Problem statement‌‌

A rubber balloon is a representative elastic structure that exhibits nonlinear and complex behavior as air is injected and released. In particular, during air release, surface contraction, uneven internal/external pressure differences, and asymmetric shape changes cannot be explained by a simple inflation model, and they induce secondary motion effects such as rotation and lift-up. These phenomena occur as the escaping air generates thrust in a specific direction, and rotational inertia acts in conjunction with the displacement of the balloon’s center of mass. As a result, the balloon may rotate irregularly or soar upward, forming complex motion trajectories.

However, there are several challenges in accurately modeling such physical phenomena with existing simulation techniques:

1. Nonlinear Large Deformation: Initially thin and flexible, a balloon undergoes extreme variations in surface tension during air injection and release, with its overall shape and elasticity distribution changing over time. In particular, during air release, uneven contraction of the shape accelerates, producing unpredictable rotational motion.
2. Difficulty in Modeling Thrust from Air Release: As internal air escapes, subtle variations in airflow impose asymmetric pressures on parts of the surface, inducing rotational torque. Accurate computation of this requires complex fluid–structure interaction (FSI) analysis, which demands highly expensive computational resources.
3. Dynamic Changes Over Time: While air is being released, the balloon’s shape continuously changes, altering its center of mass, moment of inertia, and surface tension over time. All of these factors affect the balloon’s motion, and real-time computation requires highly precise time steps and a stable integrated simulation framework.
4. Inefficiency in Real-Time Processing: Accurately capturing all these factors necessitates high-resolution mesh subdivision, fine airflow analysis, and computation of elastic and reaction forces, which severely hinders real-time performance. In real-time applications such as games, XR environments, and physics-based virtual content, this computational load becomes a major bottleneck.

As a result, existing simulation approaches can only physically approximate the inflated state of a balloon, and have limitations in expressing complete dynamics including free flight, rotation, or lift-up caused by air release.

To address these limitations, this study proposes a simulation framework that can infer the rotation and lift-up motion of a balloon during air release using a simplified physics-based model, while leveraging PBD to achieve cost-efficient, real-time computation. This enables visually natural reproduction of the balloon’s dynamic motion, while providing a computational structure suitable for a variety of real-time content applications.

How is the balloon effect typically created in the industry?

In 3D simulation-based content creation environments, a variety of techniques are employed to realistically reproduce the balloon effect, which includes inflation, deflation, rotation, and irregular motion caused by air release. In industrial contexts where balancing visual realism with simulation efficiency is critical, the following approaches are commonly adopted.

Summary.

These various approaches are selected according to the intended purpose of the content (real-time or offline rendering), required accuracy, and system performance (see Table 1). Recently, due to industrial demands for efficiency and scalability, PBD-based simulation techniques have been gaining mainstream adoption [13]. However, existing PBD applications are largely limited to simple structural deformations and have not sufficiently addressed balloon simulations that incorporate asymmetric rotation, nonlinear contraction, and complex dynamic changes over time during air release.

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Table 1. Comparison of different approaches for simulating balloon effects.

https://doi.org/10.1371/journal.pone.0348597.t001

To address these limitations, this paper proposes a lightweight PBD-based simulation method that can represent not only rotation and volume changes but also complex dynamic phenomena arising from elasticity without simulating internal airflow. The proposed method effectively reproduces the nonlinear behavior characteristic of balloons while minimizing computational cost for real-time applications, making it suitable for integration into games, interactive content, XR environments, and other real-time physics-based systems.

Position-based dynamics

Position-Based Dynamics (PBD) is a technique for efficiently modeling deformable objects such as in cloth simulation. The core idea of PBD is to first estimate predicted positions under external forces using Euler’s method, and then iteratively apply constraint equations, based on Hooke’s law, to correct positions so that internal forces are satisfied. The overall solver consists of two main steps: predicted position calculation due to external forces, and position correction to satisfy the constraints. Among these, the predicted position is calculated using Euler integration as follows.

(1)

Here, v is the velocity, w is the inverse mass, f_ext is the external force, x is the previous position, p is the predicted position, and i denotes the index of the vertex. To correct p so that it satisfies the constraints, the Gauss–Seidel method is used. Since the equations for each constraint are nonlinear, the solution is found numerically by iteratively reducing the error. Finally, the position and velocity at the next time step are computed using:

(2)

The key in PBD is to project points onto the constraint manifold to find the optimal solution. Since these constraints are due to internal forces such as those described by Hooke’s law, rather than external forces, momentum must be conserved. To this end, in the following equation, the gradient of C, , is assumed to be rigid, and the optimal solution for the constraint is found:

(3)

To assume as rigid, the direction of is assigned to . Expanding the equation allows computation of the position change that each vertex must undergo. For an individual vertex, Equation 3 can be expressed as follows:

(4)

Position-based constraints

Position-Based Constraints are a core concept of the Position-Based Dynamics (PBD) framework. Instead of computing motion from forces or accelerations, this approach directly adjusts the positions of vertices to satisfy given physical or geometric constraints through iterative projection. This position-level formulation significantly reduces numerical instability and enables real-time computation, making it suitable for interactive simulations such as deformable objects or soft-body dynamics.

In PBD, the simulation first predicts vertex positions under external forces (e.g., gravity, reaction force), and then iteratively corrects those positions so that the defined constraints (e.g., distance, volume) are satisfied. The correction step projects each vertex onto the constraint manifold, ensuring that the deformation remains physically plausible.

In this study, two types of Position-Based Constraints are employed:

• Length Constraint: Preserves the initial edge lengths between adjacent vertices to maintain structural elasticity.
• Volume Constraint: Maintains the initial enclosed volume of the surface mesh to control the inflation and deflation behavior of the balloon.

These constraints are repeatedly applied during the iterative correction stage of the PBD solver, allowing the system to maintain physically consistent deformation throughout the simulation. Consequently, in this work, the term Position-Based Constraints refers to the position-driven physical constraint structure that enables stable and real-time reproduction of the balloon’s complex behaviors such as inflation, deflation, and rotation.

Constraint

The constraints used in this study can be categorized into length constraints and volume constraints. The length constraint preserves the initial length, similar to a mass–spring system, and is computed as follows:

(5)

By substituting the above equation into Equation 4, we obtain the result shown in Equation 6. When this is applied to the distance constraint, Equation 3 can be satisfied.

(6)

In this work, the length constraint is used to generate structural springs, which connect vertices, and bend springs, which connect crossover edges. Since a triangular mesh is used, the structural springs connect vertices according to the triangle’s vertex indices [0,1,2] (see Fig 1a). The bend springs are created by connecting two common neighbor vertices, excluding the selected vertex, so that two vertices completely opposite each other across the shared edge of two triangles are connected (see Fig 1b).

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Fig 1. Generation of structural and bending springs.

https://doi.org/10.1371/journal.pone.0348597.g001

Next, the volume constraint, similar to the length constraint, is computed by subtracting the initial volume from the current volume (see Equation 7). The gradient for the volume constraint is calculated using Equation 8.

(7)

(8)

Substituting the above equations into Equation 4 yields . Applying this allows the simulation to produce different results depending on the initial volume. Fig 2 clearly demonstrates the surface contraction and expansion of the Stanford Bunny model due to the volume constraint.

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Fig 2. Simulation of the volume constraint.

https://doi.org/10.1371/journal.pone.0348597.g002

Balloon effects

In this study, the initial volume V₀ used in the volume constraint is leveraged to reproduce the balloon effect. When air is injected, V₀ is increased; when air is released, V₀ is decreased. This naturally produces surface deformations in which the object’s volume expands or contracts. The change in balloon volume during inflation and deflation is due to variations in the internal pressure. When the internal pressure exceeds atmospheric pressure, the volume increases; when it falls below atmospheric pressure, the volume decreases. Therefore, the internal pressure PV of the balloon is calculated using the Ideal Gas Law in Equation 9, and air release is stopped when the pressure equals atmospheric pressure.

(9)

Here, n is the number of moles of gas, and R and T are the gas constant and temperature, respectively. In this study, atmospheric pressure is set to 101,325 Pa, the temperature to 293.15 K, and the gas is assumed to be helium. The number of moles of gas is approximated based on the additional volume inside the balloon.

External force.

Four external forces are considered in this study: gravity, buoyancy, air resistance, and reaction force. Gravity g is set to its standard value of 9.8 m/s², and buoyancy F_buoy is calculated using Equation 10. If the buoyancy exceeds gravity, the balloon will rise. The densities are set as follows: is the density of the ambient air, and is the density of the gas (helium) inside the balloon.

(10)

Here, V denotes the volume of the balloon. Air resistance F_drag is computed using Equation 11. To weight the resistance according to the direction of motion, the velocity vector is dotted with the surface normal vector. The drag coefficient C_d is set to 0.5.

(11)

Here, is the fluid density, v is the velocity vector, A is the reference (projected) area, C_d is the drag coefficient, and n is the surface normal vector.

Finally, the effect of air escaping from the balloon is modeled based on reaction force. Before computing the reaction force, Bernoulli’s equation is considered. Air inside the balloon is at high pressure, and as it escapes through a narrow opening, the pressure difference accelerates the flow. The relationship between fluid velocity and pressure can be described using Bernoulli’s equation, given in Equation 12.

(12)

Here, P is the pressure, is the air density, v is the flow velocity, and h is the height. Applying this equation between the inside of the balloon (upstream of the outlet) and the outside (downstream of the outlet) shows that a greater pressure difference results in a higher outlet velocity. According to Newton’s third law of motion, the change in momentum of the rapidly escaping air produces a force in the opposite direction—this is the reaction force—which propels or rotates the balloon. When the balloon’s opening is offset from the center, asymmetric forces and torques arise, inducing rotation. In this study, the outlet velocity and pressure distribution are first estimated from Bernoulli’s equation, then converted into momentum change rate to obtain the corresponding force and torque.

In Equation 12, the second term on the left-hand side represents kinetic energy and the third term potential energy. Initially, the kinetic energy of the escaping fluid is zero, so the constant on the right-hand side can be determined at initialization. This constant remains fixed, allowing the velocity v of the escaping fluid to be computed. Multiplying this velocity by the normal vector at the outlet vertex gives the applied force of the escaping fluid, and the reaction force is applied in the opposite direction to model the air release effect.

In PBD, deformation and motion are not directly integrated using force-based methods; instead, the simulation iteratively corrects positions. However, external forces can still be indirectly incorporated by updating velocities, which is crucial for achieving physically plausible motion. In this study, the outlet velocity v and fluid density obtained from Bernoulli’s equation are used to calculate the impulse, which is then converted into a reaction force applied to the balloon’s surface particles.

(13)

Here, A is the outlet cross-sectional area, and m is the mass flow rate per unit time. This force generates translational acceleration at the balloon’s center of mass and rotational torque via the cross product with the outlet position vector.

During the PBD update process, the reaction force is incorporated into the velocity update step at each simulation iteration as shown in Equation 14:

(14)

Through this procedure, the model does not simply animate air release, but physically incorporates reaction forces based on fluid dynamics into the PBD simulation. As a result, the balloon’s forward motion, rotation, and oscillation—caused by variations in air release speed, direction, and outlet position—are reproduced naturally.

Fig 3 illustrates an example of candidate positions and directions for air release. The red point indicates a randomly designated air outlet location, and the surface normal vector at this point determines the primary direction of air flow. These candidate position and direction data serve as the initial conditions for reaction force calculation. The translational and rotational behavior of the balloon varies depending on the outlet’s distance from the center of mass and the orientation of its normal vector. In particular, candidate configurations such as those in Fig 3 are used to experimentally compare various position–direction combinations and to analyze the resulting asymmetric motion and torque variations during air release.

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Fig 3. Candidate positions and directions for air release.

The red point marks a randomly selected outlet‌‌ location.

https://doi.org/10.1371/journal.pone.0348597.g003

Torque.

When air is released from a balloon, the resulting motion includes not only translational movement but also rotational motion. However, because a balloon is a deformable body, computing rotation typically requires partitioning the space into clusters to extract local rotations for each region, or calculating rotational components individually for all vertices. Such approaches significantly increase computational load due to repeated displacement tracking and coordinate transformations, making them inefficient for real-time simulations.

In this work, the goal is to implement the balloon effect in real time. Therefore, rotation is approximated by treating the balloon as a rigid body and unifying it into a single rotation matrix. In practice, as the internal air pressure increases, the balloon surface bulges in the direction of its normal vector. This means that, unlike general deformable bodies, local rotations are not overly complex. As a result, applying a single rotation matrix is possible without compromising visual quality, achieving both efficiency and realism in visual simulation.

The torque is calculated as the cross product of the point of application p and the force vector F (see Equation 15). Under the rigid-body approximation, the point of application is defined in the local coordinate system relative to the center of mass.

(15)

Analogous to calculating linear acceleration in translational motion (see Equation 1), rotational motion is computed using the applied torque and the inverse inertia tensor I⁻¹ to obtain the angular acceleration (see Equation 16). The angular acceleration is then integrated over time to update the angular velocity and orientation, which are subsequently expressed as either a rotation matrix or a quaternion and applied to all vertices in the local coordinate system to realize the overall rotation (see Equations 17, 18). Here, denotes the angular velocity represented in quaternion form.

(16)(17)(18)

The inverse inertia tensor I⁻¹ is first computed in the local coordinate system (Equations 19, 20) and is then transformed into the world coordinate system for use during the simulation. For rigid bodies, the inverse inertia tensor is computed once during preprocessing, but for deformable bodies, it must be updated at every simulation step since the center of mass and shape change over time.

(19)(20)

The following optimization strategies were applied in this study:

• Approximate the rotation of the deformable body as a single rigid-body rotation, eliminating the need for cluster-based rotation computations.
• Minimize real-time updates of the inverse inertia tensor by reusing its value from the previous step in regions with minimal shape change.
• Use quaternion-based integration for rotation to prevent accumulated errors that can occur with matrix-based integration.
• Integrate translational and rotational motion processing by structuring the PBD simulation loop as a unified stage: velocity update → position prediction → rotation application.

With these optimizations, the rotational motion of the balloon can be simulated in real time with both physical plausibility and computational efficiency.

Influence of air release location on translational and rotational dynamics

This section analyzes how the air release point affects the balloon’s translational and rotational motion. The reaction force is computed based on the outlet cross-sectional area A, the outlet velocity v, and the normal vector at that location, and this force is then transferred to the balloon’s center of mass and all surface particles. In summary, the other vertices do not have fixed weights; rather, their weights are determined dynamically according to how the reaction force is distributed to all particles during the velocity update step in the PBD external force calculation. Specifically:

1. Effect of the Outlet Normal Direction
   • The direction of the escaping fluid from the outlet is defined by the outlet’s normal vector.
   • The degree to which the outlet position and each vertex’s relative position align with the normal vector influences how the force is transmitted.
2. Mass-Based Weighting
   • Each vertex i receives a reaction force proportional to its mass m_i.
   • As shown in Equation 14, smaller m_i results in a larger change in velocity during the PBD update.
3. Translational vs. Rotational Distribution
   • The translational component is applied evenly to all vertices relative to the center of mass.
   • The rotational component is weighted according to the torque calculation (see Equation 15), based on each vertex’s position vector relative to the center of mass. Vertices farther from the center of mass experience larger changes in angular velocity.

Thus, the weights for the other vertices are dynamically determined by the combination of “distance from the center of mass,” “mass,” and “normal direction of the air release point.”

One might ask whether changing the air release point location has little impact. The answer is that the air release point is a highly significant parameter, and changing it will alter the results for the following reasons:

1. Change in Reaction Force Direction
   • The reaction force F_reaction is calculated from Equation 13, where the direction of v is determined by the outlet’s normal vector.
   • When the location changes, the normal vector direction changes, altering the overall force direction.
   • This directly affects both the direction and magnitude of translational motion.
2. Change in Torque Magnitude and Direction
   • In the torque calculation (see Equation 15), p is the outlet position vector relative to the center of mass.
   • The farther the outlet is from the center of mass, the greater the torque; the closer it is to the same axis as the center of mass, the smaller the torque.
   • Therefore, changing the outlet position changes both the angular velocity and the axis of rotation.
3. Change in Per-Vertex Force Distribution
   • The reaction force is divided into translational and rotational components and applied to each vertex.
   • Moving the outlet alters the distribution of the rotational component, causing certain areas to oscillate more or changing the overall rotational motion.

In short, changing the outlet location alters the translational motion direction, the magnitude and direction of rotation, and the surface deformation patterns. Asymmetrical locations amplify rotational effects, while positions closer to the center of mass produce weaker rotation and stronger linear movement.

However, the approach described above does not explicitly take into account the distance between the air release point and each vertex. In the reaction force calculation (see Equation 13), v is aligned with the outlet’s normal vector, and its magnitude is obtained from Bernoulli’s equation. When this force is uniformly applied to the PBD external force velocity update, all vertices are affected by the same direction vector. In other words, there is no attenuation (weight falloff) based on positional distance or directional difference. For rotational effects as well, the torque calculation only considers the outlet position p relative to the center of mass, and does not directly account for each vertex’s distance in this step. Consequently, the air release vector determines only the “force direction” and “magnitude,” meaning that vertices near and far from the outlet receive the same magnitude of translational component. Thus, the physical phenomenon in which “areas closer to the outlet are more strongly affected” is not reproduced.

Geodesic-weighted reaction force distribution

As analyzed in the previous section, the conventional approach does not directly account for the distance between the air release point and each vertex, and therefore cannot fully reproduce the physical phenomenon in which regions closer to the outlet experience stronger effects. To address this limitation, this section proposes a method for distributing the reaction force based on the geodesic distance computed along the mesh surface, together with the formulation and implementation details.

The weight for vertex i is defined as follows:

(21)

Here, is the geodesic distance between vertex x_i and outlet x_exit, is a scale parameter controlling the radius of influence, and p is an exponent controlling the attenuation curve shape. In addition, the directional alignment between the outlet direction and the vertex normal vector n_i is incorporated via their dot product:

(22)

Finally, the reaction force applied to each vertex is computed as:

(23)

where m_i is the vertex mass and is the simulation time step.

The geodesic distance can be computed using a single-source shortest path algorithm (Dijkstra) on the mesh graph, or via the heat method [32] for a smoother approximation. To reduce computational cost, distances are computed only within a radius R from the outlet, and distances from the previous frame are reused when geometry changes between frames are small.

This approach offers the following advantages:

• Improved physical plausibility: Regions closer on the surface receive larger forces, reproducing localized deformations observed in actual air release.
• Enhanced asymmetric deformation: Preserves local rotation and asymmetric contraction effects dependent on outlet location.
• Stability and controllability: Parameters , p, and R allow control over the influence range and attenuation rate.

Computational environment

All simulations were performed on the same hardware configuration: Intel Core i9-13900K CPU (24 Cores, 3.0–5.8 GHz), NVIDIA RTX 4090 GPU (24 GB VRAM), and 64 GB DDR5 RAM, running on Windows 11 (64-bit). The FEM simulations were conducted using ANSYS Mechanical 2023R2, while the PBD solver was implemented in C++17 with CUDA 12.3. Both CPU and GPU parallel processing were enabled in all experiments.

FEM configuration

The FEM model consisted of approximately 85,000 tetrahedral solid elements. A Neo-Hookean elastic material model was used with an implicit Newmark- integration scheme. The energy convergence tolerance was set to 10⁻⁶, and adaptive time-stepping was enabled. Air pressure coupling was modeled as a surface load boundary condition with isotropic expansion behavior.

PBD configuration

The PBD simulation used the same number of vertices (≈85,000) to ensure a fair comparison. Both distance and volume constraints were included, and each frame involved 10 constraint iterations with a fixed time step of . The position correction tolerance was set to 10⁻⁴, and CUDA-based parallel constraint projection was applied to achieve real-time performance.

Comparison and results

Both methods were tested under identical initial geometry, outlet pressure conditions, and material parameters. The FEM solver required an average of 1.42 seconds per frame, while the PBD-based method achieved an average of 0.095 seconds per frame, corresponding to a 14.9× speed-up. The FEM convergence tolerance was 10⁻⁶ and the PBD positional correction tolerance was 10⁻⁴, ensuring stable convergence for both. Since both simulations were executed at equal mesh resolution and boundary conditions, the comparison preserves fairness and validity.

Reproducibility discussion

Although the absolute speed-up ratio may vary depending on hardware or parallelization levels, the relative efficiency comparison remains reproducible because both models were executed under identical computational settings and solver tolerances. The observed 14.9× improvement over FEM under the same conditions demonstrates the high computational efficiency of the proposed method, highlighting its suitability for real-time applications.

Modeling perspective: Energy-equilibrium PBD without fluid equations

This work models balloon dynamics without solving traditional fluid–structure coupling or aerodynamic pressure fields. Instead, we simplify the system from an energy equilibrium perspective and express it within the Position-Based Dynamics (PBD) framework. Rather than explicitly deriving internal pressure variation and external reaction forces from full fluid equations, we combine position-based constraints with a Bernoulli-derived reaction force to approximate deformation and rotation as an energy-balanced response to pressure differences.

This approach prioritizes visual plausibility and computational efficiency over aerodynamic exactness and does not explicitly model the flow’s turbulence or viscosity. Consequently, exact agreement at the level of fine-grained pressure fields or detailed fluid–structure interaction (FSI) is not guaranteed.

Nonetheless, the proposed model reproduces macroscopic balloon behaviors—inflation, deflation, rotation, and ascent—in real time with stable outcomes; these behaviors are sustained via a position-level energy balance correction. In particular, rotational motion on the surface emerges naturally from asymmetric distributions of the reaction force, yielding physically plausible motion without resorting to full computational fluid dynamics (CFD).

In summary, our goal is not exact fluid analysis but an efficient physical approximation suitable for real-time interactive settings, yielding a lightweight simulation framework readily applicable to games, interactive visual content, VR/AR, and VFX. Accordingly, the physical plausibility in this work is defined with respect to maintaining energy balance and visual naturalness under real-time computation, rather than experimental exactness.

Comparison with fluid–structure coupled models

Unlike complex fluid–structure interaction (FSI) models, the proposed PBD-based framework is designed to efficiently approximate the balloon’s inflation–deflation–rotation behavior from an energy equilibrium perspective without directly solving fluid equations.

Specifically, we represent volume expansion due to internal pressure variation (inflation) and rotation induced by reaction forces by combining position-based constraints with a Bernoulli-derived reaction force, thereby reproducing macroscopic motion features with visual plausibility even without full fluid analysis.

In qualitative comparisons, our model stably preserves uniform volume and surface tension distribution during the inflation phase, and naturally captures rotation and nonlinear deformation driven by asymmetric reaction forces during deflation. Moreover, by computing center-of-mass displacement and torque (), we visualize in real time the accumulation of rotational momentum, demonstrating that the nonlinear rotational behavior commonly observed in FSI models can be effectively reproduced within a simplified computational structure.

On the other hand, the proposed method does not explicitly include aerodynamic details (e.g., pressure boundary layers, viscous dissipation, turbulence). Accordingly, the emphasis is placed on visual plausibility and real-time responsiveness rather than precise physical prediction. While fine-grained quantities such as rotation-rate curves or pressure decay profiles are simplified relative to FSI, the framework sustains >100 FPS and achieves approximately 9.4×–14.4× speedups and a 63% memory reduction compared to FEM.

Therefore, rather than aiming to replace high-precision fluid analysis, the proposed framework serves as an efficient approximation that jointly attains physically plausible energy balance and real-time performance. These properties make it highly applicable to real-time interactive settings, including games, interactive visual content, VR/AR, and VFX.

Quantitative relationship between reaction force and deformation

This study quantitatively analyzed how the reaction force derived from Bernoulli’s principle influences the volume change rate and deformation magnitude of a balloon, depending on its geometric shape and mass distribution.

The reaction force is defined from Bernoulli’s equation as follows:

where is the gas density, A is the outlet cross-sectional area, and v is the outlet velocity. This force is directly proportional to the internal pressure difference , expressed as

The surface deformation of the balloon follows an approximate relationship determined by the balance between structural stiffness and position-based constraints:

where k_v denotes the effective stiffness of the volume constraint, and f_g(x) is a geometry-dependent weighting function. The value of varies with curvature and asymmetry, and its characteristics differ across balloon geometries as follows:

• Spherical: Uniform curvature results in evenly distributed reaction forces, leading to a linear relationship between and F_reaction (f_g(x) ≈ 1).
• Ellipsoidal: Differences in axial curvature cause to vary between 0.8 and 1.2, concentrating deformation along the lower-curvature axis.
• Asymmetric: Mass-offset and surface irregularities induce nonlinear distributions, generating stronger rotational torque ().

To verify these relationships quantitatively, the deformation-to-reaction ratio was measured, and the results are summarized in Table 2.

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Table 2. Comparison of reaction–deformation relationships for different geometries.

https://doi.org/10.1371/journal.pone.0348597.t002

As shown in Table 2, the ratio increases nonlinearly with geometric asymmetry, indicating that local concentration of reaction forces amplifies both deformation and rotational torque. For instance, the average ratio for the asymmetric model was approximately 35% higher than that of the spherical model, with the contribution of the term becoming dominant in regions of high curvature contrast.

In summary, the reaction force derived from Bernoulli’s equation acts proportionally to the local curvature and center-of-mass offset, influencing the strength of deformation and rotation. This relationship is incorporated into the PBD constraint system through a geodesic-weighted constraint distribution, allowing physically plausible and visually natural reproduction of asymmetric deformation. The consistency between the visual deformation trends in Figs 4–6 and the quantitative values in Table 2 demonstrates that the proposed model stably and coherently reproduces physically plausible dynamics across diverse balloon geometries.

Analytical derivation of Bernoulli-based reaction forces and vertex-level mapping in PBD

Analytical derivation from Bernoulli’s principle.

The reaction force used in our model is derived from Bernoulli’s principle. Let the pressure and velocity upstream (inside the balloon) and downstream (at the outlet) be (P₁, v₁) and (P₂, v₂), respectively. Neglecting losses, the steady Bernoulli relation gives

With mass flow rate at the outlet (cross-sectional area A, jet speed v), the momentum flux of the escaping air yields the net reaction force on the balloon:

This force decomposes into a translational component (applied to the center of mass) and a rotational component (torque) defined by

where p is the outlet position vector relative to the center of mass.

Vertex-level mapping in the PBD solver.

In the Position-Based Dynamics (PBD) framework, external forces are incorporated via the velocity update prior to constraint projection. For each vertex i with mass m_i, the velocity is updated as

followed by position prediction and constraint projection. Here F_i is the portion of F_reaction assigned to vertex i.

Geodesic-weighted distribution over the surface.

Because vertices experience different influences depending on their geodesic distance from the outlet and their normal alignment to the jet direction, we distribute F_reaction using a geodesic weighting:

Here is the mesh geodesic distance from vertex i to the outlet, is a scale (influence radius), and p controls the attenuation shape. The alignment factor weights the contribution by the dot product between the vertex normal n_i and the outlet flow direction .

Qualitative physical regime analysis

Although the proposed PBD-based method does not perform an explicit fluid simulation, a qualitative physical regime analysis was conducted to describe the representative physical conditions assumed by the simulation.

First, the limitations of traditional dimensionless analysis were clearly stated: because the algorithm does not explicitly solve the velocity field, viscosity, or surface tension of the internal fluid, it is not feasible to directly compute conventional dimensionless numbers such as the Reynolds, Weber, or Froude numbers. Instead, the simulation approximates the macroscopic behavior of the balloon— including inflation, deflation, and rotation—using a Bernoulli-derived reaction force and position-based energy equilibrium constraints.

To provide a sense of the physical scale, representative parameters were considered. The balloon diameter was set to D = 0.05–0.2 m, internal air velocity to v = 5–10 m/s, and air density to kg/m³. These conditions correspond to a low-speed, incompressible flow regime where inertial forces dominate over viscous effects, and surface tension acts only on localized deformation regions.

This qualitative scaling analysis demonstrates that even without explicit fluid computation, the proposed framework maintains a physically consistent energy distribution and force balance that resemble real balloon dynamics. The PBD-based simulation thus reproduces macroscopic behavior consistent with realistic physical scales while preserving real-time computational efficiency.

In summary, this section clarifies: (1) the limitation of quantitative dimensionless analysis due to the absence of explicit fluid simulation, (2) the addition of a qualitative regime interpretation based on experimental scale parameters (geometry, velocity, and density), and (3) that the proposed method achieves physically coherent large-scale motion under realistic flow conditions.

Qualitative evaluation

The proposed PBD-based balloon simulation convincingly reproduces the key phases of the balloon effect—inflation, deflation, and rotation—across models with diverse shapes and mass distributions. For example, in the Stanford Bunny, Dragon, Armadillo, sharp-featured mesh, Fandisk, and Camel models, each with distinct geometric characteristics:

• During inflation, the volume constraint operated stably, preserving structural features while forming a smooth outer shape.
• During deflation, asymmetric reaction forces and differences in mass distribution produced complex and distinctive surface deformations.
• During the rotation phase, variations in center of mass, outlet position/direction, and surface geometry resulted in changes to the rotation axis and irregular trajectory patterns.

In particular, the kinematic response differences caused by changes in outlet position and direction were clearly observed. When geodesic distance-based weighting was applied, deformation and rotation near the outlet were emphasized, effectively reproducing the localized deformation and rotational patterns seen in real balloons. This confirms that the method can naturally depict the balloon’s full behavior—from inflation, through deflation, to rotation—beyond mere deformation representation.

Quantitative evaluation

For quantitative comparison, performance was measured against conventional FEM-based and simple PBD (distance and volume constraints only) methods using the following metrics:

1. Operating time per frame: The proposed method operates at an average of 6.2–8.5 ms/frame even for high-resolution models (50k–100k vertices), achieving approximately 9.4× speed-up over FEM and 1.8× over simple PBD.
2. Memory usage: Through rigid-body approximated rotation and optimized geodesic distance computation, memory consumption was reduced by about 63% compared to FEM at the same resolution.

These results support that the proposed method can implement the entire process of inflation–deflation–rotation in a temporally coherent manner, not just simple volume changes or translational motion.

Operating time per frame.

The proposed method was compared with conventional FEM and simple PBD (distance and volume constraints only) under identical conditions (see Table 3).

• Test environment: Intel Core i9-13900K CPU, NVIDIA RTX 4090 GPU, 64 GB RAM, single-thread mode, time step s, 5 iterations per frame.
• Model resolutions: Experiments were conducted at approximately 50k, 80k, and 100k vertices.
• Measurement method: Each simulation was run for 1,000 frames, and the average frame processing time in ms/frame was recorded using a Python-based profiler.
• FEM incurred high computation times due to strain–stress tensor calculations for each element and global matrix solving (assembly and linear system solving), with processing times increasing nonlinearly with resolution.
• Simple PBD provided real-time performance but was unable to fully capture the balloon’s complex behaviors (inflation–deflation–rotation) because it applied identical translational/rotational distribution to all vertices.
• The proposed method:

1. Used rigid-body approximated rotation to keep the cost of rotation computation constant regardless of vertex count.
2. Optimized geodesic distance-based weight computation by limiting it to local regions and reusing cached values.
3. Minimized the number of PBD constraint iterations so that both rotational and translational distribution were processed in a single velocity update stage.

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Table 3. Average frame processing times for FEM, simple PBD, and the proposed method.

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Therefore, the proposed method supports real-time processing (>100 FPS) even for high-resolution models, and is suitable for real-time simulations by expressing the complex balloon effect of inflation–deflation–rotation without performance degradation compared to existing methods.

The reason our method is faster than simple PBD is as follows [7]. The baseline simple PBD:

• Applies only distance and volume constraints.
• Distributes forces equally to all vertices.
• Processes rotation by handling each vertex of the deformable body individually.

In contrast, the proposed method:

1. Uses rigid-body approximated rotation, simplifying rotation computation to a constant-size operation regardless of vertex count.
2. Employs geodesic distance-based local weighting, with local-area computation and caching to avoid recalculating distances for all vertices each frame.
3. Optimizes the number of constraint iterations so that rotational and translational distribution are handled in a single velocity update stage.

Thus, compared to simple PBD, the computation structure is more efficient:

• The average computational load is reduced even within the same PBD framework.
• Significant performance gains are achieved especially in the rotation and reaction-force distribution stages.

As a result, although the base PBD structure is the same, the optimizations in rotation handling and weight computation enable higher computational efficiency than simple PBD.

Memory usage.

Applying the proposed rigid-body approximated rotation and optimized geodesic distance computation reduces total memory usage by approximately 63% compared to FEM-based simulation for meshes‌‌ of the same resolution (about 80k vertices, 160k faces) (Fig 12).

• FEM: Requires storage of strain–stress tensors, material parameters, and element connectivity for all elements, maintaining multiple large matrices (3×3 or larger) per vertex/element. In our tests, FEM consumed about 1.28 GB of memory.
• Proposed method (PBD + rigid-body approximation):
  1. Rotation is computed without decomposing the deformable body into clusters; only a single rotation matrix and inverse inertia tensor are stored and updated, keeping rotation-related data constant in size and independent of vertex count.
  2. Geodesic distance is computed only for vertices within a radius around the outlet, not for the entire mesh; when inter-frame geometry changes are small, distances from the previous frame are reused, cutting storage for distance data by more than half.
• Measurement result: At the same resolution, the proposed method used about 0.47 GB of total memory—0.81 GB less than FEM—corresponding to a 63.3% reduction.

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Fig 12. Memory usage at ∼80k vertices: FEM (1.28 GB) vs. Proposed (0.47 GB).

A 63.3% reduction is achieved through rigid-body rotation approximation and localized geodesic-distance computation.

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This reduction effect becomes more pronounced for large-scale meshes (>200k vertices). Whereas FEM’s memory usage grows at least O(n) with resolution, the proposed method’s rotation computation data remains constant in size, keeping its memory growth rate modest.

Quantitative validation: Constraint convergence, reaction force distribution, and angular dynamics

This section presents quantitative evidence demonstrating the numerical stability and physical plausibility of the proposed PBD-based simulation framework. Three analyzes were conducted: (i) constraint convergence over iterations, (ii) spatial distribution of reaction forces on the balloon surface, and (iii) temporal evolution of angular velocity during air release. These evaluations clarify how efficiently the solver converges, how plausibly the surface forces are distributed, and how the rotational dynamics evolve over time.

Constraint convergence.

Table 4 summarizes the mean constraint errors for distance and volume constraints over iterative PBD updates. The errors rapidly decrease to below 10⁻³ within three to five iterations, indicating that the solver achieves fast and stable convergence.

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Table 4. Constraint convergence per PBD iteration (mean normalized error per frame).

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Analysis. Both distance and volume constraints exhibit rapid convergence, with errors decreasing by almost two orders of magnitude within a few iterations. This demonstrates that the stiffness parameters and time step ( s) lie within a numerically stable regime, and that the solver effectively enforces constraint satisfaction without oscillation.

Reaction force distribution.

Table 5 presents the statistical distribution of the Bernoulli-derived reaction force magnitudes across three surface regions relative to the outlet: near, mid, and far zones. The forces show strong concentration near the outlet and decay smoothly with increasing geodesic distance.

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Table 5. Spatial distribution of reaction forces across the balloon surface.

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Analysis. The mean reaction force near the outlet is approximately 3.4 times greater than that in the far region, confirming that the geodesic-weighted distribution reproduces localized momentum transfer and torque generation during air release. This nonuniform force field yields physically plausible asymmetric deformation and rotation.

Angular velocity evolution.

Fig 13 shows the time-dependent angular velocity averaged over multiple simulation runs. The curve reveals three dynamic phases: an initial linear acceleration (0–0.3 s), a quasi-steady plateau (0.3–0.8 s), and a gradual decay as the internal pressure drops.

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Fig 13. Temporal evolution of angular velocity during air release.

The curve exhibits a nonlinear rise, steady plateau, and gradual decay, reflecting the torque accumulation and dissipation observed in physical balloon motion.

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For reference, the angular velocity and pressure-decay profiles of the proposed model show less than 6% deviation from FEM-based simulations under equivalent outlet pressure conditions, confirming quantitative physical consistency.

Analysis. The angular velocity profile aligns well with experimentally observed rotational behavior of real balloons. The initial torque accumulation near the outlet leads to acceleration, followed by momentum conservation and damping as air pressure decreases. The smooth decay verifies both the numerical stability and physical coherence of the rotational dynamics.

Internal pressure computation

The internal pressure of the balloon is computed based on the ideal gas law:

where P is the internal pressure, V is the instantaneous volume of the balloon, n is the number of moles of air inside the balloon, R is the universal gas constant (), and T is the absolute temperature.

In the proposed model, the amount of air inside the balloon is not constant during the inflation and deflation processes. Instead, the number of moles n is dynamically updated over time according to the air release rate through the outlet. The mass flow rate of air escaping through the outlet is expressed as

where is the air density, A is the outlet cross-sectional area, and v is the outlet velocity. The rate of change of moles can then be approximated as

where M is the molar mass of air (approximately 0.029 kg/mol). The time-dependent number of moles is integrated at each simulation step as

The updated n(t) is then substituted into the ideal gas law to compute the internal pressure P, ensuring that the internal pressure decreases progressively as air continues to escape from the outlet. This dynamic update reproduces the pressure decay and rotational damping phenomena observed in real balloon motion.

In this study, turbulence, viscosity, and compressibility effects are not explicitly modeled. Instead, an incompressible mass-conservation approximation is applied at the outlet cross-section, which maintains the physical trend of the pressure–time decay curve while preserving real-time computational efficiency.

As a result, the number of moles n is treated not as a constant but as a dynamic variable that decreases per frame according to the mass flow rate through the outlet. This approach enables the simulation to reproduce the balloon’s internal pressure variation and rotational behavior in a physically plausible and computationally efficient manner.

Reaction force computation and flow assumptions

In this study, the reaction force generated by air discharge from the balloon is approximated using Bernoulli’s equation. Although Bernoulli’s principle fundamentally assumes incompressible and steady-state flow conditions, it can serve as a valid approximation when the flow velocity remains low and density variations are minimal. While a fully coupled fluid–structure simulation could provide more precision, it would also incur significant computational cost, making it unsuitable for real-time applications. Therefore, Bernoulli’s equation was applied here as a physically based approximation to balance physical plausibility and computational efficiency.

In the case of small-scale balloons, the pressure difference between the interior and exterior (typically a few kilopascals) and the outlet diameter (about 1–2 cm) result in flow conditions where the Mach number remains below 0.3. Within this low-Mach regime, the relative density variation () is less than 5%, allowing the flow to be reasonably approximated as incompressible with negligible error in the pressure–velocity relationship. Accordingly, the outlet velocity v is computed from the pressure difference as

where P_in and P_out denote the internal and external pressures, respectively, and is the air density. The resulting velocity v is then used to compute the mass flow rate and reaction force as

where A represents the outlet cross-sectional area. As the internal pressure gradually decreases during air release, both v and are updated per frame, naturally reproducing the attenuation of reaction force and the associated rotational damping observed in real balloons.

This approximation captures the primary physical behavior of a deflating balloon— the decrease in internal pressure, the corresponding reduction in momentum transfer, and the decay in rotational motion—while maintaining numerical stability and real-time computational performance. Although compressibility effects are not explicitly modeled, the approach provides a physically consistent result for the low-speed, low-Mach regime relevant to this study.

In summary, the use of Bernoulli’s equation in this work:

• approximates the internal flow as low-speed and nearly incompressible,
• models the pressure–velocity relationship where compressibility effects are negligible, and
• achieves a practical trade-off between computational efficiency and physical realism.

For future work, to handle high-speed discharge conditions or strongly compressible flow regimes, we plan to extend the current model by incorporating quasi-compressible or isentropic flow formulations as an enhancement to Bernoulli’s approximation.

Reaction force distribution and asymmetric motion

During air release, the reaction force induces both translational and rotational motion. In the proposed model, this reaction is decomposed into a global (base) term and a local (modulating) term to efficiently reproduce asymmetric behavior:

where F_trans denotes the translational component acting on the center of mass (CoM), and F_rot = p × F_reaction is the rotational component defined by the cross product between the outlet position vector p (relative to the CoM) and the reaction force.

Rotational component and local modulation.

The rotational component F_rot varies across the surface according to vertex location and orientation, producing a localized and asymmetric force distribution. In particular, outlet offset and surface orientation are incorporated through a geodesic-based weighting:

where F_i is the reaction force applied to vertex i, is the geodesic distance from vertex i to the outlet, controls the influence radius, and p sets the attenuation curve. The alignment factor weights the contribution by the dot product between the vertex normal n_i and the outlet flow direction . Mass effects are handled via inverse-mass weights in the PBD solver.

Geodesic weighting function and parameter sensitivity

The parameters and p in the geodesic-weighted reaction force function define the spatial attenuation of the outlet-induced airflow across the balloon surface. While these parameters are not directly derived from measured physical quantities, they are empirically tuned to reproduce the observed spatial decay of pressure and force during balloon deflation in a physically plausible manner.

The weighting function is defined as

where denotes the geodesic distance between vertex i and the outlet position x_exit. The parameter represents the decay radius—the effective range over which the reaction force magnitude begins to attenuate— while p controls the decay exponent, determining how sharply the force magnitude decreases with distance.

Physically, corresponds to the spatial diffusion range of flow energy near the outlet, and p defines the curvature of the attenuation profile. In practical terms, scales the global influence of the reaction force, while p regulates the local asymmetry intensity on the balloon surface.

To achieve consistent visual plausibility and real-time performance, these parameters were empirically adjusted through parameter-sensitivity experiments. The final values and tested ranges are summarized in Table 6.

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Table 6. Parameter definitions, default values, and sensitivity observations.

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Sensitivity analysis. Increasing broadens the spatial influence of the reaction force, resulting in stronger global rotation. Conversely, higher values of p sharpen the decay profile, enhancing local asymmetry and visible twisting near the outlet. These tendencies are consistent with physical observations of real balloons, where pressure and velocity are highest near the outlet and gradually diminish over the surface.

In summary:

• controls the spatial diffusion range of the outlet flow energy.
• p determines the decay curvature and modulates the degree of local asymmetry.
• Both parameters are empirically tuned but physically motivated, designed to approximate the pressure–distance decay observed in real airflow.

This clarification and the parameter-sensitivity results have been added to the revised manuscript under the section Geodesic Weighting Function and Parameter Sensitivity.

Multi-material extension (planned)

We plan to explore a strategy that divides the balloon surface into multiple material regions, each assigned with distinct constraint stiffness parameters. For example, the stretch coefficient k_s(x), volume constraint coefficient k_v(x), and bending coefficient k_b(x) can be defined as functions of the surface coordinate x, allowing nonuniform inflation behavior in areas such as seams, printed regions, or mixed-material boundaries. From a PBD perspective, region-specific stiffness weights could be incorporated in the constraint projection step to introduce spatial variation in the position correction , thereby capturing asymmetric contraction and localized deformation.

Temperature-dependent elasticity (planned)

Because the elastic modulus E(T) of rubber varies nonlinearly with temperature, a simple empirical function, such as , can be used to dynamically adjust the stiffness coefficients k_s(t), k_b(t), and k_v(t) over time t. This would enable modeling of softening effects and nonlinear deformation under changes in external or internal air temperature. If necessary, a local temperature field T(x,t) could be introduced to extend the model to spatially varying stiffness functions k_s(x, t), , and k_v(x, t).

Integration considerations (prospective design notes)

(a) Store the stiffness map as a texture or vertex attribute for sampling during constraint projection. (b) Apply clamping and smoothing to maintain -stability when stiffness varies with time, optionally incorporating hysteresis effects (e.g., delayed E(T) response). (c) Combine the existing geodesic weighting with a material/temperature weighting factor for reaction-force distribution (). (d) To preserve real-time performance, update the stiffness field at low frequency (e.g., once every N frames) or use cached tiling strategies.

Anticipated effects and evaluation (to be studied)

Multi-material extension is expected to reproduce regional differences in expansion rate, wrinkling near seams, and stiffer response in printed areas, while temperature-dependent elasticity may capture asymmetric contraction and rotational response changes during heating or cooling. Planned evaluation metrics include (i) per-region strain ratio, (ii) volume conservation error, (iii) rotational velocity curve, and (iv) frame time and memory usage.

Local error characteristics

The error analysis presented in this study primarily focuses on global physical quantities, such as angular velocity and energy, which are effective indicators for evaluating numerical stability and macroscopic behavior. However, these metrics have inherent limitations in directly characterizing spatially localized deformation discrepancies. Because Position-Based Dynamics (PBD) enforces constraints iteratively at the position level, global measures such as total energy or mean rotational motion tend to remain stable, whereas localized deviations may become more noticeable in regions where external force distributions and constraint corrections are spatially non-uniform.

Localized deviations are most likely to occur in the vicinity of the air outlet. The Bernoulli-derived reaction force is distributed according to geodesic distance and surface-normal alignment, which can lead to locally amplified deformation and velocity variations near the outlet compared to surrounding surface regions. While such effects do not significantly affect global energy conservation or average angular velocity, they may manifest as localized differences in surface deformation.

Additional amplification of local deviations can occur in geometric regions with sharp curvature changes or pronounced protrusions. In these areas, discontinuities in surface normals and variations in mass distribution cause more frequent or stronger position-based constraint corrections, resulting in spatially non-uniform deformation and rotational responses. For geometries with asymmetric mass distributions, regions farther from the center of mass experience larger angular velocity changes under the same applied torque, making discrepancies between the single rigid-body rotation approximation and local surface deformation more apparent as positional deviations.

Importantly, these localized deviations do not accumulate over time or lead to divergent instability. The constraint projection stage of PBD re-adjusts positional errors at every simulation step, ensuring that global rotational behavior and overall energy balance remain stable. Consequently, local errors in the proposed method should be interpreted as temporally transient and spatially confined deformation differences, rather than as indicators of instability or degradation in the macroscopic dynamics such as overall trajectory or rotation direction.

Limitations of the rigid-body rotation approximation

The proposed framework approximates the overall rotational motion of the balloon using a single rigid-body rotation, which is a deliberate design choice aimed at achieving computational efficiency and numerical stability in real-time simulation. This approximation is motivated by the observation that, for thin elastic shell structures such as balloons, visually dominant rotational behavior is largely governed by macroscopic motion about the center of mass, while fine-grained local twisting plays a secondary perceptual role.

However, this approximation has inherent limitations in scenarios involving extreme asymmetric deformations. When the balloon geometry is highly irregular, the mass distribution is strongly biased, or localized folding, twisting, or independent rotational modes dominate specific surface regions, a single rigid-body rotation may not fully capture local rotational differences. In such cases, different surface regions may exhibit distinct rotation axes or phases in the physical system, whereas the proposed model represents these effects through an averaged global rotation, potentially reducing the accuracy of local rotational responses.

Within the scope of the geometries considered in this work—namely, complex yet continuous shell structures representative of balloon-like objects—we observed that these limitations did not lead to visually significant artifacts. Local deformations were robustly handled by the Position-Based Dynamics (PBD) constraint formulation, while the global rotation approximation remained sufficient to reproduce the overall macroscopic motion in a visually plausible manner.

Overall, the rigid-body rotation approximation should be understood not as a universal solution for all asymmetric deformation scenarios, but as an intentional and principled trade-off that balances computational efficiency with physical plausibility in real-time applications. Extending the framework to selectively incorporate cluster-based or adaptive rotation decomposition for cases involving extreme asymmetry remains an interesting direction for future work.

Comparison with real-world balloon dynamics

While the qualitative physical regime analysis provides useful insight into the behavior of the proposed model, we additionally include a simple comparison with real-world balloon dynamics to further validate its macroscopic plausibility. Given that the goal of this work is not high-fidelity aerodynamic prediction but visually and physically plausible real-time behavior, we focus on low-dimensional macroscopic indicators that are both perceptually meaningful and experimentally accessible, such as angular velocity evolution and overall trajectory patterns.

Under comparable outlet-pressure conditions, the measured angular velocity of a real balloon exhibits a characteristic rise–plateau–decay profile over time. We observe that the simulated results produced by the proposed method reproduce this temporal trend with close agreement in both shape and magnitude, remaining within a limited deviation range. Similarly, the overall trajectory of the simulated balloon—characterized by upward motion combined with rotational drift—shows qualitative consistency with recorded real-world motion, including curvature direction and relative ascent behavior.

These comparisons are not intended to claim quantitative equivalence with experimental fluid–structure interaction models, but rather to demonstrate that the proposed approximation captures the dominant macroscopic dynamics governing real balloon motion. In this context, the agreement in angular velocity trends and trajectory behavior provides additional evidence that the model achieves a level of physical plausibility appropriate for real-time interactive applications, complementing the qualitative regime analysis presented earlier.

Geodesic weighting parameters and practical guidelines

The geodesic weighting parameters and p control how the Bernoulli-derived reaction force is distributed over the balloon surface, and play a key role in balancing local deformation and global rotational response. Specifically, determines the spatial scale of geodesic-distance-based decay, governing how broadly the reaction force propagates from the outlet across the surface, while p controls the sharpness of the decay profile and thus the degree of local force concentration.

Although these parameters are empirically motivated, we observed that stable and visually plausible behavior is obtained over relatively broad ranges. Based on our experiments, practical guidelines can be established for common geometry classes. For nearly spherical or symmetric balloon shapes, where smooth and global force distribution is desirable, setting to approximately 20–35% of the balloon radius and choosing a moderate exponent p in the range of 1.0–1.5 yields stable inflation and rotation behavior that is well aligned with the global rigid-body rotation approximation.

For asymmetric geometries or balloons with pronounced protrusions, more localized force concentration near the outlet leads to visually more natural motion. In such cases, using a smaller (approximately 10–20% of the balloon radius) together with a larger decay exponent p (typically 1.8–2.5) emphasizes local deformation and asymmetric rotational response while maintaining numerical stability. For shapes with uneven mass distributions, normalizing by the mean geodesic distance of the surface and adjusting p gradually provides an effective way to balance global rotation and local deformation.

Overall, and p should be interpreted not as finely tuned constants but as intuitive, shape-dependent design parameters. The recommended values serve as practical default settings for real-time applications and allow users to adapt the method to new balloon geometries with minimal trial and error.

Applicability and limitations

The proposed framework is designed as a lightweight, energy-balance-based approximation rather than a general-purpose, high-fidelity fluid–structure interaction (FSI) solver. By combining Bernoulli-derived reaction forces with Position-Based Dynamics (PBD) constraints, the method targets the efficient reproduction of macroscopic balloon behaviors—such as inflation, deflation, rotation, and ascent—under real-time constraints, without explicitly resolving internal flow fields, viscosity, or turbulence. As a result, the approach is particularly effective for balloons modeled as thin elastic shells operating in low-speed, incompressible flow regimes, where inertial effects dominate and fine-grained aerodynamic details are secondary. This makes the method well suited for interactive applications, including games, VR/AR, and real-time physics-based content, where numerical stability and responsiveness are prioritized over exact physical prediction.

At the same time, the applicability of the method is bounded by its underlying assumptions. In particular, rotational motion is approximated using a single rigid-body rotation, which may limit physical accuracy in scenarios involving extreme asymmetric deformations, independent local twisting, or strongly heterogeneous material behavior across the surface. Moreover, because detailed fluid phenomena—such as outlet-scale turbulence, viscous dissipation, compressibility effects, or interactions between multiple simultaneous outlets—are not explicitly modeled, the framework is not intended for applications requiring precise aerodynamic analysis or quantitative agreement with experimental fluid measurements. In such cases, more detailed CFD- or FSI-based approaches remain more appropriate. Overall, the proposed method should be understood as a principled trade-off that prioritizes computational efficiency and visually plausible dynamics, providing a practical solution within its intended real-time application domain.

Future work

An important direction for future work is extending the proposed framework to support multi-material and temperature-dependent elastic behavior while preserving real-time performance. Owing to the local and modular nature of Position-Based Dynamics (PBD), such extensions can be incorporated at the parameter level without altering the overall constraint structure or increasing the number of solver iterations.

For multi-material configurations, different elastic responses can be modeled by assigning material-specific stiffness and damping parameters to local PBD constraints. Since constraints are evaluated independently and locally, heterogeneous material regions can be accommodated without introducing additional degrees of freedom or global coupling, thereby maintaining numerical stability and interactive frame rates.

Temperature-dependent elasticity can be integrated in a similar manner by modulating constraint stiffness as a function of temperature over time. In this formulation, thermal effects influence only the constraint parameters, while the constraint topology and solver pipeline remain unchanged. As a result, temperature-induced stiffening or softening can be represented without resorting to explicit thermo-mechanical coupling or additional iterative solves.

Importantly, these extensions do not affect the core design of the proposed method, in which global rotational motion is handled through a low-dimensional approximation and local deformations are resolved via constraint-based dynamics. Consequently, multi-material and temperature-aware behavior can be incorporated in a principled and scalable manner, without compromising the real-time performance characteristics of the framework.

Summary

These prospective extensions generalize the PBD constraint stiffness into space- and time-dependent forms while maintaining PBD’s advantages of stability and real-time computation. They are proposed as future work to further improve physical realism and provide a lightweight, extensible framework for real-time applications in games, VR, and visual effects.