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test_helmholtz.py
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# -*- coding:utf-8 -*-
import math
import sys
import petsc4py
petsc4py.init(sys.argv)
from petsc4py import PETSc
from mpi4py import MPI
#python3 test_helmholtz.py -da_refine 4 -snes_rtol 1.0e-6 -snes_converged_reason
#python3 test_helmholtz.py 2 -snes_converged_reason -ph_no_gradient -snes_fd_function -snes_fd_color
# The quadrature points and weight of numerical integration on [-1, 1]
gausslegendre = {
1: [(0.0, 2.0)],
2: [(-0.577350269189626, 1.0), (0.577350269189626, 1.0)],
3: [(-0.774596669241483, 0.555555555555556), (0.0, 0.888888888888889),
(0.774596669241483, 0.555555555555556)]
}
symmgauss = {
1: [(1.0/3.0, 1.0/3.0, 0.5)],
3: [(1.0/6.0, 1.0/6.0, 1.0/6.0), (2.0/3.0, 1.0/6.0, 1.0/6.0), (1.0/6.0, 2.0/3.0, 1.0/6.0)],
4: [(1.0/3.0, 1.0/3.0, -27.0/96.0), (1.0/5.0, 1.0/5.0, 25.0/96.0),
(3.0/5.0, 1.0/5.0, 25.0/96.0), (1.0/5.0, 3.0/5.0, 25.0/96.0)],
}
def f_constant(x, y, p, eps):
return 1.0
def u_exact_cosines(x, y, p, eps):
return math.cos(math.pi * x) * math.cos(math.pi * y)
def f_cosine(x, y, p, eps):
uu = math.cos(math.pi*x) * math.cos(math.pi * y)
pi2 = math.pi ** 2
lapu = - 2 * pi2 * uu
if p == 2.0:
return - lapu + uu
else:
ux = - math.pi * math.sin(math.pi * x)* math.cos(math.pi * y)
uy = - math.pi * math.cos(math.pi * x)* math.sin(math.pi * y)
w = ux * ux + uy * uy + eps * eps
pi3 = pi2 * math.pi
wx = pi3 * math.sin(2 * math.pi * x)* math.cos(2 * math.pi * y)
wy = pi3 * math.cos(2 * math.pi * x)* math.sin(2 * math.pi * y)
s = (p - 2) / 2
return - s * math.pow(w,s-1) * (wx * ux + wy * uy)- math.pow(w,s) * lapu + uu
class gradRef:
def __init__(self, xi, eta):
self.xi = xi
self.eta = eta
class Helmholtz:
def __init__(self):
self.da = None
self.local_vec = None
self.quadpts = 2
self.f = None # a user defined function
self.p = 2
self.eps = 0
def FormObjectiveLocal(self, sens, Vec_au):
def ObjIntegrandRef(ff, uu, xi, eta):
du = deval(uu, xi, eta)
u = eval(uu, xi, eta)
return GradPow(hx, hy, du, self.p, 0.0)/self.p + 0.5*u*u - eval(ff,xi,eta)*u
mx, my = self.da.getSizes()
self.da.globalToLocal(Vec_au, self.local_vec)
au = self.da.getVecArray(self.local_vec)
hx = 1.0 / (mx - 1)
hy = 1.0 / (my - 1)
points_and_weight = gausslegendre[self.quadpts]
lobj = 0.0
(xs, xe), (ys, ye) = self.da.getRanges()
for j in range(ys, ye):
if j == 0:
continue
y = j * hy
for i in range(xs, xe):
if i == 0:
continue
x = i * hx
ff = [self.f(x, y, self.p, self.eps),
self.f(x-hx, y, self.p, self.eps),
self.f(x-hx, y-hy, self.p, self.eps),
self.f(x, y-hy, self.p, self.eps)]
uu = [au[i, j], au[i-1,j], au[i-1,j-1], au[i,j-1] ]
for gp_r, gw_r in points_and_weight:
for gp_s, gw_s in points_and_weight:
lobj += gw_r * gw_s * ObjIntegrandRef(ff, uu, gp_r, gp_s)
lobj *= hx * hy / 4.0
lobj = self.da.comm.tompi4py().allreduce(lobj, MPI.SUM)
return lobj
def FormFunctionLocal(self, sens, Vec_au, Vec_FF):
def IntegrandRef(L, ff, uu, xi, eta):
du = deval(uu, xi, eta)
dchiL = dchi(L, xi, eta)
return GradPow(hx,hy,du, self.p - 2.0, self.eps) \
* GradInnerProd(hx,hy,du,dchiL) \
+ (eval(uu,xi,eta) - eval(ff, xi,eta)) * chi(L,xi,eta)
mx, my = self.da.getSizes()
self.da.globalToLocal(Vec_au, self.local_vec)
au = self.da.getVecArray(self.local_vec)
FF = self.da.getVecArray(Vec_FF)
hx = 1.0 / (mx - 1)
hy = 1.0 / (my - 1)
points_and_weight = gausslegendre[self.quadpts]
li = [0, -1, -1, 0]
lj = [0, 0, -1, -1]
(xs, xe), (ys, ye) = self.da.getRanges()
for j in range(ys, ye):
for i in range(xs, xe):
FF[i,j] = 0.0
for j in range(ys, ye):
if j == 0 or j > my - 1:
continue
y = j * hy
for i in range(xs, xe):
if i == 0 or i > mx - 1:
continue
x = i * hx
ff = [self.f(x, y, self.p, self.eps),
self.f(x-hx, y, self.p, self.eps),
self.f(x-hx, y-hy, self.p, self.eps),
self.f(x, y-hy, self.p, self.eps)]
uu = [au[i,j], au[i-1,j], au[i-1,j-1], au[i,j-1]]
for l in range(4):
PP = i + li[l]
QQ = j + lj[l]
if PP >= xs and PP < xe and QQ >= ys and QQ < ye:
for gp_r,gw_r in points_and_weight:
for gp_s,gw_s in points_and_weight:
FF[PP,QQ] += 0.25 * hx * hy *gw_r* gw_s*IntegrandRef(l, ff, uu, gp_r, gp_s)
# The points of reference element
# [1.0, 1.0], [-1.0, 1.0], [-1.0, -1.0], [1.0, -1.0]
xiL = [1.0, -1.0, -1.0, 1.0]
etaL = [1.0, 1.0, -1.0, -1.0]
# The basis function on the reference element
def chi(L, xi, eta):
return 0.25 * (1.0 + xiL[L] * xi) * (1.0 + etaL[L] * eta)
def eval(v, xi, eta):
return v[0] * chi(0,xi,eta) + v[1] * chi(1,xi,eta) \
+ v[2] * chi(2,xi,eta) + v[3] * chi(3,xi,eta)
# the gradient of chi
def dchi(L, xi, eta):
return gradRef(0.25 * xiL[L] * (1.0 + etaL[L] * eta), 0.25 * etaL[L] * (1.0 + xiL[L] * xi))
# evaluate partial derivs of v(xi,eta) on reference element
def deval(v, xi, eta):
s = gradRef(0, 0)
for L in range(4):
tmp = dchi(L, xi, eta)
s.xi += v[L] * tmp.xi
s.eta += v[L] * tmp.eta
return s
def GradInnerProd(hx:float, hy:float, du:gradRef, dv:gradRef):
cx = 4.0 / (hx * hx)
cy = 4.0 / (hy * hy)
return cx * du.xi * dv.xi + cy * du.eta * dv.eta
def GradPow(hx:float, hy:float, du:gradRef, P:float, eps:float):
return math.pow(GradInnerProd(hx,hy,du,du) + eps*eps, P/2.0)
def GetVecFromFunction(da, w, func, helm):
mx, my = da.getSizes()
hx = 1.0 / (mx - 1)
hy = 1.0 / (my - 1)
aw = da.getVecArray(w)
(xs, xe), (ys, ye) = da.getRanges()
for j in range(ys, ye):
y = j * hy
for i in range(xs, xe):
x = i * hx
aw[i,j] = func(x, y, helm.p, helm.eps)
return 0
helm = Helmholtz()
problem = "COSINES"
OptDB = PETSc.Options("")
exact_init = OptDB.getBool("-ph_exact_init", False)
no_objective = OptDB.getBool("-ph_no_objective", False)
no_gradient = OptDB.getBool("-ph_no_gradient", False)
helm.eps = OptDB.getReal("-ph_eps", 0.0)
helm.p = OptDB.getReal("-ph_p", 2.0)
helm.quadpts = OptDB.getInt("-quadpts", 2)
view_f = OptDB.getBool("-view_f", False)
if helm.p < 1.0:
raise RuntimeError("p >= 1 required")
if helm.p == 1.0:
PETSc.Sys.Print("WARNING: well-posedness only known for p > 1\n")
if helm.quadpts < 1 or helm.quadpts > 3:
raise RuntimeError("quadrature points n=1,2,3 only")
da = PETSc.DMDA()
da.create(comm = PETSc.COMM_WORLD,
dim=2,
sizes=(2, 2),
proc_sizes=None,
boundary_type=(PETSc.DM.BoundaryType.NONE, PETSc.DM.BoundaryType.NONE),
stencil_type=PETSc.DMDA.StencilType.BOX,
stencil_width=1,
dof=1,
setup=False)
da.setFromOptions()
da.setUp()
da.setUniformCoordinates(0.0, 1.0, 0.0, 1.0)
helm.da = da
helm.local_vec = da.createLocalVec()
snes = PETSc.SNES().create(comm=PETSc.COMM_WORLD)
snes.setDM(da)
if not no_objective:
snes.setObjective(helm.FormObjectiveLocal)
if no_gradient:
OptDB.setValue("-snes_fd_function_eps", 0.0)
else:
snes.setFunction(helm.FormFunctionLocal)
snes.setFromOptions()
u_initial = da.createGlobalVec()
u_initial.set(0.5)
if problem == "COSINES":
if exact_init:
GetVecFromFunction(da, u_initial, u_exact_cosines, helm)
helm.f = f_cosine
elif problem == "CONSTANT":
if exact_init:
u_initial.set(1.0)
helm.f = f_constant
else:
raise RuntimeError("unknown problem type\n")
if view_f:
vf = u_initial.copy()
if problem == "CONSTANT":
vf.set(1.0)
elif problem == "COSINES":
GetVecFromFunction(da, vf, f_cosine, helm)
vf.view()
vf.destroy()
snes.solve(None, u_initial)
u = snes.getSolution()
#u.view()
u_exact = u.copy()
if problem == "CONSTANT":
u_exact.set(1.0)
elif problem == "COSINES":
GetVecFromFunction(da, u_exact, u_exact_cosines, helm)
u.axpy(-1.0, u_exact)
err = u.norm(PETSc.NormType.NORM_INFINITY)
mx, my = da.getSizes()
PETSc.Sys.Print("done on %d x %d grid with p=%.3f ...\n numerical error: |u-u_exact|_inf = %.3e\n"%(mx, my, helm.p, err))
u_exact.destroy()
u_initial.destroy()
da.destroy()
snes.destroy()