M. Repetto - polito.it...M. Repetto Introduction Orthogonal grids Unstructured meshes Uniform field...
Transcript of M. Repetto - polito.it...M. Repetto Introduction Orthogonal grids Unstructured meshes Uniform field...
Constitutiveequations
M. Repetto
Introduction
Orthogonalgrids
Unstructuredmeshes
Uniform fieldMagnetic case
Electric case
remarks
Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
Constitutive equations
M. Repetto
Politecnico di TorinoDipartimento Energia
Cell Method for multiphysics problems
Constitutiveequations
M. Repetto
Introduction
Orthogonalgrids
Unstructuredmeshes
Uniform fieldMagnetic case
Electric case
remarks
Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
Outline
1 Introduction
2 Orthogonal grids
3 Unstructured meshes
4 Uniform fieldMagnetic caseElectric caseremarks
5 Non-uniform fieldMicro-cell approachWhitney interpolation approachsymmetric Whitney
Constitutiveequations
M. Repetto
Introduction
Orthogonalgrids
Unstructuredmeshes
Uniform fieldMagnetic case
Electric case
remarks
Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
Equation classification
topological and constitutive equationsAs pointed out in the first lesson, two kinds of equationsconstraint global variables:topological equations which:
link together global variables of the same kind (sourceor configuration)do not depend on metricare exact on the given discretization
constitutive equations which:link together variables of different kind (one source withone configuration)depend on metric and on material characteristicsare approximate
Constitutiveequations
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Introduction
Orthogonalgrids
Unstructuredmeshes
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Electric case
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Whitney interpolationapproach
symmetric Whitney
electrical circuit
constraint equations
i 2v R
vi 1
R
A
B
vi
2
1
AB1
2
topological or Kirchhofflaws
i = i1 = i2 (1)vAB − v1 − v2 = 0 (2)
constitutive equations
v1 = R1i1 (3)v2 = R2i2 (4)
R1 = ρ1l1S1, R2 = ρ2
l2S2
(5)
vAB = v1 + v2 = R1i1 + R2i2 = (R1 + R2) i = Req i (6)
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Introduction
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Whitney interpolationapproach
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constitutive equations
hypothesis on field variationconstitutive equations link together the source variablecurrent i flowing through the component to theconfiguration variable v across its terminalsthe second Ohm law used to compute the value of R1and R2 has been obtained under the hypothesis of arectilinear conductor with constant conductor crosssection and uniform distribution of current density
−→J ,
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Introduction
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Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
constitutive equations
characteristics of constitutive equationsthe need to impose hypothesis on the spacedistribution of field variables derives from the definitionof macroscopic material equation which is expressed interms of field variables
−→E = ρ
−→J :
the "regular" distribution of field variables in spaceintegration is needed to link field and global variables
i =
∫S
−→J · d
−→S = JS ⇒ J =
iS
(7)
v =
∫l
−→E · d
−→l = El ⇒ E =
vl
(8)
−→E = ρ
−→J ⇒ v
l= ρ
iS⇒ v = ρ
lS
i (9)
Constitutiveequations
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Introduction
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Unstructuredmeshes
Uniform fieldMagnetic case
Electric case
remarks
Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
Orthogonal grids
two dimensional gridsamong possible ways of subdividing the problemgeometry the orthogonal one is the simplest wheneverit can be applied
in this case both primal and dual complex of cell aresharing the same "shape"
Constitutiveequations
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Introduction
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Unstructuredmeshes
Uniform fieldMagnetic case
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Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
Primal or dual mesh?
solution variableGlobal variables are associated to primal or dual spaceelementsthe choice of the variable used in the solutioninfluences the meaning of the discretization
magnetic flux↔ primal faceelectric current↔ dual face
since the shape of the cells is "regular" both in primal ordual complex, the choice of which grid defines thegeometry (for instance following material interfaces) isjust a matter of which variable is used for solution of theproblem
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Introduction
Orthogonalgrids
Unstructuredmeshes
Uniform fieldMagnetic case
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remarks
Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
Primal or dual mesh?
boundary condition on configuration variable
boundary condition on source variable
Constitutiveequations
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Introduction
Orthogonalgrids
Unstructuredmeshes
Uniform fieldMagnetic case
Electric case
remarks
Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
Orthogonal grids
three dimensional gridsthe same property holds also on three dimensionalgrids
x
y
z
primal volume
dual volume
Constitutiveequations
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Introduction
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Uniform fieldMagnetic case
Electric case
remarks
Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
Orthogonal grids
variables linked by dualityconstitutive equations link global variables on a coupleof dual geometrical entitiestwo situations are possible:
primal face↔ dual edgeprimal edge↔ dual face
usually primal cell complex define geometry so that,even if geometrically similar, the two relations aretreated in different ways
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Introduction
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Whitney interpolationapproach
symmetric Whitney
Constitutive matrix
Φ[S]←→ F [L]
B = µ0H + µ0M (10)
Bn =Φ[S]
S(11)
Ht =F [L]
LFM [L] = M · L
F =Lµ0S
Φ[S]− FM (12)
hypothesis: local uniformity offield variables andhomogeneity of material
Ht
Bn
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Orthogonal grids
material inhomogeneityif the primal face falls on the interface between twomaterials with different properties, the additivityproperty of line integral
∫L H · d l is used to treat the
constitutive equation as the series of two components,each one with its properties
R =L1
µ1S+
L2
µ2S(13)
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Constitutive matrix
u ←→ i
E = ρJ (14)
Et =uL
(15)
Jn =iS
u = ρLS
i (16)
hypothesis: local uniformity offield variables andhomogeneity of material
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Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
Orthogonal grids
material inhomogeneityif the primal edge falls on the interface between twomaterials with different properties, the constitutiverelation is treated as the parallel of two components,each one with its properties
R =
(F1
ρ1L+
F2
ρ2L
)−1
(17)
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symmetric Whitney
Orthogonal grids
building constitutive matrixas mentioned before, additivity property can be used tobuild constitutive equations breaking the dual edge orthe dual face into two parts belonging to each primalcell
Lj = Laj ∪ Lb
j (18)
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Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
Orthogonal grids
flowchart of cell by cell scheme
for i on all volumes
build local interpolation with local orientation
build local matrix M
for j on all entities in the volume
store local matrix M in global one taking into account true orientation of elements
endend
(loc)
(loc)
Constitutiveequations
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Introduction
Orthogonalgrids
Unstructuredmeshes
Uniform fieldMagnetic case
Electric case
remarks
Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
Why unstructured meshes?
square and circles
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Introduction
Orthogonalgrids
Unstructuredmeshes
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Electric case
remarks
Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
Why unstructured meshes?
Technical problemscomplex shapesuneven materialboundariesobjects of differentdimensions in the sameproblem
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Electric case
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Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
Barycentric complex
how to build a mesh dual to symplexone possible way to build a mesh dual to a symplecticone is to use barycentric mesh:
dual node← barycenter of symplex celldual edge← broken line joining two dual nodes andpassing through the barycenter of the face whichdivides themdual face← union of quadrilateral faces having asvertices: mid point of primal edge, barycenter of oneface hinged on edge, barycenter of symplex cell,barycenter of the other face hinged on edgedual volume← union of all volumes having dual facesas boundaries
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Whitney interpolationapproach
symmetric Whitney
Barycentric complex
dual entities
prim al edge connected to
node
dual edge bounding a dual face
dual node(barycenter of tetrahedron)
Constitutiveequations
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Introduction
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Unstructuredmeshes
Uniform fieldMagnetic case
Electric case
remarks
Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
Primal or dual mesh?
simplex or polytopequadrilateral orthogonalmeshes share the samecell shape for primal ordual complexsimplex mesh gives riseto a polygonal dual cellcomplex not suited fortechnical geometriessolution variable mustreside on simplex mesh
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Introduction
Orthogonalgrids
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Whitney interpolationapproach
symmetric Whitney
Building constitutive matrices
orthogonal
Et
Dn
simplex
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Introduction
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Electric case
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Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
from global to local
interpolation1 from global variables get
a local interpolation offields inside cell
2 integrate field values overgeometrical entities
3 assembly globalvariables
F =
∫λ1
H · d l +
∫λ2
H · d l
(19)
material 1
material 2
λ1~
λ2~
Constitutiveequations
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Introduction
Orthogonalgrids
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Whitney interpolationapproach
symmetric Whitney
Uniformity of field variables
hypothesisthe simplest way to build a local interpolation of fieldvariables is to consider them uniform inside the cellthis hypothesis is not always congruent with the regimeof fields variationtwo constitutive relations are considered:
magnetic flux φ on primal faces↔ magneto-motiveforce F on dual edgeselectric voltage u on primal edges↔ electriccurrent/dielectric flux on dual faces
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symmetric Whitney
uniform B
Gauss law for magneticflux
Φ1 + Φ2 + Φ3 + Φ4 = 0 (20)
Φ4 = −Φ1 − Φ2 − Φ3 (21)
a uniform B = (Bx ,By ,Bz) isconsistent with fluxsolenoidality
1
2
3
4
1
2
3
4
5
6
1
2
3
4
Constitutiveequations
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symmetric Whitney
Uniformity of field variables
area vector SSi = Aini (22)
Φi = B · Si = BxSix + BySiy + BzSiz (23)
Φ′ =
Φ1Φ2Φ3
(24) B =
BxByBz
(25)
Φ′ = [S] B (26)
B = [S]−1 Φ′ (27)
Constitutiveequations
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Uniformity of field variables
from square (3× 3) matrix [S]−1 a rectangular (3× 4) onecan be obtained by inserting a column of zeros in Φ4position so that
[S]−1 =
m11 m12 m13m21 m22 m23m31 m32 m33
(28)
[Sa]−1
=
m11 m12 m13 0m21 m22 m23 0m31 m32 m33 0
(29)
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Whitney interpolationapproach
symmetric Whitney
Uniformity of field variables
B = [Sa]−1 Φ (30)
H = νB (31)
Li = (Lix , Liy , Liz) (32)
Fi = Li · H (33)
F =[L]ν [Sa]
−1 Φ =[M(loc)
ν
]Φ (34)
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Whitney interpolationapproach
symmetric Whitney
uniform E
irrotational Ein static conditions E = −∇φ
u1 − u2 + u4 = 0 (35)
u4 = −u1 + u2 (36)
only 3 out of 6 u values areindependenta uniform E = (Ex ,Ey ,Ez) isconsistent with electric fieldirrotationality
1
2
3
4
u1 u2
u3
u4
u5
u6
Constitutiveequations
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Uniformity of field variables
edge vector L
ui = E · Li = ExLix + EyLiy + EzLiz (37)
u′ =
u1u2u3
(38) E =
ExEyEz
(39)
u′ = [L] E (40)
E = [L]−1 u′ (41)
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Whitney interpolationapproach
symmetric Whitney
Uniformity of field variables
from square (3× 3) matrix [L]−1 a rectangular (3× 6) onecan be obtained by inserting three columns of zeros inu4,u5,u6 positions so that
E = [La]−1 u (42)
D = εE (43)
Si = (Six , Siy , Siz) (44)
Ψi = Si · D (45)
Ψ =[S]ε [La]
−1 u =[M(loc)
ε
]u (46)
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symmetric Whitney
Remarks
local interpolation involves almost all global variablesdefined over the cell (φ1, φ2, φ3) or (u1,u2,u3)
local matrices containing geometrical quantities ([S]−1,[L]−1) are full matrices
resulting local matrices[M(loc)
ν
]and
[M(loc)
ε
]are full
matrices linking global variables defined over the cellglobal constitutive matrix is not anymore diagonal, likein orthogonal meshes, but has a sparse structure
Constitutiveequations
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Whitney interpolationapproach
symmetric Whitney
global matrix [Mν]
sparsity patternnon-null elements in first row of matrix [Mν ]Mν11, Mν12, Mν13, Mν14, Mν15, Mν16, Mν17
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Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
Non-uniform field
field regimessolenoidality of magnetic flux holds in any caseirrotationality of electric field holds only in staticconditions when electromagnetic induction is notpresent ∮
E · d l = −dφdt
(47)
in this case all six values of u are independent anduniform E hypothesis does not hold anymoreaffine law of variation of field variable E can beimplemented in different ways
micro-cell approachWhitney shape function approach
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Whitney interpolationapproach
symmetric Whitney
Micro-cell approach
rationalemicro-cell approach implements a computationalscheme similar to that of uniform field but fielduniformity is considered valid only on a portion of thecelladditivity of integral quantities is used to link local fieldsto global variables
primal volume V
dual volume V
~
V V~
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symmetric Whitney
Micro-cell approach
β
α
γ
δ
Micro-Cell associated to node α
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Micro-cell approach
β
α
γ
δJα
Jβk
ku(β)
ku(α)
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Micro-cell approach
ikijin
=
Skx Sky Skz
Sjx Sjy Sjz
Snx Sny Snz
J(α)x
J(α)y
J(α)z
(48)
i(α) =[Sα
]J(α) (49)
α
Jα
ku(α)
ik
ju(α)
nu(α)
in
i j
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symmetric Whitney
Micro-cell approach
electro-motive forceJ(α)
x
J(α)y
J(α)z
=[Sα
]−1
ikijin
(50)
u(α) = [Lα] ρ[Sα
]−1i(α) (51)
uk = u(α)k +u(β)
k = ρ
([Lα]
[Sα
]−1i(α)+ [Lβ]
[Sβ
]−1i(β)
)(52)
Constitutiveequations
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Orthogonalgrids
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Electric case
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Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
Whitney interpolation approach
rationaleWhitney shape function have been widely used in finiteelements to interpolate vector quantitiesthey can also be used here to interpolate field insidetetrahedra starting from global quantities on edges orfacesWhitney shape functions enforce inter-elementcontinuity of field components (normal component offlux density or tangential component of electric field)the use of Whitney shape functions gives rise tosolution matrices which are identical to the ones ofvector based finite elements
Constitutiveequations
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Non-uniformfieldMicro-cell approach
Whitney interpolationapproach
symmetric Whitney
building Whitney functions
nodal shape functions
Nα = aαx + bαy + cαz + dα (53)
β
α
γ
δ
N =0α
N =1α
N α
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symmetric Whitney
building Whitney functions
geometrical interpretation of gradient vectorthe gradient vector can be related to tetrahedronvolume
Volume =13
h1F1 ⇒ h1F1 = 3 ∗ Volume (54)
∇N1 vector is orthogonal to face opposite to node 1and makes the function goes from value 0 to 1 alongthe perpendicular direction so that:
∇N1 =1h1
−→F1
|−→F1|
=
−→F1
3 ∗ Volume(55)
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building Whitney functions
vector shape functions
wk = Nα∇Nβ − Nβ∇Nα (56)
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building constitutive matrix
vector shape functions
uk =
∫Lk
E · dl (57)
E(P) =6∑
k=1
uk wk (58)
in =
∫Sn
J · dS =
∫Sn
σ
6∑k=1
uk wk · dS = (59)
=6∑
k=1
σuk
∫Sn
wk · dS =6∑
k=1
cnkuk
Constitutiveequations
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Comparisonequilateral tetrahedron
Micro-cell
Whitney
Constitutiveequations
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symmetric Whitney
symmetric constitutive matrix
power on primal cellpower dissipated in one volume can be expressed as:
P =
∫Ω
−→E ·−→J dΩ (60)
by interpolating electric field by means of Whitney edgefunctions
−→E =
6∑k=1
uk wk ⇒ P =
∫Ω
6∑k=1
uk−→w k ·
−→J dΩ (61)
by considering that−→J would be uniform on Ω
P =6∑
k=1
uk−→J ·∫
Ω
−→w kdΩ (62)
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symmetric constitutive matrix
power on primal celldue to the geometrical property of the Whitney edgefunction, the following relation holds:
Sk =
∫Ω
−→w kdΩ (63)
the previous equation becomes:
P =6∑
k=1
uk−→J · Sk = ik (64)
so that power can be written as:
P =6∑
k=1
uk ik (65)
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symmetric constitutive matrix
power on primal cell
P =6∑
k=1
uk ik (66)
going back to the first definition of power P
P =
∫Ω
−→E ·−→J dΩ =
6∑k=1
uk
∫Ω
−→w k · σ6∑
n=1
−→w nundΩ︸ ︷︷ ︸ (67)
by comparing the two equations, the term underbracedis equal to ik
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symmetric constitutive matrix
power on primal cellk -th current tailored to cell Ω is thus written as:
ik =6∑
n=1
un
∫Ω
−→w k · σ−→w ndΩ (68)
by expressing the terms of the previous equation asmatrix element:
ik =6∑
n=1
Mknun (69)
whereMkn = σ
∫Ω
−→w k ·−→w ndΩ (70)
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symmetric constitutive matrix
constitutive matrixthe constitutive matrix obtained has a symmetricstructure and it is often referred to as Galerkin matrixsince it is formally equal to the one obtained byweighted residual (Galerkin) finite element matrix ifedge vector shape functions are usedsymmetry of the matrix allows to state somemathematical properties of the resulting system ofequations ensuring its stability when time-marchingscheme are used
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example on orthogonal tetrahedron
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example on orthogonal tetrahedron
Whitney1.0000e+00 4.4118e-01 4.4118e-01 2.9412e-02 2.9412e-02 04.4118e-01 1.0000e+00 4.4118e-01 -2.9412e-02 0 2.9412e-024.4118e-01 4.4118e-01 1.0000e+00 0 -2.9412e-02 -2.9412e-02-6.8627e-02 6.8627e-02 0 3.3333e-01 6.8627e-02 -6.8627e-02-6.8627e-02 0 6.8627e-02 6.8627e-02 3.3333e-01 6.8627e-02
0 -6.8627e-02 6.8627e-02 -6.8627e-02 6.8627e-02 3.3333e-01
Whitney symmetric1.0000e+00 5.0000e-01 5.0000e-01 4.6317e-16 4.5797e-16 05.0000e-01 1.0000e+00 5.0000e-01 -4.6317e-16 0 4.7184e-165.0000e-01 5.0000e-01 1.0000e+00 0 -4.5797e-16 -4.5797e-164.6317e-16 -4.6317e-16 0 4.0000e-01 1.0000e-01 -1.0000e-014.5797e-16 0 -4.5797e-16 1.0000e-01 4.0000e-01 1.0000e-01
0 4.7184e-16 -4.5797e-16 -1.0000e-01 1.0000e-01 4.0000e-01