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Learn the Dynamics, Enforce the Physics: Generative Latent Operators for High-Dimensional Beam Evolution

Haohan Zou, Alexander Scheinker, Yuanyuan Shi

arXiv:2610.14537v1physics.acc-phcs.LG

TL;DR

High-resolution PIC simulations make many-query accelerator workflows costly, and long-horizon 3D beam–field prediction remains challenging. GLOBE combines latent deterministic and generative dynamics with regime-specific field constraints. Across two benchmarks it improves errors over DPOT, satisfies field constraints, and cuts hours of PIC simulation to about a minute per rollout.

  • Problem

    High-resolution PIC simulations cost hours of GPU time, while efficient long-horizon prediction of 3D mesh-resolved beam–field evolution remains largely unexplored.

    Source: §1 · p. 1

  • Method

    GLOBE compresses beam–field states and combines a control-conditioned neural operator for dominant dynamics with an amplitude-normalized generative residual model and regime-specific field-constraint operators.

    Source: §1 · p. 2

  • Results

    Across two benchmarks, GLOBE lowers relative-L2 error by 9% and 13%, low-frequency spectral error by 69% and 51%, and mid-frequency spectral error by 54% and 20% versus DPOT.

    Source: §1 · p. 2

  • Takeaways & Limitations

    GLOBE satisfies field constraints and reduces hours of PIC simulation to about one minute per rollout across the evaluated benchmarks.

    Source: §6 · p. 10

    The electrostatic treatment assumes a low-energy regime where magnetic and inductive effects associated with beam self-fields are negligible.

    Source: §3 · p. 4

Abstract

from arXiv · show

Particle-in-cell (PIC) simulation is indispensable for modeling beam dynamics in particle accelerators, but its computational cost limits many-query workflows, such as configuration exploration, virtual diagnostics, and controller design. We introduce GLOBE, a physics-constrained Generative Latent Operator for Beam Evolution, modeling the nonlinear, control-dependent, multiscale evolution of 3D beam--field dynamics covering $10^7$ spatial cells. GLOBE compresses beam--field states into a compact latent space and factors their evolution into two components. A control-conditioned neural operator learns the dominant dynamics, while an amplitude-normalized generative model captures the residual distribution, recovering fine-scale structures and conditional variations. GLOBE enforces field constraints by regime-specific, parameter-free operators: a Poisson solve reconstructs the electric field in the electrostatic regime, and a spectral projection imposes Gauss's law and a divergence-free magnetic field in the electromagnetic regime. On two high-resolution 3D particle-accelerator benchmarks, bunched-beam transport and plasma-wakefield acceleration, GLOBE lowers relative-$L_2$ error by 9\% and 13\%, low-frequency spectral error by 69\% and 51\%, and mid-frequency spectral error by 54\% and 20\% relative to the strongest baseline (DPOT), satisfies field constraints, and reduces hours of PIC simulation to a minute per rollout.

1 INTRODUCTION

GLOBE targets the costly, underexplored problem of long-horizon, mesh-resolved 3D beam–field prediction. It combines latent deterministic and generative dynamics with regime-specific field constraints, improving benchmark accuracy while reducing rollout time.

1 INTRODUCTION

  • High-resolution PIC simulations cost hours of GPU time, while efficient long-horizon prediction of 3D mesh-resolved beam–field evolution remains largely unexplored.The computational burden limits workflows including configuration exploration, virtual diagnostics, and controller design.Source: §1 · p. 1
  • At each rollout step, the latest k latents and machine controls drive deterministic prediction; flow matching samples residuals before decoding and constraint enforcement.This rollout sequence is shown in Figure 1.Source: Figure 1 · p. 2, §1 · p. 2
  • GLOBE compresses beam–field states into a latent space, where a control-conditioned neural operator models dominant dynamics and an amplitude-normalized generative model captures residual distributions.The residual model recovers fine-scale structures and sampling-induced conditional variability.Source: §1 · p. 2
  • Parameter-free operators enforce field constraints by Poisson-solving the electric field in the electrostatic regime and applying a spectral Maxwell projection in the electromagnetic regime.The approach separates dynamics learning from constraint satisfaction.Source: §1 · p. 2
  • GLOBE lowers relative-L2 error by 9% and 13%, low-frequency spectral error by 69% and 51%, and mid-frequency spectral error by 54% and 20% versus DPOT across two benchmarks.It satisfies field constraints and reduces hours of PIC simulation to a minute per rollout.Source: §1 · p. 2

2 LITERATURE REVIEW

The literature covers neural operators, generative PDE models, and accelerator-specific learning, but efficient prediction of long-horizon, mesh-resolved 3D beam–field states remains underexplored.

2.1 DEEP LEARNING FOR PDE MODELING

  • Neural operators model PDE solution mappings but can miss fine scales through spectral bias; generative models represent solution distributions but make complete conditional evolution challenging and costly.Hybrid deterministic–generative methods combine these capabilities, though scaling them to high-resolution 3D systems remains difficult.Source: §2.1 · p. 3
  • Existing hybrid PDE approaches use generative refinement or correction around deterministic forecasts, but grid-size scaling and heterogeneous field amplitudes challenge high-resolution 3D applications.Repeated denoising or numerical integration can also increase generative rollout cost.Source: §2.1 · p. 3

2.2 LEARNING-BASED METHODS FOR PARTICLE ACCELERATORS

  • Accelerator-learning research includes configuration-to-metric models, tuning and control, diagnostic reconstruction, and forecasts of downstream phase-space projections, but not widespread mesh-resolved 3D state prediction.Efficient long-horizon prediction of self-consistent beam–field states remains underexplored.Source: §2.2 · p. 3

3 PRELIMINARY: BEAM DYNAMICS

Beam dynamics are self-consistent: the particle distribution determines charge and current, while the resulting fields govern particle evolution. In the low-energy regime, negligible magnetic and inductive self-field effects permit an electrostatic Vlasov–Poisson description.

3 PRELIMINARY: BEAM DYNAMICS

  • The Vlasov–Maxwell system couples the evolving phase-space distribution to self-fields generated by beam charge and current, which in turn act on the particles through the Lorentz force.The beam fields also include externally applied electric and magnetic fields.Source: §3 · pp. 3–4
  • In the low-energy regime, negligible magnetic and inductive beam self-field effects yield an approximately curl-free electric field and reduce Vlasov–Maxwell to Vlasov–Poisson.The beam magnetic field is taken as approximately zero in this regime.Source: §3 · p. 4
  • In the electrostatic regime, charge density and boundary conditions determine the potential through Poisson’s equation, and the beam electric field follows as its negative spatial gradient.Unlike the full electromagnetic case, the electrostatic beam field has no independent temporal evolution.Source: §3 · p. 4

4 PROPOSED FRAMEWORK

GLOBE models control-conditioned beam–field evolution through a compact latent representation, combining deterministic forecasting with generative residual refinement and enforcing field constraints after decoding. Its framework comprises a compressive autoencoder, latent dynamics model, and regime-specific physics operators.

4 PROPOSED FRAMEWORK

  • GLOBE encodes beam–field states and models their future conditional distribution from observed history and machine controls, rolled out autoregressively in blocks.The framework combines an autoencoder, a deterministic neural operator with a flow-matching residual model, and regime-specific field-constraint operators.Source: §4 · p. 4

4.1 COMPRESSIVE AUTOENCODER

  • The autoencoder compresses each 3D state into a structured grid latent, reducing spatial tokens from 10^7 to approximately 10^4–10^5.A 3D Haar transform separates eight subbands, while space–channel shortcuts provide cross-resolution references so learned modules fit residual corrections; reconstruction uses relative, gradient, and spectral losses and trains on perturbed latents for robustness.Source: §4.1 · pp. 4–5, Figure 2 · p. 4

4.2 LATENT DYNAMICS: DETERMINISTIC ANCHOR AND GENERATIVE RESIDUAL

  • A control-conditioned deterministic operator forecasts dominant latent evolution, while a conditional generative model learns normalized residuals to recover fine-scale structure and sampling-induced variation.The deterministic forecast anchors residual modeling; input perturbations and pushforward training address rollout error accumulation, and the generated residual is rescaled and added to the forecast.Source: §4.2 · pp. 5–6

4.3 PHYSICS-CONSTRAINED OPERATORS

  • After decoding, parameter-free, regime-specific operators enforce instantaneous field constraints by construction.For electrostatic states, GLOBE solves Poisson’s equation from predicted charge density and derives the electric field; in the electromagnetic regime, Fourier-mode projection corrects decoded predictions to satisfy Maxwell constraints in the simulation discretization.Source: §4.3 · pp. 6–7

5 EXPERIMENTS

Experiments evaluate GLOBE on high-resolution bunched-beam transport and PWFA benchmarks, comparing against matched-size baselines and testing long-horizon rollouts and component ablations. GLOBE-C improves fine-scale predictions and tracks both benchmark trajectories, while ablations assess efficiency, physical consistency, and latent-backbone choices.

5 EXPERIMENTS

  • Experiments use WarpX bunched-beam and PWFA benchmarks (464 and 504 simulations; 52 and 56 test cases), with matched-size baselines and Rel-L2 and banded spectral metrics.Generative-model errors are averaged over eight samples.Source: §5 · p. 7

5.1 QUANTITATIVE COMPARISON WITH BASELINES

  • GLOBE-D achieves the lowest deterministic Rel-L2 errors (0.144 on bunched-beam transport; 0.174 on PWFA), while GLOBE-C cuts mid-frequency error from 0.994 to 0.457 and 0.350 to 0.300, respectively.Table 1 compares GLOBE variants with baselines on both benchmarks.Source: §5.1 · pp. 7–8, Table 1 · p. 7
  • GLOBE-C uses 64.7s / 21.8s per rollout and achieves 102× / 189× speedups over WarpX, while DPOT remains fastest among deterministic baselines.Latent-space flow integration avoids full-state generative rollout; GLOBE-D uses 16.3s / 5.4s per rollout.Source: §5.1 · p. 8

5.2 LONG-HORIZON PREDICTION AND VISUALIZATION

  • Across 393-frame bunched-beam and 56-frame PWFA forecasts, GLOBE-C closely tracks beam shape, envelope diagnostics, wake cavity, witness evolution, and associated field and radius traces.The bunched-beam forecast is conditioned on eight initial frames; the PWFA forecast uses four.Source: §5.2 · p. 8, Figure 3 · p. 8, Figure 4 · p. 9

5.3 ABLATION STUDIES

  • The Haar transform reduces autoencoder training runtime by 6.2× and lowers three losses by 39%, 31%, and 19%; the shortcut further reduces losses by 16%, 7%, and 14%.The Haar transform provides the main efficiency gain, while the shortcut adds accuracy with nearly unchanged runtime.Source: §5.3 · p. 9, Table 2 · p. 9
  • On PWFA, applying the physics-constrained operator changes Rel-L2 from 0.194 to 0.191 and reduces field-constraint violations from 10^-2–10^-3 to 10^-7.The reported constraint improvement is four to five orders of magnitude with negligible accuracy loss.Source: §5.3 · p. 9, Table 3 · p. 9
  • HFS is competitive with GLOBE-D on bunched-beam transport but degrades substantially on multi-channel PWFA, where its Rel-L2 is 0.835 versus 0.174.HFS diffusion refinement further diverges on PWFA; the results support the latent Transformer backbone for strongly coupled multi-channel dynamics.Source: §5.3 · p. 9, Table 4 · p. 9

6 CONCLUSION

The conclusion presents GLOBE as a hybrid framework for high-resolution 3D accelerator beam dynamics, combining latent dynamics modeling with regime-specific field constraints. Across two benchmarks, it reports strong prediction accuracy, field-constraint satisfaction, and rollout-time reduction.

6 CONCLUSION

  • Across two particle-accelerator benchmarks, GLOBE achieves the best prediction accuracy among evaluated baselines, satisfies instantaneous field constraints, and reduces hours of PIC simulation to about one minute per rollout.The framework combines a compressed representation, a Transformer for dominant dynamics, a generative residual model, and parameter-free regime-specific operators.Source: §6 · pp. 9–10
  • The authors identify extending GLOBE across machines, control settings, and initial and boundary conditions as a future foundation-model direction.Source: §6 · p. 10
  • The conclusion also proposes using learned dynamics for beam-state estimation and control toward virtual diagnostics and real-time decision-making across accelerator applications.Source: §6 · p. 10

CONTENTS OF SUPPLEMENTARY MATERIALS

The supplementary materials list dataset descriptions, model and baseline details, and additional experiment results. They also specify the computational resource used for the experiments.

CONTENTS OF SUPPLEMENTARY MATERIALS

  • Appendix A describes the bunched-beam transport and PWFA datasets, while Appendix B details the compressive autoencoder, deterministic latent operator, generative latent residual, and physics-constrained operators.Source: DATASET · p. 16
  • Appendix C documents DPOT, UPT, and MPP baselines, and Appendix D lists amplitude-normalization and benchmark-specific supplementary experiment results.Source: DATASET · p. 16
  • All experiments were performed on a server with an Nvidia RTX Pro 6000 Blackwell Server Edition 96 GB GPU.Source: DATASET · p. 17

A DATASET GENERATION

The dataset release contains two complementary WarpX-generated 3D accelerator benchmarks, covering electrostatic bunched-beam transport and electromagnetic plasma-wakefield acceleration. The bunched-beam benchmark uses an idealized periodic focusing channel and varies beam and accelerator configurations across feasible simulations.

A DATASET GENERATION

  • The two WarpX-generated benchmarks cover bunched-beam transport in the electrostatic Vlasov-Poisson regime and PWFA in the fully electromagnetic Vlasov-Maxwell regime.Together they span different physical scales, geometries, control variables, state representations, and field treatments.Source: A DATASET GENERATION · p. 17

A.1 SPACE-CHARGE-DOMINATED BUNCHED-BEAM TRANSPORT

  • The bunched-beam benchmark models intense linac injection but is not a cell-by-cell digital twin of the LANSCE lattice.It retains strong space charge, alternating-gradient transverse focusing, longitudinal RF confinement, and controlled mismatch and centroid perturbations.Source: §A.1 · p. 17
  • At 750 keV, the beam is modeled in an idealized periodic focusing channel under electrostatic Vlasov-Poisson dynamics, with periodic longitudinal boundaries and a grounded cylindrical aperture.The learned state is charge density alone; the electrostatic self-field is reconstructed from it, while applied external fields are prescribed by machine configuration.Source: §A.1 · p. 17
  • Configurations vary five dimensionless beam-dynamics quantities, including longitudinal focusing strength, transverse tune depression, initial mismatch, and horizontal beam offset; RF phase is fixed at −30°.Source: §A.1 · p. 17
  • Machine settings are supplied as controls, whereas initial mismatch and centroid offset define the beam condition rather than additional controls; feasibility and stability checks filter configurations before simulation.Optics inversion maps design variables to WarpX inputs including quadrupole strength, RF-field amplitude, total bunch charge, and matched beam dimensions.Source: §A.1 · p. 17
  • The dataset contains 464 simulations after removing ten runs with over 20% beam loss; the train, validation, and test sets contain 364, 48, and 52 simulations.Trajectories have 401 charge-density frames on a 256×336×336 grid, with about 900 GPU hours total dataset cost.Source: §A.1 · pp. 17–18

A.2 VLASOV-MAXWELL DATASET: PLASMA-WAKEFIELD ACCELERATION (PWFA)

Together, these parts define the PWFA benchmark and describe a latent pipeline for compressing PIC states, predicting control-conditioned dynamics, and generatively modeling residuals, with emphasis on multiscale reconstruction and residual amplitude variation. They detail the generative residual model, physics-constrained electromagnetic projection, and architectures and training setups used for comparison.

A.2 VLASOV-MAXWELL DATASET: PLASMA-WAKEFIELD ACCELERATION (PWFA)

  • The PWFA dataset contains 456 physically feasible configurations and 504 simulations, including controlled perturbation, driver-only, extreme-regime, and repeated-seed subsets.The benchmark models self-consistent driver–witness beam and plasma evolution in the Vlasov–Maxwell regime; it uses 399 training, 49 validation, and 56 test simulations.Source: §A.2 · pp. 18–19

B.1 COMPRESSIVE AUTOENCODER

  • The PWFA autoencoder compresses 10-channel states by 16×, using a 3D Haar transform and reconstruction losses that target field values, spatial gradients, and intermediate-frequency structure.Latent perturbation training also improves decoder robustness to prediction errors during autoregressive rollout.Source: §B.1 · pp. 20–22

B.2 DETERMINISTIC LATENT NEURAL OPERATOR.

  • A full-attention Transformer predicts multiple latent steps jointly, using spatial Fourier features and adaLN-Zero to condition its dynamics on machine controls.Zero-initialized residual gates make each block begin as an identity mapping and learn control-dependent updates.Source: §B.2 · p. 23

B.3 GENERATIVE LATENT RESIDUAL.

  • Amplitude normalization scales residuals across channels, forecast blocks, and spatial locations before conditional flow matching, then restores their predicted amplitudes at inference.The conditional scale is predicted from information available at inference because the future residual is unknown at deployment.Source: §B.3 · pp. 24–25
  • A two-sample energy-score regularizer complements flow matching, while 16-step Euler integration generates normalized residuals that are rescaled and added to the latent forecast.The regularizer encourages agreement with the target while discouraging an under-dispersed conditional ensemble.Source: §B.3 · p. 25
  • The deterministic operator and generative residual model share a Transformer backbone, with GLOBE-C using 12 blocks of width 512 and 10 residual blocks of width 384.Both use eight attention heads and an MLP ratio of 4.Source: §B.3 · p. 25, Figure 5 · p. 25

B.4 PHYSICS-CONSTRAINED OPERATOR

  • The Vlasov–Maxwell projection corrects charge density and electric fields jointly, enforcing Gauss’s law in the interior while leaving the boundary shell unconstrained.At zero frequency, the correction acts through charge density, avoiding a forced charge-neutrality condition.Source: §B.4 · pp. 27–28

C.1 DENOISING OPERATOR TRANSFORMER (DPOT)

  • DPOT uses patch embeddings, temporal aggregation, and Fourier-attention blocks, and is adapted at matched model size to predict h-step futures from k-step histories.It is trained with a relative-L2 objective.Source: §C.1 · p. 29

C.2 UNIVERSAL PHYSICS TRANSFORMER (UPT)

  • UPT encodes states into latent tokens, propagates them with a Transformer, and decodes queried locations; training predicts one future state and evaluates loss on sampled grid points.For PWFA, 163296 query points are sampled for loss evaluation.Source: §C.2 · p. 29

C.3 MULTIPLE PHYSICS PRETRAINING (MPP)

  • MPP uses an Axial Vision Transformer with factorized temporal and spatial attention, and predicts only the next state from k-step histories.Its backbone is instantiated at a size comparable to GLOBE.Source: §C.3 · p. 29

C.4 P3D

  • P3D combines convolutional feature extraction with windowed attention in a hierarchical 3D encoder–decoder, evaluated as deterministic and generative baselines.P3D-G applies conditional flow matching directly to full 3D states; Table 5 describes an amplitude-normalization ablation for GLOBE-C, with GLOBE-D shown for reference.Source: §C.4 · pp. 29–30, Table 5 · p. 31

D.1 AMPLITUDE-NORMALIZATION ABLATION

Amplitude normalization improves generative refinement by balancing residual scales across locations and forecast dimensions. Removing window- and token-level scaling sharply worsens spectral error on bunched-beam transport and multiple errors on PWFA.

D.1 AMPLITUDE-NORMALIZATION ABLATION

  • On bunched-beam transport, σr-only scaling increases mid-band spectral error from 0.457 to 0.894, near the deterministic GLOBE-D value of 0.994.Without window- and token-level scaling, larger-amplitude residuals dominate the objective and weaker fine-scale variations near the bunch boundary receive insufficient weight.Source: §D.1 · p. 31
  • On PWFA, σr-only scaling raises Rel-L2 from 0.194 to 0.266 and low/mid/high spectral errors from 0.077/0.300/0.618 to 0.231/4.32/6.66.Factorized normalization brings residual targets to comparable, approximately unit-RMS scales across channels, windows, and tokens, improving flow-matching conditioning.Source: §D.1 · p. 31
  • Across both benchmarks, factorized normalization supports stable and effective generative refinement, whereas σr-only scaling limits refinement over deterministic prediction.On PWFA, residual magnitudes vary by orders of magnitude across field channels, forecast windows, and spatial locations.Source: §D.1 · p. 31

D.2 SPACE-CHARGE-DOMINATED BUNCHED-BEAM TRANSPORT

Representative bunched-beam rollouts span mismatches, charge levels, focusing strengths, and space-charge regimes. GLOBE tracks bulk envelope evolution while generative refinement restores mid-band density texture that deterministic predictions smooth or suppress.

D.2 SPACE-CHARGE-DOMINATED BUNCHED-BEAM TRANSPORT

  • The trajectory views combine transverse projections and beam moments, longitudinal projections, and y = 0 slices; the mid-band display isolates 0.168 < |k| ≤0.246 cycles/cell.The band-pass view highlights scales where deterministic regression smooths the PIC realization and generative refinement recovers structure.Source: §D.2 · p. 31
  • Run 392’s 20% initial envelope mismatch produces sustained transverse oscillations, with σx increasing about 42% and σz contracting about 5%; refinement restores mid-band texture.The deterministic forecast captures bulk beam shape and envelope evolution but smooths realization-dependent structure.Source: §D.2 · p. 31, Figure 6 · p. 32, Figure 7 · p. 32
  • Run 244 remains well confined while its transverse aspect ratio changes from about 2.3 to 0.5; generative residuals restore fluctuations similar in extent and amplitude to the reference.The deterministic forecast recovers the smooth envelope but largely suppresses mid-band PIC texture.Source: §D.2 · pp. 31–32, Figure 9 · p. 33, Figure 10 · p. 34
  • Run 389’s nearly matched beam grows transversely roughly 5%; GLOBE reproduces recurring shape changes and nearly stationary longitudinal extent, while refinement recovers missing mid-band texture.Its deterministic density field is visibly smoother than the WarpX realization despite accurate macroscopic evolution.Source: §D.2 · p. 32
  • Run 87, the highest-charge and weakest-RF-confinement case, has σz ≈4.5 mm; GLOBE tracks its oscillations and longitudinal extent while refinement restores mid-band structure.Projected densities remain accurate throughout the rollout.Source: §D.2 · p. 33
  • Run 343’s strong space charge drives about 35% transverse envelope growth and 8.5% longitudinal compression; GLOBE captures the overall changes, though deviations accumulate later.Density errors are larger and more spatially organized than in preceding cases, yet generative residuals recover a substantial fraction of suppressed mid-band structure.Source: §D.2 · pp. 33–34

D.3 PWFA VISUALIZATION

The PWFA visualizations compare wake evolution, trajectory-level observables, representative field predictions, and relative-error maps across operating regimes. Additional bunched-beam views contrast smooth bulk-density forecasts with generative recovery of realization-dependent structure.

PWFA visualization

  • The PWFA diagnostics pair wake snapshots and trajectory observables with field comparisons and channel-wise error maps across quasi-linear to strongly nonlinear regimes.The representative field quantities include Ez, Ex, cBy, and Wx = Ex − cBy.Source: §D.3 · p. 34

Bunched-beam visualizations

  • Bunched-beam visualizations show ground-truth and predicted transverse and longitudinal evolution, while generative refinement recovers realization-dependent mid-band structure beyond the smooth bulk forecast.The views include beam-size evolution and density projections for selected runs.Source: Figure 11 · p. 35, Figure 12 · p. 35, Figure 13 · p. 36

PWFA visualization

  • Whole-field Rel-L2 errors over Run 312 are 0.129/0.132/0.144 for deterministic, ensemble-mean, and sampled predictions, respectively.The visual diagnostics also compare wake evolution, trajectory observables, field predictions, and channel-wise relative errors.Source: §D.3 · pp. 34–35, Figure 21 · p. 41, Figure 22 · p. 42
  • Run 355 combines an overloaded witness (ηload = 1.23) with about 14% driver depletion; its largest discrepancies occur near the rear cavity and witness.The predicted fields preserve phase and spatial organization, while the error maps localize the remaining discrepancies where loading most reshapes the wake.Source: §D.3 · pp. 35–36, Figure 24 · p. 42, Figure 25 · p. 43
  • Run 271’s high-charge, underloaded case maintains the dominant nearly axisymmetric cavity structure, with relative errors concentrated mainly near the driver and rear sheath.The selected case has the strongest peak field; the visualizations compare wake evolution, representative fields, and relative-error maps.Source: §D.3 · p. 36, Figure 27 · p. 43, Figure 28 · p. 44
  • Run 27’s high-energy, lightly loaded wake and Run 118’s quasi-linear wake both retain aligned field perturbations through rollout, despite their distinct regimes.Run 118 lacks a well-developed blowout cavity; its 7 pC witness is at the diagnostic floor, so transverse-size error is not reported.Source: §D.3 · pp. 36–37, Figure 30 · p. 44, Figure 33 · p. 45

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