Multiscale Stabilized Control Volume Finite Element Method for … · 2017. 8. 1. · Multiscale Stabilized Control Volume Finite Element Method for Advection-Diffusion Kara Peterson

Multiscale Stabilized Control Volume Finite ElementMethod for Advection-Diffusion

Kara Peterson Pavel Bochev Mauro PeregoSuzey Gao

Sandia National Laboratories

FEF 2017

CCRCenter for Computing Research

Sandia National Laboratories is a multi-mission laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary ofLockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

SAND 2017-3512C 1

Application Driver

Semiconductor electrical transport simulation

Scaled semiconductor drift-diffusion equations

Poisson equation ∇ · (λ2∇φ) + (p− n + C) = 0

Electron continuity equation∂n

∂t−∇ · Jn + R(φ, n, p) = 0

Hole continuity equation∂p

∂t+∇ · Jp + R(φ, n, p) = 0

Electric field E = −∇φElectron current density Jn = µnEn +Dn∇n

Hole current density Jp = µpEn−Dp∇p

source drain

gate

substrate

Nd Nd

Na

Ωox

Ωsi

Vg = 2 V

Vd = 1 V

Vs = 0

Vsub = 0

↑ ↑drift diffusion

Require a numerical scheme that is accurate and stable in the strong drift regime(Dn µnE, Dp µpE)

SAND 2017-3512C 2

Application Driver





∂t−∇ · Jn + R(φ, n, p) = 0


∂t+∇ · Jp + R(φ, n, p) = 0



source drain

gate

substrate

Nd Nd

Na

Ωox

Ωsi

Vg = 2 V

Vd = 1 V

Vs = 0

Vsub = 0



SAND 2017-3512C 3

Application Driver





∂t−∇ · Jn + R(φ, n, p) = 0


∂t+∇ · Jp + R(φ, n, p) = 0



source drain

gate

substrate

Nd Nd

Na

Ωox

Ωsi

Vg = 2 V

Vd = 1 V

Vs = 0

Vsub = 0



SAND 2017-3512C 4

Numerical DiscretizationControl Volume Finite Element Method (CVFEM)

Electron continuity equation

∂n

∂t−∇ · J + R = 0

J = un +D∇nu = µE

Finite element approximation of theelectron densitynh(x, t) =

∑j

nj(t)Nj(x)

Integrate over control volume and applythe divergence theorem

SAND 2017-3512C 5



∂n

∂t−∇ · J + R = 0


Ci

vi


∑j

nj(t)Nj(x)


∫Ci

∂nh

∂tdV−

∫∂Ci

J(nh)·~n dS+

∫Ci

R(nh) dV = 0

SAND 2017-3512C 6



∂n

∂t−∇ · J + R = 0


Ci

vi


∑j

nj(t)Nj(x)


∫Ci

∂nh

∂tdV−

∫∂Ci

J(nh)·~n dS+

∫Ci

R(nh) dV = 0

J(nh) =∑j

nj(t) (µENj +D∇Nj)

Nodal approximation for J(nh) can lead toinstabilities in strong drift regime.

SAND 2017-3512C 7



∂n

∂t−∇ · J + R = 0


Ci

vi


∑j

nj(t)Nj(x)


∫Ci

∂nh

∂tdV−

∫∂Ci

J(nh)·~n dS+

∫Ci

R(nh) dV = 0

J(nh) =∑j

nj(t) (µENj +D∇Nj)

Want a stabilized approximation for J thatincludes information on drift.

SAND 2017-3512C 8

Scharfetter-Gummel Upwinding

On edge eij solve 1-d boundary value problem forconstant Jij

dJij

ds= 0; Jij = µEijn(s) +D

dn(s)

ds

n(0) = ni and n(hij) = nj

Jij =aijD

hij(nj (coth(aij) + 1)− ni (coth(aij)− 1))

where aij =hijEijµ

2D, Eij = − (ψj−ψi)

hij

Ci

vi

vj

Jij

On structured grids, Jij is a goodestimate of J · ~n on ∂Cij

∫∂Ci

Jn · ~n dS ≈∑

∂Cij∈∂Ci

Jij |∂Cij |

D. L. Scharfetter and H. K Gummel, Large-signal analysis of a silicon read diode oscillator, IEEE Transactions onElectron Devices 16, 64-77, 1969.

SAND 2017-3512C 9



dJij


dn(s)

ds


Jij =aijD


where aij =hijEijµ


hij

Ci

vi

vj

Jij

On structured grids, Jij is a goodestimate of J · ~n on ∂Cij

∫∂Ci

Jn · ~n dS ≈∑

∂Cij∈∂Ci

Jij |∂Cij |


SAND 2017-3512C 10



dJij


dn(s)

ds


Jij =aijD


where aij =hijEijµ


hij

Ci

∂Cij

vi

vj

Jij

On structured grids, Jij is a goodestimate of J · ~n on ∂Cij∫

∂Ci

Jn · ~n dS ≈∑

∂Cij∈∂Ci

Jij |∂Cij |


SAND 2017-3512C 11



dJij


dn(s)

ds


Jij =aijD


where aij =hijEijµ


hij

Ci

∂Cij

∂Cik

∂Cil

∂Cim

vi

vj

vk

vl

vm

Jij

Jik

Jil

Jim


∂Ci

Jn · ~n dS ≈∑

∂Cij∈∂Ci

Jij |∂Cij |


SAND 2017-3512C 12



dJij


dn(s)

ds


Jij =aijD


where aij =hijEijµ


hij

Ci

vi

vj

Jijn n


∂Ci

Jn · ~n dS ≈∑

∂Cij∈∂Ci

Jij |∂Cij |


SAND 2017-3512C 13



dJij


dn(s)

ds


Jij =aijD


where aij =hijEijµ


hij

Ci

vi

vj

Jijn n

On unstructured grids, Jij is no longer agood estimate of J · ~n∫

∂Ci

Jn · ~n dS ≈∑

∂Cij∈∂Ci

Jij |∂Cij |


SAND 2017-3512C 14



dJij


dn(s)

ds


Jij =aijD


where aij =hijEijµ


hij

Ci

vi

vj

vm

vpJij

Jmi

Jpm

Jjp

JE

JE

On unstructured grids, Jij is no longer agood estimate of J · ~n∫

∂Ci

Jn · ~n dS ≈∑

∂Cij∈∂Ci

Jij |∂Cij |


SAND 2017-3512C 15

Multi-dimensional S-G Upwinding

Idea: Use H(curl)-conforming finite elements to expand edge current density intoprimary cell

Nodal space, GhD(Ω), and edge element space, ChD(Ω), belong toan exact sequence

given Ni ∈ GhD(Ω), then∇Ni ∈ C

hD(Ω)

∇Ni =∑

eij∈E(vi)

σij−→W ij , σij = ±1

In the limit of carrier drift velocity µE = 0,

limµE→0

Jij =D(nj − ni)

hij

JE =∑

eij∈E(Ω)

D(nj−ni)−→W ij =

∑vi∈V (Ω)

Dnj∇Nj = J(nh

)

Exponentially fitted current density

JE(x) =∑eij

hijJij−→W ij(x)

JE

JE

vi

vj

vm

vp

Jij

Jmi

Jpm

Jjp

Ci

∫eij

−→W ij · trsdl = δ

rsij

P. Bochev, K. Peterson, X. Gao A new control-volume finite element method for the stable and accurate solution ofthe drift-diffusion equations on general unstructured grids, CMAME, 254, pp. 126-145, 2013.

SAND 2017-3512C 16

Convergence on Structured Grids

Steady-state manufactured solution

−∇ · J +R = 0 in Ωn = g on ΓD

n(x, y) = x3 − y2

µE = (− sinπ/6, cosπ/6)

CVFEM-SG FVM-SGL2 error H1 error L2 error H1 error

Grid D = 1× 10−3

32 0.4373E-02 0.7620E-01 0.4364E-02 0.7572E-0164 0.2108E-02 0.4954E-01 0.2107E-02 0.4937E-01

128 0.9870E-03 0.3089E-01 0.9870E-03 0.3084E-01Rate 1.095 0.681 1.094 0.679Grid D = 1× 10−5

32 0.4732E-02 0.7897E-01 0.4723E-02 0.7850E-0164 0.2517E-02 0.5477E-01 0.2515E-02 0.5460E-01

128 0.1298E-02 0.3834E-01 0.1298E-02 0.3828E-01Rate 0.955 0.514 0.955 0.514

CVFEM-SG control volume finite element method with multi-dimensional S-G upwindingFVM-SG finite volume method with 1-d S-G upwinding

SAND 2017-3512C 17

Robust on Unstructured Grids

Manufactured linear solution

Grid CVFEM-SG FVM-SG Grid CVFEM-SG FVM-SG

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SAND 2017-3512C 18

Charon

To solve coupled drift-diffusion equations CVFEM-SG has been implementedin Sandia’s Charon code

Electrical transport simulation code for semiconductor devices, solvingPDE-based nonlinear equations

Built with Trilinos libraries (https://github.com/trilinos/Trilinos)that provide

Framework and residual-based assembly (Panzer, Phalanx)Linear and Nonlinear solvers (Belos, Nox, ML, etc)Temporal and spatial discretization (Tempus, Intrepid, Shards)Automatic differentiation (Sacado)Advanced manycore performance portability (Kokkos)

Developers: Suzey Gao, Gary Hennigan, Larry Musson, Andy Huang

SAND 2017-3512C 19

PN Diode

PN Diode coupled drift-diffusion equations

∇ · (ε0εsi∇ψ) = −q(p− n +Nd −Na) in Ω

−∇ · Jn + R(ψ, n, p) = 0 in Ω

∇ · Jp + R(ψ, n, p) = 0 in Ω

R(ψ, n, p) =np−n2

iτp(n+ni)+τn(p+ni)

+(cnn + cpp)(np− n2i )

PN Diode

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Compare with SUPG Stabilized FEM∫Ω

(µE +D∇n) · ∇ψdV +

∫ΩRψdV +

∫Ωτ(µE · ∇n)(µE · ∇ψ)dV = 0 ∀ψ

τ =1

√uTGu

(cothα−

1

α

)α =

√uTGu

D‖G‖u = µE

SAND 2017-3512C 20

PN DiodeMesh Dependence Study

hx = 0.01µm hx = 0.02µm hx = 0.05µm

0 0.2 0.4 0.6 0.8 1

−15

−10

−5

0

5

10

15

x [µ m]

Ele

ctric

al P

oten

tial i

n un

it of

kBT

/q

SUPG−FEMCVFEM−SG

0 0.2 0.4 0.6 0.8 1

−15

−10

−5

0

5

10

15

x [µ m]

Ele

ctric

al P

oten

tial i

n un

it of

kBT

/q


0 0.2 0.4 0.6 0.8 1

−15

−10

−5

0

5

10

15

x [µ m]

Ele

ctric

al P

oten

tial i

n un

it of

kBT

/q


0 0.2 0.4 0.6 0.8 110

−14

10−12

10−10

10−8

10−6

10−4

10−2

100

x [µ m]

Car

rier

Den

sity

[x 1

017 c

m−

3 ]

0 0.2 0.4 0.6 0.8 110

−14

10−12

10−10

10−8

10−6

10−4

10−2

100

x [µ m]

Car

rier

Den

sity

[x 1

018 c

m−

3 ]

0 0.2 0.4 0.6 0.8 110

−14

10−12

10−10

10−8

10−6

10−4

10−2

100

x [µ m]

Car

rier

Den

sity

[x 1

017 c

m−

3 ]

SAND 2017-3512C 21

PN DiodeStrong Drift Case

Na = Nd = 1.0× 1018cm−3, Va = −1.5V

0 0.2 0.4 0.6 0.8 1−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

x [µ m]

Ele

ctric

al P

oten

tial i

n un

it of

kBT

/q


FEM-SUPG solution develops undershoots and becomes negative in junctionregion, while CVFEM-SG exhibits only minimal undershoots and values

remain positive.

SAND 2017-3512C 22

Multi-scale Stabilized CVFEM

Divide each element into 4 bilinear (Q1)sub-elements

Define control volumes around each sub-cellnode

Compute 2nd order Jij at each macroelement edge

Use 2nd order H(curl) basis to evaluate JEat control volume integration points

JE(nh) =∑

eij∈E(Ω)

hijJij−→W ij

K

K1

K2

K3

K4

Bochev, Peterson, Perego "A multi-scale control-volume finite element method for advection-diffusionequations",IJNMF Vol. 77, Issue 11, pp. 641-667 (2015).

SAND 2017-3512C 23






JE(nh) =∑

eij∈E(Ω)

hijJij−→W ij

K


SAND 2017-3512C 24






JE(nh) =∑

eij∈E(Ω)

hijJij−→W ij

K

J04

J41

J78

J85

J36 J62

J07

J73

J48

J86

J15

J52JE


SAND 2017-3512C 25

Multi-scale Stabilized CVFEM2nd Order Edge Current Density

Solve 1-d boundary value problem along acompound edge for a linear J(s) = a+ bs

J(s) = µEsn(s) +Ddn

dsn(0) = ni, n(hs/2) = nk and n(hs) = nj

Jik = J(hs/4) Jkj = J(3hs/4)

Jkj

Jik

ni

nk

nj

Edge current density

Jik = Φ(ni, nk) + γ(ni, nj , nk)

Jkj = Φ(nk, nj) + γ(ni, nj , nk)

Φ(ni, nk) = aDh

(nk (coth(a) + 1)− ni (coth(a)− 1))

γ(ni, nj , nk) = Dh

(a coth(a)− 1)(ni(coth(a)− 1)− 2nk coth(a) + nj(coth(a) + 1)

)

SAND 2017-3512C 26






Jkj

Jik

ni

nk

nj




Φ(ni, nk) = aDh




)

SAND 2017-3512C 27






Jkj

Jik

ni

nk

nj




Φ(ni, nk) = aDh




)

SAND 2017-3512C 28


Subedge fluxes are a sum ofScharfetter-Gummel fluxes and ahigher-order correction term

High-order correction contributes ananti-diffusive flux

In pure advection limit (D → 0) methodonly modifies downstream segment


Jkj

Jik

ni

nk

nj

K

J04

J41

J78

J85

J36 J62

J07

J73

J48

J86

J15

J52JE

For assembly, edge basisfunctions are used only locally sono global edge data structuresare required

JE(nh) =∑

eij∈E(Ω)

hijJij−→W ij

SAND 2017-3512C 29

Manufactured Solution

−∇ · J(n) = R in Ω

J(n) = (D∇n+ µEn) in Ω

n = g on Γ

n(x, y) = x3 − y2

µE = (− sinπ/6, cosπ/6)

CVFEM-MS CVFEM-SG FEM-SUPGL2 error H1 error L2 error H1 error L2 error H1 error

Grid∗ ε = 1× 10−3

32 1.57e-3 6.05e-2 4.24e-3 7.48e-2 2.09e-4 3.61e-264 3.93e-4 2.89e-2 2.07e-3 4.91e-2 4.85e-5 1.80e-2128 8.98e-5 1.24e-2 9.78e-4 3.07e-2 1.11e-5 9.02e-3Rate 2.06 1.14 1.06 0.642 2.12 1.00Grid ε = 1× 10−5

32 1.69e-3 6.60e-2 4.73e-3 7.90e-2 2.30e-4 3.61e-264 4.54e-4 3.45e-2 2.52e-3 5.48e-2 5.78e-5 1.80e-2128 1.18e-4 1.76e-2 1.30e-3 3.83e-2 1.45e-5 9.02e-3Rate 1.92 0.955 0.933 0.521 1.99 1.00∗ For CVFEM-MS the size corresponds sub-elements rather than macro-elements.

SAND 2017-3512C 30

Skew Advection Test

−∇ · J(n) = R in Ω


n = g on Γ

g =

0 on ΓL ∪ ΓT ∪ (ΓB ∩ x ≤ 0.5)1 on ΓR ∪ (ΓB ∩ x > 0.5)

µE = (− sinπ/6, cosπ/6) D = 1.0× 10−5

CVFEM-MS CVFEM-SG SUPG

min = -0.0445 min = 0.00 min = -0.0459

max = 1.077 max = 1.004 max = 1.251

SAND 2017-3512C 31

Double Glazing Test

−∇ · J(n) = R in Ω


n = g on Γ

D = 1.0× 10−5

g =

0 on ΓL ∪ ΓT ∪ (ΓB ∩ x ≤ 0.5)1 on ΓR ∪ (ΓB ∩ x > 0.5)

µE =

(2(2y − 1)(1− (2x− 1)2)−2(2x− 1)(1− (2y − 1)2)

)CVFEM-MS CVFEM-SG SUPG

SAND 2017-3512C 32

Conclusions

Stabilization using an edge-element lifting of edge current densities offers astable and robust method for solving drift-diffusion equations

Works on unstructured grids

Does not require heuristic stabilization parameters

Although not provably monotone, violations of solution bounds are less than for acomparable scheme with SUPG stabilization

Can achieve 2nd-order convergence with multi-scale approach

Future workInvestigate modifications to achieve a monotone schemeImplement 2nd-order scheme in CharonMore detailed comparison of methods for full drift-diffusion equations

SAND 2017-3512C 33

Documents

Multiscale Stabilized Control Volume Finite Element Method for … · 2017. 8. 1. · Multiscale Stabilized Control Volume Finite Element Method for Advection-Diffusion Kara Peterson