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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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


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remarks



symmetric Whitney


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 ,


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney


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)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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"


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney


boundary condition on configuration variable

boundary condition on source variable


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney

Orthogonal grids

three dimensional gridsthe same property holds also on three dimensionalgrids

x

y

z

primal volume

dual volume


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney

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)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney

Constitutive matrix

u ←→ i

E = ρJ (14)

Et =uL

(15)

Jn =iS

u = ρLS

i (16)

hypothesis: local uniformity offield variables andhomogeneity of material


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney

Why unstructured meshes?

square and circles


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney

Why unstructured meshes?

Technical problemscomplex shapesuneven materialboundariesobjects of differentdimensions in the sameproblem


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney

Barycentric complex

dual entities

prim al edge connected to

node

dual edge bounding a dual face

dual node(barycenter of tetrahedron)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney


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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney

Building constitutive matrices

orthogonal

Et

Dn

simplex


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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~


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney


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)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

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symmetric Whitney


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)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney


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)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney


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)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney


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)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

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symmetric Whitney

global matrix [Mν]

sparsity patternnon-null elements in first row of matrix [Mν ]Mν11, Mν12, Mν13, Mν14, Mν15, Mν16, Mν17


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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~


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney

Micro-cell approach

β

α

γ

δ

Micro-Cell associated to node α


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney

Micro-cell approach

β

α

γ

δJα

Jβk

ku(β)

ku(α)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

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symmetric Whitney

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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney

building Whitney functions

nodal shape functions

Nα = aαx + bαy + cαz + dα (53)

β

α

γ

δ

N =0α

N =1α

N α


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

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symmetric Whitney


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)


M. Repetto

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symmetric Whitney


vector shape functions

wk = Nα∇Nβ − Nβ∇Nα (56)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

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symmetric Whitney

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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney

Comparisonequilateral tetrahedron

Micro-cell

Whitney


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



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)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


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symmetric Whitney


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)


M. Repetto

Introduction

Orthogonalgrids

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symmetric Whitney


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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


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symmetric Whitney


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)


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney


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


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney

example on orthogonal tetrahedron


M. Repetto

Introduction

Orthogonalgrids

Unstructuredmeshes


Electric case

remarks



symmetric Whitney

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

m. repetto - polito.it...m. repetto introduction orthogonal grids unstructured meshes uniform ﬁeld...

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