{ "scope": "Bounded supplement; not all scientific formulas", "equations": [ { "id": "M01", "domain": "Mathematics", "title": "Scale coordinate", "equation": "L = 10^k m; k = log10(L / 1 m)", "assumptions_units": "L is a positive physical length in metres; k is dimensionless. A view coordinate does not prescribe mesh spacing or instrument accuracy.", "verification": "Check units and positivity; choose governing model independently.", "reference": "BIPM" }, { "id": "M02", "domain": "Mathematics", "title": "Nondimensionalization", "equation": "x* = x/L0; t* = t/t0; u* = u/U0", "assumptions_units": "L0, t0 and U0 are positive reference scales with the units of their variables.", "verification": "Report reference values with every dimensionless tolerance.", "reference": "BIPM" }, { "id": "M03", "domain": "Mathematics", "title": "Linear system and forward error", "equation": "A x = b; r = b - A xh; ||x-xh|| <= ||A^(-1)|| ||r||", "assumptions_units": "A is nonsingular; compatible induced norm. xh is an approximation. Residual alone does not bound error without stability information.", "verification": "Estimate conditioning and verify an independent residual.", "reference": "NUM" }, { "id": "M04", "domain": "Mathematics", "title": "Newton iteration", "equation": "J(xk) delta = -F(xk); x(k+1) = xk + alpha delta", "assumptions_units": "F differentiable; local nonsingular Jacobian J; alpha in (0,1]. Global convergence is not automatic.", "verification": "Check scaled residual, step, line search and domain constraints.", "reference": "NUM" }, { "id": "M05", "domain": "Mathematics", "title": "Weak Poisson problem", "equation": "Integral_Omega k grad(u).grad(v) dV = Integral_Omega f v dV + Integral_GammaN g v dS", "assumptions_units": "Solve -div(k grad u)=f; u prescribed on GammaD; k grad(u).n=g on GammaN; k bounded and uniformly positive.", "verification": "Coercivity with appropriate Dirichlet boundary; pure Neumann needs compatibility and a fixed mean.", "reference": "FEM" }, { "id": "M06", "domain": "Mathematics", "title": "Galerkin quasi-optimality", "equation": "a(u-uh,vh)=0; ||u-uh||V <= (M/alpha) inf_vh ||u-vh||V", "assumptions_units": "Continuous bilinear form a with continuity M and coercivity alpha>0; conforming Vh subset V.", "verification": "Applies to this coercive setting, not every nonlinear or saddle-point model.", "reference": "FEM" }, { "id": "M07", "domain": "Mathematics", "title": "Residual estimator", "equation": "eta^2 = sum_K hK^2 ||f+div(k grad uh)||K^2 + sum_e he ||jump(k grad uh.n)||e^2", "assumptions_units": "Illustrative diffusion estimator with matching boundary residuals; coefficient jumps require weighting. Norms are L2.", "verification": "Reliability constants, data oscillation and regularity must be documented.", "reference": "FEM" }, { "id": "M08", "domain": "Mathematics", "title": "Richardson order estimate", "equation": "p = log(|(q3-q2)/(q2-q1)|)/log(r)", "assumptions_units": "h3=r h2=r^2 h1, r>1; asymptotic leading error q(h)=q_exact+C h^p and nonzero differences.", "verification": "Use at least three refinements; do not use near cancellation or outside asymptotic range.", "reference": "NUM" }, { "id": "M09", "domain": "Mathematics", "title": "Mixed-method stability", "equation": "inf_qh sup_vh b(vh,qh)/(||vh||V ||qh||Q) >= beta0 > 0", "assumptions_units": "Incompressible mixed FEM needs an inf-sup stable pair on the actual mesh family, or consistent stabilization.", "verification": "Taylor-Hood is a method choice, not a universal proof for all geometries.", "reference": "FEM" }, { "id": "M10", "domain": "Mathematics", "title": "Uncertainty propagation", "equation": "uc^2(y) = sum_i sum_j (df/dxi)(df/dxj) cov(xi,xj)", "assumptions_units": "First-order propagation for y=f(x); include covariance and measurement model inputs.", "verification": "For strong nonlinearity propagate distributions and record sampling settings.", "reference": "NIST" }, { "id": "M11", "domain": "Mathematics", "title": "Zero-failure reliability bound", "equation": "p_lower = alpha^(1/n); n >= ceil(log(alpha)/log(p_target))", "assumptions_units": "n independent representative Bernoulli trials, all successful; one-sided confidence 1-alpha. For p_target=0.99 and alpha=0.05, n=299.", "verification": "A lower bound on one defined pass probability is not total engineering feasibility.", "reference": "DERIVED" }, { "id": "M12", "domain": "Mathematics", "title": "Bounded integer search", "equation": "D(x1,...,xn)=0; xi in {-M,...,M}; cases=(2M+1)^n", "assumptions_units": "Finite M and integer polynomial D; exhaustive finite search is decidable.", "verification": "Timeout means unknown; finite absence does not prove unbounded absence.", "reference": "NUM" }, { "id": "M13", "domain": "Mathematics", "title": "Constrained optimization", "equation": "min_x f(x), subject to g(x)<=0 and h(x)=0", "assumptions_units": "Geometry, manufacturing and reliability constraints must be explicit.", "verification": "Local stationary points are not global certificates for nonconvex problems.", "reference": "NUM" }, { "id": "M14", "domain": "Mathematics", "title": "Linear reduced model", "equation": "u approximately V a; (V^T A V)a=V^T b", "assumptions_units": "Columns of V form a selected reduced basis; A and b refer to a discretized problem.", "verification": "Validate outside training samples and retain a residual-based estimator.", "reference": "FEM" }, { "id": "P01", "domain": "Physics", "title": "Mass conservation", "equation": "partial_t rho + div(rho v) = s_m", "assumptions_units": "rho kg/m^3; velocity v m/s; source s_m kg/(m^3 s).", "verification": "Integrate flux and storage independently at all scale interfaces.", "reference": "CONT" }, { "id": "P02", "domain": "Physics", "title": "Incompressible momentum", "equation": "rho(partial_t v + (v.grad)v) = -grad p + mu laplacian v + f; div v=0", "assumptions_units": "Newtonian constant-density fluid; mu Pa s; f force per volume N/m^3. Specify initial and boundary data.", "verification": "Monitor momentum residual and divergence; validate mesh/time refinement.", "reference": "CONT" }, { "id": "P03", "domain": "Physics", "title": "Heat transport", "equation": "rho cp(partial_t T+v.grad T)=div(k grad T)+Q", "assumptions_units": "T K; cp J/(kg K); k W/(m K); Q W/m^3. Constant-property sensible heat approximation.", "verification": "Include latent heat, radiation and thermo-mechanical work when material conditions require.", "reference": "CONT" }, { "id": "P04", "domain": "Physics", "title": "Closed-system first law", "equation": "Delta E = Q_in - W_out", "assumptions_units": "E J includes relevant internal, kinetic and potential energy; heat Q and work W signed as stated.", "verification": "Energy conservation checks the balance residual, not whether Delta E is near zero.", "reference": "THERMO" }, { "id": "P05", "domain": "Physics", "title": "Entropy balance", "equation": "Delta S = Integral delta Q/T_boundary + S_gen; S_gen >= 0", "assumptions_units": "Closed system may exchange heat; isolated system has zero heat and mass exchange.", "verification": "A closed system can lose entropy; test production including transfers.", "reference": "THERMO" }, { "id": "P06", "domain": "Physics", "title": "Scaled balance residual", "equation": "rE = |Delta E-Q_in+W_out| / max(E_ref,|Delta E|,|Q_in|,|W_out|)", "assumptions_units": "E_ref>0 J chosen and reported; rE dimensionless.", "verification": "Tolerance derived from discretization and metrology; 1e-12 J is not universal.", "reference": "DERIVED" }, { "id": "P07", "domain": "Physics", "title": "Small-strain elasticity", "equation": "eps = (grad u + grad u^T)/2; sigma = lambda tr(eps) I + 2 mu eps", "assumptions_units": "u m; stress Pa; isotropic linear elastic solid; infinitesimal strain.", "verification": "For stable ordinary compressible material E>0 and -1=3 impossibility threshold; finite-basis diagonalization is possible but can scale prohibitively.", "reference": "QM" }, { "id": "P11", "domain": "Physics", "title": "Kohn-Sham equations", "equation": "[-hbar^2 laplacian/(2 me)+v_ext+v_H+v_xc] psi_i=epsilon_i psi_i; n=sum_i fi |psi_i|^2", "assumptions_units": "fi occupation; n electrons/m^3; normalization Integral n dV=Ne.", "verification": "Converge basis, k mesh and SCF; PBE does not provide a universal bounded model error.", "reference": "QE" }, { "id": "P12", "domain": "Physics", "title": "Kohn-Sham energy", "equation": "E[n]=Ts[n]+Integral v_ext n dV+EH[n]+Exc[n]+E_NN", "assumptions_units": "Electronic ground-state functional with nuclear contribution for total energy; convention must be consistent.", "verification": "Converged electronic numerics do not prove agreement with experimental properties.", "reference": "QE" }, { "id": "P13", "domain": "Physics", "title": "Velocity Verlet", "equation": "r_new=r+v dt+a dt^2/2; v_new=v+(a+a_new)dt/2", "assumptions_units": "a=F/m evaluated from chosen force model before and after position update.", "verification": "Time-step refinement, energy drift and force-model validation required.", "reference": "MD" }, { "id": "P14", "domain": "Physics", "title": "Pair-potential virial stress", "equation": "sigma = -(sum_i mi ci tensor ci + 0.5 sum_i sum_j!=i rij tensor fij)/V", "assumptions_units": "Tension-positive convention; ci peculiar velocity relative to local mean; rij points j to i; fij force on i from j; pair forces.", "verification": "Pressure tensor has opposite sign; many-body potentials require appropriate virial decomposition.", "reference": "MD" }, { "id": "P15", "domain": "Physics", "title": "Continuum validity indicators", "equation": "Kn=lambda_mfp/L; Re=rho U L/mu; Ma=U/c_sound", "assumptions_units": "Mean free path lambda_mfp and characteristic length L in m.", "verification": "Select slip, kinetic or compressible models by regime; scale alone is insufficient.", "reference": "CONT" }, { "id": "P16", "domain": "Physics", "title": "Maxwell equations", "equation": "div D=rho_e; div B=0; curl E=-partial_t B; curl H=J+partial_t D", "assumptions_units": "SI fields plus constitutive relations, e.g. D=eps E, B=mu H where applicable.", "verification": "Frequency-dependent, anisotropic or nonlinear media need corresponding constitutive laws.", "reference": "EM" }, { "id": "P17", "domain": "Physics", "title": "Diffraction-limited imaging", "equation": "d approximately lambda/(2 NA)", "assumptions_units": "Lateral Abbe scale under the stated imaging assumptions; wavelength and NA matter.", "verification": "Virtual pixels do not establish molecular optical resolution or fabrication capability.", "reference": "EM" }, { "id": "P18", "domain": "Physics", "title": "Ideal gas density", "equation": "rho=p M/(R T)", "assumptions_units": "p Pa; molar mass M kg/mol; R J/(mol K); T K. R is universal, not the specific gas constant in this expression.", "verification": "Use real-gas equations where required; local weather and gas temperature are inputs.", "reference": "THERMO" }, { "id": "S01", "domain": "Materials", "title": "Fick transport", "equation": "J=-D grad c; partial_t c=div(D grad c)+s", "assumptions_units": "c mol/m^3; D m^2/s; J mol/(m^2 s); s mol/(m^3 s).", "verification": "State phase, temperature, concentration dependence and interface conditions.", "reference": "MAT" }, { "id": "S02", "domain": "Materials", "title": "Arrhenius kinetics", "equation": "D=D0 exp(-Ea/(R T))", "assumptions_units": "Ea J/mol and R J/(mol K); use Ea/(kB T) only for energy per particle.", "verification": "Fit D0,Ea with uncertainty; do not mix molar and particle activation energies.", "reference": "MAT" }, { "id": "S03", "domain": "Materials", "title": "Allen-Cahn evolution", "equation": "F[c]=Integral (f(c)+kappa |grad c|^2/2)dV; partial_t c=-L delta F/delta c", "assumptions_units": "Nonconserved order parameter, L>=0, suitable boundaries.", "verification": "Check nonincreasing free energy without external work; coefficients need calibration.", "reference": "MAT" }, { "id": "S04", "domain": "Materials", "title": "Cahn-Hilliard evolution", "equation": "mu_c=df/dc-kappa laplacian c; partial_t c=div(M grad mu_c)", "assumptions_units": "Conserved field c, mobility M>=0; no-flux closed domain conserves integral c.", "verification": "Check mass conservation and energy decay; boundary conditions on c and mu_c.", "reference": "MAT" }, { "id": "S05", "domain": "Materials", "title": "Linear homogenization", "equation": "C_eff = volume_average(C(x):A(x)); eps(x)=A(x):eps_macro", "assumptions_units": "Representative volume element and scale separation; A is strain localization tensor.", "verification": "Converge RVE size and ensure macro-micro work consistency.", "reference": "MAT" }, { "id": "S06", "domain": "Materials", "title": "Hill-Mandel condition", "equation": "average(sigma:delta eps)=sigma_macro:delta eps_macro", "assumptions_units": "Work equivalence under admissible micro-boundary conditions.", "verification": "Track interface tractions and displacements; simple parameter copying is insufficient.", "reference": "MAT" }, { "id": "S07", "domain": "Materials", "title": "Thermal strain", "equation": "eps_th=Integral_Tref^T alpha(theta) dtheta|Scalar for isotropic thermal expansion; tensor form for anisotropy.|Constrained uniaxial elastic magnitude E ", "assumptions_units": "eps_th", "verification": " is a special case, not universal stress.", "reference": "MAT" }, { "id": "S08", "domain": "Materials", "title": "J2 yield criterion", "equation": "sigma_eq=sqrt(3 s:s/2); f=sigma_eq-sigma_y<=0", "assumptions_units": "s=sigma-tr(sigma)I/3; isotropic ductile yield model; Pa.", "verification": "Composite failure requires laminate allowables and appropriate failure criteria.", "reference": "MAT" }, { "id": "S09", "domain": "Materials", "title": "Fracture mechanics", "equation": "KI=Y sigma sqrt(pi a); require KI MgNH4PO4.6H2O", "assumptions_units": "Moles product cannot exceed the limiting molar supply of Mg, N or P.", "verification": "pH speciation, activities, kinetics and fertilizer demand must be quantified independently.", "reference": "MAT" }, { "id": "E17", "domain": "Engineering", "title": "Root-zone storage", "equation": "dS/dt=P_eff+I-ET-R-D", "assumptions_units": "All water terms as depth per time; storage S depth; R runoff; D drainage.", "verification": "Water exported into atmosphere is not guaranteed to return to the same basin.", "reference": "FAO" }, { "id": "E18", "domain": "Engineering", "title": "Radiative and latent energy", "equation": "q_rad=eps sigmaSB(Ts^4-Ta^4); q_lat=m_dot Lv", "assumptions_units": "Radiative exchange approximation; q_rad W/m^2; q_lat W if m_dot kg/s.", "verification": "Air mixing, humidity and advection bound evaporative cooling; visual mist proves no temperature reduction.", "reference": "THERMO" }, { "id": "E19", "domain": "Engineering", "title": "Cost per accepted product", "equation": "C_unit=(C_material+C_machine+C_energy+C_labour+C_QA+C_overhead)/N_accepted", "assumptions_units": "All costs in one currency and price date; N_accepted>0; avoid double-counting energy in machine rates.", "verification": "Use quotes, yield and ranges; no product price without geometry, material, quantity and process.", "reference": "DERIVED" }, { "id": "E20", "domain": "Engineering", "title": "Levelized water cost", "equation": "LCOW=sum_t Ct/(1+r)^t / sum_t Vt/(1+r)^t", "assumptions_units": "Costs Ct include investment, operations and retirement net of stated residual value; Vt sold water; discount r.", "verification": "Specify currency, base year, real/nominal consistency and financing assumptions.", "reference": "DERIVED" }, { "id": "E21", "domain": "Engineering", "title": "Process capability", "equation": "Cp=(USL-LSL)/(6s); Cpk=min(USL-xbar,xbar-LSL)/(3s)", "assumptions_units": "Stable process; s estimates standard deviation; limits share measurement units.", "verification": "Normal-process assumptions needed for simple tail probability interpretation; measurement uncertainty matters.", "reference": "NIST" }, { "id": "E22", "domain": "Engineering", "title": "CAM scallop screening", "equation": "h=R-sqrt(R^2-(s/2)^2)", "assumptions_units": "Ball cutter radius R; step-over s<=2R; ideal local flat surface.", "verification": "Real geometry, tool deflection, chatter, collision and post-processor require verification.", "reference": "DERIVED" }, { "id": "E23", "domain": "Engineering", "title": "Additive volumetric energy proxy", "equation": "Ev=P/(v h t)", "assumptions_units": "P W; scan speed v mm/s; hatch h mm; layer t mm; Ev J/mm^3.", "verification": "A proxy, not process qualification: equal Ev can produce different melt dynamics.", "reference": "AM" }, { "id": "E24", "domain": "Engineering", "title": "Product reuse metric", "equation": "reuse_rate=N_qualified_reused/N_total", "assumptions_units": "Define item granularity, requirements equivalence, version and qualified context.", "verification": "98% is a hypothesis until measured; identical shape does not mean identical qualification.", "reference": "DERIVED" }, { "id": "E25", "domain": "Engineering", "title": "Quantum variational objective", "equation": "E(theta)=; H=sum_j cj Pj", "assumptions_units": "Normalized trial state; Pauli operators Pj; cj in energy units; finite-basis encoded Hamiltonian.", "verification": "Ideal variational bound does not remove noisy measurement bias; no universal QPU acceleration claim.", "reference": "QM" }, { "id": "E26", "domain": "Engineering", "title": "Cable wave speed and impedance", "equation": "ct=sqrt(H/muL); Z=sqrt(H muL)", "assumptions_units": "Ideal taut string, tension H N, mass per length muL kg/m; ct m/s; mechanical impedance Z N s/m.", "verification": "Neglects bending stiffness and nonlinear geometry; use measured broadband dynamics for real corridors.", "reference": "CONT" }, { "id": "E27", "domain": "Engineering", "title": "Ideal terminal wave reflection", "equation": "Gamma=(Zload-Zline)/(Zload+Zline)", "assumptions_units": "Consistent mechanical wave convention and scalar impedances; energy reflection is abs(Gamma)^2 for real positive impedances.", "verification": "Matching one ideal line does not prove whole-network damping or resilience.", "reference": "CONT" }, { "id": "E28", "domain": "Engineering", "title": "Bulk charge relaxation and sheet resistance", "equation": "tau=epsilon/sigma_e=epsilon rho_e; R_sheet=rho_e/t", "assumptions_units": "Homogeneous Ohmic bulk medium; rho_e ohm m; t m; R_sheet ohms per square.", "verification": "Do not combine metal bulk resistivity with polymer permittivity to certify a multilayer lightning response.", "reference": "EM" }, { "id": "E29", "domain": "Engineering", "title": "Joule heating", "equation": "E_J=Integral I(t)^2 R(t) dt", "assumptions_units": "Current A, resistance ohm, time s, energy J; applies to resistive dissipation.", "verification": "High-frequency current crowding, contact resistance, phase change and breakdown need coupled modelling.", "reference": "EM" }, { "id": "E30", "domain": "Engineering", "title": "Carnot upper bound", "equation": "eta_th <= 1-Tcold/Thot; P_e=eta_th Qdot_hot", "assumptions_units": "Reservoir temperatures K; bound for heat-engine conversion between two thermal reservoirs.", "verification": "Real geothermal cycle, pump loads, reservoir decline and availability reduce net performance.", "reference": "THERMO" }, { "id": "E31", "domain": "Engineering", "title": "Lorentz force and resistive MHD", "equation": "f=J cross B; J=sigma_e(E+v cross B); q_J=J.J/sigma_e", "assumptions_units": "Simple isotropic Ohmic MHD, f N/m3, J A/m2, B T, q_J W/m3.", "verification": "Electrode chemistry, Hall effects if relevant, magnetic power and hydraulic losses require additional models.", "reference": "EM" }, { "id": "E32", "domain": "Engineering", "title": "Inductive energy and discharge", "equation": "E=0.5 L I^2; P=-dE/dt; Inew=sqrt(Iold^2-2 Delta E/L)", "assumptions_units": "Constant positive inductance L H; ideal released energy Delta E between 0 and Eold.", "verification": "If requested energy exceeds storage, reject the request; silently clamping current to zero violates delivered-energy accounting.", "reference": "EM" }, { "id": "E33", "domain": "Engineering", "title": "Circular toroid inductance", "equation": "L=mu0 N^2 (R-sqrt(R^2-a^2))", "assumptions_units": "Ideal air-core continuously wound toroid, circular cross-section radius aP.", "verification": "Lake hypsometry, salt accumulation and seasonal climate prevent a universal fixed area prediction.", "reference": "DERIVED" }, { "id": "E41", "domain": "Engineering", "title": "Latent heat transport", "equation": "P_latent=rho_w Q_evap Lv", "assumptions_units": "rho kg/m3, Q m3/s, Lv J/kg; P W.", "verification": "Energy transfer into water vapour does not prescribe a regional temperature change or create net cooling of the planet.", "reference": "THERMO" }, { "id": "E42", "domain": "Engineering", "title": "Atmospheric column moisture balance", "equation": "partial_t W + div(Fq)=E-P; P=E-partial_t W-div(Fq)", "assumptions_units": "W kg/m2; Fq=(1/g) Integral q v dp, kg/(m s); E,P kg/(m2 s); outward flux divergence convention.", "verification": "The source image 164 has the wrong sign for outward divergence and omits storage; cloud-water terms may also matter.", "reference": "ATM" }, { "id": "E43", "domain": "Engineering", "title": "Soil carbon stock dynamics", "equation": "dC/dt=I-k C; C(t)=I/k+(C0-I/k) exp(-k t)", "assumptions_units": "Constant I stock/time and k>0 inverse time; C stock on a stated area/depth basis.", "verification": "For wt%, convert stock using bulk density and sampled depth; calibrated dynamics do not follow from assumed asymptotes.", "reference": "DERIVED" }, { "id": "E44", "domain": "Engineering", "title": "Periodic soil thermal penetration", "equation": "d=sqrt(2 alpha/omega); alpha=k/(rho cp); Tamp(z)=Tamp(0) exp(-z/d)", "assumptions_units": "Homogeneous semi-infinite diffusion model with sinusoidal boundary, omega rad/s.", "verification": "Surface temperature amplitude is imposed here; changing d alone does not prove a smaller surface amplitude.", "reference": "CONT" }, { "id": "E45", "domain": "Engineering", "title": "Spectral index and salinity regression", "equation": "I=(rho_a-rho_b)/(rho_a+rho_b); EC=a0 exp(a1 I)+a2 rho_c/rho_d", "assumptions_units": "Reflectances dimensionless; denominators nonzero; a0,a2 carry EC units; a1 dimensionless.", "verification": "Empirical source-specific index, not universal root-zone retrieval. Requires ground truth, calibration and uncertainty.", "reference": "DERIVED" }, { "id": "E46", "domain": "Engineering", "title": "Soil carbon to CO2 mass", "equation": "mCO2=mC MCO2/MC approximately (44.01/12.01) mC", "assumptions_units": "Mass of carbon and CO2 consistent; molar masses g/mol; chemical conversion only.", "verification": "Sequestration claims need baseline, leakage, permanence, sampling depth and lifecycle emissions.", "reference": "DERIVED" }, { "id": "E47", "domain": "Engineering", "title": "Pile shaft resistance screening", "equation": "Rs=Integral shaft tau_s dA; for uniform shaft Rs=pi D L tau_s", "assumptions_units": "D,L m; shaft resistance tau_s Pa; Rs N for one pile.", "verification": "Geotechnical factors, installation, rock joints, group effects and tension loading require project-specific design.", "reference": "CONT" }, { "id": "E48", "domain": "Engineering", "title": "Linear laminate mechanics", "equation": "[N;M]=[A B;B D] [eps0;kappa]-[Nthermal;Mthermal]", "assumptions_units": "Classical lamination assumptions; membrane resultants N N/m and moments M N; stiffness from oriented ply data.", "verification": "Strength, damage, transverse shear, creep, joints and pressure-pipe qualification require further models.", "reference": "MAT" }, { "id": "E49", "domain": "Engineering", "title": "Explicit heat-equation benchmark", "equation": "theta_i^(n+1)=theta_i^n+Fo(theta_(i-1)^n-2theta_i^n+theta_(i+1)^n); Fo=alpha dt/dx^2<=1/2", "assumptions_units": "Uniform 1D constant-diffusivity grid, forward Euler centred differences, fixed end temperatures.", "verification": "Analytic sine-mode reference theta=exp(-pi^2 alpha t/L^2) sin(pi x/L) enables a convergence test.", "reference": "NUM" }, { "id": "E50", "domain": "Engineering", "title": "Ideal net-lift envelope sizing", "equation": "V_per_length=m_per_length(1+margin)/(rho_air-rho_gas); Denv=sqrt(4 V_per_length/pi)", "assumptions_units": "Circular continuous-envelope approximation; V_per_length m2; complete carried line mass kg/m; positive density difference.", "verification": "Envelope/tether/ballast mass and wind cannot be omitted in a final sizing calculation.", "reference": "DERIVED" }, { "id": "M15", "domain": "Mathematics", "title": "Lax-Milgram existence statement", "equation": "a(u,v)=l(v) for all v in V; norm(u)<=norm(l)/alpha", "assumptions_units": "Real Hilbert space V; bounded coercive bilinear a with coercivity alpha>0; continuous linear functional l.", "verification": "Existence and uniqueness under these hypotheses; not a statement about arbitrary PDEs.", "reference": "FEM" }, { "id": "M16", "domain": "Mathematics", "title": "Gronwall bound", "equation": "y(t)<=a+Integral_0^t b(s)y(s)ds implies y(t)<=a exp(Integral_0^t b(s)ds)", "assumptions_units": "a>=0; y,b nonnegative and appropriately integrable.", "verification": "A priori amplification estimate; a large bound is not a useful numerical error certificate.", "reference": "NUM" }, { "id": "M17", "domain": "Mathematics", "title": "Contraction iteration", "equation": "norm(Tx-Ty)<=q norm(x-y); norm(x*-xk)<=q^k norm(x1-x0)/(1-q)", "assumptions_units": "0<=q<1; complete metric/normed setting and invariant iteration set.", "verification": "Prove contraction on the actual domain before asserting convergence.", "reference": "NUM" }, { "id": "M18", "domain": "Mathematics", "title": "SVD regularization", "equation": "A=U Sigma V^T; x_lambda=sum_i [sigma_i/(sigma_i^2+lambda^2)] (ui^T b) vi", "assumptions_units": "Tikhonov penalty lambda^2 norm(x)^2 with lambda>0, Euclidean norms.", "verification": "Regularization introduces bias; choose lambda by data uncertainty and validation.", "reference": "NUM" }, { "id": "M19", "domain": "Mathematics", "title": "Preconditioned conjugate gradients", "equation": "A x=b; error_A(k)<=2[(sqrt(kappa)-1)/(sqrt(kappa)+1)]^k error_A(0)", "assumptions_units": "Symmetric positive definite A and suitable SPD preconditioner; kappa of preconditioned operator.", "verification": "Exact-arithmetic bound; monitor finite-precision residual and actual convergence.", "reference": "NUM" }, { "id": "M20", "domain": "Mathematics", "title": "Generalized eigenproblem", "equation": "K phi=omega^2 M phi; phi_i^T M phi_j=delta_ij", "assumptions_units": "Symmetric stiffness K, SPD mass M, conservative linear vibration problem.", "verification": "Constraints remove rigid modes as intended; damping and gyroscopic effects alter formulation.", "reference": "FEM" }, { "id": "M21", "domain": "Mathematics", "title": "Adjoint sensitivity", "equation": "R(u,p)=0; R_u^T lambda=J_u^T; dJ/dp=J_p-lambda^T R_p", "assumptions_units": "Differentiable residual and objective with locally invertible state Jacobian.", "verification": "Check against finite differences; nonsmooth contact or topology changes need special handling.", "reference": "NUM" }, { "id": "M22", "domain": "Mathematics", "title": "Interval Newton enclosure", "equation": "N(X)=m-[F'(X)]^(-1)F(m)", "assumptions_units": "Interval Jacobian over box X, point m in X; an appropriate interval linear solver is needed.", "verification": "Inclusion and uniqueness conditions are method-specific; interval overestimation can prevent certification.", "reference": "NUM" }, { "id": "M23", "domain": "Mathematics", "title": "Bayesian calibration", "equation": "p(theta|d) proportional to p(d|theta) p(theta)", "assumptions_units": "Specify prior, likelihood, noise and model discrepancy; posterior is conditional on these choices.", "verification": "Do not equate posterior confidence with externally validated physical truth.", "reference": "NUM" }, { "id": "M24", "domain": "Mathematics", "title": "Monte Carlo standard error", "equation": "SE(mean)=s/sqrt(N)", "assumptions_units": "Independent identically distributed samples with finite variance; correlated chains need effective sample size.", "verification": "Record seeds and convergence; deterministic reproducibility does not eliminate sampling uncertainty.", "reference": "NUM" }, { "id": "M25", "domain": "Mathematics", "title": "Gaussian chance constraint", "equation": "Pr(g<=0)>=1-alpha is equivalent to mu_g+z_(1-alpha) sigma_g<=0", "assumptions_units": "Exactly Gaussian scalar g with known mean/standard deviation; z is standard-normal quantile.", "verification": "Approximate for non-Gaussian nonlinear responses; include parameter estimation uncertainty.", "reference": "NUM" }, { "id": "M26", "domain": "Mathematics", "title": "Topology-optimization interpolation", "equation": "E(rho)=Emin+rho^p(E0-Emin); 0<=rho<=1", "assumptions_units": "SIMP interpolation, penalization p>1; positive Emin; density-filter/projection definition required.", "verification": "Mesh dependence, minimum feature, buckling, fatigue and manufacturing constraints need explicit treatment.", "reference": "FEM" }, { "id": "M27", "domain": "Mathematics", "title": "Discrete conservation at interfaces", "equation": "sum_cells Delta conserved_quantity + Integral boundary_flux dt = sources", "assumptions_units": "Every internal face uses equal and opposite fluxes in a common unit convention.", "verification": "Check global and local budgets; interpolation between nonmatching meshes must preserve transfer quantities.", "reference": "DERIVED" }, { "id": "M28", "domain": "Mathematics", "title": "CFL condition for upwind transport", "equation": "C=abs(a) dt/dx<=1", "assumptions_units": "1D linear advection, uniform grid, explicit first-order upwind scheme.", "verification": "This bound is scheme-specific; not a universal timestep criterion for all PDE solvers.", "reference": "NUM" }, { "id": "M29", "domain": "Mathematics", "title": "Energy-stable backward Euler diffusion", "equation": "(norm(u_new)^2-norm(u_old)^2)/(2dt)+k norm(grad u_new)^2<=0", "assumptions_units": "Homogeneous diffusion, zero source, compatible homogeneous boundaries, k>0 and suitable inner product.", "verification": "Unconditional linear stability is not unconditional time accuracy.", "reference": "FEM" }, { "id": "M30", "domain": "Mathematics", "title": "Dimensionless scalar error budget", "equation": "e_total <= e_discretization+e_iteration+e_input+e_model", "assumptions_units": "Triangle-inequality bound only if every term bounds the same quantity in compatible units/norms.", "verification": "Unknown model discrepancy must remain unknown; do not assign it a convenient zero.", "reference": "DERIVED" }, { "id": "P19", "domain": "Physics", "title": "One-dimensional quantum box", "equation": "En=n^2 pi^2 hbar^2/(2m L^2); psi_n=sqrt(2/L) sin(n pi x/L)", "assumptions_units": "Infinite potential walls, 00, n>=0.", "verification": "Real molecular vibrations require force constants and anharmonic/coupled-mode corrections.", "reference": "QM" }, { "id": "P21", "domain": "Physics", "title": "Born-Oppenheimer force", "equation": "F_A=-partial E_electronic_total/partial R_A", "assumptions_units": "Adiabatic ground-state potential-energy surface, consistent total energy.", "verification": "Incomplete basis may need Pulay corrections; nonadiabatic transitions need additional dynamics.", "reference": "QE" }, { "id": "P22", "domain": "Physics", "title": "Transition-state rate estimate", "equation": "k(T)=kappa(T) (kB T/h) exp(-Delta G_dagger/(R T))", "assumptions_units": "Molar activation free energy J/mol; transmission coefficient kappa; appropriate standard-state conventions.", "verification": "Recrossing, tunnelling and multiple pathways require treatment; activation data cannot be invented.", "reference": "THERMO" }, { "id": "P23", "domain": "Physics", "title": "Boltzmann distribution", "equation": "p_i=exp(-Ei/(kB T))/Z; Z=sum_i exp(-Ei/(kB T))", "assumptions_units": "Canonical equilibrium, states and degeneracy specified; T>0.", "verification": "Non-equilibrium processes and strongly driven manufacturing are outside this equilibrium statement.", "reference": "THERMO" }, { "id": "P24", "domain": "Physics", "title": "Fluctuation-dissipation Langevin model", "equation": "m dv=-gamma v dt+F dt+sqrt(2 gamma kB T) dW", "assumptions_units": "Markovian white-noise bath, gamma kg/s, W Wiener process; chosen stochastic convention.", "verification": "Validate thermostat and time integration; seed reproducibility is not independent replication.", "reference": "MD" }, { "id": "P25", "domain": "Physics", "title": "Boltzmann transport", "equation": "partial_t f+v.grad_x f+a.grad_v f=C[f]", "assumptions_units": "Distribution f in phase space; collision operator requires a physical model.", "verification": "For rarefied gases choose collision cross sections, boundaries and statistical error controls.", "reference": "CONT" }, { "id": "P26", "domain": "Physics", "title": "Radiative transfer", "equation": "dI_lambda/ds=-kappa_lambda I_lambda+j_lambda", "assumptions_units": "No scattering in this simplified ray form; intensity, absorption and emission use consistent spectral units.", "verification": "Scattering, wavelength integration and view geometry must be added when relevant.", "reference": "EM" }, { "id": "P27", "domain": "Physics", "title": "Thermoelastic coupling", "equation": "sigma=C:(eps-eps_th); rho u_tt=div sigma+f", "assumptions_units": "Small-strain elastic mechanics with thermal eigenstrain; C units Pa.", "verification": "Temperature-dependent properties, plastic dissipation and geometric nonlinearities require extension.", "reference": "CONT" }, { "id": "P28", "domain": "Physics", "title": "General finite-strain balances", "equation": "F=I+Grad u; J=det F>0; rho0=rho J; Div P+rho0 b=rho0 u_tt", "assumptions_units": "Reference-coordinate deformation gradient F, first Piola stress P, mass density rho0.", "verification": "Constitutive law must be objective and thermodynamically admissible; prevent inverted elements.", "reference": "CONT" }, { "id": "P29", "domain": "Physics", "title": "Newtonian gravitation and field", "equation": "laplacian Phi=4 pi G rho; a=-grad Phi", "assumptions_units": "Weak-field slow-motion approximation; boundary conditions and mass distribution specified.", "verification": "A scale slider cannot extend Newtonian validity into relativistic regimes.", "reference": "GRAV" }, { "id": "P30", "domain": "Physics", "title": "Einstein field equation", "equation": "G_mu_nu+Lambda g_mu_nu=(8 pi G/c^4) T_mu_nu", "assumptions_units": "Metric-sign and curvature conventions specified consistently; stress-energy tensor conserved covariantly.", "verification": "Included as a domain boundary, not a solved cosmological manufacturing engine.", "reference": "GRAV" }, { "id": "P31", "domain": "Physics", "title": "Friedmann expansion equation", "equation": "H^2=(8 pi G/3)rho+Lambda c^2/3-k c^2/a^2", "assumptions_units": "Homogeneous isotropic cosmology; rho mass-equivalent density; choose a in length units and k dimensionless.", "verification": "Requires cosmological parameters and assumptions; unrelated to increasing machine resolution.", "reference": "GRAV" }, { "id": "P32", "domain": "Physics", "title": "Thermal diffusion analytic benchmark", "equation": "theta(x,t)=sin(pi x/L) exp(-alpha pi^2 t/L^2)", "assumptions_units": "1D rod, fixed zero excess temperature at endpoints, initial unit sine, constant alpha>0.", "verification": "Substitute in theta_t=alpha theta_xx; compare numerical solution with exact profile.", "reference": "DERIVED" }, { "id": "S13", "domain": "Materials", "title": "Chemical potential", "equation": "mu_i=(partial G/partial n_i)_(T,p,nj)", "assumptions_units": "Gibbs energy J, amount n_i mol; chemical potential J/mol.", "verification": "Standard states and reference energies must match across phases.", "reference": "THERMO" }, { "id": "S14", "domain": "Materials", "title": "Ideal mixing free energy", "equation": "Delta G_mix=R T sum_i n_i ln(x_i)", "assumptions_units": "Ideal solution with mole fractions x_i>0 and sum x_i=1.", "verification": "Real alloys/electrolytes require excess Gibbs-energy/activity terms.", "reference": "THERMO" }, { "id": "S15", "domain": "Materials", "title": "Phase equilibrium", "equation": "T_alpha=T_beta; p_alpha=p_beta; mu_i_alpha=mu_i_beta", "assumptions_units": "Equilibrium phases able to exchange heat, volume and relevant components.", "verification": "Metastability and kinetic barriers may prevent equilibrium microstructure during manufacture.", "reference": "MAT" }, { "id": "S16", "domain": "Materials", "title": "Classical nucleation", "equation": "Delta G(r)=4 pi r^2 gamma+(4 pi/3)r^3 Delta gv; r*=-2 gamma/Delta gv", "assumptions_units": "Spherical homogeneous nucleus, gamma>0 J/m2, driving Delta gv<0 J/m3.", "verification": "Heterogeneous nucleation, anisotropy and atomistic scales can invalidate continuum capillarity.", "reference": "MAT" }, { "id": "S17", "domain": "Materials", "title": "Critical nucleation barrier", "equation": "Delta G*=16 pi gamma^3/[3(Delta gv)^2]", "assumptions_units": "Same assumptions as S16; units J.", "verification": "Nucleation rate also depends on a kinetic prefactor and site population.", "reference": "MAT" }, { "id": "S18", "domain": "Materials", "title": "Diffusion length scaling", "equation": "ell_rms=sqrt(2 d D t)", "assumptions_units": "Isotropic Brownian diffusion in d dimensions, constant D and no boundaries/drift.", "verification": "Not the same as a guaranteed penetration depth or interface thickness.", "reference": "MAT" }, { "id": "S19", "domain": "Materials", "title": "Reaction-diffusion species balance", "equation": "partial_t c_i+div(c_i v)=div(D_i grad c_i)+sum_r nu_ir R_r", "assumptions_units": "Concentrations mol/m3, stoichiometric coefficients nu_ir, reaction rates mol/(m3 s).", "verification": "Enforce elemental and charge conservation; electroneutral or Poisson coupling as appropriate.", "reference": "MAT" }, { "id": "S20", "domain": "Materials", "title": "Nernst-Planck flux", "equation": "J_i=-D_i grad c_i-z_i D_i F c_i grad(phi)/(R T)+c_i v", "assumptions_units": "Dilute isothermal ionic transport; z_i charge number, Faraday F C/mol.", "verification": "Concentrated brines require nonideal electrochemical potentials and coupled transport.", "reference": "GEOCHEM" }, { "id": "S21", "domain": "Materials", "title": "Faraday electrolysis balance", "equation": "m=eta_F M I t/(z F)", "assumptions_units": "M kg/mol, I A, t s, z electrons per product unit, eta_F current efficiency.", "verification": "Mass yield does not include energy or separation cost; side reactions and purity need testing.", "reference": "EM" }, { "id": "S22", "domain": "Materials", "title": "Butler-Volmer relation", "equation": "j=j0[exp(alpha_a F eta/(R T))-exp(-alpha_c F eta/(R T))]", "assumptions_units": "Simple one-electron activation-controlled form, current densities A/m2, overpotential eta V.", "verification": "Mass-transfer limitations, multistep kinetics, surface changes and concentrated electrolytes require extension.", "reference": "GEOCHEM" }, { "id": "S23", "domain": "Materials", "title": "Crystal plasticity kinematics", "equation": "F=Fe Fp; Lp=dot(Fp) inverse(Fp)=sum_s gamma_dot_s m_s tensor n_s", "assumptions_units": "Slip directions/normals in the intermediate configuration; multiplicative finite-strain decomposition.", "verification": "Slip systems, hardening and texture must be material-specific and calibrated.", "reference": "MAT" }, { "id": "S24", "domain": "Materials", "title": "Rate-dependent slip rule", "equation": "gamma_dot_s=gamma0 abs(tau_s/g_s)^n sign(tau_s)", "assumptions_units": "Empirical power-law slip, positive resistance g_s, rate gamma0 1/s.", "verification": "Parameters depend on material, temperature and strain-rate regime; not universal.", "reference": "MAT" }, { "id": "S25", "domain": "Materials", "title": "Orowan plastic shear rate", "equation": "gamma_dot=rho_mobile b v_dislocation", "assumptions_units": "Dislocation density 1/m2, Burgers vector magnitude m, mean glide speed m/s.", "verification": "Needs slip-system resolution, mobile fraction and velocity law.", "reference": "MAT" }, { "id": "S26", "domain": "Materials", "title": "Hall-Petch relation", "equation": "sigma_y=sigma0+kHP/sqrt(d_grain)", "assumptions_units": "Empirical grain-size strengthening law with calibrated kHP and valid grain-size range.", "verification": "Do not extrapolate to arbitrarily small grains; mechanisms and inverse trends may change.", "reference": "MAT" }, { "id": "S27", "domain": "Materials", "title": "Norton creep law", "equation": "eps_dot=A sigma^n exp(-Q/(R T))", "assumptions_units": "Empirical uniaxial steady-creep law; A units depend on stress units; Q J/mol.", "verification": "Calibration domain, transient creep, rupture and multiaxiality must be addressed.", "reference": "MAT" }, { "id": "S28", "domain": "Materials", "title": "Linear damage accumulation", "equation": "D=sum_i n_i/N_i", "assumptions_units": "Miner rule for specified cycle classes and material fatigue data.", "verification": "Load sequence, mean stress, environment and nonlinear damage can invalidate D=1 predictions.", "reference": "MAT" }, { "id": "S29", "domain": "Materials", "title": "Moving heat-source balance", "equation": "rho cp T_t=div(k grad T)+q(x-vt)-rho Lm f_dot_liquid", "assumptions_units": "Simplified enthalpy balance with latent heat Lm J/kg and liquid fraction.", "verification": "Laser absorption, melt convection, radiation and evaporation determine process adequacy.", "reference": "MAT" }, { "id": "S30", "domain": "Materials", "title": "Thermal residual-stress screening", "equation": "sigma_thermal approximately E alpha Delta T/(1-nu)", "assumptions_units": "Ideal isotropic equal biaxial in-plane restraint, plane stress and uniform cooling; sign by convention.", "verification": "Not a universal welding/additive stress formula; actual constraints and plasticity matter.", "reference": "MAT" }, { "id": "E51", "domain": "Engineering", "title": "Rigid-body pose and tool transform", "equation": "p_world=R p_tool+t; R^T R=I; det R=1", "assumptions_units": "SE(3) pose with rotation R and translation t in consistent length units.", "verification": "Calibrate robot/tool/workpiece frames; display accuracy does not establish pose accuracy.", "reference": "ROBOT" }, { "id": "E52", "domain": "Engineering", "title": "Robot Jacobian and velocity", "equation": "v_task=J(q) q_dot", "assumptions_units": "Joint coordinates and task-velocity convention fixed; angular and linear components distinguished.", "verification": "Check singular values, joint limits, collisions and acceleration/torque capacity.", "reference": "ROBOT" }, { "id": "E53", "domain": "Engineering", "title": "Kinematic error propagation", "equation": "Sigma_pose approximately J Sigma_joint J^T+Sigma_calibration", "assumptions_units": "First-order covariance model with consistent coordinates; omitted cross-covariances assumed zero.", "verification": "Near singularity and nonlinear errors require more complete propagation.", "reference": "NIST" }, { "id": "E54", "domain": "Engineering", "title": "State-space control", "equation": "x_dot=A x+B u; y=C x+D u", "assumptions_units": "Linearized model about a specified operating point, units and sampling stated.", "verification": "A stable nominal model does not prove robustness to delays, saturation or unmodelled dynamics.", "reference": "CONTROL" }, { "id": "E55", "domain": "Engineering", "title": "Discrete Kalman measurement update", "equation": "K=Pminus H^T(H Pminus H^T+R)^(-1); xplus=xminus+K(y-H xminus)", "assumptions_units": "Linear observation model and nonsingular innovation covariance; classical optimality assumes Gaussian noise.", "verification": "Bias, correlation and changing noise models require diagnostics; filter confidence is conditional.", "reference": "CONTROL" }, { "id": "E56", "domain": "Engineering", "title": "Geometric chord tolerance", "equation": "delta=R[1-cos(pi/N)]", "assumptions_units": "Regular N-sided circular approximation, N>=3, radius R, maximum radial chord deviation delta.", "verification": "Select tessellation by a specified geometric tolerance; manufacturing uncertainty is additional.", "reference": "DERIVED" }, { "id": "E57", "domain": "Engineering", "title": "CNC material removal rate", "equation": "MRR=a_p a_e v_f", "assumptions_units": "Axial depth a_p mm, radial engagement a_e mm, feed v_f mm/min; MRR mm3/min for a simple milling estimate.", "verification": "Tool force, spindle power, chatter, wear and actual engagement limit productivity.", "reference": "DERIVED" }, { "id": "E58", "domain": "Engineering", "title": "Yield and scrap propagation", "equation": "N_started=ceil(N_required/Y); C_accepted=C_total/N_accepted", "assumptions_units": "0