def addKernels(generator, aderdg):
numberOf3DBasisFunctions = aderdg.numberOf3DBasisFunctions()
numberOfQuantities = aderdg.numberOfQuantities()
## Point sources
mStiffnessTensor = Tensor('stiffnessTensor', (3, 3, 3, 3))
mSlip = Tensor('mSlip', (3, ))
mNormal = Tensor('mNormal', (3, ))
mArea = Scalar('mArea')
basisFunctionsAtPoint = Tensor('basisFunctionsAtPoint',
(numberOf3DBasisFunctions, ))
mInvJInvPhisAtSources = Tensor('mInvJInvPhisAtSources',
(numberOf3DBasisFunctions, ))
JInv = Scalar('JInv')
generator.add(
'computeMInvJInvPhisAtSources', mInvJInvPhisAtSources['k'] <=
JInv * aderdg.db.M3inv['kl'] * basisFunctionsAtPoint['l'])
#extract the moment tensors entries in SeisSol ordering (xx, yy, zz, xy, yz, xz)
assert (numberOfQuantities >= 6)
momentToNRF_spp = np.zeros((numberOfQuantities, 3, 3))
momentToNRF_spp[0, 0, 0] = 1
momentToNRF_spp[1, 1, 1] = 1
momentToNRF_spp[2, 2, 2] = 1
momentToNRF_spp[3, 0, 1] = 1
momentToNRF_spp[4, 1, 2] = 1
momentToNRF_spp[5, 0, 2] = 1
momentToNRF = Tensor('momentToNRF', (numberOfQuantities, 3, 3),
spp=momentToNRF_spp)
momentNRFKernel = momentToNRF['tpq'] * mArea * mStiffnessTensor[
'pqij'] * mSlip['i'] * mNormal['j']
if aderdg.Q.hasOptDim():
sourceNRF = aderdg.Q['kt'] <= aderdg.Q['kt'] + mInvJInvPhisAtSources[
'k'] * momentNRFKernel * aderdg.oneSimToMultSim['s']
else:
sourceNRF = aderdg.Q['kt'] <= aderdg.Q[
'kt'] + mInvJInvPhisAtSources['k'] * momentNRFKernel
generator.add('sourceNRF', sourceNRF)
momentFSRM = Tensor('momentFSRM', (numberOfQuantities, ))
stfIntegral = Scalar('stfIntegral')
if aderdg.Q.hasOptDim():
sourceFSRM = aderdg.Q[
'kp'] <= aderdg.Q['kp'] + stfIntegral * mInvJInvPhisAtSources[
'k'] * momentFSRM['p'] * aderdg.oneSimToMultSim['s']
else:
sourceFSRM = aderdg.Q['kp'] <= aderdg.Q[
'kp'] + stfIntegral * mInvJInvPhisAtSources['k'] * momentFSRM['p']
generator.add('sourceFSRM', sourceFSRM)
## Receiver output
QAtPoint = OptionalDimTensor('QAtPoint', aderdg.Q.optName(),
aderdg.Q.optSize(), aderdg.Q.optPos(),
(numberOfQuantities, ))
evaluateDOFSAtPoint = QAtPoint[
'p'] <= aderdg.Q['kp'] * basisFunctionsAtPoint['k']
generator.add('evaluateDOFSAtPoint', evaluateDOFSAtPoint)
def addInit(self, generator):
fluxScale = Scalar('fluxScale')
computeFluxSolverLocal = self.AplusT['ij'] <= fluxScale * self.Tinv[
'ki'] * self.QgodLocal['kq'] * self.starMatrix(
0)['ql'] * self.T['jl']
generator.add('computeFluxSolverLocal', computeFluxSolverLocal)
computeFluxSolverNeighbor = self.AminusT[
'ij'] <= fluxScale * self.Tinv['ki'] * self.QgodNeighbor[
'kq'] * self.starMatrix(0)['ql'] * self.T['jl']
generator.add('computeFluxSolverNeighbor', computeFluxSolverNeighbor)
QFortran = Tensor(
'QFortran',
(self.numberOf3DBasisFunctions(), self.numberOfQuantities()))
multSimToFirstSim = Tensor('multSimToFirstSim', (self.Q.optSize(), ),
spp={(0, ): '1.0'})
if self.Q.hasOptDim():
copyQToQFortran = QFortran[
'kp'] <= self.Q['kp'] * multSimToFirstSim['s']
else:
copyQToQFortran = QFortran['kp'] <= self.Q['kp']
generator.add('copyQToQFortran', copyQToQFortran)
stiffnessTensor = Tensor('stiffnessTensor', (3, 3, 3, 3))
direction = Tensor('direction', (3, ))
christoffel = Tensor('christoffel', (3, 3))
computeChristoffel = christoffel[
'ik'] <= stiffnessTensor['ijkl'] * direction['j'] * direction['l']
generator.add('computeChristoffel', computeChristoffel)
def addTime(self, generator):
qShape = (self.numberOf3DBasisFunctions(), self.numberOfQuantities())
dQ0 = OptionalDimTensor('dQ(0)',
self.Q.optName(),
self.Q.optSize(),
self.Q.optPos(),
qShape,
alignStride=True)
power = Scalar('power')
derivatives = [dQ0]
generator.add('derivativeTaylorExpansion(0)',
self.I['kp'] <= power * dQ0['kp'])
for i in range(1, self.order):
derivativeSum = Add()
if self.sourceMatrix():
derivativeSum += derivatives[-1]['kq'] * self.sourceMatrix(
)['qp']
for j in range(3):
derivativeSum += self.db.kDivMT[j][self.t(
'kl')] * derivatives[-1]['lq'] * self.starMatrix(j)['qp']
derivativeSum = DeduceIndices(
self.Q['kp'].indices).visit(derivativeSum)
derivativeSum = EquivalentSparsityPattern().visit(derivativeSum)
dQ = OptionalDimTensor('dQ({})'.format(i),
self.Q.optName(),
self.Q.optSize(),
self.Q.optPos(),
qShape,
spp=derivativeSum.eqspp(),
alignStride=True)
generator.add('derivative({})'.format(i),
dQ['kp'] <= derivativeSum)
generator.add('derivativeTaylorExpansion({})'.format(i),
self.I['kp'] <= self.I['kp'] + power * dQ['kp'])
derivatives.append(dQ)
def addTime(self, generator):
qShape = (self.numberOf3DBasisFunctions(), self.numberOfQuantities())
dQ = [OptionalDimTensor('dQ({})'.format(d), self.Q.optName(), self.Q.optSize(), self.Q.optPos(), qShape, alignStride=True) for d in range(self.order)]
dQext = [OptionalDimTensor('dQext({})'.format(d), self.Q.optName(), self.Q.optSize(), self.Q.optPos(), self._qShapeExtended, alignStride=True) for d in range(self.order)]
dQane = [OptionalDimTensor('dQane({})'.format(d), self.Q.optName(), self.Q.optSize(), self.Q.optPos(), self._qShapeAnelastic, alignStride=True) for d in range(self.order)]
power = Scalar('power')
derivativeTaylorExpansionEla = lambda d: (self.I['kp'] <= self.I['kp'] + power * dQ[d]['kp']) if d > 0 else (self.I['kp'] <= power * dQ[0]['kp'])
derivativeTaylorExpansionAne = lambda d: (self.Iane['kpm'] <= self.Iane['kpm'] + power * dQane[d]['kpm']) if d > 0 else (self.Iane['kpm'] <= power * dQane[0]['kpm'])
def derivative(kthDer):
derivativeSum = Add()
for j in range(3):
derivativeSum += self.db.kDivMT[j][self.t('kl')] * dQ[kthDer-1]['lq'] * self.db.star[j]['qp']
return derivativeSum
generator.addFamily('derivative', parameterSpaceFromRanges(range(1,self.order)), lambda d: [
dQext[d]['kp'] <= derivative(d),
dQ[d]['kp'] <= dQext[d]['kq'] * self.selectEla['qp'] + dQane[d-1]['kqm'] * self.E['qmp'],
dQane[d]['kpm'] <= self.w['m'] * dQext[d]['kq'] * self.selectAne['qp'] + dQane[d-1]['kpl'] * self.W['lm']
])
generator.addFamily('derivativeTaylorExpansion', simpleParameterSpace(self.order), lambda d: [
derivativeTaylorExpansionEla(d),
derivativeTaylorExpansionAne(d)
])
generator.addFamily('derivativeTaylorExpansionEla', simpleParameterSpace(self.order), derivativeTaylorExpansionEla)
def addKernels(generator, aderdg):
numberOf3DBasisFunctions = aderdg.numberOf3DBasisFunctions()
numberOfQuantities = aderdg.numberOfQuantities()
## Point sources
mInvJInvPhisAtSources = Tensor('mInvJInvPhisAtSources', (numberOf3DBasisFunctions,))
momentNRF = Tensor('momentNRF', (numberOfQuantities,), spp=np.array([1]*6 + [0]*(numberOfQuantities-6), dtype=bool))
if aderdg.Q.hasOptDim():
sourceNRF = aderdg.Q['kp'] <= aderdg.Q['kp'] - mInvJInvPhisAtSources['k'] * momentNRF['p'] * aderdg.oneSimToMultSim['s']
else:
sourceNRF = aderdg.Q['kp'] <= aderdg.Q['kp'] - mInvJInvPhisAtSources['k'] * momentNRF['p']
generator.add('sourceNRF', sourceNRF)
momentFSRM = Tensor('momentFSRM', (numberOfQuantities,))
stfIntegral = Scalar('stfIntegral')
if aderdg.Q.hasOptDim():
sourceFSRM = aderdg.Q['kp'] <= aderdg.Q['kp'] + stfIntegral * mInvJInvPhisAtSources['k'] * momentFSRM['p'] * aderdg.oneSimToMultSim['s']
else:
sourceFSRM = aderdg.Q['kp'] <= aderdg.Q['kp'] + stfIntegral * mInvJInvPhisAtSources['k'] * momentFSRM['p']
generator.add('sourceFSRM', sourceFSRM)
## Receiver output
basisFunctionsAtPoint = Tensor('basisFunctions', (numberOf3DBasisFunctions,))
QAtPoint = OptionalDimTensor('QAtPoint', aderdg.Q.optName(), aderdg.Q.optSize(), aderdg.Q.optPos(), (numberOfQuantities,))
evaluateDOFSAtPoint = QAtPoint['p'] <= aderdg.Q['kp'] * basisFunctionsAtPoint['k']
generator.add('evaluateDOFSAtPoint', evaluateDOFSAtPoint)
def addKernels(generator, aderdg, include_tensors, matricesDir, dynamicRuptureMethod):
easi_ident_map = np.stack([np.eye(aderdg.numberOfQuantities())] * aderdg.numberOf2DBasisFunctions(), axis=2)
assert(easi_ident_map.shape ==
(aderdg.numberOfQuantities(), aderdg.numberOfQuantities(), aderdg.numberOf2DBasisFunctions()))
easi_ident_map = Tensor('easiIdentMap',
easi_ident_map.shape,
easi_ident_map,
alignStride=False)
easi_boundary_constant = Tensor('easiBoundaryConstant',
(aderdg.numberOfQuantities(), aderdg.numberOf2DBasisFunctions()),
alignStride=False)
easi_boundary_map = Tensor('easiBoundaryMap',
(aderdg.numberOfQuantities(), aderdg.numberOfQuantities(), aderdg.numberOf2DBasisFunctions(),),
alignStride=False)
create_easi_boundary_ghost_cells = (
aderdg.INodal['la'] <= easi_boundary_map['abl'] * aderdg.INodal['lb'] + easi_ident_map['abl'] * easi_boundary_constant['bl']
)
generator.add('createEasiBoundaryGhostCells', create_easi_boundary_ghost_cells)
projectToNodalBoundary = lambda j: aderdg.INodal['kp'] <= aderdg.db.V3mTo2nFace[j]['km'] * aderdg.I['mp']
generator.addFamily('projectToNodalBoundary',
simpleParameterSpace(4),
projectToNodalBoundary)
projectToNodalBoundaryRotated = lambda j: aderdg.INodal['kp'] <= aderdg.db.V3mTo2nFace[j]['kl'] \
* aderdg.I['lm'] \
* aderdg.Tinv['pm']
generator.addFamily('projectToNodalBoundaryRotated',
simpleParameterSpace(4),
projectToNodalBoundaryRotated)
# To be used as Tinv in flux solver - this way we can save two rotations for
# Dirichlet boundary, as ghost cell dofs are already rotated
identity_rotation = np.double(aderdg.transformation_spp())
identity_rotation[0:9, 0:9] = np.eye(9)
identity_rotation = Tensor('identityT',
aderdg.transformation_spp().shape,
identity_rotation,
)
include_tensors.add(identity_rotation)
aderdg.INodalUpdate = OptionalDimTensor('INodalUpdate',
aderdg.INodal.optName(),
aderdg.INodal.optSize(),
aderdg.INodal.optPos(),
(aderdg.numberOf2DBasisFunctions(), aderdg.numberOfQuantities()),
alignStride=True)
factor = Scalar('factor')
updateINodal = aderdg.INodal['kp'] <= aderdg.INodal['kp'] + factor * aderdg.INodalUpdate['kp']
generator.add('updateINodal', updateINodal)
def addKernels(generator, aderdg, matricesDir, dynamicRuptureMethod):
if dynamicRuptureMethod == 'quadrature':
numberOfPoints = (aderdg.order+1)**2
elif dynamicRuptureMethod == 'cellaverage':
numberOfPoints = int(4**math.ceil(math.log(aderdg.order*(aderdg.order+1)/2,4)))
else:
raise ValueError('Unknown dynamic rupture method.')
clones = dict()
# Load matrices
db = parseJSONMatrixFile('{}/dr_{}_matrices_{}.json'.format(matricesDir, dynamicRuptureMethod, aderdg.order), clones, alignStride=aderdg.alignStride, transpose=aderdg.transpose)
# Determine matrices
# Note: This does only work because the flux does not depend on the mechanisms in the case of viscoelastic attenuation
godShape = (aderdg.numberOfQuantities(), aderdg.numberOfQuantities())
godunovMatrix = Tensor('godunovMatrix', godShape)
fluxSolverShape = (aderdg.numberOfQuantities(), aderdg.numberOfExtendedQuantities())
fluxSolver = Tensor('fluxSolver', fluxSolverShape)
gShape = (numberOfPoints, aderdg.numberOfQuantities())
godunovState = OptionalDimTensor('godunovState', aderdg.Q.optName(), aderdg.Q.optSize(), aderdg.Q.optPos(), gShape, alignStride=True)
generator.add('rotateGodunovStateLocal', godunovMatrix['qp'] <= aderdg.Tinv['kq'] * aderdg.QgodLocal['kp'])
generator.add('rotateGodunovStateNeighbor', godunovMatrix['qp'] <= aderdg.Tinv['kq'] * aderdg.QgodNeighbor['kp'])
fluxScale = Scalar('fluxScale')
generator.add('rotateFluxMatrix', fluxSolver['qp'] <= fluxScale * aderdg.starMatrix(0)['qk'] * aderdg.T['pk'])
def godunovStateGenerator(i,h):
target = godunovState['kp']
term = db.V3mTo2n[i,h][aderdg.t('kl')] * aderdg.Q['lq'] * godunovMatrix['qp']
if h == 0:
return target <= term
return target <= target + term
godunovStatePrefetch = lambda i,h: godunovState
generator.addFamily('godunovState', simpleParameterSpace(4,4), godunovStateGenerator, godunovStatePrefetch)
nodalFluxGenerator = lambda i,h: aderdg.extendedQTensor()['kp'] <= aderdg.extendedQTensor()['kp'] + db.V3mTo2nTWDivM[i,h][aderdg.t('kl')] * godunovState['lq'] * fluxSolver['qp']
nodalFluxPrefetch = lambda i,h: aderdg.I
generator.addFamily('nodalFlux', simpleParameterSpace(4,4), nodalFluxGenerator, nodalFluxPrefetch)
def addKernels(generator, aderdg, matricesDir, dynamicRuptureMethod):
if dynamicRuptureMethod == 'quadrature':
numberOfPoints = (aderdg.order + 1)**2
elif dynamicRuptureMethod == 'cellaverage':
numberOfPoints = int(4**math.ceil(
math.log(aderdg.order * (aderdg.order + 1) / 2, 4)))
else:
raise ValueError('Unknown dynamic rupture method.')
clones = dict()
# Load matrices
db = parseJSONMatrixFile('{}/dr_{}_matrices_{}.json'.format(
matricesDir, dynamicRuptureMethod, aderdg.order),
clones,
alignStride=aderdg.alignStride,
transpose=aderdg.transpose)
db.update(
parseJSONMatrixFile('{}/resample_{}.json'.format(
matricesDir, aderdg.order)))
# Determine matrices
# Note: This does only work because the flux does not depend on the mechanisms in the case of viscoelastic attenuation
trans_inv_spp_T = aderdg.transformation_inv_spp().transpose()
TinvT = Tensor('TinvT', trans_inv_spp_T.shape, spp=trans_inv_spp_T)
flux_solver_spp = aderdg.flux_solver_spp()
fluxSolver = Tensor('fluxSolver',
flux_solver_spp.shape,
spp=flux_solver_spp)
gShape = (numberOfPoints, aderdg.numberOfQuantities())
QInterpolated = OptionalDimTensor('QInterpolated',
aderdg.Q.optName(),
aderdg.Q.optSize(),
aderdg.Q.optPos(),
gShape,
alignStride=True)
generator.add('transposeTinv', TinvT['ij'] <= aderdg.Tinv['ji'])
fluxScale = Scalar('fluxScale')
generator.add(
'rotateFluxMatrix', fluxSolver['qp'] <=
fluxScale * aderdg.starMatrix(0)['qk'] * aderdg.T['pk'])
def interpolateQGenerator(i, h):
return QInterpolated['kp'] <= db.V3mTo2n[i, h][aderdg.t(
'kl')] * aderdg.Q['lq'] * TinvT['qp']
interpolateQPrefetch = lambda i, h: QInterpolated
generator.addFamily('evaluateAndRotateQAtInterpolationPoints',
simpleParameterSpace(4, 4), interpolateQGenerator,
interpolateQPrefetch)
nodalFluxGenerator = lambda i, h: aderdg.extendedQTensor()[
'kp'] <= aderdg.extendedQTensor()['kp'] + db.V3mTo2nTWDivM[i, h][
aderdg.t('kl')] * QInterpolated['lq'] * fluxSolver['qp']
nodalFluxPrefetch = lambda i, h: aderdg.I
generator.addFamily('nodalFlux', simpleParameterSpace(4, 4),
nodalFluxGenerator, nodalFluxPrefetch)
return {db.resample}
def addKernels(generator, aderdg, include_tensors, matricesDir, dynamicRuptureMethod):
easi_ident_map = np.stack([np.eye(aderdg.numberOfQuantities())] * aderdg.numberOf2DBasisFunctions(), axis=2)
assert(easi_ident_map.shape ==
(aderdg.numberOfQuantities(), aderdg.numberOfQuantities(), aderdg.numberOf2DBasisFunctions()))
easi_ident_map = Tensor('easiIdentMap',
easi_ident_map.shape,
easi_ident_map,
alignStride=False)
easi_boundary_constant = Tensor('easiBoundaryConstant',
(aderdg.numberOfQuantities(), aderdg.numberOf2DBasisFunctions()),
alignStride=False)
easi_boundary_map = Tensor('easiBoundaryMap',
(aderdg.numberOfQuantities(), aderdg.numberOfQuantities(), aderdg.numberOf2DBasisFunctions(),),
alignStride=False)
create_easi_boundary_ghost_cells = (
aderdg.INodal['la'] <= easi_boundary_map['abl'] * aderdg.INodal['lb'] + easi_ident_map['abl'] * easi_boundary_constant['bl']
)
generator.add('createEasiBoundaryGhostCells', create_easi_boundary_ghost_cells)
projectToNodalBoundary = lambda j: aderdg.INodal['kp'] <= aderdg.db.V3mTo2nFace[j]['km'] * aderdg.I['mp']
generator.addFamily('projectToNodalBoundary',
simpleParameterSpace(4),
projectToNodalBoundary)
projectToNodalBoundaryRotated = lambda j: aderdg.INodal['kp'] <= aderdg.db.V3mTo2nFace[j]['kl'] \
* aderdg.I['lm'] \
* aderdg.Tinv['pm']
generator.addFamily('projectToNodalBoundaryRotated',
simpleParameterSpace(4),
projectToNodalBoundaryRotated)
# To be used as Tinv in flux solver - this way we can save two rotations for
# Dirichlet boundary, as ghost cell dofs are already rotated
identity_rotation = np.double(aderdg.transformation_spp())
identity_rotation[0:9, 0:9] = np.eye(9)
identity_rotation = Tensor('identityT',
aderdg.transformation_spp().shape,
identity_rotation,
)
include_tensors.add(identity_rotation)
project2nFaceTo3m = tensor_collection_from_constant_expression(
base_name='project2nFaceTo3m',
expressions=lambda i: aderdg.db.rDivM[i]['jk'] * aderdg.db.V2nTo2m['kl'],
group_indices=range(4),
target_indices='jl')
aderdg.db.update(project2nFaceTo3m)
selectZDisplacementFromQuantities = np.zeros(aderdg.numberOfQuantities())
selectZDisplacementFromQuantities[8] = 1
selectZDisplacementFromQuantities= Tensor('selectZDisplacementFromQuantities',
selectZDisplacementFromQuantities.shape,
selectZDisplacementFromQuantities,
)
selectZDisplacementFromDisplacements = np.zeros(3)
selectZDisplacementFromDisplacements[2] = 1
selectZDisplacementFromDisplacements = Tensor('selectZDisplacementFromDisplacements',
selectZDisplacementFromDisplacements.shape,
selectZDisplacementFromDisplacements,
)
aderdg.INodalDisplacement = OptionalDimTensor('INodalDisplacement',
aderdg.Q.optName(),
aderdg.Q.optSize(),
aderdg.Q.optPos(),
(aderdg.numberOf2DBasisFunctions(),),
alignStride=True)
displacement = OptionalDimTensor('displacement',
aderdg.Q.optName(),
aderdg.Q.optSize(),
aderdg.Q.optPos(),
(aderdg.numberOf3DBasisFunctions(), 3),
alignStride=True)
dt = Scalar('dt')
displacementAvgNodal = lambda side: aderdg.INodalDisplacement['i'] <= \
aderdg.db.V3mTo2nFace[side]['ij'] * aderdg.I['jk'] * selectZDisplacementFromQuantities['k'] \
+ dt * aderdg.db.V3mTo2nFace[side]['ij'] * displacement['jk'] * selectZDisplacementFromDisplacements['k']
generator.addFamily('displacementAvgNodal',
simpleParameterSpace(4),
displacementAvgNodal)
aderdg.INodalUpdate = OptionalDimTensor('INodalUpdate',
aderdg.INodal.optName(),
aderdg.INodal.optSize(),
aderdg.INodal.optPos(),
(aderdg.numberOf2DBasisFunctions(), aderdg.numberOfQuantities()),
alignStride=True)
factor = Scalar('factor')
updateINodal = aderdg.INodal['kp'] <= aderdg.INodal['kp'] + factor * aderdg.INodalUpdate['kp']
generator.add('updateINodal', updateINodal)
def addTime(self, generator, targets):
super().addTime(generator, targets)
stiffnessValues = [
self.db.kDivMT[d].values_as_ndarray() for d in range(3)
]
fullShape = (self.numberOf3DBasisFunctions(),
self.numberOf3DBasisFunctions())
qShape = (self.numberOf3DBasisFunctions(), self.numberOfQuantities())
stpShape = (self.numberOf3DBasisFunctions(), self.numberOfQuantities(),
self.order)
quantityShape = (self.numberOfQuantities(), self.numberOfQuantities())
spaceTimePredictorRhs = OptionalDimTensor('spaceTimePredictorRhs',
self.Q.optName(),
self.Q.optSize(),
self.Q.optPos(),
stpShape,
alignStride=True)
spaceTimePredictor = OptionalDimTensor('spaceTimePredictor',
self.Q.optName(),
self.Q.optSize(),
self.Q.optPos(),
stpShape,
alignStride=True)
testRhs = OptionalDimTensor('testRhs',
self.Q.optName(),
self.Q.optSize(),
self.Q.optPos(),
stpShape,
alignStride=True)
testLhs = OptionalDimTensor('testLhs',
self.Q.optName(),
self.Q.optSize(),
self.Q.optPos(),
stpShape,
alignStride=True)
timestep = Scalar('timestep')
G = {10: Scalar('Gk'), 11: Scalar('Gl'), 12: Scalar('Gm')}
## Compute the index range for basis functions of a certain degree
#
# The basis functions are ordered with increasing degree, i.e. the first
# basis function has degree 0, the next three basis functions have degree
# 1, the next six basis functions have degree 2 and so forth.
# This method computes the indices Bn_lower, Bn_upper, such that
# forall Bn_lower =< i < Bn_upper: degree(phi_i) == n
#
# @param n The desired polynomial degree
def modeRange(n):
Bn_lower = choose(n - 1 + 3, 3)
Bn_upper = choose(n + 3, 3)
return (Bn_lower, Bn_upper)
## Compute a matrix, which filters out all basis function of degree n
#
# Compute a matrix M, such that for any DOF vector Q, M*Q only contains
# the parts of Q, which correspond to basis functions of degree n
#
# @param n The desired polynomial degree
def selectModes(n):
Bn_1, Bn = modeRange(n)
selectModesSpp = np.zeros(fullShape)
selectModesSpp[Bn_1:Bn, Bn_1:Bn] = np.eye(Bn - Bn_1)
return Tensor('selectModes({})'.format(n),
fullShape,
spp=selectModesSpp)
## Compute a matrix, which slices out one quantity
#
# @param quantityNumber The number of the quantity, which is sliced out
def selectQuantity(quantityNumber):
selectSpp = np.zeros(
(self.numberOfQuantities(), self.numberOfQuantities()))
selectSpp[quantityNumber, quantityNumber] = 1
return Tensor('selectQuantity({})'.format(quantityNumber),
selectSpp.shape,
spp=selectSpp)
## Compute a matrix, which slides out one quantity from the upper triangular
# part of the source matrix
#
# Note: G = E - diag(E)
#
# @param quantityNumber The number of the quantity, which is sliced out
def selectQuantityG(quantityNumber):
selectSpp = np.zeros(
(self.numberOfQuantities(), self.numberOfQuantities()))
#The source matrix G only contains values at (o-4, o)
selectSpp[quantityNumber - 4, quantityNumber] = 1
return Tensor('selectQuantityG({})'.format(quantityNumber),
selectSpp.shape,
spp=selectSpp)
## Zinv(o) = $(Z - E^*_{oo} * I)^{-1}$
#
# @param o Index as described above
def Zinv(o):
return Tensor('Zinv({})'.format(o), (self.order, self.order))
## Submatrix of the stiffness matrix
#
# The stiffness matrix for derivatives in direction d, where only the
# contribution from basis functions of degree n is considered
#
# @param d The direction in which derivatives are taken 0 =< d < 3
# @param n The polynomial degree n, which is taken into account
def kSub(d, n):
Bn_1, Bn = modeRange(n)
stiffnessSpp = np.zeros(fullShape)
stiffnessSpp[:, Bn_1:Bn] = -stiffnessValues[d][:, Bn_1:Bn]
return Tensor('kDivMTSub({},{})'.format(d, n),
fullShape,
spp=stiffnessSpp)
kernels = list()
kernels.append(
spaceTimePredictorRhs['kpt'] <= self.Q['kp'] * self.db.wHat['t'])
for n in range(self.order - 1, -1, -1):
for o in range(self.numberOfQuantities() - 1, -1, -1):
kernels.append(
spaceTimePredictor['kpt'] <= spaceTimePredictor['kpt'] +
selectModes(n)['kl'] * selectQuantity(o)['pq'] *
spaceTimePredictorRhs['lqu'] * Zinv(o)['ut'])
#G only has one relevant non-zero entry in each iteration, so we make it a scalar
#G[o] = E[o-4, o] * timestep
#In addition E only has non-zero entries, if o > 10
if o >= 10:
kernels.append(
spaceTimePredictorRhs['kpt'] <=
spaceTimePredictorRhs['kpt'] + G[o] *
selectQuantityG(o)['pv'] * selectQuantity(o)['vq'] *
selectModes(n)['kl'] * spaceTimePredictor['lqt'])
if n > 0:
derivativeSum = spaceTimePredictorRhs['kpt']
for d in range(3):
derivativeSum += kSub(d, n)['kl'] * spaceTimePredictor[
'lqt'] * self.starMatrix(d)['qp']
kernels.append(spaceTimePredictorRhs['kpt'] <= derivativeSum)
kernels.append(self.I['kp'] <= timestep * spaceTimePredictor['kpt'] *
self.db.timeInt['t'])
generator.add('spaceTimePredictor', kernels)
# Test to see if the kernel actually solves the system of equations
# This part is not used in the time kernel, but for unit testing
deltaSppLarge = np.eye(self.numberOfQuantities())
deltaLarge = Tensor('deltaLarge',
deltaSppLarge.shape,
spp=deltaSppLarge)
deltaSppSmall = np.eye(self.order)
deltaSmall = Tensor('deltaSmall',
deltaSppSmall.shape,
spp=deltaSppSmall)
minus = Scalar('minus')
lhs = deltaLarge['oq'] * self.db.Z['uk'] * spaceTimePredictor['lqk']
lhs += minus * self.sourceMatrix(
)['qo'] * deltaSmall['uk'] * spaceTimePredictor['lqk']
generator.add('stpTestLhs', testLhs['lou'] <= lhs)
rhs = self.Q['lo'] * self.db.wHat['u']
for d in range(3):
rhs += minus * self.starMatrix(
d)['qo'] * self.db.kDivMT[d]['lm'] * spaceTimePredictor['mqu']
generator.add('stpTestRhs', testRhs['lou'] <= rhs)
