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Andrews' Squeezing Mechanism DAE Work-Precision Diagrams

This is a benchmark of Andrews' squeezing mechanism, a classic index-3 DAE from the IVP Test Set (andrews.f). The system models the motion of 7 rigid bodies connected by joints without friction.

Mathematical form:$K \frac{dy}{dt} = \phi(y)$ where $y = (q, \dot{q}, \ddot{q}, \lambda) \in \mathbb{R}^{27}$ consists of 7 position coordinates, 7 velocities, 7 acceleration-like variables, and 6 Lagrange multipliers.

The underlying constrained mechanical system is:

\[M(q)\ddot{q} = f(q, \dot{q}) - G^T(q)\lambda, \quad 0 = g(q)\]

We benchmark three formulations of this problem:

  1. Mass-Matrix ODE Form:$K \cdot dy/dt = \phi(y)$ with $K = \textrm{diag}(I_{14}, 0_{13})$, solved with ODE solvers that handle singular mass matrices.
  2. DAE Residual Form:$F(\dot{y}, y, t) = K \cdot \dot{y} - \phi(y) = 0$, solved with dedicated DAE solvers (IDA from Sundials).
  3. MTK Index-Reduced ODE: The full index-3 system defined symbolically with position-level constraints, automatically index-reduced by structural_simplify.

Characteristics:

  • 14 differential equations + 13 algebraic equations = 27 total
  • Index-3 DAE (position-level constraints)
  • Time interval: $t \in [0, 0.03]$
  • 7×7 configuration-dependent mass matrix $M(q)$
  • 6 holonomic constraints $g(q) = 0$
  • The mechanism rotates dramatically: $\beta$ goes from $-0.062$ to $+15.8$ rad

This problem has a special structure exploited by the Fortran RADAU5 code: $y_i' = y_{i+7}$ for $i = 1,\ldots,14$ (positions track velocities, velocities track accelerations). The M1/M2 parameters tell RADAU5 about this structure so it only needs to solve a 13×13 (not 27×27) nonlinear system at each step.

Reference: Giles (1978), Manning (1981); Fortran source from http://www.dm.uniba.it/~testset/

using OrdinaryDiffEq, Sundials, DiffEqDevTools, ModelingToolkit, Plots
using ODEInterfaceDiffEq, LinearAlgebra
using ModelingToolkit: t_nounits as t, D_nounits as D

Parameters

Physical parameters of the 7-body mechanism (from andrews.f):

# Body masses
const am_m1 = 0.04325;  const am_m2 = 0.00365;  const am_m3 = 0.02373
const am_m4 = 0.00706;  const am_m5 = 0.07050;  const am_m6 = 0.00706
const am_m7 = 0.05498

# Fixed pivot coordinates
const am_xa = -0.06934;  const am_ya = -0.00227
const am_xb = -0.03635;  const am_yb = 0.03273
const am_xc = 0.014;     const am_yc = 0.072

# Spring constant
const am_c0 = 4530.0

# Moments of inertia
const am_i1 = 2.194e-6;  const am_i2 = 4.410e-7;  const am_i3 = 5.255e-6
const am_i4 = 5.667e-7;  const am_i5 = 1.169e-5;  const am_i6 = 5.667e-7
const am_i7 = 1.912e-5

# Link lengths and geometry (d, e, u renamed to avoid Julia conflicts)
const am_d  = 28e-3;     const am_da = 0.0115
const am_e  = 0.02;      const am_ea = 0.01421
const am_rr = 7e-3;      const am_ra = 9.2e-4;    const am_l0 = 0.07785
const am_ss = 35e-3;     const am_sa = 0.01874;   const am_sb = 0.01043
const am_sc = 18e-3;     const am_sd = 0.02
const am_ta = 0.02308;   const am_tb = 9.16e-3
const am_u  = 0.04;      const am_ua = 0.01228;   const am_ub = 4.49e-3
const am_zf = 0.02;      const am_zt = 0.04;      const am_fa = 0.01421
const am_mom = 33e-3
0.033

Initial Conditions

Consistent initial values from andrews.f. Positions $q_0$ satisfy the constraints $g(q_0) = 0$, with zero initial velocities and consistent initial accelerations and multipliers.

y0 = zeros(27)
# Positions q (angles of 7 rigid bodies: β, Θ, γ, Φ, δ, Ω, ε)
y0[1]  = -0.0617138900142764496358948458001
y0[2]  =  0.0
y0[3]  =  0.455279819163070380255912382449
y0[4]  =  0.222668390165885884674473185609
y0[5]  =  0.487364979543842550225598953530
y0[6]  = -0.222668390165885884674473185609
y0[7]  =  1.23054744454982119249735015568
# Velocities dq/dt (all zero)
# y0[8:14] = 0
# Accelerations (algebraic, from M*a = f - G^T*λ at t=0)
y0[15] =  14222.4439199541138705911625887
y0[16] = -10666.8329399655854029433719415
# y0[17:21] = 0
# Lagrange multipliers
y0[22] =  98.5668703962410896057654982170
y0[23] = -6.12268834425566265503114393122
# y0[24:27] = 0

println("Initial positions q₀ = ", y0[1:7])
println("Initial multipliers λ₀ = ", y0[22:27])
Initial positions q₀ = [-0.06171389001427645, 0.0, 0.45527981916307037, 0.2
2266839016588588, 0.48736497954384256, -0.22266839016588588, 1.230547444549
8212]
Initial multipliers λ₀ = [98.56687039624109, -6.122688344255662, 0.0, 0.0, 
0.0, 0.0]

Reference Solution at t = 0.03

Computed by PSIDE on Cray C90 with double precision (from andrews.fsolut):

y_ref = zeros(27)
y_ref[1]  =  0.1581077119629904e+2
y_ref[2]  = -0.1575637105984298e+2
y_ref[3]  =  0.4082224013073101e-1
y_ref[4]  = -0.5347301163226948e+0
y_ref[5]  =  0.5244099658805304e+0
y_ref[6]  =  0.5347301163226948e+0
y_ref[7]  =  0.1048080741042263e+1
y_ref[8]  =  0.1139920302151208e+4
y_ref[9]  = -0.1424379294994111e+4
y_ref[10] =  0.1103291221937134e+2
y_ref[11] =  0.1929337464421385e+2
y_ref[12] =  0.5735699284790808e+0
y_ref[13] = -0.1929337464421385e+2
y_ref[14] =  0.3231791658026955e+0
y_ref[15] = -0.2463176316945196e+5
y_ref[16] =  0.5185037701610329e+5
y_ref[17] =  0.3241025686413781e+6
y_ref[18] =  0.5667493645176213e+6
y_ref[19] =  0.1674362929479361e+5
y_ref[20] = -0.5667493645176222e+6
y_ref[21] =  0.9826520791458422e+4
y_ref[22] =  0.1991753333731910e+3
y_ref[23] = -0.2975531228015052e+2
y_ref[24] =  0.2306654119098399e+2
y_ref[25] =  0.3145271365475927e+2
y_ref[26] =  0.2264249232082739e+2
y_ref[27] =  0.1161740700019673e+2
11.61740700019673

Right-Hand Side

The complete right-hand side $\phi(y)$ for the mass-matrix ODE form $K \cdot dy/dt = \phi(y)$. The 27 variables decompose into 14 differential and 13 algebraic components. The special block structure ($y_i' = y_{i+7}$ for $i = 1,\ldots,14$) is exploited by RADAU5 via the DIMOFIND1VAR, DIMOFIND2VAR, DIMOFIND3VAR parameters passed through the radau5() wrapper.

function andrews_rhs_full!(du, u, p, t)
    sibe = sin(u[1]); cobe = cos(u[1]); sith = sin(u[2]); cth  = cos(u[2])
    siga = sin(u[3]); coga = cos(u[3]); siph = sin(u[4]); coph = cos(u[4])
    side = sin(u[5]); cde  = cos(u[5]); siom = sin(u[6]); coom = cos(u[6])
    siep = sin(u[7]); coep = cos(u[7])
    sibeth = sin(u[1]+u[2]); cobeth = cos(u[1]+u[2])
    siphde = sin(u[4]+u[5]); cophde = cos(u[4]+u[5])
    siomep = sin(u[6]+u[7]); coomep = cos(u[6]+u[7])
    bep = u[8]; thp = u[9]; php = u[11]; dep = u[12]; omp = u[13]; epp = u[14]

    # 7×7 configuration-dependent mass matrix
    M = zeros(7,7)
    M[1,1] = am_m1*am_ra^2 + am_m2*(am_rr^2 - 2*am_da*am_rr*cth + am_da^2) + am_i1 + am_i2
    M[2,1] = am_m2*(am_da^2 - am_da*am_rr*cth) + am_i2; M[2,2] = am_m2*am_da^2 + am_i2
    M[3,3] = am_m3*(am_sa^2+am_sb^2) + am_i3; M[4,4] = am_m4*(am_e-am_ea)^2 + am_i4
    M[5,4] = am_m4*((am_e-am_ea)^2 + am_zt*(am_e-am_ea)*siph) + am_i4
    M[5,5] = am_m4*(am_zt^2 + 2*am_zt*(am_e-am_ea)*siph + (am_e-am_ea)^2) +
             am_m5*(am_ta^2+am_tb^2) + am_i4 + am_i5
    M[6,6] = am_m6*(am_zf-am_fa)^2 + am_i6
    M[7,6] = am_m6*((am_zf-am_fa)^2 - am_u*(am_zf-am_fa)*siom) + am_i6
    M[7,7] = am_m6*((am_zf-am_fa)^2 - 2*am_u*(am_zf-am_fa)*siom + am_u^2) +
             am_m7*(am_ua^2+am_ub^2) + am_i6 + am_i7
    for j in 2:7, i in 1:j-1; M[i,j] = M[j,i]; end

    # Spring force
    xd = am_sd*coga+am_sc*siga+am_xb; yd = am_sd*siga-am_sc*coga+am_yb
    lang = sqrt((xd-am_xc)^2+(yd-am_yc)^2); force = -am_c0*(lang-am_l0)/lang
    fx = force*(xd-am_xc); fy = force*(yd-am_yc)

    # Force vector
    f = zeros(7)
    f[1] = am_mom - am_m2*am_da*am_rr*thp*(thp+2*bep)*sith
    f[2] = am_m2*am_da*am_rr*bep^2*sith
    f[3] = fx*(am_sc*coga-am_sd*siga) + fy*(am_sd*coga+am_sc*siga)
    f[4] = am_m4*am_zt*(am_e-am_ea)*dep^2*coph
    f[5] = -am_m4*am_zt*(am_e-am_ea)*php*(php+2*dep)*coph
    f[6] = -am_m6*am_u*(am_zf-am_fa)*epp^2*coom
    f[7] = am_m6*am_u*(am_zf-am_fa)*omp*(omp+2*epp)*coom

    # Constraint Jacobian G(q): 6×7
    G = zeros(6,7)
    G[1,1] = -am_rr*sibe+am_d*sibeth; G[1,2] = am_d*sibeth; G[1,3] = -am_ss*coga
    G[2,1] = am_rr*cobe-am_d*cobeth; G[2,2] = -am_d*cobeth; G[2,3] = -am_ss*siga
    G[3,1] = -am_rr*sibe+am_d*sibeth; G[3,2] = am_d*sibeth
    G[3,4] = -am_e*cophde; G[3,5] = -am_e*cophde+am_zt*side
    G[4,1] = am_rr*cobe-am_d*cobeth; G[4,2] = -am_d*cobeth
    G[4,4] = -am_e*siphde; G[4,5] = -am_e*siphde-am_zt*cde
    G[5,1] = -am_rr*sibe+am_d*sibeth; G[5,2] = am_d*sibeth
    G[5,6] = am_zf*siomep; G[5,7] = am_zf*siomep-am_u*coep
    G[6,1] = am_rr*cobe-am_d*cobeth; G[6,2] = -am_d*cobeth
    G[6,6] = -am_zf*coomep; G[6,7] = -am_zf*coomep-am_u*siep

    # Constraints g(q)
    gg = zeros(6)
    gg[1] = am_rr*cobe-am_d*cobeth-am_ss*siga-am_xb
    gg[2] = am_rr*sibe-am_d*sibeth+am_ss*coga-am_yb
    gg[3] = am_rr*cobe-am_d*cobeth-am_e*siphde-am_zt*cde-am_xa
    gg[4] = am_rr*sibe-am_d*sibeth+am_e*cophde-am_zt*side-am_ya
    gg[5] = am_rr*cobe-am_d*cobeth-am_zf*coomep-am_u*siep-am_xa
    gg[6] = am_rr*sibe-am_d*sibeth-am_zf*siomep+am_u*coep-am_ya

    # Assemble: rows 1-14 are dq/dt = v, dv/dt = a
    for i in 1:14; du[i] = u[i+7]; end
    # Rows 15-21: 0 = -f + M*a + G^T*λ
    for i in 1:7
        du[14+i] = -f[i]
        for j in 1:7; du[14+i] += M[i,j]*u[14+j]; end
        for j in 1:6; du[14+i] += G[j,i]*u[21+j]; end
    end
    # Rows 22-27: 0 = g(q)
    for i in 1:6; du[21+i] = gg[i]; end
    nothing
end
andrews_rhs_full! (generic function with 1 method)

Problem Collection

Mass-Matrix ODE Formulation

The 27-variable first-order form $K \cdot dy/dt = \phi(y)$ where $K = \text{diag}(I_{14}, 0_{13})$:

K_mat = Matrix(Diagonal(vcat(ones(14), zeros(13))))
mmf = ODEFunction(andrews_rhs_full!, mass_matrix = K_mat)
tspan = (0.0, 0.03)
prob_mm = ODEProblem(mmf, y0, tspan)
ODEProblem with uType Vector{Float64} and tType Float64. In-place: true
Non-trivial mass matrix: true
timespan: (0.0, 0.03)
u0: 27-element Vector{Float64}:
 -0.06171389001427645
  0.0
  0.45527981916307037
  0.22266839016588588
  0.48736497954384256
 -0.22266839016588588
  1.2305474445498212
  0.0
  0.0
  0.0
  ⋮
  0.0
  0.0
  0.0
 98.56687039624109
 -6.122688344255662
  0.0
  0.0
  0.0
  0.0

DAE Residual Formulation

The same system as $F(\dot{y}, y, t) = K \cdot \dot{y} - \phi(y) = 0$:

function andrews_dae!(res, du, u, p, t)
    f = similar(u)
    andrews_rhs_full!(f, u, p, t)
    res .= K_mat * du - f
    nothing
end

du0 = zeros(27)
du0[1:7]  .= y0[8:14]    # d(positions)/dt = velocities (all zero)
du0[8:14] .= y0[15:21]   # d(velocities)/dt = accelerations
differential_vars = [trues(14); falses(13)]
prob_dae = DAEProblem(andrews_dae!, du0, y0, tspan,
                      differential_vars = differential_vars)

# Verify DAE consistency at initial conditions
f_check = similar(y0)
andrews_rhs_full!(f_check, y0, nothing, 0.0)
println("DAE residual norm at IC: ", norm(K_mat * du0 - f_check))
DAE residual norm at IC: 2.3870227603277585e-15

MTK Index-Reduced Formulation

ModelingToolkit can automatically reduce the DAE index via structural_simplify. We define the full system symbolically — 7 kinematic equations (positions track velocities), 7 dynamics equations (with configuration-dependent mass matrix), and 6 position-level holonomic constraints — for a total of 20 equations with 20 unknowns (7 positions, 7 velocities, 6 multipliers).

@variables begin
    p1(t) = y0[1]     # β
    p2(t) = y0[2]     # Θ
    p3(t) = y0[3]     # γ
    p4(t) = y0[4]     # Φ
    p5(t) = y0[5]     # δ
    p6(t) = y0[6]     # Ω
    p7(t) = y0[7]     # ε
    w1(t) = 0.0       # dβ/dt
    w2(t) = 0.0       # dΘ/dt
    w3(t) = 0.0       # dγ/dt
    w4(t) = 0.0       # dΦ/dt
    w5(t) = 0.0       # dδ/dt
    w6(t) = 0.0       # dΩ/dt
    w7(t) = 0.0       # dε/dt
    lm1(t) = y0[22]
    lm2(t) = y0[23]
    lm3(t) = 0.0
    lm4(t) = 0.0
    lm5(t) = 0.0
    lm6(t) = 0.0
end

# Symbolic compound angles
s12 = sin(p1 + p2);  c12 = cos(p1 + p2)
s45 = sin(p4 + p5);  c45 = cos(p4 + p5)
s67 = sin(p6 + p7);  c67 = cos(p6 + p7)

# 7×7 mass matrix M(q) entries (symmetric, block-sparse)
mm11_s = am_m1*am_ra^2 + am_m2*(am_rr^2 - 2*am_da*am_rr*cos(p2) + am_da^2) + am_i1 + am_i2
mm21_s = am_m2*(am_da^2 - am_da*am_rr*cos(p2)) + am_i2
mm22_s = am_m2*am_da^2 + am_i2
mm33_s = am_m3*(am_sa^2 + am_sb^2) + am_i3
mm44_s = am_m4*(am_e - am_ea)^2 + am_i4
mm54_s = am_m4*((am_e - am_ea)^2 + am_zt*(am_e - am_ea)*sin(p4)) + am_i4
mm55_s = am_m4*(am_zt^2 + 2*am_zt*(am_e - am_ea)*sin(p4) + (am_e - am_ea)^2) +
         am_m5*(am_ta^2 + am_tb^2) + am_i4 + am_i5
mm66_s = am_m6*(am_zf - am_fa)^2 + am_i6
mm76_s = am_m6*((am_zf - am_fa)^2 - am_u*(am_zf - am_fa)*sin(p6)) + am_i6
mm77_s = am_m6*((am_zf - am_fa)^2 - 2*am_u*(am_zf - am_fa)*sin(p6) + am_u^2) +
         am_m7*(am_ua^2 + am_ub^2) + am_i6 + am_i7

# M(q) × D(v) — configuration-dependent mass matrix × acceleration vector
Mdv = [
    mm11_s*D(w1) + mm21_s*D(w2),
    mm21_s*D(w1) + mm22_s*D(w2),
    mm33_s*D(w3),
    mm44_s*D(w4) + mm54_s*D(w5),
    mm54_s*D(w4) + mm55_s*D(w5),
    mm66_s*D(w6) + mm76_s*D(w7),
    mm76_s*D(w6) + mm77_s*D(w7),
]

# Spring force (depends only on γ = p3)
xd_s = am_sd*cos(p3) + am_sc*sin(p3) + am_xb
yd_s = am_sd*sin(p3) - am_sc*cos(p3) + am_yb
lang_s = sqrt((xd_s - am_xc)^2 + (yd_s - am_yc)^2)
force_s = -am_c0 * (lang_s - am_l0) / lang_s
fx_s = force_s * (xd_s - am_xc)
fy_s = force_s * (yd_s - am_yc)

# Force vector f(q, dq/dt)
F_s = [
    am_mom - am_m2*am_da*am_rr*w2*(w2 + 2*w1)*sin(p2),
    am_m2*am_da*am_rr*w1^2*sin(p2),
    fx_s*(am_sc*cos(p3) - am_sd*sin(p3)) + fy_s*(am_sd*cos(p3) + am_sc*sin(p3)),
    am_m4*am_zt*(am_e - am_ea)*w5^2*cos(p4),
    -am_m4*am_zt*(am_e - am_ea)*w4*(w4 + 2*w5)*cos(p4),
    -am_m6*am_u*(am_zf - am_fa)*w7^2*cos(p6),
    am_m6*am_u*(am_zf - am_fa)*w6*(w6 + 2*w7)*cos(p6),
]

# Constraint Jacobian G(q): 6×7 — nonzero entries only
gp11_s = -am_rr*sin(p1) + am_d*s12;  gp12_s =  am_d*s12;  gp13_s = -am_ss*cos(p3)
gp21_s =  am_rr*cos(p1) - am_d*c12;  gp22_s = -am_d*c12;  gp23_s = -am_ss*sin(p3)
gp31_s = -am_rr*sin(p1) + am_d*s12;  gp32_s =  am_d*s12
gp34_s = -am_e*c45;  gp35_s = -am_e*c45 + am_zt*sin(p5)
gp41_s =  am_rr*cos(p1) - am_d*c12;  gp42_s = -am_d*c12
gp44_s = -am_e*s45;  gp45_s = -am_e*s45 - am_zt*cos(p5)
gp51_s = -am_rr*sin(p1) + am_d*s12;  gp52_s =  am_d*s12
gp56_s =  am_zf*s67;  gp57_s = am_zf*s67 - am_u*cos(p7)
gp61_s =  am_rr*cos(p1) - am_d*c12;  gp62_s = -am_d*c12
gp66_s = -am_zf*c67;  gp67_s = -am_zf*c67 - am_u*sin(p7)

# G^T × λ for each DOF
GTlam = [
    gp11_s*lm1 + gp21_s*lm2 + gp31_s*lm3 + gp41_s*lm4 + gp51_s*lm5 + gp61_s*lm6,
    gp12_s*lm1 + gp22_s*lm2 + gp32_s*lm3 + gp42_s*lm4 + gp52_s*lm5 + gp62_s*lm6,
    gp13_s*lm1 + gp23_s*lm2,
    gp34_s*lm3 + gp44_s*lm4,
    gp35_s*lm3 + gp45_s*lm4,
    gp56_s*lm5 + gp66_s*lm6,
    gp57_s*lm5 + gp67_s*lm6,
]

# 6 position constraints g(q) = 0
g_cons = [
    am_rr*cos(p1) - am_d*c12 - am_ss*sin(p3) - am_xb,
    am_rr*sin(p1) - am_d*s12 + am_ss*cos(p3) - am_yb,
    am_rr*cos(p1) - am_d*c12 - am_e*s45 - am_zt*cos(p5) - am_xa,
    am_rr*sin(p1) - am_d*s12 + am_e*c45 - am_zt*sin(p5) - am_ya,
    am_rr*cos(p1) - am_d*c12 - am_zf*c67 - am_u*sin(p7) - am_xa,
    am_rr*sin(p1) - am_d*s12 - am_zf*s67 + am_u*cos(p7) - am_ya,
]

# Assemble 20 equations: 7 kinematic + 7 dynamics + 6 constraints
pvec = [p1, p2, p3, p4, p5, p6, p7]
wvec = [w1, w2, w3, w4, w5, w6, w7]
eqs = vcat(
    [D(pvec[i]) ~ wvec[i] for i in 1:7],
    [Mdv[i] ~ F_s[i] - GTlam[i] for i in 1:7],
    [0 ~ g_cons[i] for i in 1:6],
)

@mtkbuild sys = ODESystem(eqs, t)
prob_mtk = ODEProblem(sys, [], tspan; warn_initialize_determined = false)
println("MTK index-reduced: $(length(ModelingToolkit.unknowns(sys))) states ",
        "(from 20 original)")
MTK index-reduced: 23 states (from 20 original)

Reference Solution

High-accuracy reference computed using RADAU5 with Hessenberg index hints (14, 7, 6) and an initial step-size hint dt=1e-6 (mapped to Fortran INITIALSS). The tightest reliable tolerance for the DiffEq wrapper on this index-3 problem is 3e-8.

const radau5_alg = radau5(DIMOFIND1VAR=14, DIMOFIND2VAR=7, DIMOFIND3VAR=6)

ref_sol = solve(prob_mm, radau5_alg; abstol=3e-8, reltol=3e-8, dt=1e-6,
                maxiters=Int(1e6))
println("Reference retcode: ", ref_sol.retcode)
println("NaN in reference? ", any(isnan, ref_sol.u[end]))
Reference retcode: Success
NaN in reference? false
probs = [prob_mm, prob_dae, prob_mtk]
refs  = [ref_sol, ref_sol, ref_sol];

Solution Plots

plot(ref_sol; idxs=[1,2,3,4,5,6,7],
     label=["q₁ (β)" "q₂ (Θ)" "q₃ (γ)" "q₄ (Φ)" "q₅ (δ)" "q₆ (Ω)" "q₇ (ε)"],
     title="Andrews' Mechanism: Position Angles",
     xlabel="t", ylabel="angle (rad)",
     layout=(4,2), size=(800,800), lw=1.5)

plot(ref_sol; idxs=[8,9,10,11,12,13,14],
     label=["v₁" "v₂" "v₃" "v₄" "v₅" "v₆" "v₇"],
     title="Andrews' Mechanism: Velocities",
     xlabel="t", ylabel="angular velocity",
     layout=(4,2), size=(800,800), lw=1.5)

plot(ref_sol; idxs=[22,23,24,25,26,27],
     label=["λ₁" "λ₂" "λ₃" "λ₄" "λ₅" "λ₆"],
     title="Andrews' Mechanism: Lagrange Multipliers",
     xlabel="t", ylabel="λ",
     layout=(3,2), size=(800,600), lw=1.5)

Verification against PSIDE Reference

names_ref = ["q₁(β)", "q₂(Θ)", "q₃(γ)", "q₄(Φ)", "q₅(δ)", "q₆(Ω)", "q₇(ε)"]
xs = ref_sol.u[end]
println("=== Verification at t = 0.03 (positions) ===")
println("Variable    | PSIDE Reference         | RADAU5 Solution         | Rel Error")
println("-"^85)
for i in 1:7
    ref_val = y_ref[i]
    our_val = xs[i]
    relerr = abs(ref_val) > 0 ? abs((our_val - ref_val) / ref_val) : abs(our_val)
    status = relerr < 1e-6 ? "✓" : (relerr < 1e-3 ? "~" : "✗")
    println("$(rpad(names_ref[i], 12))| $(lpad(string(ref_val), 24)) | $(lpad(string(round(our_val, sigdigits=15)), 24)) | $(round(relerr, sigdigits=3)) $status")
end
=== Verification at t = 0.03 (positions) ===
Variable    | PSIDE Reference         | RADAU5 Solution         | Rel Error
---------------------------------------------------------------------------
----------
q₁(β)       |        15.81077119629904 |         15.8107711215518 | 4.73e-9
 ✓
q₂(Θ)       |       -15.75637105984298 |        -15.7563709664431 | 5.93e-9
 ✓
q₃(γ)       |      0.04082224013073101 |        0.040822239407278 | 1.77e-8
 ✓
q₄(Φ)       |      -0.5347301163226948 |       -0.534730117587806 | 2.37e-9
 ✓
q₅(δ)       |       0.5244099658805303 |         0.52440996584292 | 7.17e-1
1 ✓
q₆(Ω)       |       0.5347301163226948 |        0.534730117587805 | 2.37e-9
 ✓
q₇(ε)       |        1.048080741042263 |         1.04808074102107 | 2.02e-1
1 ✓

Constraint Satisfaction

The 6 position constraints $g(q) = 0$ must be satisfied throughout the integration. We check their violation at the final time and along the full trajectory:

function eval_constraints(q)
    sibe = sin(q[1]); cobe = cos(q[1])
    sibeth = sin(q[1] + q[2]); cobeth = cos(q[1] + q[2])
    siga = sin(q[3]); coga = cos(q[3])
    siphde = sin(q[4] + q[5]); cophde = cos(q[4] + q[5])
    side = sin(q[5]); cde = cos(q[5])
    siomep = sin(q[6] + q[7]); coomep = cos(q[6] + q[7])
    siep = sin(q[7]); coep = cos(q[7])
    return [
        am_rr*cobe - am_d*cobeth - am_ss*siga - am_xb,
        am_rr*sibe - am_d*sibeth + am_ss*coga - am_yb,
        am_rr*cobe - am_d*cobeth - am_e*siphde - am_zt*cde - am_xa,
        am_rr*sibe - am_d*sibeth + am_e*cophde - am_zt*side - am_ya,
        am_rr*cobe - am_d*cobeth - am_zf*coomep - am_u*siep - am_xa,
        am_rr*sibe - am_d*sibeth - am_zf*siomep + am_u*coep - am_ya,
    ]
end

g_final = eval_constraints(xs[1:7])
println("Position constraints at t = 0.03:")
for i in 1:6
    println("  g[$i] = $(g_final[i])")
end
println("Max |g|: $(maximum(abs, g_final))")
Position constraints at t = 0.03:
  g[1] = 6.938893903907228e-18
  g[2] = -4.163336342344337e-17
  g[3] = 1.3877787807814457e-17
  g[4] = -4.2934406030425976e-17
  g[5] = 1.3877787807814457e-17
  g[6] = -3.599551212651875e-17
Max |g|: 4.2934406030425976e-17
max_g = [maximum(abs, eval_constraints(u[1:7])) for u in ref_sol.u]
plot(ref_sol.t, max_g, yscale = :log10,
    title = "Constraint Violation Along Trajectory",
    xlabel = "t", ylabel = "max|g(q)|",
    lw = 2, legend = false, ylims = (1e-16, 1e-6))

Solver Performance on Each Formulation

Andrews' squeezing mechanism is an index-3 DAE, significantly harder than the index-1 or index-2 systems encountered in most SciMLBenchmarks. We test each formulation with standard solvers to demonstrate this challenge.

Mass-Matrix ODE Form

The singular mass matrix $K = \text{diag}(I_{14}, 0_{13})$ creates difficulties for standard implicit methods. Without DIMOFIND index hints, solvers cannot exploit the Hessenberg structure. Rosenbrock-W methods fail due to singular Jacobian.

println("=== Mass-Matrix ODE Form ===")
for (name, alg) in [("Rodas5P(autodiff=false)", Rodas5P(autodiff = false)),
                     ("FBDF(autodiff=false)",    FBDF(autodiff = false)),
                     ("radau()",                 radau()),
                     ("radau5()",                radau5())]
    print("  $name: ")
    try
        sol = solve(prob_mm, alg, reltol = 1e-6, abstol = 1e-6, maxiters = 100000)
        has_nan = any(isnan, sol.u[end])
        println("retcode = $(sol.retcode), NaN = $has_nan")
    catch e
        println("failed ($(typeof(e)))")
    end
end
=== Mass-Matrix ODE Form ===
  Rodas5P(autodiff=false): retcode = MaxIters, NaN = false
  FBDF(autodiff=false): retcode = Unstable, NaN = false
  radau():  EXIT OF RADAU AT X=        0.0000E+00
  MATRIX IS REPEATEDLY SINGULAR, IER=                   22
retcode = Unstable, NaN = false
  radau5():  EXIT OF RADAU5 AT X=        0.0000E+00
  MATRIX IS REPEATEDLY SINGULAR, IER=                   22
retcode = Unstable, NaN = false

DAE Residual Form

IDA (Sundials) is designed for index-1 DAEs and cannot handle the index-3 structure of Andrews' mechanism:

println("\n=== DAE Residual Form ===")
print("  IDA: ")
try
    sol_dae = solve(prob_dae, IDA(), reltol = 1e-6, abstol = 1e-6)
    println("retcode = $(sol_dae.retcode), t_final = $(sol_dae.t[end])")
catch e
    println("failed ($(typeof(e)))")
end

print("  IDA (init_all=false): ")
try
    sol_dae2 = solve(prob_dae, IDA(init_all = false), reltol = 1e-6, abstol = 1e-6)
    println("retcode = $(sol_dae2.retcode), t_final = $(sol_dae2.t[end])")
catch e
    println("failed ($(typeof(e)))")
end
=== DAE Residual Form ===
  IDA: retcode = ConvergenceFailure, t_final = 0.0
  IDA (init_all=false): retcode = ConvergenceFailure, t_final = 0.0

MTK Index-Reduced Form

After structural_simplify, the system has fewer states but the numerical conditioning of the reduced system may still be challenging:

println("\n=== MTK Index-Reduced Form ===")
for (name, alg) in [("Rodas5P", Rodas5P()),
                     ("FBDF",    FBDF()),
                     ("Rodas4P", Rodas4P())]
    print("  $name: ")
    try
        sol_mtk = solve(prob_mtk, alg, reltol = 1e-6, abstol = 1e-6, maxiters = 100000)
        println("retcode = $(sol_mtk.retcode), t_final = $(sol_mtk.t[end])")
    catch e
        println("failed ($(typeof(e)))")
    end
end
=== MTK Index-Reduced Form ===
  Rodas5P: retcode = Unstable, t_final = 0.0
  FBDF: retcode = Unstable, t_final = 0.0
  Rodas4P: retcode = Unstable, t_final = 0.0

Work-Precision Diagrams

We use WorkPrecisionSet from DiffEqDevTools to benchmark RADAU5 with Hessenberg index hints on the mass-matrix ODE formulation.

The dt=1e-6 initial step-size hint is required — without it, the Fortran RADAU5 code fails at certain tolerance levels on this stiff index-3 system.

abstols = [3e-5, 1e-5, 3e-6, 3e-7, 1e-7]
reltols = [3e-5, 1e-5, 3e-6, 3e-7, 1e-7]

setups = [Dict(:prob_choice => 1, :alg => radau5_alg)]

wp = WorkPrecisionSet(probs, abstols, reltols, setups;
    save_everystep = false, appxsol = refs, maxiters = Int(1e5), numruns = 3,
    dt = 1e-6)
plot(wp; title = "Andrews' Mechanism WPD")

Conclusion

Andrews' squeezing mechanism is one of the most challenging standard DAE benchmarks. This index-3 problem reveals fundamental solver limitations:

  • Mass-matrix ODE: Standard implicit solvers (Rosenbrock-W, BDF) fail due to singular Jacobian from the zero mass-matrix block.
  • DAE residual: IDA (Sundials) cannot handle index-3 DAEs (designed for index-1 only).
  • MTK index-reduced:structural_simplify produces a numerically challenging system that standard solvers struggle with.
  • RADAU5 with DIMOFIND: The only reliable approach. The Hessenberg index dimensions (14, 7, 6) are essential for correct error estimation and step size control on this index-3 system.

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","andrews_mechanism.jmd")

Computer Information:

Julia Version 1.10.11
Commit a2b11907d7b (2026-03-09 14:59 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`
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⌅ [c3572dad] Sundials v4.28.0
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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`

And the full manifest:

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  [19f34311] SciMLJacobianOperators v0.1.12
  [a6db7da4] SciMLLogging v1.9.1
  [c0aeaf25] SciMLOperators v1.15.1
  [431bcebd] SciMLPublic v1.0.1
  [53ae85a6] SciMLStructures v1.10.0
  [6c6a2e73] Scratch v1.3.0
  [efcf1570] Setfield v1.1.2
  [992d4aef] Showoff v1.0.3
  [777ac1f9] SimpleBufferStream v1.2.0
  [727e6d20] SimpleNonlinearSolve v2.11.0
  [699a6c99] SimpleTraits v0.9.5
  [a2af1166] SortingAlgorithms v1.2.2
  [0a514795] SparseMatrixColorings v0.4.24
  [276daf66] SpecialFunctions v2.7.1
  [860ef19b] StableRNGs v1.0.4
  [aedffcd0] Static v1.3.1
  [0d7ed370] StaticArrayInterface v1.9.0
  [90137ffa] StaticArrays v1.9.18
  [1e83bf80] StaticArraysCore v1.4.4
  [82ae8749] StatsAPI v1.8.0
  [2913bbd2] StatsBase v0.34.10
  [4c63d2b9] StatsFuns v1.5.2
  [7792a7ef] StrideArraysCore v0.5.8
  [69024149] StringEncodings v0.3.7
  [09ab397b] StructArrays v0.7.2
⌅ [c3572dad] Sundials v4.28.0
  [2efcf032] SymbolicIndexingInterface v0.3.46
⌅ [19f23fe9] SymbolicLimits v0.2.3
⌅ [d1185830] SymbolicUtils v3.32.0
⌅ [0c5d862f] Symbolics v6.58.0
  [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 v0.1.13
  [781d530d] TruncatedStacktraces v1.4.0
  [5c2747f8] URIs v1.6.1
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [1986cc42] Unitful v1.28.0
  [a7c27f48] Unityper v0.1.6
  [41fe7b60] Unzip v0.2.0
  [81def892] VersionParsing v1.3.0
  [44d3d7a6] Weave v0.10.12
  [ddb6d928] YAML v0.4.16
  [c2297ded] ZMQ v1.5.1
  [6e34b625] Bzip2_jll v1.0.9+0
  [83423d85] Cairo_jll v1.18.5+1
  [655fdf9c] DASKR_jll v1.0.1+0
  [ee1fde0b] Dbus_jll v1.16.2+0
  [2702e6a9] EpollShim_jll v0.0.20230411+1
  [2e619515] Expat_jll v2.7.3+0
⌃ [b22a6f82] FFMPEG_jll v8.0.1+0
  [a3f928ae] Fontconfig_jll v2.17.1+0
  [d7e528f0] FreeType2_jll v2.13.4+0
  [559328eb] FriBidi_jll v1.0.17+0
  [0656b61e] GLFW_jll v3.4.1+0
  [d2c73de3] GR_jll v0.73.24+0
  [b0724c58] GettextRuntime_jll v0.22.4+0
  [61579ee1] Ghostscript_jll v9.55.1+0
  [020c3dae] Git_LFS_jll v3.7.0+0
  [f8c6e375] Git_jll v2.53.0+0
  [7746bdde] Glib_jll v2.86.3+0
  [3b182d85] Graphite2_jll v1.3.15+0
  [2e76f6c2] HarfBuzz_jll v8.5.1+0
  [1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
  [aacddb02] JpegTurbo_jll v3.1.4+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 v2.41.3+0
  [89763e89] Libtiff_jll v4.7.2+0
  [38a345b3] Libuuid_jll v2.41.3+0
  [856f044c] MKL_jll v2025.2.0+0
  [c771fb93] ODEInterface_jll v0.0.2+0
  [e7412a2a] Ogg_jll v1.3.6+0
  [9bd350c2] OpenSSH_jll v10.2.1+0
  [458c3c95] OpenSSL_jll v3.5.5+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.6.1+0
  [36c8627f] Pango_jll v1.57.0+0
⌅ [30392449] Pixman_jll v0.44.2+0
  [c0090381] Qt6Base_jll v6.10.2+1
  [629bc702] Qt6Declarative_jll v6.10.2+1
  [ce943373] Qt6ShaderTools_jll v6.10.2+1
  [6de9746b] Qt6Svg_jll v6.10.2+0
  [e99dba38] Qt6Wayland_jll v6.10.2+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 v1.24.0+0
  [ffd25f8a] XZ_jll v5.8.2+0
  [f67eecfb] Xorg_libICE_jll v1.1.2+0
  [c834827a] Xorg_libSM_jll v1.2.6+0
  [4f6342f7] Xorg_libX11_jll v1.8.13+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 v1.3.8+0
  [d091e8ba] Xorg_libXfixes_jll v6.0.2+0
  [a51aa0fd] Xorg_libXi_jll v1.8.3+0
  [d1454406] Xorg_libXinerama_jll v1.1.7+0
  [ec84b674] Xorg_libXrandr_jll v1.5.6+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [cc61e674] Xorg_libxkbfile_jll v1.2.0+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+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 v3.13.1+0
  [0ac62f75] libass_jll v0.17.4+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 v1.6.55+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+1
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.13.0+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`