[petsc-users] Additiional DoF per cell
Matthew Knepley
knepley at gmail.com
Fri May 8 14:12:55 CDT 2026
On Fri, May 8, 2026 at 2:54 PM Noam T. <dontbugthedevs at proton.me> wrote:
> Code attached.
>
I think you misunderstand the definition of the element. I do not think you
want 3 components for your pressure. You want 1 component. It is a scalar
variable. In my original example, I had a single component. Components
create tensor spaces.
If you are using discontinuous P1, then it has 3 unknowns, which is what I
think you are trying to specify. You should change the space you are using
if you want to change the number of variables.
Does that clear it up?
Thanks,
Matt
> Noticed from PetscSectionView:
>
> PetscSectionSym Object: 1 MPI process
> type: label
> Label 'depth'
> Symmetry for stratum value 0 (0 dofs per point): no symmetries
> Symmetry for stratum value 1 (0 dofs per point): no symmetries
> Symmetry for stratum value 2 (12 dofs per point):
> <---------------------- 12 DoF ?
> Orientation range: [-4, 4)
> Symmetry for stratum value -1 (0 dofs per point): no symmetries
>
> Thanks,
> Noam
>
> On Friday, May 8th, 2026 at 3:58 PM, Noam T. <dontbugthedevs at proton.me>
> wrote:
>
> This is likely to be a Section construction problem again. The Section
> fields are separated by DMPlexVecGetClosure(), sp you must not have defined
> the fields in the Section. If you put these calls into the code that I sent
> you, what do you get?
>
>
> If I use the exact same arrays/values for constructing the section, I get
> the same closure arrangement in the code you provided. Not surprising I
> guess? A seemingly random number of coordinates / zeroes / more coordinates.
>
> I checked example 14 (
> https://urldefense.us/v3/__https://petsc.org/release/src/dm/impls/plex/tutorials/ex14.c.html__;!!G_uCfscf7eWS!dBe0zmoP1EXuDC3Da6esPOb65iny79VkOAgA01sP84NE6TyW10KDbA-RSHT2uMS6hvXHo--Yav85WxiwA5OT$ ) for
> constructing the section. These are the values I used:
>
> - n_fields = 2
>
> - numDof [n_fields*(dim+1)] = numDof[2*(2+1)] = [2 0 0 0 0 N]
>
> The u field is defined in the nodes, with 2 components (u, v), hence the
> first "2". Then no components for edges/faces.
> For the p field, what I am trying to simulate is having nc (independent)
> DoF, sort of like having nc scalar fields, if that makes sense? Hence using:
>
> - N = nc
> - Also using the flag "-pres_petscspace_degree 0"
>
> But I'm not sure about those values.
>
> Then for simplicity numBC = 0 and all other arguments are NULL (will deal
> with boundary conditions later).
>
> With these arguments, I get the same as before e.g.
>
> -- nc = 3 (total dimension = 11)
> closure = y1 x2 y2 x3 y3 0 0 0 x0 y0 x1
>
> Thanks,
> Noam
>
>
> On Friday, May 8th, 2026 at 1:25 PM, Matthew Knepley <knepley at gmail.com>
> wrote:
>
> On Fri, May 8, 2026 at 8:01 AM Noam T. <dontbugthedevs at proton.me> wrote:
>
>> Moving on to the values of the closure.
>>
>> Initially I create a vector for the DM with DMCreateLocal/GlobalVector,
>> which is used for e.g. the solution. Following the example before, it will
>> have size 8 (coordinates) + nc (for the p field).
>>
>> I am getting a seemingly random order when I test with different values
>> of nc. Say, I initialize the vector (VecSetValues) with the coordinates
>> x_i, y_i . The rest nc components are set to zero. The closure returns:
>>
>> -- nc = 1
>> closure = y0 x1 y1 x2 y2 x3 y3 0 x0
>>
>> -- nc = 3
>> closure = y1 x2 y2 x3 y3 0 0 0 x0 y0 x1
>>
>> -- nc = 7
>> closure = y3 0 0 0 0 0 0 0 x0 y0 x1 y1 x2 y2
>>
>> Looking at that output:
>> - The first value is always a y coordinate (second component of u field).
>> - The position of the nc = 0 components change (starting with nc = 8, all
>> zero values come first, followed by the coordinates).
>>
>> Looking again at ex77.c (
>> https://urldefense.us/v3/__https://petsc.org/release/src/snes/tutorials/ex77.c.html__;!!G_uCfscf7eWS!dBe0zmoP1EXuDC3Da6esPOb65iny79VkOAgA01sP84NE6TyW10KDbA-RSHT2uMS6hvXHo--Yav85W1bcMCjb$ ), it says the
>> closure stacks up the fields, which aligns with the output above (all u
>> values, then all p values), but which is the starting point seems arbitrary.
>>
>
> This is likely to be a Section construction problem again. The Section
> fields are separated by DMPlexVecGetClosure(), sp you must not have defined
> the fields in the Section. If you put these calls into the code that I sent
> you, what do you get?
>
> Thanks,
>
> Matt
>
>> The vector is initialized as [x0 y0 .... x3 y3 0 0 ... 0], which may not
>> be correct to start with.
>>
>> I noticed I could instead use DMPlexVecGetClosureAtDepth( ..., depth = 0,
>> ...) so that I only get the u field DoF. But I will need to deal with the p
>> field values as well at some point.
>>
>> Thanks,
>> Noam
>>
>> On Friday, May 8th, 2026 at 11:05 AM, Matthew Knepley <knepley at gmail.com>
>> wrote:
>>
>> On Fri, May 8, 2026 at 5:08 AM Noam T. <dontbugthedevs at proton.me> wrote:
>>
>>> On the plus side, using the flags you provided in the previous email
>>>
>>> -dm_plex_simplex 0 -mech_petscspace_degree 1 -pres_petscspace_degree 1
>>> -pres_petscspace_poly_tensor 0 -pres_petscdualspace_lagrange_continuity 0
>>>
>>> the output of -ds_view is identical to yours, so setting up the fields
>>> should be correct.
>>> Thanks. I forgot about the flag "-[]_petscdualspace_lagrange_continuity
>>> 0" so as to have a discontinuous field.
>>>
>>> However, the closure size was still wrong.
>>>
>>> But, I figured out the reason: the manual set up of the section.
>>> For the argument "numDof" in DMPlexCreateSection(), I was computing the
>>> number of DoF with PetscSectionGetDof() (which was fine until now, since
>>> there was a single field), instead of PetscSectionGetFieldDof().
>>> So as you previously suggested
>>>
>>> [...] It sounds like somehow you have a copy of your fields.
>>>
>>>
>>> the number of DoF per field were indeed "doubled" with the sum of both
>>> fields.
>>>
>>
>> Excellent. Let me know if you have any more issues.
>>
>> Thanks,
>>
>> Matt
>>
>>> Thanks,
>>> Noam
>>>
>>> On Wednesday, May 6th, 2026 at 2:06 PM, Matthew Knepley <
>>> knepley at gmail.com> wrote:
>>>
>>> Okay, this is a problem with specifying the space I believe. You want a
>>> discontinuous space for pressure.
>>>
>>> Here is what I get
>>>
>>> Discrete System with 2 fields
>>> cell total dim 11 total comp 3
>>> Field Q1 FEM 2 components (implicit) (Nq 4 Nqc 1) 1-jet
>>> PetscFE Object: Q1 (mech_) 1 MPI process
>>> type: vector
>>> Vector Finite Element in 2 dimensions with 2 components
>>> PetscSpace Object: Q1 (mech_) 1 MPI process
>>> type: sum
>>> Space in 2 variables with 2 components, size 8
>>> Sum space of 2 concatenated subspaces (all identical)
>>> PetscSpace Object: Q1 (mech_sumcomp_) 1 MPI process
>>> type: tensor
>>> Space in 2 variables with 1 components, size 4
>>> Tensor space of 2 subspaces (all identical)
>>> PetscSpace Object: sum component tensor component
>>> (mech_sumcomp_tensorcomp_) 1 MPI process
>>> type: poly
>>> Space in 1 variables with 1 components, size 2
>>> Polynomial space of degree 1
>>> PetscDualSpace Object: Q1 (mech_) 1 MPI process
>>> type: sum
>>> Dual space with 2 components, size 8
>>> Sum dual space of 2 concatenated subspaces (all identical)
>>> PetscDualSpace Object: Q1 1 MPI process
>>> type: lagrange
>>> Dual space with 1 components, size 4
>>> Continuous tensor Lagrange dual space
>>> Quadrature on a quadrilateral of order 3 on 4 points (dim 2)
>>> Field Q1 FEM 1 component (implicit) (Nq 4 Nqc 1) 1-jet
>>> PetscFE Object: Q1 (pres_) 1 MPI process
>>> type: basic
>>> Basic Finite Element in 2 dimensions with 1 components
>>> PetscSpace Object: Q1 (pres_) 1 MPI process
>>> type: poly
>>> Space in 2 variables with 1 components, size 3
>>> Polynomial space of degree 1
>>> PetscDualSpace Object: Q1 (pres_) 1 MPI process
>>> type: lagrange
>>> Dual space with 1 components, size 3
>>> Discontinuous Lagrange dual space
>>> Quadrature on a quadrilateral of order 3 on 4 points (dim 2)
>>> Weak Form System with 2 fields
>>>
>>> I have attached the code I used.
>>>
>>> Thanks,
>>>
>>> Matt
>>>
>>> On Wed, May 6, 2026 at 9:32 AM Noam T. <dontbugthedevs at proton.me> wrote:
>>>
>>>>
>>>> Field p FEM 3 components (implicit) (Nq 4 Nqc 1) 1-jet
>>>>
>>>>
>>>> 1. Why does pressure have 3 components in 2D?
>>>>
>>>>
>>>> I used nc = 3 when calling PetscFECreateDefault() for the pressure
>>>> field. Confusion between wanting 3 addditional DoF and components.
>>>>
>>>> With nc = 1 the closure size is 18 instead.
>>>>
>>>> 2. Did you look at the incompressible example SNES ex69? It sets up
>>>> this exact element.
>>>>
>>>>
>>>> I changed some of my code following the example and got rid of some
>>>> errors. But I'll have a more thorough look again.
>>>>
>>>> 3. Please send me that code that sets up your DM. It sounds like
>>>> somehow you have a copy of your fields.
>>>>
>>>> Thanks,
>>>>
>>>>
>>>> For reference, -ds_view with only one (displacements) fields shows:
>>>>
>>>> ---
>>>> Discrete System with 1 fields
>>>> cell total dim 8 total comp 2
>>>> Field u FEM 2 components (implicit) (Nq 4 Nqc 1) 1-jet
>>>> PetscFE Object: u (disp_) 1 MPI process
>>>> type: vector
>>>> Vector Finite Element in 2 dimensions with 2 components
>>>> PetscSpace Object: Q1 (disp_) 1 MPI process
>>>> type: sum
>>>> Space in 2 variables with 2 components, size 8
>>>> Sum space of 2 concatenated subspaces (all identical)
>>>> PetscSpace Object: Q1 (disp_sumcomp_) 1 MPI process
>>>> type: tensor
>>>> Space in 2 variables with 1 components, size 4
>>>> Tensor space of 2 subspaces (all identical)
>>>> PetscSpace Object: sum component tensor component
>>>> (disp_sumcomp_tensorcomp_) 1 MPI process
>>>> type: poly
>>>> Space in 1 variables with 1 components, size 2
>>>> Polynomial space of degree 1
>>>> PetscDualSpace Object: Q1 (disp_) 1 MPI process
>>>> type: sum
>>>> Dual space with 2 components, size 8
>>>> Sum dual space of 2 concatenated subspaces (all identical)
>>>> PetscDualSpace Object: Q1 1 MPI process
>>>> type: lagrange
>>>> Dual space with 1 components, size 4
>>>> Continuous tensor Lagrange dual space
>>>> Quadrature on a quadrilateral of order 3 on 4 points (dim 2)
>>>> Weak Form System with 1 fields
>>>> ---
>>>>
>>>> and -dm_view
>>>>
>>>> ---
>>>> DM Object: 1 MPI process
>>>> type: plex
>>>> DM_0x55c117db1b00_2 in 2 dimensions:
>>>> Number of 0-cells per rank: 4
>>>> Number of 1-cells per rank: 4
>>>> Number of 2-cells per rank: 1
>>>> Labels:
>>>> celltype: 3 strata with value/size (0 (4), 1 (4), 4 (1))
>>>> depth: 3 strata with value/size (0 (4), 1 (4), 2 (1))
>>>> Cell Sets: 1 strata with value/size (1 (1))
>>>> surf: 1 strata with value/size (1 (1))
>>>> Face Sets: 2 strata with value/size (2 (1), 3 (1))
>>>> edge_right: 1 strata with value/size (3 (1))
>>>> edge_left: 1 strata with value/size (2 (1))
>>>> Vertex Sets: 4 strata with value/size (4 (1), 5 (1), 6 (1), 7 (1))
>>>> P1: 1 strata with value/size (4 (1))
>>>> P2: 1 strata with value/size (5 (1))
>>>> P3: 1 strata with value/size (6 (1))
>>>> P4: 1 strata with value/size (7 (1))
>>>> Field u:
>>>> adjacency FEM
>>>> /// With the second field, the two lines below are added
>>>> Field p:
>>>> adjacency FEM
>>>> ---
>>>>
>>>> DM set up
>>>>
>>>> DM :: dm_mesh
>>>> PetscFE :: u_FE, p_FE
>>>> PetscInt :: dim, dim_u_FE, tdim
>>>> PetscDS :: dm_ds
>>>>
>>>>
>>>> DMCreate(PETSC_COMM_WORLD, dm_mesh)
>>>> DMSetType(dm_mesh, DMPLEX)
>>>> DMGetDimension(dm_mesh, dim) // dim = 2
>>>> DMSetFromOptions(dm_mesh) // this was called twice by mistake, now
>>>> removed; no changes that I can see
>>>>
>>>> -- using flags
>>>>
>>>> -dm_plex_filename QUAD1.msh
>>>> -dm_plex_interpolate 1
>>>> -dm_plex_gmsh_use_generic
>>>> -dm_plex_gmsh_use_regions
>>>> -dm_plex_gmsh_multiple_tags
>>>> -dm_plex_gmsh_mark_vertices
>>>>
>>>> -- some operations with labels
>>>>
>>>> // u field
>>>> PetscFECreateDefault(PETSC_COMM_SELF, dim, dim, PETSC_FALSE, "mech_",
>>>> PETSC_DETERMINE, u_FE)
>>>> PetscFEGetDimension(u_FE, dim_u_FE) // dim_u_FE = 8
>>>> DMAddField(dm_mesh, NULL, (PetscObject)u_FE)
>>>>
>>>> // p field
>>>> PetscFECreateDefault(PETSC_COMM_SELF, dim , 1, PETSC_FALSE, "pres_",
>>>> PETSC_DETERMINE, p_FE)
>>>> PetscFEGetDimension(p_FE, dim_p_FE) // dim_p_FE = 1
>>>> DMAddField(dm_mesh, NULL, (PetscObject)p_FE)
>>>>
>>>> DMCreateDS(dm_mesh)
>>>> DMGetDS(dm_mesh, dm_DS)
>>>> PetscDSGetTotalDimension(dm_DS, tdim) // tdim
>>>>
>>>> with flags
>>>>
>>>> -disp_petscdualspace_lagrange_node_type equispaced
>>>> -disp_petscdualspace_lagrange_node_endpoints 1
>>>>
>>>>
>>>> I believe these are all the DM related calls.
>>>>
>>>> Thanks,
>>>> Noam
>>>>
>>>> On Wednesday, May 6th, 2026 at 11:39 AM, Matthew Knepley <
>>>> knepley at gmail.com> wrote:
>>>>
>>>> On Wed, May 6, 2026 at 6:56 AM Noam T. <dontbugthedevs at proton.me>
>>>> wrote:
>>>>
>>>>> Hello,
>>>>>
>>>>> That is wrong. Are you sure about 22?
>>>>>
>>>>>
>>>>> That's what I get, from the argument "csize" in DMPlexGetVecClosure().
>>>>>
>>>>> Here's the output of PetscDSView:
>>>>>
>>>>> Discrete System with 2 fields
>>>>> cell total dim 11 total comp 5
>>>>> Field u FEM 2 components (implicit) (Nq 4 Nqc 1) 1-jet
>>>>> PetscFE Object: u (mech_) 1 MPI process
>>>>> type: vector
>>>>> Vector Finite Element in 2 dimensions with 2 components
>>>>> PetscSpace Object: Q1 (mech_) 1 MPI process
>>>>> type: sum
>>>>> Space in 2 variables with 2 components, size 8
>>>>> Sum space of 2 concatenated subspaces (all identical)
>>>>> PetscSpace Object: Q1 (mech_sumcomp_) 1 MPI process
>>>>> type: tensor
>>>>> Space in 2 variables with 1 components, size 4
>>>>> Tensor space of 2 subspaces (all identical)
>>>>> PetscSpace Object: sum component tensor component
>>>>> (mech_sumcomp_tensorcomp_) 1 MPI process
>>>>> type: poly
>>>>> Space in 1 variables with 1 components, size 2
>>>>> Polynomial space of degree 1
>>>>> PetscDualSpace Object: Q1 (mech_) 1 MPI process
>>>>> type: sum
>>>>> Dual space with 2 components, size 8
>>>>> Sum dual space of 2 concatenated subspaces (all identical)
>>>>> PetscDualSpace Object: Q1 1 MPI process
>>>>> type: lagrange
>>>>> Dual space with 1 components, size 4
>>>>> Continuous tensor Lagrange dual space
>>>>> Quadrature on a quadrilateral of order 3 on 4 points (dim 2)
>>>>> Field p FEM 3 components (implicit) (Nq 4 Nqc 1) 1-jet
>>>>>
>>>>
>>>> 1. Why does pressure have 3 components in 2D?
>>>>
>>>>> PetscFE Object: p (pres_) 1 MPI process
>>>>> type: vector
>>>>> Vector Finite Element in 2 dimensions with 3 components
>>>>> PetscSpace Object: Q0 (pres_) 1 MPI process
>>>>> type: sum
>>>>> Space in 2 variables with 3 components, size 3
>>>>> Sum space of 3 concatenated subspaces (all identical)
>>>>> PetscSpace Object: Q0 (pres_sumcomp_) 1 MPI process
>>>>> type: tensor
>>>>> Space in 2 variables with 1 components, size 1
>>>>> Tensor space of 2 subspaces (all identical)
>>>>> PetscSpace Object: sum component tensor component
>>>>> (pres_sumcomp_tensorcomp_) 1 MPI process
>>>>> type: poly
>>>>> Space in 1 variables with 1 components, size 1
>>>>> Polynomial space of degree 0
>>>>> PetscDualSpace Object: Q0 (pres_) 1 MPI process
>>>>> type: sum
>>>>> Dual space with 3 components, size 3
>>>>> Sum dual space of 3 concatenated subspaces (all identical)
>>>>> PetscDualSpace Object: Q0 1 MPI process
>>>>> type: lagrange
>>>>> Dual space with 1 components, size 1
>>>>> Discontinuous tensor Lagrange dual space
>>>>> Quadrature on a quadrilateral of order 3 on 4 points (dim 2)
>>>>> Weak Form System with 2 fields
>>>>>
>>>>> using as the second field (dim = 2, nc = 3, prefix = "pres") and
>>>>> setting the FE object name to "p".
>>>>>
>>>>> The entry "cell total dim 10" agrees with the output of
>>>>> PetscFEGetTotalDimension(); "comp = 5" I assume is 2 (u) + 3 (p)
>>>>>
>>>>> However, the closure of a Vec is still 22.
>>>>>
>>>>
>>>> 2. Did you look at the incompressible example SNES ex69? It sets up
>>>> this exact element.
>>>>
>>>> 3. Please send me that code that sets up your DM. It sounds like
>>>> somehow you have a copy of your fields.
>>>>
>>>> Thanks,
>>>>
>>>> Matt
>>>>
>>>>> On Tuesday, May 5th, 2026 at 1:58 PM, Matthew Knepley <
>>>>> knepley at gmail.com> wrote:
>>>>>
>>>>> On Tue, May 5, 2026 at 9:06 AM Noam T. via petsc-users <
>>>>> petsc-users at mcs.anl.gov> wrote:
>>>>>
>>>>>> Hello,
>>>>>>
>>>>>> I am trying to work with a "mixed" FE discretization, where besides
>>>>>> the usual displacements DoF in nodes, there is an additional field (e.g.
>>>>>> pressure) that is also part of the system. This additional field has a
>>>>>> certain number of additional dof : p0, p1, p2...
>>>>>>
>>>>>> Looking at example 77 (
>>>>>> https://urldefense.us/v3/__https://petsc.org/release/src/snes/tutorials/ex77.c.html__;!!G_uCfscf7eWS!dBe0zmoP1EXuDC3Da6esPOb65iny79VkOAgA01sP84NE6TyW10KDbA-RSHT2uMS6hvXHo--Yav85W1bcMCjb$
>>>>>> <https://urldefense.us/v3/__https://petsc.org/release/src/snes/tutorials/ex77.c.html__;!!G_uCfscf7eWS!aZNOxZZeoHYqe6lRZV0wHMVVTu3YIEfc1Sr-dca7xiGVfG3enULFD4_g6_fxWIa7A_Cu9LcXU-MuxRj4sOwDRyWqvsdGdrEd$>)
>>>>>> this seems to be handled with an additional field, added to the DM. I've
>>>>>> tried so, but then I am getting from the DM arrays whose size/contents are
>>>>>> not what I expected.
>>>>>>
>>>>>> For example, a four-noded Q1 element (PetscFE created with a 2x2
>>>>>> Gauss rule for quadrature), with just one field, a call to
>>>>>> DMPlexGetVecClosure() gives me an array with 8 entries (say, the
>>>>>> coordinates of the initial mesh): x = x0, y0, .... x3, y3
>>>>>>
>>>>>> Then add the new field:
>>>>>>
>>>>>> PetscFECreateDefault(..., nc = 3, ..., p_FE) /* not sure about the
>>>>>> value of nc here */
>>>>>> DMAddField(dm, ..., p_FE)
>>>>>>
>>>>>> The total dimension, from PetscDSGetTotalDimension(), is now 11 (8 +
>>>>>> 3). This results in a closure of size 22.
>>>>>>
>>>>>
>>>>> That is wrong. Are you sure about 22?
>>>>>
>>>>>> However, what I am looking for is a closure of size 8 + 3 i.e. the
>>>>>> original 8 DoF at the nodes, plus exactly 3 DoF (for the cell, so to speak)
>>>>>> p0, p1, p2, so that in a system with "block" matrices of the form
>>>>>>
>>>>>> [K_uu, K_up | K_pu, K_pp] { u | p } = RHS
>>>>>>
>>>>>> the unknowns {p} has size 3 per cell. Is this possible? I tried some
>>>>>> combinations of dim / nc for the new PetscFE, but when creating the DM
>>>>>> section I get errors e.g.
>>>>>>
>>>>>
>>>>> This is what you should get. I definitely have examples that do this.
>>>>> For example, here is Q1-P0 (I think that is what you are suggesting) for
>>>>> incompressible Stokes
>>>>>
>>>>>
>>>>> https://urldefense.us/v3/__https://gitlab.com/petsc/petsc/-/blob/main/src/snes/tutorials/ex69.c?ref_type=heads*L3454__;Iw!!G_uCfscf7eWS!dBe0zmoP1EXuDC3Da6esPOb65iny79VkOAgA01sP84NE6TyW10KDbA-RSHT2uMS6hvXHo--Yav85Wx2HPV3Z$
>>>>>
>>>>> Thanks,
>>>>>
>>>>> Matt
>>>>>
>>>>>> "point X has a number of DoF not divisible by 2 field components"
>>>>>>
>>>>>> Is this a hint that I should have 3 x dim new DoF, and simply not
>>>>>> deal with entries that I don't need? Or I am not setting up the section
>>>>>> properly (works with just one field)?
>>>>>>
>>>>>> Thank you,
>>>>>> Noam.
>>>>>>
>>>>>
>>>>>
>>>>> --
>>>>> What most experimenters take for granted before they begin their
>>>>> experiments is infinitely more interesting than any results to which their
>>>>> experiments lead.
>>>>> -- Norbert Wiener
>>>>>
>>>>> https://urldefense.us/v3/__https://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!dBe0zmoP1EXuDC3Da6esPOb65iny79VkOAgA01sP84NE6TyW10KDbA-RSHT2uMS6hvXHo--Yav85W67yMHOl$
>>>>> <https://urldefense.us/v3/__http://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!dBe0zmoP1EXuDC3Da6esPOb65iny79VkOAgA01sP84NE6TyW10KDbA-RSHT2uMS6hvXHo--Yav85Wy2dVqX9$ >
>>>>>
>>>>>
>>>>>
>>>>
>>>> --
>>>> What most experimenters take for granted before they begin their
>>>> experiments is infinitely more interesting than any results to which their
>>>> experiments lead.
>>>> -- Norbert Wiener
>>>>
>>>> https://urldefense.us/v3/__https://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!dBe0zmoP1EXuDC3Da6esPOb65iny79VkOAgA01sP84NE6TyW10KDbA-RSHT2uMS6hvXHo--Yav85W67yMHOl$
>>>> <https://urldefense.us/v3/__http://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!dBe0zmoP1EXuDC3Da6esPOb65iny79VkOAgA01sP84NE6TyW10KDbA-RSHT2uMS6hvXHo--Yav85Wy2dVqX9$ >
>>>>
>>>>
>>>>
>>>
>>> --
>>> What most experimenters take for granted before they begin their
>>> experiments is infinitely more interesting than any results to which their
>>> experiments lead.
>>> -- Norbert Wiener
>>>
>>> https://urldefense.us/v3/__https://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!dBe0zmoP1EXuDC3Da6esPOb65iny79VkOAgA01sP84NE6TyW10KDbA-RSHT2uMS6hvXHo--Yav85W67yMHOl$
>>> <https://urldefense.us/v3/__http://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!dBe0zmoP1EXuDC3Da6esPOb65iny79VkOAgA01sP84NE6TyW10KDbA-RSHT2uMS6hvXHo--Yav85Wy2dVqX9$ >
>>>
>>>
>>>
>>
>> --
>> What most experimenters take for granted before they begin their
>> experiments is infinitely more interesting than any results to which their
>> experiments lead.
>> -- Norbert Wiener
>>
>> https://urldefense.us/v3/__https://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!dBe0zmoP1EXuDC3Da6esPOb65iny79VkOAgA01sP84NE6TyW10KDbA-RSHT2uMS6hvXHo--Yav85W67yMHOl$
>> <https://urldefense.us/v3/__http://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!dBe0zmoP1EXuDC3Da6esPOb65iny79VkOAgA01sP84NE6TyW10KDbA-RSHT2uMS6hvXHo--Yav85Wy2dVqX9$ >
>>
>>
>>
>
> --
> What most experimenters take for granted before they begin their
> experiments is infinitely more interesting than any results to which their
> experiments lead.
> -- Norbert Wiener
>
> https://urldefense.us/v3/__https://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!dBe0zmoP1EXuDC3Da6esPOb65iny79VkOAgA01sP84NE6TyW10KDbA-RSHT2uMS6hvXHo--Yav85W67yMHOl$
> <https://urldefense.us/v3/__http://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!dBe0zmoP1EXuDC3Da6esPOb65iny79VkOAgA01sP84NE6TyW10KDbA-RSHT2uMS6hvXHo--Yav85Wy2dVqX9$ >
>
>
>
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--
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener
https://urldefense.us/v3/__https://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!dBe0zmoP1EXuDC3Da6esPOb65iny79VkOAgA01sP84NE6TyW10KDbA-RSHT2uMS6hvXHo--Yav85W67yMHOl$ <https://urldefense.us/v3/__http://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!dBe0zmoP1EXuDC3Da6esPOb65iny79VkOAgA01sP84NE6TyW10KDbA-RSHT2uMS6hvXHo--Yav85Wy2dVqX9$ >
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