Combined Lagrangian-Eulerian Approach for Accurate Advection Toshiya HACHISUKA The University of Tokyo Introduction Grid-based fluid simulations using the Navier-Stokes equations have become popular for computer animations of fluid flow. While grid-based methods have some advantages as compared with particle-based methods, they also have a major disadvantage called “numerical diffusion”. Numerical diffusion is the numerical errors that occurs in calculation of advection equations. This error causes mass dissipation and motion damping of fluids, so the detail of fluid animation is filtered out. Both of the artifacts are visually distracting artifacts in fluid animations, such as a shrinking water or a rising smoke without small vortices. Results Demo application is available at Mapping Functions In this poster, we propose a new algorithm to accurately calculate advection equations. The method can calculate highly detailed advection of arbitrary fields including fields without boundaries. This property is important because almost all physical quantities do not have specific boundaries, like the density field of smoke or velocity fields. The method can be used for any application involving advection phenomena because it is general solver of advection equations. The following list shows some features of the proposed method. Provides concept of mapping functions of advection Greatly reduces numerical diffusion Can be applied to any type of fields GPU friendly Easy to implement Algorithm Overview The proposed method mainly consists of the advection phase and the non- advection phase. In the advection phase, the method calculates the current field from the initial field by using the combined method of lagrangian method and eulerian method. In the non-advection phase, influences of non-advection terms are added back to the initial field by using the mapping functions of advection equations. The main feature of the method is that the method retains the initial field and warps the initial fields into the current advected field at every time-step. Initial fields, velocity fields and the current fields are stored in fixed grid. Mapping functions are also stored at each grid point as mass-less particles. Advection Phase Initial Field Mapping Function G Current Field Non-Advection Terms Mapping Function F Advection Phase Non-Advection Phase The algorithm updates the mapping functions of advection equations, not the advected field itself. It calculates both the mapping function F and the inverse mapping function G. The update process of mapping function is performed by tracing mass-less particles that have positions as values of mapping functions. The process is done once per time-step. Flow Direction Forward Tracing of Particles Backward Tracing of Particles This process is full-Lagrangian method (i.e. flow using particles), so numerical diffusion associated with Eulerian methods does not exist to the mapping functions. Note that the method has no problem for time-space varying velocity fields (even if the velocity field self-advected, the method can be applied without problems). Theoretical basis of our method does not impose restriction for the type of velocity fields (in practice, it works). In the advection phase, the method warps initial field into the current advected field. This can be done by performing “texture mapping” of initial fields using inverse mapping function G as texture coordinates. The order of the method is determined by the order of filtering function for texture mapping. If you use cubic filtering function, the method becomes 3rd-order in space. The method is not restricted to 1st-order in space. Remapping In the non-advection phase, the method adds non-advection terms into the initial field. This can be done by performing "texture mapping" using mapping function F as texture coordinates, then adding the mapped current non-advection terms. Non-Advection Phase Map Initial Field Current Field Interpolate at G Future work Applying the method to various fluid phenomena Developing more consistent remapping process Modifying the method to mesh-less calculation Map & Add Non-advection terms include any effect like source term, pressure term and boundary condition (include moving, arbitrary one). The method can handle almost all processes which can be handled by previous semi-Lagrangian methods. Current Field This process is needed at every several time-steps in practice, to prevent extreme stretch of the mapping. Remapping process copies the current field into the initial field and initializes the mapping functions by the position of grid points. Our method causes numerical diffusion only at this process. Note that even if you perform remapping at every other time-step, numerical diffusion is reduced compared with previous grid-based method like semi-Lagrangian methods. This process can be viewed as a extension of semi-Lagrangian methods. While semi- Lagrangian methods use the upstream positions by one time-step, our method uses the upstream positions by several time-step. The advantage over particle-based methods, which have no numerical diffusion, is that the fields are defined continuously. Note that each grid point of the field is associated with exactly one particle of the mapping functions. Calculate Non-Advection Terms Initial Field The method is easy to implement if you have already implemented semi-Lagrangian methods because the required processes are similar. Moreover, the method is suitable for GPU implementation because fundamental process is the same as texture mapping. As a preliminary result, we have implemented 2D/3D smoke simulation using the Navier-Stokes equations. Our method produces highly detailed fluid motion/appearance because it greatly prevents numerical diffusion. In addition, computation time per frame is almost the same because calculation of advection is not the bottleneck of fluid simulation. We also have implemented the method on GPUs. Interactive demo application is available on the above website. Semi-Lagrangian Method Our Method Since our method always constructs the current field from the initial field, the error in current field is not accumulated. In contrast, since previous Eulerian methods for advection equations uses the previous fields to calculate the current field, the error is accumulated and becomes numerical diffusion.