From d4f813a6ee56a98afcc778bb3b08410f7ada4ccd Mon Sep 17 00:00:00 2001
From: Martin Bauer <martin.bauer@fau.de>
Date: Thu, 8 Mar 2018 15:01:46 +0100
Subject: [PATCH] Finite differences
- merged finite difference functions into one function
- put derivative operators from lbmpy into pystencils
---
derivative.py | 473 +++++++++++++++++++++++++++++++++++++++++++
finitedifferences.py | 70 ++++++-
2 files changed, 537 insertions(+), 6 deletions(-)
create mode 100644 derivative.py
diff --git a/derivative.py b/derivative.py
new file mode 100644
index 000000000..af097ccb5
--- /dev/null
+++ b/derivative.py
@@ -0,0 +1,473 @@
+import sympy as sp
+from collections import namedtuple, defaultdict
+from pystencils.sympyextensions import normalizeProduct, prod
+
+
+def defaultDiffSortKey(d):
+ return str(d.ceIdx), str(d.label)
+
+
+class DiffOperator(sp.Expr):
+ """
+ Un-applied differential, i.e. differential operator
+ Its args are:
+ - label: the differential is w.r.t to this label / variable.
+ This label is mainly for display purposes (its the subscript) and to distinguish DiffOperators
+ If the label is '-1' no subscript is displayed
+ - ceIdx: expansion order index in the Chapman Enskog expansion. It is displayed as superscript.
+ and not displayed if set to '-1'
+ The DiffOperator behaves much like a variable with special name. Its main use is to be applied later, using the
+ DiffOperator.apply(expr, arg) which transforms 'DiffOperator's to applied 'Diff's
+ """
+ is_commutative = True
+ is_number = False
+ is_Rational = False
+
+ def __new__(cls, label=-1, ceIdx=-1, **kwargs):
+ return sp.Expr.__new__(cls, sp.sympify(label), sp.sympify(ceIdx), **kwargs)
+
+ @property
+ def label(self):
+ return self.args[0]
+
+ @property
+ def ceIdx(self):
+ return self.args[1]
+
+ def _latex(self, printer, *args):
+ result = "{\partial"
+ if self.ceIdx >= 0:
+ result += "^{(%s)}" % (self.ceIdx,)
+ if self.label != -1:
+ result += "_{%s}" % (self.label,)
+ result += "}"
+ return result
+
+ @staticmethod
+ def apply(expr, argument):
+ """
+ Returns a new expression where each 'DiffOperator' is replaced by a 'Diff' node.
+ Multiplications of 'DiffOperator's are interpreted as nested application of differentiation:
+ i.e. DiffOperator('x')*DiffOperator('x') is a second derivative replaced by Diff(Diff(arg, x), t)
+ """
+ def handleMul(mul):
+ args = normalizeProduct(mul)
+ diffs = [a for a in args if isinstance(a, DiffOperator)]
+ if len(diffs) == 0:
+ return mul
+ rest = [a for a in args if not isinstance(a, DiffOperator)]
+ diffs.sort(key=defaultDiffSortKey)
+ result = argument
+ for d in reversed(diffs):
+ result = Diff(result, label=d.label, ceIdx=d.ceIdx)
+ return prod(rest) * result
+
+ expr = expr.expand()
+ if expr.func == sp.Mul or expr.func == sp.Pow:
+ return handleMul(expr)
+ elif expr.func == sp.Add:
+ return expr.func(*[handleMul(a) for a in expr.args])
+ else:
+ return expr
+
+
+class Diff(sp.Expr):
+ """
+ Sympy Node representing a derivative. The difference to sympy's built in differential is:
+ - shortened latex representation
+ - all simplifications have to be done manually
+ - each Diff has a Chapman Enskog expansion order index: 'ceIdx'
+ """
+ is_number = False
+ is_Rational = False
+
+ def __new__(cls, argument, label=-1, ceIdx=-1, **kwargs):
+ if argument == 0:
+ return sp.Rational(0, 1)
+ return sp.Expr.__new__(cls, argument.expand(), sp.sympify(label), sp.sympify(ceIdx), **kwargs)
+
+ @property
+ def is_commutative(self):
+ anyNonCommutative = any(not s.is_commutative for s in self.atoms(sp.Symbol))
+ if anyNonCommutative:
+ return False
+ else:
+ return True
+
+ def getArgRecursive(self):
+ """Returns the argument the derivative acts on, for nested derivatives the inner argument is returned"""
+ if not isinstance(self.arg, Diff):
+ return self.arg
+ else:
+ return self.arg.getArgRecursive()
+
+ def changeArgRecursive(self, newArg):
+ """Returns a Diff node with the given 'newArg' instead of the current argument. For nested derivatives
+ a new nested derivative is returned where the inner Diff has the 'newArg'"""
+ if not isinstance(self.arg, Diff):
+ return Diff(newArg, self.label, self.ceIdx)
+ else:
+ return Diff(self.arg.changeArgRecursive(newArg), self.label, self.ceIdx)
+
+ def splitLinear(self, functions):
+ """
+ Applies linearity property of Diff: i.e. 'Diff(c*a+b)' is transformed to 'c * Diff(a) + Diff(b)'
+ The parameter functions is a list of all symbols that are considered functions, not constants.
+ For the example above: functions=[a, b]
+ """
+ constant, variable = 1, 1
+
+ if self.arg.func != sp.Mul:
+ constant, variable = 1, self.arg
+ else:
+ for factor in normalizeProduct(self.arg):
+ if factor in functions or isinstance(factor, Diff):
+ variable *= factor
+ else:
+ constant *= factor
+
+ if isinstance(variable, sp.Symbol) and variable not in functions:
+ return 0
+
+ if isinstance(variable, int) or variable.is_number:
+ return 0
+ else:
+ return constant * Diff(variable, label=self.label, ceIdx=self.ceIdx)
+
+ @property
+ def arg(self):
+ """Expression the derivative acts on"""
+ return self.args[0]
+
+ @property
+ def label(self):
+ """Subscript, usually the variable the Diff is w.r.t. """
+ return self.args[1]
+
+ @property
+ def ceIdx(self):
+ """Superscript, used as the Chapman Enskog order index"""
+ return self.args[2]
+
+ def _latex(self, printer, *args):
+ result = "{\partial"
+ if self.ceIdx >= 0:
+ result += "^{(%s)}" % (self.ceIdx,)
+ if self.label != -1:
+ result += "_{%s}" % (printer.doprint(self.label),)
+
+ contents = printer.doprint(self.arg)
+ if isinstance(self.arg, int) or isinstance(self.arg, sp.Symbol) or self.arg.is_number or self.arg.func == Diff:
+ result += " " + contents
+ else:
+ result += " (" + contents + ") "
+
+ result += "}"
+ return result
+
+ def __str__(self):
+ return "D(%s)" % self.arg
+
+
+# ----------------------------------------------------------------------------------------------------------------------
+
+def derivativeTerms(expr):
+ """
+ Returns set of all derivatives in an expression
+ this is different from `expr.atoms(Diff)` when nested derivatives are in the expression,
+ since this function only returns the outer derivatives
+ """
+ result = set()
+
+ def visit(e):
+ if isinstance(e, Diff):
+ result.add(e)
+ else:
+ for a in e.args:
+ visit(a)
+ visit(expr)
+ return result
+
+
+def collectDerivatives(expr):
+ """Rewrites expression into a sum of distinct derivatives with prefactors"""
+ return expr.collect(derivativeTerms(expr))
+
+
+def createNestedDiff(*args, arg=None):
+ """Shortcut to create nested derivatives"""
+ assert arg is not None
+ args = sorted(args, reverse=True)
+ res = arg
+ for i in args:
+ res = Diff(res, i)
+ return res
+
+
+def expandUsingLinearity(expr, functions=None, constants=None):
+ """
+ Expands all derivative nodes by applying Diff.splitLinear
+ :param expr: expression containing derivatives
+ :param functions: sequence of symbols that are considered functions and can not be pulled before the derivative.
+ if None, all symbols are viewed as functions
+ :param constants: sequence of symbols which are considered constants and can be pulled before the derivative
+ """
+ if functions is None:
+ functions = expr.atoms(sp.Symbol)
+ if constants is not None:
+ functions.difference_update(constants)
+
+ if isinstance(expr, Diff):
+ arg = expandUsingLinearity(expr.arg, functions)
+ if hasattr(arg, 'func') and arg.func == sp.Add:
+ result = 0
+ for a in arg.args:
+ result += Diff(a, label=expr.label, ceIdx=expr.ceIdx).splitLinear(functions)
+ return result
+ else:
+ diff = Diff(arg, label=expr.label, ceIdx=expr.ceIdx)
+ if diff == 0:
+ return 0
+ else:
+ return diff.splitLinear(functions)
+ else:
+ newArgs = [expandUsingLinearity(e, functions) for e in expr.args]
+ result = sp.expand(expr.func(*newArgs) if newArgs else expr)
+ return result
+
+
+def fullDiffExpand(expr, functions=None, constants=None):
+ if functions is None:
+ functions = expr.atoms(sp.Symbol)
+ if constants is not None:
+ functions.difference_update(constants)
+
+ def visit(e):
+ e = e.expand()
+
+ if e.func == Diff:
+ result = 0
+ diffArgs = {'label': e.label, 'ceIdx': e.ceIdx}
+ diffInner = e.args[0]
+ diffInner = visit(diffInner)
+ for term in diffInner.args if diffInner.func == sp.Add else [diffInner]:
+ independentTerms = 1
+ dependentTerms = []
+ for factor in normalizeProduct(term):
+ if factor in functions or isinstance(factor, Diff):
+ dependentTerms.append(factor)
+ else:
+ independentTerms *= factor
+ for i in range(len(dependentTerms)):
+ dependentTerm = dependentTerms[i]
+ otherDependentTerms = dependentTerms[:i] + dependentTerms[i+1:]
+ processedDiff = normalizeDiffOrder(Diff(dependentTerm, **diffArgs))
+ result += independentTerms * prod(otherDependentTerms) * processedDiff
+ return result
+ else:
+ newArgs = [visit(arg) for arg in e.args]
+ return e.func(*newArgs) if newArgs else e
+
+ if isinstance(expr, sp.Matrix):
+ return expr.applyfunc(visit)
+ else:
+ return visit(expr)
+
+
+def normalizeDiffOrder(expression, functions=None, constants=None, sortKey=defaultDiffSortKey):
+ """Assumes order of differentiation can be exchanged. Changes the order of nested Diffs to a standard order defined
+ by the sorting key 'sortKey' such that the derivative terms can be further simplified """
+ def visit(expr):
+ if isinstance(expr, Diff):
+ nodes = [expr]
+ while isinstance(nodes[-1].arg, Diff):
+ nodes.append(nodes[-1].arg)
+
+ processedArg = visit(nodes[-1].arg)
+ nodes.sort(key=sortKey)
+
+ result = processedArg
+ for d in reversed(nodes):
+ result = Diff(result, label=d.label, ceIdx=d.ceIdx)
+ return result
+ else:
+ newArgs = [visit(e) for e in expr.args]
+ return expr.func(*newArgs) if newArgs else expr
+
+ expression = expandUsingLinearity(expression.expand(), functions, constants).expand()
+ return visit(expression)
+
+
+def expandUsingProductRule(expr):
+ """Fully expands all derivatives by applying product rule"""
+ if isinstance(expr, Diff):
+ arg = expandUsingProductRule(expr.args[0])
+ if arg.func == sp.Add:
+ newArgs = [Diff(e, label=expr.label, ceIdx=expr.ceIdx)
+ for e in arg.args]
+ return sp.Add(*newArgs)
+ if arg.func not in (sp.Mul, sp.Pow):
+ return Diff(arg, label=expr.label, ceIdx=expr.ceIdx)
+ else:
+ prodList = normalizeProduct(arg)
+ result = 0
+ for i in range(len(prodList)):
+ preFactor = prod(prodList[j] for j in range(len(prodList)) if i != j)
+ result += preFactor * Diff(prodList[i], label=expr.label, ceIdx=expr.ceIdx)
+ return result
+ else:
+ newArgs = [expandUsingProductRule(e) for e in expr.args]
+ return expr.func(*newArgs) if newArgs else expr
+
+
+def combineUsingProductRule(expr):
+ """Inverse product rule"""
+
+ def exprToDiffDecomposition(expr):
+ """Decomposes a sp.Add node containing CeDiffs into:
+ diffDict: maps (label, ceIdx) -> [ (preFactor, argument), ... ]
+ i.e. a partial(b) ( a is prefactor, b is argument)
+ in case of partial(a) partial(b) two entries are created (0.5 partial(a), b), (0.5 partial(b), a)
+ """
+ DiffInfo = namedtuple("DiffInfo", ["label", "ceIdx"])
+
+ class DiffSplit:
+ def __init__(self, preFactor, argument):
+ self.preFactor = preFactor
+ self.argument = argument
+
+ def __repr__(self):
+ return str((self.preFactor, self.argument))
+
+ assert isinstance(expr, sp.Add)
+ diffDict = defaultdict(list)
+ rest = 0
+ for term in expr.args:
+ if isinstance(term, Diff):
+ diffDict[DiffInfo(term.label, term.ceIdx)].append(DiffSplit(1, term.arg))
+ else:
+ mulArgs = normalizeProduct(term)
+ diffs = [d for d in mulArgs if isinstance(d, Diff)]
+ factor = prod(d for d in mulArgs if not isinstance(d, Diff))
+ if len(diffs) == 0:
+ rest += factor
+ else:
+ for i, diff in enumerate(diffs):
+ allButCurrent = [d for j, d in enumerate(diffs) if i != j]
+ preFactor = factor * prod(allButCurrent) * sp.Rational(1, len(diffs))
+ diffDict[DiffInfo(diff.label, diff.ceIdx)].append(DiffSplit(preFactor, diff.arg))
+
+ return diffDict, rest
+
+ def matchDiffSplits(own, other):
+ ownFac = own.preFactor / other.argument
+ otherFac = other.preFactor / own.argument
+
+ if sp.count_ops(ownFac) > sp.count_ops(own.preFactor) or sp.count_ops(otherFac) > sp.count_ops(other.preFactor):
+ return None
+
+ newOtherFactor = ownFac - otherFac
+ return newOtherFactor
+
+ def processDiffList(diffList, label, ceIdx):
+ if len(diffList) == 0:
+ return 0
+ elif len(diffList) == 1:
+ return diffList[0].preFactor * Diff(diffList[0].argument, label, ceIdx)
+
+ result = 0
+ matches = []
+ for i in range(1, len(diffList)):
+ matchResult = matchDiffSplits(diffList[i], diffList[0])
+ if matchResult is not None:
+ matches.append((i, matchResult))
+
+ if len(matches) == 0:
+ result += diffList[0].preFactor * Diff(diffList[0].argument, label, ceIdx)
+ else:
+ otherIdx, matchResult = sorted(matches, key=lambda e: sp.count_ops(e[1]))[0]
+ newArgument = diffList[0].argument * diffList[otherIdx].argument
+ result += (diffList[0].preFactor / diffList[otherIdx].argument) * Diff(newArgument, label, ceIdx)
+ if matchResult == 0:
+ del diffList[otherIdx]
+ else:
+ diffList[otherIdx].preFactor = matchResult * diffList[0].argument
+ result += processDiffList(diffList[1:], label, ceIdx)
+ return result
+
+ expr = expr.expand()
+ if isinstance(expr, sp.Add):
+ diffDict, rest = exprToDiffDecomposition(expr)
+ for (label, ceIdx), diffList in diffDict.items():
+ rest += processDiffList(diffList, label, ceIdx)
+ return rest
+ else:
+ newArgs = [combineUsingProductRule(e) for e in expr.args]
+ return expr.func(*newArgs) if newArgs else expr
+
+
+def replaceDiff(expr, replacementDict):
+ """replacementDict: maps variable (label) to a new Differential operator"""
+
+ def visit(e):
+ if isinstance(e, Diff):
+ if e.label in replacementDict:
+ return DiffOperator.apply(replacementDict[e.label], visit(e.arg))
+ newArgs = [visit(arg) for arg in e.args]
+ return e.func(*newArgs) if newArgs else e
+
+ return visit(expr)
+
+
+def zeroDiffs(expr, label):
+ """Replaces all differentials with the given label by 0"""
+ def visit(e):
+ if isinstance(e, Diff):
+ if e.label == label:
+ return 0
+ newArgs = [visit(arg) for arg in e.args]
+ return e.func(*newArgs) if newArgs else e
+ return visit(expr)
+
+
+def evaluateDiffs(expr, var=None):
+ """Replaces Diff nodes by sp.diff , the free variable is either the label (if var=None) otherwise
+ the specified var"""
+ if isinstance(expr, Diff):
+ if var is None:
+ var = expr.label
+ return sp.diff(evaluateDiffs(expr.arg, var), var)
+ else:
+ newArgs = [evaluateDiffs(arg, var) for arg in expr.args]
+ return expr.func(*newArgs) if newArgs else expr
+
+
+def functionalDerivative(functional, v, constants=None):
+ """
+ Computes functional derivative of functional with respect to v using Euler-Lagrange equation
+
+ .. math ::
+
+ \frac{\delta F}{\delta v} =
+ \frac{\partial F}{\partial v} - \nabla \cdot \frac{\partial F}{\partial \nabla v}
+
+ - assumes that gradients are represented by Diff() node (from Chapman Enskog module)
+ - Diff(Diff(r)) represents the divergence of r
+ - the constants parameter is a list with symbols not affected by the derivative. This is used for simplification
+ of the derivative terms.
+ """
+ diffs = functional.atoms(Diff)
+ nonDiffPart = functional.subs({d: sp.Dummy() for d in diffs})
+
+ partialF_partialV = sp.diff(nonDiffPart, v)
+
+ gradientPart = 0
+ for diffObj in diffs:
+ if diffObj.args[0] != v:
+ continue
+ dummy = sp.Dummy()
+ partialF_partialGradV = functional.subs(diffObj, dummy).diff(dummy).subs(dummy, diffObj)
+ gradientPart += Diff(partialF_partialGradV, label=diffObj.label, ceIdx=diffObj.ceIdx)
+
+ result = partialF_partialV - gradientPart
+ return expandUsingLinearity(result, constants=constants)
diff --git a/finitedifferences.py b/finitedifferences.py
index 9f1152d15..9dd141666 100644
--- a/finitedifferences.py
+++ b/finitedifferences.py
@@ -1,7 +1,10 @@
import numpy as np
import sympy as sp
+
+from pystencils.equationcollection import EquationCollection
from pystencils.field import Field
from pystencils.transformations import fastSubs
+from pystencils.derivative import Diff
def grad(var, dim=3):
@@ -185,10 +188,14 @@ class Advection(sp.Function):
def advection(advectedScalar, velocityField, idx=None):
"""Advection term: divergence( velocityField * advectedScalar )"""
+ if isinstance(advectedScalar, Field):
+ firstArg = advectedScalar.center
+ elif isinstance(advectedScalar, Field.Access):
+ firstArg = advectedScalar
+ else:
+ raise ValueError("Advected scalar has to be a pystencils Field or Field.Access")
- assert isinstance(advectedScalar, Field), "Advected scalar has to be a pystencils.Field"
-
- args = [advectedScalar.center, velocityField if not isinstance(velocityField, Field) else velocityField.center]
+ args = [firstArg, velocityField if not isinstance(velocityField, Field) else velocityField.center]
if idx is not None:
args.append(idx)
return Advection(*args)
@@ -236,8 +243,14 @@ class Diffusion(sp.Function):
def diffusion(scalar, diffusionCoeff, idx=None):
- assert isinstance(scalar, Field), "Advected scalar has to be a pystencils.Field"
- args = [scalar.center, diffusionCoeff if not isinstance(diffusionCoeff, Field) else diffusionCoeff.center]
+ if isinstance(scalar, Field):
+ firstArg = scalar.center
+ elif isinstance(scalar, Field.Access):
+ firstArg = scalar
+ else:
+ raise ValueError("Diffused scalar has to be a pystencils Field or Field.Access")
+
+ args = [firstArg, diffusionCoeff if not isinstance(diffusionCoeff, Field) else diffusionCoeff.center]
if idx is not None:
args.append(idx)
return Diffusion(*args)
@@ -261,7 +274,12 @@ class Transient(sp.Function):
def transient(scalar, idx=None):
- args = [scalar.center]
+ if isinstance(scalar, Field):
+ args = [scalar.center]
+ elif isinstance(scalar, Field.Access):
+ args = [scalar]
+ else:
+ raise ValueError("Scalar has to be a pystencils Field or Field.Access")
if idx is not None:
args.append(idx)
return Transient(*args)
@@ -272,6 +290,13 @@ class Discretization2ndOrder:
self.dx = dx
self.dt = dt
+ @staticmethod
+ def __diffOrder(e):
+ if not isinstance(e, Diff):
+ return 0
+ else:
+ return 1 + Discretization2ndOrder.__diffOrder(e.args[0])
+
def _discretize_diffusion(self, expr):
result = 0
for c in range(expr.dim):
@@ -296,11 +321,44 @@ class Discretization2ndOrder:
return self._discretize_diffusion(e)
elif isinstance(e, Advection):
return self._discretize_advection(e)
+ elif isinstance(e, Diff):
+ return self._discretize_diff(e)
else:
newArgs = [self._discretizeSpatial(a) for a in e.args]
return e.func(*newArgs) if newArgs else e
+ def _discretize_diff(self, e):
+ order = self.__diffOrder(e)
+ if order == 1:
+ fa = e.args[0]
+ index = e.label
+ return (fa.neighbor(index, 1) - fa.neighbor(index, -1)) / (2 * self.dx)
+ elif order == 2:
+ indices = sorted([e.label, e.args[0].label])
+ fa = e.args[0].args[0]
+ if indices[0] == indices[1] and all(i >= 0 for i in indices):
+ result = (-2 * fa + fa.neighbor(indices[0], -1) + fa.neighbor(indices[0], +1))
+ elif indices[0] == indices[1]:
+ result = 0
+ for d in range(fa.field.spatialDimensions):
+ result += (-2 * fa + fa.neighbor(d, -1) + fa.neighbor(d, +1))
+ else:
+ assert all(i >= 0 for i in indices)
+ offsets = [(1, 1), [-1, 1], [1, -1], [-1, -1]]
+ result = sum(o1*o2 * fa.neighbor(indices[0], o1).neighbor(indices[1], o2) for o1, o2 in offsets) / 4
+ return result / (self.dx**2)
+ else:
+ raise NotImplementedError("Term contains derivatives of order > 2")
+
def __call__(self, expr):
+ if isinstance(expr, list):
+ return [self(e) for e in expr]
+ elif isinstance(expr, sp.Matrix):
+ return expr.applyfunc(self.__call__)
+ elif isinstance(expr, EquationCollection):
+ return expr.copy(mainEquations=[e for e in expr.mainEquations],
+ subexpressions=[e for e in expr.subexpressions])
+
transientTerms = expr.atoms(Transient)
if len(transientTerms) == 0:
return self._discretizeSpatial(expr)
--
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