diff --git a/.github/workflows/nightly.yaml b/.github/workflows/nightly.yaml
index d285185..160a674 100644
--- a/.github/workflows/nightly.yaml
+++ b/.github/workflows/nightly.yaml
@@ -30,3 +30,9 @@ jobs:
cd OpenSn_Logo_CAD && mpirun -np 8 ../opensn/build/python/opensn -i opensn.py && cd ..
cd Urban_Source && mpirun -np 96 ../opensn/build/python/opensn -i urban_source.py && cd ..
cd HEU_MET_FAST_003 && mpirun -np 12 ../opensn/build/python/opensn -i HEU_MET_FAST_003.py && cd ..
+ cd Six_1g_spherical_benchmarks && mpirun -np 12 ../opensn/build/python/opensn -i Problem_1.py && cd ..
+ cd Six_1g_spherical_benchmarks && mpirun -np 12 ../opensn/build/python/opensn -i Problem_2.py && cd ..
+ cd Six_1g_spherical_benchmarks && mpirun -np 12 ../opensn/build/python/opensn -i Problem_3.py && cd ..
+ cd Six_1g_spherical_benchmarks && mpirun -np 12 ../opensn/build/python/opensn -i Problem_4.py && cd ..
+ cd Six_1g_spherical_benchmarks && mpirun -np 12 ../opensn/build/python/opensn -i Problem_5.py && cd ..
+ cd Six_1g_spherical_benchmarks && mpirun -np 12 ../opensn/build/python/opensn -i Problem_6.py && cd ..
diff --git a/README.md b/README.md
index 076f786..ab2bd5c 100644
--- a/README.md
+++ b/README.md
@@ -11,3 +11,5 @@ A short description of each example is given below:
3D fixed-source problem that models a source in an urban setting
- [HEU_MET_FAST_003](./HEU_MET_FAST_003/README.md):
3D criticality benchmark problem
+- [Six 1-g Spherical Benchmarks](./Six_1g_spherical_benchmarks/README.md):
+ Six 3D spherical benchmark problems
diff --git a/Six_1g_spherical_benchmarks/Problem_1.png b/Six_1g_spherical_benchmarks/Problem_1.png
new file mode 100644
index 0000000..ea88733
Binary files /dev/null and b/Six_1g_spherical_benchmarks/Problem_1.png differ
diff --git a/Six_1g_spherical_benchmarks/Problem_1.py b/Six_1g_spherical_benchmarks/Problem_1.py
new file mode 100644
index 0000000..0069453
--- /dev/null
+++ b/Six_1g_spherical_benchmarks/Problem_1.py
@@ -0,0 +1,195 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+"""
+Problem 1
+Chang, B. (2021a, February 19).
+Six 1D polar transport test problems for the deterministic and monte-carlo method.
+LLNL-TR-819680
+https://www.osti.gov/servlets/purl/1769096
+"""
+
+import sys
+import os
+import numpy as np
+
+if "opensn_console" not in globals():
+ from mpi4py import MPI
+ size = MPI.COMM_WORLD.size
+ rank = MPI.COMM_WORLD.rank
+ # Append parent directory to locate the pyopensn modules
+ sys.path.append(os.path.abspath(os.path.join(os.path.dirname(__file__), "../../../../../")))
+ from pyopensn.mesh import FromFileMeshGenerator, PETScGraphPartitioner
+ from pyopensn.xs import MultiGroupXS
+ from pyopensn.aquad import GLCProductQuadrature3DXYZ
+ from pyopensn.solver import DiscreteOrdinatesProblem, SteadyStateSolver
+ from pyopensn.fieldfunc import FieldFunctionInterpolationVolume
+ from pyopensn.logvol import SphereLogicalVolume, BooleanLogicalVolume
+
+if __name__ == "__main__":
+
+ meshgen = FromFileMeshGenerator(
+ filename="./meshes/single_sphere.msh",
+ partitioner=PETScGraphPartitioner(type='parmetis'),
+ )
+ grid = meshgen.Execute()
+
+ # Get measured volumes
+ volumes_per_block = grid.ComputeVolumePerBlockID()
+
+ # Define radii per block-ID
+ radii = {1: 1.0}
+
+ # Sort block IDs by increasing radius
+ block_ids = sorted(volumes_per_block.keys())
+ if rank == 0:
+ print(block_ids)
+
+ # Check for missing radii
+ missing = [blk for blk in block_ids if blk not in radii]
+ if missing:
+ raise KeyError(f"No radius provided for block IDs: {missing}")
+
+ # Compute exact volumes
+ exact_volumes_per_block = {}
+ prev_R = 0.0
+ for blk in block_ids:
+ R = radii[blk]
+ exact_volumes_per_block[blk] = (4.0 / 3.0) * np.pi * (R ** 3 - prev_R ** 3)
+ prev_R = R
+
+ # Build the ratios array
+ ratios = np.array([
+ exact_volumes_per_block[blk] / volumes_per_block[blk]
+ for blk in block_ids
+ ])
+
+ if rank == 0:
+ for idx, blk in enumerate(block_ids):
+ print(f"Block {blk}: measured = {volumes_per_block[blk]:.11e}, "
+ f"exact = {exact_volumes_per_block[blk]:.11e}, "
+ f"ratio = {ratios[idx]:.6f}")
+
+ # Create and modify XS
+ num_groups = 1
+ xs_mat = MultiGroupXS()
+ sigt = 1.0
+ c = 0.0
+ xs_mat.CreateSimpleOneGroup(sigma_t=sigt, c=c)
+
+ # Boundary source
+ bsrc = [1.0]
+
+ # Angular quadrature
+ nazimu = 4
+ npolar = 2
+ pquad = GLCProductQuadrature3DXYZ(npolar, nazimu)
+
+ # Solver
+ phys = DiscreteOrdinatesProblem(
+ mesh=grid,
+ num_groups=num_groups,
+ groupsets=[
+ {
+ "groups_from_to": (0, num_groups - 1),
+ "angular_quadrature": pquad,
+ "inner_linear_method": "petsc_gmres",
+ "angle_aggregation_type": "single",
+ "angle_aggregation_num_subsets": 1,
+ "l_max_its": 500,
+ "l_abs_tol": 1.0e-6,
+ "gmres_restart_interval": 30,
+ },
+ ],
+ xs_map=[
+ {"block_ids": [1], "xs": xs_mat},
+ ],
+ options={
+ "scattering_order": 0,
+ "boundary_conditions": [
+ {"name": "xmin", "type": "isotropic", "group_strength": bsrc},
+ {"name": "xmax", "type": "isotropic", "group_strength": bsrc},
+ {"name": "ymin", "type": "isotropic", "group_strength": bsrc},
+ {"name": "ymax", "type": "isotropic", "group_strength": bsrc},
+ {"name": "zmin", "type": "isotropic", "group_strength": bsrc},
+ {"name": "zmax", "type": "isotropic", "group_strength": bsrc},
+ ],
+ },
+ )
+
+ ss_solver = SteadyStateSolver(lbs_problem=phys)
+ ss_solver.Initialize()
+ ss_solver.Execute()
+
+ # Export
+ fflist = phys.GetFieldFunctions()
+
+ # Compute the average flux over a logical volume
+ def average_flx_logvol(logvol):
+ ffvol = FieldFunctionInterpolationVolume()
+ ffvol.SetOperationType("avg")
+ ffvol.SetLogicalVolume(logvol)
+ ffvol.AddFieldFunction(fflist[0])
+ ffvol.Initialize()
+ ffvol.Execute()
+ avgval = ffvol.GetValue()
+ return avgval
+
+ # Compute the average flux over spherical shells
+ def compute_average_flux(N_logvols, r_max):
+ r_vals = np.linspace(0, r_max, N_logvols + 1)
+ avg_flx = np.zeros(N_logvols)
+ for i in range(N_logvols):
+ if i != 0:
+ inner_vol = SphereLogicalVolume(r=r_vals[i])
+ outer_vol = SphereLogicalVolume(r=r_vals[i + 1])
+ logvol = BooleanLogicalVolume(parts=[
+ {"op": True, "lv": outer_vol},
+ {"op": False, "lv": inner_vol},
+ ])
+ else:
+ logvol = SphereLogicalVolume(r=r_vals[i + 1])
+ avg_flx[i] = average_flx_logvol(logvol)
+ return r_vals, avg_flx
+
+ # Perform the calculation
+ N_logvols = 15
+ r_vals, avg_flx = compute_average_flux(N_logvols, 1)
+ if rank == 0:
+ for r, v in zip(r_vals[1:], avg_flx):
+ print("end-radius: {:.2f}, avg-value: {:.6f}".format(r, v))
+
+ # Compute the analytical answer
+ from scipy.special import exp1
+
+ def E2(x):
+ return np.exp(-x) - x * exp1(x)
+
+ def get_phi(r, psi, r_max, sig):
+ term1 = (np.exp(-sig * (r_max - r)) - np.exp(-sig * (r_max + r))) / sig
+ term2 = (r + r_max) * E2(sig * (r_max - r))
+ term3 = (r_max - r) * E2(sig * (r_max + r))
+ phi = psi / (2 * r) * (term1 + term2 - term3)
+ return phi
+
+ if rank == 0:
+ r_max = radii[1]
+ eps = 1e-3
+ psi = 2 * np.pi
+ r_fine = np.linspace(0.0 + eps, r_max - eps, 100)
+ exact_phi = np.zeros_like(r_fine)
+ for i, r in enumerate(r_fine):
+ exact_phi[i] = get_phi(r, psi, r_max, sigt)
+
+ import matplotlib.pyplot as plt
+
+ plt.figure()
+ plt.plot(r_fine, exact_phi, label='Exact')
+ plt.step(r_vals, np.append(avg_flx, avg_flx[-1]), where='post',
+ label='Radial Average Flux in Spherical Shells')
+ plt.xlabel("Radius")
+ plt.ylabel("Average Flux")
+ plt.title("Flux as a function of radius")
+ plt.grid(True)
+ plt.legend()
+ plt.savefig("Problem_1.png", dpi=300, bbox_inches='tight')
+ plt.show()
diff --git a/Six_1g_spherical_benchmarks/Problem_2.png b/Six_1g_spherical_benchmarks/Problem_2.png
new file mode 100644
index 0000000..10fbc3a
Binary files /dev/null and b/Six_1g_spherical_benchmarks/Problem_2.png differ
diff --git a/Six_1g_spherical_benchmarks/Problem_2.py b/Six_1g_spherical_benchmarks/Problem_2.py
new file mode 100644
index 0000000..0069453
--- /dev/null
+++ b/Six_1g_spherical_benchmarks/Problem_2.py
@@ -0,0 +1,195 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+"""
+Problem 1
+Chang, B. (2021a, February 19).
+Six 1D polar transport test problems for the deterministic and monte-carlo method.
+LLNL-TR-819680
+https://www.osti.gov/servlets/purl/1769096
+"""
+
+import sys
+import os
+import numpy as np
+
+if "opensn_console" not in globals():
+ from mpi4py import MPI
+ size = MPI.COMM_WORLD.size
+ rank = MPI.COMM_WORLD.rank
+ # Append parent directory to locate the pyopensn modules
+ sys.path.append(os.path.abspath(os.path.join(os.path.dirname(__file__), "../../../../../")))
+ from pyopensn.mesh import FromFileMeshGenerator, PETScGraphPartitioner
+ from pyopensn.xs import MultiGroupXS
+ from pyopensn.aquad import GLCProductQuadrature3DXYZ
+ from pyopensn.solver import DiscreteOrdinatesProblem, SteadyStateSolver
+ from pyopensn.fieldfunc import FieldFunctionInterpolationVolume
+ from pyopensn.logvol import SphereLogicalVolume, BooleanLogicalVolume
+
+if __name__ == "__main__":
+
+ meshgen = FromFileMeshGenerator(
+ filename="./meshes/single_sphere.msh",
+ partitioner=PETScGraphPartitioner(type='parmetis'),
+ )
+ grid = meshgen.Execute()
+
+ # Get measured volumes
+ volumes_per_block = grid.ComputeVolumePerBlockID()
+
+ # Define radii per block-ID
+ radii = {1: 1.0}
+
+ # Sort block IDs by increasing radius
+ block_ids = sorted(volumes_per_block.keys())
+ if rank == 0:
+ print(block_ids)
+
+ # Check for missing radii
+ missing = [blk for blk in block_ids if blk not in radii]
+ if missing:
+ raise KeyError(f"No radius provided for block IDs: {missing}")
+
+ # Compute exact volumes
+ exact_volumes_per_block = {}
+ prev_R = 0.0
+ for blk in block_ids:
+ R = radii[blk]
+ exact_volumes_per_block[blk] = (4.0 / 3.0) * np.pi * (R ** 3 - prev_R ** 3)
+ prev_R = R
+
+ # Build the ratios array
+ ratios = np.array([
+ exact_volumes_per_block[blk] / volumes_per_block[blk]
+ for blk in block_ids
+ ])
+
+ if rank == 0:
+ for idx, blk in enumerate(block_ids):
+ print(f"Block {blk}: measured = {volumes_per_block[blk]:.11e}, "
+ f"exact = {exact_volumes_per_block[blk]:.11e}, "
+ f"ratio = {ratios[idx]:.6f}")
+
+ # Create and modify XS
+ num_groups = 1
+ xs_mat = MultiGroupXS()
+ sigt = 1.0
+ c = 0.0
+ xs_mat.CreateSimpleOneGroup(sigma_t=sigt, c=c)
+
+ # Boundary source
+ bsrc = [1.0]
+
+ # Angular quadrature
+ nazimu = 4
+ npolar = 2
+ pquad = GLCProductQuadrature3DXYZ(npolar, nazimu)
+
+ # Solver
+ phys = DiscreteOrdinatesProblem(
+ mesh=grid,
+ num_groups=num_groups,
+ groupsets=[
+ {
+ "groups_from_to": (0, num_groups - 1),
+ "angular_quadrature": pquad,
+ "inner_linear_method": "petsc_gmres",
+ "angle_aggregation_type": "single",
+ "angle_aggregation_num_subsets": 1,
+ "l_max_its": 500,
+ "l_abs_tol": 1.0e-6,
+ "gmres_restart_interval": 30,
+ },
+ ],
+ xs_map=[
+ {"block_ids": [1], "xs": xs_mat},
+ ],
+ options={
+ "scattering_order": 0,
+ "boundary_conditions": [
+ {"name": "xmin", "type": "isotropic", "group_strength": bsrc},
+ {"name": "xmax", "type": "isotropic", "group_strength": bsrc},
+ {"name": "ymin", "type": "isotropic", "group_strength": bsrc},
+ {"name": "ymax", "type": "isotropic", "group_strength": bsrc},
+ {"name": "zmin", "type": "isotropic", "group_strength": bsrc},
+ {"name": "zmax", "type": "isotropic", "group_strength": bsrc},
+ ],
+ },
+ )
+
+ ss_solver = SteadyStateSolver(lbs_problem=phys)
+ ss_solver.Initialize()
+ ss_solver.Execute()
+
+ # Export
+ fflist = phys.GetFieldFunctions()
+
+ # Compute the average flux over a logical volume
+ def average_flx_logvol(logvol):
+ ffvol = FieldFunctionInterpolationVolume()
+ ffvol.SetOperationType("avg")
+ ffvol.SetLogicalVolume(logvol)
+ ffvol.AddFieldFunction(fflist[0])
+ ffvol.Initialize()
+ ffvol.Execute()
+ avgval = ffvol.GetValue()
+ return avgval
+
+ # Compute the average flux over spherical shells
+ def compute_average_flux(N_logvols, r_max):
+ r_vals = np.linspace(0, r_max, N_logvols + 1)
+ avg_flx = np.zeros(N_logvols)
+ for i in range(N_logvols):
+ if i != 0:
+ inner_vol = SphereLogicalVolume(r=r_vals[i])
+ outer_vol = SphereLogicalVolume(r=r_vals[i + 1])
+ logvol = BooleanLogicalVolume(parts=[
+ {"op": True, "lv": outer_vol},
+ {"op": False, "lv": inner_vol},
+ ])
+ else:
+ logvol = SphereLogicalVolume(r=r_vals[i + 1])
+ avg_flx[i] = average_flx_logvol(logvol)
+ return r_vals, avg_flx
+
+ # Perform the calculation
+ N_logvols = 15
+ r_vals, avg_flx = compute_average_flux(N_logvols, 1)
+ if rank == 0:
+ for r, v in zip(r_vals[1:], avg_flx):
+ print("end-radius: {:.2f}, avg-value: {:.6f}".format(r, v))
+
+ # Compute the analytical answer
+ from scipy.special import exp1
+
+ def E2(x):
+ return np.exp(-x) - x * exp1(x)
+
+ def get_phi(r, psi, r_max, sig):
+ term1 = (np.exp(-sig * (r_max - r)) - np.exp(-sig * (r_max + r))) / sig
+ term2 = (r + r_max) * E2(sig * (r_max - r))
+ term3 = (r_max - r) * E2(sig * (r_max + r))
+ phi = psi / (2 * r) * (term1 + term2 - term3)
+ return phi
+
+ if rank == 0:
+ r_max = radii[1]
+ eps = 1e-3
+ psi = 2 * np.pi
+ r_fine = np.linspace(0.0 + eps, r_max - eps, 100)
+ exact_phi = np.zeros_like(r_fine)
+ for i, r in enumerate(r_fine):
+ exact_phi[i] = get_phi(r, psi, r_max, sigt)
+
+ import matplotlib.pyplot as plt
+
+ plt.figure()
+ plt.plot(r_fine, exact_phi, label='Exact')
+ plt.step(r_vals, np.append(avg_flx, avg_flx[-1]), where='post',
+ label='Radial Average Flux in Spherical Shells')
+ plt.xlabel("Radius")
+ plt.ylabel("Average Flux")
+ plt.title("Flux as a function of radius")
+ plt.grid(True)
+ plt.legend()
+ plt.savefig("Problem_1.png", dpi=300, bbox_inches='tight')
+ plt.show()
diff --git a/Six_1g_spherical_benchmarks/Problem_3.png b/Six_1g_spherical_benchmarks/Problem_3.png
new file mode 100644
index 0000000..23ecfc6
Binary files /dev/null and b/Six_1g_spherical_benchmarks/Problem_3.png differ
diff --git a/Six_1g_spherical_benchmarks/Problem_3.py b/Six_1g_spherical_benchmarks/Problem_3.py
new file mode 100644
index 0000000..ed100e2
--- /dev/null
+++ b/Six_1g_spherical_benchmarks/Problem_3.py
@@ -0,0 +1,159 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+"""
+Problem 3
+Chang, B. (2021a, February 19).
+Six 1D polar transport test problems for the deterministic and monte-carlo method.
+LLNL-TR-819680
+https://www.osti.gov/servlets/purl/1769096
+"""
+
+import sys
+import os
+import numpy as np
+
+if "opensn_console" not in globals():
+ from mpi4py import MPI
+ size = MPI.COMM_WORLD.size
+ rank = MPI.COMM_WORLD.rank
+ sys.path.append(os.path.abspath(os.path.join(os.path.dirname(__file__), "../../../../../")))
+ from pyopensn.mesh import FromFileMeshGenerator, PETScGraphPartitioner
+ from pyopensn.xs import MultiGroupXS
+ from pyopensn.aquad import GLCProductQuadrature3DXYZ
+ from pyopensn.solver import DiscreteOrdinatesProblem, SteadyStateSolver
+ from pyopensn.fieldfunc import FieldFunctionInterpolationVolume
+ from pyopensn.logvol import SphereLogicalVolume, BooleanLogicalVolume
+ from pyopensn.source import VolumetricSource
+
+if __name__ == "__main__":
+
+ meshgen = FromFileMeshGenerator(
+ filename="./meshes/three_region_sphere.msh",
+ partitioner=PETScGraphPartitioner(type='parmetis'),
+ )
+ grid = meshgen.Execute()
+
+ # Create XS
+ num_groups = 1
+ xs_void = MultiGroupXS()
+ xs_void.CreateSimpleOneGroup(sigma_t=0.0, c=0.0)
+
+ # Volumetric source
+ mg_src = VolumetricSource(block_ids=[2], group_strength=[1.0])
+
+ # Angular quadrature
+ nazimu = 4
+ npolar = 2
+ pquad = GLCProductQuadrature3DXYZ(npolar, nazimu)
+
+ # Solver
+ phys = DiscreteOrdinatesProblem(
+ mesh=grid,
+ num_groups=num_groups,
+ groupsets=[
+ {
+ "groups_from_to": (0, num_groups - 1),
+ "angular_quadrature": pquad,
+ "inner_linear_method": "petsc_gmres",
+ "angle_aggregation_type": "single",
+ "angle_aggregation_num_subsets": 1,
+ "l_max_its": 500,
+ "l_abs_tol": 1.0e-6,
+ "gmres_restart_interval": 30,
+ },
+ ],
+ xs_map=[
+ {"block_ids": [1, 2, 3], "xs": xs_void},
+ ],
+ options={
+ "scattering_order": 0,
+ "volumetric_sources": [mg_src],
+ "boundary_conditions": [
+ {"name": "xmin", "type": "vacuum"},
+ {"name": "xmax", "type": "vacuum"},
+ {"name": "ymin", "type": "vacuum"},
+ {"name": "ymax", "type": "vacuum"},
+ {"name": "zmin", "type": "vacuum"},
+ {"name": "zmax", "type": "vacuum"},
+ ],
+ },
+ )
+
+ ss_solver = SteadyStateSolver(lbs_problem=phys)
+ ss_solver.Initialize()
+ ss_solver.Execute()
+
+ # Export
+ fflist = phys.GetFieldFunctions()
+
+ # Compute the average flux over a logical volume
+ def average_flx_logvol(logvol):
+ ffvol = FieldFunctionInterpolationVolume()
+ ffvol.SetOperationType("avg")
+ ffvol.SetLogicalVolume(logvol)
+ ffvol.AddFieldFunction(fflist[0])
+ ffvol.Initialize()
+ ffvol.Execute()
+ avgval = ffvol.GetValue()
+ return avgval
+
+ # Compute the average flux over spherical shells
+ def compute_average_flux(N_logvols, r_max):
+ r_vals = np.linspace(0, r_max, N_logvols + 1)
+ avg_flx = np.zeros(N_logvols)
+ for i in range(N_logvols):
+ if i != 0:
+ inner_vol = SphereLogicalVolume(r=r_vals[i])
+ outer_vol = SphereLogicalVolume(r=r_vals[i + 1])
+ logvol = BooleanLogicalVolume(parts=[
+ {"op": True, "lv": outer_vol},
+ {"op": False, "lv": inner_vol},
+ ])
+ else:
+ logvol = SphereLogicalVolume(r=r_vals[i + 1])
+ avg_flx[i] = average_flx_logvol(logvol)
+ return r_vals, avg_flx
+
+ # Perform the calculation
+ N_logvols = 15
+ r_vals, avg_flx = compute_average_flux(N_logvols, 1)
+ if rank == 0:
+ for r, v in zip(r_vals[1:], avg_flx):
+ print("end-radius: {:.2f}, avg-value: {:.6f}".format(r, v))
+
+ # Compute the analytical answer
+ def get_phi(r, q, a, b):
+ a2 = a ** 2
+ b2 = b ** 2
+ r2 = r ** 2
+ term1 = q * (b + (b2 - r2) / (2 * r) * np.log((b + r) / np.abs(r - b)))
+ term2 = q * (a + (a2 - r2) / (2 * r) * np.log((a + r) / np.abs(r - a)))
+ return term1 - term2
+
+ if rank == 0:
+ q = 0.5
+ a = 1.0 / 3.0
+ b = 2.0 / 3.0
+ eps = 1e-3
+ r_fine = np.linspace(0.0 + eps, 1.0 - eps, 100)
+ exact_phi = np.zeros_like(r_fine)
+ for i, r in enumerate(r_fine):
+ exact_phi[i] = get_phi(r, q, a, b)
+ print(r_fine[i], exact_phi[i])
+
+ import matplotlib.pyplot as plt
+
+ plt.figure()
+ plt.plot(r_fine, exact_phi, label='Exact')
+ plt.step(
+ r_vals, np.append(avg_flx, avg_flx[-1]),
+ where='post',
+ label='Radial Average Flux in Spherical Shells'
+ )
+ plt.xlabel("Radius")
+ plt.ylabel("Average Flux")
+ plt.title("Flux as a function of radius")
+ plt.grid(True)
+ plt.legend()
+ plt.savefig("Problem_3.png", dpi=300, bbox_inches='tight')
+ plt.show()
diff --git a/Six_1g_spherical_benchmarks/Problem_4.png b/Six_1g_spherical_benchmarks/Problem_4.png
new file mode 100644
index 0000000..baad3fb
Binary files /dev/null and b/Six_1g_spherical_benchmarks/Problem_4.png differ
diff --git a/Six_1g_spherical_benchmarks/Problem_4.py b/Six_1g_spherical_benchmarks/Problem_4.py
new file mode 100644
index 0000000..68d1a47
--- /dev/null
+++ b/Six_1g_spherical_benchmarks/Problem_4.py
@@ -0,0 +1,175 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+"""
+Problem 4
+Chang, B. (2021a, February 19).
+Six 1D polar transport test problems for the deterministic and monte-carlo method.
+LLNL-TR-819680
+https://www.osti.gov/servlets/purl/1769096
+"""
+
+import sys
+import os
+import numpy as np
+
+if "opensn_console" not in globals():
+ from mpi4py import MPI
+ size = MPI.COMM_WORLD.size
+ rank = MPI.COMM_WORLD.rank
+ sys.path.append(os.path.abspath(os.path.join(os.path.dirname(__file__), "../../../../../")))
+ from pyopensn.mesh import FromFileMeshGenerator, PETScGraphPartitioner
+ from pyopensn.xs import MultiGroupXS
+ from pyopensn.aquad import GLCProductQuadrature3DXYZ
+ from pyopensn.solver import DiscreteOrdinatesProblem, SteadyStateSolver
+ from pyopensn.fieldfunc import FieldFunctionInterpolationVolume
+ from pyopensn.logvol import SphereLogicalVolume, BooleanLogicalVolume
+
+if __name__ == "__main__":
+
+ meshgen = FromFileMeshGenerator(
+ filename="./meshes/two_region_sphere.msh",
+ partitioner=PETScGraphPartitioner(type='parmetis'),
+ )
+ grid = meshgen.Execute()
+
+ # Create XS
+ num_groups = 1
+ xs_void = MultiGroupXS()
+ xs_void.CreateSimpleOneGroup(sigma_t=0.0, c=0.0)
+ xs_mat = MultiGroupXS()
+ xs_mat.CreateSimpleOneGroup(sigma_t=1.0, c=0.0)
+
+ # Boundary source
+ bsrc = [1.0]
+
+ # Angular quadrature
+ nazimu = 4
+ npolar = 2
+ pquad = GLCProductQuadrature3DXYZ(npolar, nazimu)
+
+ # Solver
+ phys = DiscreteOrdinatesProblem(
+ mesh=grid,
+ num_groups=num_groups,
+ groupsets=[
+ {
+ "groups_from_to": (0, num_groups - 1),
+ "angular_quadrature": pquad,
+ "inner_linear_method": "petsc_gmres",
+ "angle_aggregation_type": "single",
+ "angle_aggregation_num_subsets": 1,
+ "l_max_its": 500,
+ "l_abs_tol": 1.0e-6,
+ "gmres_restart_interval": 30,
+ },
+ ],
+ xs_map=[
+ {"block_ids": [1], "xs": xs_mat},
+ {"block_ids": [2], "xs": xs_void},
+ ],
+ options={
+ "scattering_order": 0,
+ "boundary_conditions": [
+ {"name": "xmin", "type": "isotropic", "group_strength": bsrc},
+ {"name": "xmax", "type": "isotropic", "group_strength": bsrc},
+ {"name": "ymin", "type": "isotropic", "group_strength": bsrc},
+ {"name": "ymax", "type": "isotropic", "group_strength": bsrc},
+ {"name": "zmin", "type": "isotropic", "group_strength": bsrc},
+ {"name": "zmax", "type": "isotropic", "group_strength": bsrc},
+ ],
+ },
+ )
+
+ ss_solver = SteadyStateSolver(lbs_problem=phys)
+ ss_solver.Initialize()
+ ss_solver.Execute()
+
+ # Export
+ fflist = phys.GetFieldFunctions()
+
+ # Compute the average flux over a logical volume
+ def average_flx_logvol(logvol):
+ ffvol = FieldFunctionInterpolationVolume()
+ ffvol.SetOperationType("avg")
+ ffvol.SetLogicalVolume(logvol)
+ ffvol.AddFieldFunction(fflist[0])
+ ffvol.Initialize()
+ ffvol.Execute()
+ avgval = ffvol.GetValue()
+ return avgval
+
+ # Compute the average flux over spherical shells
+ def compute_average_flux(N_logvols, r_max):
+ r_vals = np.linspace(0, r_max, N_logvols + 1)
+ avg_flx = np.zeros(N_logvols)
+ for i in range(N_logvols):
+ if i != 0:
+ inner_vol = SphereLogicalVolume(r=r_vals[i])
+ outer_vol = SphereLogicalVolume(r=r_vals[i + 1])
+ logvol = BooleanLogicalVolume(parts=[
+ {"op": True, "lv": outer_vol},
+ {"op": False, "lv": inner_vol},
+ ])
+ else:
+ logvol = SphereLogicalVolume(r=r_vals[i + 1])
+ avg_flx[i] = average_flx_logvol(logvol)
+ return r_vals, avg_flx
+
+ # Perform the calculation
+ N_logvols = 15
+ r_vals, avg_flx = compute_average_flux(N_logvols, 1)
+ if rank == 0:
+ for r, v in zip(r_vals[1:], avg_flx):
+ print("end-radius: {:.2f}, avg-value: {:.6f}".format(r, v))
+
+ # Compute the analytical answer
+ from scipy.special import exp1
+ from scipy.integrate import quad
+
+ def E2(x):
+ return np.exp(-x) - x * exp1(x)
+
+ def get_phi(r, psi, a, sig):
+ if r <= a:
+ term1 = (np.exp(-sig * (a - r)) - np.exp(-sig * (a + r))) / sig
+ term2 = (r + a) * E2(sig * (a - r))
+ term3 = (a - r) * E2(sig * (a + r))
+ phi = psi / (2 * r) * (term1 + term2 - term3)
+ else:
+ a2 = a ** 2
+ r2 = r ** 2
+ sqrt1 = np.sqrt(1 - a2 / r2)
+
+ def integrand(x):
+ return np.exp(-2 * sig * np.sqrt(a2 - r2 * (1 - x ** 2)))
+
+ phi = psi * (sqrt1 + 1 + quad(integrand, sqrt1, 1)[0])
+ return phi
+
+ if rank == 0:
+ psi_0 = 2 * np.pi
+ a = 0.5
+ sigt = 1.0
+ eps = 1e-3
+ r_fine = np.linspace(0.0 + eps, 1.0 - eps, 100)
+ exact_phi = np.zeros_like(r_fine)
+ for i, r in enumerate(r_fine):
+ exact_phi[i] = get_phi(r, psi_0, a, sigt)
+ print(r_fine[i], exact_phi[i])
+
+ import matplotlib.pyplot as plt
+
+ plt.figure()
+ plt.plot(r_fine, exact_phi, label='Exact')
+ plt.step(
+ r_vals, np.append(avg_flx, avg_flx[-1]),
+ where='post',
+ label='Radial Average Flux in Spherical Shells'
+ )
+ plt.xlabel("Radius")
+ plt.ylabel("Average Flux")
+ plt.title("Flux as a function of radius")
+ plt.grid(True)
+ plt.legend()
+ plt.savefig("Problem_4.png", dpi=300, bbox_inches='tight')
+ plt.show()
diff --git a/Six_1g_spherical_benchmarks/Problem_5.png b/Six_1g_spherical_benchmarks/Problem_5.png
new file mode 100644
index 0000000..40c5f65
Binary files /dev/null and b/Six_1g_spherical_benchmarks/Problem_5.png differ
diff --git a/Six_1g_spherical_benchmarks/Problem_5.py b/Six_1g_spherical_benchmarks/Problem_5.py
new file mode 100644
index 0000000..7a7562e
--- /dev/null
+++ b/Six_1g_spherical_benchmarks/Problem_5.py
@@ -0,0 +1,198 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+"""
+Problem 5
+Chang, B. (2021a, February 19).
+Six 1D polar transport test problems for the deterministic and monte-carlo method.
+LLNL-TR-819680
+https://www.osti.gov/servlets/purl/1769096
+"""
+
+import sys
+import os
+import numpy as np
+
+if "opensn_console" not in globals():
+ from mpi4py import MPI
+ size = MPI.COMM_WORLD.size
+ rank = MPI.COMM_WORLD.rank
+ sys.path.append(os.path.abspath(os.path.join(os.path.dirname(__file__), "../../../../../")))
+ from pyopensn.mesh import FromFileMeshGenerator, PETScGraphPartitioner
+ from pyopensn.xs import MultiGroupXS
+ from pyopensn.aquad import GLCProductQuadrature3DXYZ
+ from pyopensn.solver import DiscreteOrdinatesProblem, SteadyStateSolver
+ from pyopensn.fieldfunc import FieldFunctionInterpolationVolume
+ from pyopensn.logvol import SphereLogicalVolume, BooleanLogicalVolume
+
+if __name__ == "__main__":
+
+ meshgen = FromFileMeshGenerator(
+ filename="./meshes/three_region_sphere.msh",
+ partitioner=PETScGraphPartitioner(type='parmetis'),
+ )
+ grid = meshgen.Execute()
+
+ # Create XS
+ num_groups = 1
+ xs_void = MultiGroupXS()
+ xs_void.CreateSimpleOneGroup(sigma_t=0.0, c=0.0)
+ xs_mat = MultiGroupXS()
+ xs_mat.CreateSimpleOneGroup(sigma_t=1.0, c=0.0)
+
+ # Boundary source
+ bsrc = [1.0]
+
+ # Angular quadrature
+ nazimu = 4
+ npolar = 2
+ pquad = GLCProductQuadrature3DXYZ(npolar, nazimu)
+
+ # Solver
+ phys = DiscreteOrdinatesProblem(
+ mesh=grid,
+ num_groups=num_groups,
+ groupsets=[
+ {
+ "groups_from_to": (0, num_groups - 1),
+ "angular_quadrature": pquad,
+ "inner_linear_method": "petsc_gmres",
+ "angle_aggregation_type": "single",
+ "angle_aggregation_num_subsets": 1,
+ "l_max_its": 500,
+ "l_abs_tol": 1.0e-6,
+ "gmres_restart_interval": 30,
+ },
+ ],
+ xs_map=[
+ {"block_ids": [2], "xs": xs_mat},
+ {"block_ids": [1, 3], "xs": xs_void},
+ ],
+ options={
+ "scattering_order": 0,
+ "boundary_conditions": [
+ {"name": "xmin", "type": "isotropic", "group_strength": bsrc},
+ {"name": "xmax", "type": "isotropic", "group_strength": bsrc},
+ {"name": "ymin", "type": "isotropic", "group_strength": bsrc},
+ {"name": "ymax", "type": "isotropic", "group_strength": bsrc},
+ {"name": "zmin", "type": "isotropic", "group_strength": bsrc},
+ {"name": "zmax", "type": "isotropic", "group_strength": bsrc},
+ ],
+ },
+ )
+
+ ss_solver = SteadyStateSolver(lbs_problem=phys)
+ ss_solver.Initialize()
+ ss_solver.Execute()
+
+ # Export
+ fflist = phys.GetFieldFunctions()
+
+ # Compute the average flux over a logical volume
+ def average_flx_logvol(logvol):
+ ffvol = FieldFunctionInterpolationVolume()
+ ffvol.SetOperationType("avg")
+ ffvol.SetLogicalVolume(logvol)
+ ffvol.AddFieldFunction(fflist[0])
+ ffvol.Initialize()
+ ffvol.Execute()
+ avgval = ffvol.GetValue()
+ return avgval
+
+ # Compute the average flux over spherical shells
+ def compute_average_flux(N_logvols, r_max):
+ r_vals = np.linspace(0, r_max, N_logvols + 1)
+ avg_flx = np.zeros(N_logvols)
+ for i in range(N_logvols):
+ if i != 0:
+ inner_vol = SphereLogicalVolume(r=r_vals[i])
+ outer_vol = SphereLogicalVolume(r=r_vals[i + 1])
+ logvol = BooleanLogicalVolume(parts=[
+ {"op": True, "lv": outer_vol},
+ {"op": False, "lv": inner_vol},
+ ])
+ else:
+ logvol = SphereLogicalVolume(r=r_vals[i + 1])
+ avg_flx[i] = average_flx_logvol(logvol)
+ return r_vals, avg_flx
+
+ # Perform the calculation
+ N_logvols = 15
+ r_vals, avg_flx = compute_average_flux(N_logvols, 1)
+ if rank == 0:
+ for r, v in zip(r_vals[1:], avg_flx):
+ print("end-radius: {:.2f}, avg-value: {:.6f}".format(r, v))
+
+ # Compute the analytical answer
+ from scipy.integrate import quad
+
+ def get_phi(r, psi, a, b, sig):
+ a2 = a ** 2
+ b2 = b ** 2
+ r2 = r ** 2
+ phi = 0.0
+ if r > b:
+ phi += 1 + np.sqrt(1 - b2 / r2)
+
+ def integrand_1(x):
+ return np.exp(-2 * sig * np.sqrt(b2 - r2 * (1 - x ** 2)))
+
+ phi += quad(integrand_1, np.sqrt(1 - b2 / r2), np.sqrt(1 - a2 / r2))[0]
+
+ def integrand_2(x):
+ return np.exp(-2 * sig * (
+ np.sqrt(b2 - r2 * (1 - x ** 2)) - np.sqrt(a2 - r2 * (1 - x ** 2)))
+ )
+
+ phi += quad(integrand_2, np.sqrt(1 - a2 / r2), 1.)[0]
+
+ elif r < a:
+ def integrand_3(x):
+ return np.exp(-sig * (
+ np.sqrt(b2 - r2 * (1 - x ** 2)) - np.sqrt(a2 - r2 * (1 - x ** 2)))
+ )
+ phi += quad(integrand_3, -1, 1)[0]
+
+ else:
+ def integrand_4(x):
+ return np.exp(-sig * (
+ r * x + np.sqrt(b2 - r2 * (1 - x ** 2)) - 2 * np.sqrt(a2 - r2 * (1 - x ** 2)))
+ )
+
+ phi += quad(integrand_4, np.sqrt(1 - a2 / r2), 1)[0]
+
+ def integrand_5(x):
+ return np.exp(-sig * (r * x + np.sqrt(b2 - r2 * (1 - x ** 2))))
+
+ phi += quad(integrand_5, -1, np.sqrt(1 - a2 / r2))[0]
+
+ phi *= psi
+ return phi
+
+ if rank == 0:
+ psi_0 = 2 * np.pi
+ a = 1.0 / 3.0
+ b = 2.0 / 3.0
+ sigt = 1.0
+ eps = 1e-3
+ r_fine = np.linspace(0.0 + eps, 1.0 - eps, 100)
+ exact_phi = np.zeros_like(r_fine)
+ for i, r in enumerate(r_fine):
+ exact_phi[i] = get_phi(r, psi_0, a, b, sigt)
+ print(r_fine[i], exact_phi[i])
+
+ import matplotlib.pyplot as plt
+
+ plt.figure()
+ plt.plot(r_fine, exact_phi, label='Exact')
+ plt.step(
+ r_vals, np.append(avg_flx, avg_flx[-1]),
+ where='post',
+ label='Radial Average Flux in Spherical Shells'
+ )
+ plt.xlabel("Radius")
+ plt.ylabel("Average Flux")
+ plt.title("Flux as a function of radius")
+ plt.grid(True)
+ plt.legend()
+ plt.savefig("Problem_5.png", dpi=300, bbox_inches='tight')
+ plt.show()
diff --git a/Six_1g_spherical_benchmarks/Problem_6.png b/Six_1g_spherical_benchmarks/Problem_6.png
new file mode 100644
index 0000000..da57a52
Binary files /dev/null and b/Six_1g_spherical_benchmarks/Problem_6.png differ
diff --git a/Six_1g_spherical_benchmarks/Problem_6.py b/Six_1g_spherical_benchmarks/Problem_6.py
new file mode 100644
index 0000000..d62c24d
--- /dev/null
+++ b/Six_1g_spherical_benchmarks/Problem_6.py
@@ -0,0 +1,176 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+"""
+Problem 6
+Chang, B. (2021a, February 19).
+Six 1D polar transport test problems for the deterministic and monte-carlo method.
+LLNL-TR-819680
+https://www.osti.gov/servlets/purl/1769096
+"""
+
+import sys
+import os
+import numpy as np
+
+if "opensn_console" not in globals():
+ from mpi4py import MPI
+ size = MPI.COMM_WORLD.size
+ rank = MPI.COMM_WORLD.rank
+ sys.path.append(os.path.abspath(os.path.join(os.path.dirname(__file__), "../../../../../")))
+ from pyopensn.mesh import FromFileMeshGenerator, PETScGraphPartitioner
+ from pyopensn.xs import MultiGroupXS
+ from pyopensn.aquad import GLCProductQuadrature3DXYZ
+ from pyopensn.solver import DiscreteOrdinatesProblem, SteadyStateSolver
+ from pyopensn.fieldfunc import FieldFunctionInterpolationVolume
+ from pyopensn.logvol import SphereLogicalVolume, BooleanLogicalVolume
+ from pyopensn.source import VolumetricSource
+
+if __name__ == "__main__":
+
+ meshgen = FromFileMeshGenerator(
+ filename="./meshes/two_region_sphere.msh",
+ partitioner=PETScGraphPartitioner(type='parmetis'),
+ )
+ grid = meshgen.Execute()
+
+ # Create XS
+ num_groups = 1
+ xs_void = MultiGroupXS()
+ xs_void.CreateSimpleOneGroup(sigma_t=0.0, c=0.0)
+ xs_mat = MultiGroupXS()
+ xs_mat.CreateSimpleOneGroup(sigma_t=1.0, c=0.0)
+
+ # Volumetric source
+ mg_src = VolumetricSource(block_ids=[1], group_strength=[1.0])
+
+ # Angular quadrature
+ nazimu = 4
+ npolar = 2
+ pquad = GLCProductQuadrature3DXYZ(npolar, nazimu)
+
+ # Solver
+ phys = DiscreteOrdinatesProblem(
+ mesh=grid,
+ num_groups=num_groups,
+ groupsets=[
+ {
+ "groups_from_to": (0, num_groups - 1),
+ "angular_quadrature": pquad,
+ "inner_linear_method": "petsc_gmres",
+ "angle_aggregation_type": "single",
+ "angle_aggregation_num_subsets": 1,
+ "l_max_its": 500,
+ "l_abs_tol": 1.0e-6,
+ "gmres_restart_interval": 30,
+ },
+ ],
+ xs_map=[
+ {"block_ids": [1], "xs": xs_mat},
+ {"block_ids": [2], "xs": xs_void},
+ ],
+ options={
+ "scattering_order": 0,
+ "volumetric_sources": [mg_src],
+ "boundary_conditions": [
+ {"name": "xmin", "type": "vacuum"},
+ {"name": "xmax", "type": "vacuum"},
+ {"name": "ymin", "type": "vacuum"},
+ {"name": "ymax", "type": "vacuum"},
+ {"name": "zmin", "type": "vacuum"},
+ {"name": "zmax", "type": "vacuum"},
+ ],
+ },
+ )
+
+ ss_solver = SteadyStateSolver(lbs_problem=phys)
+ ss_solver.Initialize()
+ ss_solver.Execute()
+
+ # Export
+ fflist = phys.GetFieldFunctions()
+
+ # Compute the average flux over a logical volume
+ def average_flx_logvol(logvol):
+ ffvol = FieldFunctionInterpolationVolume()
+ ffvol.SetOperationType("avg")
+ ffvol.SetLogicalVolume(logvol)
+ ffvol.AddFieldFunction(fflist[0])
+ ffvol.Initialize()
+ ffvol.Execute()
+ avgval = ffvol.GetValue()
+ return avgval
+
+ # Compute the average flux over spherical shells
+ def compute_average_flux(N_logvols, r_max):
+ r_vals = np.linspace(0, r_max, N_logvols + 1)
+ avg_flx = np.zeros(N_logvols)
+ for i in range(N_logvols):
+ if i != 0:
+ inner_vol = SphereLogicalVolume(r=r_vals[i])
+ outer_vol = SphereLogicalVolume(r=r_vals[i + 1])
+ logvol = BooleanLogicalVolume(parts=[
+ {"op": True, "lv": outer_vol},
+ {"op": False, "lv": inner_vol},
+ ])
+ else:
+ logvol = SphereLogicalVolume(r=r_vals[i + 1])
+ avg_flx[i] = average_flx_logvol(logvol)
+ return r_vals, avg_flx
+
+ # Perform the calculation
+ N_logvols = 15
+ r_vals, avg_flx = compute_average_flux(N_logvols, 1)
+ if rank == 0:
+ for r, v in zip(r_vals[1:], avg_flx):
+ print("end-radius: {:.2f}, avg-value: {:.6f}".format(r, v))
+
+ # Compute the analytical answer
+ from scipy.special import exp1
+ from scipy.integrate import quad
+
+ def E2(x):
+ return np.exp(-x) - x * exp1(x)
+
+ def get_phi(r, q, a, sig):
+ if r <= a:
+ term1 = (np.exp(-sig * (a - r)) - np.exp(-sig * (a + r))) / sig
+ term2 = (a + r) * E2(sig * (a - r))
+ term3 = (a - r) * E2(sig * (a + r))
+ phi = q * (2 - (1 / (2 * r)) * (term1 + term2 - term3))
+ else:
+ a2 = a ** 2
+ r2 = r ** 2
+
+ def integrand(x):
+ return np.exp(-2 * sig * np.sqrt(a2 - r2 * (1 - x ** 2)))
+
+ term = quad(integrand, np.sqrt(1 - a2 / r2), 1)[0]
+ phi = q * (1 - np.sqrt(1 - (a2 / r2)) - term)
+ return phi
+
+ if rank == 0:
+ q = 0.5
+ a = 0.5
+ sigt = 1.0
+ eps = 1e-3
+ r_fine = np.linspace(0.0 + eps, 1.0 - eps, 100)
+ exact_phi = np.zeros_like(r_fine)
+ for i, r in enumerate(r_fine):
+ exact_phi[i] = get_phi(r, q, a, sigt)
+
+ import matplotlib.pyplot as plt
+
+ plt.figure()
+ plt.plot(r_fine, exact_phi, label='Exact')
+ plt.step(
+ r_vals, np.append(avg_flx, avg_flx[-1]),
+ where='post',
+ label='Radial Average Flux in Spherical Shells'
+ )
+ plt.xlabel("Radius")
+ plt.ylabel("Average Flux")
+ plt.title("Flux as a function of radius")
+ plt.grid(True)
+ plt.legend()
+ plt.savefig("Problem_6.png", dpi=300, bbox_inches='tight')
+ plt.show()
diff --git a/Six_1g_spherical_benchmarks/README.md b/Six_1g_spherical_benchmarks/README.md
new file mode 100644
index 0000000..94d9b56
--- /dev/null
+++ b/Six_1g_spherical_benchmarks/README.md
@@ -0,0 +1,94 @@
+## Six 1-group Spherical Benchmarks
+
+This is a suite of 6 one-group spherical benchmarks taken from the publicly available LLNL technical report:
+
+> Chang, B. (2021)
+> *Six 1D Polar Transport Test Problems for the Deterministic and Monte-Carlo Method*
+> LLNL-TR-819680
+> https://www.osti.gov/servlets/purl/1769096
+
+Each problem involves angular transport in 1D spherical geometry and is solved here using 3D tetrahedral meshes in Gmsh, as a verification exercise. Although exact 1D solutions exist, this approach exercises a general-purpose 3D solver's ability to reproduce spherically symmetric transport behavior.
+
+The scalar flux $\phi(r)$ is computed as the angular integral of the angular flux $\psi(r, \mu)$, and is post-processed using concentric logical spherical shells. The numerical solutions are therefore piecewise-constant averages over these shells.
+
+The exact analytical scalar flux is also plotted on each figure.
+
+---
+
+### Problem 1: Interior Dirichlet
+
+This test problem models angular flux $\psi_0$ injected uniformly at the outer boundary of a sphere of radius $r_0$, propagating inward through a homogeneous, purely absorbing medium with constant cross-section $\sigma$. The analytic solution is derived from the method of characteristics and integrated to yield the scalar flux $\phi(r)$, involving exponential integral functions.
+
+The Python OpenSn input file for this example can be found here: [Problem_1.py](./Problem_1.py).
+
+
+ 
+ Solution for Problem-1 (Interior Dirichlet)
+
+
+---
+
+### Problem 2: Radiating Ball
+
+This test describes a spherical source of radius $a$ emitting particles into a vacuum. The scalar flux profile has two regions: inside the ball (source active) and outside the ball (free streaming of escaped flux). The benchmark exercises both source handling and angular cutoff effects in vacuum. In this example, $a=0.5$.
+
+The Python OpenSn input file for this example can be found here: [Problem_2.py](./Problem_2.py).
+
+
+ 
+ Solution for Problem-2 (Radiating Ball)
+
+
+---
+
+### Problem 3: Radiating Shell
+
+Here, a spherical shell located between radii $a$ and $b$ emits radiation into a surrounding vacuum. The scalar flux solution combines two contributions: one from a full radiating ball of radius $b$, and another subtracted term representing a non-emitting core of radius $a$. This benchmark checks multiple-region source handling.
+In this example, $a=1/3$ and $b=2/3$.
+
+The Python OpenSn input file for this example can be found here: [Problem_3.py](./Problem_3.py).
+
+
+ 
+ Solution for Problem-3 (Radiating Shell)
+
+
+---
+
+### Problem 4: Pacman
+
+This test features a spherical absorber embedded in a vacuum. The incoming boundary flux is attenuated only inside a smaller radius $a$, where absorption occurs. The scalar flux solution includes a closed-form expression inside the absorber and a semi-analytic expression outside, showing a sharp transition near $r = a$. In this example, $a=0.5$.
+
+The Python OpenSn input file for this example can be found here: [Problem_4.py](./Problem_4.py).
+
+
+ 
+ Solution for Problem-4 (Pacman)
+
+
+---
+
+### Problem 5: Barrier
+
+This benchmark simulates shielding by a shell located between $r = a$ and $r = b$, surrounded by vacuum on both sides. An incoming flux is partially absorbed by the shell before reaching the center, then reflected outward. This problem exercises both inward and outward transport with attenuation.
+In this example, $a=1/3$ and $b=2/3$.
+
+The Python OpenSn input file for this example can be found here: [Problem_5.py](./Problem_5.py).
+
+
+ 
+ Solution for Problem-5 (Barrier)
+
+
+---
+
+### Problem 6: Radiating Absorber
+
+In this final test, a uniformly emitting sphere of radius $a$ radiates into surrounding vacuum, but also absorbs internally. The scalar flux is derived using both analytic and semi-analytic expressions. This test evaluates internal source + attenuation behavior and is useful for checking accuracy of source treatment in absorptive regions. In this example, $a=0.5$.
+
+The Python OpenSn input file for this example can be found here: [Problem_6.py](./Problem_6.py).
+
+
+ 
+ Solution for Problem-6 (Radiating Absorber)
+
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diff --git a/Six_1g_spherical_benchmarks/meshes/three_region_sphere.msh b/Six_1g_spherical_benchmarks/meshes/three_region_sphere.msh
new file mode 100644
index 0000000..44386b5
--- /dev/null
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