This page was generated from unit-1.1-poisson/poisson.ipynb.
1.1 First NGSolve example¶
Let us solve the Poisson problem of finding \(u\) satisfying
\[\begin{split}\begin{aligned}
-\Delta u & = f && \text { in the unit square},
\\
u & = 0 && \text{ on the bottom and right parts of the boundary},
\\
\frac{\partial u }{\partial n } & = 0
&& \text{ on the remaining boundary parts}.
\end{aligned}\end{split}\]
Quick steps to solution:¶
1. Import NGSolve and Netgen Python modules:¶
[1]:
from ngsolve import *
from ngsolve.webgui import Draw
from netgen.geom2d import unit_square
2. Generate an unstructured mesh¶
[2]:
mesh = Mesh(unit_square.GenerateMesh(maxh=0.2))
mesh.nv, mesh.ne # number of vertices & elements
[2]:
(38, 54)
Here we prescribed a maximal mesh-size of 0.2 using the
maxhflag.The mesh can be viewed by switching to the
Meshtab in the Netgen GUI.
3. Declare a finite element space:¶
[3]:
fes = H1(mesh, order=2, dirichlet="bottom|right")
fes.ndof # number of unknowns in this space
[3]:
129
Python’s help system displays further documentation.
[4]:
help(fes)
Help on H1 in module ngsolve.comp object:
class H1(FESpace)
| An H1-conforming finite element space.
|
| The H1 finite element space consists of continuous and
| element-wise polynomial functions. It uses a hierarchical (=modal)
| basis built from integrated Legendre polynomials on tensor-product elements,
| and Jaboci polynomials on simplicial elements.
|
| Boundary values are well defined. The function can be used directly on the
| boundary, using the trace operator is optional.
|
| The H1 space supports variable order, which can be set individually for edges,
| faces and cells.
|
| Internal degrees of freedom are declared as local dofs and are eliminated
| if static condensation is on.
|
| The wirebasket consists of all vertex dofs. Optionally, one can include the
| first (the quadratic bubble) edge basis function, or all edge basis functions
| into the wirebasket.
|
| Keyword arguments can be:
| order: int = 1
| order of finite element space
| complex: bool = False
| Set if FESpace should be complex
| dirichlet: regexpr
| Regular expression string defining the dirichlet boundary.
| More than one boundary can be combined by the | operator,
| i.e.: dirichlet = 'top|right'
| dirichlet_bbnd: regexpr
| Regular expression string defining the dirichlet bboundary,
| i.e. points in 2D and edges in 3D.
| More than one boundary can be combined by the | operator,
| i.e.: dirichlet_bbnd = 'top|right'
| dirichlet_bbbnd: regexpr
| Regular expression string defining the dirichlet bbboundary,
| i.e. points in 3D.
| More than one boundary can be combined by the | operator,
| i.e.: dirichlet_bbbnd = 'top|right'
| definedon: Region or regexpr
| FESpace is only defined on specific Region, created with mesh.Materials('regexpr')
| or mesh.Boundaries('regexpr'). If given a regexpr, the region is assumed to be
| mesh.Materials('regexpr').
| dim: int = 1
| Create multi dimensional FESpace (i.e. [H1]^3)
| dgjumps: bool = False
| Enable discontinuous space for DG methods, this flag is needed for DG methods,
| since the dofs have a different coupling then and this changes the sparsity
| pattern of matrices.
| low_order_space: bool = True
| Generate a lowest order space together with the high-order space,
| needed for some preconditioners.
| order_policy: ORDER_POLICY = ORDER_POLICY.OLDSTYLE
| CONSTANT .. use the same fixed order for all elements,
| NODAL ..... use the same order for nodes of same shape,
| VARIABLE ... use an individual order for each edge, face and cell,
| OLDSTYLE .. as it used to be for the last decade
| wb_withedges: bool = true(3D) / false(2D)
| use lowest-order edge dofs for BDDC wirebasket
| wb_fulledges: bool = false
| use all edge dofs for BDDC wirebasket
|
| Method resolution order:
| H1
| FESpace
| NGS_Object
| pybind11_builtins.pybind11_object
| builtins.object
|
| Methods defined here:
|
| __getstate__(...)
| __getstate__(self: ngsolve.comp.FESpace) -> tuple
|
| __init__(...)
| __init__(self: ngsolve.comp.H1, mesh: ngsolve.comp.Mesh, autoupdate: bool = False, **kwargs) -> None
|
| __setstate__(...)
| __setstate__(self: ngsolve.comp.H1, arg0: tuple) -> None
|
| ----------------------------------------------------------------------
| Static methods defined here:
|
| __flags_doc__(...) from builtins.PyCapsule
| __flags_doc__() -> dict
|
| ----------------------------------------------------------------------
| Data descriptors defined here:
|
| __dict__
|
| ----------------------------------------------------------------------
| Methods inherited from FESpace:
|
| ApplyM(...)
| ApplyM(self: ngsolve.comp.FESpace, vec: ngsolve.la.BaseVector, rho: ngsolve.fem.CoefficientFunction = None, definedon: ngsolve.comp.Region = None) -> None
|
| Apply mass-matrix. Available only for L2-like spaces
|
| ConvertL2Operator(...)
| ConvertL2Operator(self: ngsolve.comp.FESpace, l2space: ngsolve.comp.FESpace) -> BaseMatrix
|
| CouplingType(...)
| CouplingType(self: ngsolve.comp.FESpace, dofnr: int) -> ngsolve.comp.COUPLING_TYPE
|
|
| Get coupling type of a degree of freedom.
|
| Parameters:
|
| dofnr : int
| input dof number
|
| CreateDirectSolverCluster(...)
| CreateDirectSolverCluster(self: ngsolve.comp.FESpace, **kwargs) -> list
|
| Elements(...)
| Elements(self: ngsolve.comp.FESpace, VOL_or_BND: ngsolve.comp.VorB = <VorB.VOL: 0>) -> ngsolve.comp.FESpaceElementRange
|
|
| Returns an iterable range of elements.
|
| Parameters:
|
| VOL_or_BND : ngsolve.comp.VorB
| input VOL, BND, BBND,...
|
| FinalizeUpdate(...)
| FinalizeUpdate(self: ngsolve.comp.FESpace) -> None
|
| finalize update
|
| FreeDofs(...)
| FreeDofs(self: ngsolve.comp.FESpace, coupling: bool = False) -> pyngcore.BitArray
|
|
|
| Return BitArray of free (non-Dirichlet) dofs\n
| coupling=False ... all free dofs including local dofs\n
| coupling=True .... only element-boundary free dofs
|
| Parameters:
|
| coupling : bool
| input coupling
|
| GetDofNrs(...)
| GetDofNrs(*args, **kwargs)
| Overloaded function.
|
| 1. GetDofNrs(self: ngsolve.comp.FESpace, ei: ngsolve.comp.ElementId) -> tuple
|
|
|
| Parameters:
|
| ei : ngsolve.comp.ElementId
| input element id
|
|
|
| 2. GetDofNrs(self: ngsolve.comp.FESpace, ni: ngsolve.comp.NodeId) -> tuple
|
|
|
| Parameters:
|
| ni : ngsolve.comp.NodeId
| input node id
|
| GetDofs(...)
| GetDofs(self: ngsolve.comp.FESpace, region: ngsolve.comp.Region) -> pyngcore.BitArray
|
|
| Returns all degrees of freedom in given region.
|
| Parameters:
|
| region : ngsolve.comp.Region
| input region
|
| GetFE(...)
| GetFE(self: ngsolve.comp.FESpace, ei: ngsolve.comp.ElementId) -> object
|
|
| Get the finite element to corresponding element id.
|
| Parameters:
|
| ei : ngsolve.comp.ElementId
| input element id
|
| GetOrder(...)
| GetOrder(self: ngsolve.comp.FESpace, nodeid: ngsolve.comp.NodeId) -> int
|
| return order of node.
| by now, only isotropic order is supported here
|
| GetTrace(...)
| GetTrace(self: ngsolve.comp.FESpace, arg0: ngsolve.comp.FESpace, arg1: ngsolve.la.BaseVector, arg2: ngsolve.la.BaseVector, arg3: bool) -> None
|
| GetTraceTrans(...)
| GetTraceTrans(self: ngsolve.comp.FESpace, arg0: ngsolve.comp.FESpace, arg1: ngsolve.la.BaseVector, arg2: ngsolve.la.BaseVector, arg3: bool) -> None
|
| HideAllDofs(...)
| HideAllDofs(self: ngsolve.comp.FESpace, component: object = <ngsolve.ngstd.DummyArgument>) -> None
|
| set all visible coupling types to HIDDEN_DOFs (will be overwritten by any Update())
|
| InvM(...)
| InvM(self: ngsolve.comp.FESpace, rho: ngsolve.fem.CoefficientFunction = None) -> BaseMatrix
|
| Mass(...)
| Mass(self: ngsolve.comp.FESpace, rho: ngsolve.fem.CoefficientFunction = None, definedon: Optional[ngsolve.comp.Region] = None) -> BaseMatrix
|
| ParallelDofs(...)
| ParallelDofs(self: ngsolve.comp.FESpace) -> ngsolve.la.ParallelDofs
|
| Return dof-identification for MPI-distributed meshes
|
| Prolongation(...)
| Prolongation(self: ngsolve.comp.FESpace) -> ngmg::Prolongation
|
| Return prolongation operator for use in multi-grid
|
| Range(...)
| Range(self: ngsolve.comp.FESpace, arg0: int) -> ngsolve.la.DofRange
|
| deprecated, will be only available for ProductSpace
|
| SetCouplingType(...)
| SetCouplingType(*args, **kwargs)
| Overloaded function.
|
| 1. SetCouplingType(self: ngsolve.comp.FESpace, dofnr: int, coupling_type: ngsolve.comp.COUPLING_TYPE) -> None
|
|
| Set coupling type of a degree of freedom.
|
| Parameters:
|
| dofnr : int
| input dof number
|
| coupling_type : ngsolve.comp.COUPLING_TYPE
| input coupling type
|
|
|
| 2. SetCouplingType(self: ngsolve.comp.FESpace, dofnrs: ngsolve.ngstd.IntRange, coupling_type: ngsolve.comp.COUPLING_TYPE) -> None
|
|
| Set coupling type for interval of dofs.
|
| Parameters:
|
| dofnrs : Range
| range of dofs
|
| coupling_type : ngsolve.comp.COUPLING_TYPE
| input coupling type
|
| SetDefinedOn(...)
| SetDefinedOn(self: ngsolve.comp.FESpace, region: ngsolve.comp.Region) -> None
|
|
| Set the regions on which the FESpace is defined.
|
| Parameters:
|
| region : ngsolve.comp.Region
| input region
|
| SetOrder(...)
| SetOrder(*args, **kwargs)
| Overloaded function.
|
| 1. SetOrder(self: ngsolve.comp.FESpace, element_type: ngsolve.fem.ET, order: int) -> None
|
|
|
| Parameters:
|
| element_type : ngsolve.fem.ET
| input element type
|
| order : object
| input polynomial order
|
|
| 2. SetOrder(self: ngsolve.comp.FESpace, nodeid: ngsolve.comp.NodeId, order: int) -> None
|
|
|
| Parameters:
|
| nodeid : ngsolve.comp.NodeId
| input node id
|
| order : int
| input polynomial order
|
| SolveM(...)
| SolveM(self: ngsolve.comp.FESpace, vec: ngsolve.la.BaseVector, rho: ngsolve.fem.CoefficientFunction = None, definedon: ngsolve.comp.Region = None) -> None
|
|
| Solve with the mass-matrix. Available only for L2-like spaces.
|
| Parameters:
|
| vec : ngsolve.la.BaseVector
| input right hand side vector
|
| rho : ngsolve.fem.CoefficientFunction
| input CF
|
| TestFunction(...)
| TestFunction(self: ngsolve.comp.FESpace) -> object
|
| Return a proxy to be used as a testfunction for :any:`Symbolic Integrators<symbolic-integrators>`
|
| TnT(...)
| TnT(self: ngsolve.comp.FESpace) -> Tuple[object, object]
|
| Return a tuple of trial and testfunction
|
| TraceOperator(...)
| TraceOperator(self: ngsolve.comp.FESpace, tracespace: ngsolve.comp.FESpace, average: bool) -> BaseMatrix
|
| TrialFunction(...)
| TrialFunction(self: ngsolve.comp.FESpace) -> object
|
| Return a proxy to be used as a trialfunction in :any:`Symbolic Integrators<symbolic-integrators>`
|
| Update(...)
| Update(self: ngsolve.comp.FESpace) -> None
|
| update space after mesh-refinement
|
| UpdateDofTables(...)
| UpdateDofTables(self: ngsolve.comp.FESpace) -> None
|
| update dof-tables after changing polynomial order distribution
|
| __eq__(...)
| __eq__(self: ngsolve.comp.FESpace, space: ngsolve.comp.FESpace) -> bool
|
| __mul__(...)
| __mul__(self: ngsolve.comp.FESpace, arg0: ngsolve.comp.FESpace) -> ngcomp::CompoundFESpace
|
| __pow__(...)
| __pow__(self: ngsolve.comp.FESpace, arg0: int) -> ngcomp::CompoundFESpaceAllSame
|
| __str__(...)
| __str__(self: ngsolve.comp.FESpace) -> str
|
| __timing__(...)
| __timing__(self: ngsolve.comp.FESpace) -> object
|
| ----------------------------------------------------------------------
| Static methods inherited from FESpace:
|
| __special_treated_flags__(...) from builtins.PyCapsule
| __special_treated_flags__() -> dict
|
| ----------------------------------------------------------------------
| Readonly properties inherited from FESpace:
|
| components
| deprecated, will be only available for ProductSpace
|
| couplingtype
|
| dim
| multi-dim of FESpace
|
| globalorder
| query global order of space
|
| is_complex
|
| loembedding
|
| lospace
|
| mesh
| mesh on which the FESpace is created
|
| ndof
| number of degrees of freedom
|
| ndofglobal
| global number of dofs on MPI-distributed mesh
|
| type
| type of finite element space
|
| ----------------------------------------------------------------------
| Data and other attributes inherited from FESpace:
|
| __hash__ = None
|
| ----------------------------------------------------------------------
| Readonly properties inherited from NGS_Object:
|
| __memory__
|
| ----------------------------------------------------------------------
| Data descriptors inherited from NGS_Object:
|
| name
|
| ----------------------------------------------------------------------
| Static methods inherited from pybind11_builtins.pybind11_object:
|
| __new__(*args, **kwargs) from pybind11_builtins.pybind11_type
| Create and return a new object. See help(type) for accurate signature.
4. Declare test function, trial function, and grid function¶
Test and trial function are symbolic objects - called
ProxyFunctions- that help you construct bilinear forms (and have no space to hold solutions).GridFunctions, on the other hand, represent functions in the finite element space and contains memory to hold coefficient vectors.
[5]:
u = fes.TrialFunction() # symbolic object
v = fes.TestFunction() # symbolic object
gfu = GridFunction(fes) # solution
Alternately, you can get both the trial and test variables at once:
[6]:
u, v = fes.TnT()
5. Define and assemble linear and bilinear forms:¶
[7]:
a = BilinearForm(fes, symmetric=True)
a += grad(u)*grad(v)*dx
a.Assemble()
f = LinearForm(fes)
f += x*v*dx
f.Assemble()
[7]:
<ngsolve.comp.LinearForm at 0x7f79dcaa2c70>
You can examine the linear system in more detail:
[8]:
print(f.vec)
0.000333333
0.00888667
0.00633333
0.000772552
0.00388886
0.00745593
0.0102799
0.0115007
0.0150668
0.0169176
0.0154171
0.0183849
0.0212923
0.0109064
0.00750505
0.00352697
0.000376105
0.00262832
0.00051516
0.00266077
0.0117876
0.0178168
0.0223235
0.0204184
0.0318376
0.0272957
0.029483
0.0190557
0.0227156
0.00905798
0.00529671
0.00779569
0.020463
0.0157845
0.0216524
0.0157801
0.0172011
0.019586
-6.66667e-05
-3.33333e-05
-0.000525777
-0.000575773
-0.00109151
-0.0008
-0.000766667
-9.18288e-05
-1.88053e-05
-0.000121132
-0.000238755
-0.000215067
-0.000426062
-0.000372909
-0.000551384
-0.000691744
-0.000485002
-0.000796725
-0.000909071
-0.000911696
-0.000942778
-0.000763411
-0.00110769
-0.0012761
-0.000653931
-0.00142763
-0.00133086
-0.000654795
-0.00125029
-0.00125106
-0.0016032
-0.00147642
-0.000475024
-0.00142506
-0.0010807
-0.00045191
-0.000903122
-0.000886228
-0.000204734
-0.00073304
-0.000494315
-0.00034316
-0.000246364
-1.88053e-05
-7.52211e-05
-2.5758e-05
-0.000173161
-0.000213128
-0.000357643
-2.5758e-05
-0.000103032
-0.000329228
-0.000194844
-0.00064017
-0.000422578
-0.000560041
-0.000834752
-0.000785222
-0.00070094
-0.000921418
-0.00105231
-0.000949261
-0.00104196
-0.00117992
-0.00101029
-0.000970793
-0.00116359
-0.000974281
-0.000925954
-0.00105527
-0.00099741
-0.000847599
-0.00087727
-0.000615988
-0.000891989
-0.000762229
-0.000931528
-0.000320916
-0.000490049
-0.0003756
-0.000511811
-0.000491896
-0.000762552
-0.000835836
-0.000773654
-0.000642566
-0.00076271
-0.00081234
-0.000855256
-0.000808305
-0.000755427
[9]:
print(a.mat)
Row 0: 0: 1 4: -0.5 19: -0.5 38: -0.0833333 39: -0.0833333 49: 0.166667
Row 1: 1: 0.828444 7: -0.208707 8: -0.210132 23: -0.409604 40: -0.034013 41: -0.0342543 42: -0.0698066 58: 0.0687976 60: 0.0692764
Row 2: 2: 1 11: -0.5 12: -0.5 43: -0.0833333 44: -0.0833333 68: 0.166667
Row 3: 3: 0.907169 15: -0.407338 16: -0.376463 30: -0.123368 45: -0.020713 46: 0.000151649 47: -0.130634 80: 0.0886026 82: 0.0625923
Row 4: 0: -0.5 4: 1.8918 5: -0.369703 19: -0.170695 20: -0.851406 38: 0 39: 0.0833333 48: -0.0553087 49: -0.169926 50: -0.0900664 52: 0.116926 89: 0.115041
Row 5: 4: -0.369703 5: 1.76479 6: -0.315355 20: -0.373272 21: -0.70646 48: -0.0302454 50: 0.0918626 51: -0.0537908 52: -0.12557 53: -0.0845258 55: 0.10635 91: 0.0959192
Row 6: 5: -0.315355 6: 1.74934 7: -0.271622 21: -0.444807 22: -0.71756 51: -0.0387141 53: 0.0912733 54: -0.0577312 55: -0.114421 56: -0.0806907 57: 0.103002 94: 0.0972825
Row 7: 1: -0.208707 6: -0.271622 7: 1.82575 22: -0.375475 23: -0.969947 40: -0.0837431 42: 0.118528 54: -0.0420473 56: 0.0873176 57: -0.123185 58: -0.0553164 97: 0.0984466
Row 8: 1: -0.210132 8: 1.88178 9: -0.354044 23: -1.05651 24: -0.261086 41: -0.0829256 42: 0.117948 59: -0.0322254 60: -0.0463111 61: -0.152168 63: 0.0912327 100: 0.104449
Row 9: 8: -0.354044 9: 1.7442 10: -0.268795 24: -0.640588 25: -0.480773 59: -0.0552752 61: 0.114283 62: -0.0341102 63: -0.105026 64: -0.0962885 66: 0.0789093 101: 0.0975081
Row 10: 9: -0.268795 10: 1.77359 11: -0.272339 25: -0.832083 26: -0.400373 62: -0.0686401 64: 0.113439 65: -0.0327251 66: -0.0788029 67: -0.11543 69: 0.078115 104: 0.104044
Row 11: 2: -0.5 10: -0.272339 11: 1.97625 12: -0.118504 26: -1.0854 43: 0 44: 0.0833333 65: -0.0698849 67: 0.115275 68: -0.194349 69: -0.0651405 71: 0.130766
Row 12: 2: -0.5 11: -0.118504 12: 1.80573 13: -0.26228 26: -0.248259 27: -0.676685 43: 0.0833333 44: 0 68: -0.119671 69: 0.0560888 70: -0.0411842 71: -0.0913473 72: -0.0487517 74: 0.0848976 107: 0.0766349
Row 13: 12: -0.26228 13: 1.81175 14: -0.428891 27: -0.813095 28: -0.307486 70: -0.0605924 72: 0.104306 73: -0.0361696 74: -0.0587915 75: -0.146405 77: 0.107651 109: 0.0900017
Row 14: 13: -0.428891 14: 1.79855 15: -0.254586 28: -0.343721 29: -0.771353 73: -0.0404916 75: 0.111973 76: -0.0578574 77: -0.142183 78: -0.0592263 79: 0.100288 111: 0.0874967
Row 15: 3: -0.407338 14: -0.254586 15: 1.75967 29: -0.52416 30: -0.573581 45: -0.0625066 47: 0.130396 76: -0.0447659 78: 0.0871969 79: -0.0755213 80: -0.110484 115: 0.0756844
Row 16: 3: -0.376463 16: 2.08248 17: -0.374644 30: -1.33137 46: -0.111099 47: 0.173843 81: -0.110796 82: -0.125185 84: 0.173237
Row 17: 16: -0.374644 17: 1.78398 18: -0.43722 30: -0.336404 31: -0.30274 35: -0.332973 81: -0.000151649 82: 0.0625923 83: -0.000386213 84: -0.0860908 85: -0.104715 86: -0.105986 88: 0.0732563 117: 0.0795659 119: 0.0819158
Row 18: 17: -0.43722 18: 2.01666 19: -0.441855 31: -1.13758 83: -0.0944124 85: 0.167282 87: -0.0951848 88: -0.146513 90: 0.168827
Row 19: 0: -0.5 4: -0.170695 18: -0.441855 19: 1.80929 20: -0.457008 31: -0.239727 38: 0.0833333 39: 0 49: -0.122284 50: 0.0674001 87: 0.000386213 88: 0.0732563 89: -0.0687899 90: -0.11086 92: 0.0775578
Row 20: 4: -0.851406 5: -0.373272 19: -0.457008 20: 3.55115 21: -0.621608 31: -0.660832 33: -0.587027 48: 0.0855542 49: 0.125543 50: -0.0691963 52: -0.106394 53: 0.0830523 89: -0.156773 90: 0.107398 91: -0.0811964 92: -0.0858253 93: -0.0924734 96: 0.101746 118: 0.0885658
Row 21: 5: -0.70646 6: -0.444807 20: -0.621608 21: 3.49846 22: -0.562802 32: -0.707916 33: -0.454863 51: 0.0925049 52: 0.115038 53: -0.0897997 55: -0.102651 56: 0.0842808 91: -0.106796 93: 0.095359 94: -0.0895218 95: -0.0779072 96: -0.1164 99: 0.0990414 120: 0.0968518
Row 22: 6: -0.71756 7: -0.375475 21: -0.562802 22: 3.51385 23: -0.705204 24: -0.539682 32: -0.613124 54: 0.0997784 55: 0.110723 56: -0.0909077 57: -0.112022 58: 0.0748223 94: -0.10743 95: 0.090508 97: -0.055892 98: -0.119863 99: -0.0995268 100: 0.0986036 102: 0.111206
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Row 73: 13: -0.0361696 14: -0.0404916 28: 0.0766612 73: 0.0370358 75: -0.0101229 77: -0.00904239
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Row 76: 14: -0.0578574 15: -0.0447659 29: 0.102623 76: 0.0362636 78: -0.0111915 79: -0.0144643
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Row 81: 16: -0.110796 17: -0.000151649 30: 0.110948 81: 0.0433471 82: -3.79121e-05 84: -0.027699
Row 82: 3: 0.0625923 16: -0.125185 17: 0.0625923 30: 0 46: 3.79121e-05 47: -0.015686 81: -3.79121e-05 82: 0.08677 84: -0.0156102
Row 83: 17: -0.000386213 18: -0.0944124 31: 0.0947986 83: 0.0419172 85: -0.0236031 88: -9.65532e-05
Row 84: 16: 0.173237 17: -0.0860908 30: -0.181455 35: 0.0943089 81: -0.027699 82: -0.0156102 84: 0.0809032 86: -0.0176647 117: -0.00591256
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Row 87: 18: -0.0951848 19: 0.000386213 31: 0.0947986 87: 0.0421103 88: 9.65532e-05 90: -0.0237962
Row 88: 17: 0.0732563 18: -0.146513 19: 0.0732563 31: 0 83: -9.65532e-05 85: -0.0182175 87: 9.65532e-05 88: 0.0840274 90: -0.0184106
Row 89: 4: 0.115041 19: -0.0687899 20: -0.156773 31: 0.110522 49: -0.021648 50: -0.0071123 89: 0.0754328 90: -0.0175452 92: -0.0100852
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Row 93: 20: -0.0924734 21: 0.095359 31: 0.0964375 33: -0.099323 91: -0.0107108 92: -0.0141199 93: 0.0724086 96: -0.0131289 118: -0.00998942
Row 94: 6: 0.0972825 21: -0.0895218 22: -0.10743 32: 0.0996696 55: -0.0154655 56: -0.00885509 94: 0.0726881 95: -0.011392 99: -0.0135254
Row 95: 21: -0.0779072 22: 0.090508 32: -0.0955243 33: 0.0829234 94: -0.011392 95: 0.0728544 96: -0.012489 99: -0.011235 120: -0.00824182
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Row 97: 7: 0.0984466 22: -0.055892 23: -0.125663 24: 0.0831084 57: -0.0194787 58: -0.00513297 97: 0.0747722 98: -0.0119371 100: -0.00884002
Row 98: 22: -0.119863 23: 0.110992 24: -0.0824026 32: 0.0912736 97: -0.0119371 98: 0.0730531 99: -0.00866359 100: -0.0158109 102: -0.0141548
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Row 105: 25: -0.113567 26: 0.105121 34: -0.119067 36: 0.127513 104: -0.0151484 105: 0.0739576 106: -0.0146185 108: -0.0111318 125: -0.0172599
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Row 107: 12: 0.0766349 26: -0.0517256 27: -0.122692 34: 0.0977827 71: -0.0178992 72: -0.00125957 107: 0.0767377 108: -0.0127739 110: -0.0116718
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Row 109: 13: 0.0900017 27: -0.140104 28: -0.0443342 34: 0.0944361 74: -0.00376953 75: -0.0187309 109: 0.075597 110: -0.00731402 112: -0.016295
Row 110: 26: 0.0933104 27: -0.0999664 28: 0.0825994 34: -0.0759434 107: -0.0116718 108: -0.0116558 109: -0.00731402 110: 0.0730463 112: -0.0133358
Row 111: 14: 0.0874967 28: -0.0325616 29: -0.127439 35: 0.0725036 77: -0.0176754 78: -0.00419882 111: 0.07731 113: -0.0141843 116: -0.00394157
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Row 113: 28: -0.137443 29: 0.14018 35: -0.117127 37: 0.114391 111: -0.0141843 113: 0.0756342 114: -0.0150976 116: -0.0208607 127: -0.0135001
Row 114: 28: -0.0818333 34: 0.116869 35: 0.0925899 37: -0.127626 112: -0.0168088 113: -0.0150976 114: 0.0735814 126: -0.0124084 127: -0.00804992
Row 115: 15: 0.0756844 29: -0.0798592 30: -0.0621923 35: 0.0663671 79: -0.00827257 80: -0.0106485 115: 0.07587 116: -0.00727552 117: -0.00931627
Row 116: 28: 0.0992092 29: -0.171739 30: 0.117398 35: -0.0448683 111: -0.00394157 113: -0.0208607 115: -0.00727552 116: 0.0776524 117: -0.022074
Row 117: 17: 0.0795659 29: 0.125561 30: -0.144212 35: -0.0609153 84: -0.00591256 86: -0.0139789 115: -0.00931627 116: -0.022074 117: 0.0762219
Row 118: 20: 0.0885658 31: -0.0828523 33: -0.0824973 35: 0.0767838 92: -0.012152 93: -0.00998942 118: 0.0733347 119: -0.0084723 123: -0.0107237
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Row 120: 21: 0.0968518 32: -0.104609 33: -0.0854633 37: 0.0932201 95: -0.00824182 96: -0.0159711 120: 0.0728975 122: -0.013124 124: -0.010181
Row 121: 24: 0.0726606 32: -0.0441856 36: -0.105576 37: 0.0771011 102: -0.0123468 103: -0.00581837 121: 0.0760864 122: -0.0140472 128: -0.00522804
Row 122: 32: -0.126391 33: 0.104058 36: 0.131018 37: -0.108685 120: -0.013124 121: -0.0140472 122: 0.074178 124: -0.0128904 128: -0.0187073
Row 123: 31: 0.114404 33: -0.14111 35: -0.0937792 37: 0.120485 118: -0.0107237 119: -0.0178774 123: 0.0742553 124: -0.0127212 127: -0.0174
Row 124: 32: 0.0922859 33: -0.0798048 35: 0.0791278 37: -0.0916088 120: -0.010181 122: -0.0128904 123: -0.0127212 124: 0.0733774 127: -0.00706078
Row 125: 25: 0.0918406 34: -0.0603016 36: -0.117284 37: 0.0857449 105: -0.0172599 106: -0.00570025 125: 0.0739873 126: -0.0120611 128: -0.00937515
Row 126: 28: 0.0804999 34: -0.109524 36: 0.108134 37: -0.0791105 112: -0.00771657 114: -0.0124084 125: -0.0120611 126: 0.0733425 128: -0.0149725
Row 127: 28: 0.0862002 33: 0.0978433 35: -0.0604428 37: -0.123601 113: -0.0135001 114: -0.00804992 123: -0.0174 124: -0.00706078 127: 0.0738296
Row 128: 32: 0.0957414 34: 0.0973906 36: -0.134719 37: -0.0584127 121: -0.00522804 122: -0.0187073 125: -0.00937515 126: -0.0149725 128: 0.0743913
6. Solve the system:¶
[10]:
gfu.vec.data = \
a.mat.Inverse(freedofs=fes.FreeDofs()) * f.vec
Draw(gfu)
[10]:
BaseWebGuiScene
The Dirichlet boundary condition constrains some degrees of freedom. The argument fes.FreeDofs() indicates that only the remaining “free” degrees of freedom should participate in the linear solve.
You can examine the coefficient vector of solution if needed:
[11]:
print(gfu.vec)
0
0
0
0.0923051
0
0
0
0
0
0
0
0
0.0578976
0.0863397
0.095413
0.0944941
0.088859
0.0779511
0.059639
0.0330669
0.0361131
0.0364559
0.0317036
0.0185143
0.0384776
0.0431703
0.0476901
0.0763835
0.0896404
0.0935756
0.0902834
0.0619423
0.05721
0.0634411
0.072785
0.0832449
0.0629249
0.0775277
0
-0.00576212
0
0
0.0208108
0
-0.0350834
0.00265432
-0.00721308
-0.00488766
0
-0.00992425
-0.00630809
0
-0.0163826
-0.0115972
0
-0.0146117
-0.019397
-0.00629918
-0.021289
0
-0.0255422
-0.00978494
0
-0.0385158
-0.0136376
0
-0.026807
-0.018579
-0.0333669
-0.024961
-0.0242793
-0.00449407
-0.0124615
-0.0145086
-0.00594865
-0.0117808
-0.00555001
-0.00890522
-0.00588952
-0.00539555
-0.00706629
-0.00787172
0.0025744
-0.00832323
-0.00587822
-0.00475479
0.00342728
-0.00825162
0.00390131
0.00421041
-0.00475943
-0.00353907
-0.00993744
-0.009953
-0.00708995
-0.0135307
-0.0114551
-0.0078331
-0.0254921
-0.0071086
-0.00325064
-0.0089176
-0.0210183
-0.00522333
-0.0030572
-0.0144307
-0.0182813
-0.0155115
-0.0215164
-0.0172692
-0.00414894
-0.00783019
-0.0132532
-0.016378
-0.00903592
-0.00194529
-0.00888682
-0.00574139
-0.00227548
-0.0105216
-0.00871044
-0.0118078
-0.00808258
-0.0113092
-0.0118318
-0.00416845
-0.0163461
-0.00951852
-0.0109196
Ways to interact with NGSolve¶
A jupyter notebook (like this one) gives you one way to interact with NGSolve. When you have a complex sequence of tasks to perform, the notebook may not be adequate.
You can write an entire python module in a text editor and call python on the command line. (A script of the above is provided in
poisson.py.)python3 poisson.pyIf you want the Netgen GUI, then use
netgenon the command line:netgen poisson.pyYou can then ask for a python shell from the GUI’s menu options (Solve -> Python shell).
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