This is a corrigendum for the article “Wavelet Fluids” published in ACM Trans. Graph. 44, 6, Article 270 (December 2025), 17 pages.

This paper introduces a novel wavelet-based framework for simulating both single-phase (e.g., smoke) and two-phase (e.g., bubbly water) flows, featuring unified boundary condition handling for free surfaces and solid obstacles. In liquid simulations, conventional pressure projection methods enforce zero-pressure Dirichlet conditions at free surfaces by solving a simplified pressure Poisson equation. However, these approaches neglect air-phase incompressibility, leading to artificial bubble collapse. Stream function methods overcome this limitation by solving a density-variable vector potential Poisson equation, ensuring incompressibility in both simulated and unsimulated regions while maintaining divergence-free liquid phases independent of solver accuracy. Yet, they triple the linear system's dimensionality and exhibit poor convergence near solid boundaries. The fundamental limitation of both methods stems from their governing equations: singularities emerge as density approaches extreme values. The pressure Poisson equation becomes ill-conditioned when density nears zero (air phase), compromising air-phase incompressibility, while the vector potential equation degrades as density approaches infinity (solid phase), impeding solid-boundary convergence. To address these singularities, we first propose a novel decomposition where zero and infinite densities are well-defined. We then reformulate this decomposition as a fixed-point iteration using density-agnostic curl-free and divergence-free projections, eliminating the need for linear system solves. The error equation is derived, and a necessary and sufficient convergence condition is established. Building on this, we develop an iterative algorithm that efficiently solves the fixed-point problem through alternating wavelet-based non-orthogonal curl-free and divergence-free projections. Additionally, we investigate orthogonal curl-free projections (e.g., Fourier methods) and their complementary divergence-free counterparts, providing a comprehensive comparison between wavelet and Fourier approaches. Our method simultaneously computes pressure and stream functions, retaining the incompressibility benefits of stream function approaches while resolving their computational inefficiencies and solid-boundary convergence issues. Experiments demonstrate our framework's ability to efficiently simulate complex two-phase phenomena, such as the glugging effect during water pouring and multi-liquid-region interactions across zero-density air.

Abstract We introduce an efficient technique for recovering the vector potential in wavelet space to simulate pointwise incompressible fluids. This technique ensures that fluid velocities remain divergence‐free at any point within the fluid domain and preserves local volume during the simulation. Divergence‐free wavelets are utilized to calculate the wavelet coefficients of the vector potential, resulting in a smooth vector potential with enhanced accuracy, even when the input velocities exhibit some degree of divergence. This enhanced accuracy eliminates the need for additional computational time to achieve a specific accuracy threshold, as fewer iterations are required for the pressure Poisson solver. Additionally, in 3D, since the wavelet transform is taken in‐place, only the memory for storing the vector potential is required. These two features make the method remarkably efficient for recovering vector potential for fluid simulation. Furthermore, the method can handle various boundary conditions during the wavelet transform, making it adaptable for simulating fluids with Neumann and Dirichlet boundary conditions. Our approach is highly parallelizable and features a time complexity of O(n), allowing for seamless deployment on GPUs and yielding remarkable computational efficiency. Experiments demonstrate that, taking into account the time consumed by the pressure Poisson solver, the method achieves an approximate 2x speedup on GPUs compared to state‐of‐the‐art vector potential recovery techniques while maintaining a precision level of 10−6 when single float precision is employed. The source code of ‘Wavelet Potentials’ can be found in https://github.com/yours321dog/WaveletPotentials .

Creating visualizations of multiple volumetric density fields is demanding in virtual reality (VR) applications, which often include divergent volumetric density distributions mixed with geometric models and physics-based simulations. Real-time rendering of such complex environments poses significant challenges for rendering quality and performance. This article presents a novel scheme for efficient real-time rendering of varying translucent volumetric density fields with global illumination (GI) effects on high-resolution binocular VR displays. Our scheme proposes creative solutions to address three challenges involved in the target problem. First, to tackle the doubled heavy workloads of binocular ray marching, we explore the anti-aliasing principles and more advanced potentials of ray marching on interior cube-map faces, and propose a coupled ray-marching technique that converges to multi-resolution cube maps with interleaved adaptive sampling. Second, we devise a fully dynamic ambient GI approximation method that leverages spherical-harmonics (SH) transform information of the phase function to reduce the huge amount of ray sampling required for GI while ensuring fidelity. The method catalyzes spatial ray-marching reuse and adaptive temporal accumulation. Third, we deploy a two-phase ray-tracing algorithm with a tiled k-buffer to achieve fast processing of order-independent transparency (OIT) for multiple volume instances. Consequently, high-quality and high-performance real-time dynamic volume rendering can be achieved under constrained budgets controlled by developers. As our solution supports mixed mesh-volume rendering, the test results prove the practical usefulness of our approach for high-resolution binocular VR rendering on hybrid multi-volumetric and geometric environments.

Abstract In this paper, we propose a novel second‐order explicit midpoint method to address the issue of energy loss and vorticity dissipation in Eulerian fluid simulation. The basic idea is to explicitly compute the pressure gradient at the middle time of each time step and apply it to the velocity field after advection. Theoretically, our solver can achieve higher accuracy than the first‐order solvers at similar computational cost. On the other hand, our method is twice and even faster than the implicit second‐order solvers at the cost of a small loss of accuracy. We have carried out a large number of 2D, 3D and numerical experiments to verify the effectiveness and availability of our algorithm.

Abstract Physically plausible fracture animation is a challenging topic in computer graphics. Most of the existing approaches focus on the fracture of isotropic materials. We proposed a frame‐field method for the design of anisotropic brittle fracture patterns. In this case, the material anisotropy is determined by two parts: anisotropic elastic deformation and anisotropic damage mechanics. For the elastic deformation, we reformulate the constitutive model of hyperelastic materials to achieve anisotropy by adding additional energy density functions in particular directions. For the damage evolution, we propose an improved phase‐field fracture method to simulate the anisotropy by designing a deformation‐aware second‐order structural tensor. These two parts can present elastic anisotropy and fractured anisotropy independently, or they can be well coupled together to exhibit rich crack effects. To ensure the flexibility of simulation, we further introduce a frame‐field concept to assist in setting local anisotropy, similar to the fiber orientation of textiles. For the discretization of the deformable object, we adopt a novel Material Point Method(MPM) according to its fracture‐friendly nature. We also give some design criteria for anisotropic models through comparative analysis. Experiments show that our anisotropic method is able to be well integrated with the MPM scheme for simulating the dynamic fracture behavior of anisotropic materials.

Abstract Physically based deformation simulation has been studied for many years in computer graphics. In order to simulate more complex geometric models and better meet the designer's requirements, many anisotropic approaches have been proposed in recent years. However, most of the approaches focus on simulating orthotropic models. In comparison with orthotropic models, nonorthotropic ones allow the objects to have anisotropic behaviors along nonorthogonal directions. In this paper, we introduce an improved approach to simulate nonorthotropic geometric models under large deformation. The improvements are mainly twofold. First, a frame field is specified on a given undeformed object, that is, each point of the object is equipped with a frame. In each local frame, we construct three independent vectors and form a nonorthogonal coordinate. Second, we design the deformation properties along each axis in the local nonorthogonal coordinate to get a local constitutive model. The final nonorthotropic model is generated by transforming the designed model from local nonorthogonal coordinates to the global standard Cartesian coordinate. To improve the stability, we introduce a time‐varying method to simultaneously track the local coordinates reorientation by pushing forward the original frame field to the deformed frame field. Experiments show that the deformation simulation using the designed nonorthotropic models exhibits anisotropic behaviors along different directions and are more stable than previous methods.

Abstract We introduce a new biorthogonal wavelet approach to creating a water‐tight surface defined by an implicit function, from a finite set of oriented points. Our approach aims at addressing problems with previous wavelet methods which are not resilient to missing or nonuniformly sampled data. To address the problems, our approach has two key elements. First, by applying a three‐dimensional partial integration, we derive a new integral formula to compute the wavelet coefficients without requiring the implicit function to be an indicator function. It can be shown that the previously used formula is a special case of our formula when the integrated function is an indicator function. Second, a simple yet general method is proposed to construct smooth wavelets with small support. With our method, a family of wavelets can be constructed with the same support size as previously used wavelets while having one more degree of continuity. Experiments show that our approach can robustly produce results comparable to those produced by the Fourier and Poisson methods, regardless of the input data being noisy, missing or nonuniform. Moreover, our approach does not need to compute global integrals or solve large linear systems.

Abstract In this paper, a novel method with good numerical stability is proposed from the perspective of energy preserving to alleviate the numerical dissipations in the advection step of Eulerian fluid simulation. The main idea is to measure the vorticity loss during advection, calculate the lost angular kinetic energy with a proposed scheme, and then synthesize a high‐frequency incompressible details field to compensate the lost energy in a way that is consistent with Kolmogorov's theory, which prevents the synthetic details from interfering with the existing fluid flow. The method works independently of the advection scheme and can be easily combined with other advection schemes to enhance the effect. It adds only 5% to 10% of the computational overhead while producing convincing fluid details without changing the overall behavior of the original flow.

Abstract Gradient‐domain compositing has been widely used to create a seamless composite with gradient close to a composite gradient field generated from one or more registered images. The key to this problem is to solve a Poisson equation, whose unknown variables can reach the size of the composite if no region of interest is drawn explicitly, thus making both the time and memory cost expensive in processing multi‐megapixel images. In this paper, we propose an approximate projection method based on biorthogonal Multiresolution Analyses (MRA) to solve the Poisson equation. Unlike previous Poisson equation solvers which try to converge to the accurate solution with iterative algorithms, we use biorthogonal compactly supported curl‐free wavelets as the fundamental bases to approximately project the composite gradient field onto a curl‐free vector space. Then, the composite can be efficiently recovered by applying a fast inverse wavelet transform. Considering an n‐pixel composite, our method only requires 2n of memory for all vector fields and is more efficient than state‐of‐the‐art methods while achieving almost identical results. Specifically, experiments show that our method gains a 5× speedup over the streaming multigrid in certain cases.