Fracture-GS: Dynamic Fracture Simulation with Physics-Integrated Gaussian Splatting
Authors: Xiaogang Wang
Hongyu Wu
Wenfeng Song
Kai Xu
Venue: ICLR 2026 Poster (2026)
Keywords: 3D vision
Physics-based Simulation
Primary area: applications to robotics, autonomy, planning

Abstract: This paper presents a unified framework for simulating and visualizing dynamic fracture phenomena in extreme mechanical collisions using multi-view image inputs. While existing methods primarily address elastic deformations at contact surfaces, they fail to capture the complex physics of extreme collisions, often producing non-physical artifacts and material adhesion at fracture interfaces. Our approach integrates two key innovations: (1) an enhanced Collision Material Point Method (Collision-MPM) with momentum-conserving interface forces derived from normalized mass distributions, which effectively eliminates unphysical adhesion in fractured solids; and (2) a fracture-aware 3D Gaussian continuum representation that enables physically plausible rendering without post-processing. The framework operates through three main stages: First, performing implicit reconstruction of collision objects from multi-view images while sampling both surface and internal particles and simultaneously learning surface particle Gaussian properties via splatting; Second, high-fidelity collision resolution using our improved Collision-MPM formulation; Third, dynamic fracture tracking with Gaussian attribute optimization for fracture surfaces rendering. Through comprehensive testing, our framework demonstrates significant improvements over existing methods in handling diverse scenarios, including homogeneous materials, heterogeneous composites, and complex multi-body collisions. The results confirm superior physical accuracy, while maintaining computational efficiency for rendering.

Published as a conference paper at ICLR 2026
FRACTURE-GS: DYNAMIC FRACTURE SIMULATION WITH PHYSICS-INTEGRATED GAUSSIAN SPLATTING
Xiaogang Wang*"", Hongyu W Wenfeng Song", Kai Xu
College of Computer and Information Science, Southwest University
State Key Laboratory of Virtual Reality Technology and Systems, QRI & SCSE, Beihang University
"School of Computer, Beijing Information Science and Technology University
*Institute of Al for Industries, Chinese Academy of Sciences
ABSTRACT
This paper presents a unified framework for simulating and visualizing dynamic fracture phenomena in extreme mechanical collisions using multi-view image in- puts. While existing methods primarily address elastic deformations at contact surfaces, they fail to capture the complex physics of extreme collisions, often producing non-physical artifacts and material adhesion at fracture interfaces. Our approach integrates two key innovations: (1) an enhanced Collision Material Point Method (Collision-MPM) with nomentum-connervvivn
interface forces derived from normalized mass distributions, which effectively eliminates unphysical ad- hesion in fractured solids; and (2) fracture-aware 3D Gaussian continuum repre- sentation that enables physically plausible rendering without post-procsssing. The framework operates through three main stages: First, performing implicit recon- struction of collision objects from multi-view images while sampling both surface and internal particles and simultaneously learning surface particle Gaussian prop- erties via splatting: Second, high-fidelity collision resolution using improved Collision-MPHN formulation; Third, dynamic fracture tracking with Gaussian at- tribute optimization for fracture surfaces rendering. Through comprehensive test- ing, our framework demonstrates significant improvements over existing methods in handling diverse scenarios, including homogeneous materials, heterogeneous composites, and complex multi-body collisions. The results confirm superior physical accuracy, while maintaining computational efficiency for rendering.
INTRODUCTION
Dynamic fracture simulation stands at the intersection of computational physics and computer graphics, enabling realistic modeling of material failure across diverse applications. While physics- based methods like the Material Point Method (MPM) have advanced significantly since their in- troduction (Stomakhin al. 2013), critical gaps remain in handling extreme mechanical collisions rendering pipelines.
and achieving seamless simulation-to- to-ddeitt----ddriinng
Recent advances in explicit scene representation, particularly 3D Gaussian splatting (Kerbl et al, 2023 have revolutionized real-time rendering capabilities. Building upon this foundation, several studies have successfully integrated physical simulation with Gaussian representations, including (Borycki et al. [2024] Cai al. (2024] Feng al., 2024 Xie al.) 2024 Zhang et al. 2205). Among these, (Xie et al. (0224) established a significant milestone by coupling 3D Gaussians with MPM simulations, demonstrating remarkable adaptability across various material types. Parallel develop- ments include (Cai et al. 2024)'s 's phhysics parameter estimation through Gaussian differentiability and (Borycki al.) (024)) GASP framework for point-wise physical attribute embedding. While these approaches have made substantial progress in bridging the simulation-rendering gap. their applicability remains constrained to moderate mechanical conditions and specific material classes. A critical limitation emerges when addressing extreme mechanical collisions, such as the high- energy fragmentation observed in brittle materials (Wolper et al. 2019). Current methodologies face two fundamental challenges in these scenarios: (I) unphysical adhesion artifacts in MPM sim- ulations, and (2) inadequate fracture surface representation for rendering. These challenges become
"Corresponding authors: wangxi aogggiiiggggggsssw. www.sweeeddu.... edu..h, abybua. edu.cn

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Published as a conference paper at ICLR 2026
Single Object
Multi-Objeet
Figure I: Two extreme mechanical collision scenarios are simulated using our proposed framework: (1) single-object impacts against wind, and (2) multi-object collision interactions with complex fracture dynamics. Notably, the flowerpot object comprises heterogeneous materials (with distinct properties for leaves, stems, and the pot itself), and the table object consists of legs and tabletop, while other teapot is modeled as homogeneous materials.
particularly apparent in high-energy impact scenarios like brittle material fragmentation or multi- body collisions.
To address these limitations, we introduce Fracture-GS, a unified framework for simulating and vi- sualizing dynamic fracture phenomena in extreme mechanical collisions using multi-view image inputs. First, the Signed Distance Function (SDF) of the object is constructed from multi-view im- ages to implicitly represent the volumetric geometry of the object, followed by sampling both surface and internal particles within the SDF-constrained domain to ensure spatial coherence; Meanwhile, surface particles learn Gaussian attributes using isotropic kernels. Subsequently, we proposed an enhanced collision-MPP with dynamic fracttre. It can effectively alleviate the non- non-physical adhesion phenomenon caused Then, all fracture particles are tracked through a hardening- aware tracking criterion defined by (Wolper et al.] (019). Finally, based the tracking fracture particles, regenerate their Gaussian attributes through the proposed fracture particles Gaussian optimization strategy, enabling high-quality rendering, shown in Figure In summary, key contributions are:
which used to perform extreme collisions between multiple objects
by
efffciently
A unified physics-rendering framework that combines our enhanced Collision-MPM with fracture-aware 3D Gaussian representations, enabling high-fidelity simulation and visual- ization of extreme mechanical collisions across diverse materials.
An improved Collision-MPM formulation that introduces momentum--cnsserving interface forces derived from normalized mass distributions, effectively eliminating the non-physical adhesion artifacts prevalent in conventional MPM simulations.
A dynamic fracture tracking and rendering pipeline that identifies fracture surfaces through hardening parameter and reconstructs Gaussian attributes via minimal-volume enclosing ellipsoid (MVEE) optimization.
2 PRELIMINARY AND RELATED WORK
2.1 3D GAUSSIAN SPLATTING AND DYNAMIC SCENE RECONSTRUCTION
3D Gaussian Splatting (3DGS) a fast neural rendering method that primarily represents volumetric scenes using collection of anisotropic 3D Gaussian kernels. Each Gaussian kernel is defined by set of differentiable parameters (Ip; pp Ap: Cp), where xp denotes the spatial position, dp represents opacity, Ap is the covariance matrix, and Cp is the view-dependent color function. The covariance matrix Ap can be further decomposed into scale s, and rotation тp components, which control the spatial distribution and orientation of the Gaussian kernel, respectively. The color function Cp achieves view iewedeeendeent characteristics through spherical harmonics, capturing complex lighting and material effects. The rendering process involves projecting (splatting) the 3D Gaussian kernels onto 2D image plane, incorporating viewpoint transformation, opacity blending, and depth sorting. The final color of the i-th pixel is computed using the following formula:
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R-1 C - G- KEI
Cack (d,)
II (1 (1-o)) oT) 2=1
(I)
Here, stands for the set of Gaussian kernels, 2% and jj represent the opacity of the k-th and j-th Gaussian kernels, respectively. d, means the viewing direction from the camera to the i-th pixel, and eu (d,) indicates the color of the Gaussian kernels in the viewing direction d,, Since 3D Gaussian explicitly represents the scene, its learning and rendering speeds are generally faster than those of 3D reconstruction methods based on NeRE, As result, a wide range of applications based on 3D Gaussian have emerged.
Dynamic 3D scene reconstruction has long been challenging problem, aiming reconstruct dy- namic scenes from various representations such as videos and images. The introduction of Neural Radiance Fields (NeRF) has significantly advanced this field, leading to series of dynamic scene reconstruction methods that build upon NeRF's framework. These methods focus on addressing challenges such as non-ideal input conditions, including sparse views and motion blur, enhance reconstruction quality. Notable works in this domain include (Pumarola et al. 202T), which handles dynamic scenes by incorporating temporal information, and (Fang et al. (022). which improves ef- ficiency through time-aware neural voxels. More recently, (Song et al. 2023) proposed a streamable dynamic scene representation that decomposes neural radiance fields for efficient reconstruction and rendering.
Another approach to scene representation is based on 3D Gaussian Splats (3DGS), which explicitly represents 3D scenes using a set of Gaussian kernels and achieves fast rendering through Gaussian splatting. Due to its high efficiency and interpretabilitty, several dynamic scene reconstruction meth- ods based on 3DGS have been proposed, such as (Lin et al., [024 Huang et al., [024] [w et al.]
2024 Yang et al.) 2024 (unn et al. 2024 Zhang et al., 2024 Dahmani et al. 2024). Among these, (Lin et al. 2024) incorporates DDDM (Deformable Dynamie Model) into the optimization pro- cess of Gaussians, eliminating the need to reconstruct Gaussian for each frame and directly guiding Gaussian deformation using DDDM. On the other hand, (Huang et al., (2024) learns control points for Gaussian and uses small number of control points to govern the motion of the entire Gaussian set, achieving efficient reconstruction of dynamic motion processes. (Dahmani et al.) (024) divides the dynamic sequence into different windows based on the motion number, and train dynamic Gaus- sian models for different windows, together with different canonical spaces and deformation fields.
2.2 MATERIAL POINT METHOD AND PHYSICS- BASSED GAUSSIAN APPROACHES
The Material Point Method (MPM) is numerical approach based on a hybrid Eulerian Mulerian-Lagrangiann framework, used to solve governing equations in continuum mechanics and facilitate bidirectional information transfer between particles and grids. Letp E {Pa, denote the classification criterion for distinguishing particle subsets. Following established methodologies, the process adheres to the MLS-MPM framework (Hu et al. 2018). which is divided into the following three stages:
Particle-to-Grid (P2G) Stage: The particle mass m, and velocity Up are interpolated to neighboring grid nodes through a weighted projection scheme governed by basis functions:
m =
Σ w mps
(2)
m 10 = > Wh my (1)) + (o -()))
where wii represents the interpolation kernel (c.g., quadratic B-spline) evaluated at particle position Xp for grid node i, P is the set of active particles, and m; and e; are the aggregated mass and velocity at grid node i, respectively.
Grid Update Stage: Grid velocities are advanced by solving the discrete momentum con- servation equations through an explicit forward Euler integration scheme:
m (v
v!") -t ft
+ A fert.
(4)
f f:= Σ Vo
xz))
(5)
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where Ax denotes the grid size, f is the grid force calculated from the particle volume Vp, Cauchy stress op, and positions xi and zy, and fq is the external force (typically gravity).
Grid-to-Parte (G2P) Stage: The updated grid velocities oft are mapped back to La- grangian particles to update their kinematic states for the next time step + 1. This transfer is achieved through interpolation:
p
Σ
up
a+1
(6)
4
n+1
(2
p
Ar
After obtaining the updated particle velocities, the particle positions are advanced through an explicit time integration scheme:
n++ = + Atuptt
(8)
The Material Point Method (MPM) has emerged as a powerful tool for simulating complex physical phenomena, combining the advantages of both Lagrangian and Eulerian approaches. Recent ad- vancements in MPM simulations have significantly expanded its applications and performance opti- mization. (Hu et al. 2018) introduces a variant of the MPM based on Moving Least Squares (MLS), referred to as MLS -PMM, for simulating complex physical phenomena involving displacement dis- continuities and bidirectional rigid-body coupling. By incorporating the Compatible Particle-In- In-Cell (CPIC) algorithm, this method enables the handling of discontinuities in material points, infinitely thin boundaries, and bidirectional coupling with rigid bodies. As a result, it is capable of simulating material cutting, dynamic open boundaries, and interactions between rigid and deformable bodies. (Wolper et al. 2019) integrates a phase field into the MPM to develop a crack- sacckttacking approach known PFF-MPM Additionally, it proposes incompressible plastic flow rule that maintains constant volume during plastic stress projection. Meanwhile, (Fang et al. (020) proposed a novel framework for fluid-solid coupling using IQ-MPM. This method combines "ghost matrix" operator splitting scheme with weak-form governing equations to achieve stable and efficient coupling un- der the CFL time step constraint. supports discrete consistency with hybrid Lagrangian Eulerian solvers and uses an interface quadrature (IQ) technique to handle free-slip boundaries, avoiding the "stickiness"" issues traditional MPM implementations. Moreover, the effectiveness of employing GPUs to enhance the computational efficiency of MPM implementations has been well-documented in several studies, such as (Gao al.) 2018, Hu et al., 2019 Qiu al. 2023). However, these meth- ods all require post-processing of the simulation results to achieve high-quality rendering effects.
auuleiiannn
The integration of Gaussian-based techniques into physical simulations leverages the rendering ef- ficiency of Gaussian, thereby eliminating the need for post-processing with dedicated rendering engines such as Houdini after the completion of the physical simulation, and directly yielding re- sults with Gaussian splatting. Specifically, (Xie et al. 2024) incorporates Gaussian kernels into the dynamic simulation process of MPM, enabling continuum mechanics simulations based on 3D Gaussian kernels and achieving real-time rendering in simple scene simulations. (Zhang et al. (025) uses diffusion images to obtain prior motion videos of objects, simulates the motion process based on MPM, leverages the differentiability of MPM to learn the material field in specified regions, and completes the forward simulation process generate dynamic videos. (Cai et al. 2024) aims to guide the learning of physical properties of objects using 3D Gaussian splats (3DGS). It first recon- structs the static Gaussians in the initial state and learns the deformation models of Gaussians based on dynamic inputs, optimizing the initial velocities and physical parameters (Young's 's modulus and Poisson's 's ratio) of the static Gaussians through differentiable MPM. (Tan et al., 2024) generates high-quality, physics-basee based videos from a single image. However, none of these methods address the extreme cases where objects undergo fragmentation, which is the primary focus of this work.
3 METHOD
In this section, we introduce Fracture-GS, unified framework for simulating and visualizing dy- namic fracture phenomena in extreme mechanical collisions using multi-view image inputs. As illustrated in Figure 2 we first reconstruct the geometry of colliding objects using multi-view im- ages combined with existing implicit 3D reconstruction algorithms (Xiao et al., 2024), followed by sampling both surface and internal particles within the SDD-constrrieed domain to ensure spatial
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Cellision Damage Dynamics
Implicit Reconstruction
1g Gg
&&
A0
Fracture Particles
Surface and Internal Particles Sampling
Pg
Callision
MPM
Gaussian
&& Surface Gaussian Generation
Ph.
Gp
Optimization
Fb Ag
Multi-view Images
Collision-MPM
Va
Ma F?" Că cemp Vin
mi 12
Yield Surface
FP
P2G
Update
C2
G2P
Cg
Ca
Aarria
Collision
V. Cg
me
V
my
Mlly
Ve
Update
Grid
Fb
C2
G
Ch
Yield Surface
C3
Pip
cempp
igrini
n+1
Fg 2+1
Figure 2: Pipeline. The object is implicitly reconstructed from multi-view images, followed by sam- pling both surface and internal particles. Surface particles Iearn Gaussian attributes using isotropic kernels. Next, the sampled particles undergo extreme mechanical collision simulation with dy- namic fracture using enhanced Collision-MPM. Finally, fracture particles tracked and their Gaussian attributes are optimized through our proposed Fracture Partiele Gaussian Optimization strategy, enabling high-quality rendering of the simulation results. For Collision-MPMM the key parameters are highlighted in red. The yield surface determines whether a particle enters the plastic region, triggering return mapping procedure project stress back to the yield surface and up- date the particle's s deformation gradient. Parameters in the blue bottom plate are computed in the Lagrangian coordinate system, while those in the gray bottom plate are computed in the Eulerian coordinate system.
coherence; Subsequently, surface particles Gaussian kernels are trained using 3D Gaussian Splat- ting from input images. Then, proposed enhanced collision-MPM. which is used to perform extreme collisions between multiple objects with dynamic fracture. can effectively alleviate the non-physical adhesion phenomenon caused by MLS-MPM. Finally, enhance the visual realism of the mechanical simulation, all fracture particles are tracked through a hardening-aware tracking criterion defined by (Wolper et al. 2019), based on the tracking fracture particles (as shown in Fig- ure 3 (right)), we can efffciently regenerate their Gaussian attributes through the proposed fracture particles Gaussian optimization strategy, enabling high-quality rendering.
The all particle attributes of the colliding objects Pa {mai Va, Ca, Fa, Vas Fay 0a) and P. {m,, V,, Ch, Fb, Ub, Ib, 0,} include mass (m)) volume (V), deformation gradient (F), velocity gra- dient (C), velocity (v), position (r)) and elastoplastic parameters (0), where 0 comprises Young's modulus (E), Poisson's 's ratio (y), hardening tracking parameters (a), cohesion coefficient (8), and hardening factor (E).
3.1 COLLISION-MPM
Although the Material Point Method (MPM) itself solve the problem of collisions between objects, its performance has limitations. For instance, in fluid-solid coupling simulations, fluid particles may unnaturally adhere solid surfaces-a phenomenon similarly observed in multi-body fracture simulations, where fragmented solids exhibit unphysical adhesion to neighboring objects. To mitigate these artifacts, we propose a Collision-MPP framework, integrating the collision force Yan et al.) 2018) into the MPM framework. This method, originally designed to prevent fluid-solid interpenetration, ensures collision realism across material interfaces.
In Collision-MPP framework, the particle information of two objects is independently trans- ferred to the grid. First, after completing the P2G and Grid Update processes, the mass distributions Ai and îea of object particles P. and P, grid node G; are calculated. Simultaneously, the grid velocities ug n+1 replaced with is semp, Subsequently, the interface direction tendencies nia and
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Particlss (nwrd.12)
Particess
ondy
Grid code Gy (1o Mo 10
Figure 3: (left) Illustration of the mass distributions of particles Pa and at grid node Gt, (right)
Tracking fracture particles through hardening factor
at grid node G; for object particles Pa and Po are determined:
úia Σp. ma Vwia (2; xx)
(9)
En. ma Vwia (x; 2%²)
Ep, my Vwis - x%6)
ûg6
(10)
En, my VWnn - 27,)
As shown in Figure 3 (left), 7ia and ΩGo illustrate the mass distributions of particles Pa and P. on grid node G, We utilize these distributions to compute the interface direction tendency, which can also be interpreted as the tendency of the contact surface's normal direction:
Aia Rub
Faia
T44
Aija
fall
(1I)
The computation of collision forces is conditionally activated based on relative velocity analysis at material interfaces. Specifically, collision forces are generated at grid node G, only if the inequality (uta temmp Vib temp) nia particles on grid node G; after the grid update. The collision forces are derived from the momentum conservation principle during the collision process:
satisfied, where vio temp and wib denote the velocities of Pa and P.
temp Piu m
temp
Pub
S
(12)
(m
m
f S u(Si maa)
(13)
Here, ff represents the collision force on grid node i, m and m denote the mass contributions of Pa and Ps particles on grid node G₁, respectively, Pia temp = m " tempp and temp m temp rep- resent the momenta of P. and Ps partieles grid node G., respectively. y constant controlling the magnitude of the collision force. fia, collision and fit, collision denote the collision forces acting on P. and P. particles on grid node G, After computing the collision forces, they are integrated into the grid velocity update:
f At, nn+ temp Vib
fg
Jemp + Uy
(14)
m
m
where +++1 and un-1 represent the updated velocities of P. and P. particles grid node G;, respectively.
3.2 FRACTURE PARTICLES GAUSSIAN OPTIMIZATION (FPGO).
3.2.1 CONTINUUM MECHANICS AND CONSTITUTIVE MODEL.
To simulate the dynamic behavior of an elastic- plastic object, is essential to solve the conservation equations for momentum and mass:
Dv
Dp
= .
+ v=0
(15)
Dt
Dt

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Here, p denotes density, represents the velocity field, and f is external force. The Cauchy stress tensor, denoted by o, is given by: a det(F] oF
OviF"1 p=" where (fF) is the strain energy density function (or constitutive model), which describes the relationship between stress and strain material. The total deformation gradient, F, is decomposed into elastic and plastic components: F F= FB enabling the simulation of plastic deformation. In this work, we use the NACC model in (Wolper et al. 2019). It extends CCC (Coherent Cam Clay) model with non-associated flow rules to better simulate plastic deformation while maintaining volume during plastic projection, and introduced four plastic parameters a, B, E, MM to control the simulation effect the plastic model.
gi
gi
9 im
gin
Im
In
9m
In
9m
In
(a)
(b)
(c)
Figure 4: Illustration of Fracture Particles Gaussian Optimization
3.2.2 FRACTURE TRACKING AND RENDERING
The enhanced Collision-MPM framework successfully simulates extreme mechanical collisions with dynamic fracture phenomena. While one might intuitively consider directly rendering using the initially learned Gaussian attributes of each surface particles, this naive approach leads to signif- icant non -hyysicl artifacts at fracture interfaces (highlighted by red boxes in Figure 5)
The underlying mechanism of these artifacts can be explained as follows: As shown in Figure 1 con- sider three Gaussian particles {gi, Im: 9n in initial configuration. When fracture occurs particle gr. causing displacement relative to 9m (iguuee4b)) the increased interparticle distance reduces or eliminates the Gaussian overlap region between 9i and gm. thereby disrupting the continuity of the rendering field and generating visual artifacts (As demonstrated in Figure 4(a) and (b)).
To address this challenge, we propose a novel fracture- aware Gaussian attribute optimization strat- egy consisting of four key components. First, leveraging the hardening parameter from the NACC constitutive model (Wolper et al. 2019), we dynamically identify fractured particles at each timestep (visualized green particles in Figure 4). Taking particle gi example (Figure 4 our tracking begins when its hardening parameter exceeds the hardening parameter threshold a.
For each identified fractured particle g.. we perform neighborhood analysis within a radius de to locate adjacent intact particles 9. This adaptive search range ensures proper coverage of potential interaction zones while maintaining computational efficiency.
The core optimization involves Gaussian cloning and attribute reconstruction. We first compute the minimal-volume enclosing ellipsoid (MVEE) for the overlap regions between 9: and its neighbors Ims In- generating two new Gaussian particles Iim and 9in to replace the original 9:- The attribute assignment follows two principles: (1) optical properties including opacity and color are directly inherited from g; to maintain visual consistency; (2) spatial parameters are recomputed through:
(Hneww Enew> } MVEE
(16)
where Peross denotes the original overlap region (Implementation details are provided in Appendix A.4
During final rendering, we implement an occlusion-aware sampling scheme: if a pixel's rendering path contains multiple optimized particles (gim. Isn), only the nearest particle contributes to shading. This prevents overcounting while preserving physical correctness. As demonstrated in Fig our approach generates physically plausible transitional particles that maintain both visual continuity across fracture surfaces and mechanical accuracy in collision regions.

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4 EXPERIMENTS
4.1 EXPERIMENTAL DATA AND PHYSICAL PARAMETER SETTINGS
To comprehensively validate the effectiveness of our experiments, we selected three objects for col- lision (More experimental results are provided in the supplementary material): Ficus, Teapot and Table. Among these, the Fieus plant and the table are heterogeneous material objects. Specifically, the Ficus plant consists of leaves, branches, and a ceramic pot, each made of different materials, while the table has tabletop and legs constructed from distinct materials. The teapot is homoge- neous in material composition. The physical parameters for all objects are detailed in Appendix A.I For specific implementation details, please refer to A.2 Additional experimental results are provided in A.5 and Appendix A.6
Method
PSNR
LPIPS 4
FID
FSF
PhysGaussian
17.2
0.43
120.92
1.5
GIC
16.8
0.45
129.33
1.2
Ours (w/o FPGO)
17.0
0.45
123.63
1.5
Ours (w/o C-MPM)
20.6
0.35
93.05
3.1
Ours
21.1
0.29
90.75
3.5
Figure 5: Integrated comparison showing both qualitative visualizations (left) and quantitative met-
ries (right), with our method achieving the best results.
4.2 COMPARISON WITH STATE-OF-THE-A METHODS
We compare our method with two state-of-the-arr Gaussian splatting based simulation frameworks. First, PhysGaussian (Xie et al., 2024) is a physics-integrated Gaussian framework that simulates and renders mechanical behaviors under external forces from multi-view inputs. Second, GIC (Cai et al. 2024), originally designed for material property estimation from videos, is adapted for compar- ison by utilizing only its forward simulation component with given material parameters, analogous to PhysGaussian. All methods employ identical initialization conditions to ensure fair comparison: the same static 3D Gaussian reconstruction pipeline following GIC's methodology; the same NACC constitutive model
adopted for physical simulation, but the above comparison methods did not include FPGO and used the MLS-MPM: and identical initial conditions and material parameters. We use PSNR (Hore & Ziou| 2010). LPIPS (Zhang et al. 2018)) and FID (Heusel al. (017)) as primary metrics to evaluate reconstruction quality. Due to the absence of ground truth for dynamic fracture sequences, we employ self- self-referencing evaluation scheme using established image quality metrics. The specific implementation details of our self-referencing metric calculation are provided Appendix A.3
User Study. We also conducted human evaluation to assess simulation fidelity, following meth- ods from prior work (Liu et al.) 2025 Wei et al., 2024). Ten participants with varying experience in simulation and vision rated to Fracture Simulation Fidelity (FSF), checking was realistic and expected. Rendered videos of simulations
presented in random order, with participants rating each on a five-point scale (1 poor, excellent). Mean scores appear in Figure 5 (right).
Results. Quantitative and qualitative results in Figure I and 5 Our method significantly outper- forms other methods in GS-based simulation. Competing methods often produce artifacts due to inadequate neglected handling of fracture surfaces and collision adhesion, which degrades simu- lation quality. Although our approach excels simulation, the object's 's FID is high due to reliance on relies on training view interpolation for Gaussian restoration and cannot perform Gaussian re- construction on hidden areas. Future work will explore 3D Al-based texture generative inpainting to improve this.
Effect of Collision-MPPM (C-MPM) and Fracture Particles Gaussian Optimization (FPGO). To validate C-MPM effectiveness, we conduct ablation studies comparing against the conven- tional MLS-MPM approach. As demonstrated in Figure 6 the right column reveals significant
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w/ Collision Force
w/o Collision Force
Figure 6: Collision-MPP effectively resolves the non-physical adhesion artifacts multi-body collisions. As demonstrated in the red box regions, wood fragments from the fractured table exhibit natural separation behavior rather than adhering unnaturally to the teapot surface.
non-physical adhesion artifacts (highlighted in red boxes) MLS-MPM simulations, whereas our method successfully eliminates these unrealistic phenomena through face forces derived from normalized mass distributions. Regarding the FPGO module, quantitative experiments in Figure 5 demonstrate its critical advantages. Both baseline methods PhyGaussian and GIC directly render using initially learned Gaussian attributes, resulting in various visual ar- tifacts fracture interfaces. In contrast, our dynamic attribute optimization strategy significantly enhances visual realism by reconstructing Gaussian properties through MVEE-based interpolation.
siveenttumcooservinn inter-
Momentum
[-4]
[=2
F6
F12
Figure 7: Momentum conservation during a collision between two objects. The total momentum (black curve) remains constant throughout the simulation, demonstrating strict adherence to the conservation law, while the individual momenta of the objects exchange during the impact.
Momentum Conservation Validation. To rigorously validate that our simulation framework strictly adheres to law momentum conservation, we designed a controlled experiment. This experiment involves two objects-2 teapot and bowl-propelled towards each other with initial velocities in a environment free from external influences such as gravity and friction. The figure tracks the system's 's evolution over time (t to t 20). It visually demonstrates that while indi- vidual momenta change during the collision interval (t = 5 to Rs 15), the "Total Momentum" curve remains constant in both magnitude and direction throughout the entire sequence. This constant to- tal momentum can be clearly demonstrated from the figure that our simulation method accurately maintains the total linear momentum of the system.
Energy Stability Analysis. To quantitatively validate the concern regarding non-physical energy growth and numerical instability, we conduct thorough energy evolution analysis throughout representative simulation involving the collision and fracture of a teapot on a table surface. As illustrated in Figure 8 track the kinetic, elastic, and gravitational potential energy components for both the teapot and the table individually, as well as for the combined system over the first 50 frames. The results demonstrate that the total energy of the system remains strictly bounded and does not exhibit any anomalous increase. Energy is transferred in a physically consistent manner: kinetic energy converts into elastic deformation energy upon impact, and part of it is dissipated through

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fracture processes, while gravitational potential energy varies accordingly with object height. The smooth transitions and the absence of energy blow-up confirm that our method inherently prevents non-physical energy accumulation. This energy behavior serves effective unit test, validating that our simulation not only captures complex dynamic and fracture phenomena but also maintains numerical stability under severe contact and deformation conditions.
Table Surface Particle Energy (First 50 Frames)
Tonal Energy
Kinctc Energr
Elestic Energy
Energy
Graviyy Fotential
30
Frome
Teapot Energy Frames)
20
Energy
Energy 01
Enengy
Gravity Fotencial Energy
Frame
Table Surface Teapot Particle Energy (First 50 Frames)
20
Total Breryy
Kinbic Energy
15
Elastic Energy
Potential
10
Frama
F-
F00
20
1=30
1-40
50
Figure 8: Energy evolution and qualitative visualization during the teapot-table collision and fracture simulation. The top and middle rows plot the kinetic, elastic, and gravitational potential energy com- ponents for the table surface, the teapot, and the combined system over the first 50 frames, showing bounded total energy without non-physical growth. The bottom row provides corresponding qual- itative visualizations at key frames (t=0, 10.0,,00.003000 10.20.33.440.50 depicting the physical progression of the collision and fracture process that correlates with the energy transitions observed the graphs.
5 CONCLUSION
This paper presents unified framework for simulating and rendering extreme mechanical collisions with dynamic fracture effects. The framework demonstrates robust performance across various sce- narios, including high-velocity impacts and heterogeneous material fractures. Qualitative and quan- titative evaluations show significant improvements over existing methods in both physical accuracy and rendering quality.
Current limitations include high computational demands restricting real-time performance in com- plex scenes, the need for manual parameter setting, and shortcomings in the evaluation ap- proach-such as indirect quantitative metrics for this task-all of which will be addressed in fu- ture work: implementing GPU optimization and adaptive timestepping for faster computation, de- veloping learning-bar based methods for automatic parameter estimation. and establishing specialized evaluation metrics to enhance the framework's practicality in virtual prototyping and visual effects production.
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ACKNOWLEDGEMENTS
This work was supported in part by the Chongqing Natural Science Foundation (CSTB2024NSCQ- MSX1026), in part by the Fundamental Research Funds for the Central Universities (No. SWU- KT25012), in part by the Key R&D Program of Shandong Province under grant 2025TSGC- CZZB0165, and in part by the NSFC (62572062).
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Tianyuan Zhang, Hong-Xing Xing Yu, Rundi Wu, Brandon Y Feng, Changxi Zheng, Noah Snavely, Jiajun Wu, and William Freeman. Physdreamer: Physics-based interaction with 3d objects via video generation. In European Conference on Computer Vision, pp. 388-406. Springer, 2025.
A APPENDIX
A.1 MATERIAL PARAMETERS
Table ! summarizes the material parameters employed in our collision simulations, including Young's 's modulus (E), Poisson's ratio (v), density, and non-associated flow rule parameters (o. (,C, M). These physically-based values were assigned to each object component to validate our framework's 's capability handling diverse material properties. The selected parameters reflect realistic material contrasts, enabling quantitative analysis of how mechanical properties-sur as the stiffness variation between flexible leaves and rigid branches-influence fracture propagation patterns across all experiments.
Table I: Material Parameters
Scene
E (MPa)
V
Density
NACC (ax, 3,C, M)
Bowl
5x104
0.46
2
0.98, 0.5, 1.
Ficus leaf
8x104
0.39
0.6
0.94, 2, 3, 2.36
Ficus branch
1x106
0.39
5
0.94, 2, 2.36
Ficus pot
2x104
0.39
2
0.98, 0.5, 2.36
Teapot
5x105
0.46
0.98, 0.5, 2,36
Table top
105
0.39
0.99, 0.5, 2.36
Table leg
1x10s
0.39
1000
0.94, 2, 3, 2.36
A.2 EXPERIMENTS DETAIL
Here, provide additional details regarding the experiments. These include collisions between single object and rigid surface, collisions among multiple objects, as well as experiments on interactions between individual objects and rigid surface under varying physical parameters. We also present an ablation study on the impact of fracture particle tracking.
Input Resolution Sampling: The input resolution starts at 1024 x 1024. For each collision object, we sample 200,000 surface points and 100,000 interior points. Simulation Setup: Building upon Warp, the simulation executed on an 18-core Intel Xeon Gold 5220 CPU and an NVIDIA GefFrce RTX 3090 GPU, achieving 100-frame sequences for each collisions scene.
Implementation Details. Following established practice in GIC (Cai et al. 2024), our simulation pipeline begins with isotropic Gaussian reconstruction for visual surface representation. For phys- ical discretization, we employ an SDF-based voxelization strategy: after constructing a volumetric grid and identifying interior voxels via SDF filtering, perform uniform random sampling inside each interior voxel using a controllable density parameter N,, (samples per voxel). This preprocess- ing step decouples visual quality from simulation discretization while ensuring physically plausible material sampling. To evaluate the influence of the particle sampling density o, the simulation results, we set multiple sets of internal sampling parameters (N₂ 0, 5, 10, 15, 20, 50, 100, 200), and the visualization results are shown in Figure
A.3 METRIC COMPUTATION FOR FRACTURE ASSESSMENT
To quantitatively evaluate the visual plausibility of fracture propagation in the absence of ground truth dynamic sequences, we designed a self-referencing assessment protocol using three estab- lished image quality metrics: PSNR, LPIPS, and FID. Our evaluation strategy focuses measuring how faithfully each method maintains visual continuity during the fracture process. For each frac- ture event, identify the last frame before fracture initiation (detected via the hardening parameter a) as the reference frame J representing the intact object's appearance. We then analyze the sub- sequent 5 frames where fracture propagation becomes fully visible, ensuring this window captures
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eritical fracture dynamics while maintaining visual comparability to the reference. The core princi- ple underlying our metric design that high-quality fracture rendering should appear as a coherent extension of the original material. Accordingly, sistency with the pre-fracture state, LPIPS to assess perceptual similarity to the intact appearance, and FID evaluate distributional similarity pre-fracture rendering. Methods that produce visual artifacts or unnatural fracture surfaces consequently exhibit large deviations from Leff resulting in degraded metric scores that reflect their reduced visual plausibility.
compute PSNR measure pixel-level con-
It should be noted, however, that this represents pragmatic compromise in the absence of more principled evaluation methodologies. We regard the development of dedicated metrics for physics- based rendering as important direction for future work.
A.4
MINIMAL-VOUUME ENCLOSING ELLIPSOID (MVEB)
A.4.1
PROBLEM DEFINITION
Given two spheres in 3D space with centers C₁ C2 and radii r1; r2, we aim to compute the minimal- volume enclosing ellipsoid (MVEE) of their intersection region. Step I: Intersection Conditions and Geometric Parameters Intersection Criteria. The spheres intersect if:
m - r2 < d <rri +2, where d= C - C2ll
2 rı + 12, the spheres are disjoint; if d S In r2/. one sphere entirely contained within the other.
Key Geometric Properties. The intersection region is a lens-shaped volume bounded by two
spherical caps. Its properties include:
SSymmtry axis: The line connecting C₁ and C2 (unit vector u C CzCCi,
Maximal width: Perpendicular to u, determined by the radius of the circle of intersection.
Step 2: Analytical Estimation of the Enclosing Ellipsoid
Ellipsoid Center. The center the MVEE is approximated as a weighted midpoint along the symmetry axis:
c-C1+ 11+12'
(C2 C1)
This heuristic prioritizes the larger sphere's influence. Ellipsoid Axes. The ellipsoid has three principal axes:
Major axis (aligned with u):
12 d
2. Minor axes (perpendicular to u): Lengths = c, given by the radius of the intersection circle:
b b=c= V - r3
dc +
2d
Orientation Matrix. The ellipsoid's rotation matrix R. is constructed from the orthonormal basis:
R iu v w], where v uux
A.5 COMPUTATIONAL EFFICIENCY
We analyze the computational overhead of our fracture tracking mechanism by comparing the per- frame rendering time for three core scenarios in main paper, with and without this feature enabled. As detailed in Table the incorporation of fracture tracking introduces moderate and consistent computational cost, increasing rendering time by approximately 20-80% across the scenes. This overhead is attributed the additional steps of dynamically updating the Gaussian splatting model to reflect new fracture surfaces and collisions in each frame.
A.6 ABLATION STUDIES
Effect of Fracture Particle Tracking and Gaussian Generation. During the fracture process, some internal particles become exposed and visible. To achieve higher visual realism, we need to quickly locate these particles and generate corresponding Gaussian visual attributes. Based on the
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Table 2: Per-frame Gaussian rendering time.
Scene
w/o fracture tracking
w/ fracture tracking
Ficus
49.19 ms
62.78 ms
Tеapot
6.13 ms
10.87 ms
Teapot & Table
27.15 ms
29.85 85 ms
Crack Visulization
Collision Dynamies
Figure 9: Collisions with three distinct initial hardening factors.
hardening tracking parameters (a), we can efficiently track relevant internal particles the fracture surface and rapidly generate their Gaussian visual attributes through interpolation with neighboring external surface particles, enabling high-quality rendering. Additionally, large number of isolated particles may be generated during collisions. We utilize the fracture particle tracking mechanism exclude these fine particles from Gaussian rendering. This selective exclusion ensures that only particles meeting specific material integrity criteria participate in the final rendering, thereby maintaining the physical accuracy while improving computational efficiency.
Single-Object Analysis. To systematically evaluate our method's sensitivity to material properties, conduct extensive collision experiments individual objects with varying plasticity parameters, as shown in Figure 9 Our results demonstrate that increasing the initial hardening factor systemati- cally enhances object fragmentation and promotes more extensive crack propagation, validating our method's ability to capture material-dependent fracture behaviors.
Multi-Object Interactions. To assess performance complex scenarios, we execute multiple chal- lenging multi-body interaction experiments, shown in Figure I0 These include bowl-teapot col- lisions, bowl-table impacts, and ficus-table interactions. These experiments demonstrate our frame- work's robustness in handling heterogeneous material compositions and complex contact dynamics across diverse object categories.
Real-world Data Validation. To further validate the generalizability our approach, we extend our evaluation to real-world data from the DTU dataset. As shown in Figure 11 and [2] our pipeline suc- cessfully performs Gaussian reconstruction and initialization from real images, followed by physi- cally plausible fracture simulation. These results demonstrate our method's 's robustness when applied to real-world captured data, confirming its practical applicability beyond synthetic environments.
Analysis of sampling point density. To assess the influence of particle sampling density on simulation outcomes, we conducted a systematic sweep of internal sampling parameters (N, 0, 5, 10. 15, 20, 50. 100, 200). Using the highest density configuration (N, 200) as a convergence reference, we evaluated trajectory deviations across all sampling conditions (Figure 12] and and Table The results reveal a fundamental characteristic MPM simulations: particle sampling density in- trinsically influences the resolved mechanical response. As shown in quantitative analysis and
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"Bowl & Teupot"
"Bowl & Taable"
"Picus Table"
Figure 10: Collisions between different objects: "Bowl & Teapot", "Bowl & Table", "Ficus & Table".
Figure 11: Real-world data validation of our method: results for three different real-world objects.
qualitative visualizations (Figure 12 and Table 3). variations sampling density naturally lead to differences in fracture patterns and collision dynamics-this thins is an inherent property of particle-based methods where discretization density affects solution convergence.
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Interior Point Density
X-corddinate Error
Y-coordinate Error
Z-coordinate Error
0
0.1998
0.1479
0.0535
5
0.1936
0.1433
0.0474
10
0.1884
0.1360
0.0445
15
0.1772
0.1279
0.0435
20
0.1786
0.1224
0.0412
50
0.1713
0,1191
0.0337
100
0.0935
0.0749
0.0315
Table 3: Coordinate errors different interior point densities.
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N, 0
N,
N, 10
N₂ = 15
N₂ = 20
No 50
N, 100
200
5
1-
1=15
1-20
1-25
t-30
Figure 12: Visual comparison of simulation and rendering results under different internal sampling densities (N,) across three representative timesteps.
18


Reviews:
[{"review_id": "oODGr9r0g0", "reviewer": "Reviewer_sTDw", "summary": "This paper generalizes Gaussian splatting to handle deformation and fracture simulation. This is done by integrating Gaussian platting with deformation and fracture simulation handled by the material point method. The proposed approach builds on a number of recent papers at the intersection of these topics, namely \"Physics-integrated 3D gaussians for generative dynamics\" by Xie et al. (2024) and \"Gaussian-informed continuum for physical property identification and simulation\" by Cai et al. (2024). The key difference is that this paper adds support for simulating fractures. The equations for doing so are spelled out in detail, and the authors benchmark their results on a number of reasonable looking examples. In particular, the authors include not only quantitative benchmarks, but also a human evaluation study to assess simulation fidelity.\n\nAs a disclaimer, I largely work on topics outside of this area, but did write a few papers earlier in my career on related topics. Therefore, my views on state-of-the-art methods and benchmarking are likely not to be up to date, so I will concentrate more on factors such as soundness and readability as opposed to performance comparisons, which I hope other reviewers will be more able to contribute to.", "questions": "My main question is not to the authors, but rather to other reviewers in the discussion phase, who are hopefully more familiar with state-of-the-art in this space. The question is: are the authors' results SOTA compared to what is done today? If not, how far are they? In asking this, I encourage the AC to _not_ interpret this question as a demand for uniformly-SOTA results: the paper may well still be publication-worthy even if there are limitations - instead, I would simply like to know where this paper stands compared to other papers published very recently that I am less familiar with because they are too recent.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "I see the following key strengths:\n* **Important topic.** Being able to simulate physics in a Gaussian splatting like framework is a useful technical capability to build, because over time this might allow us to build interpretable world models for domains like robotics.\n* **Approach is interpretable and makes sense.** Combining a material description built in the framework of Gaussian splatting - where rather than visual parameters like opacity, the authors propose a larger set of parameters including mass, volume, elasticity parameters such as Young's modulus and Poisson ratio, and others to do with deformation and fracture simulation. This is a reasonably intuitive way to bring together these methods. While I have not read the prior papers on non-fracture physics simulation via Gaussian splatting, I believe the new part here are the parameters needed to support fractures, and pinning this down is a reasonable contribution.\n* **Key equations are all stated in the paper, with only details deferred to appendix.** The equations used to simulate how the Gaussians evolve over time are spelled out in some detail. I expect a reader familiar with this literature will not have trouble figuring out precisely what the authors are doing purely from the paper, that is without having to do detailed digging in the appendix or code.\n* **Experiments include both quantitative comparisons and human evaluation.** Following my original disclaimer, due to not working in related areas for some time, I am less equipped to assess whether or not the results are state-of-the-art, and if so how strong is the degree of improvement compared to prior papers. However, the problems tested, such as simulation of a potted plant breaking, seem reasonable, as does the the larger evaluation structure, given the authors include both quantitative comparisons and human evaluations.", "weaknesses": "I am concerned about the following:\n* **Sloppy writing with typos, unresolved references, and other evidence of work done in a rush.** For example, many of the citations such as Stomakhin et al. (2013) should be in parentheses, meaning (Stomakhin et al., 2013). Similar for Wolper et al. (2019). Please fix these. For another example, Section 2.3 has a section titled \"Physics Simulation based on Gaussian\". To avoid grammar mistakes that distract the reader, \"Gaussian\" should be changed to most likely \"Gaussians\" or something otherwise correct sounding. In the end, there is also a text \"zhang2018unreasonable\" where the authors forgot to add a citation command. This paper was clearly written up in a big rush, and for this reason alone I am open to the idea of rejecting the work to give the authors more time to polish the results before publication and presentation at a conference. This is the main reason I mark the work borderline: the contribution seems good, but it is very important for the review process incentivizes quality, and not rushed submissions.\n* **Prior work section interrupts the paper's flow.** I would merge Section 2 with Section 3, moving the two subsections inside the corresponding parts of Section 3. The problem right now is that it is difficult to understand what the prior work is doing before the equations are introduced. For instance, on my first read, I was unable to properly understand the differences between this paper and prior papers also working with the material point method, due to not being immediately familiar with it. This became more clear once I read that section and realized that this method's broad structure is very similar to contact dynamics simulation methods I have worked extensively with in the past. At this point, I had to go back and re-read the prior work section. A modified structure such as the one I propose above would make it possible to avoid backtracking and read the paper all in one go, even if the reader is familiar with neighboring work rather than the precise approach used.\n* **Diversity of experiments.** The authors' method is only tested on two models - a potted plant, and a desk. While the experiments themselves look reasonable, this seems much less in terms of comprehensiveness compared to what I am used to seeing in computer graphics papers. I hypothesize the relatively-small number of examples is related to the paper's writing issues mentioned earlier: the authors may have wanted to test more, but ran out of time before getting everything implemented. I suspect a re-submitted version would contain results closer to the typical level of comprehensiveness in this area.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5HoshvcCU6", "reviewer": "Reviewer_og4k", "summary": "The paper proposes Fracture‑GS, a pipeline that (i) reconstructs objects from multi‑view images and assigns surface Gaussians, (ii) runs an enhanced Collision‑MPM intended to avoid non‑physical adhesion in impacts, and (iii) performs fracture‑aware Gaussian optimization (FPGO) that clones/reshapes Gaussians near newly exposed fracture surfaces via a minimal‑volume enclosing ellipsoid heuristic. The system is demonstrated on three scenarios (Ficus, Teapot, Table) with qualitative renderings and a small quantitative table.", "questions": "- What is the benefit of Gaussians over point particles? You already restrict to isotropic Gaussians on the surface, and FPGO’s shape comes from sphere overlaps. That suggests a point‑splat baseline could be nearly identical but simpler. Please quantify rendering quality, temporal stability, and runtime against:\n(i) MPM + point sprites / disk splats;\n(ii) MPM + sphere splats with the same nearest‑only occlusion rule;\n(iii) MPM + mesh (e.g., marching cubes) + rasterization.\n\n- What’s different from treating Gaussians as points and running standard MPM? If the Collision‑MPM and fracture rule run on the same particle set regardless of the renderer, then the scientific contribution is not physics‑integrated Gaussians but a contact heuristic and a rendering trick. Spell out where Gaussians influence the physics (if at all) and justify the coupling.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "+ The paper tackles a real pain point: adhesion artifacts in grid‑based MPM around contacts. The visual before/after in Fig. 6 makes the problem clear.\n\n+ The pipeline is pragmatic and easy to follow, with implementation details and timings.", "weaknesses": "- Theoretical issues: Section 4.2.1 writes the plasticity split as F = F_E + F_P and uses a non‑standard Cauchy‑stress expression. The correct finite strain plasticity should be F = F_E @ F_P; addition will lead to wrong stresses. It’s unclear what the code actually uses. This is a correctness issue, not just notation.\n\n- The contact model is a heuristic, frictionless, and under‑explained. Normals come from differences of normalized mass‑weighted kernel gradients, then a momentum‑conserving node force is applied and projected only along the normal with a constant \\mu. There is no friction cone, no restitution, and no complementarity constraints. The paper asserts conservation but shows no momentum/energy plots or penetration statistics.\n\n- I am also confused about the motivation for using Gaussian points as the representation. Most of the physics here is standard MLS‑MPM on particles. Surface particles just carry isotropic Gaussians for rendering, and FPGO computes shapes by fitting an ellipsoid to the overlap of two spheres before duplicating the particle. That’s a rendering heuristic, not physics. The paper never quantifies what Gaussians buy you over simply rendering particles as points/splats or extracting a mesh from the simulation. If I treat Gaussians as points and run a vanilla MPM, do I lose anything besides SH‑based view‑dependent color?\n\n- Appendix A.3 gives a closed‑form approximation to the MVEE of a sphere–sphere intersection and uses it to set the new Gaussian’s mean/covariance. There is no actual optimization loop, proof of minimality, or ablation against simpler choices (e.g., just reuse the parent particle, or refit an anisotropic Gaussian from neighbors). Yet this step is claimed to drive the gains in Fig. 5.\n\n- Comparisons are only against PhysGaussian and GIC, not against fracture‑capable MPM/contact baselines; metrics are image‑space (PSNR/LPIPS/FID) plus a tiny user study (N = 10). There’s no physics validation: no momentum/energy traces, no penetration depths, no fragment size distributions, no restitution. If the key claim is momentum‑conserving collision forces that remove adhesion, I need physics diagnostics.\n\n- Sensitivity to particle sampling is unaddressed. Change the interior density, or its near‑boundary distribution, and those nodewise estimates can change materially. There's no ablation for particle‑per‑cell, interior/surface ratios, or grid resolution.\n\n- The number of scenes shown across the paper is extremely limited. The generalization ability of the proposed method is a large concern.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "AecX0eUFes", "reviewer": "Reviewer_JoU4", "summary": "The paper proposes Fracture-GS, a unified system that integrates physics-based dynamic fracture simulation with 3D Gaussian splatting rendering to realistically model high-energy collisions and material breakage. Traditional methods either simulate deformation without handling fracture or rely on post-processing for visualization, often producing non-physical adhesion artifacts and discontinuous fracture surfaces. Fracture-GS addresses these limitations through two core components: an enhanced Collision Material Point Method (Collision-MPM) and a Fracture Particle Gaussian Optimization (FPGO) strategy. The Collision-MPM introduces momentum-conserving interface forces based on normalized mass distributions, effectively preventing unrealistic adhesion in multi-body collisions. Meanwhile, FPGO leverages hardening-aware fracture tracking and minimal-volume enclosing ellipsoid (MVEE) optimization to regenerate Gaussian particle attributes dynamically, ensuring smooth, physically consistent fracture rendering.\n\nThe framework operates from multi-view image reconstruction to high-fidelity simulation and real-time visualization, handling both homogeneous and heterogeneous materials under extreme impact. Experiments on scenes such as colliding teapots, tables, and plants show that Fracture-GS delivers significantly higher physical realism and visual quality than prior Gaussian-based approaches like PhysGaussian and GIC. Quantitative metrics (PSNR, LPIPS, FID) and human evaluation confirm its superior fidelity, while ablation studies demonstrate the critical roles of both the Collision-MPM and FPGO modules. Overall, Fracture-GS offers a physically accurate, visually coherent, and computationally efficient solution for simulating and rendering complex fracture dynamics directly from image-based inputs.", "questions": "The followings are the questions:\n1. How does Fracture-GS scale with respect to the number of particles and fracture events? Could GPU parallelization or adaptive time-stepping make the system closer to real-time performance?\n2. The method relies on several material-dependent parameters (e.g., Young’s modulus, hardening factor, cohesion). How sensitive is the simulation to these values, and could a learning-based or automatic parameter estimation approach reduce manual tuning?\n3. Have you evaluated the framework using real captured multi-view data with noise or incomplete coverage? How robust is the reconstruction and simulation pipeline in such cases?\n4. Can the same physics-integrated Gaussian framework be extended to simulate fluids, soft bodies, or granular materials under fracture or mixing conditions?\n5. When handling high-energy impacts or very thin fracture surfaces, how does the momentum-conserving collision formulation maintain numerical stability? Are there cases where it fails or diverges?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The main strengths of Fracture-GS lie in its innovative integration of physics-based simulation and Gaussian rendering. The framework unifies dynamic fracture simulation with photorealistic visualization, allowing collisions and material breakage to be simulated and rendered directly from multi-view inputs without post-processing. Its enhanced Collision-MPM formulation introduces momentum-conserving interface forces that eliminate the common adhesion artifacts seen in standard MPM methods, resulting in more realistic object separation and fracture behavior. This physically grounded improvement enables the system to handle both homogeneous and heterogeneous materials under extreme impact conditions with high accuracy.\n\nA second major strength is the Fracture Particle Gaussian Optimization (FPGO) module, which uses hardening-aware fracture tracking and ellipsoid-based Gaussian reconstruction to maintain visual smoothness across newly formed fracture surfaces. This innovation ensures physical plausibility and visual coherence even in complex fragmentation scenes. Combined with strong empirical validation—demonstrating clear advantages over prior approaches like PhysGaussian and GIC in both quantitative metrics and human perception tests—Fracture-GS establishes a new standard for integrating continuum mechanics with 3D Gaussian splatting. Its blend of physical realism, visual fidelity, and computational efficiency makes it a significant contribution to the field of physics-integrated neural rendering.", "weaknesses": "The main weaknesses of Fracture-GS stem from its computational cost and scalability limits. The enhanced Collision-MPM and fracture-aware Gaussian optimization (FPGO) introduce additional per-frame overhead for fracture tracking and Gaussian reconstruction, making the system too slow for real-time or large-scale simulations. While the method achieves high-quality results, it is currently best suited for offline rendering and controlled experimental setups.\n\nAnother limitation is the framework’s parameter sensitivity and narrow validation scope. It relies on manually tuned material parameters such as elasticity and hardening factors, which may limit generalization to new materials or real-world data. The experiments focus mainly on clean, synthetic collisions, without testing robustness under noisy or incomplete multi-view inputs. Overall, the method is conceptually strong but computationally heavy and parameter-dependent, leaving room for improvement in automation, generalization, and efficiency.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YnhmIppLUd", "reviewer": "Reviewer_yz5d", "summary": "The authors propose an extension to the PhysGaussian framework to handle fractures. In their simulation, they use separate velocity fields for different objects and apply collision forces to the nodes that are commonly affected. This contact model can eliminate the sticky artifacts when two objects collide. The NACC plasticity model is employed to simulate fracture behavior. However, this improvement of MPM simulation is novel. The contact model seems identical to the one in [1]. For rendering, the hardening factor is used to track crack propagation and to split Gaussian kernels accordingly. This handling is kind of novel and clever.\n\n[1] Wretborn, J., Armiento, R. and Museth, K., 2017. Animation of crack propagation by means of an extended multi-body solver for the material point method. Computers & Graphics, 69, pp.131-139.", "questions": "- How image metrics are computed? I assume there is no ground truth for simulated results. If only the reconstruction error is compared, it is not fair. I would expect that all method can start from the same initial Gaussian Splatting representation? Are there any modifications to GS reconstruction to increase the image metrics?\n\n- How is $\\mu$ in Eq.13 set? I would expect that a too small value can cause penetration and a too large value can cause simulation explosion?\n\n- How to make sure the total energy (kinematics energy + elasticity energy + gravity potential) does not increase to cause instability.  Unit tests on energy should be included.\n\n- The plant in Fig. 5 should not be plastic. How different materials are assigned?", "limitations": "", "rating": 2, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The handling of fractures in GS-based MPM simulation is indeed a neat solution.", "weaknesses": "- The contact model is claimed to be one of contributions. However, the model seems identical to the one in [1]. A clarification on difference and novelty is needed.\n\n- The experiment is insufficient. Only 3 examples are provided for comparison and the user study. The conclusion is not significant enough.\n\n- The simulation scenes are overly idealized. The presented examples do not fully demonstrate the strength of Gaussian splatting in reconstructing real-world data. A traditional MPM simulation and rendering pipeline should be able to handle better.\n\n- A limitation is that the framework seems quite bound to NACC or similar plasticity model that has hardening mechanism. However, MPM supports many commonly used plasticity models.\n\n- Some unprofessional typos: \n  - Line 256: Figure 7?\n  - Line 344: The deformation gradient is not additive.\n  - Line 369: \"and\" is in math mode\n  - Line 425: zhang2018unreasonable?\n\n[1] Wretborn, J., Armiento, R. and Museth, K., 2017. Animation of crack propagation by means of an extended multi-body solver for the material point method. Computers & Graphics, 69, pp.131-139.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)