cs.CVOct 3, 2026

A differentiable Lagrangian-coupled 3D Gaussian Splatting-SPH model for forward simulation and inverse analysis in solid mechanics

Authors: Tian Xu, Soroush Atashi, Tianju Xue

Organizations: Department of Civil and Environmental Engineering, Hong Kong University of Science and Technology

Read PDF
Ask AI
Cite

Copy citation

DOINot available for this paper

Abstract

Recent advances in generative world models have increased interest in digital models that reproduce both the appearance of real objects and their response to physical interaction. Three-dimensional reconstruction techniques, including 3D Gaussian Splatting, capture detailed surface geometry and appearance from images and videos. However, extending these representations beyond plausible animation to mechanically interpretable models for constitutive behavior, boundary conditions, and inverse parameter identification remains less explored. In this work, a differentiable Lagrangian-coupled 3DGS-smoothed particle hydrodynamics (SPH) model is proposed for forward simulation and inverse analysis of deformable solids. The observed object is first reconstructed from multi-view calibrated visual dataset as a 3DGS rendering model. An envelope-based procedure then generates an independent SPH support for the solid-mechanics model, avoiding the direct use of rendering primitives as mechanical particles. A reference-configuration Lagrangian transfer maps SPH deformation to Gaussian positions and covariances, thereby coupling the physical model and the image observation model while preserving a differentiable computational path. The SPH formulation supports linear elastic, hyperelastic, and Kelvin--Voigt viscoelastic responses, together with fixed, free, and Robin-type boundary conditions. Numerical studies validate the SPH response against finite-element results, assess accuracy and efficiency against a conventional model using Gaussian centers as surface SPH particles, and demonstrate forward simulations on beam, bridge, and liver-shaped examples. Inverse analyses further estimate constitutive and boundary parameters from rendered deformation observations, including noisy cases, demonstrating the feasibility of the proposed model for mechanics-based parameter identification from image data.

Figures & tables

Figure 1: Workflow of the proposed Lagrangian-coupled 3DGS-SPH model for forward simulation and inverse analysis. (a) Reference multiview images; (b) prescribed scene configuration; (c) constitutive and boundary parameters; (d) 3DGS rendering model and SPH physical model coupled through the proposed Lagrangian transfer; (e) simulated deformation frames; (f) inverse analysis using the discrepancy between simulated and observed frames; and (g) observed deformation frames. Brown arrows indicate model construction, red arrows indicate forward simulation, and dashed blue arrows indicate inverse analysis, including gradient propagation through the coupled model for parameter estimation.
Figure 2: Envelope-based generation of the SPH support from a reconstructed Gaussian model. Expanded Gaussian ellipsoids first define a finite-thickness shell. After gap closing and interior filling, occupied grid-cell centers are retained as surface and interior SPH supports.
Figure 3: Schematic support relation in the total-Lagrangian SPH discretization. The approximation at a reference support uses neighboring supports within the compact kernel radius.
Figure 4: Reference-configuration support relation for the Lagrangian SPH-GS transfer. Each Gaussian center selects neighboring SPH supports within the transfer kernel radius, forming a fixed bipartite graph for deformation transfer.
Figure 5: Schematic setting for inverse analysis. The observed image sequence is used to identify selected physical parameters, including elastic modulus, Robin boundary stiffness, and, for dynamic viscoelastic cases, Kelvin–Voigt viscosity parameters.
GeometrySim. typeMaterial modelB.C. setting
BeamQuasi-staticCorotated linearRobin: weak, strong, heterogeneous
BeamQuasi-staticneo-HookeanRobin: weak, heterogeneous
BeamDynamicCorotated linear + Kelvin–VoigtWeak Robin
BeamDynamicneo-Hookean + Kelvin–VoigtWeak Robin
BridgeQuasi-staticCorotated linearFixed lower band and central top load
Liver-shapedDynamicCorotated linear + Kelvin–VoigtFixed lower region, Robin region, and end load
Table 1: Design of the forward mechanics experiments.
Figure 6: Multiview reconstruction of the undeformed speckle-textured beam used in the forward simulations. The upper row shows four reference images from Dr\mathfrak{D}_{\mathrm{r}} , and the lower row shows the corresponding renderings generated by the reconstructed 10,000-kernel 3DGS model.
Figure 7: Quasi-static corotated-linear beam response under different Robin support fields. Subfigure (a) show the boundary condition settings of the beam case. Subfigures (b)–(d) compare weak, strong, and heterogeneous Robin support, respectively. The upper row shows the rendering images of final deformation, and the lower row shows the corresponding SPH particle deformation colored by downward displacement w=−uzw=-u_{z} .
Elastic modelRobin supportkRk_{R} ( s−2\mathrm{s^{-2}} )Final wloadw_{\mathrm{load}} (m)Final wRw_{R} (m)
corotated linearweak10−210^{-2}0.52440.1478
corotated linearstrong10410^{4}0.042653.21×10−53.21\times 10^{-5}
corotated linearheterogeneous104,10210^{4},10^{2}0.081045.03×10−35.03\times 10^{-3}
neo-Hookeanheterogeneous104,10210^{4},10^{2}0.081015.03×10−35.03\times 10^{-3}
neo-Hookeanweak10−210^{-2}0.52450.1475
Table 2: Quasi-static forward response for the beam. Downward displacement is w=−uzw=-u_{z} .
Figure 8: Dynamic bending response of the beam for different viscous coefficients. The gray curve is the numerically damped quasi-static reference; CL denotes the corotated-linear model and KV denotes Kelvin–Voigt viscosity. Increasing ηs=ηb\eta_{s}=\eta_{b} reduces the oscillation amplitude.
Elastic modelViscous modelFinal wloadw_{\mathrm{load}} (m)Peak wloadw_{\mathrm{load}} (m)Final max⁡∥u∥\max\|\bm{u}\| (m)time (s)
corotated linearnone0.91881.03690.991611.15
corotated linearKV, ηs=ηb=1000 Pa⋅s\eta_{s}=\eta_{b}=1000\,\mathrm{Pa}\cdot\mathrm{s}0.77010.96770.818637.83
corotated linearKV, ηs=ηb=5000 Pa⋅s\eta_{s}=\eta_{b}=5000\,\mathrm{Pa}\cdot\mathrm{s}0.55400.76020.579471.11
corotated linearKV, ηs=ηb=10000 Pa⋅s\eta_{s}=\eta_{b}=10000\,\mathrm{Pa}\cdot\mathrm{s}0.52670.61730.550192.68
neo-HookeanKV, ηs=ηb=5000 Pa⋅s\eta_{s}=\eta_{b}=5000\,\mathrm{Pa}\cdot\mathrm{s}0.55480.76280.580472.55
Table 3: Dynamic forward response with weak Robin support.
Figure 9: Multiview reconstruction of the undeformed bridge (a–d) and liver-shaped object (e–h). For each object, the upper row shows reference images from its dataset Dr\mathfrak{D}_{\mathrm{r}} , and the lower row shows the corresponding renderings generated by the reconstructed 3DGS model.
GeometryEE (Pa)ν\nuρ0\rho_{0} ( kg⋅m−3\mathrm{kg}\cdot\mathrm{m^{-3}} )ηs=ηb\eta_{s}=\eta_{b} ( Pa⋅s\mathrm{Pa}\cdot\mathrm{s} )Load (N)
Bridge10510^{5}0.2024000500
Liver-shaped10410^{4}0.45106030008
Table 4: Material parameters and loading for the bridge and liver-shaped forward simulations.
Figure 10: Boundary configurations for the bridge and liver-shaped forward simulations. The bridge case uses a lower fixed band and a central loaded patch on the top layer. The liver-shaped case uses surface supports for the lower fixed region, the adjacent Robin region, and the loaded end region.
Figure 11: Forward deformation responses for the bridge and liver-shaped examples. Initial and deformed Gaussian renderings are shown together with the corresponding mean downward displacement histories. (a) The bridge case illustrates a quasi-static elastic response on a multi-member structure. (b) The liver-shaped case illustrates a transient viscoelastic response with both loaded-region and Robin-region displacement curves.
GeometrySPH particlesGaussiansFinal wloadw_{\mathrm{load}} (m)Peak wloadw_{\mathrm{load}} (m)time (s)
Bridge2381113100.051160.0639232.85
Liver-shaped2677100000.114340.19272123.28
Table 5: Forward simulation results for the bridge and liver-shaped cases. Final and peak wloadw_{\mathrm{load}} denote the mean loaded-region displacement at t=20 st=20\,\mathrm{s} and its maximum over the simulated interval.
GeometryUnknown parametersMaterial model
BeamEEcorotated linear
BridgeEEneo-Hookean
BeamE,ηs,ηbE,\eta_{s},\eta_{b}corotated linear + Kelvin–Voigt
BeamkRk_{R}corotated linear
BeamE,kRE,k_{R}corotated linear / neo-Hookean
Liver-shapedE,kRE,k_{R} , then ηs,ηb\eta_{s},\eta_{b}neo-Hookean + Kelvin–Voigt
Table 6: Design of the inverse-identification experiments.
Figure 12: Iteration histories for constitutive identification under prescribed boundary conditions. (a) Image-data mismatch objective Jo\mathcal{J}_{\mathrm{o}} normalized by its initial value. (b) Elastic modulus relative to its reference. (c) Recovered Kelvin–Voigt coefficients in the five-frame beam test. Horizontal dotted lines denote the reference values.
Figure 13: Forward-evaluated states for constitutive identification. Columns show the initial estimate, recovered state, and synthetic reference. Rows correspond to the corotated linear beam, neo-Hookean bridge, and the last observed frame of the Kelvin–Voigt beam.
Figure 14: Joint identification of material parameters and boundary conditions for the beam. (a,c) Normalized image-data mismatch for the corotated-linear and neo-Hookean models. (b,d) Corresponding Young’s modulus estimates normalized by the reference value. Particlewise and RBF boundary representations use the same scene configuration. The corotated-linear particlewise history reports best-so-far evaluations.
Figure 15: Final rendered states for joint identification of material parameters and boundary conditions for the beam. Columns show initialization, recovery, and reference. Rows compare particlewise and RBF Robin fields under corotated linear and neo-Hookean elasticity.
Figure 16: Joint identification of material parameters and boundary conditions for the beam using an image with 1% salt-and-pepper noise. (a,b) Clean reference and noisy observation. (c,d) Simulated images with the initial and recovered parameters. (e) Image MSE between the simulated image and noisy observation during optimization; the dashed line denotes the MSE between the clean reference and noisy observation. (f) Elastic modulus estimates and reference value. (g,h) Reference, initial, and recovered Robin stiffness and final vertical displacement over the Robin region. Particles in (g,h) are ordered by increasing reference xx -coordinate from the fixed-end side to the loaded-end side.
Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 1: Reference validation of the solid SPH discretization. Panels (a)–(d) show the final deformed configurations of the FEM reference, the corrected SPH model (ours), the corrected surface-dense SPH model, and the uncorrected SPH model, respectively. Panel (e) compares the section-averaged displacement magnitude along the normalized axial coordinate ξ=(Xx−Xmin⁡)/(Xmax⁡−Xmin⁡)\xi=(X_{x}-X_{\min})/(X_{\max}-X_{\min}) .
ModelSupports/nodesMaximum ∥u∥2\|\bm{u}\|_{2} (m)∥u∥2\|\bm{u}\|_{2} NRMSE
FEM reference4961 nodes0.7183–
Corrected SPH4000 supports0.73434.27%4.27\%
Surface-enriched corrected SPH5000 supports0.79008.16%8.16\%
Uncorrected SPH4000 supports0.954318.9%18.9\%
Appendix
Table 1: Comparison of the FEM reference and SPH models for the linear beam benchmark.
Figure 2: Physical discretizations and B.C. assignments used in the transfer experiment. (a) Co-located model, in which every surface support coincides with a Gaussian center and is therefore not overplotted. (b) Proposed Lagrangian-coupled model with an independently discretized SPH domain; Gaussian centers are omitted to expose its support distribution. Panels (c) and (d) show the fixed, weak Robin-supported, loaded, and free supports selected in the two models.
Figure 3: Rendering and Gaussian-displacement agreement between the co-located construction and the proposed 3DGS–SPH model with Lagrangian SPH–GS transfer. The comparison uses matched deformation amplitudes. (a) Co-located model. (b) Proposed model. (c) Pixelwise mean absolute RGB difference between (a) and (b), with the color scale truncated at the 99.5th percentile. (d) Global relative L2L_{2} Gaussian-displacement error ε2(t)\varepsilon_{2}(t) of the proposed model over simulation time. (e) Pointwise normalized displacement errors ege_{g} at five representative states. Both displacement-error measures use the co-located response as the baseline; boxes denote the interquartile range, center lines the median, and whiskers the 5th and 95th percentiles.
Parameter (unit)Material/B.C. fieldEvaluationReferenceDim.Relative error
EE (Pa)corotated linear50001000014.6788×10−24.6788\times 10^{-2}
EE (Pa)neo-Hookean50001000016.3272×10−36.3272\times 10^{-3}
kRk_{R} ( s−2\mathrm{s^{-2}} )Robin stiffness field10001001057.4073×10−37.4073\times 10^{-3}
(ηs,ηb)(\eta_{s},\eta_{b}) ( Pa⋅s\mathrm{Pa}\cdot\mathrm{s} )Kelvin–Voigt1000500025.0778×10−35.0778\times 10^{-3}
Appendix
Table 2: AD–FD comparison for the image-mismatch term Jo\mathcal{J}_{\mathrm{o}} in Eq. ( 24 ). Evaluation values specify where derivatives are computed; reference values generate the reference images. Dim. denotes the parameter dimension.
Figure 4: Automatic differentiation validation for the beam 3DGS–SPH coupled pipeline. Relative errors between AD and FD directional derivatives of Jo\mathcal{J}_{\mathrm{o}} are evaluated through the complete SPH rollout, SPH–GS transfer, covariance update, and differentiable rendering path. Here EC-LE_{\mathrm{C\text{-}L}} and EN-HE_{\mathrm{N\text{-}H}} denote Young’s-modulus checks for the corotated-linear and neo-Hookean material models, respectively.
SymbolDefinition
g=1,…,Gg=1,\ldots,GGaussian index and number of rendered Gaussians.
i,j=1,…,Ni,j=1,\ldots,NSPH support indices and number of physical supports.
μg0,μgn\bm{\mu}_{g}^{0},\bm{\mu}_{g}^{n}Reference and current Gaussian centers.
Σg0,Σgn\bm{\Sigma}_{g}^{0},\bm{\Sigma}_{g}^{n}Reference and current Gaussian covariances.
Xi,xin,vin\bm{X}_{i},\bm{x}_{i}^{n},\bm{v}_{i}^{n}Reference position, current position, and velocity of SPH support ii .
mi,ΔVi,ρ0m_{i},\Delta V_{i},\rho_{0}Support mass, reference volume, and reference density.
Appendix
Table 3: Principal notation used in the coupled formulation.

Explore similar work

CardsList
Aug 24, 2026cs.CV

LagrangeGS: Non-Conservative Lagrangian System on Dynamic 3D Gaussian Splatting

Dynamic 3D Gaussian Splatting (3DGS) achieves photorealistic reconstruction of time-varying scenes, and recent physics-aware extensions improve extrapolation by explicitly predicting velocity fields. However, these extensions merely fit vector fields to visual deformations without satisfying Lagrangian mechanics, leading to three major issues: (i) physically inconsistent trajectories, (ii) lack of time-reversibility, and (iii) geometric collapse during long-term extrapolation. In this paper, we propose LagrangeGS, which formulates dynamic 3DGS as a non-conservative Lagrangian system. While this Lagrangian formulation fundamentally solves (i), a direct application of general LNNs to dynamic 3DGS requires a large velocity-Hessian inversion for millions of Gaussian particles. To overcome this computational bottleneck, we approximate the velocity-Hessian as an identity matrix, decoupling particle dynamics for computational tractability. For (ii), we restrict the non-conservative forces to be explicitly time independent, enabling consistent backward integration. Finally, to address (iii), we introduce local rigid alignment that regularizes particle trajectories. Extensive evaluations on dynamic scene benchmarks demonstrate that LagrangeGS enables stable long-term extrapolation, consistent time reversal, and counterfactual physics-based editing without retraining.

Shogo Sato, Takuhiro Kaneko, Shoichiro Takeda +4

Human Informatics Laboratories NTT, Inc. Kanagawa, Japan

3D Gaussian SplattingDynamic Scene ReconstructionDynamical Systems
AskPDF
May 11, 2026cs.CV

PG-3DGS: Optimizing 3D Gaussian Splatting to Satisfy Physics Objectives

Recent advances in Gaussian Splatting have enabled fast, high-fidelity 3D scene generation, yet these methods remain purely visual and lack an understanding of how shapes behave in the physical world. We introduce Physics-Guided 3D Gaussian Splatting (PG-3DGS), a framework that couples differentiable physics simulation with 3D Gaussian representations to generate 3D structures satisfying physics functionalities. By allowing physical objectives to guide the shape optimization process alongside visual losses, our approach produces geometries that are not only photometrically accurate but also physically functional. The model learns to adjust shapes so that the generated objects exhibit physically meaningful behaviors, for example, teapots that can pour and airplanes that can generate lift, without sacrificing visual quality. Experiments on pouring and aerodynamic lift tasks show that PG-3DGS improves physical functionality while preserving visual quality. In addition to simulation gains, bench-top physical lift tests with 3D-printed aircraft (Cessna, B-2 Spirit, and paper plane) under identical airflow conditions show higher scale-measured lift for PG-3DGS, generated structures than an appearance-matching baseline in all three cases. Our unified framework connects appearance-based reconstruction with physics-based reasoning, enabling end-to-end generation of 3D structures that both look realistic and function correctly.

Zachary Lee, Maxwell Jacobson, Yexiang Xue

Department of Computer Science, Purdue University, West Lafayette, 47907, IN, United States

3D Gaussian Splatting3D Shape GenerationFunctional 3D Scene Generation
AskPDF
Jul 28, 2026cs.CV

Spline-Based Boundary Representations for Sparse View Reconstruction and Simulation Using Isogeometric Analysis

Image-based reconstruction aims to recover three-dimensional geometry from images. Recent advances have enabled the recovery of visually detailed models, yet their representations are not well-suited for numerical simulation. Simulation frameworks typically require explicit, watertight, and smooth geometries to ensure numerical robustness and accuracy, properties that surfaces extracted from image-based reconstructions lack. We propose FORGE-SIM, a method to directly reconstruct a multi-patch B-spline boundary representation from sparse posed RGB images without manual intervention. By optimizing the spline representation itself, our approach produces compact, smooth, and watertight geometries that are natively compatible with both Computer Aided Design and simulation workflows. Additionally, we introduce a strategy to project observation-derived fields, such as a thermal state and semantic information, onto the reconstructed models in the same spline basis, enabling immediate use in simulation. We demonstrate that the obtained models are of sufficiently high quality to enable thermal simulation and modal analysis. By unifying image-based reconstruction and simulation-ready modeling within a single optimization framework, this work removes a long-standing barrier between computer vision and numerical analysis. We anticipate that it will enable new workflows for simulation-driven design, inspection, and digital twin applications.

Davor Dobrota, Vsevolod Skorokhodov, Chenghao Xu +2

Schindler Lab EPFL, Schindler AG, EPFL Innovation Park, Lausanne, 1015, Vaud, Switzerland. · Chair of Numerical Modelling and Simulation, ´Ecole polytechnique f´ed´erale de Lausanne, Station 8, Lausanne, 1015, Vaud, Switzerland. · Intelligent Maintenance and Operating Systems Lab, ´Ecole polytechnique f´ed´erale de Lausanne, Station 18, Lausanne, 1015, Vaud, Switzerland.

3D ReconstructionParametric CAD ModelingSparse-View 3D Reconstruction
AskPDF