Part III: Fluid-Driven Rigid-Body Motion
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Introduction and scope
Regularized immersed boundary methods couple Eulerian fluid variables to a separately represented structure through regularized interpolation and its adjoint force spreading [1 ] . Parts I and II of this series [2 , 3 ] analyzed one conforming Cartesian realization under arbitrary changes of local enrichments, first for fixed interfaces and then for prescribed rigid motion. This paper makes the rigid motion an unknown. Translation and rotation are determined by Newton’s equations, and the numerical body configuration therefore differs from the exact one. A convergence argument must control that difference jointly with the fluid error.
The passage from prescribed to computed motion requires more than inserting a numerical trajectory into the estimates of Part II. There the fictitious interior forcing was chosen to reproduce a known rigid acceleration. Here that acceleration is unknown. Surface constraints alone do not identify the interior fluid momentum with the displaced rigid momentum. Fictitious-domain formulations enforce interior rigidity and account for displaced-fluid inertia [4 Section 4] ; the interior-inertia issue is also explicit in regularized rigid-body IB formulations [5 Section II] . Accordingly, we add regularized volume moments to the regularized surface moments of Part II. The discrete interior is only approximately rigid. Its defect is estimated rather than silently identified with zero.
There is substantial prior analysis of fluid-driven rigid motion. San Martín et al. proved convergence of a fixed-mesh Lagrange–Galerkin method [6 ] . Legendre and Takahashi derived joint fluid, rigid-velocity, and configuration error estimates in an ALE formulation [7 Theorem 3.5 and Section 4] . Discontinuous-density global formulations [8 ] and free-body penalization convergence [9 ] provide other established approaches. In a fully Eulerian unfitted setting, von Wahl and Richter analyzed a coupled vector-valued parabolic equation and translational-motion ODE, including dynamic domain error [10 ] . Their model omits the fluid divergence constraint and body rotation, but its comparison of numerical and exact geometry is a direct methodological precedent. Thus neither coupled rigid-body convergence nor geometry comparison is claimed to originate here.
Our energy argument builds on the usual cancellation of interface power in multiplier formulations [11 , 12 ] . The background kinetic inner product uses positive excess mass and inertia. These assumptions restrict the present analysis to sufficiently heavy bodies; they do not describe a physical instability of lighter bodies. An additional skew term is needed because the fluid equation uses the Eulerian Stokes derivative, not the Navier–Stokes material acceleration. We derive that term explicitly from the rigid interior. Its power vanishes, but omitting it would change the translational momentum equation.
The numerical spaces and transfer principles retain the architecture of Parts I–II. Dynamic-space Stokes projection estimates [13 ] , stable adaptive transfers [14 ] , and moving-boundary remeshing analysis [15 ] supply the relevant context. We use exact orthogonal projection in a fixed fluid–body mass inner product. Every admissible local enrichment retains the entire coarse Cartesian streamfunction space. A constructed coarse comparison then removes any factor counting refinement and coarsening events. The method does not include an error-based marking rule, and the coarse grid itself is refined in a convergence sequence. Optimal adaptive fictitious-domain and regularized-forcing results [16 , 17 ] address different questions.
Regularized IB convergence results of Mori and Liu–Mori [18 , 19 , 20 ] , and prescribed-velocity Stokes estimates of Li et al. [21 ] , motivate careful separation of bulk accuracy from boundary enforcement. Our constraints use exact continuum convolution and cell moments, rather than a particular sampled delta stencil. Regularization estimates [22 ] and Sobolev trace and approximation results [23 ] are used explicitly in the defect bounds. The background velocity spaces are generated by conforming bicubic Hermite streamfunctions [24 ] ; the theorem is not a convergence theorem for a conventional staggered-grid AMR solver. Adaptive IB computations [25 ] and the dynamically adaptive DLM–IB framework of Zeng et al. [26 ] predate this analysis and use different implementations.
For moving fluid equations, evolving-space formulations [27 , 28 ] , Eulerian unfitted analyses [29 , 30 , 31 ] , and ALE error estimates [32 ] provide further context. We use an auxiliary volumepreserving map to compare the exact and computed body configurations. This map is part of the proof, not a transport step in the algorithm. In particular, the square-root continuity of a trace with respect to interface displacement is not by itself a closed estimate for an unknown trajectory. Geometry alignment must first produce residuals linear in the configuration error.
Computational studies also illustrate why boundary and force consistency deserve separate attention. He et al. developed an immersed pressure treatment to mitigate penetration through thin flexible structures in multiphase flow [33 ] . Zheng et al. studied prescribed flapping near a free surface [34 ] . IB calculations of backward-facing-step flow [35 ] , airfoil transition [36 ] , and flexible canopies [37 ] give additional application context. These turbulent, multiphase, or deformable configurations are outside the present model; their numerical observations are not premises of our proofs.
The result concerns one planar rigid body on an interval with positive wall clearance, positive excess mass and inertia, and a sufficiently regular exact Stokes–Newton solution. It is a numerical analysis conditional on that solution, not a new general existence theorem. The distinction between energy error after geometry alignment and energy error at the same Eulerian points is retained throughout: moving a gradient jump can weaken the latter estimate. Exact integration is assumed in the theorem. Numerical quadrature, finite-precision constraints, and selected algebraic identities are checked independently in the computational and Lean companions.
A physical rigid-body model and its whole-box extension
Exterior Stokes flow coupled to Newton’s equations
Let Ω = ( 0 , 1 ) 2 \Omega=(0,1)^2 Ω = ( 0 , 1 ) 2 B ∗ ⋐ R 2 B_\ast\Subset\mathbb{R}^2 B ∗ ⋐ R 2 C 6 C^6 C 6 ∫ B ∗ x d x = 0 \int_{B_\ast}x\,\mathrm{d}x=0 ∫ B ∗ x d x = 0
J = ( 0 − 1 1 0 ) , Q ( θ ) = ( cos θ − sin θ sin θ cos θ ) . J=\begin{pmatrix}0&-1\\1&0\end{pmatrix},\qquad Q(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}. J = ( 0 1 − 1 0 ) , Q ( θ ) = ( cos θ sin θ − sin θ cos θ ) . For a pose q = ( c , θ ) q=(c,\theta) q = ( c , θ ) B ( q ) = c + Q ( θ ) B ∗ B(q)=c+Q(\theta)B_\ast B ( q ) = c + Q ( θ ) B ∗ F ( q ) = Ω ∖ B ( q ) ‾ F(q)=\Omega\setminus\overline{B(q)} F ( q ) = Ω ∖ B ( q ) Γ ( q ) = ∂ B ( q ) \Gamma(q)=\partial B(q) Γ ( q ) = ∂ B ( q ) n q n_q n q [ 0 , T ] [0,T] [ 0 , T ]
Let the fluid density and viscosity be ρ f > 0 \rho_f>0 ρ f > 0 μ > 0 \mu>0 μ > 0 c c c m s > 0 m_s>0 m s > 0 I s > 0 I_s>0 I s > 0 ξ = ( V , ω ) \xi=(V,\omega) ξ = ( V , ω )
R q ξ ( x ) = V + ω J ( x − c ) , b ( t , x ) = R q ( t ) ξ ( t ) ( x ) . \mathcal{R}_q\xi(x)=V+\omega J(x-c),\qquad b(t,x)=\mathcal{R}_{q(t)}\xi(t)(x). R q ξ ( x ) = V + ω J ( x − c ) , b ( t , x ) = R q ( t ) ξ ( t ) ( x ) . The pose satisfies
c ˙ = V , θ ˙ = ω . (1) \dot{c}=V,\qquad\dot{\theta}=\omega. \tag*{(1)} c ˙ = V , θ ˙ = ω . (1) The exterior fluid equations, with an Eulerian time derivative, are
ρ f ∂ t u − div σ = f in F ( q ( t ) ) , div u = 0 , σ = 2 μ D ( u ) − p I , D ( u ) = 1 2 ( ∇ u + ∇ u T ) , u = 0 on ∂ Ω , u = b on Γ ( q ( t ) ) . (2) \begin{aligned}
\rho_f\partial_tu-\operatorname{div}\sigma&=f &&\text{in }F(q(t)),\\
\operatorname{div}u&=0,\qquad\sigma=2\mu D(u)-pI,\qquad D(u)=\frac{1}{2}(\nabla u+\nabla u^{\mathsf T}),\\
u&=0 &&\text{on }\partial\Omega,\\
u&=b &&\text{on }\Gamma(q(t)). \tag*{(2)}
\end{aligned} ρ f ∂ t u − div σ div u u u = f = 0 , σ = 2 μ D ( u ) − p I , D ( u ) = 2 1 ( ∇ u + ∇ u T ) , = 0 = b in F ( q ( t )) , on ∂ Ω , on Γ ( q ( t )) . (2) Here f ( t , x ) f(t,x) f ( t , x ) F ( q ( t ) ) F(q(t)) F ( q ( t )) F e ( t ) F_e(t) F e ( t ) T e ( t ) T_e(t) T e ( t )
m s V ˙ = F e + ∫ Γ ( q ) σ n q d s , I s ω ˙ = T e + ∫ Γ ( q ) J ( x − c ) ⋅ σ n q d s . (3) \begin{aligned}
m_s\dot{V} &= F_e+\int_{\Gamma(q)}\sigma n_q\,\mathrm{d}s,\\
I_s\dot{\omega} &= T_e+\int_{\Gamma(q)}J(x-c)\cdot\sigma n_q\,\mathrm{d}s. \tag*{(3)}
\end{aligned} m s V ˙ I s ω ˙ = F e + ∫ Γ ( q ) σ n q d s , = T e + ∫ Γ ( q ) J ( x − c ) ⋅ σ n q d s . (3) The fluid traction in these equations is the exterior stress acting on the body. We do not replace it by an uncorrected force spread over a fictitious interior.
Define the mass and inertia of the fluid displaced by the body, and their excess-body counterparts, by
m f = ρ f ∣ B ∗ ∣ , I f = ρ f ∫ B ∗ ∣ x ∣ 2 d x , m e = m s − m f > 0 , I e = I s − I f > 0. (4) \begin{aligned}
m_f &= \rho_f\lvert B_*\rvert, & I_f &= \rho_f\int_{B_*}\lvert x\rvert^2\,\mathrm{d}x,\\
m_e &= m_s-m_f>0, & I_e &= I_s-I_f>0. \tag*{(4)}
\end{aligned} m f m e = ρ f ∣ B ∗ ∣ , = m s − m f > 0 , I f I e = ρ f ∫ B ∗ ∣ x ∣ 2 d x , = I s − I f > 0. (4) The strict inequalities are assumptions of the method analyzed here. For a homogeneous body they follow from a solid density larger than the fluid density. They make the discrete background-domain mass form positive even before interior rigidity is accurately enforced. The result does not cover zero or negative excess inertia.
The fixed positive product space
Let
U = { v ∈ H 0 1 ( Ω ) 2 : div v = 0 } , ∥ v ∥ U = ∥ ∇ v ∥ L 2 ( Ω ) , V = U × R 2 × R . U=\{v\in H^1_0(\Omega)^2:\operatorname{div}v=0\},\qquad\lVert v\rVert_U=\lVert\nabla v\rVert_{L^2(\Omega)},\qquad V=U\times\mathbb{R}^2\times\mathbb{R}. U = { v ∈ H 0 1 ( Ω ) 2 : div v = 0 } , ∥ v ∥ U = ∥ ∇ v ∥ L 2 ( Ω ) , V = U × R 2 × R . The ambient kinetic-energy space is H = L 2 ( Ω ) 2 × R 2 × R \mathbb{H}=L^2(\Omega)^2\times\mathbb{R}^2\times\mathbb{R} H = L 2 ( Ω ) 2 × R 2 × R X = ( u , V , ω ) X=(u,V,\omega) X = ( u , V , ω ) Y = ( v , W , η ) Y=(v,W,\eta) Y = ( v , W , η )
( X , Y ) M = ρ f ( u , v ) + m e V ⋅ W + I e ω η , ∥ X ∥ M 2 = ( X , X ) M . (5) (X,Y)_M=\rho_f(u,v)+m_eV\cdot W+I_e\omega\eta,\qquad\lVert X\rVert_M^2=(X,X)_M. \tag*{(5)} ( X , Y ) M = ρ f ( u , v ) + m e V ⋅ W + I e ω η , ∥ X ∥ M 2 = ( X , X ) M . (5) The inner product is independent of the pose and of every mesh and moment partition. The viscous and gyroscopic forms are
a ( u , v ) = 2 μ ( D ( u ) , D ( v ) ) , G ( ϖ ; X , Y ) = m f ϖ ( J V ) ⋅ W . (6) a(u,v)=2\mu(D(u),D(v)),\qquad\mathcal{G}(\varpi;X,Y)=m_f\varpi(JV)\cdot W. \tag*{(6)} a ( u , v ) = 2 μ ( D ( u ) , D ( v )) , G ( ϖ ; X , Y ) = m f ϖ ( J V ) ⋅ W . (6) For u , v ∈ U u,v\in U u , v ∈ U a ( u , v ) = μ ( ∇ v , ∇ v ) a(u,v)=\mu(\nabla v,\nabla v) a ( u , v ) = μ ( ∇ v , ∇ v ) G ( ϖ ; X , Y ) = − G ( ϖ ; Y , X ) \mathcal{G}(\varpi;X,Y)=-\mathcal{G}(\varpi;Y,X) G ( ϖ ; X , Y ) = − G ( ϖ ; Y , X ) G ( ϖ ; X , X ) = 0 \mathcal{G}(\varpi;X,X)=0 G ( ϖ ; X , X ) = 0
Extend the physical fluid velocity into the body by u = b u=b u = b
K ( q ) = { ( v , W , η ) ∈ V : v = R q ( W , η ) a.e. in B ( q ) } . (7) K(q)=\{(v,W,\eta)\in V:v=R_q(W,\eta)\ \text{a.e. in }B(q)\}. \tag*{(7)} K ( q ) = {( v , W , η ) ∈ V : v = R q ( W , η ) a.e. in B ( q )} . (7) The extension belongs to U U U K ( q ( t ) ) K(q(t)) K ( q ( t )) Y Y Y
L ( t , q ; Y ) = ∫ F ( q ) f ( t , x ) ⋅ v ( x ) d x + F e ( t ) ⋅ W + T e ( t ) η . (8) \mathcal{L}(t,q;Y)=\int_{F(q)}f(t,x)\cdot v(x)\,\mathrm{d}x+F_e(t)\cdot W+T_e(t)\eta. \tag*{(8)} L ( t , q ; Y ) = ∫ F ( q ) f ( t , x ) ⋅ v ( x ) d x + F e ( t ) ⋅ W + T e ( t ) η . (8) Proposition 2.1 (The Eulerian fictitious-inertia correction). For a sufficiently regular collision-free solution of (2) –(3) , the rigid extension satisfies
( ∂ t X , Y ) M + a ( u , v ) + G ( ω ; X , Y ) = L ( t , q ; Y ) , Y ∈ K ( q ( t ) ) . (9) (\partial_tX,Y)_M+a(u,v)+\mathcal{G}(\omega;X,Y)=\mathcal{L}(t,q;Y),\qquad Y\in K(q(t)). \tag*{(9)} ( ∂ t X , Y ) M + a ( u , v ) + G ( ω ; X , Y ) = L ( t , q ; Y ) , Y ∈ K ( q ( t )) . (9) Conversely, the sufficiently regular constrained formulation (1) , (9) gives the exterior Stokes and physical Newton equations, with the exterior pressure and traction understood in their usual weak sense.
Proof. Since c ′ = V c' = V c ′ = V
∂ t b = V ˙ + ω ˙ J ( x − c ) − ω J V . (10) \partial_t b = \dot{V} + \dot{\omega}J(x-c) - \omega JV. \tag*{(10)} ∂ t b = V ˙ + ω ˙ J ( x − c ) − ω J V . (10) For a rigid interior test v = W + η J ( x − c ) v = W + \eta J(x-c) v = W + η J ( x − c )
ρ f ∫ B ( q ) ∂ t b ⋅ v d x = m f ( V ˙ − ω J V ) ⋅ W + I f ω ˙ η . (11) \rho_f \int_{B(q)} \partial_t b \cdot v\,\mathrm{d}x = m_f(\dot{V} - \omega JV)\cdot W + I_f\dot{\omega}\eta. \tag*{(11)} ρ f ∫ B ( q ) ∂ t b ⋅ v d x = m f ( V ˙ − ω J V ) ⋅ W + I f ω ˙ η . (11) Thus the whole-box fluid inertia, the excess body inertia, and G ( ω ; X , Y ) \mathcal{G}(\omega;X,Y) G ( ω ; X , Y ) m s V ˙ ⋅ W + I s ω ˙ η m_s\dot{V}\cdot W + I_s\dot{\omega}\eta m s V ˙ ⋅ W + I s ω ˙ η ∫ Γ σ n q ⋅ R q ( W , η ) \int_{\Gamma}\sigma n_q\cdot\mathcal{R}_q(W,\eta) ∫ Γ σ n q ⋅ R q ( W , η ) (3) proves (9) .
Conversely, tests with zero rigid component and fluid support in the exterior give the Stokes equation. Any rigid test velocity admits a solenoidal, outer-zero extension: multiply its affine/quadratic streamfunction by a cutoff equal to one near the body and take its curl. Subtracting the exterior fluid weak identity for these tests from (9) , and using (11) , gives both equations in (3) .
The extra term m f ω J V m_f\omega JV m f ω J V [4 Section 4] , cannot be transferred unchanged after deleting advection. For the present planar rigid motion, this additional term is gyroscopic and contributes no kinetic-energy work.
Volume and surface reactions
For consistency estimates we need an identity on the full test space, not only on exactly rigid tests. Under the same regularity, choose the interior pressure to be zero and set
λ B = − ρ f ∂ t b in B ( q ) , λ Γ = σ n q on Γ ( q ) . (12) \lambda_B = -\rho_f\partial_t b \text{ in } B(q), \qquad\lambda_\Gamma= \sigma n_q \text{ on } \Gamma(q). \tag*{(12)} λ B = − ρ f ∂ t b in B ( q ) , λ Γ = σ n q on Γ ( q ) . (12) Then, for every Y = ( v , W , η ) ∈ V Y=(v,W,\eta)\in\mathcal{V} Y = ( v , W , η ) ∈ V ζ = ( W , η ) \zeta=(W,\eta) ζ = ( W , η )
( ∂ t X , Y ) M + a ( u , v ) + G ( ω ; X , Y ) + ∫ B ( q ) λ B ⋅ ( v − R q ζ ) d x + ∫ Γ ( q ) λ Γ ⋅ ( γ q v − R q ζ ) d s = L ( t , q ; Y ) . (13) \begin{aligned}
(\partial_t X,Y)_M + a(u,v) + \mathcal{G}(\omega;X,Y) + \int_{B(q)} \lambda_B\cdot(v-\mathcal{R}_q\zeta)\,\mathrm{d}x \\
{}+ \int_{\Gamma(q)} \lambda_\Gamma\cdot(\gamma_qv-\mathcal{R}_q\zeta)\,\mathrm{d}s &= \mathcal{L}(t,q;Y). \tag*{(13)}
\end{aligned} ( ∂ t X , Y ) M + a ( u , v ) + G ( ω ; X , Y ) + ∫ B ( q ) λ B ⋅ ( v − R q ζ ) d x + ∫ Γ ( q ) λ Γ ⋅ ( γ q v − R q ζ ) d s = L ( t , q ; Y ) . (13) Indeed, the volume reaction cancels the fictitious interior fluid inertia in the fluid equation. In the rigid equation its opposite work restores the displaced-fluid inertia in (11) ; the gyroscopic term restores the remaining m f ω J V m_f\omega JV m f ω J V
Surface velocity constraints alone do not ensure that an unsteady fictitious interior has the rigid angular momentum used in (11) ; this issue is also discussed by Kallemov et al. [5 ] . The discrete method therefore uses both volume and surface moments. At a finite resolution its interior rigidity is approximate, and its positive energy must be interpreted as a surrogate, as quantified in Proposition 3.3.
Physical energy of the continuous model
For an exactly rigid extended state, the mass energy is precisely the physical kinetic energy:
1 2 ∥ X ∥ M 2 = ρ f 2 ∫ F ( q ) ∣ u ∣ 2 d x + m s 2 ∣ V ∣ 2 + I s 2 ω 2 . (14) \frac{1}{2}\lVert X\rVert_M^2 = \frac{\rho_f}{2}\int_{F(q)} |u|^2\,dx + \frac{m_s}{2}|V|^2 + \frac{I_s}{2}\omega^2. \tag*{(14)} 2 1 ∥ X ∥ M 2 = 2 ρ f ∫ F ( q ) ∣ u ∣ 2 d x + 2 m s ∣ V ∣ 2 + 2 I s ω 2 . (14) Testing (9) by X X X
d d t 1 2 ∥ X ∥ M 2 + a ( u , u ) = L ( t , q ; X ) . (15) \frac{d}{dt}\frac{1}{2}\lVert X\rVert_M^2 + a(u,u) = \mathcal{L}(t,q;X). \tag*{(15)} d t d 2 1 ∥ X ∥ M 2 + a ( u , u ) = L ( t , q ; X ) . (15) The moving exterior does not leave an unaccounted kinetic-energy flux in this identity. On the interface u = b u=b u = b
∫ Γ ( q ) ∣ b ∣ 2 b ⋅ n q d s = ∫ B ( q ) div ( ∣ b ∣ 2 b ) d x = 0 , \int_{\Gamma(q)} |b|^2 b\cdot n_q\,\mathrm{d}s = \int_{B(q)} \operatorname{div}(|b|^2b)\,\mathrm{d}x = 0, ∫ Γ ( q ) ∣ b ∣ 2 b ⋅ n q d s = ∫ B ( q ) div ( ∣ b ∣ 2 b ) d x = 0 , because div b = 0 \operatorname{div} b=0 div b = 0 b ⋅ ( ∇ b ) b = b ⋅ ( ω J ) b = 0 b\cdot(\nabla b)b=b\cdot(\omega J)b=0 b ⋅ ( ∇ b ) b = b ⋅ ( ω J ) b = 0 D ( u ) = 0 D(u)=0 D ( u ) = 0 a ( u , u ) = 2 μ ∫ F ( q ) ∣ D ( u ) ∣ 2 a(u,u)=2\mu\int_{F(q)}|D(u)|^2 a ( u , u ) = 2 μ ∫ F ( q ) ∣ D ( u ) ∣ 2
Regularized volume and surface constraints
The surface-only constraints of Parts I–II are completed by interior moments. This is a change of the discrete method, not an inference that surface no-slip makes the fictitious interior rigid. Interior constraints are standard in fictitious-domain rigid-body formulations [4 , 5 ] ; the particular regularized moments and their defects are specified here.
Let ξ = ( V , ω ) \xi=(V,\omega) ξ = ( V , ω ) R q ξ ( x ) = V + ω J ( x − c ) R_q\xi(x)=V+\omega J(x-c) R q ξ ( x ) = V + ω J ( x − c ) v ∈ U v\in U v ∈ U E v Ev E v J ε v = ρ ε ∗ E v J_\varepsilon v=\rho_\varepsilon*Ev J ε v = ρ ε ∗ E v ρ \rho ρ Ω \Omega Ω C \mathcal{C} C B ( q ) B(q) B ( q ) κ \kappa κ P \mathcal{P} P Γ ( q ) \Gamma(q) Γ ( q ) ℓ \ell ℓ
∥ z − z C ∥ L 2 ( C ) ≤ C v o l κ ∥ ∇ z ∥ L 2 ( C ) , z C = ∣ C ∣ − 1 ∫ C z . \lVert z-z_C\rVert_{L^2(C)}\le C_{\mathrm{vol}}\kappa\lVert\nabla z\rVert_{L^2(C)},\qquad z_C=|C|^{-1}\int_C z. ∥ z − z C ∥ L 2 ( C ) ≤ C vol κ ∥ ∇ z ∥ L 2 ( C ) , z C = ∣ C ∣ − 1 ∫ C z . with one constant for all volume cells and runs. Convex cells have such a diameter bound even when they are thin, by the Payne–Weinberger inequality [38 ] . In particular, a convex reference body intersected with Cartesian squares, followed by the rigid map, provides admissible partitions. For a nonconvex smooth body one may use transported reference partitions satisfying this inequality. Denote volume and surface cell-average projectors by Π C \Pi_{\mathcal{C}} Π C Π P \Pi_{\mathcal{P}} Π P
In the retained-coarse Cartesian velocity space X h , A X_{h,\mathcal{A}} X h , A
K h ( q ) = { ( v h , ζ ) ∈ X h , A × R 3 : Π C ( J ε v h − R q ζ ) = 0 , Π P γ q ( J ε v h − R q ζ ) = 0 } . (16) K_h(q)=\left\{(v_h,\zeta)\in X_{h,\mathcal{A}}\times\mathbb{R}^3:\ \Pi_{\mathcal{C}}(J_\varepsilon v_h-R_q\zeta)=0,\quad
\Pi_{\mathcal{P}}\gamma_q(J_\varepsilon v_h-R_q\zeta)=0\right\}. \tag*{(16)} K h ( q ) = { ( v h , ζ ) ∈ X h , A × R 3 : Π C ( J ε v h − R q ζ ) = 0 , Π P γ q ( J ε v h − R q ζ ) = 0 } . (16) The parameters A , ε , C , P \mathcal{A},\varepsilon,\mathcal{C},\mathcal{P} A , ε , C , P ε ˉ \bar{\varepsilon} ε ˉ κ ˉ \bar{\kappa} κ ˉ ℓ ˉ \bar{\ell} ℓ ˉ n = 0 n=0 n = 0
The coarse lift L h , q ζ L_{h,q}\zeta L h , q ζ Lemma 4.1 equals R q ζ R_q\zeta R q ζ
∥ L h , q ζ ∥ U ≤ C ∣ ζ ∣ . \lVert L_{h,q}\zeta\rVert_U\le C|\zeta|. ∥ L h , q ζ ∥ U ≤ C ∣ ζ ∣. Consequently ( L h , q ζ , ζ ) ∈ K h ( q ) (L_{h,q}\zeta,\zeta) \in K_h(q) ( L h , q ζ , ζ ) ∈ K h ( q ) ζ \zeta ζ Y h = ( v h , ζ ) ∈ K h ( q ) Y_h=(v_h,\zeta) \in K_h(q) Y h = ( v h , ζ ) ∈ K h ( q ) w h = v h − L h , q ζ w_h=v_h-L_{h,q}\zeta w h = v h − L h , q ζ ∥ Y h ∥ V = ∥ v h ∥ U + ∣ ζ ∣ \|Y_h\|_V=\|v_h\|_U+|\zeta| ∥ Y h ∥ V = ∥ v h ∥ U + ∣ ζ ∣
Lemma 3.1 (Interior rigidity and physical surface defects). Uniformly over the admissible configurations and partitions,
∥ v h − R q ζ ∥ L 2 ( B ( q ) ) ≤ C ( ε + κ ) ∥ Y h ∥ V , (17) \|v_h-R_q\zeta\|_{L^2(B(q))} \le C(\varepsilon+\kappa)\|Y_h\|_V, \tag*{(17)} ∥ v h − R q ζ ∥ L 2 ( B ( q )) ≤ C ( ε + κ ) ∥ Y h ∥ V , (17) ∥ γ q v h − R q ζ ∥ L 2 ( Γ ( q ) ) ≤ C ( ε + ℓ ) ∥ Y h ∥ V . (18) \|\gamma_qv_h-R_q\zeta\|_{L^2(\Gamma(q))} \le C(\sqrt{\varepsilon}+\sqrt{\ell})\|Y_h\|_V. \tag*{(18)} ∥ γ q v h − R q ζ ∥ L 2 ( Γ ( q )) ≤ C ( ε + ℓ ) ∥ Y h ∥ V . (18) Proof. Polynomial reproduction by the lift and the zero first moment of the kernel give Π C J ε w h = 0 \Pi_C\mathcal{J}_\varepsilon w_h=0 Π C J ε w h = 0 Π P γ q J ε w h = 0 \Pi_P\gamma_q\mathcal{J}_\varepsilon w_h=0 Π P γ q J ε w h = 0 [22 Lemmas 3–4] imply
∥ w h − J ε w h ∥ L 2 ( Ω ) ≤ C ε ∥ w h ∥ U , ∥ ∇ J ε w h ∥ L 2 ( R 2 ) ≤ ∥ w h ∥ U . \|w_h-\mathcal{J}_\varepsilon w_h\|_{L^2(\Omega)}\le C\varepsilon\|w_h\|_U,\qquad\|\nabla\mathcal{J}_\varepsilon w_h\|_{L^2(\mathbb{R}^2)}\le\|w_h\|_U. ∥ w h − J ε w h ∥ L 2 ( Ω ) ≤ C ε ∥ w h ∥ U , ∥∇ J ε w h ∥ L 2 ( R 2 ) ≤ ∥ w h ∥ U . Apply the volume-cell Poincare inequality to J ε w h \mathcal{J}_\varepsilon w_h J ε w h ∥ w h ∥ L 2 ( B ( q ) ) ≤ C ( ε + κ ) ∥ w h ∥ U \|w_h\|_{L^2(B(q))}\le C(\varepsilon+\kappa)\|w_h\|_U ∥ w h ∥ L 2 ( B ( q )) ≤ C ( ε + κ ) ∥ w h ∥ U B ( q ) B(q) B ( q ) w h = v h − R q ζ w_h=v_h-R_q\zeta w h = v h − R q ζ
The multiplicative trace estimate on a uniform tubular neighborhood gives ∥ γ q ( w h − J ε w h ) ∥ L 2 ( Γ ( q ) ) ≤ C ε ∥ w h ∥ U \|\gamma_q(w_h-\mathcal{J}_\varepsilon w_h)\|_{L^2(\Gamma(q))}\le C\sqrt{\varepsilon}\|w_h\|_U ∥ γ q ( w h − J ε w h ) ∥ L 2 ( Γ ( q )) ≤ C ε ∥ w h ∥ U H 1 / 2 H^{1/2} H 1/2 J ε w h \mathcal{J}_\varepsilon w_h J ε w h ∥ γ q J ε w h ∥ L 2 ( Γ ( q ) ) ≤ C ℓ ∥ w h ∥ U \|\gamma_q\mathcal{J}_\varepsilon w_h\|_{L^2(\Gamma(q))}\le C\sqrt{\ell}\|w_h\|_U ∥ γ q J ε w h ∥ L 2 ( Γ ( q )) ≤ C ℓ ∥ w h ∥ U [3 ] ; see also Ern and Guermond [23 Theorem 3.10, Lemmas 3.26 and 12.15] . Finally ∥ w h ∥ U ≤ C ∥ Y h ∥ V \|w_h\|_U\le C\|Y_h\|_V ∥ w h ∥ U ≤ C ∥ Y h ∥ V □ \square □
Lemma 3.2 (Uniform control of rigid test velocities). There is κ 0 > 0 \kappa_0>0 κ 0 > 0 κ ≤ κ 0 \kappa\le\kappa_0 κ ≤ κ 0 ( v h , ζ ) ∈ K h ( q ) (v_h,\zeta)\in K_h(q) ( v h , ζ ) ∈ K h ( q )
∣ ζ ∣ ≤ C ∥ v h ∥ L 2 ( Ω ) ≤ C ∥ v h ∥ U , ∥ ( v h , ζ ) ∥ V ≤ C ∥ v h ∥ U . |\zeta|\le C\|v_h\|_{L^2(\Omega)}\le C\|v_h\|_U,\qquad\|(v_h,\zeta)\|_V\le C\|v_h\|_U. ∣ ζ ∣ ≤ C ∥ v h ∥ L 2 ( Ω ) ≤ C ∥ v h ∥ U , ∥ ( v h , ζ ) ∥ V ≤ C ∥ v h ∥ U . Proof. Since the reference centroid is zero,
∥ R q ( W , η ) ∥ L 2 ( B ( q ) ) 2 = ∣ B ∗ ∣ ∣ W ∣ 2 + ( ∫ B ∗ ∣ x ∣ 2 d x ) ∣ η ∣ 2 ≥ c B 2 ∣ ( W , η ) ∣ 2 \|R_q(W,\eta)\|_{L^2(B(q))}^2=|B_*||W|^2+\left(\int_{B_*}|x|^2\,dx\right)|\eta|^2\ge c_B^2|(W,\eta)|^2 ∥ R q ( W , η ) ∥ L 2 ( B ( q )) 2 = ∣ B ∗ ∣∣ W ∣ 2 + ( ∫ B ∗ ∣ x ∣ 2 d x ) ∣ η ∣ 2 ≥ c B 2 ∣ ( W , η ) ∣ 2 for a fixed c B > 0 c_B>0 c B > 0
∥ Π C R q ζ ∥ L 2 ( B ( q ) ) = ∥ Π C J ε v h ∥ L 2 ( B ( q ) ) ≤ ∥ v h ∥ L 2 ( Ω ) . \|\Pi_C R_q\zeta\|_{L^2(B(q))}=\|\Pi_C\mathcal{J}_\varepsilon v_h\|_{L^2(B(q))}\le\|v_h\|_{L^2(\Omega)}. ∥ Π C R q ζ ∥ L 2 ( B ( q )) = ∥ Π C J ε v h ∥ L 2 ( B ( q )) ≤ ∥ v h ∥ L 2 ( Ω ) . The cell Poincare inequality applied to the affine rigid field gives ∥ ( I − Π C ) R q ζ ∥ L 2 ( B ( q ) ) ≤ C κ ∣ η ∣ \|(I-\Pi_C)R_q\zeta\|_{L^2(B(q))}\le C\kappa|\eta| ∥ ( I − Π C ) R q ζ ∥ L 2 ( B ( q )) ≤ C κ ∣ η ∣ c B ∣ ζ ∣ ≤ ∥ v h ∥ 0 + C κ ∣ ζ ∣ c_B|\zeta|\le\|v_h\|_0+C\kappa|\zeta| c B ∣ ζ ∣ ≤ ∥ v h ∥ 0 + C κ ∣ ζ ∣ C κ 0 ≤ c B / 2 C\kappa_0\le c_B/2 C κ 0 ≤ c B /2 □ \square □
Proposition 3.3 (What the discrete energy represents). For a joint state X h = ( u h , V h , ω h ) ∈ K h ( q ) X_h=(u_h,V_h,\omega_h)\in K_h(q) X h = ( u h , V h , ω h ) ∈ K h ( q ) E M ( X h ) = 1 2 ∥ X h ∥ M 2 E_M(X_h)=\frac{1}{2}\|X_h\|_M^2 E M ( X h ) = 2 1 ∥ X h ∥ M 2 § 2 , and define
E p h y s ( X h ; q ) = ρ f 2 ∥ u h ∥ L 2 ( F ( q ) ) 2 + m s 2 ∣ V h ∣ 2 + I s 2 ∣ ω h ∣ 2 . E_{\mathrm{phys}}(X_h;q)=\frac{\rho_f}{2}\|u_h\|_{L^2(F(q))}^2+\frac{m_s}{2}|V_h|^2+\frac{I_s}{2}|\omega_h|^2. E phys ( X h ; q ) = 2 ρ f ∥ u h ∥ L 2 ( F ( q )) 2 + 2 m s ∣ V h ∣ 2 + 2 I s ∣ ω h ∣ 2 . Then
E M ( X h ) − E p h y s ( X h ; q ) = ρ f 2 ∫ B ( q ) ( u h − R q ξ h ) ⋅ ( u h + R q ξ h ) d x . (19) E_M(X_h)-E_{\mathrm{phys}}(X_h;q)=\frac{\rho_f}{2}\int_{B(q)}(u_h-R_q\xi_h)\cdot(u_h+R_q\xi_h)\,dx. \tag*{(19)} E M ( X h ) − E phys ( X h ; q ) = 2 ρ f ∫ B ( q ) ( u h − R q ξ h ) ⋅ ( u h + R q ξ h ) d x . (19) In particular, bounds on max n ∥ X h n ∥ M \max_n\|X_h^n\|_M max n ∥ X h n ∥ M ∑ n τ n ∥ X h n ∥ V 2 \sum_n\tau_n\|X_h^n\|_V^2 ∑ n τ n ∥ X h n ∥ V 2
∑ n τ n ∣ E M ( X h n ) − E p h y s ( X h n ; q n ) ∣ ≤ C ( ε ˉ + κ ˉ ) . \sum_n\tau_n|E_M(X_h^n)-E_{\mathrm{phys}}(X_h^n;q_n)|\le C(\bar{\varepsilon}+\bar{\kappa}). n ∑ τ n ∣ E M ( X h n ) − E phys ( X h n ; q n ) ∣ ≤ C ( ε ˉ + κ ˉ ) . This statement alone is time-integrated; it does not assert uniform nodal convergence of the two energies.
Proof. The reference centroid is zero and rigid maps preserve area and polar moment. Thus
ρ f ∫ B ( q ) ∣ R q ξ h ∣ 2 = m f ∣ V h ∣ 2 + I f ∣ ω h ∣ 2 . \rho_f \int_{B(q)} |R_q \xi_h|^2 = m_f |V_h|^2 + I_f |\omega_h|^2. ρ f ∫ B ( q ) ∣ R q ξ h ∣ 2 = m f ∣ V h ∣ 2 + I f ∣ ω h ∣ 2 . Subtract the two definitions, use the excess masses m e = m s − m f m_e = m_s - m_f m e = m s − m f I e = I s − I f I_e = I_s - I_f I e = I s − I f (17) , and the positive mass norm give the asserted time-integrated estimate.
Joint recovery and alignment of two rigid configurations
Fix a compact family of collision-free configurations containing a neighborhood of the exact trajectory. All bodies in this family have a common positive clearance from the outer wall. Constants in this section are uniform on this family. Angles are represented by real continuous lifts, rather than reduced modulo 2 π 2\pi 2 π
Retained Cartesian spaces and a joint lift
For h = 1 / M h = 1/M h = 1/ M S h ⊂ H 0 2 ( Ω ) S_h \subset H^2_0(\Omega) S h ⊂ H 0 2 ( Ω ) ψ \psi ψ ψ x \psi_x ψ x ψ y \psi_y ψ y ψ x y \psi_{xy} ψ x y K K K ∂ K \partial K ∂ K B h ( K ) B_h(K) B h ( K )
X h , A = curl ( S h + ∑ K ∈ A B h ( K ) ) , curl ψ = ( ψ y , − ψ x ) . X_{h,A} = \operatorname{curl}\left(S_h + \sum_{K\in A} B_h(K)\right), \qquad\operatorname{curl}\psi= (\psi_y,-\psi_x). X h , A = curl ( S h + K ∈ A ∑ B h ( K ) ) , curl ψ = ( ψ y , − ψ x ) . Every configuration contains curl S h \operatorname{curl} S_h curl S h U U U [3 , 24 ] . Its coarse interpolant satisfies
∥ I h ψ − ψ ∥ H j ≤ C h 4 − j ∥ ψ ∥ H 4 , j = 1 , 2. (20) \|I_h\psi-\psi\|_{H^j} \le Ch^{4-j}\|\psi\|_{H^4}, \qquad j=1,2. \tag*{(20)} ∥ I h ψ − ψ ∥ H j ≤ C h 4 − j ∥ ψ ∥ H 4 , j = 1 , 2. (20) Choose a fixed χ ∈ C c 5 ( Ω ) \chi\in C^5_c(\Omega) χ ∈ C c 5 ( Ω ) x ∗ = ( 1 / 2 , 1 / 2 ) x_*=(1/2,1/2) x ∗ = ( 1/2 , 1/2 ) r = x − x ∗ r=x-x_* r = x − x ∗ ζ = ( W , η ) \zeta=(W,\eta) ζ = ( W , η )
β q ( ζ ) = W − η J ( c − x ∗ ) , \beta_q(\zeta)=W-\eta J(c-x_*), β q ( ζ ) = W − η J ( c − x ∗ ) , p q , ζ ( x ) = β q , 1 ( ζ ) r 2 − β q , 2 ( ζ ) r 1 − 1 2 η ∣ r ∣ 2 , p_{q,\zeta}(x)=\beta_{q,1}(\zeta)r_2-\beta_{q,2}(\zeta)r_1-\frac{1}{2}\eta|r|^2, p q , ζ ( x ) = β q , 1 ( ζ ) r 2 − β q , 2 ( ζ ) r 1 − 2 1 η ∣ r ∣ 2 , L q ζ = curl ( χ p q , ζ ) , L h , q ζ = curl I h ( χ p q , ζ ) . (21) L_q\zeta=\operatorname{curl}(\chi p_{q,\zeta}), \qquad L_{h,q}\zeta=\operatorname{curl} I_h(\chi p_{q,\zeta}). \tag*{(21)} L q ζ = curl ( χ p q , ζ ) , L h , q ζ = curl I h ( χ p q , ζ ) . (21) This fixes the additive normalization before multiplication by the cutoff. Direct differentiation gives curl p q , ζ = R q ζ \operatorname{curl}p_{q,\zeta}=R_q\zeta curl p q , ζ = R q ζ
Lemma 4.1 (A common joint rigid lift). For sufficiently small h + ε ˉ h+\bar{\varepsilon} h + ε ˉ L h , q ζ L_{h,q}\zeta L h , q ζ R q ζ R_q\zeta R q ζ ( L h , q ζ , ζ ) ∈ K h ( q ) (L_{h,q}\zeta,\zeta)\in K_h(q) ( L h , q ζ , ζ ) ∈ K h ( q ) q q q
∥ L h , q ζ ∥ U ≤ C ∣ ζ ∣ , ∥ ( L h , q − L q ) ζ ∥ U ≤ C h 2 ∣ ζ ∣ , ∥ ( L h , q − L q ) ζ ∥ 0 ≤ C h 3 ∣ ζ ∣ . \|L_{h,q}\zeta\|_U \le C|\zeta|,\qquad\|(L_{h,q}-L_q)\zeta\|_U \le Ch^2|\zeta|,\qquad\|(L_{h,q}-L_q)\zeta\|_0 \le Ch^3|\zeta|. ∥ L h , q ζ ∥ U ≤ C ∣ ζ ∣ , ∥ ( L h , q − L q ) ζ ∥ U ≤ C h 2 ∣ ζ ∣ , ∥ ( L h , q − L q ) ζ ∥ 0 ≤ C h 3 ∣ ζ ∣. For a Lipschitz path q ( t ) q(t) q ( t ) ζ ∈ H 1 ( 0 , T ; R 3 ) \zeta\in H^1(0,T;\mathbb{R}^3) ζ ∈ H 1 ( 0 , T ; R 3 )
∥ ∂ t [ ( L h , q ( t ) − L q ( t ) ) ζ ( t ) ] ∥ L t 2 L x 2 ≤ C h 3 ( 1 + ∥ q ′ ∥ L ∞ ) ∥ ζ ∥ H t 1 . (22) \left\|\partial_t\left[(L_{h,q(t)}-L_{q(t)})\zeta(t)\right]\right\|_{L^2_tL^2_x}
\le Ch^3\left(1+\|q'\|_{L^\infty}\right)\|\zeta\|_{H^1_t}. \tag*{(22)} ∂ t [ ( L h , q ( t ) − L q ( t ) ) ζ ( t ) ] L t 2 L x 2 ≤ C h 3 ( 1 + ∥ q ′ ∥ L ∞ ) ∥ ζ ∥ H t 1 . (22) Proof. Every cell meeting a kernel support lies within ε ˉ + 2 h \bar{\varepsilon}+\sqrt{2}h ε ˉ + 2 h χ p q , ζ = p q , ζ \chi_{pq,\zeta}=p_{q,\zeta} χ pq , ζ = p q , ζ (20) ; the coefficients of p q ( t ) , ζ ( t ) p_q(t),\zeta(t) p q ( t ) , ζ ( t ) H 1 H^1 H 1 (22) . Only the fixed interpolant I h I_h I h
Volume-preserving configuration alignment
The comparison follows the cutoff solenoidal-flow construction of Legendre and Takahashi [7 Section 4.1] , together with the transformed rigid velocities in their Section 4.2. We give its realization and the continuous-time bounds needed for the present Cartesian recovery. For two sufficiently close configurations q = ( c , θ ) q=(c,\theta) q = ( c , θ ) q ^ = ( c ^ , θ ^ ) \widehat{q}=(\widehat{c},\widehat{\theta}) q = ( c , θ ) d q = ∣ c ^ − c ∣ + ∣ θ ^ − θ ∣ d_q=|\widehat{c}-c|+|\widehat{\theta}-\theta| d q = ∣ c − c ∣ + ∣ θ − θ ∣ O = Q ( θ ^ − θ ) O=Q(\widehat{\theta}-\theta) O = Q ( θ − θ ) d q \sqrt{d_q} d q
Lemma 4.2 (Constructive solenoidal alignment). There is a smooth volume-preserving diffeomorphism Ψ = Ψ ( q , q ^ ) : Ω → Ω \Psi=\Psi(q,\widehat{q}):\Omega\to\Omega Ψ = Ψ ( q , q ) : Ω → Ω
Ψ ( x ) = c ^ + O ( x − c ) on a fixed-width neighborhood of B ( q ) ‾ . (23) \Psi(x)=\widehat{c}+O(x-c)\quad\text{on a fixed-width neighborhood of }\overline{B(q)}. \tag*{(23)} Ψ ( x ) = c + O ( x − c ) on a fixed-width neighborhood of B ( q ) . (23) For every fixed finite k k k d q d_q d q
∥ Ψ − i d ∥ W k , ∞ + ∥ Ψ − 1 − i d ∥ W k , ∞ ≤ C k d q . (24) \|\Psi-\mathrm{id}\|_{W^{k,\infty}}+\|\Psi^{-1}-\mathrm{id}\|_{W^{k,\infty}}\le C_kd_q. \tag*{(24)} ∥Ψ − id ∥ W k , ∞ + ∥ Ψ − 1 − id ∥ W k , ∞ ≤ C k d q . (24) For Lipschitz paths q ( t ) , q ^ ( t ) q(t),\widehat{q}(t) q ( t ) , q ( t )
∥ ∂ t Ψ ∥ W k , ∞ ≤ C k ( d q + ∣ q ^ ′ − q ′ ∣ ) a.e. (25) \|\partial_t\Psi\|_{W^{k,\infty}}\le C_k\left(d_q+|\widehat{q}'-q'|\right)\quad\text{a.e.} \tag*{(25)} ∥ ∂ t Ψ ∥ W k , ∞ ≤ C k ( d q + ∣ q ′ − q ′ ∣ ) a.e. (25) The Piola map
( T Ψ v ) ( Ψ ( x ) ) = D Ψ ( x ) v ( x ) (26) (T_\Psi v)(\Psi(x))=D\Psi(x)v(x) \tag*{(26)} ( T Ψ v ) ( Ψ ( x )) = D Ψ ( x ) v ( x ) (26) preserves U U U q q q q ^ \widehat{q} q ( V , ω ) ↦ ( O V , ω ) (V,\omega)\mapsto(OV,\omega) ( V , ω ) ↦ ( O V , ω )
Proof. Let Δ c = c ^ − c \Delta c=\widehat{c}-c Δ c = c − c Δ θ = θ ^ − θ \Delta\theta=\widehat{\theta}-\theta Δ θ = θ − θ c s = c + s Δ c c_s=c+s\Delta c c s = c + s Δ c θ s = θ + s Δ θ \theta_s=\theta+s\Delta\theta θ s = θ + s Δ θ 0 ≤ s ≤ 1 0\le s\le1 0 ≤ s ≤ 1 s s s Δ c + Δ θ J ( x − c s ) \Delta c+\Delta\theta J(x-c_s) Δ c + Δ θ J ( x − c s ) (21) , but use a separate auxiliary cutoff χ A ∈ C ∞ ( Ω ) \chi_A\in C^\infty(\Omega) χ A ∈ C ∞ ( Ω ) H s H_s H s C 5 C^5 C 5 Ψ \Psi Ψ ∂ s Ψ s = H s ∘ Ψ s \partial_s\Psi_s=H_s\circ\Psi_s ∂ s Ψ s = H s ∘ Ψ s Ψ 0 = i d \Psi_0=\mathrm{id} Ψ 0 = id d q d_q d q (23) . The field H s H_s H s det D Ψ s = 1 \det D\Psi_s=1 det D Ψ s = 1
All spatial derivatives of H s H_s H s C k d q C_kd_q C k d q ∥ ∂ t H s ∥ W k , ∞ ≤ C k ( d q + ∣ q ^ ′ − q ′ ∣ ) \|\partial_tH_s\|_{W^{k,\infty}}\le C_k(d_q+|\widehat{q}'-q'|) ∥ ∂ t H s ∥ W k , ∞ ≤ C k ( d q + ∣ q ′ − q ′ ∣ ) 0 ≤ s ≤ 1 0\le s\le1 0 ≤ s ≤ 1 (24) –(25) . These estimates use the geometric clearance, not a regularization width. Finally, change of variables in the distributional divergence identity proves that (26) preserves solenoidality. Near the body D Ψ = O D\Psi=O D Ψ = O O J = J O OJ=JO O J = J O
O R q ( V , ω ) ( x ) = O V + ω J ( Ψ ( x ) − c ^ ) . OR_q(V,\omega)(x)=OV+\omega J(\Psi(x)-\widehat{c}). O R q ( V , ω ) ( x ) = O V + ω J ( Ψ ( x ) − c ) . This proves the claimed rigid trace and interior transformation.
Recovery of a whole rigid interior
Let q ( t ) q(t) q ( t ) X ~ ( t ) = ( u ~ ( t ) , ξ ~ ( t ) ) ∈ K ( q ( t ) ) \widetilde{X}(t) = (\widetilde{u}(t),\widetilde{\xi}(t)) \in K(q(t)) X ( t ) = ( u ( t ) , ξ ( t )) ∈ K ( q ( t )) ξ ~ ∈ H t 1 \widetilde{\xi} \in H^1_t ξ ∈ H t 1 q q q w = u ~ − L q ξ ~ w = \widetilde{u} - L_q\widetilde{\xi} w = u − L q ξ w ∈ H t 1 L x 2 w \in H^1_t L^2_x w ∈ H t 1 L x 2 W 3 , ∞ W^{3,\infty} W 3 , ∞ W 3 , ∞ W^{3,\infty} W 3 , ∞ L t 2 L^2_t L t 2
Lemma 4.3 (Joint common recovery). Put δ = 4 ( ε ~ + 2 h ) \delta= 4(\widetilde{\varepsilon}+\sqrt{2}h) δ = 4 ( ε + 2 h ) For small δ \delta δ Z h ( t ) = ( z h ( t ) , ξ ~ ( t ) ) Z_h(t) = (z_h(t),\widetilde{\xi}(t)) Z h ( t ) = ( z h ( t ) , ξ ( t ))
sup t ∥ z h − u ~ ∥ U + ∥ ∂ t ( z h − u ~ ) ∥ L t 2 L x 2 ≤ C δ , sup t ∥ z h − u ~ ∥ 0 ≤ C δ 3 / 2 . (27) \sup_t \|z_h-\widetilde{u}\|_U+\|\partial_t(z_h-\widetilde{u})\|_{L^2_tL^2_x}\le C\sqrt{\delta},\qquad\sup_t\|z_h-\widetilde{u}\|_0\le C\delta^{3/2}. \tag*{(27)} t sup ∥ z h − u ∥ U + ∥ ∂ t ( z h − u ) ∥ L t 2 L x 2 ≤ C δ , t sup ∥ z h − u ∥ 0 ≤ C δ 3/2 . (27) The constant depends on the displayed regularity bounds, ∥ ξ ~ ∥ H 1 \|\widetilde{\xi}\|_{H^1} ∥ ξ ∥ H 1 ∥ q ′ ∥ L ∞ \|q'\|_{L^\infty} ∥ q ′ ∥ L ∞ q q q
Proof. Let curl ψ = w \operatorname{curl}\psi=w curl ψ = w w = 0 w=0 w = 0 ψ \psi ψ c ψ ( t ) c_\psi(t) c ψ ( t ) d ( t , x ) d(t,x) d ( t , x ) χ 0 \chi_0 χ 0 [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] [ − 2 , 2 ] [-2,2] [ − 2 , 2 ]
ψ δ = ψ − χ 0 ( d / δ ) ( ψ − c ψ ) near the interface , z h = curl I h ψ δ + L h , q ξ ~ . \psi_\delta=\psi-\chi_0(d/\delta)(\psi-c_\psi)\quad\text{near the interface},\qquad z_h=\operatorname{curl} I_h\psi_\delta+L_{h,q}\widetilde{\xi}. ψ δ = ψ − χ 0 ( d / δ ) ( ψ − c ψ ) near the interface , z h = curl I h ψ δ + L h , q ξ . The streamfunction is unchanged away from the interface tube and is constant throughout the body and the δ \delta δ Lemma 4.1 completes membership.
For completeness, the width-dependent estimates are the moving collar estimates of Part II [3 ] , and require only a Lipschitz configuration path. Writing r = ψ − c ψ r=\psi-c_\psi r = ψ − c ψ r = O ( d 2 ) r=O(d^2) r = O ( d 2 ) ∇ r = O ( d ) \nabla r=O(d) ∇ r = O ( d ) r t = O ( d ) r_t=O(d) r t = O ( d ) c ψ ′ = ψ t ∣ Γ ( t ) c_\psi'=\psi_t|_{\Gamma(t)} c ψ ′ = ψ t ∣ Γ ( t ) ∇ ψ ∣ Γ ( t ) = 0 \nabla\psi|_{\Gamma(t)}=0 ∇ ψ ∣ Γ ( t ) = 0
( ψ δ − ψ ) t = − χ 0 ( d / δ ) r t − δ − 1 χ 0 ′ ( d / δ ) d t r . (\psi_\delta-\psi)_t=-\chi_0(d/\delta)r_t-\delta^{-1}\chi_0'(d/\delta)d_t r. ( ψ δ − ψ ) t = − χ 0 ( d / δ ) r t − δ − 1 χ 0 ′ ( d / δ ) d t r . The support has area O ( δ ) O(\delta) O ( δ )
∥ ψ δ − ψ ∥ H 2 = O ( δ ) , ∥ ψ δ − ψ ∥ H 1 = O ( δ 3 / 2 ) , ∥ ( ψ δ − ψ ) t ∥ L t 2 H x 1 = O ( δ ) . \|\psi_\delta-\psi\|_{H^2}=O(\sqrt{\delta}),\qquad\|\psi_\delta-\psi\|_{H^1}=O(\delta^{3/2}),\qquad\|(\psi_\delta-\psi)_t\|_{L^2_tH^1_x}=O(\sqrt{\delta}). ∥ ψ δ − ψ ∥ H 2 = O ( δ ) , ∥ ψ δ − ψ ∥ H 1 = O ( δ 3/2 ) , ∥ ( ψ δ − ψ ) t ∥ L t 2 H x 1 = O ( δ ) . The corresponding H 4 H^4 H 4 ψ δ \psi_\delta ψ δ ψ δ , t \psi_{\delta,t} ψ δ , t O ( 1 + δ − 3 / 2 ) O(1+\delta^{-3/2}) O ( 1 + δ − 3/2 ) L t ∞ L^\infty_t L t ∞ O ( 1 + δ − 5 / 2 ) O(1+\delta^{-5/2}) O ( 1 + δ − 5/2 ) L t 2 L^2_t L t 2 d t = − ( c ′ + θ ′ J ( x − c ) ) ⋅ ∇ d d_t=-(c'+\theta'J(x-c))\cdot\nabla d d t = − ( c ′ + θ ′ J ( x − c )) ⋅ ∇ d q ′ q' q ′ q ′ ′ q'' q ′′ h 2 δ − 3 / 2 h^2\delta^{-3/2} h 2 δ − 3/2 H 1 H^1 H 1 h 3 δ − 5 / 2 h^3\delta^{-5/2} h 3 δ − 5/2 L 2 L^2 L 2 C δ C\sqrt{\delta} C δ Lemma 4.1 are smaller. This proves (27) . The moving-cutoff derivative has been included explicitly; no commutation of a time-dependent recovery map with differentiation is asserted. □ \square □
Lemma 4.4 (Alignment errors at the same physical points). Suppose u ∈ U u\in U u ∈ U W 2 , ∞ W^{2,\infty} W 2 , ∞ Γ ( q ) \Gamma(q) Γ ( q ) For small d q d_q d q
∥ T q u − u ∥ 0 ≤ C d q ∥ u ∥ H 1 , ∥ T q u − u ∥ U ≤ C d q ( ∥ u ∥ W 2 , ∞ ( B ( q ) ) + ∥ u ∥ W 2 , ∞ ( F ( q ) ) ) . (28) \|\mathcal{T}_q u-u\|_0\le C d_q\|u\|_{H^1},\qquad\|\mathcal{T}_q u-u\|_U\le C\sqrt{d_q}\left(\|u\|_{W^{2,\infty}(B(q))}+\|u\|_{W^{2,\infty}(F(q))}\right). \tag*{(28)} ∥ T q u − u ∥ 0 ≤ C d q ∥ u ∥ H 1 , ∥ T q u − u ∥ U ≤ C d q ( ∥ u ∥ W 2 , ∞ ( B ( q )) + ∥ u ∥ W 2 , ∞ ( F ( q )) ) . (28) Proof. Integrating u u u O ( d q ) O(d_q) O ( d q ) L 2 L^2 L 2 D Ψ − I D\Psi-I D Ψ − I O ( d q ) O(d_q) O ( d q ) C d q C d_q C d q O ( d q ) O(d_q) O ( d q ) O ( d q ) O(d_q) O ( d q )
A coupled Cartesian scheme and its exact energy law
Known geometry at each linear solve
Choose 0 = t 0 < ⋯ < t N = T 0=t_0<\cdots<t_N=T 0 = t 0 < ⋯ < t N = T τ n = t n − t n − 1 \tau_n=t_n-t_{n-1} τ n = t n − t n − 1 q 0 q_0 q 0 X 0 X_0 X 0 X h 0 = P M 0 X 0 X_h^0=P_M^0X_0 X h 0 = P M 0 X 0 q 0 q_0 q 0 n ≥ 1 n\ge1 n ≥ 1
c n = c n − 1 + τ n V h n − 1 , θ n = θ n − 1 + τ n ω h n − 1 . (29) c_n=c_{n-1}+\tau_n V_h^{n-1}, \qquad\theta_n=\theta_{n-1}+\tau_n\omega_h^{n-1}. \tag*{(29)} c n = c n − 1 + τ n V h n − 1 , θ n = θ n − 1 + τ n ω h n − 1 . (29) The angle is represented as a real-valued continuous angle. The piecewise affine pose reconstruction has derivative q ˙ h = ( V h n − 1 , ω h n − 1 ) \dot q_h=(V_h^{n-1},\omega_h^{n-1}) q ˙ h = ( V h n − 1 , ω h n − 1 ) n n n
At q n q_n q n
K h n = K h ( q n ) , P M n : H ⟶ K h n \mathbb{K}_h^n=\mathbb{K}_h(q_n), \qquad P_M^n:\mathbb{H}\longrightarrow\mathbb{K}_h^n K h n = K h ( q n ) , P M n : H ⟶ K h n for the current joint space of (16) and its orthogonal projector in (5) . All the indicated parameters may change at every time step. The projection is in the combined fluid and rigid-body energy metric. The same physical bulk mass integrals are used in every configuration.
Let F n F^n F n F e n F_e^n F e n T e n T_e^n T e n current exterior region:
L n ( Y ) = ∫ F ( q n ) F n ( x ) ⋅ v ( x ) d x + F e n ⋅ W + T e n η . (30) \mathcal{L}^n(Y)=\int_{F(q_n)}F^n(x)\cdot v(x)\,\mathrm{d}x+F_e^n\cdot W+T_e^n\eta. \tag*{(30)} L n ( Y ) = ∫ F ( q n ) F n ( x ) ⋅ v ( x ) d x + F e n ⋅ W + T e n η . (30) The scheme finds X h n = ( u h n , V h n , ω h n ) ∈ K h n X_h^n=(u_h^n,V_h^n,\omega_h^n)\in\mathbb{K}_h^n X h n = ( u h n , V h n , ω h n ) ∈ K h n
( X h n − P M n X h n − 1 τ n , Y h ) M + a ( u h n , v h ) + G ( ω h n − 1 ; X h n , Y h ) = L n ( Y h ) , Y h ∈ K h n . (31) \left(\frac{X_h^n-P_M^nX_h^{n-1}}{\tau_n},Y_h\right)_M+a(u_h^n,v_h)+G(\omega_h^{n-1};X_h^n,Y_h)=\mathcal{L}^n(Y_h), \qquad Y_h\in\mathbb{K}_h^n. \tag*{(31)} ( τ n X h n − P M n X h n − 1 , Y h ) M + a ( u h n , v h ) + G ( ω h n − 1 ; X h n , Y h ) = L n ( Y h ) , Y h ∈ K h n . (31) The previous angular velocity is a known coefficient of a skew operator, whereas the current translational velocity is unknown in that operator. Both fluid and rigid velocities are solved together.
For a current test, orthogonality gives ( P M n X h n − 1 , Y h ) M = ( X h n − 1 , Y h ) M (P_M^nX_h^{n-1},Y_h)_M=(X_h^{n-1},Y_h)_M ( P M n X h n − 1 , Y h ) M = ( X h n − 1 , Y h ) M (31) can be assembled directly using the cross-space mass pairing with the full previous state, without performing a separate projection solve. For each admissible current geometry the linear system has a unique joint state: its bilinear form evaluated on ( Y h , Y h ) (Y_h,Y_h) ( Y h , Y h ) τ n − 1 ∥ Y h ∥ M 2 + a ( v h , v h ) > 0 \tau_n^{-1}\lVert Y_h\rVert_M^2+a(v_h,v_h)>0 τ n − 1 ∥ Y h ∥ M 2 + a ( v h , v h ) > 0 Y h ≠ 0 Y_h\ne0 Y h = 0
The geometric algorithm is considered until a configuration leaves the collision-free region used by the analysis. Stability below holds on every admissible prefix. The convergence argument closes the geometry bound against the assumed collision-free exact branch for sufficiently small discretization parameters; collision avoidance is not an extra conclusion of the energy identity alone.
Cancellation of the coupled reactions
Opposite fluid and structural work is the standard multiplier energy mechanism [11 , 12 ] . Here it is applied to both regularized volume and surface moments in the same positive fluid–body mass space. Let b h n ( Y h , Λ h ) b_h^n(Y_h,\Lambda_h) b h n ( Y h , Λ h ) J ε n v h − R q n ( W h , η h ) \mathcal{J}_{\varepsilon}^n v_h-\mathcal{R}q^n(W_h,\eta_h) J ε n v h − R q n ( W h , η h ) X h , A n × R 3 X_{h,A_n}\times\mathbb{R}^3 X h , A n × R 3
( X h n − X h n − 1 τ n , Y h ) M + a ( u h n , v h ) + G ( ω h n − 1 ; X h n , Y h ) + b h n ( Y h , Λ h n ) = L n ( Y h ) , (32) \left(\frac{X_h^n-X_h^{n-1}}{\tau_n},Y_h\right)_M+a(u_h^n,v_h)+G(\omega_h^{n-1};X_h^n,Y_h)+b_h^n(Y_h,\Lambda_h^n)=\mathcal{L}^n(Y_h), \tag*{(32)} ( τ n X h n − X h n − 1 , Y h ) M + a ( u h n , v h ) + G ( ω h n − 1 ; X h n , Y h ) + b h n ( Y h , Λ h n ) = L n ( Y h ) , (32) b h n ( X h n , Υ h ) = 0 for all Υ h . b_h^n(X_h^n,\Upsilon_h)=0 \qquad\text{for all }\Upsilon_h. b h n ( X h n , Υ h ) = 0 for all Υ h . The minus rigid velocity in each moment is essential: the same reaction acts with opposite fluid and body work. Testing by X h n X_h^n X h n
Exact identity and bounds independent of mesh history
Theorem 5.1 (Coupled discrete energy). Every step of (31) satisfies
∥ X h n ∥ M 2 − ∥ X h n − 1 ∥ M 2 + ∥ X h n − X h n − 1 ∥ M 2 + 2 τ n a ( u h n , u h n ) = 2 τ n L n ( X h n ) . (33) \lVert X_h^n\rVert_M^2-\lVert X_h^{n-1}\rVert_M^2+\lVert X_h^n-X_h^{n-1}\rVert_M^2+2\tau_n a(u_h^n,u_h^n)=2\tau_n\mathcal{L}^n(X_h^n). \tag*{(33)} ∥ X h n ∥ M 2 − ∥ X h n − 1 ∥ M 2 + ∥ X h n − X h n − 1 ∥ M 2 + 2 τ n a ( u h n , u h n ) = 2 τ n L n ( X h n ) . (33) Moreover,
∥ X h n − X h n − 1 ∥ M 2 = ∥ X h n − P M n X h n − 1 ∥ M 2 + ∥ ( I − P M n ) X h n − 1 ∥ M 2 . (34) \lVert X_h^n-X_h^{n-1}\rVert_M^2=\lVert X_h^n-P_M^nX_h^{n-1}\rVert_M^2+\lVert(I-P_M^n)X_h^{n-1}\rVert_M^2. \tag*{(34)} ∥ X h n − X h n − 1 ∥ M 2 = ∥ X h n − P M n X h n − 1 ∥ M 2 + ∥( I − P M n ) X h n − 1 ∥ M 2 . (34) Let ℓ f n ∈ U ′ \ell_f^n\in U' ℓ f n ∈ U ′ (30) , and put
A N = ∑ n = 1 N τ n ( ∣ F e n ∣ 2 m e + ∣ T e n ∣ 2 I e ) 1 / 2 , B N = ∥ X h 0 ∥ M 2 + μ − 1 ∑ n = 1 N τ n ∥ ℓ f n ∥ U ′ 2 . A_N=\sum_{n=1}^{N}\tau_n\left(\frac{\lvert F_e^n\rvert^2}{m_e}+\frac{\lvert T_e^n\rvert^2}{I_e}\right)^{1/2},\qquad B_N=\lVert X_h^0\rVert_M^2+\mu^{-1}\sum_{n=1}^{N}\tau_n\lVert\ell_f^n\rVert_{U'}^2. A N = n = 1 ∑ N τ n ( m e ∣ F e n ∣ 2 + I e ∣ T e n ∣ 2 ) 1/2 , B N = ∥ X h 0 ∥ M 2 + μ − 1 n = 1 ∑ N τ n ∥ ℓ f n ∥ U ′ 2 . Then
max 0 ≤ n ≤ N ∥ X h n ∥ M 2 + ∑ n = 1 N ∥ X h n − X h n − 1 ∥ M 2 + μ ∑ n = 1 N τ n ∥ u h n ∥ U 2 ≤ C ( B N + A N 2 ) . (35) \max_{0\le n\le N}\lVert X_h^n\rVert_M^2+\sum_{n=1}^{N}\lVert X_h^n-X_h^{n-1}\rVert_M^2+\mu\sum_{n=1}^{N}\tau_n\lVert u_h^n\rVert_U^2\le C(B_N+A_N^2). \tag*{(35)} 0 ≤ n ≤ N max ∥ X h n ∥ M 2 + n = 1 ∑ N ∥ X h n − X h n − 1 ∥ M 2 + μ n = 1 ∑ N τ n ∥ u h n ∥ U 2 ≤ C ( B N + A N 2 ) . (35) with an absolute constant C C C
Proof. Test (31) with X h n X_h^n X h n 2 ( X − Y , X ) M = ∥ X ∥ M 2 − ∥ Y ∥ M 2 + ∥ X − Y ∥ M 2 2(X-Y,X)_M=\lVert X\rVert_M^2-\lVert Y\rVert_M^2+\lVert X-Y\rVert_M^2 2 ( X − Y , X ) M = ∥ X ∥ M 2 − ∥ Y ∥ M 2 + ∥ X − Y ∥ M 2 (33) . The vector ( I − P M n ) X h n − 1 (I-P_M^n)X_h^{n-1} ( I − P M n ) X h n − 1 X h n − P M n X h n − 1 X_h^n-P_M^nX_h^{n-1} X h n − P M n X h n − 1 (34) .
Write β n = ( ∣ F e n ∣ 2 / m e + ∣ T e n ∣ 2 / I e ) 1 / 2 \beta_n=(\lvert F_e^n\rvert^2/m_e+\lvert T_e^n\rvert^2/I_e)^{1/2} β n = (∣ F e n ∣ 2 / m e + ∣ T e n ∣ 2 / I e ) 1/2
∣ F e n ⋅ V h n + T e n ω h n ∣ ≤ β n ∥ X h n ∥ M . \lvert F_e^n\cdot V_h^n+T_e^n\omega_h^n\rvert\le\beta_n\lVert X_h^n\rVert_M. ∣ F e n ⋅ V h n + T e n ω h n ∣ ≤ β n ∥ X h n ∥ M . Use a ( u h n , u h n ) = μ ∥ u h n ∥ U 2 a(u_h^n,u_h^n)=\mu\lVert u_h^n\rVert_U^2 a ( u h n , u h n ) = μ ∥ u h n ∥ U 2 k k k
∥ X h k ∥ M 2 + ∑ n = 1 k ∥ X h n − X h n − 1 ∥ M 2 + μ ∑ n = 1 k τ n ∥ u h n ∥ U 2 ≤ B N + 2 ∑ n = 1 k τ n β n ∥ X h n ∥ M . (36) \lVert X_h^k\rVert_M^2+\sum_{n=1}^{k}\lVert X_h^n-X_h^{n-1}\rVert_M^2+\mu\sum_{n=1}^{k}\tau_n\lVert u_h^n\rVert_U^2\le B_N+2\sum_{n=1}^{k}\tau_n\beta_n\lVert X_h^n\rVert_M. \tag*{(36)} ∥ X h k ∥ M 2 + n = 1 ∑ k ∥ X h n − X h n − 1 ∥ M 2 + μ n = 1 ∑ k τ n ∥ u h n ∥ U 2 ≤ B N + 2 n = 1 ∑ k τ n β n ∥ X h n ∥ M . (36) Let Z = max 0 ≤ n ≤ N ∥ X h n ∥ M Z = \max_{0 \le n \le N} \|X_h^n\|_M Z = max 0 ≤ n ≤ N ∥ X h n ∥ M Z 2 ≤ B N Z^2 \le B_N Z 2 ≤ B N (36) to obtain Z 2 ≤ B N + 2 A N Z Z^2 \le B_N + 2A_NZ Z 2 ≤ B N + 2 A N Z
Z ≤ A N + A N 2 + B N ≤ 2 A N + B N . Z \le A_N + \sqrt{A_N^2 + B_N} \le2A_N + \sqrt{B_N}. Z ≤ A N + A N 2 + B N ≤ 2 A N + B N . Use this bound in (36) with k = N k=N k = N (35) . The argument is an algebraic maximum estimate and does not introduce a time-step condition through a discrete Gronwall denominator.
For exact ambient time averages, ∥ f h n ∥ U ′ ≤ C ∥ f n ∥ L 2 ( Ω ) \|f_h^n\|_{U'} \le C\|f^n\|_{L^2(\Omega)} ∥ f h n ∥ U ′ ≤ C ∥ f n ∥ L 2 ( Ω ) C ∥ f ∥ L 2 ( 0 , T ; L 2 ( Ω ) ) 2 C\|f\|_{L^2(0,T;L^2(\Omega))}^2 C ∥ f ∥ L 2 ( 0 , T ; L 2 ( Ω )) 2 A N A_N A N L 1 L^1 L 1 max n ( ∣ V h n ∣ + ∣ ω h n ∣ ) \max_n(|V_h^n|+|\omega_h^n|) max n ( ∣ V h n ∣ + ∣ ω h n ∣ ) (29) .
Corollary 5.2 (Physical constraint defects). Suppose the data bounds in Theorem 5.1 are uniform over a sequence of admissible computations. Then
( ∑ n τ n ∥ u h n − R q n ξ h n ∥ L 2 ( B ( q n ) ) 2 ) 1 / 2 ≤ C ( ε ˉ + κ ˉ ) , (37) \left(\sum_n \tau_n \|u_h^n-R_{q_n}\xi_h^n\|_{L^2(B(q_n))}^2\right)^{1/2} \le C(\bar{\varepsilon}+\bar{\kappa}), \tag*{(37)} ( n ∑ τ n ∥ u h n − R q n ξ h n ∥ L 2 ( B ( q n )) 2 ) 1/2 ≤ C ( ε ˉ + κ ˉ ) , (37) ( ∑ n τ n ∥ γ q n u h n − R q n ξ h n ∥ L 2 ( Γ ( q n ) ) 2 ) 1 / 2 ≤ C ( ε ˉ + ℓ ˉ ) . (38) \left(\sum_n \tau_n \|\gamma_{q_n}u_h^n-R_{q_n}\xi_h^n\|_{L^2(\Gamma(q_n))}^2\right)^{1/2} \le C(\sqrt{\bar{\varepsilon}}+\sqrt{\bar{\ell}}). \tag*{(38)} ( n ∑ τ n ∥ γ q n u h n − R q n ξ h n ∥ L 2 ( Γ ( q n )) 2 ) 1/2 ≤ C ( ε ˉ + ℓ ˉ ) . (38) Here bars denote the maxima of the indicated widths over the run. The surrogate-to-physical kinetic-energy discrepancy in (19) tends to zero in the weighted L 1 L^1 L 1 ε ˉ + κ ˉ → 0 \bar{\varepsilon}+\bar{\kappa}\to0 ε ˉ + κ ˉ → 0
Proof. The stability bound controls u h u_h u h L 2 ( U ) L^2(U) L 2 ( U ) ξ h \xi_h ξ h m e , I e m_e,I_e m e , I e ∑ n τ n ∥ X h n ∥ V 2 \sum_n\tau_n\|X_h^n\|_{\mathbb{V}}^2 ∑ n τ n ∥ X h n ∥ V 2
Corollary 5.3 (Defects relative to the actual interpolated motion). Under the uniform bounds of Corollary 5.2, suppose also that the piecewise affine pose path remains in the fixed admissible configuration family. For t ∈ ( t n − 1 , t n ] t\in(t_{n-1},t_n] t ∈ ( t n − 1 , t n ]
U h ( t ) = u h n , b h p a t h ( t , x ) = V h n − 1 + ω h n − 1 J ( x − c h ( t ) ) . U_h(t)=u_h^n,\qquad b_h^{\mathrm{path}}(t,x)=V_h^{n-1}+\omega_h^{n-1}J(x-c_h(t)). U h ( t ) = u h n , b h path ( t , x ) = V h n − 1 + ω h n − 1 J ( x − c h ( t )) . The latter is the material velocity of the reconstructed body, because q ˙ h = ( V h n − 1 , ω h n − 1 ) \dot q_h=(V_h^{n-1},\omega_h^{n-1}) q ˙ h = ( V h n − 1 , ω h n − 1 )
( ∫ 0 T ∥ U h − b h p a t h ∥ L 2 ( B ( q h ( t ) ) ) 2 d t ) 1 / 2 ≤ C ( ε ˉ + κ ˉ + τ max ) , (39) \left(\int_0^T \|U_h-b_h^{\mathrm{path}}\|_{L^2(B(q_h(t)))}^2\,\mathrm{d}t\right)^{1/2}
\le C(\bar{\varepsilon}+\bar{\kappa}+\sqrt{\tau_{\max}}), \tag*{(39)} ( ∫ 0 T ∥ U h − b h path ∥ L 2 ( B ( q h ( t ))) 2 d t ) 1/2 ≤ C ( ε ˉ + κ ˉ + τ m a x ) , (39) ( ∫ 0 T ∥ γ q h ( t ) U h − b h p a t h ∥ L 2 ( Γ ( q h ( t ) ) ) 2 d t ) 1 / 2 ≤ C ( ε ˉ + ℓ ˉ + τ max ) . (40) \left(\int_0^T \|\gamma_{q_h(t)}U_h-b_h^{\mathrm{path}}\|_{L^2(\Gamma(q_h(t)))}^2\,\mathrm{d}t\right)^{1/2}
\le C(\sqrt{\bar{\varepsilon}}+\sqrt{\bar{\ell}}+\sqrt{\tau_{\max}}). \tag*{(40)} ( ∫ 0 T ∥ γ q h ( t ) U h − b h path ∥ L 2 ( Γ ( q h ( t ))) 2 d t ) 1/2 ≤ C ( ε ˉ + ℓ ˉ + τ m a x ) . (40) The constants may depend on the uniform data bound and the fixed admissible family, but not on the mesh history. The nodal defects in (37) –(38) use the newly solved velocity ξ h n \xi_h^n ξ h n
Proof. Let w n = u h n − L h , q n ξ h n w_n=u_h^n-L_{h,q_n}\xi_h^n w n = u h n − L h , q n ξ h n q h q_h q h R t , n R_{t,n} R t , n B ( q n ) B(q_n) B ( q n ) B ( q h ( t ) ) B(q_h(t)) B ( q h ( t )) C τ n C\tau_n C τ n w n w_n w n
∥ E w n ∘ R t , n − E w n ∥ L 2 ( R 2 ) ≤ C τ n ∥ w n ∥ U . \|Ew_n\circ R_{t,n}-Ew_n\|_{L^2(\mathbb{R}^2)}\le C\tau_n\|w_n\|_U. ∥ E w n ∘ R t , n − E w n ∥ L 2 ( R 2 ) ≤ C τ n ∥ w n ∥ U . The corresponding H 1 H^1 H 1 C ∥ w n ∥ U C\lVert w_n\rVert_U C ∥ w n ∥ U Γ ( q n ) \Gamma(q_n) Γ ( q n ) C τ n ∥ w n ∥ U C\sqrt{\tau_n}\lVert w_n\rVert_U C τ n ∥ w n ∥ U [3 ] . The rigid map preserves volume and arclength. Combining these shift estimates with the current volume and surface defects bounds w n w_n w n C ( ϵ n + κ n + τ n ) ∥ X h n ∥ V C(\epsilon_n+\kappa_n+\tau_n)\lVert X_h^n\rVert_V C ( ϵ n + κ n + τ n ) ∥ X h n ∥ V C ( ϵ n + ℓ n + τ n ) ∥ X h n ∥ V C(\sqrt{\epsilon_n}+\sqrt{\ell_n}+\sqrt{\tau_n})\lVert X_h^n\rVert_V C ( ϵ n + ℓ n + τ n ) ∥ X h n ∥ V
The common lift reproduces its rigid affine field on every body in the admissible family, including the intermediate body. At the same physical point its difference from the path velocity obeys
∣ L h , q n ξ h n − b p a t h ( t ) ∣ ≤ C ∣ ξ h n − ξ n − 1 ∣ + C τ n on B ( q h ( t ) ) ∪ Γ ( q h ( t ) ) . \left\lvert L_{h,q_n}\xi_h^n-b^{\mathrm{path}}(t)\right\rvert\le C\left\lvert\xi_h^n-\xi^{n-1}\right\rvert+C\tau_n \quad\text{on } B(q_h(t))\cup\Gamma(q_h(t)). L h , q n ξ h n − b path ( t ) ≤ C ξ h n − ξ n − 1 + C τ n on B ( q h ( t )) ∪ Γ ( q h ( t )) . Indeed, the first term accounts for the change of rigid velocity and the second for ∣ c n − c h ( t ) ∣ ≤ C τ n \lvert c_n-c_h(t)\rvert\le C\tau_n ∣ c n − c h ( t )∣ ≤ C τ n
∑ n τ n ∣ ξ h n − ξ h n − 1 ∣ 2 ≤ C τ max ∑ n ∥ X h n − X h n − 1 ∥ M 2 ≤ C τ max . \sum_n \tau_n\left\lvert\xi_h^n-\xi_h^{n-1}\right\rvert^2\le C\tau_{\max}\sum_n\left\lVert X_h^n-X_h^{n-1}\right\rVert_M^2\le C\tau_{\max}. n ∑ τ n ξ h n − ξ h n − 1 2 ≤ C τ m a x n ∑ X h n − X h n − 1 M 2 ≤ C τ m a x . Integrate over each interval, sum, and use ∑ n τ n ∥ X h n ∥ V 2 ≤ C \sum_n\tau_n\lVert X_h^n\rVert_V^2\le C ∑ n τ n ∥ X h n ∥ V 2 ≤ C τ max \tau_{\max} τ m a x T T T τ max \sqrt{\tau_{\max}} τ m a x
The discrete dissipation is 2 μ ∫ Ω ∣ D ( u h ) ∣ 2 2\mu\int_{\Omega}\lvert D(u_h)\rvert^2 2 μ ∫ Ω ∣ D ( u h ) ∣ 2
Closing the fluid, rigid-velocity, and configuration errors
The theorem is conditional on a sufficiently regular physical solution, not on convergence of numerical forces or trajectories. Geometry is an unknown of the discrete scheme. The comparison below is constructed on that computed geometry, and its discrepancy from the exact geometry is estimated together with the velocity error. The comparison and residual strategy has precedents in coupled geometry analyses [7 , 10 ] and changing-space Stokes estimates [13 ] . The constructions and bounds below specify the regularized volume and surface consistency terms and the retained-coarse transfer argument needed for the present method.
Explicit assumptions on the exact branch and loads
Assume that (2)--(3) has a solution on [ 0 , T ] [0,T] [ 0 , T ]
u ∈ H 1 ( 0 , T ; L 2 ( Ω ) 2 ) , X ( t ) ∈ K ( q ( t ) ) , ξ ∈ H 1 ( 0 , T ; R 3 ) , M 0 : = ess sup t ( ∥ u ( t ) ∥ W 4 , ∞ ( B ( q ( t ) ) ) + ∥ u ( t ) ∥ W 4 , ∞ ( F ( q ( t ) ) ) ) < ∞ , M 1 ( t ) : = ∥ u t ( t ) ∥ W 3 , ∞ ( B ( q ( t ) ) ) + ∥ u t ( t ) ∥ W 3 , ∞ ( F ( q ( t ) ) ) ∈ L 2 ( 0 , T ) . (41) \begin{aligned}
u &\in H^1(0,T;L^2(\Omega)^2), \qquad X(t)\in K(q(t)), \qquad\xi\in H^1(0,T;\mathbb{R}^3),\\
M_0 &:= \operatorname*{ess\,sup}_{t}\left(\lVert u(t)\rVert_{W^{4,\infty}(B(q(t)))}+\lVert u(t)\rVert_{W^{4,\infty}(F(q(t)))}\right)<\infty,\\
M_1(t)&:=\lVert u_t(t)\rVert_{W^{3,\infty}(B(q(t)))}+\lVert u_t(t)\rVert_{W^{3,\infty}(F(q(t)))}\in L^2(0,T). \tag*{(41)}
\end{aligned} u M 0 M 1 ( t ) ∈ H 1 ( 0 , T ; L 2 ( Ω ) 2 ) , X ( t ) ∈ K ( q ( t )) , ξ ∈ H 1 ( 0 , T ; R 3 ) , := t ess sup ( ∥ u ( t ) ∥ W 4 , ∞ ( B ( q ( t ))) + ∥ u ( t ) ∥ W 4 , ∞ ( F ( q ( t ))) ) < ∞ , := ∥ u t ( t ) ∥ W 3 , ∞ ( B ( q ( t ))) + ∥ u t ( t ) ∥ W 3 , ∞ ( F ( q ( t ))) ∈ L 2 ( 0 , T ) . (41) The time derivative is Eulerian and may have different side traces. The extra spatial derivative compared with Part II is used when differentiating the configuration alignment: that derivative contains ∇ u \nabla u ∇ u
f ∈ L 2 ( 0 , T ; W 1 , ∞ ( Ω ) 2 ) , F e , T e ∈ L 2 ( 0 , T ) , Λ B 2 : = ∫ 0 T ∥ λ B ( t ) ∥ L 2 ( B ( q ( t ) ) ) 2 d t < ∞ , Λ Γ 2 : = ∫ 0 T ∥ λ ( t ) ∥ L 2 ( Γ ( q ( t ) ) ) 2 d t < ∞ . (42) \begin{aligned}
f&\in L^2(0,T;W^{1,\infty}(\Omega)^2), \qquad F_e,T_e\in L^2(0,T),\\
\Lambda_B^2&:=\int_0^T\lVert\lambda_B(t)\rVert_{L^2(B(q(t)))}^2\,dt<\infty,\qquad
\Lambda_\Gamma^2:=\int_0^T\lVert\lambda(t)\rVert_{L^2(\Gamma(q(t)))}^2\,dt<\infty. \tag*{(42)}
\end{aligned} f Λ B 2 ∈ L 2 ( 0 , T ; W 1 , ∞ ( Ω ) 2 ) , F e , T e ∈ L 2 ( 0 , T ) , := ∫ 0 T ∥ λ B ( t ) ∥ L 2 ( B ( q ( t ))) 2 d t < ∞ , Λ Γ 2 := ∫ 0 T ∥ λ ( t ) ∥ L 2 ( Γ ( q ( t ))) 2 d t < ∞. (42) Here F e F_e F e (12) already has this regularity under (41) . The surface-traction bound is a separate assumption on the exact solution. A mere ambient L t 2 U ′ L^2_t U' L t 2 U ′
The primary load choice is the exact ambient time average in (30) . More generally, let a bar denote each ambient time average and define the data error before computing the trajectory by
E L 2 = ∑ n τ n ( ∥ f n − f ˉ n ∥ L 2 ( Ω ) 2 + ∣ F e n − F ˉ e n ∣ 2 + ∣ T e n − T ˉ e n ∣ 2 ) . (43) E_L^2=\sum_n\tau_n\left(\lVert f^n-\bar f^n\rVert_{L^2(\Omega)}^2+\lvert F_e^n-\bar F_e^n\rvert^2+\lvert T_e^n-\bar T_e^n\rvert^2\right). \tag*{(43)} E L 2 = n ∑ τ n ( ∥ f n − f ˉ n ∥ L 2 ( Ω ) 2 + ∣ F e n − F ˉ e n ∣ 2 + ∣ T e n − T ˉ e n ∣ 2 ) . (43) If L ˉ n \bar L_n L ˉ n F ( q n ) F(q_n) F ( q n ) Lemma 3.2 and Poincare’s inequality bound the squared time sum of sup Y h ≠ 0 ∣ L n ( Y h ) − L ˉ n ( Y h ) ∣ / ∥ v h ∥ U \sup_{Y_h\ne0}\lvert L^n(Y_h)-\bar L_n(Y_h)\rvert/\lVert v_h\rVert_U sup Y h = 0 ∣ L n ( Y h ) − L ˉ n ( Y h )∣ / ∥ v h ∥ U C E L 2 CE_L^2 C E L 2 E L E_L E L Theorem 5.1 . Exact time averages have E L = 0 E_L=0 E L = 0 f ∈ H 1 ( 0 , T ; L 2 ( Ω ) 2 ) f\in H^1(0,T;L^2(\Omega)^2) f ∈ H 1 ( 0 , T ; L 2 ( Ω ) 2 ) F e , T e ∈ H 1 ( 0 , T ) F_e,T_e\in H^1(0,T) F e , T e ∈ H 1 ( 0 , T ) f n = f ( t n ) f^n=f(t_n) f n = f ( t n ) F e n = F e ( t n ) F_e^n=F_e(t_n) F e n = F e ( t n ) T e n = T e ( t n ) T_e^n=T_e(t_n) T e n = T e ( t n )
E L ≤ C τ max ( ∥ f t ∥ L t 2 L x 2 2 + ∥ F e ′ ∥ L t 2 2 + ∥ T e ′ ∥ L t 2 2 ) . E_L\le C\tau_{\max}\left(\lVert f_t\rVert_{L^2_tL^2_x}^2+\lVert F'_e\rVert_{L^2_t}^2+\lVert T'_e\rVert_{L^2_t}^2\right). E L ≤ C τ m a x ( ∥ f t ∥ L t 2 L x 2 2 + ∥ F e ′ ∥ L t 2 2 + ∥ T e ′ ∥ L t 2 2 ) . This follows from the fundamental theorem of calculus, Jensen’s inequality, and Lemma 3.2 ; the fluid integral in both compared loads is on the same current domain F ( q n ) F(q_n) F ( q n )
r h 2 = h + ε ˉ + ℓ ˉ + κ ˉ 2 + τ max + E L 2 , τ max = max n τ n . (44) r_h^2=h+\bar\varepsilon+\bar\ell+\bar\kappa^2+\tau_{\max}+E_L^2,\qquad\tau_{\max}=\max_n\tau_n. \tag*{(44)} r h 2 = h + ε ˉ + ℓ ˉ + κ ˉ 2 + τ m a x + E L 2 , τ m a x = n max τ n . (44) All constants below may depend on the exact branch and the fixed physical parameters, including the strictly positive excess inertias, but not on a refinement history or on inverse regularization widths.
The mapped identity on all ambient tests
On an admissible prefix of the computation, align the exact pose to the piecewise affine computed pose using Lemma 4.2 :
Ψ ( t ) = Ψ ( q ( t ) , q h ( t ) ) , O ( t ) = Q ( θ h ( t ) − θ ( t ) ) , A = D Ψ , \Psi(t)=\Psi(q(t),q_h(t)),\qquad O(t)=Q(\theta_h(t)-\theta(t)),\qquad A=D\Psi, Ψ ( t ) = Ψ ( q ( t ) , q h ( t )) , O ( t ) = Q ( θ h ( t ) − θ ( t )) , A = D Ψ , u ~ = T Ψ u , ξ ~ = ( O V , ω ) , X ~ = ( u ~ , ξ ~ ) , d ( t ) = ∣ q h ( t ) − q ( t ) ∣ . (45) \widetilde u=T_\Psi u,\qquad\widetilde\xi=(OV,\omega),\qquad\widetilde X=(\widetilde u,\widetilde\xi),\qquad d(t)=\lvert q_h(t)-q(t)\rvert. \tag*{(45)} u = T Ψ u , ξ = ( O V , ω ) , X = ( u , ξ ) , d ( t ) = ∣ q h ( t ) − q ( t )∣ . (45) The norms used for poses are equivalent finite-dimensional norms. The comparison belongs to K ( q h ( t ) ) K(q_h(t)) K ( q h ( t )) q h q_h q h
Lemma 6.1 (An explicit geometric commutator). For every Y = ( v , W , η ) ∈ V Y=(v,W,\eta)\in V Y = ( v , W , η ) ∈ V
( ∂ t X ~ , Y ) M + a ( u ~ , v ) + G ( ω ; X ~ , Y ) + ∫ B ( q h ) λ ~ B ⋅ ( v − R q h ζ ) d x + ∫ Γ ( q h ) λ ~ Γ ⋅ ( γ q h v − R q h ζ ) d s = L ( t , q h ; Y ) + E ( t ; Y ) , (46) (\partial_t\widetilde X,Y)_M+a(\widetilde u,v)+\mathcal{G}(\omega;\widetilde X,Y)+\int_{B(q_h)}\widetilde\lambda_B\cdot(v-R_{q_h}\zeta)\,dx
+\int_{\Gamma(q_h)}\widetilde\lambda_\Gamma\cdot(\gamma_{q_h}v-R_{q_h}\zeta)\,ds
=\mathcal{L}(t,q_h;Y)+\mathcal{E}(t;Y), \tag*{(46)} ( ∂ t X , Y ) M + a ( u , v ) + G ( ω ; X , Y ) + ∫ B ( q h ) λ B ⋅ ( v − R q h ζ ) d x + ∫ Γ ( q h ) λ Γ ⋅ ( γ q h v − R q h ζ ) d s = L ( t , q h ; Y ) + E ( t ; Y ) , (46) where ζ = ( W , η ) \zeta=(W,\eta) ζ = ( W , η ) L 2 L^2 L 2 a ∗ ( t ) ∈ L 2 ( 0 , T ) a_*(t)\in L^2(0,T) a ∗ ( t ) ∈ L 2 ( 0 , T )
∣ E ( t ; Y ) ∣ ≤ C ( a ∗ ( t ) d ( t ) + ∣ q h ′ ( t ) − q ′ ( t ) ∣ ) ∥ Y ∥ V . (47) \lvert\mathcal{E}(t;Y)\rvert\le C\left(a_*(t)d(t)+\lvert q'_h(t)-q'(t)\rvert\right)\lVert Y\rVert_V. \tag*{(47)} ∣ E ( t ; Y )∣ ≤ C ( a ∗ ( t ) d ( t ) + ∣ q h ′ ( t ) − q ′ ( t )∣ ) ∥ Y ∥ V . (47) One may take a constant multiple of 1 + ∥ u t ( t ) ∥ 0 + ∥ f ( t ) ∥ W 1 , ∞ + ∣ F e ( t ) ∣ 1+\lVert u_t(t)\rVert_0+\lVert f(t)\rVert_{W^{1,\infty}}+\lvert F_e(t)\rvert 1 + ∥ u t ( t ) ∥ 0 + ∥ f ( t ) ∥ W 1 , ∞ + ∣ F e ( t )∣ a ∗ a_* a ∗
Proof. In (13) , use the inverse Piola fluid test ( T Ψ − 1 v ) ( x ) = A ( x ) − 1 v ( Ψ ( x ) ) (T_{\Psi}^{-1}v)(x)=A(x)^{-1}v(\Psi(x)) ( T Ψ − 1 v ) ( x ) = A ( x ) − 1 v ( Ψ ( x )) ( O T W , η ) (O^{\mathsf T}W,\eta) ( O T W , η ) A = O A=O A = O Ψ \Psi Ψ O T ( v − R q h ζ ) ∘ Ψ O^{\mathsf T}(v-R_{q_h}\zeta)\circ\Psi O T ( v − R q h ζ ) ∘ Ψ λ ~ ∘ Ψ = O λ \widetilde{\lambda}\circ\Psi=O\lambda λ ∘ Ψ = O λ
Put z = Ψ t ∘ Ψ − 1 z=\Psi_t\circ\Psi^{-1} z = Ψ t ∘ Ψ − 1
u ~ t − ( A − T u t ) ∘ Ψ − 1 = ( ( A − A − T ) u t + A t u ) ∘ Ψ − 1 − ∇ u ~ z . (48) \widetilde{u}_t-(A^{-\mathsf T}u_t)\circ\Psi^{-1}=\left((A-A^{-\mathsf T})u_t+A_tu\right)\circ\Psi^{-1}-\nabla\widetilde{u}\,z. \tag*{(48)} u t − ( A − T u t ) ∘ Ψ − 1 = ( ( A − A − T ) u t + A t u ) ∘ Ψ − 1 − ∇ u z . (48) The first term is bounded in L 2 L^2 L 2 C d ∥ u t ∥ 0 Cd\lVert u_t\rVert_0 C d ∥ u t ∥ 0 C ( d + ∣ q h ′ − q ′ ∣ ) ∥ u ∥ H 1 C(d+\lvert q_h'-q'\rvert)\lVert u\rVert_{H^1} C ( d + ∣ q h ′ − q ′ ∣) ∥ u ∥ H 1 (24) –(25) . For the body mass, ( O V ) t = O V t + O t V (OV)_t=OV_t+O_tV ( O V ) t = O V t + O t V O t = ( θ h ′ − ω ) J O O_t=(\theta_h'-\omega)JO O t = ( θ h ′ − ω ) J O m e ( O t V ) ⋅ W m_e(O_tV)\cdot W m e ( O t V ) ⋅ W O J = J O OJ=JO O J = J O
( J V ) ⋅ O T W = ( J O V ) ⋅ W . (JV)\cdot O^{\mathsf T}W=(JOV)\cdot W. ( J V ) ⋅ O T W = ( J O V ) ⋅ W . In particular, the interior acceleration is not replaced by the acceleration of a body moving with ξ ~ \widetilde{\xi} ξ
∂ t b ~ − O ∂ t b ∘ Ψ − 1 = O t V − ω J ( c h ′ − O V ) . (49) \partial_t\widetilde{b}-O\partial_tb\circ\Psi^{-1}=O_tV-\omega J(c_h'-OV). \tag*{(49)} ∂ t b − O ∂ t b ∘ Ψ − 1 = O t V − ω J ( c h ′ − O V ) . (49) This is part of (48) , and is controlled by the pose and pose-speed errors; it has not been omitted.
For the viscous form, change variables using det A = 1 \det A=1 det A = 1 A v Av A v A − 1 v ∘ Ψ A^{-1}v\circ\Psi A − 1 v ∘ Ψ A − I A-I A − I A − 1 − I A^{-1}-I A − 1 − I D A DA D A L ∞ L^\infty L ∞ O ( d ) O(d) O ( d )
∣ a ( T Ψ u , v ) − a ( u , T Ψ − 1 v ) ∣ ≤ C d ∥ u ∥ H 1 ∥ v ∥ H 1 . \left|a(T_{\Psi}u,v)-a(u,T_{\Psi}^{-1}v)\right|\le Cd\lVert u\rVert_{H^1}\lVert v\rVert_{H^1}. a ( T Ψ u , v ) − a ( u , T Ψ − 1 v ) ≤ C d ∥ u ∥ H 1 ∥ v ∥ H 1 . This also follows componentwise for the symmetric-gradient form; no assumption of a global H 2 H^2 H 2
The transformed fluid load is integrated on precisely F ( q h ) F(q_h) F ( q h ) ( A − T f ) ∘ Ψ − 1 (A^{-\mathsf T}f)\circ\Psi^{-1} ( A − T f ) ∘ Ψ − 1 f f f L 2 L^2 L 2 C d ∥ f ∥ W 1 , ∞ Cd\lVert f\rVert_{W^{1,\infty}} C d ∥ f ∥ W 1 , ∞ O F e O F_e O F e F e F_e F e C d ∣ F e ∣ Cd\lvert F_e\rvert C d ∣ F e ∣ (46) and (47) . In particular no convergence or bounded inverse estimate for a discrete multiplier was used.
The numerical speed is uniformly bounded by Theorem 5.1 . On a sufficiently close admissible prefix, the preceding alignment consequently has bounded derivatives independent of all resolutions. Formula (48) , the side regularity in (41) , and the chain rule show that u ~ \widetilde{u} u W 3 , ∞ W^{3,\infty} W 3 , ∞ W 3 , ∞ W^{3,\infty} W 3 , ∞ L t 2 L_t^2 L t 2 u u u ∇ u ~ z \nabla\widetilde{u}\,z ∇ u z ξ ~ ∈ H t 1 \widetilde{\xi}\in H_t^1 ξ ∈ H t 1 ξ ′ \xi' ξ ′ θ h ′ − ω \theta_h'-\omega θ h ′ − ω Lemma 4.3 therefore constructs
Z h ( t ) = ( z h ( t ) , ξ ~ ( t ) ) , H h ( t ) = X ~ ( t ) − Z h ( t ) = ( η h ( t ) , 0 , 0 ) , Z h ( t n ) ∈ K h n , (50) Z_h(t)=(z_h(t),\widetilde{\xi}(t)),\qquad H_h(t)=\widetilde{X}(t)-Z_h(t)=(\eta_h(t),0,0),\qquad Z_h(t_n)\in\mathbb{K}_h^n, \tag*{(50)} Z h ( t ) = ( z h ( t ) , ξ ( t )) , H h ( t ) = X ( t ) − Z h ( t ) = ( η h ( t ) , 0 , 0 ) , Z h ( t n ) ∈ K h n , (50) with the uniform estimates (27) . No numerical acceleration bound enters their constants.
Consistency on the current joint space
Let b ‾ n = τ n − 1 ∫ t n − 1 t n b ( s ) d s \overline{b}^{\,n}=\tau_n^{-1}\int_{t_{n-1}}^{t_n}b(s)\,\mathrm{d}s b n = τ n − 1 ∫ t n − 1 t n b ( s ) d s (46) with the current geometry and the lagged gyroscopic coefficient.
Lemma 6.2 (Current-geometry residual bounds). On every sufficiently close admissible prefix there exist nonnegative k n , s n k_n,s_n k n , s n
∑ n τ n k n ≤ C , ∑ n τ n s n 2 ≤ C r h 2 . (51) \sum_n \tau_n k_n \le C,\qquad\sum_n \tau_n s_n^2 \le C r_h^2. \tag*{(51)} n ∑ τ n k n ≤ C , n ∑ τ n s n 2 ≤ C r h 2 . (51) For e n = X h n − Z h ( t n ) e^n=X_h^n-Z_h(t_n) e n = X h n − Z h ( t n ) E n = ∥ e n ∥ M 2 E_n=\lVert e^n\rVert_M^2 E n = ∥ e n ∥ M 2 P n = ∣ q n − q ( t n ) ∣ 2 P_n=\lvert q_n-q(t_n)\rvert^2 P n = ∣ q n − q ( t n ) ∣ 2
( e n − P M n e n − 1 τ n , Y h ) M + a ( e n , v h ) + G ( ω h n − 1 ; e n , Y h ) = R n ( Y h ) , (52) \left(\frac{e^n-P_M^n e^{n-1}}{\tau_n},Y_h\right)_M+a(e^n,v_h)+\mathcal{G}(\omega_h^{n-1};e^n,Y_h)=\mathcal{R}_n(Y_h), \tag*{(52)} ( τ n e n − P M n e n − 1 , Y h ) M + a ( e n , v h ) + G ( ω h n − 1 ; e n , Y h ) = R n ( Y h ) , (52) where, for all Y h = ( v h , ζ ) ∈ K h n Y_h=(v_h,\zeta)\in\mathbb{K}_h^n Y h = ( v h , ζ ) ∈ K h n
∣ R n ( Y h ) ∣ ≤ C ( k n E n − 1 + P n − 1 + s n ) ∥ v h ∥ U . (53) \lvert\mathcal{R}_n(Y_h)\rvert\le C\left(\sqrt{k_n}\sqrt{E_{n-1}+P_{n-1}}+s_n\right)\lVert v_h\rVert_U. \tag*{(53)} ∣ R n ( Y h )∣ ≤ C ( k n E n − 1 + P n − 1 + s n ) ∥ v h ∥ U . (53) Proof. Orthogonality removes the current projection inside a current test, including for the previous comparison Z h ( t n − 1 ) Z_h(t_{n-1}) Z h ( t n − 1 ) Z h Z_h Z h − E ‾ n -\overline{\mathcal{E}}^{\,n} − E n ( ∂ t H h n , Y h ) M (\partial_t H_h^{n},Y_h)_M ( ∂ t H h n , Y h ) M a ( η h ( t n ) , v h ) a(\eta_h(t_n),v_h) a ( η h ( t n ) , v h ) − a ( u ~ ( t n ) − u ~ n , v h ) -a(\widetilde{u}(t_n)-\widetilde{u}^{\,n},v_h) − a ( u ( t n ) − u n , v h )
G ( ω ; X ~ , Y h ) ‾ n − G ( ω h n − 1 ; X ~ ( t n ) , Y h ) . (54) \overline{\mathcal{G}(\omega;\widetilde{X},Y_h)}^{\,n}-\mathcal{G}(\omega_h^{n-1};\widetilde{X}(t_n),Y_h). \tag*{(54)} G ( ω ; X , Y h ) n − G ( ω h n − 1 ; X ( t n ) , Y h ) . (54) Here the averaged load in the mapped equation uses q h ( s ) q_h(s) q h ( s ) q n q_n q n
We bound these terms explicitly. The configuration speed is uniformly bounded. A rigid shift over one interval has displacement O ( τ n ) O(\tau_n) O ( τ n ) L 2 L^2 L 2 H 1 H^1 H 1
∥ v h − R q h ( s ) ζ ∥ L 2 ( B ( q h ( s ) ) ) ≤ C ( ε n + κ n + τ n ) ∥ Y h ∥ V . (55) \lVert v_h-\mathcal{R}_{q_h(s)}\zeta\rVert_{L^2(B(q_h(s)))}\le C(\varepsilon_n+\kappa_n+\tau_n)\lVert Y_h\rVert_V. \tag*{(55)} ∥ v h − R q h ( s ) ζ ∥ L 2 ( B ( q h ( s ))) ≤ C ( ε n + κ n + τ n ) ∥ Y h ∥ V . (55) Indeed, pull the integral back by the rigid map from B ( q n ) B(q_n) B ( q n ) v h v_h v h C τ n ∥ v h ∥ H 1 C\tau_n\lVert v_h\rVert_{H^1} C τ n ∥ v h ∥ H 1 C τ n ∣ ζ ∣ C\tau_n\lvert\zeta\rvert C τ n ∣ ζ ∣ (17) . The analogous surface estimate is
∥ γ q h ( s ) v h − R q h ( s ) ζ ∥ L 2 ( Γ ( q h ( s ) ) ) ≤ C ( ε n + ℓ n + τ n ) ∥ Y h ∥ V . (56) \lVert\gamma_{q_h(s)}v_h-\mathcal{R}_{q_h(s)}\zeta\rVert_{L^2(\Gamma(q_h(s)))}\le C(\sqrt{\varepsilon_n}+\sqrt{\ell_n}+\sqrt{\tau_n})\lVert Y_h\rVert_V. \tag*{(56)} ∥ γ q h ( s ) v h − R q h ( s ) ζ ∥ L 2 ( Γ ( q h ( s ))) ≤ C ( ε n + ℓ n + τ n ) ∥ Y h ∥ V . (56) For the shifted trace, integration on normal fibers gives a C τ n ∥ v h ∥ H 1 C\sqrt{\tau_n}\lVert v_h\rVert_{H^1} C τ n ∥ v h ∥ H 1 H 1 / 2 H^{1/2} H 1/2 [3 ] . The normal and tangential maps have uniform Jacobians on this admissible family. Applying (18) proves (56) . Pairing with the two mapped reactions and Jensen’s inequality gives total squared residual at most
C ( ε ‾ + κ ‾ + τ max ) 2 Λ B 2 + C ( ε ‾ + ℓ ‾ + τ max ) Λ Γ 2 . C(\overline{\varepsilon}+\overline{\kappa}+\tau_{\max})^2\Lambda_B^2+C(\overline{\varepsilon}+\overline{\ell}+\tau_{\max})\Lambda_\Gamma^2. C ( ε + κ + τ m a x ) 2 Λ B 2 + C ( ε + ℓ + τ m a x ) Λ Γ 2 . Thus time averaging never replaces a reaction on one configuration by a reaction on another without a displacement estimate.
The symmetric difference of F ( q h ( s ) ) F(q_h(s)) F ( q h ( s )) F ( q n ) F(q_n) F ( q n ) O ( τ n ) O(\tau_n) O ( τ n ) C τ n ∥ f ( s ) ∥ L ∞ ∥ v h ∥ 0 C\sqrt{\tau_n}\lVert f(s)\rVert_{L^\infty}\lVert v_h\rVert_0 C τ n ∥ f ( s ) ∥ L ∞ ∥ v h ∥ 0 C τ max ∥ f ∥ L t 2 L x ∞ 2 C\tau_{\max}\lVert f\rVert_{L_t^2L_x^\infty}^2 C τ m a x ∥ f ∥ L t 2 L x ∞ 2 E L 2 E_L^2 E L 2 C δ C\delta C δ (27) . For the viscous endpoint/average term, the swept-strip proof of Part II gives
∥ u ~ ( t ) − u ~ ( s ) ∥ U ≤ C ∣ t − s ∣ + C ∫ s t ∥ u ~ r ∥ W 3 , ∞ ( B ( q h ( r ) ) ) ⊕ W 3 , ∞ ( F ( q h ( r ) ) ) d r . \|\widetilde{u}(t)-\widetilde{u}(s)\|_{U} \le C\sqrt{|t-s|}+C\int_s^t \|\widetilde{u}_r\|_{W^{3,\infty}(B(q_h(r)))\oplus W^{3,\infty}(F(q_h(r)))}\,\mathrm{d}r. ∥ u ( t ) − u ( s ) ∥ U ≤ C ∣ t − s ∣ + C ∫ s t ∥ u r ∥ W 3 , ∞ ( B ( q h ( r ))) ⊕ W 3 , ∞ ( F ( q h ( r ))) d r . The strip has area O ( ∣ t − s ∣ ) O(|t-s|) O ( ∣ t − s ∣ ) C τ max C\tau_{\max} C τ m a x
To close the geometric residual rather than assume it small, on s ∈ [ t n − 1 , t n ] s\in[t_{n-1},t_n] s ∈ [ t n − 1 , t n ]
d ( s ) ≤ P n − 1 + C τ n , d(s)\le\sqrt{P_{n-1}}+C\tau_n, d ( s ) ≤ P n − 1 + C τ n , ∣ q h ′ ( s ) − q ′ ( s ) ∣ ≤ C E n − 1 + C P n − 1 + ∣ ξ ( s ) − ξ ( t n − 1 ) ∣ . (57) |q_h'(s)-q'(s)|\le C\sqrt{E_{n-1}}+C\sqrt{P_{n-1}}+|\xi(s)-\xi(t_{n-1})|. \tag*{(57)} ∣ q h ′ ( s ) − q ′ ( s ) ∣ ≤ C E n − 1 + C P n − 1 + ∣ ξ ( s ) − ξ ( t n − 1 ) ∣. (57) The second line uses ξ ~ ( t n − 1 ) = ( O ( t n − 1 ) V ( t n − 1 ) , ω ( t n − 1 ) ) \widetilde{\xi}(t_{n-1})=(O(t_{n-1})V(t_{n-1}),\omega(t_{n-1})) ξ ( t n − 1 ) = ( O ( t n − 1 ) V ( t n − 1 ) , ω ( t n − 1 )) q h ′ = ξ h n − 1 q_h'=\xi_h^{n-1} q h ′ = ξ h n − 1 ξ h \xi_h ξ h (47) and (57) imply the past-error contribution in (53) with k n = C ( 1 + a ∗ n ) k_n=C(1+a_*^{n}) k n = C ( 1 + a ∗ n ) C τ max 2 ( ∥ a ∗ ∥ L 2 2 + ∥ ξ ′ ∥ L 2 2 ) C\tau_{\max}^2(\|a_*\|_{L^2}^2+\|\xi'\|_{L^2}^2) C τ m a x 2 ( ∥ a ∗ ∥ L 2 2 + ∥ ξ ′ ∥ L 2 2 )
Finally, split (54) at G ( ω ( t n ) ; X ~ ( t n ) , Y h ) \mathcal{G}(\omega(t_n);\widetilde{X}(t_n),Y_h) G ( ω ( t n ) ; X ( t n ) , Y h ) C τ max 2 C\tau_{\max}^2 C τ m a x 2 ω \omega ω ξ ~ \widetilde{\xi} ξ H t 1 H_t^1 H t 1
∣ ω ( t n ) − ω h n − 1 ∣ ≤ C E n − 1 + ∣ ω ( t n ) − ω ( t n − 1 ) ∣ . |\omega(t_n)-\omega_h^{n-1}|\le C\sqrt{E_{n-1}}+|\omega(t_n)-\omega(t_{n-1})|. ∣ ω ( t n ) − ω h n − 1 ∣ ≤ C E n − 1 + ∣ ω ( t n ) − ω ( t n − 1 ) ∣. The last term again contributes at most C τ max 2 ∥ ω ′ ∥ L 2 2 C\tau_{\max}^2\|\omega'\|_{L^2}^2 C τ m a x 2 ∥ ω ′ ∥ L 2 2 ∥ Y h ∥ V \|Y_h\|_V ∥ Y h ∥ V C ∥ v h ∥ U C\|v_h\|_U C ∥ v h ∥ U κ ˉ \bar{\kappa} κ ˉ (51) –(53) .
The coupled recursion and its consequence
Theorem 6.3 (Conditional convergence with an unknown trajectory). Assume (4) , the stated body and partition regularity, and (41) –(42) . Initialize with the exact pose q 0 = q ( 0 ) q_0=q(0) q 0 = q ( 0 ) X M 0 = P M 0 X ( 0 ) X_M^0=P_M^0X(0) X M 0 = P M 0 X ( 0 ) r h r_h r h T T T
max n ( ∣ q n − q ( t n ) ∣ + ∣ ξ ~ h n − ξ ( t n ) ∣ + ∥ u h n − u ( t n ) ∥ 0 ) + ( ∑ n τ n ∥ u h n − T Ψ ( t n ) u ( t n ) ∥ U 2 ) 1 / 2 ≤ C r h . (58) \max_n\left(|q_n-q(t_n)|+|\widetilde{\xi}_h^n-\xi(t_n)|+\|u_h^n-u(t_n)\|_0\right)
+\left(\sum_n\tau_n\|u_h^n-\mathcal{T}_{\Psi(t_n)}u(t_n)\|_U^2\right)^{1/2}\le Cr_h. \tag*{(58)} n max ( ∣ q n − q ( t n ) ∣ + ∣ ξ h n − ξ ( t n ) ∣ + ∥ u h n − u ( t n ) ∥ 0 ) + ( n ∑ τ n ∥ u h n − T Ψ ( t n ) u ( t n ) ∥ U 2 ) 1/2 ≤ C r h . (58) The same-point physical energy seminorm satisfies the generally weaker estimate
( ∑ n τ n ∥ u h n − u ( t n ) ∥ U 2 ) 1 / 2 ≤ C ( r h + r h ) . (59) \left(\sum_n\tau_n\|u_h^n-u(t_n)\|_U^2\right)^{1/2}\le C(r_h+\sqrt{r_h}). \tag*{(59)} ( n ∑ τ n ∥ u h n − u ( t n ) ∥ U 2 ) 1/2 ≤ C ( r h + r h ) . (59) In particular it converges strongly. No ratio between the time step, mesh size, and regularization widths is required, and there is no factor counting enrichment changes. The theorem gives no multiplier or force convergence estimate.
Proof. Test (52) by e n e^n e n
E n − E n − 1 + ∥ e n − e n − 1 ∥ M 2 + 2 μ τ n ∥ e u n ∥ U 2 = 2 τ n R n ( e n ) . E_n-E_{n-1}+\|e^n-e^{n-1}\|_M^2+2\mu\tau_n\|e_u^n\|_U^2=2\tau_n\mathcal{R}_n(e^n). E n − E n − 1 + ∥ e n − e n − 1 ∥ M 2 + 2 μ τ n ∥ e u n ∥ U 2 = 2 τ n R n ( e n ) . Use (53) and Young’s inequality to absorb one viscous term. After changing constants,
E n − E n − 1 + ∥ e n − e n − 1 ∥ M 2 + μ τ n ∥ e u n ∥ U 2 ≤ C τ n k n ( E n − 1 + P n − 1 ) + C τ n s n 2 . (60) E_n-E_{n-1}+\|e^n-e^{n-1}\|_M^2+\mu\tau_n\|e_u^n\|_U^2\le C\tau_n k_n(E_{n-1}+P_{n-1})+C\tau_n s_n^2. \tag*{(60)} E n − E n − 1 + ∥ e n − e n − 1 ∥ M 2 + μ τ n ∥ e u n ∥ U 2 ≤ C τ n k n ( E n − 1 + P n − 1 ) + C τ n s n 2 . (60) All error terms on the right are from the previous step. In particular the lagged gyroscopic coefficient leaves no current error term: G ( ω n − 1 ; e n , e n ) = 0 G(\omega^{n-1};e^n,e^n)=0 G ( ω n − 1 ; e n , e n ) = 0
The exact kinematic truncation defect is
r q n = q ( t n ) − q ( t n − 1 ) − τ n ξ ( t n − 1 ) , ∑ n ∣ r q n ∣ 2 τ n ≤ τ max 2 ∥ ξ ′ ′ ∥ L 2 ( 0 , T ) 2 . r_q^n=q(t_n)-q(t_{n-1})-\tau_n\xi(t_{n-1}),\qquad\sum_n\frac{|r_q^n|^2}{\tau_n}\leq\tau_{\max}^2\|\xi''\|_{L^2(0,T)}^2. r q n = q ( t n ) − q ( t n − 1 ) − τ n ξ ( t n − 1 ) , n ∑ τ n ∣ r q n ∣ 2 ≤ τ m a x 2 ∥ ξ ′′ ∥ L 2 ( 0 , T ) 2 . Subtracting the exact pose increment from (29) and using ∣ ξ h n − 1 − ξ ( t n − 1 ) ∣ ≤ C E n − 1 + C P n − 1 |\xi_h^{n-1}-\xi(t_{n-1})|\leq C\sqrt{E_{n-1}}+C\sqrt{P_{n-1}} ∣ ξ h n − 1 − ξ ( t n − 1 ) ∣ ≤ C E n − 1 + C P n − 1 τ n ≤ 1 \tau_n\leq1 τ n ≤ 1
P n ≤ ( 1 + C τ n ) P n − 1 + C τ n E n − 1 + C τ n − 1 ∣ r q n ∣ 2 . (61) P_n\leq(1+C\tau_n)P_{n-1}+C\tau_n E_{n-1}+C\tau_n^{-1}|r_q^n|^2. \tag*{(61)} P n ≤ ( 1 + C τ n ) P n − 1 + C τ n E n − 1 + C τ n − 1 ∣ r q n ∣ 2 . (61) This is simply Young’s inequality applied to the pose increment; it does not assume that the computed path is already convergent.
Set A n = E n + P n A_n=E_n+P_n A n = E n + P n k n k_n k n
A n + ∥ e n − e n − 1 ∥ M 2 + μ τ n ∥ e n ∥ U 2 ≤ ( 1 + C τ n k n ) A n − 1 + C τ n s n 2 + C τ n − 1 ∣ r q n ∣ 2 . (62) A_n+\|e^n-e^{n-1}\|_M^2+\mu\tau_n\|e^n\|_U^2\leq(1+C\tau_n k_n)A_{n-1}+C\tau_n s_n^2+C\tau_n^{-1}|r_q^n|^2. \tag*{(62)} A n + ∥ e n − e n − 1 ∥ M 2 + μ τ n ∥ e n ∥ U 2 ≤ ( 1 + C τ n k n ) A n − 1 + C τ n s n 2 + C τ n − 1 ∣ r q n ∣ 2 . (62) Iteration uses ∏ j ( 1 + C τ j k j ) ≤ exp ( C ∑ j τ j k j ) \prod_j(1+C\tau_jk_j)\leq\exp(C\sum_j\tau_jk_j) ∏ j ( 1 + C τ j k j ) ≤ exp ( C ∑ j τ j k j ) (51) , even though its individual coefficients need not be uniformly bounded. Applying the same argument after adding the accumulated nonnegative increment and viscous terms gives
max n A n + ∑ n ∥ e n − e n − 1 ∥ M 2 + μ ∑ n τ n ∥ e u n ∥ U 2 ≤ C ( A 0 + r h 2 ) . (63) \max_n A_n+\sum_n\|e^n-e^{n-1}\|_M^2+\mu\sum_n\tau_n\|e_u^n\|_U^2\leq C(A_0+r_h^2). \tag*{(63)} n max A n + n ∑ ∥ e n − e n − 1 ∥ M 2 + μ n ∑ τ n ∥ e u n ∥ U 2 ≤ C ( A 0 + r h 2 ) . (63) At initialization the poses agree and Ψ ( 0 ) = i d \Psi(0)=\mathrm{id} Ψ ( 0 ) = id Z h ( 0 ) ∈ K h 0 Z_h(0)\in\mathbb{K}_h^0 Z h ( 0 ) ∈ K h 0 e 0 = P M 0 ( X ( 0 ) − Z h ( 0 ) ) e^0=P_M^0(X(0)-Z_h(0)) e 0 = P M 0 ( X ( 0 ) − Z h ( 0 )) A 0 ≤ C δ 3 ≤ C r h 2 A_0\leq C\delta^3\leq Cr_h^2 A 0 ≤ C δ 3 ≤ C r h 2
The argument has so far been made on a sufficiently close admissible prefix, to justify the alignment. Choose two nested pose neighborhoods of the exact branch, with positive clearance on the larger one. The energy bound supplies a fixed bound for each numerical pose increment. For small τ max \tau_{\max} τ m a x (63) makes the pose error smaller than half the exit threshold when r h r_h r h T T T
The recovery bound now gives the aligned fluid energy estimate. The rigid error satisfies ∣ ξ h n − ξ ( t n ) ∣ ≤ C ∥ e n ∥ M + C P n |\xi_h^n-\xi(t_n)|\leq C\|e^n\|_M+C\sqrt{P_n} ∣ ξ h n − ξ ( t n ) ∣ ≤ C ∥ e n ∥ M + C P n L 2 L^2 L 2 (58) . Its U U U
( ∑ n τ n ∥ T Ψ ( t n ) u ( t n ) − u ( t n ) ∥ U 2 ) 1 / 2 ≤ C ( max n ∣ q n − q ( t n ) ∣ ) 1 / 2 . \left(\sum_n\tau_n\|\mathcal{T}\Psi(t_n)u(t_n)-u(t_n)\|_U^2\right)^{1/2}\leq C\left(\max_n|q_n-q(t_n)|\right)^{1/2}. ( n ∑ τ n ∥ T Ψ ( t n ) u ( t n ) − u ( t n ) ∥ U 2 ) 1/2 ≤ C ( n max ∣ q n − q ( t n ) ∣ ) 1/2 . which proves (59) .
Corollary 6.4 (Recovery of physical interior inertia and dissipation). Under Theorem 6.3 ,
max n ∥ u h n − R q n ξ h n ∥ L 2 ( B ( q n ) ) ≤ C r h , ∑ n τ n ∥ D ( u h n ) ∥ L 2 ( B ( q n ) ) 2 ≤ C r h 2 . \max_n\|u_h^n-R_{q_n}\xi_h^n\|_{L^2(B(q_n))}\leq Cr_h,\qquad\sum_n\tau_n\|D(u_h^n)\|_{L^2(B(q_n))}^2\leq Cr_h^2. n max ∥ u h n − R q n ξ h n ∥ L 2 ( B ( q n )) ≤ C r h , n ∑ τ n ∥ D ( u h n ) ∥ L 2 ( B ( q n )) 2 ≤ C r h 2 . The nodal surrogate-to-physical kinetic-energy difference is at most C r h Cr_h C r h
Proof. The aligned exact velocity equals R q n ξ ~ ( t n ) \mathcal{R}_{q_n}\widetilde{\xi}(t_n) R q n ξ ( t n ) B ( q n ) B(q_n) B ( q n ) ∣ ξ h n − ξ ~ ( t n ) ∣ |\xi_h^n-\widetilde{\xi}(t_n)| ∣ ξ h n − ξ ( t n ) ∣ (19) and the uniform discrete mass bound. Unlike the time-integrated statement obtained from constraints alone, this nodal assertion uses the convergence theorem.
Corollary 6.5 (Vanishing joint transfer losses). Under Theorem 6.3,
∑ n ( ∥ X h n − P M n X h n − 1 ∥ M 2 + ∥ ( I − P M n ) X h n − 1 ∥ M 2 ) = ∑ n ∥ X h n − X h n − 1 ∥ M 2 ≤ C r h 2 . (64) \sum_n\left(\lVert X_h^n-P_M^nX_h^{n-1}\rVert_M^2+\lVert(I-P_M^n)X_h^{n-1}\rVert_M^2\right)
=\sum_n\lVert X_h^n-X_h^{n-1}\rVert_M^2\leq Cr_h^2. \tag*{(64)} n ∑ ( ∥ X h n − P M n X h n − 1 ∥ M 2 + ∥( I − P M n ) X h n − 1 ∥ M 2 ) = n ∑ ∥ X h n − X h n − 1 ∥ M 2 ≤ C r h 2 . (64) Thus neither the accumulated transfer loss nor the accumulated implicit-step increment loss survives the convergence limit.
Proof. The increment term retained in (63) controls ∑ n ∥ e n − e n − 1 ∥ M 2 \sum_n\lVert e^n-e^{n-1}\rVert_M^2 ∑ n ∥ e n − e n − 1 ∥ M 2 C r h 2 Cr_h^2 C r h 2 H t 1 H H_t^1\mathbb{H} H t 1 H (27) gives the same bound for Z h Z_h Z h
∑ n ∥ Z h ( t n ) − Z h ( t n − 1 ) ∥ M 2 ≤ τ max ∥ ∂ t Z h ∥ L t 2 M 2 ≤ C τ max . \sum_n\lVert Z_h(t_n)-Z_h(t_{n-1})\rVert_M^2\leq\tau_{\max}\lVert\partial_tZ_h\rVert_{L_t^2M}^2\leq C\tau_{\max}. n ∑ ∥ Z h ( t n ) − Z h ( t n − 1 ) ∥ M 2 ≤ τ m a x ∥ ∂ t Z h ∥ L t 2 M 2 ≤ C τ m a x . Use X h n = e n + Z h ( t n ) X_h^n=e^n+Z_h(t_n) X h n = e n + Z h ( t n ) τ max ≤ r h 2 \tau_{\max}\leq r_h^2 τ m a x ≤ r h 2 (34) . No bound on the number of changes of the joint constraint space is used.
For the piecewise constant velocity reconstruction and the piecewise affine pose, the same convergence conclusions hold on [ 0 , T ] [0,T] [ 0 , T ] H t 1 L x 2 H_t^1L_x^2 H t 1 L x 2 U U U ∣ t − s ∣ \sqrt{|t-s|} ∣ t − s ∣ C τ max C\sqrt{\tau_{\max}} C τ m a x L 2 ( 0 , T ; H 1 ( Ω ) 2 ) L^2(0,T;H^1(\Omega)^2) L 2 ( 0 , T ; H 1 ( Ω ) 2 ) L t ∞ H x 2 L_t^\infty H_x^2 L t ∞ H x 2 C r h Cr_h C r h H 1 H^1 H 1
Minimal verification with an unknown body trajectory
The computations test the coupled implementation, the energy identity, and the response of a freely computed rigid body. They are deliberately small manufactured-solution tests in the sense of Salari and Knupp [39 ] ; they do not constitute an asymptotic-rate study or a validation against experimental fluid–structure data. In particular, the reference trajectory below is never assigned during time stepping. All poses after initialization are obtained from the preceding computed body velocity through (29) .
A continuous Stokes–Newton manufactured solution
The reference body is an ellipse with semiaxes a = 0.19 a=0.19 a = 0.19 b = 0.115 b=0.115 b = 0.115 Ω = ( 0 , 1 ) 2 \Omega=(0,1)^2 Ω = ( 0 , 1 ) 2 T = 0.5 T=0.5 T = 0.5 ρ f = 1 \rho_f=1 ρ f = 1 ρ s = 4 \rho_s=4 ρ s = 4 μ = 0.1 \mu=0.1 μ = 0.1
m s = 4 π a b , I s = π a b ( a 2 + b 2 ) , m e = 3 π a b , I e = 3 π a b ( a 2 + b 2 ) 4 . m_s=4\pi ab,\qquad I_s=\pi ab(a^2+b^2),\qquad m_e=3\pi ab,\qquad I_e=\frac{3\pi ab(a^2+b^2)}{4}. m s = 4 π ab , I s = π ab ( a 2 + b 2 ) , m e = 3 π ab , I e = 4 3 π ab ( a 2 + b 2 ) . Thus m e ≃ 0.2059314 m_e \simeq0.2059314 m e ≃ 0.2059314 I e ≃ 0.002539392 I_e \simeq0.002539392 I e ≃ 0.002539392
c ∗ ( t ) = ( 0.5 + 0.04 sin ( 2 t ) , 0.5 + 0.03 sin ( 3 t ) ) , V ∗ ( t ) = ( 0.08 cos ( 2 t ) , 0.09 cos ( 3 t ) ) , θ ∗ ( t ) = 0.6 sin ( 2 t ) , ω ∗ ( t ) = 1.2 cos ( 2 t ) . (65) \begin{aligned}
c_*(t) &= (0.5 + 0.04\sin(2t),\, 0.5 + 0.03\sin(3t)),\\
V_*(t) &= (0.08\cos(2t),\, 0.09\cos(3t)),\\
\theta_*(t) &= 0.6\sin(2t),\qquad\omega_*(t) = 1.2\cos(2t). \tag*{(65)}
\end{aligned} c ∗ ( t ) V ∗ ( t ) θ ∗ ( t ) = ( 0.5 + 0.04 sin ( 2 t ) , 0.5 + 0.03 sin ( 3 t )) , = ( 0.08 cos ( 2 t ) , 0.09 cos ( 3 t )) , = 0.6 sin ( 2 t ) , ω ∗ ( t ) = 1.2 cos ( 2 t ) . (65) A star denotes the reference solution, not an input to the numerical pose update. With z = Q ( θ ∗ ) T ( x − c ∗ ) z = Q(\theta_*)^{\mathsf{T}}(x-c_*) z = Q ( θ ∗ ) T ( x − c ∗ )
ϕ ( t , x ) = z 1 2 a 2 + z 2 2 b 2 − 1 , W ( x ) = x 1 2 ( 1 − x 1 ) 2 x 2 2 ( 1 − x 2 ) 2 , α ( t ) = 0.3 + 0.2 sin ( 2 t ) . \phi(t,x) = \frac{z_1^2}{a^2} + \frac{z_2^2}{b^2} - 1,\qquad W(x) = x_1^2(1-x_1)^2x_2^2(1-x_2)^2,\qquad\alpha(t) = 0.3 + 0.2\sin(2t). ϕ ( t , x ) = a 2 z 1 2 + b 2 z 2 2 − 1 , W ( x ) = x 1 2 ( 1 − x 1 ) 2 x 2 2 ( 1 − x 2 ) 2 , α ( t ) = 0.3 + 0.2 sin ( 2 t ) . We use curl ψ = ( ∂ 2 ψ , − ∂ 1 ψ ) \operatorname{curl}\psi= (\partial_2\psi,-\partial_1\psi) curl ψ = ( ∂ 2 ψ , − ∂ 1 ψ )
u ∗ ( t , x ) = g ∗ ( t , x ) + curl { α ( t ) W ( x ) [ ϕ ( t , x ) ] + 2 } , p ∗ = 0. (66) u_*(t,x) = g_*(t,x) + \operatorname{curl}\{\alpha(t)W(x)[\phi(t,x)]_+^2\},\qquad p_* = 0. \tag*{(66)} u ∗ ( t , x ) = g ∗ ( t , x ) + curl { α ( t ) W ( x ) [ ϕ ( t , x ) ] + 2 } , p ∗ = 0. (66) The fixed-cutoff rigid lift is
g ∗ = curl [ χ ( x ) { β ∗ , 1 X 2 − β ∗ , 2 X 1 − 1 2 ω ∗ ∣ X ∣ 2 } ] , X = x − ( 0.5 , 0.5 ) , β ∗ = V ∗ − ω ∗ J ( c ∗ − ( 0.5 , 0.5 ) ) . g_* = \operatorname{curl}\left[\chi(x)\left\{\beta_{*,1}X_2-\beta_{*,2}X_1-\frac{1}{2}\omega_*|X|^2\right\}\right],
\qquad
X = x-(0.5,0.5),\qquad
\beta_* = V_*-\omega_*J(c_*-(0.5,0.5)). g ∗ = curl [ χ ( x ) { β ∗ , 1 X 2 − β ∗ , 2 X 1 − 2 1 ω ∗ ∣ X ∣ 2 } ] , X = x − ( 0.5 , 0.5 ) , β ∗ = V ∗ − ω ∗ J ( c ∗ − ( 0.5 , 0.5 )) . The radial cutoff equals one for ∣ X ∣ ≤ 0.40 |X|\le0.40 ∣ X ∣ ≤ 0.40 ∣ X ∣ ≥ 0.49 |X|\ge0.49 ∣ X ∣ ≥ 0.49
χ = 1 − 2772 ∫ 0 s r 5 ( 1 − r ) 5 d r , s = ∣ X ∣ 2 − 0.40 2 0.49 2 − 0.40 2 . \chi= 1 - 2772\int_0^s r^5(1-r)^5\,\mathrm{d}r,\qquad
s = \frac{|X|^2-0.40^2}{0.49^2-0.40^2}. χ = 1 − 2772 ∫ 0 s r 5 ( 1 − r ) 5 d r , s = 0.4 9 2 − 0.4 0 2 ∣ X ∣ 2 − 0.4 0 2 . This cutoff is C 5 C^5 C 5 (66) retains nonzero exterior shear at the interface. In particular, this is not a zero-traction manufactured test.
The exterior force is constructed from the smooth ambient field
u ~ ∗ ( t , x ) = g ∗ ( t , x ) + curl { α ( t ) W ( x ) ϕ ( t , x ) 2 } , f = ρ f ∂ t u ~ ∗ − μ Δ u ~ ∗ . (67) \widetilde{u}_*(t,x) = g_*(t,x) + \operatorname{curl}\{\alpha(t)W(x)\phi(t,x)^2\},\qquad
f = \rho_f\partial_t\widetilde{u}_*-\mu\Delta\widetilde{u}_*. \tag*{(67)} u ∗ ( t , x ) = g ∗ ( t , x ) + curl { α ( t ) W ( x ) ϕ ( t , x ) 2 } , f = ρ f ∂ t u ∗ − μ Δ u ∗ . (67) The positive-part operation is absent from this ambient extension. In every discrete load integral, f ( t n , x ) f(t_n,x) f ( t n , x ) computed exterior F ( q h n ) F(q_h^n) F ( q h n )
On the exact interface, n ∗ = ∇ ϕ / ∣ ∇ ϕ ∣ n_*=\nabla\phi/|\nabla\phi| n ∗ = ∇ ϕ /∣∇ ϕ ∣
σ ∗ n ∗ = − 2 μ α W ∣ ∇ ϕ ∣ 2 J n ∗ . (68) \sigma_*n_*=-2\mu\alpha W|\nabla\phi|^2Jn_*. \tag*{(68)} σ ∗ n ∗ = − 2 μα W ∣∇ ϕ ∣ 2 J n ∗ . (68) The external body data are the continuous physical Newton loads
F e = m s V ˙ ∗ − ∫ Γ ( q ∗ ) σ ∗ n ∗ d s , T e = I s ω ˙ ∗ − ∫ Γ ( q ∗ ) J ( x − c ∗ ) ⋅ σ ∗ n ∗ d s . F_e=m_s\dot{V}_*-\int_{\Gamma(q_*)}\sigma_*n_*\,\mathrm{d}s,\qquad
T_e=I_s\dot{\omega}_*-\int_{\Gamma(q_*)}J(x-c_*)\cdot\sigma_*n_*\,\mathrm{d}s. F e = m s V ˙ ∗ − ∫ Γ ( q ∗ ) σ ∗ n ∗ d s , T e = I s ω ˙ ∗ − ∫ Γ ( q ∗ ) J ( x − c ∗ ) ⋅ σ ∗ n ∗ d s . They depend on time alone and are evaluated without numerical fluid coefficients. The stress integrals use 128 equally spaced angular points; the parametrized integrands reduce to smooth trigonometric polynomials and the reported independent stress checks agree to roundoff. The fluid and body forcing therefore come from a continuous PDE–ODE solution, rather than from a discrete matrix residual. Endpoint evaluation of these data is an approximation of the temporal load in (43) ; it contributes a load consistency error and is not identified with exact time averaging.
Discrete constraints, transfer and diagnostics
The coarse Cartesian spacing is h = 1 / M h = 1/M h = 1/ M ε = h / 2 \varepsilon= h/2 ε = h /2 h h h h h h 2 h \sqrt{2}h 2 h
Both constraint families are assembled as C ( q h n ) u h n − R ( q h n ) ξ h n = 0 C(q_h^n)u_h^n - R(q_h^n)\xi_h^n = 0 C ( q h n ) u h n − R ( q h n ) ξ h n = 0 m f ω h n − 1 J m_f\omega_h^{n-1}J m f ω h n − 1 J 10 − 12 10^{-12} 1 0 − 12 M M M ( u ∗ ( 0 ) , ξ ∗ ( 0 ) ) (u_*(0), \xi_*(0)) ( u ∗ ( 0 ) , ξ ∗ ( 0 )) M = 24 M = 24 M = 24 1.10 × 10 − 3 1.10 \times10^{-3} 1.10 × 1 0 − 3
Mass and stiffness integrals are polynomial-exact with four-point Gauss rules on Cartesian half-cells. The default kernel, surface, volume, regular-load, curved-cut-load and sharp-trace rules have orders 6, 4, 4, 6, 10, 10, respectively. The stronger-quadrature control raises these to 10, 8, 6, 10, 16, 16. Curved load integration is split at exact conic intersections, including the moving computed ellipse. The interior-defect diagnostic uses a separate order-eight rule (order twelve in the stronger control). It is not used to impose the constraints or solve the equations.
The reported final fluid errors compare u h ( T ) u_h(T) u h ( T ) u ∗ ( T ) u_*(T) u ∗ ( T ) B ( q ∗ ( T ) ) B(q_*(T)) B ( q ∗ ( T )) unaligned diagnostics , not the geometry-aligned error in (58). Body position, angle, translation-velocity and angular-velocity errors are absolute. We also record the physical defects
S 2 = ∑ n τ ∥ u h n − R q h n ξ h n ∥ L 2 ( Γ ( q h n ) ) 2 , D 2 = ∑ n τ ∥ u h n − R q h n ξ h n ∥ L 2 ( B ( q h n ) ) 2 . S^2 = \sum_n \tau\left\|u_h^n - R_{q_h^n}\xi_h^n\right\|_{L^2(\Gamma(q_h^n))}^2,\qquad D^2 = \sum_n \tau\left\|u_h^n - R_{q_h^n}\xi_h^n\right\|_{L^2(B(q_h^n))}^2. S 2 = n ∑ τ u h n − R q h n ξ h n L 2 ( Γ ( q h n )) 2 , D 2 = n ∑ τ u h n − R q h n ξ h n L 2 ( B ( q h n )) 2 . These are nodal sums, not exact continuous-time integrals. Sharp surface traces are integrated after splitting the ellipse at all Cartesian polynomial crossings. A separate order-eight versus order-sixteen interior integration at the same saved M = 24 M = 24 M = 24 3.02 × 10 − 4 3.02 \times10^{-4} 3.02 × 1 0 − 4
Refinement and controlled changes
The archive contains eight runs. The three spatial cases use M = 12 , 24 , 48 M = 12, 24, 48 M = 12 , 24 , 48 N = 80 N = 80 N = 80 τ = T / N = 0.00625 \tau= T/N = 0.00625 τ = T / N = 0.00625 M = 24 M = 24 M = 24 N = 160 N = 160 N = 160
Figure 1. A computed translating and rotating body. Left: the final M = 24 M = 24 M = 24
Table 1 and Figure 2 show decreasing fluid errors and interior/surface defects. The coarsest fluid error is substantial; the narrow fixed lift transition is only poorly resolved at M = 12 M = 12 M = 12 M = 24 M = 24 M = 24 M = 24 M = 24 M = 24
Table 1. Errors at T = 0.5 T = 0.5 T = 0.5 N = 80 N = 80 N = 80
Figure 2. Finite-resolution observations at fixed N = 80 N = 80 N = 80
Table 2. Changes from the dynamic M = 24 M = 24 M = 24 N = 80 N = 80 N = 80 T T T L 2 L^2 L 2
The fixed-active-basis run has exactly zero mesh changes; it still uses the computed moving volume and surface moments. The stronger-quadrature control changes the final fluid velocity by relative 5.90 × 10 − 5 5.90 \times10^{-5} 5.90 × 1 0 − 5
Δ f ( t , x ) = 30 sin ( π t / T ) curl W ( x ) \Delta f(t,x) = 30\sin(\pi t/T)\operatorname{curl} W(x) Δ f ( t , x ) = 30 sin ( π t / T ) curl W ( x ) to the ambient fluid force, again integrated over the computed exterior, and leave F e , T e F_e,T_e F e , T e 1.37 × 10 − 3 1.37 \times10^{-3} 1.37 × 1 0 − 3 1.58 × 10 − 1 1.58 \times10^{-1} 1.58 × 1 0 − 1 Figure 3 displays this response. It provides a direct check that the trajectory is a coupled unknown: changing only fluid forcing changes body motion. No exact-solution errors are reported for this perturbed problem or for the unforced problem.
Figure 3. Computed body velocities with unchanged external body force and torque but a changed fluid load. The dashed curve is the manufactured reference for the baseline problem only; it is not an exact solution for the perturbed problem.
Energy and independent implementation audits
For the unforced run, the discrete joint energy decreases from 4.23 × 10 − 1 4.23 \times10^{-1} 4.23 × 1 0 − 1 3.80 × 10 − 5 3.80 \times10^{-5} 3.80 × 1 0 − 5 Figure 4 plots this energy, the separately estimated physical energy, and the transfer loss. The exact algebraic identity concerns 1 2 ∥ X h n ∥ M 2 \frac{1}{2}\lVert X_h^n\rVert_M^2 2 1 ∥ X h n ∥ M 2 1.59 × 10 − 6 1.59 \times10^{-6} 1.59 × 1 0 − 6
Figure 4. Unforced coupled dynamics under repeated space changes. Joint energy decreases; the physical-energy estimate is visually close but has a different finite-resolution definition. Orthogonal projection removes energy, and the cumulative balance closes to floating-point accuracy.
The maximum absolute per-step energy residual is 1.15 × 10 − 14 1.15 \times10^{-14} 1.15 × 1 0 − 14 2.13 × 10 − 13 2.13 \times10^{-13} 2.13 × 1 0 − 13
E h n = E h 0 + ∑ j = 1 n τ a ( u h j , u h j ) + 1 2 ∑ j = 1 n ∥ X h j − X h j − 1 ∥ M 2 − ∑ j = 1 n τ L h j ( X h j ) = 0. E_h^n = E_h^0 + \sum_{j=1}^{n} \tau a(u_h^j,u_h^j) + \frac{1}{2}\sum_{j=1}^{n}\lVert X_h^j - X_h^{j-1}\rVert_M^2 - \sum_{j=1}^{n}\tau\mathcal{L}_h^j(X_h^j) = 0. E h n = E h 0 + j = 1 ∑ n τ a ( u h j , u h j ) + 2 1 j = 1 ∑ n ∥ X h j − X h j − 1 ∥ M 2 − j = 1 ∑ n τ L h j ( X h j ) = 0. The maximum gyro power is 1.05 × 10 − 19 1.05 \times10^{-19} 1.05 × 1 0 − 19 4.39 × 10 − 15 4.39 \times10^{-15} 4.39 × 1 0 − 15 5.94 × 10 − 14 5.94 \times10^{-14} 5.94 × 1 0 − 14 2.78 × 10 − 17 2.78 \times10^{-17} 2.78 × 1 0 − 17 5.55 × 10 − 17 5.55 \times10^{-17} 5.55 × 1 0 − 17
Three standalone audit programs supplement the saved-state audit. First, exterior stresses reconstructed directly from analytic velocity gradients recover (68) ; independent interior integration recovers the Eulerian fictitious-inertia correction, physical Newton balance, and zero rigid Reynolds energy flux. Second, a small joint step agrees with a direct dense mixed-system solve to relative 1.35 × 10 − 14 1.35 \times10^{-14} 1.35 × 1 0 − 14
Relation to the prescribed-motion framework and limitations
The present construction shares the background Stokes operator, retained Cartesian coarse space, regularized moment enforcement, orthogonal transfer, and moving-collar recovery with Parts I–II. Its continuous model reduces to prescribed motion when the rigid trajectory is supplied externally. The discrete construction has an additional ingredient: volume moments are needed to recover the physical rigid interior when its acceleration is unknown. Freezing the body velocities therefore gives an interior-completed prescribed-motion method, not literally the surface-only scheme of Part II. This distinction prevents an incorrect assertion that surface no-slip alone legitimizes subtracting displaced-fluid mass. The positive excess-mass restrictions make the ambient kinetic form an inner product on independent fluid and rigid variables. They permit a current-space orthogonal projector with constants independent of regularization and mesh history. The physical energy of an exact rigid interior can remain positive outside this parameter regime; the present restriction is a property of the chosen proof and discrete ambient formulation. Our constants may depend on inverse excess mass and inertia. They provide no uniform statement as either quantity approaches zero.
The state-space energy is an exact discrete identity. At finite resolution it includes interior fluid motion that is only approximately rigid. Proposition 3.3 quantifies its difference from the physical exterior-fluid and rigid-body kinetic energy. Conversely, the gyroscopic correction is invisible to that energy identity: it performs zero work. Energy tests alone cannot detect its omission. The momentum equations and the independent manufactured-force audit are consequently necessary parts of verification.
The numerical geometry is genuinely computed. The proof does not assume that it equals the exact trajectory, nor does it assume convergence of a surface multiplier to infer the body acceleration. Instead, interior and surface reactions enter through consistency defects of coupled test functions, and an alignment map compares the two configurations. The mapping is used only in the analysis; the Eulerian mesh is not transported. The discrete geometry update and lagged angular coefficient are specified parts of the algorithm, not an unresolved nonlinear monolithic solve.
A smooth exact solution with a positive clearance is assumed on the interval of interest. This paper does not prove existence of that solution for arbitrary weak data or establish a collision law. The convergence argument controls the computed clearance by closeness to that exact configuration. It also requires the stated spatial regularity of an ambient extension of the fluid load. An arbitrary load discontinuity across a separately prescribed moving surface would require additional geometric consistency analysis.
Geometry-aligned energy convergence and same-point Eulerian energy convergence answer different questions. A small displacement of an interface across which the velocity gradient jumps produces a strip of gradient mismatch. Its L 2 L^2 L 2
All theoretical constraints, kernels, and integrals are exact. The computations use quadrature and finite-precision linear algebra; they verify selected behavior and report those additional errors. There is no pressure estimate, full multiplier convergence theorem, adaptive marking result, or adaptive optimality result. The refined patch may change arbitrarily, but refinement of the retained coarse grid is part of the limiting sequence. Multiple bodies, general Navier–Stokes convection, and lighter-body ambient formulations require further analysis.
Machine-checked algebra and scope of verification
The accompanying Lean project contains 48 kernel-checked theorems: 35 unchanged declarations inherited from Part II and 13 additions for the coupled formulation. The complete project was compiled with Lean 4.19.0 and a pinned Mathlib revision. The six proof modules were then checked separately and their axiom dependencies printed. Every declaration passed; the reported foundations are only propext, Classical.choice, and Quot.sound. There are no proof admissions or custom axioms. The compiler output, dependency lockfile, source hashes, and exact commands are included in the repository. Lean and Mathlib provide the verification framework [40 , 41 ] .
The nine additions in Coupled.lean construct a genuine Hilbert product using square-root mass coordinates, identify its inner product and squared norm, and prove positive definiteness under positive excess mass and inertia. The fluid Hilbert norm may already contain ρ f \rho_f ρ f ( ρ f u , m e V , I e ω ) (\sqrt{\rho_f}\,u,\sqrt{m_e}\,V,\sqrt{I_e}\,\omega) ( ρ f u , m e V , I e ω )
The four additions in Recurrence.lean combine energy and pose-error inequalities and establish a discrete Gronwall estimate with cumulative dissipation and the factor exp ( ∑ n γ n ) \exp(\sum_n \gamma_n) exp ( ∑ n γ n ) 0 < τ n ≤ 1 0 < \tau_n \le1 0 < τ n ≤ 1
These checks establish the indicated algebra and finite-sum arguments, not a formal proof of the full PDE convergence theorem. The physical Eulerian inertia derivation, regularized volume and surface estimates, pose alignment, moving-collar recovery, regularity assumptions, and clearance bootstrap remain ordinary mathematical proofs. In particular, zero gyroscopic work does not prove that the gyroscopic coefficient has been derived correctly from the physical equations. The numerical checks supply a further, finite-resolution assessment of the implemented scheme; neither compiler acceptance nor those computations replace the analytical hypotheses of the convergence theorem.
AI use, human understanding, and verification limits
The research direction, scope, and revision requests were provided by Sida He. OpenAI ChatGPT/Codex agents generated most of the mathematical derivations, manuscript text, numerical implementation, and Lean formalization, and performed automated literature searches and internal checks. This work is therefore disclosed as primarily AI-generated text, rather than merely language editing.
At the preparation date of this version, the author reports that he does not understand the mathematical proofs and has not independently verified them. The intended Hexagon disclosure is Human understanding: no parts , with respect to the proofs. No independent human expert verification or external peer review of the proofs is claimed. Internal reviews by additional AI agents do not constitute independent human mathematical review.
The Lean checks described above cover selected algebraic identities and finite-step arguments only. They do not formalize the full PDE convergence theorems or establish their analytic hypotheses. Numerical experiments are implementation checks and illustrations, not proofs. The original proof source, dependency pins, numerical results, and recorded build logs are retained in the companion reproducibility material. This preparation revision does not claim a new Lean compilation or a new numerical experiment. The manuscript is offered for critical scrutiny; its correctness and novelty are not certified by these disclosures or by repository acceptance.
Relationship to companion manuscripts. The other parts of this series are Part I [2 ] , and Part II [3 ] . They are cited as unpublished companion manuscripts; no public identifier or acceptance is claimed. Each part has its own hypotheses and scope. The series is not asserted to be a chain of special-case theorems.
Availability of reproducibility material. A separate versioned package has been prepared with the Lean source, dependency lockfiles, numerical code, and archived results. No public repository URL or archive identifier has been assigned to that package at the preparation date of this manuscript. The mathematical source submitted to Hexagon does not itself contain the Lean project.
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