown id: tags:
To get some more interesting results here in the notebook we put multiple blocks onto our single notebook process. This makes not much sense for real simulations, but for testing and demonstration purposes this is useful.
%% Cell type:code id: tags:
``` python
from waLBerla import createUniformBlockGrid
from pystencils.datahandling import ParallelDataHandling
blocks = createUniformBlockGrid(blocks=(2,1,2), cellsPerBlock=(20, 10, 20),
oneBlockPerProcess=False, periodic=(1, 0, 0))
dh = ParallelDataHandling(blocks)
field = dh.add_array('field')
for block in dh.iterate():
print(block.offset, block['field'].shape)
```
%% Output
(-1, -1, -1)[0][PROGRESS]------(0.025 sec) Initializing SetupBlockForest:
[0] - AABB: [ <0,0,0>, <40,10,40> ]
[0] - forest size (root blocks / blocks on the initial grid): 2 x 1 x 2
[0] - periodicity: true x false x false
(22, 12, 22)
(19, -1, -1) (22, 12, 22)
(-1, -1, 19) (22, 12, 22)
(19, -1, 19) (22, 12, 22)
[0][PROGRESS]------(0.025 sec) Initializing SetupBlockForest: Allocating root blocks ...
[0][PROGRESS]------(0.025 sec) Initializing SetupBlockForest: Setting up neighborhood information for each block (with respect to periodicity) ...
[0][PROGRESS]------(0.025 sec) Initializing SetupBlockForest: Assigning workload, memory requirements, and SUIDs to blocks ...
[0][PROGRESS]------(0.025 sec) Initializing SetupBlockForest: finished!
[0] The following block structure has been created:
[0] - AABB: [ <0,0,0>, <40,10,40> ]
[0] - initial decomposition: 2 x 1 x 2 (= forest size)
[0] - periodicity: true x false x false
[0] - number of blocks discarded from the initial grid: 0 (= 0 %)
[0] - number of levels: 1
[0] - tree ID digits: 3 (-> block ID bytes = 1)
[0] - total number of blocks: 4
[0] - blocks have not yet been distributed to processes
[0][PROGRESS]------(0.025 sec) Balancing SetupBlockForest: Creating a process distribution for 1 process(es) ...
[0][PROGRESS]------(0.025 sec) Balancing SetupBlockForest: process distribution to 1 process(es) finished!
[0] - number of worker processes: 1
[0] - number of empty buffer processes: 0
[0] - buffer processes are inserted into the process network: no
[0] The resulting block structure looks like as follows:
[0] - AABB: [ <0,0,0>, <40,10,40> ]
[0] - initial decomposition: 2 x 1 x 2 (= forest size)
[0] - periodicity: true x false x false
[0] - number of blocks discarded from the initial grid: 0 (= 0 %)
[0] - number of levels: 1
[0] - tree ID digits: 3 (-> block ID bytes = 1)
[0] - total number of blocks: 4
[0] - number of processes: 1 (1 worker process(es) / 0 empty buffer process(es))
[0] - buffer processes are inserted into the process network: no
[0] - process ID bytes: 0
[0] - blocks/memory/workload per process:
[0] + blocks:
[0] - min = 4
[0] - max = 4
[0] - avg = 4
[0] - stdDev = 0
[0] - relStdDev = 0
[0] + memory:
[0] - min = 4
[0] - max = 4
[0] - avg = 4
[0] - stdDev = 0
[0] - relStdDev = 0
[0] + workload:
[0] - min = 4
[0] - max = 4
[0] - avg = 4
[0] - stdDev = 0
[0] - relStdDev = 0
[0][PROGRESS]------(0.025 sec) Adding block data ("cell bounding box")
[0][PROGRESS]------(0.026 sec) Adding block data ("field")
%% Cell type:markdown id: tags:
Now we see that we have four blocks with (20, 10, 20) block each, and the global domain is (40, 10, 40) big.
All subblock also have a ghost layer around them, which is used to synchronize with their neighboring blocks (over the network). For ghost layer synchronization the same `synchronization_function` is used that we used above for periodic boundaries, because copying between blocks and copying the ghost layer for periodicity uses the same mechanism.
%% Cell type:code id: tags:
``` python
dh.gather_array('field').shape
```
%% Output
$\displaystyle \left( 40, \ 10, \ 40\right)$
(40, 10, 40)
%% Cell type:markdown id: tags:
### 3.2. Parallel example
To illustrate this in more detail, lets run a simple kernel on a parallel domain. waLBerla can handle 3D domains only, so we choose a z-size of 1.
%% Cell type:code id: tags:
``` python
blocks = createUniformBlockGrid(blocks=(2,4,1), cellsPerBlock=(20, 10, 1),
oneBlockPerProcess=False, periodic=(1, 1, 0))
dh = ParallelDataHandling(blocks, dim=2)
src_field = dh.add_array('src')
dst_field = dh.add_array('dst')
update_rule = [ps.Assignment(lhs=dst_field[0, 0 ],
rhs=(src_field[1, 0] + src_field[-1, 0] +
src_field[0, 1] + src_field[0, -1]) / 4)]
# 3. compile update rule to function
kernel_function = ps.create_kernel(update_rule).compile()
```
%% Output
[0][PROGRESS]------(0.119 sec) Initializing SetupBlockForest:
[0] - AABB: [ <0,0,0>, <40,40,1> ]
[0] - forest size (root blocks / blocks on the initial grid): 2 x 4 x 1
[0] - periodicity: true x true x false
[0][PROGRESS]------(0.119 sec) Initializing SetupBlockForest: Allocating root blocks ...
[0][PROGRESS]------(0.119 sec) Initializing SetupBlockForest: Setting up neighborhood information for each block (with respect to periodicity) ...
[0][PROGRESS]------(0.119 sec) Initializing SetupBlockForest: Assigning workload, memory requirements, and SUIDs to blocks ...
[0][PROGRESS]------(0.119 sec) Initializing SetupBlockForest: finished!
[0] The following block structure has been created:
[0] - AABB: [ <0,0,0>, <40,40,1> ]
[0] - initial decomposition: 2 x 4 x 1 (= forest size)
[0] - periodicity: true x true x false
[0] - number of blocks discarded from the initial grid: 0 (= 0 %)
[0] - number of levels: 1
[0] - tree ID digits: 4 (-> block ID bytes = 1)
[0] - total number of blocks: 8
[0] - blocks have not yet been distributed to processes
[0][PROGRESS]------(0.119 sec) Balancing SetupBlockForest: Creating a process distribution for 1 process(es) ...
[0][PROGRESS]------(0.119 sec) Balancing SetupBlockForest: process distribution to 1 process(es) finished!
[0] - number of worker processes: 1
[0] - number of empty buffer processes: 0
[0] - buffer processes are inserted into the process network: no
[0] The resulting block structure looks like as follows:
[0] - AABB: [ <0,0,0>, <40,40,1> ]
[0] - initial decomposition: 2 x 4 x 1 (= forest size)
[0] - periodicity: true x true x false
[0] - number of blocks discarded from the initial grid: 0 (= 0 %)
[0] - number of levels: 1
[0] - tree ID digits: 4 (-> block ID bytes = 1)
[0] - total number of blocks: 8
[0] - number of processes: 1 (1 worker process(es) / 0 empty buffer process(es))
[0] - buffer processes are inserted into the process network: no
[0] - process ID bytes: 0
[0] - blocks/memory/workload per process:
[0] + blocks:
[0] - min = 8
[0] - max = 8
[0] - avg = 8
[0] - stdDev = 0
[0] - relStdDev = 0
[0] + memory:
[0] - min = 8
[0] - max = 8
[0] - avg = 8
[0] - stdDev = 0
[0] - relStdDev = 0
[0] + workload:
[0] - min = 8
[0] - max = 8
[0] - avg = 8
[0] - stdDev = 0
[0] - relStdDev = 0
[0][PROGRESS]------(0.119 sec) Adding block data ("cell bounding box")
[0][PROGRESS]------(0.119 sec) Adding block data ("src")
[0][PROGRESS]------(0.120 sec) Adding block data ("dst")
%% Cell type:markdown id: tags:
Now lets initialize the arrays. To do this we can get arrays (meshgrids) from the block with the coordinates of the local cells. We use this to initialize a circular shape.
%% Cell type:code id: tags:
``` python
dh.fill('src', 0.0)
for block in dh.iterate(ghost_layers=False, inner_ghost_layers=False):
x, y = block.midpoint_arrays
inside_sphere = (x -20)**2 + (y-25)**2 < 8 ** 2
block['src'][inside_sphere] = 1.0
plt.scalar_field( dh.gather_array('src') );
```
%% Output
%% Cell type:markdown id: tags:
Now we can run our compute kernel on this data as usual. We just have to make sure that the field is synchronized after every step, i.e. that the ghost layers are correctly updated.
%% Cell type:code id: tags:
``` python
sync = dh.synchronization_function(['src'])
for i in range(40):
sync()
dh.run_kernel(kernel_function)
dh.swap('src', 'dst')
```
%% Cell type:code id: tags:
``` python
plt.scalar_field( dh.gather_array('src') );
```
%% Output
doc/notebooks/04_tutorial_advection_diffusion.ipynb 0 → 100644
View file @ 5600b6b6
This source diff could not be displayed because it is too large. You can view the blob instead.
doc/notebooks/05_tutorial_phasefield_spinodal_decomposition.ipynb 0 → 100644
View file @ 5600b6b6
This source diff could not be displayed because it is too large. You can view the blob instead.
doc/notebooks/06_tutorial_phasefield_dentritic_growth.ipynb 0 → 100644
View file @ 5600b6b6
%% Cell type:code id: tags:
``` python
from pystencils.session import *
import pystencils as ps
sp.init_printing()
frac = sp.Rational
```
%% Cell type:markdown id: tags:
# Tutorial 06: Phase-field simulation of dentritic solidification
This is the second tutorial on phase field methods with pystencils. Make sure to read the previous tutorial first.
In this tutorial we again implement a model described in **Programming Phase-Field Modelling** by S. Bulent Biner.
This time we implement the model from chapter 4.7 that describes dentritic growth. So get ready for some beautiful snowflake pictures.
We start again by adding all required arrays fields. This time we explicitly store the change of the phase variable φ in time, since the dynamics is calculated using an Allen-Cahn formulation where a term $\partial_t \phi$ occurs.
%% Cell type:code id: tags:
``` python
dh = ps.create_data_handling(domain_size=(300, 300), periodicity=True,
default_target=ps.Target.CPU)
φ_field = dh.add_array('phi', latex_name='φ')
φ_field_tmp = dh.add_array('phi_temp', latex_name='φ_temp')
φ_delta_field = dh.add_array('phidelta', latex_name='φ_D')
t_field = dh.add_array('T')
t_field_tmp = dh.add_array('T_tmp')
```
%% Cell type:markdown id: tags:
This model has a lot of parameters that are created here in a symbolic fashion.
%% Cell type:code id: tags:
``` python
ε, m, δ, j, θzero, α, γ, Teq, κ, τ = sp.symbols("ε m δ j θ_0 α γ T_eq κ τ")
εb = sp.Symbol("\\bar{\\epsilon}")
φ = φ_field.center
φ_tmp = φ_field_tmp.center
T = t_field.center
def f(φ, m):
return φ**4 / 4 - (frac(1, 2) - m/3) * φ**3 + (frac(1,4)-m/2)*φ**2
free_energy_density = ε**2 / 2 * (ps.fd.Diff(φ,0)**2 + ps.fd.Diff(φ,1)**2 ) + f(φ, m)
free_energy_density
```
%% Output
$\displaystyle \frac{{φ}_{(0,0)}^{4}}{4} - {φ}_{(0,0)}^{3} \left(\frac{1}{2} - \frac{m}{3}\right) + {φ}_{(0,0)}^{2} \left(\frac{1}{4} - \frac{m}{2}\right) + \frac{ε^{2} \left({\partial_{0} {φ}_{(0,0)}}^{2} + {\partial_{1} {φ}_{(0,0)}}^{2}\right)}{2}$
4 2 ⎛ 2 2⎞
φ_C 3 ⎛1 m⎞ 2 ⎛1 m⎞ ε ⋅⎝D(φ[0,0]) + D(φ[0,0]) ⎠
──── - φ_C ⋅⎜─ - ─⎟ + φ_C ⋅⎜─ - ─⎟ + ────────────────────────────
4 ⎝2 3⎠ ⎝4 2⎠ 2
%% Cell type:markdown id: tags:
The free energy is again composed of a bulk and interface part.
%% Cell type:code id: tags:
``` python
plt.figure(figsize=(7,4))
plt.sympy_function(f(φ, m=1), x_values=(-1.05, 1.5) )
plt.xlabel("φ")
plt.title("Bulk free energy");
```
%% Output
%% Cell type:markdown id: tags:
Compared to last tutorial, this bulk free energy has also two minima, but at different values.
Another complexity of the model is its anisotropy. The gradient parameter $\epsilon$ depends on the interface normal.
%% Cell type:code id: tags:
``` python
def σ(θ):
return 1 + δ * sp.cos(j * (θ - θzero))
θ = sp.atan2(ps.fd.Diff(φ, 1), ps.fd.Diff(φ, 0))
ε_val = εb * σ(θ)
ε_val
```
%% Output
$\displaystyle \bar{\epsilon} \left(δ \cos{\left(j \left(- θ_{0} + \operatorname{atan_{2}}{\left({\partial_{1} {φ}_{(0,0)}},{\partial_{0} {φ}_{(0,0)}} \right)}\right) \right)} + 1\right)$
\bar{\epsilon}⋅(δ⋅cos(j⋅(-θ₀ + atan2(D(φ[0,0]), D(φ[0,0])))) + 1)
%% Cell type:code id: tags:
``` python
def m_func(T):
return (α / sp.pi) * sp.atan(γ * (Teq - T))
```
%% Cell type:markdown id: tags:
However, we just insert these parameters into the free energy formulation before doing the functional derivative, to make the dependence of $\epsilon$ on $\nabla \phi$ explicit.
%% Cell type:code id: tags:
``` python
fe = free_energy_density.subs({
m: m_func(T),
ε: εb * σ(θ),
})
dF_dφ = ps.fd.functional_derivative(fe, φ)
dF_dφ = ps.fd.expand_diff_full(dF_dφ, functions=[φ])
dF_dφ
```
%% Output
$\displaystyle {φ}_{(0,0)}^{3} - \frac{{φ}_{(0,0)}^{2} α \operatorname{atan}{\left({T}_{(0,0)} γ - T_{eq} γ \right)}}{\pi} - \frac{3 {φ}_{(0,0)}^{2}}{2} + \frac{{φ}_{(0,0)} α \operatorname{atan}{\left({T}_{(0,0)} γ - T_{eq} γ \right)}}{\pi} + \frac{{φ}_{(0,0)}}{2} - \bar{\epsilon}^{2} δ^{2} \cos^{2}{\left(j θ_{0} - j \operatorname{atan_{2}}{\left({\partial_{1} {φ}_{(0,0)}},{\partial_{0} {φ}_{(0,0)}} \right)} \right)} {\partial_{0} {\partial_{0} {φ}_{(0,0)}}} - \bar{\epsilon}^{2} δ^{2} \cos^{2}{\left(j θ_{0} - j \operatorname{atan_{2}}{\left({\partial_{1} {φ}_{(0,0)}},{\partial_{0} {φ}_{(0,0)}} \right)} \right)} {\partial_{1} {\partial_{1} {φ}_{(0,0)}}} - 2 \bar{\epsilon}^{2} δ \cos{\left(j θ_{0} - j \operatorname{atan_{2}}{\left({\partial_{1} {φ}_{(0,0)}},{\partial_{0} {φ}_{(0,0)}} \right)} \right)} {\partial_{0} {\partial_{0} {φ}_{(0,0)}}} - 2 \bar{\epsilon}^{2} δ \cos{\left(j θ_{0} - j \operatorname{atan_{2}}{\left({\partial_{1} {φ}_{(0,0)}},{\partial_{0} {φ}_{(0,0)}} \right)} \right)} {\partial_{1} {\partial_{1} {φ}_{(0,0)}}} - \bar{\epsilon}^{2} {\partial_{0} {\partial_{0} {φ}_{(0,0)}}} - \bar{\epsilon}^{2} {\partial_{1} {\partial_{1} {φ}_{(0,0)}}}$
2 2
3 φ_C ⋅α⋅atan(T_C⋅γ - T_eq⋅γ) 3⋅φ_C φ_C⋅α⋅atan(T_C⋅γ - T_eq⋅γ) φ_C 2 2 2
φ_C - ─────────────────────────── - ────── + ────────────────────────── + ─── - \bar{\epsilon} ⋅δ ⋅cos (j⋅θ₀ - j⋅atan2(
π 2 π 2
2 2 2
D(φ[0,0]), D(φ[0,0])))⋅D(D(φ[0,0])) - \bar{\epsilon} ⋅δ ⋅cos (j⋅θ₀ - j⋅atan2(D(φ[0,0]), D(φ[0,0])))⋅D(D(φ[0,0])) - 2⋅\ba
2 2
r{\epsilon} ⋅δ⋅cos(j⋅θ₀ - j⋅atan2(D(φ[0,0]), D(φ[0,0])))⋅D(D(φ[0,0])) - 2⋅\bar{\epsilon} ⋅δ⋅cos(j⋅θ₀ - j⋅atan2(D(φ[0,0])
2 2
, D(φ[0,0])))⋅D(D(φ[0,0])) - \bar{\epsilon} ⋅D(D(φ[0,0])) - \bar{\epsilon} ⋅D(D(φ[0,0]))
%% Cell type:markdown id: tags:
Then we insert all the numeric parameters and discretize:
%% Cell type:code id: tags:
``` python
discretize = ps.fd.Discretization2ndOrder(dx=0.03, dt=1e-5)
parameters = {
τ: 0.0003,
κ: 1.8,
εb: 0.01,
δ: 0.02,
γ: 10,
j: 6,
α: 0.9,
Teq: 1.0,
θzero: 0.2,
sp.pi: sp.pi.evalf()
}
parameters
```
%% Output
$\displaystyle \left\{ \pi : 3.14159265358979, \ T_{eq} : 1.0, \ \bar{\epsilon} : 0.01, \ j : 6, \ α : 0.9, \ γ : 10, \ δ : 0.02, \ θ_{0} : 0.2, \ κ : 1.8, \ τ : 0.0003\right\}$
{π: 3.14159265358979, T_eq: 1.0, \bar{\epsilon}: 0.01, j: 6, α: 0.9, γ: 10, δ: 0.02, θ₀: 0.2, κ: 1.8, τ: 0.0003}
%% Cell type:code id: tags:
``` python
dφ_dt = - dF_dφ / τ
φ_eqs = ps.simp.sympy_cse_on_assignment_list([ps.Assignment(φ_delta_field.center,
discretize(dφ_dt.subs(parameters)))])
φ_eqs.append(ps.Assignment(φ_tmp, discretize(ps.fd.transient(φ) - φ_delta_field.center)))
temperature_evolution = -ps.fd.transient(T) + ps.fd.diffusion(T, 1) + κ * φ_delta_field.center
temperature_eqs = [
ps.Assignment(t_field_tmp.center, discretize(temperature_evolution.subs(parameters)))
]
```
%% Cell type:markdown id: tags:
When creating the kernels we pass as target (which may be 'Target.CPU' or 'Target.GPU') the default target of the target handling. This enables to switch to a GPU simulation just by changing the parameter of the data handling.
The rest is similar to the previous tutorial.
%% Cell type:code id: tags:
``` python
φ_kernel = ps.create_kernel(φ_eqs, cpu_openmp=True, target=dh.default_target).compile()
temperature_kernel = ps.create_kernel(temperature_eqs, target=dh.default_target).compile()
```
%% Cell type:code id: tags:
``` python
def timeloop(steps=200):
φ_sync = dh.synchronization_function(['phi'])
temperature_sync = dh.synchronization_function(['T'])
dh.all_to_gpu() # this does nothing when running on CPU
for t in range(steps):
φ_sync()
dh.run_kernel(φ_kernel)
dh.swap(φ_field.name, φ_field_tmp.name)
temperature_sync()
dh.run_kernel(temperature_kernel)
dh.swap(t_field.name, t_field_tmp.name)
dh.all_to_cpu()
return dh.gather_array('phi')
def init(nucleus_size=np.sqrt(5)):
for b in dh.iterate():
x, y = b.cell_index_arrays
x, y = x-b.shape[0]//2, y-b.shape[0]//2
bArr = (x**2 + y**2) < nucleus_size**2
b['phi'].fill(0)
b['phi'][(x**2 + y**2) < nucleus_size**2] = 1.0
b['T'].fill(0.0)
def plot():
plt.subplot(1,3,1)
plt.scalar_field(dh.gather_array('phi'))
plt.title("φ")
plt.colorbar()
plt.subplot(1,3,2)
plt.title("T")
plt.scalar_field(dh.gather_array('T'))
plt.colorbar()
plt.subplot(1,3,3)
plt.title("∂φ")
plt.scalar_field(dh.gather_array('phidelta'))
plt.colorbar()
```
%% Cell type:code id: tags:
``` python
ps.show_code(φ_kernel)
```
%% Output
%% Cell type:code id: tags:
``` python
timeloop(10)
init()
plot()
print(dh)
```
%% Output
Name| Inner (min/max)| WithGl (min/max)
----------------------------------------------------
T| ( 0, 0)| ( 0, 0)
T_tmp| ( 0, 0)| ( 0, 0)
phi| ( 0, 1)| ( 0, 1)
phi_temp| ( 0, 0)| ( 0, 0)
phidelta| ( 0, 0)| ( 0, 0)
%% Cell type:code id: tags:
``` python
result = None
if 'is_test_run' not in globals():
ani = ps.plot.scalar_field_animation(timeloop, rescale=True, frames=600)
result = ps.jupyter.display_as_html_video(ani)
result
```
%% Cell type:code id: tags:
``` python
assert np.isfinite(dh.max('phi'))
assert np.isfinite(dh.max('T'))
```
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%% Cell type:code id: tags:
``` python
from pystencils.session import *
```
%% Cell type:markdown id: tags:
# Demo: Assignment collections and simplification
## Assignment collections
The assignment collection class helps to formulate and simplify assignments for numerical kernels.
An ``AssignmentCollection`` is an ordered collection of assignments, together with an optional ordered collection of subexpressions, that are required to evaluate the main assignments. There are various simplification rules available that operate on ``AssignmentCollection``s.
%% Cell type:markdown id: tags:
We start by defining some stencil update rule. Here we also use the *pystencils* ``Field``, note however that the assignment collection module works purely on the *sympy* level.
%% Cell type:code id: tags:
``` python
a,b,c = sp.symbols("a b c")
f = ps.fields("f(2) : [2D]")
g = ps.fields("g(2) : [2D]")
a1 = ps.Assignment(g[0,0](1), (a**2 +b) * f[0,1] + \
(a**2 - c) * f[1,0] + \
(a**2 - 2*c) * f[-1,0] + \
(a**2) * f[0, -1])
a2 = ps.Assignment(g[0,0](0), (c**2 +b) * f[0,1] + \
(c**2 - c) * f[1,0] + \
(c**2 - 2*c) * f[-1,0] + \
(c**2 - a**2) * f[0, -1])
ac = ps.AssignmentCollection([a1, a2], subexpressions=[])
ac
```
%% Output
AssignmentCollection: g_C^0, g_C^1 <- f(f_N^0, b, f_S^0, f_E^0, a, f_W^0, c)
%% Cell type:markdown id: tags:
*sympy* operations can be applied on an assignment collection: In this example we first expand the collection, then look for common subexpressions.
%% Cell type:code id: tags:
``` python
expand_all = ps.simp.apply_to_all_assignments(sp.expand)
expandedEc = expand_all(ac)
```
%% Cell type:code id: tags:
``` python
ac_cse = ps.simp.sympy_cse(expandedEc)
ac_cse
```
%% Output
AssignmentCollection: g_C^0, g_C^1 <- f(f_N^0, b, f_S^0, f_E^0, a, f_W^0, c)
%% Cell type:markdown id: tags:
Symbols occuring in assignment collections are classified into 3 categories:
- ``free_symbols``: symbols that occur in right-hand-sides but never on left-hand-sides
- ``bound_symbols``: symbols that occur on left-hand-sides
- ``defined_symbols``: symbols that occur on left-hand-sides of a main assignment
%% Cell type:code id: tags:
``` python
ac_cse.free_symbols
```
%% Output
$\displaystyle \left\{{f}_{(1,0)}^{0}, {f}_{(0,1)}^{0}, {f}_{(0,-1)}^{0}, {f}_{(-1,0)}^{0}, a, b, c\right\}$
{f_E__0, f_N__0, f_S__0, f_W__0, a, b, c}
%% Cell type:code id: tags:
``` python
ac_cse.bound_symbols
```
%% Output
$\displaystyle \left\{{g}_{(0,0)}^{0}, {g}_{(0,0)}^{1}, \xi_{0}, \xi_{1}, \xi_{2}, \xi_{3}\right\}$
{g_C__0, g_C__1, ξ₀, ξ₁, ξ₂, ξ₃}
%% Cell type:code id: tags:
``` python
ac_cse.defined_symbols
```
%% Output
$\displaystyle \left\{{g}_{(0,0)}^{0}, {g}_{(0,0)}^{1}\right\}$
{g_C__0, g_C__1}
%% Cell type:markdown id: tags:
Assignment collections can be splitted up, and merged together. For splitting, a list of symbols that occur on the left-hand-side in the main assignments has to be passed. The returned assignment collection only contains these main assignments together with all necessary subexpressions.
%% Cell type:code id: tags:
``` python
ac_f0 = ac_cse.new_filtered([g(0)])
ac_f1 = ac_cse.new_filtered([g(1)])
ac_f1
```
%% Output
AssignmentCollection: g_C^1, <- f(f_N^0, b, f_S^0, f_E^0, a, f_W^0, c)
%% Cell type:markdown id: tags:
Note here that $\xi_4$ is no longer part of the subexpressions, since it is not used in the main assignment of $f_C^1$.
If we merge both collections together, we end up with the original collection.
%% Cell type:code id: tags:
``` python
ac_f0.new_merged(ac_f1)
```
%% Output
AssignmentCollection: g_C^0, g_C^1 <- f(f_N^0, b, f_S^0, f_E^0, a, f_W^0, c)
%% Cell type:markdown id: tags:
There is also a method that inserts all subexpressions into the main assignments. This is the inverse operation of common subexpression elimination.
%% Cell type:code id: tags:
``` python
assert sp.simplify(ac_f0.new_without_subexpressions().main_assignments[0].rhs - a2.rhs) == 0
ac_f0.new_without_subexpressions()
```
%% Output
AssignmentCollection: g_C^0, <- f(f_N^0, b, f_S^0, f_E^0, a, f_W^0, c)
%% Cell type:markdown id: tags:
To evaluate an assignment collection, use the ``lambdify`` method. It is very similar to *sympy*s ``lambdify`` function.
%% Cell type:code id: tags:
``` python
evalFct = ac_cse.lambdify([f[0,1], f[1,0]], # new parameters of returned function
fixed_symbols={a:1, b:2, c:3, f[0,-1]: 4, f[-1,0]: 5}) # fix values of other symbols
evalFct(2,1)
```
%% Output
$\displaystyle \left\{ {g}_{(0,0)}^{0} : 75, \ {g}_{(0,0)}^{1} : -17\right\}$
{g_C__0: 75, g_C__1: -17}
%% Cell type:markdown id: tags:
lambdify is rather slow for evaluation. The intended way to evaluate an assignment collection is *pystencils* i.e. create a fast kernel, that applies the update at every site of a structured grid. The collection can be directly passed to the `create_kernel` function.
%% Cell type:code id: tags:
``` python
func = ps.create_kernel(ac_cse).compile()
```
%% Cell type:markdown id: tags:
## Simplification Strategies
In above examples, we already applied simplification rules to assignment collections. Simplification rules are functions that take, as a single argument, an assignment collection and return an modified/simplified copy of it. The ``SimplificationStrategy`` class holds a list of simplification rules and can apply all of them in the specified order. Additionally it provides useful printing and reporting functions.
We start by creating a simplification strategy, consisting of the expand and CSE simplifications we have already applied above:
%% Cell type:code id: tags:
``` python
strategy = ps.simp.SimplificationStrategy()
strategy.add(ps.simp.apply_to_all_assignments(sp.expand))
strategy.add(ps.simp.sympy_cse)
```
%% Cell type:markdown id: tags:
This strategy can be applied to any assignment collection:
%% Cell type:code id: tags:
``` python
strategy(ac)
```
%% Output
AssignmentCollection: g_C^0, g_C^1 <- f(f_N^0, b, f_S^0, f_E^0, a, f_W^0, c)
%% Cell type:markdown id: tags:
The strategy can also print the simplification results at each stage.
The report contains information about the number of operations after each simplification as well as the runtime of each simplification routine.
%% Cell type:code id: tags:
``` python
strategy.create_simplification_report(ac)
```
%% Output
<pystencils.simp.simplificationstrategy.SimplificationStrategy.create_simplification_report.<locals>.Report at 0x147de3e90>
%% Cell type:markdown id: tags:
The strategy can also print the full collection after each simplification...
%% Cell type:code id: tags:
``` python
strategy.show_intermediate_results(ac)
```
%% Output
<pystencils.simp.simplificationstrategy.SimplificationStrategy.show_intermediate_results.<locals>.IntermediateResults at 0x147e09c90>
%% Cell type:markdown id: tags:
... or only specific assignments for better readability
%% Cell type:code id: tags:
``` python
strategy.show_intermediate_results(ac, symbols=[g(1)])
```
%% Output
<pystencils.simp.simplificationstrategy.SimplificationStrategy.show_intermediate_results.<locals>.IntermediateResults at 0x1265a1b90>
%% Cell type:code id: tags:
``` python
```
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%% Cell type:code id: tags:
``` python
from pystencils.session import *
import timeit
%load_ext Cython
```
%% Cell type:markdown id: tags:
# Demo: Benchmark numpy, Cython, pystencils
In this notebook we compare and benchmark different ways of implementing a simple stencil kernel in Python.
Our simple example computes the average of the four neighbors in 2D and stores it in a second array. To prevent out-of-bounds accesses, we skip the cells at the border and compute values only in the range `[1:-1, 1:-1]`
%% Cell type:markdown id: tags:
## Implementations
The first implementation is a pure Python implementation:
%% Cell type:code id: tags:
``` python
def avg_pure_python(src, dst):
for x in range(1, src.shape[0] - 1):
for y in range(1, src.shape[1] - 1):
dst[x, y] = (src[x + 1, y] + src[x - 1, y] +
src[x, y + 1] + src[x, y - 1]) / 4
```
%% Cell type:markdown id: tags:
Obviously, this will be a rather slow version, since the loops are written directly in Python.
Next, we use *numpy* functions to delegate the looping to numpy. The first version uses the `roll` function to shift the array by one element in each direction. This version has to allocate a new array for each accessed neighbor.
%% Cell type:code id: tags:
``` python
def avg_numpy_roll(src, dst):
neighbors = [np.roll(src, axis=a, shift=s) for a in (0, 1) for s in (-1, 1)]
np.divide(sum(neighbors), 4, out=dst)
```
%% Cell type:markdown id: tags:
Using views, we can get rid of the additional copies:
%% Cell type:code id: tags:
``` python
def avg_numpy_slice(src, dst):
dst[1:-1, 1:-1] = src[2:, 1:-1] + src[:-2, 1:-1] + \
src[1:-1, 2:] + src[1:-1, :-2]
```
%% Cell type:markdown id: tags:
To further optimize the kernel we switch to Cython, to get a compiled C version.
%% Cell type:code id: tags:
``` python
%%cython
import cython
@cython.boundscheck(False)
@cython.wraparound(False)
def avg_cython(object[double, ndim=2] src, object[double, ndim=2] dst):
cdef int xs, ys, x, y
xs, ys = src.shape
for x in range(1, xs - 1):
for y in range(1, ys - 1):
dst[x, y] = (src[x + 1, y] + src[x - 1, y] +
src[x, y + 1] + src[x, y - 1]) / 4
```
%% Cell type:markdown id: tags:
If available, we also try the numba just-in-time compiler
%% Cell type:code id: tags:
``` python
try:
from numba import jit
@jit(nopython=True)
def avg_numba(src, dst):
dst[1:-1, 1:-1] = src[2:, 1:-1] + src[:-2, 1:-1] + \
src[1:-1, 2:] + src[1:-1, :-2]
except ImportError:
pass
```
%% Cell type:markdown id: tags:
And finally we also create a *pystencils* version of the same stencil code:
%% Cell type:code id: tags:
``` python
src, dst = ps.fields("src, dst: [2D]")
update = ps.Assignment(dst[0,0],
(src[1, 0] + src[-1, 0] + src[0, 1] + src[0, -1]) / 4)
avg_pystencils = ps.create_kernel(update).compile()
```
%% Cell type:code id: tags:
``` python
all_implementations = {
'pure Python': avg_pure_python,
'numpy roll': avg_numpy_roll,
'numpy slice': avg_numpy_slice,
'pystencils': avg_pystencils,
}
if 'avg_cython' in globals():
all_implementations['Cython'] = avg_cython
if 'avg_numba' in globals():
all_implementations['numba'] = avg_numba
```
%% Cell type:markdown id: tags:
## Benchmark functions
We implement a short function to get in- and output arrays of a given shape and to measure the runtime.
%% Cell type:code id: tags:
``` python
def get_arrays(shape):
in_arr = np.random.rand(*shape)
out_arr = np.empty_like(in_arr)
return in_arr, out_arr
def do_benchmark(func, shape):
in_arr, out_arr = get_arrays(shape)
func(src=in_arr, dst=out_arr) # warmup
timer = timeit.Timer('f(src=src, dst=dst)', globals={'f': func, 'src': in_arr, 'dst': out_arr})
calls, time_taken = timer.autorange()
return time_taken / calls
```
%% Cell type:markdown id: tags:
## Comparison
%% Cell type:code id: tags:
``` python
plot_order = ['pystencils', 'Cython', 'numba', 'numpy slice', 'numpy roll', 'pure Python']
plot_order = [p for p in plot_order if p in all_implementations]
def bar_plot(*shape):
names = plot_order
runtimes = tuple(do_benchmark(all_implementations[name], shape) for name in names)
speedups = tuple(runtime / min(runtimes) for runtime in runtimes)
y_pos = np.arange(len(names))
labels = tuple(f"{name} ({round(speedup, 1)} x)" for name, speedup in zip(names, speedups))
plt.text(0.5, 0.5, f"Size {shape}", horizontalalignment='center', fontsize=16,
verticalalignment='center', transform=plt.gca().transAxes)
plt.barh(y_pos, runtimes, log=True)
plt.yticks(y_pos, labels);
plt.xlabel('Runtime of single iteration')
plt.figure(figsize=(8, 8))
plt.subplot(3, 1, 1)
bar_plot(32, 32)
plt.subplot(3, 1, 2)
bar_plot(128, 128)
plt.subplot(3, 1, 3)
bar_plot(2048, 2048)
```
%% Output
%% Cell type:markdown id: tags:
All runtimes are plotted logarithmically. Numbers next to the labels show how much slower the version is than the fastest one.
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%% Cell type:code id: tags:
``` python
from pystencils.session import *
```
%% Cell type:markdown id: tags:
# Demo: Working with derivatives
## Overview
This notebook demonstrates how to formulate continuous differential operators in *pystencils* and automatically derive finite difference stencils from them.
Instead of using the built-in derivatives in *sympy*, *pystencils* comes with its own derivative objects. They represent spatial derivatives of pystencils fields.
%% Cell type:code id: tags:
``` python
f = ps.fields("f: [2D]")
first_derivative_x = ps.fd.diff(f, 0)
first_derivative_x
```
%% Output
$\displaystyle {\partial_{0} {f}_{(0,0)}}$
D(f[0,0])
%% Cell type:markdown id: tags:
This object is the derivative of the field $f$ with respect to the first spatial coordinate $x$. To get a finite difference approximation a discretization strategy is required:
%% Cell type:code id: tags:
``` python
discretize_2nd_order = ps.fd.Discretization2ndOrder(dx=sp.symbols("h"))
discretize_2nd_order(first_derivative_x)
```
%% Output
$\displaystyle \frac{{f}_{(1,0)} - {f}_{(-1,0)}}{2 h}$
f_E - f_W
─────────
2⋅h
%% Cell type:markdown id: tags:
Strictly speaking, derivative objects act on *field accesses*, not *fields*, that why there is a $(0,0)$ index at the field:
%% Cell type:code id: tags:
``` python
first_derivative_x
```
%% Output
$\displaystyle {\partial_{0} {f}_{(0,0)}}$
D(f[0,0])
%% Cell type:markdown id: tags:
Sometimes it might be useful to specify derivatives at an offset e.g.
%% Cell type:code id: tags:
``` python
derivative_offset = ps.fd.diff(f[0, 1], 0)
derivative_offset, discretize_2nd_order(derivative_offset)
```
%% Output
$\displaystyle \left( {\partial_{0} {f}_{(0,1)}}, \ \frac{{f}_{(1,1)} - {f}_{(-1,1)}}{2 h}\right)$
⎛ f_NE - f_NW⎞
⎜D(f[0,1]), ───────────⎟
⎝ 2⋅h ⎠
%% Cell type:markdown id: tags:
Another example with second order derivatives:
%% Cell type:code id: tags:
``` python
laplacian = ps.fd.diff(f, 0, 0) + ps.fd.diff(f, 1, 1)
laplacian
```
%% Output
$\displaystyle {\partial_{0} {\partial_{0} {f}_{(0,0)}}} + {\partial_{1} {\partial_{1} {f}_{(0,0)}}}$
D(D(f[0,0])) + D(D(f[0,0]))
%% Cell type:code id: tags:
``` python
discretize_2nd_order(laplacian)
```
%% Output
$\displaystyle \frac{- 2 {f}_{(0,0)} + {f}_{(1,0)} + {f}_{(-1,0)}}{h^{2}} + \frac{- 2 {f}_{(0,0)} + {f}_{(0,1)} + {f}_{(0,-1)}}{h^{2}}$
-2⋅f_C + f_E + f_W -2⋅f_C + f_N + f_S
────────────────── + ──────────────────
2 2
h h
%% Cell type:markdown id: tags:
## Working with derivative terms
No automatic simplifications are done on derivative terms i.e. linearity relations or product rule are not applied automatically.
%% Cell type:code id: tags:
``` python
f, g = ps.fields("f, g :[2D]")
c = sp.symbols("c")
δ = ps.fd.diff
expr = δ( δ(f, 0) + δ(g, 0) + c + 5 , 0)
expr
```
%% Output
$\displaystyle {\partial_{0} (c + {\partial_{0} {f}_{(0,0)}} + {\partial_{0} {g}_{(0,0)}} + 5) }$
D(c + Diff(f_C, 0, -1) + Diff(g_C, 0, -1) + 5)
%% Cell type:markdown id: tags:
This nested term can not be discretized automatically.
%% Cell type:code id: tags:
``` python
try:
discretize_2nd_order(expr)
except ValueError as e:
print(e)
```
%% Output
Only derivatives with field or field accesses as arguments can be discretized
%% Cell type:markdown id: tags:
### Linearity
The following function expands all derivatives exploiting linearity:
%% Cell type:code id: tags:
``` python
ps.fd.expand_diff_linear(expr)
```
%% Output
$\displaystyle {\partial_{0} c} + {\partial_{0} {\partial_{0} {f}_{(0,0)}}} + {\partial_{0} {\partial_{0} {g}_{(0,0)}}}$
D(c) + D(D(f[0,0])) + D(D(g[0,0]))
%% Cell type:markdown id: tags:
The symbol $c$ that was included is interpreted as a function by default.
We can control the simplification behaviour by specifying all functions or all constants:
%% Cell type:code id: tags:
``` python
ps.fd.expand_diff_linear(expr, functions=(f[0,0], g[0, 0]))
```
%% Output
$\displaystyle {\partial_{0} {\partial_{0} {f}_{(0,0)}}} + {\partial_{0} {\partial_{0} {g}_{(0,0)}}}$
D(D(f[0,0])) + D(D(g[0,0]))
%% Cell type:code id: tags:
``` python
ps.fd.expand_diff_linear(expr, constants=[c])
```
%% Output
$\displaystyle {\partial_{0} {\partial_{0} {f}_{(0,0)}}} + {\partial_{0} {\partial_{0} {g}_{(0,0)}}}$
D(D(f[0,0])) + D(D(g[0,0]))
%% Cell type:markdown id: tags:
The expanded term can then be discretized:
%% Cell type:code id: tags:
``` python
discretize_2nd_order(ps.fd.expand_diff_linear(expr, constants=[c]))
```
%% Output
$\displaystyle \frac{- 2 {f}_{(0,0)} + {f}_{(1,0)} + {f}_{(-1,0)}}{h^{2}} + \frac{- 2 {g}_{(0,0)} + {g}_{(1,0)} + {g}_{(-1,0)}}{h^{2}}$
-2⋅f_C + f_E + f_W -2⋅g_C + g_E + g_W
────────────────── + ──────────────────
2 2
h h
%% Cell type:markdown id: tags:
### Product rule
The next cells show how to apply product rule and its reverse:
%% Cell type:code id: tags:
``` python
expr = δ(f[0, 0] * g[0, 0], 0 )
expr
```
%% Output
$\displaystyle {\partial_{0} ({f}_{(0,0)} {g}_{(0,0)}) }$
D(f_C*g_C)
%% Cell type:code id: tags:
``` python
expanded_expr = ps.fd.expand_diff_products(expr)
expanded_expr
```
%% Output
$\displaystyle {f}_{(0,0)} {\partial_{0} {g}_{(0,0)}} + {g}_{(0,0)} {\partial_{0} {f}_{(0,0)}}$
f_C⋅D(g[0,0]) + g_C⋅D(f[0,0])
%% Cell type:code id: tags:
``` python
recombined_expr = ps.fd.combine_diff_products(expanded_expr)
recombined_expr
```
%% Output
$\displaystyle {\partial_{0} ({f}_{(0,0)} {g}_{(0,0)}) }$
D(f_C*g_C)
%% Cell type:code id: tags:
``` python
assert recombined_expr == expr
```
%% Cell type:markdown id: tags:
### Evaluate derivatives
Arguments of derivative need not be to be fields, only when trying to discretize them.
The next cells show how to transform them to *sympy* derivatives and evaluate them.
%% Cell type:code id: tags:
``` python
k = sp.symbols("k")
expr = δ(k**3 + 2 * k, 0 )
expr
```
%% Output
$\displaystyle {\partial_{0} (k^{3} + 2 k) }$
D(k**3 + 2*k)
%% Cell type:code id: tags:
``` python
ps.fd.evaluate_diffs(expr, var=k)
```
%% Output
$\displaystyle 3 k^{2} + 2$
2
3⋅k + 2
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%% Cell type:code id: tags:
``` python
import psutil
from pystencils.session import *
import shutil
```
%% Cell type:markdown id: tags:
# Demo: Finite differences - 2D wave equation
In this tutorial we show how to use the finite difference module of *pystencils* to solve a 2D wave equations. The time derivative is discretized by a simple forward Euler method.
%% Cell type:markdown id: tags:
$$ \frac{\partial^2 u}{\partial t^2} = \mbox{div} \left( q(x,y) \nabla u \right)$$
%% Cell type:markdown id: tags:
We begin by creating three *numpy* arrays for the current, the previous and the next timestep. Actually we will see later that two fields are enough, but let's keep it simple. From these *numpy* arrays we create *pystencils* fields to formulate our update rule.
%% Cell type:code id: tags:
``` python
size = (60, 70) # domain size
u_arrays = [np.zeros(size), np.zeros(size), np.zeros(size)]
u_fields = [ps.Field.create_from_numpy_array("u%s" % (name,), arr)
for name, arr in zip(["0", "1", "2"], u_arrays)]
# Nicer display for fields
for i, field in enumerate(u_fields):
field.latex_name = "u^{(%d)}" % (i,)
```
%% Cell type:markdown id: tags:
*pystencils* contains already simple rules to discretize the a diffusion term. The time discretization is done manually.
%% Cell type:code id: tags:
``` python
discretize = ps.fd.Discretization2ndOrder()
def central2nd_time_derivative(fields):
f_next, f_current, f_last = fields
return (f_next[0, 0] - 2 * f_current[0, 0] + f_last[0, 0]) / discretize.dt**2
rhs = ps.fd.diffusion(u_fields[1], 1)
wave_eq = sp.Eq(central2nd_time_derivative(u_fields), discretize(rhs))
wave_eq = sp.simplify(wave_eq)
wave_eq
```
%% Output
$\displaystyle \frac{{u^{(0)}}_{(0,0)} - 2 {u^{(1)}}_{(0,0)} + {u^{(2)}}_{(0,0)}}{dt^{2}} = \frac{- 4 {u^{(1)}}_{(0,0)} + {u^{(1)}}_{(1,0)} + {u^{(1)}}_{(0,1)} + {u^{(1)}}_{(0,-1)} + {u^{(1)}}_{(-1,0)}}{dx^{2}}$
u_0_C - 2⋅u_1_C + u_2_C -4⋅u_1_C + u_1_E + u_1_N + u_1_S + u_1_W
─────────────────────── = ────────────────────────────────────────
2 2
dt dx
%% Cell type:markdown id: tags:
The explicit Euler scheme is now obtained by solving above equation with respect to $u_C^{next}$.
%% Cell type:code id: tags:
``` python
u_next_C = u_fields[-1][0, 0]
update_rule = ps.Assignment(u_next_C, sp.solve(wave_eq, u_next_C)[0])
update_rule
```
%% Output
$\displaystyle {u^{(2)}}_{(0,0)} \leftarrow \frac{- 4 {u^{(1)}}_{(0,0)} dt^{2} + {u^{(1)}}_{(1,0)} dt^{2} + {u^{(1)}}_{(0,1)} dt^{2} + {u^{(1)}}_{(0,-1)} dt^{2} + {u^{(1)}}_{(-1,0)} dt^{2} + dx^{2} \left(- {u^{(0)}}_{(0,0)} + 2 {u^{(1)}}_{(0,0)}\right)}{dx^{2}}$
2 2 2 2 2 2
- 4⋅u_1_C⋅dt + u_1_E⋅dt + u_1_N⋅dt + u_1_S⋅dt + u_1_W⋅dt + dx ⋅(-u_0_C + 2⋅u_1_C)
u_2_C := ──────────────────────────────────────────────────────────────────────────────────────
2
dx
%% Cell type:markdown id: tags:
Before creating the kernel, we substitute numeric values for $dx$ and $dt$.
Then a kernel is created just like in the last tutorial.
%% Cell type:code id: tags:
``` python
update_rule = update_rule.subs({discretize.dx: 0.1, discretize.dt: 0.05})
ast = ps.create_kernel(update_rule)
kernel = ast.compile()
ps.show_code(ast)
```
%% Output
%% Cell type:code id: tags:
``` python
ast.get_parameters()
```
%% Output
[_data_u0, _data_u1, _data_u2]
%% Cell type:code id: tags:
``` python
ast.fields_accessed
```
%% Output
{u0: double[60,70], u1: double[60,70], u2: double[60,70]}
%% Cell type:markdown id: tags:
To run simulation a suitable initial condition and boundary treatment is required. We chose an initial condition which is zero at the borders of the domain. The outermost layer is not changed by the update kernel, so we have an implicit homogenous Dirichlet boundary condition.
%% Cell type:code id: tags:
``` python
X,Y = np.meshgrid( np.linspace(0, 1, size[1]), np.linspace(0,1, size[0]))
Z = np.sin(2*X*np.pi) * np.sin(2*Y*np.pi)
# Initialize the previous and current values with the initial function
np.copyto(u_arrays[0], Z)
np.copyto(u_arrays[1], Z)
# The values for the next timesteps do not matter, since they are overwritten
u_arrays[2][:, :] = 0
```
%% Cell type:markdown id: tags:
One timestep now consists of applying the kernel once, then shifting the arrays.
%% Cell type:code id: tags:
``` python
def run(timesteps=1):
for t in range(timesteps):
kernel(u0=u_arrays[0], u1=u_arrays[1], u2=u_arrays[2])
u_arrays[0], u_arrays[1], u_arrays[2] = u_arrays[1], u_arrays[2], u_arrays[0]
return u_arrays[2]
```
%% Cell type:markdown id: tags:
Lets create an animation of the solution:
%% Cell type:code id: tags:
``` python
if shutil.which("ffmpeg") is not None:
ani = plt.surface_plot_animation(run, zlim=(-1, 1))
ps.jupyter.display_as_html_video(ani)
else:
print("No ffmpeg installed")
```
%% Output
No ffmpeg installed
%% Cell type:code id: tags:
``` python
assert np.isfinite(np.max(u_arrays[2]))
```
%% Cell type:markdown id: tags:
__Runing on GPU__
We can also run the same kernel on the GPU, by using the ``cupy`` package.
%% Cell type:code id: tags:
``` python
try:
import cupy
except ImportError:
cupy=None
print('No cupy installed')
res = None
if cupy:
gpu_ast = ps.create_kernel(update_rule, target=ps.Target.GPU)
gpu_kernel = gpu_ast.compile()
res = ps.show_code(gpu_ast)
res
```
%% Output
%% Cell type:markdown id: tags:
The run function has to be changed now slightly, since the data has to be transfered to the GPU first, then the kernel can be executed, and in the end the data has to be transfered back
%% Cell type:code id: tags:
``` python
if cupy:
def run_on_gpu(timesteps=1):
# Transfer arrays to GPU
gpuArrs = [cupy.asarray(cpu_array) for cpu_array in u_arrays]
for t in range(timesteps):
gpu_kernel(u0=gpuArrs[0], u1=gpuArrs[1], u2=gpuArrs[2])
gpuArrs[0], gpuArrs[1], gpuArrs[2] = gpuArrs[1], gpuArrs[2], gpuArrs[0]
# Transfer arrays to CPU
for gpuArr, cpuArr in zip(gpuArrs, u_arrays):
cpuArr[:] = gpuArr.get()
assert np.isfinite(np.max(u_arrays[2]))
```
%% Cell type:code id: tags:
``` python
if cupy:
run_on_gpu(400)
```
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API Reference
=============
.. toctree::
:maxdepth: 3
kernel_compile_and_call.rst
enums.rst
simplifications.rst
datahandling.rst
configuration.rst
field.rst
stencil.rst
finite_differences.rst
plot.rst
ast.rst
doc/sphinx/ast.rst 0 → 100644
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*********************************************
For developers: AST Nodes and Transformations
*********************************************
AST Nodes
=========
.. automodule:: pystencils.astnodes
:members:
Transformations
===============
.. automodule:: pystencils.transformations
:members:
doc/sphinx/configuration.rst 0 → 100644
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*************
Configuration
*************
.. automodule:: pystencils.cpu.cpujit
\ No newline at end of file
doc/sphinx/datahandling.rst 0 → 100644
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************
DataHandling
************
.. autoclass:: pystencils.datahandling.DataHandling
:members:
\ No newline at end of file
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************
Enumerations
************
.. automodule:: pystencils.enums
:members:
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*****
Field
*****
.. automodule:: pystencils.field
:members:
\ No newline at end of file
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******************
Finite Differences
******************
.. automodule:: pystencils.fd
:members:
doc/sphinx/kernel_compile_and_call.rst 0 → 100644
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*****************************************
Creating and calling kernels from Python
*****************************************
Creating kernels
----------------
.. autofunction:: pystencils.create_kernel
.. autoclass:: pystencils.CreateKernelConfig
:members:
.. autofunction:: pystencils.kernelcreation.create_domain_kernel
.. autofunction:: pystencils.kernelcreation.create_indexed_kernel
.. autofunction:: pystencils.kernelcreation.create_staggered_kernel
Code printing
-------------
.. autofunction:: pystencils.show_code
GPU Indexing
-------------
.. autoclass:: pystencils.gpu.AbstractIndexing
:members:
.. autoclass:: pystencils.gpu.BlockIndexing
:members:
.. autoclass:: pystencils.gpu.LineIndexing
:members:
doc/sphinx/plot.rst 0 → 100644
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**********************
Plotting and Animation
**********************
.. automodule:: pystencils.plot
:members:
doc/sphinx/pystencils.bib 0 → 100644
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