NAND Gate Differential-Algebraic Equation (DAE) Work-Precision Diagrams
using OrdinaryDiffEq, DiffEqDevTools, ModelingToolkit, ODEInterfaceDiffEq,
Plots, Sundials, DASSL, DASKR
using LinearAlgebra
using ModelingToolkit: t_nounits as t, D_nounits as DProblem Parameters
const RGS = 4.0
const RGD = 4.0
const RBS = 10.0
const RBD = 10.0
const CGS = 6e-5
const CGD = 6e-5
const CBD = 2.4e-5
const CBS = 2.4e-5
const C9 = 5e-5
const DELTA = 0.02
const CURIS = 1e-14
const VTH = 25.85
const VDD = 5.0
const VBB = -2.5
const VT0_DEPL = -2.43
const CGAMMA_DEPL = 0.2
const PHI_DEPL = 1.28
const BETA_DEPL = 5.35e-4
const VT0_ENH = 0.2
const CGAMMA_ENH = 0.035
const PHI_ENH = 1.01
const BETA_ENH = 1.748e-30.001748Input Signal Functions
function pulse(t, t_start, v_low, t_rise, v_high, t_high, t_fall, t_period)
t_mod = mod(t, t_period)
if t_mod < t_start
return v_low
elseif t_mod < t_start + t_rise
return v_low + (v_high - v_low) * (t_mod - t_start) / t_rise
elseif t_mod < t_start + t_rise + t_high
return v_high
elseif t_mod < t_start + t_rise + t_high + t_fall
return v_high - (v_high - v_low) * (t_mod - t_start - t_rise - t_high) / t_fall
else
return v_low
end
end
V1(t) = pulse(t, 0.0, 0.0, 5.0, 5.0, 5.0, 5.0, 20.0)
V2(t) = pulse(t, 0.0, 0.0, 15.0, 5.0, 15.0, 5.0, 40.0)
function V1_derivative(t)
t_mod = mod(t, 20.0)
if 0.0 < t_mod < 5.0
return 1.0
elseif 10.0 < t_mod < 15.0
return -1.0
else
return 0.0
end
end
function V2_derivative(t)
t_mod = mod(t, 40.0)
if 0.0 < t_mod < 15.0
return 1.0/15.0
elseif 20.0 < t_mod < 35.0
return -1.0/15.0
else
return 0.0
end
endV2_derivative (generic function with 1 method)MOSFET Model Functions
function gdsp(ned, vds, vgs, vbs)
if ned == 1
vt0, cgamma, phi, beta = VT0_DEPL, CGAMMA_DEPL, PHI_DEPL, BETA_DEPL
else
vt0, cgamma, phi, beta = VT0_ENH, CGAMMA_ENH, PHI_ENH, BETA_ENH
end
phi_vbs = max(phi - vbs, 1e-12)
phi_safe = max(phi, 1e-12)
vte = vt0 + cgamma * (sqrt(phi_vbs) - sqrt(phi_safe))
if vgs - vte <= 0.0
return 0.0
elseif 0.0 < vgs - vte <= vds
return -beta * (vgs - vte)^2 * (1.0 + DELTA * vds)
elseif 0.0 < vds < vgs - vte
return -beta * vds * (2.0 * (vgs - vte) - vds) * (1.0 + DELTA * vds)
else
return 0.0
end
end
function gdsm(ned, vds, vgd, vbd)
if ned == 1
vt0, cgamma, phi, beta = VT0_DEPL, CGAMMA_DEPL, PHI_DEPL, BETA_DEPL
else
vt0, cgamma, phi, beta = VT0_ENH, CGAMMA_ENH, PHI_ENH, BETA_ENH
end
phi_vbd = max(phi - vbd, 1e-12)
phi_safe = max(phi, 1e-12)
vte = vt0 + cgamma * (sqrt(phi_vbd) - sqrt(phi_safe))
if vgd - vte <= 0.0
return 0.0
elseif 0.0 < vgd - vte <= -vds
return beta * (vgd - vte)^2 * (1.0 - DELTA * vds)
elseif 0.0 < -vds < vgd - vte
return -beta * vds * (2.0 * (vgd - vte) + vds) * (1.0 - DELTA * vds)
else
return 0.0
end
end
function ids(ned, vds, vgs, vbs, vgd, vbd)
if vds > 0.0
return gdsp(ned, vds, vgs, vbs)
elseif vds == 0.0
return 0.0
else
return gdsm(ned, vds, vgd, vbd)
end
end
function ibs(vbs)
if vbs <= 0.0
return -CURIS * (exp(vbs / VTH) - 1.0)
else
return 0.0
end
end
function ibd(vbd)
if vbd <= 0.0
return -CURIS * (exp(vbd / VTH) - 1.0)
else
return 0.0
end
endibd (generic function with 1 method)DAE System Definition
function nand_rhs!(f, y, p, t)
v1 = V1(t)
v2 = V2(t)
v1d = V1_derivative(t)
v2d = V2_derivative(t)
y1, y2, y3, y4, y5, y6, y7, y8, y9, y10, y11, y12, y13, y14 = y
f[1] = -(y1 - y5) / RGS - ids(1, y2 - y1, y5 - y1, y3 - y5, y5 - y2, y4 - VDD)
f[2] = -(y2 - VDD) / RGD + ids(1, y2 - y1, y5 - y1, y3 - y5, y5 - y2, y4 - VDD)
f[3] = -(y3 - VBB) / RBS + ibs(y3 - y5)
f[4] = -(y4 - VBB) / RBD + ibd(y4 - VDD)
f[5] = -(y5 - y1) / RGS - ibs(y3 - y5) - (y5 - y7) / RGD - ibd(y9 - y5)
f[6] = CGS * v1d - (y6 - y10) / RGS - ids(2, y7 - y6, v1 - y6, y8 - y10, v1 - y7, y9 - y5)
f[7] = CGD * v1d - (y7 - y5) / RGD + ids(2, y7 - y6, v1 - y6, y8 - y10, v1 - y7, y9 - y5)
f[8] = -(y8 - VBB) / RBS + ibs(y8 - y10)
f[9] = -(y9 - VBB) / RBD + ibd(y9 - y5)
f[10] = -(y10 - y6) / RGS - ibs(y8 - y10) - (y10 - y12) / RGD - ibd(y14 - y10)
f[11] = CGS * v2d - y11 / RGS - ids(2, y12 - y11, v2 - y11, y13, v2 - y12, y14 - y10)
f[12] = CGD * v2d - (y12 - y10) / RGD + ids(2, y12 - y11, v2 - y11, y13, v2 - y12, y14 - y10)
f[13] = -(y13 - VBB) / RBS + ibs(y13)
f[14] = -(y14 - VBB) / RBD + ibd(y14 - y10)
return nothing
end
# Mass matrix (singular is fine!)
dirMassMatrix = [
CGS 0 0 0 0 0 0 0 0 0 0 0 0 0
0 CGD 0 0 0 0 0 0 0 0 0 0 0 0
0 0 CBS 0 0 0 0 0 0 0 0 0 0 0
0 0 0 CBD 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 CGS 0 0 0 0 0 0 0 0
0 0 0 0 0 0 CGD 0 0 0 0 0 0 0
0 0 0 0 0 0 0 CBS 0 0 0 0 0 0
0 0 0 0 0 0 0 0 CBD 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 CGS 0 0 0
0 0 0 0 0 0 0 0 0 0 0 CGD 0 0
0 0 0 0 0 0 0 0 0 0 0 0 CBS 0
0 0 0 0 0 0 0 0 0 0 0 0 0 CBD
]
# Initial conditions
y0 = [5.0, 5.0, VBB, VBB, 5.0, 3.62385, 5.0, VBB, VBB, 3.62385, 0.0, 3.62385, VBB, VBB]
tspan = (0.0, 80.0)
# Mass matrix problem (original approach)
mmf = ODEFunction(nand_rhs!, mass_matrix=dirMassMatrix)
mmprob = ODEProblem(mmf, y0, tspan)
# DAEProblem version using direct DAE formulation
function nand_dae!(out, du, u, p, t)
v1 = V1(t)
v2 = V2(t)
v1d = V1_derivative(t)
v2d = V2_derivative(t)
y1, y2, y3, y4, y5, y6, y7, y8, y9, y10, y11, y12, y13, y14 = u
dy1, dy2, dy3, dy4, dy5, dy6, dy7, dy8, dy9, dy10, dy11, dy12, dy13, dy14 = du
# Differential equations: M*dy/dt - f = 0
# Convert from mass matrix form: M*dy/dt = f => M*dy/dt - f = 0
out[1] = CGS * dy1 - (-(y1 - y5) / RGS - ids(1, y2 - y1, y5 - y1, y3 - y5, y5 - y2, y4 - VDD))
out[2] = CGD * dy2 - (-(y2 - VDD) / RGD + ids(1, y2 - y1, y5 - y1, y3 - y5, y5 - y2, y4 - VDD))
out[3] = CBS * dy3 - (-(y3 - VBB) / RBS + ibs(y3 - y5))
out[4] = CBD * dy4 - (-(y4 - VBB) / RBD + ibd(y4 - VDD))
# Algebraic equations: g(y) = 0
out[5] = -(y5 - y1) / RGS - ibs(y3 - y5) - (y5 - y7) / RGD - ibd(y9 - y5)
out[6] = CGS * dy6 - (CGS * v1d - (y6 - y10) / RGS - ids(2, y7 - y6, v1 - y6, y8 - y10, v1 - y7, y9 - y5))
out[7] = CGD * dy7 - (CGD * v1d - (y7 - y5) / RGD + ids(2, y7 - y6, v1 - y6, y8 - y10, v1 - y7, y9 - y5))
out[8] = CBS * dy8 - (-(y8 - VBB) / RBS + ibs(y8 - y10))
out[9] = CBD * dy9 - (-(y9 - VBB) / RBD + ibd(y9 - y5))
# Algebraic equation: g(y) = 0
out[10] = -(y10 - y6) / RGS - ibs(y8 - y10) - (y10 - y12) / RGD - ibd(y14 - y10)
out[11] = CGS * dy11 - (CGS * v2d - y11 / RGS - ids(2, y12 - y11, v2 - y11, y13, v2 - y12, y14 - y10))
out[12] = CGD * dy12 - (CGD * v2d - (y12 - y10) / RGD + ids(2, y12 - y11, v2 - y11, y13, v2 - y12, y14 - y10))
out[13] = CBS * dy13 - (-(y13 - VBB) / RBS + ibs(y13))
out[14] = CBD * dy14 - (-(y14 - VBB) / RBD + ibd(y14 - y10))
return nothing
end
# Create DAE problem with automatic initialization
# Let IDA determine consistent initial derivatives automatically
du0_dae = zeros(14)
daeprob = DAEProblem(nand_dae!, du0_dae, y0, tspan)
# Generate reference solutions
ref_sol = solve(mmprob, Rodas5P(), abstol=1e-12, reltol=1e-12, tstops=0.0:5.0:80.0)
dae_ref_sol = solve(daeprob, DASKR.daskr(), abstol=1e-10, reltol=1e-10)
probs = [mmprob, daeprob]
refs = [ref_sol, dae_ref_sol]2-element Vector{SciMLBase.AbstractODESolution{Float64, 2, Vector{Vector{Fl
oat64}}}}:
SciMLBase.ODESolution{Float64, 2, Vector{Vector{Float64}}, Nothing, Nothin
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Problem{Vector{Float64}, Tuple{Float64, Float64}, true, SciMLBase.NullParam
eters, SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, typeof(Main.va
r"##WeaveSandBox#225".nand_rhs!), Matrix{Float64}, Nothing, Nothing, Nothin
g, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing,
typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}, Ba
se.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardODEPro
blem}, OrdinaryDiffEqRosenbrock.Rodas5P{0, ADTypes.AutoForwardDiff{nothing,
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dinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limit
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Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), No
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Float64}, SciMLBase.TimeGradientWrapper{true, SciMLBase.ODEFunction{true,
SciMLBase.FullSpecialize, typeof(Main.var"##WeaveSandBox#225".nand_rhs!), M
atrix{Float64}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothi
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lve.LinearCache{Matrix{Float64}, Vector{Float64}, Vector{Float64}, SciMLBas
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thing}, LinearSolve.InvPreconditioner{LinearAlgebra.Diagonal{Float64, Vecto
r{Float64}}}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Float64, Bo
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Config{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 7,
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rfaceForwardDiffExt.ForwardDiffTwoArgJacobianPrep{Nothing, ForwardDiff.Jaco
bianConfig{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64,
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tiationInterfaceForwardDiffExt.ForwardDiffTwoArgDerivativePrep{Nothing, For
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le{}}}, Float64, OrdinaryDiffEqRosenbrock.Rodas5P{0, ADTypes.AutoForwardDif
f{nothing, ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}}, Nothing
, typeof(OrdinaryDiffEqCore.DEFAULT_PRECS), Val{:forward}(), true, nothing,
typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.tri
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thing, Nothing, Nothing}([[5.0, 5.0, -2.5, -2.5, 5.0, 3.62385, 5.0, -2.5, -
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37444083e-9, 6.140180116257942e-9, 1.2577445462253577e-8, 2.531668589372688
e-8 … 75.9656227034349, 76.03275439714066, 76.11377726596815, 76.21497589
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591712, -2.5, -2.5], [5.00000000002622, 5.0, -2.5, -2.5, 5.0000000687241455
, 3.623850137423838, 5.0000001374220915, -2.5, -2.5, 3.6238500733057575, 9.
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5, -2.5, 3.623850146547945, 1.8304524201617467e-8, 3.6238500184162845, -2.5
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999999999999747, 4.999999999997678, -0.16064741111952058, 4.999999999997678
, -2.499999999999991, -2.4999999999999747, -0.16064741111952405, 2.16975020
0998369e-46, -0.16064741111952058, -2.4999999999999907, -2.499999999999991]
, [4.999999999997699, 4.9999999999999805, -2.499999999999975, -2.4999999999
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999999991, -2.499999999999975, -0.16064741111952596, 1.3969261109800321e-46
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999999997678, -0.16064741111952344, 4.999999999997678, -2.499999999999991,
-2.499999999999975, -0.1606474111195269, 1.0942414667594956e-46, -0.1606474
1111952344, -2.4999999999999907, -2.499999999999991], [4.999999999997699, 4
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997678, -2.499999999999991, -2.499999999999975, -0.16064741111952743, 9.445
681557259906e-47, -0.16064741111952396, -2.4999999999999907, -2.49999999999
9991], [4.999999999997699, 4.9999999999999805, -2.499999999999975, -2.49999
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99999999997699, 4.9999999999999805, -2.499999999999975, -2.499999999999975,
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6064741111952396, -2.4999999999999907, -2.499999999999991], [4.999999999997
706, 4.9999999999999805, -2.499999999999975, -2.499999999999975, 5.00000000
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76007080079e-8, 1.3744831085205078e-7, 2.747774124145508e-7 … 79.99976337
013346, 79.99989614579006, 79.99996253361836, 79.99999572753251, 79.9999998
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se.LinearInterpolation{Vector{Float64}, Vector{Vector{Float64}}}([0.0, 1.19
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07032e-9, 1.7285346984863283e-8, 3.4451484680175785e-8, 6.878376007080079e-
8, 1.3744831085205078e-7, 2.747774124145508e-7 … 79.99976337013346, 79.99
989614579006, 79.99996253361836, 79.99999572753251, 79.99999987677178, 79.9
9999990918771, 79.99999997401957, 79.99999998212355, 79.99999999833152, 80.
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3.62385000119209, 5.00000000119209, -2.5, -2.5, 3.6238500006357746, 7.9472
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-2.5, -2.5, 5.000000001132478, 3.623850002264969, 5.000000002264969, -2.5,
-2.5, 3.6238500012079786, 1.5099734684785147e-10, 3.623850000151006, -2.5,
-2.5], [5.000000000000028, 5.0, -2.5, -2.5, 5.000000002205362, 3.6238500044
10717, 5.000000004410715, -2.5, -2.5, 3.623850002352388, 2.940458653338863e
-10, 3.6238500002940754, -2.5, -2.5], [5.000000000000105, 5.0, -2.5, -2.5,
5.000000004351129, 3.6238500087021794, 5.000000008702173, -2.5, -2.5, 3.623
8500046412065, 5.801377865837543e-10, 3.6238500005802505, -2.5, -2.5], [5.0
00000000000415, 5.0, -2.5, -2.5, 5.0000000086426635, 3.6238500172849597, 5.
000000017284933, -2.5, -2.5, 3.6238500092188435, 1.1523011669264668e-9, 3.6
238500011527437, -2.5, -2.5], [5.000000000001647, 5.0, -2.5, -2.5, 5.000000
017225733, 3.623850034449948, 5.0000000344498385, -2.5, -2.5, 3.62385001837
4117, 2.2965460848376502e-9, 3.623850002298303, -2.5, -2.5], [5.00000000000
6565, 5.0, -2.5, -2.5, 5.000000034391871, 3.623850068777632, 5.000000068777
1955, -2.5, -2.5, 3.6238500366846638, 4.584708596387163e-9, 3.6238500045917
12, -2.5, -2.5], [5.00000000002622, 5.0, -2.5, -2.5, 5.0000000687241455, 3.
623850137423838, 5.0000001374220915, -2.5, -2.5, 3.6238500733057575, 9.1597
24696880534e-9, 3.6238500091876933, -2.5, -2.5], [5.000000000104774, 5.0, -
2.5, -2.5, 5.000000137388696, 3.623850274679622, 5.000000274672638, -2.5, -
2.5, 3.623850146547945, 1.8304524201617467e-8, 3.6238500184162845, -2.5, -2
.5] … [4.999999999997699, 4.99999999999998, -2.4999999999999747, -2.49999
99999999747, 4.999999999997678, -0.16064741111952058, 4.999999999997678, -2
.499999999999991, -2.4999999999999747, -0.16064741111952405, 2.169750200998
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5, 4.999999999997678, -0.1606474111195225, 4.999999999997678, -2.4999999999
99991, -2.499999999999975, -0.16064741111952596, 1.3969261109800321e-46, -0
.1606474111195225, -2.4999999999999907, -2.499999999999991], [4.99999999999
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99997678, -0.16064741111952344, 4.999999999997678, -2.499999999999991, -2.4
99999999999975, -0.1606474111195269, 1.0942414667594956e-46, -0.16064741111
952344, -2.4999999999999907, -2.499999999999991], [4.999999999997699, 4.999
9999999999805, -2.499999999999975, -2.499999999999975, 4.999999999997678, -
0.1606474111195239, 4.999999999997678, -2.499999999999991, -2.4999999999999
75, -0.16064741111952738, 9.612877096469394e-47, -0.1606474111195239, -2.49
99999999999907, -2.499999999999991], [4.999999999997699, 4.9999999999999805
, -2.499999999999975, -2.499999999999975, 4.999999999997678, -0.16064741111
952396, 4.999999999997678, -2.499999999999991, -2.499999999999975, -0.16064
741111952743, 9.44950928400666e-47, -0.16064741111952396, -2.49999999999999
07, -2.499999999999991], [4.999999999997699, 4.9999999999999805, -2.4999999
99999975, -2.499999999999975, 4.999999999997678, -0.16064741111952396, 4.99
9999999997678, -2.499999999999991, -2.499999999999975, -0.16064741111952743
, 9.448233145335228e-47, -0.16064741111952396, -2.4999999999999907, -2.4999
99999999991], [4.999999999997699, 4.9999999999999805, -2.499999999999975, -
2.499999999999975, 4.999999999997678, -0.16064741111952396, 4.9999999999976
78, -2.499999999999991, -2.499999999999975, -0.16064741111952743, 9.4456815
57259906e-47, -0.16064741111952396, -2.4999999999999907, -2.499999999999991
], [4.999999999997699, 4.9999999999999805, -2.499999999999975, -2.499999999
999975, 4.999999999997678, -0.16064741111952396, 4.999999999997678, -2.4999
99999999991, -2.499999999999975, -0.16064741111952743, 9.445362619519935e-4
7, -0.16064741111952396, -2.4999999999999907, -2.499999999999991], [4.99999
9999997699, 4.9999999999999805, -2.499999999999975, -2.499999999999975, 4.9
99999999997678, -0.16064741111952396, 4.999999999997678, -2.499999999999991
, -2.499999999999975, -0.16064741111952743, 9.444724787114847e-47, -0.16064
741111952396, -2.4999999999999907, -2.499999999999991], [4.999999999997706,
4.9999999999999805, -2.499999999999975, -2.499999999999975, 5.000000000831
9185, -0.16064740945105146, 5.000000001666151, -2.499999999999991, -2.49999
9999999975, -0.16064741022967194, 1.112309979429854e-10, -0.160647411008285
45, -2.4999999999999907, -2.499999999999991]], false), true, 0, nothing, Sc
iMLBase.ReturnCode.Success, nothing)Generate Reference Solution and Plot
plot(ref_sol, title="NAND Gate Circuit - Node Potentials (Mass Matrix)",
xlabel="Time", ylabel="Voltage (V)", legend=:outertopright)
plot(dae_ref_sol, title="NAND Gate Circuit - Node Potentials (DAE)",
xlabel="Time", ylabel="Voltage (V)", legend=:outertopright)
Omissions
Some solvers are omitted due to performance or stability issues at certain tolerances.
Work-Precision Benchmarks
High Tolerances
abstols = 1.0 ./ 10.0 .^ (5:8)
reltols = 1.0 ./ 10.0 .^ (1:4)
setups = [
Dict(:prob_choice => 1, :alg=>Rodas4()),
Dict(:prob_choice => 1, :alg=>FBDF()),
Dict(:prob_choice => 1, :alg=>QNDF()),
Dict(:prob_choice => 1, :alg=>radau()),
Dict(:prob_choice => 1, :alg=>RadauIIA5()),
Dict(:prob_choice => 2, :alg=>IDA()),
Dict(:prob_choice => 2, :alg=>DASKR.daskr())
]
wp = WorkPrecisionSet(probs, abstols, reltols, setups;
save_everystep=false, appxsol=refs,
maxiters=Int(1e5), numruns=10,
tstops=0.0:5.0:80.0)
plot(wp, title="NAND Gate DAE - Work-Precision (High Tolerances)")DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.6002616966073D+02
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.3372659064979D+02
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.2002280781277D+02
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
abstols = 1.0 ./ 10.0 .^ (6:8)
reltols = 1.0 ./ 10.0 .^ (2:4)
setups = [
Dict(:prob_choice => 1, :alg=>Rosenbrock23()),
Dict(:prob_choice => 1, :alg=>Rodas4()),
Dict(:prob_choice => 1, :alg=>Rodas5P()),
Dict(:prob_choice => 1, :alg=>FBDF()),
Dict(:prob_choice => 2, :alg=>IDA()),
Dict(:prob_choice => 2, :alg=>DASKR.daskr())
]
wp = WorkPrecisionSet(probs, abstols, reltols, setups;
save_everystep=false, appxsol=refs,
maxiters=Int(1e5), numruns=10,
tstops=0.0:5.0:80.0)
plot(wp, title="NAND Gate DAE - Work-Precision (Medium Tolerances)")DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.6002616966073D+02
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.3372659064979D+02
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.2002280781277D+02
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
Timeseries Errors
abstols = 1.0 ./ 10.0 .^ (5:8)
reltols = 1.0 ./ 10.0 .^ (1:4)
setups = [
Dict(:prob_choice => 1, :alg=>Rosenbrock23()),
Dict(:prob_choice => 1, :alg=>Rodas4()),
Dict(:prob_choice => 1, :alg=>FBDF()),
Dict(:prob_choice => 1, :alg=>QNDF()),
Dict(:prob_choice => 1, :alg=>radau()),
Dict(:prob_choice => 1, :alg=>RadauIIA5()),
Dict(:prob_choice => 2, :alg=>IDA())
]
wp = WorkPrecisionSet(probs, abstols, reltols, setups; error_estimate=:l2,
save_everystep=false, appxsol=refs,
maxiters=Int(1e5), numruns=10,
tstops=0.0:5.0:80.0)
plot(wp, title="NAND Gate DAE - Timeseries Errors (High Tolerances)")
abstols = 1.0 ./ 10.0 .^ (6:8)
reltols = 1.0 ./ 10.0 .^ (2:4)
setups = [
Dict(:prob_choice => 1, :alg=>Rosenbrock23()),
Dict(:prob_choice => 1, :alg=>Rodas4()),
Dict(:prob_choice => 1, :alg=>Rodas5P()),
Dict(:prob_choice => 1, :alg=>FBDF()),
Dict(:prob_choice => 2, :alg=>IDA()),
Dict(:prob_choice => 2, :alg=>DASKR.daskr())
]
wp = WorkPrecisionSet(probs, abstols, reltols, setups; error_estimate=:l2,
save_everystep=false, appxsol=refs,
maxiters=Int(1e5), numruns=10,
tstops=0.0:5.0:80.0)
plot(wp, title="NAND Gate DAE - Timeseries Errors (Medium Tolerances)")DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.6002616966073D+02
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.3372659064979D+02
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.2002280781277D+02
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
Low Tolerances (High Accuracy)
This measures performance when high accuracy is needed.
abstols = 1.0 ./ 10.0 .^ (7:12)
reltols = 1.0 ./ 10.0 .^ (4:9)
setups = [
Dict(:prob_choice => 1, :alg=>Rodas5P()),
Dict(:prob_choice => 1, :alg=>Rodas4()),
Dict(:prob_choice => 1, :alg=>FBDF()),
Dict(:prob_choice => 1, :alg=>QNDF()),
Dict(:prob_choice => 1, :alg=>radau()),
Dict(:prob_choice => 1, :alg=>RadauIIA5()),
Dict(:prob_choice => 2, :alg=>IDA()),
Dict(:prob_choice => 2, :alg=>DASKR.daskr())
]
wp = WorkPrecisionSet(probs, abstols, reltols, setups;
save_everystep=false, appxsol=refs,
maxiters=Int(1e5), numruns=10,
tstops=0.0:5.0:80.0)
plot(wp, title="NAND Gate DAE - Work-Precision (Low Tolerances)")DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.2001247231433D+02
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.1467053150785D+02
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.9999970978524D+01
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.5000417061503D+01
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.4184044592402D+01
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.2682405091706D+01
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
wp = WorkPrecisionSet(probs, abstols, reltols, setups; error_estimate=:l2,
save_everystep=false, appxsol=refs,
maxiters=Int(1e5), numruns=10,
tstops=0.0:5.0:80.0)
plot(wp, title="NAND Gate DAE - Timeseries Errors (Low Tolerances)")DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.2001247231433D+02
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.1467053150785D+02
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.9999970978524D+01
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.5000417061503D+01
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.4184044592402D+01
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
DASKR-- AT CURRENT T (=R1) 500 STEPS
In above message, R1 = 0.2682405091706D+01
DASKR-- TAKEN ON THIS CALL BEFORE REACHING TOUT
Analysis of Key Nodes
# Original 14-variable system: y1, y2, y3, y4, y5, y6, y7, y8, y9, y10, y11, y12, y13, y14
key_nodes = [1, 5, 6, 10, 11, 12] # Representative nodes from different parts of the circuit
node_names = ["Node 1", "Node 5", "Node 6", "Node 10", "Node 11", "Node 12"]
p_nodes = plot()
for (i, node) in enumerate(key_nodes)
plot!(ref_sol.t, [u[node] for u in ref_sol.u],
label=node_names[i], linewidth=2)
end
plot!(p_nodes, title="NAND Gate - Key Node Potentials",
xlabel="Time (s)", ylabel="Voltage (V)", legend=:outertopright)
Conclusion
Appendix
These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.
To locally run this benchmark, do the following commands:
using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/DAE","NANDGateProblem.jmd")Computer Information:
Julia Version 1.10.10
Commit 95f30e51f41 (2025-06-27 09:51 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 128 × AMD EPYC 7502 32-Core Processor
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 128 default, 0 interactive, 64 GC (on 128 virtual cores)
Environment:
JULIA_CPU_THREADS = 128
JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953
Package Information:
Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/DAE/Project.toml`
[165a45c3] DASKR v2.9.1
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[09606e27] ODEInterfaceDiffEq v3.13.4
⌃ [1dea7af3] OrdinaryDiffEq v6.98.0
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[31c91b34] SciMLBenchmarks v0.1.3
⌃ [90137ffa] StaticArrays v1.9.13
[c3572dad] Sundials v4.28.0
[10745b16] Statistics v1.10.0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.And the full manifest:
Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/DAE/Manifest.toml`
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⌃ [0c5d862f] Symbolics v6.43.0
[3783bdb8] TableTraits v1.0.1
[bd369af6] Tables v1.12.1
[ed4db957] TaskLocalValues v0.1.3
[62fd8b95] TensorCore v0.1.1
[8ea1fca8] TermInterface v2.0.0
[1c621080] TestItems v1.0.0
[8290d209] ThreadingUtilities v0.5.5
[a759f4b9] TimerOutputs v0.5.29
[3bb67fe8] TranscodingStreams v0.11.3
⌃ [410a4b4d] Tricks v0.1.10
[781d530d] TruncatedStacktraces v1.4.0
[5c2747f8] URIs v1.6.1
[3a884ed6] UnPack v1.0.2
[1cfade01] UnicodeFun v0.4.1
⌃ [1986cc42] Unitful v1.23.1
[45397f5d] UnitfulLatexify v1.7.0
[a7c27f48] Unityper v0.1.6
[41fe7b60] Unzip v0.2.0
[81def892] VersionParsing v1.3.0
[897b6980] WeakValueDicts v0.1.0
[44d3d7a6] Weave v0.10.12
[ddb6d928] YAML v0.4.14
[c2297ded] ZMQ v1.4.1
[6e34b625] Bzip2_jll v1.0.9+0
[83423d85] Cairo_jll v1.18.5+0
[655fdf9c] DASKR_jll v1.0.0+0
[ee1fde0b] Dbus_jll v1.16.2+0
[2702e6a9] EpollShim_jll v0.0.20230411+1
[2e619515] Expat_jll v2.6.5+0
⌅ [b22a6f82] FFMPEG_jll v4.4.4+1
[a3f928ae] Fontconfig_jll v2.16.0+0
[d7e528f0] FreeType2_jll v2.13.4+0
[559328eb] FriBidi_jll v1.0.17+0
[0656b61e] GLFW_jll v3.4.0+2
[d2c73de3] GR_jll v0.73.17+0
[b0724c58] GettextRuntime_jll v0.22.4+0
⌃ [f8c6e375] Git_jll v2.50.0+0
[7746bdde] Glib_jll v2.84.3+0
[3b182d85] Graphite2_jll v1.3.15+0
[2e76f6c2] HarfBuzz_jll v8.5.1+0
⌃ [1d5cc7b8] IntelOpenMP_jll v2025.0.4+0
[aacddb02] JpegTurbo_jll v3.1.1+0
[c1c5ebd0] LAME_jll v3.100.3+0
[88015f11] LERC_jll v4.0.1+0
[1d63c593] LLVMOpenMP_jll v18.1.8+0
[dd4b983a] LZO_jll v2.10.3+0
[e9f186c6] Libffi_jll v3.4.7+0
[7e76a0d4] Libglvnd_jll v1.7.1+1
[94ce4f54] Libiconv_jll v1.18.0+0
[4b2f31a3] Libmount_jll v2.41.0+0
[89763e89] Libtiff_jll v4.7.1+0
[38a345b3] Libuuid_jll v2.41.0+0
⌃ [856f044c] MKL_jll v2025.0.1+1
⌃ [c771fb93] ODEInterface_jll v0.0.1+0
[e7412a2a] Ogg_jll v1.3.6+0
[9bd350c2] OpenSSH_jll v10.0.1+0
[458c3c95] OpenSSL_jll v3.5.1+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
[91d4177d] Opus_jll v1.5.2+0
[36c8627f] Pango_jll v1.56.3+0
⌅ [30392449] Pixman_jll v0.44.2+0
[c0090381] Qt6Base_jll v6.8.2+1
[629bc702] Qt6Declarative_jll v6.8.2+1
[ce943373] Qt6ShaderTools_jll v6.8.2+1
[e99dba38] Qt6Wayland_jll v6.8.2+1
[f50d1b31] Rmath_jll v0.5.1+0
⌅ [fb77eaff] Sundials_jll v5.2.2+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
⌃ [a2964d1f] Wayland_jll v1.23.1+2
⌅ [02c8fc9c] XML2_jll v2.13.6+1
[ffd25f8a] XZ_jll v5.8.1+0
[f67eecfb] Xorg_libICE_jll v1.1.2+0
[c834827a] Xorg_libSM_jll v1.2.6+0
[4f6342f7] Xorg_libX11_jll v1.8.12+0
[0c0b7dd1] Xorg_libXau_jll v1.0.13+0
[935fb764] Xorg_libXcursor_jll v1.2.4+0
[a3789734] Xorg_libXdmcp_jll v1.1.6+0
[1082639a] Xorg_libXext_jll v1.3.7+0
[d091e8ba] Xorg_libXfixes_jll v6.0.1+0
[a51aa0fd] Xorg_libXi_jll v1.8.3+0
[d1454406] Xorg_libXinerama_jll v1.1.6+0
[ec84b674] Xorg_libXrandr_jll v1.5.5+0
[ea2f1a96] Xorg_libXrender_jll v0.9.12+0
[c7cfdc94] Xorg_libxcb_jll v1.17.1+0
[cc61e674] Xorg_libxkbfile_jll v1.1.3+0
[e920d4aa] Xorg_xcb_util_cursor_jll v0.1.5+0
[12413925] Xorg_xcb_util_image_jll v0.4.1+0
[2def613f] Xorg_xcb_util_jll v0.4.1+0
[975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
[0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
[c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
[35661453] Xorg_xkbcomp_jll v1.4.7+0
[33bec58e] Xorg_xkeyboard_config_jll v2.44.0+0
[c5fb5394] Xorg_xtrans_jll v1.6.0+0
[8f1865be] ZeroMQ_jll v4.3.6+0
[3161d3a3] Zstd_jll v1.5.7+1
[35ca27e7] eudev_jll v3.2.14+0
[214eeab7] fzf_jll v0.61.1+0
⌃ [a4ae2306] libaom_jll v3.11.0+0
⌅ [0ac62f75] libass_jll v0.15.2+0
[1183f4f0] libdecor_jll v0.2.2+0
[2db6ffa8] libevdev_jll v1.13.4+0
[f638f0a6] libfdk_aac_jll v2.0.4+0
[36db933b] libinput_jll v1.28.1+0
[b53b4c65] libpng_jll v1.6.50+0
[a9144af2] libsodium_jll v1.0.21+0
[f27f6e37] libvorbis_jll v1.3.8+0
[009596ad] mtdev_jll v1.1.7+0
[1317d2d5] oneTBB_jll v2022.0.0+0
⌅ [1270edf5] x264_jll v2021.5.5+0
⌅ [dfaa095f] x265_jll v3.5.0+0
[d8fb68d0] xkbcommon_jll v1.9.2+0
[0dad84c5] ArgTools v1.1.1
[56f22d72] Artifacts
[2a0f44e3] Base64
[ade2ca70] Dates
[8ba89e20] Distributed
[f43a241f] Downloads v1.6.0
[7b1f6079] FileWatching
[9fa8497b] Future
[b77e0a4c] InteractiveUtils
[4af54fe1] LazyArtifacts
[b27032c2] LibCURL v0.6.4
[76f85450] LibGit2
[8f399da3] Libdl
[37e2e46d] LinearAlgebra
[56ddb016] Logging
[d6f4376e] Markdown
[a63ad114] Mmap
[ca575930] NetworkOptions v1.2.0
[44cfe95a] Pkg v1.10.0
[de0858da] Printf
[3fa0cd96] REPL
[9a3f8284] Random
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization
[1a1011a3] SharedArrays
[6462fe0b] Sockets
[2f01184e] SparseArrays v1.10.0
[10745b16] Statistics v1.10.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test
[cf7118a7] UUIDs
[4ec0a83e] Unicode
[e66e0078] CompilerSupportLibraries_jll v1.1.1+0
[deac9b47] LibCURL_jll v8.4.0+0
[e37daf67] LibGit2_jll v1.6.4+0
[29816b5a] LibSSH2_jll v1.11.0+1
[c8ffd9c3] MbedTLS_jll v2.28.2+1
[14a3606d] MozillaCACerts_jll v2023.1.10
[4536629a] OpenBLAS_jll v0.3.23+4
[05823500] OpenLibm_jll v0.8.5+0
[efcefdf7] PCRE2_jll v10.42.0+1
[bea87d4a] SuiteSparse_jll v7.2.1+1
[83775a58] Zlib_jll v1.2.13+1
[8e850b90] libblastrampoline_jll v5.11.0+0
[8e850ede] nghttp2_jll v1.52.0+1
[3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.