Advances in computer graphics continue to revolutionize the production of animation tools for mimicking complicated contacts among objects such as penguins and deferred colliding elastic spheres and plastic cylinders, which are mentioned in the paper. Lagrangian methods are known to support the unrestrained simulation of a variety of object domains while curtailing the time-step limitation that can perturb Eulerian simulations. Conversely, Eulerian simulations depend on static essential features of a spatial discretization of objects to circumvent the exposure of extremely distorted objects. But how should an Eulerian solid simulator be rooted in a Lagrangian grid to capitalize on the advantages of both simulation methods?

Recognizing the advantages and downsides of both simulation methods, the authors build an Eulerian-on-Lagrangian solid simulator that avoids the entangled distortions in large objects without requiring any explicit object discretization. They succinctly present, discuss, and evaluate algorithms for (1) simulating the Lagrangian rigid object motions using the deformed Eulerian object to derive the spatial velocity at any time and point in the object domain; (2) taking advantage of redundancies in the space of advections to efficiently compute the Eulerian momentum as an object transforms from the spatial to intermediate velocity; (3) solving quadratic programs of the spatial velocities of bumping objects with limits and dynamic time steps to decide on the overall momentum, prior to applying a least-squares method to optimally compute the applicable Eulerian and Lagrangian velocities; and (4) mirroring “plasticity without smoothing of the plastic deformation in an Eulerian context.”

The Eulerian-on-Lagrangian solid simulator was applied in the simulation of colliding spheres, plastic cylinders, and penguins. The results reveal the freedom of objects to roam and interact without wasting much computer memory. However, the runtime for the simulator depends on sequentially processing the objects. This constrains the performance of the simulation of several biological tissues and large fluid flows. Nevertheless, the authors provide a procedure for reconstructing the surfaces of the object materials to transform the meshes into the spatial domains for adaptation in accurate simulations. The presentation of the various numerical solutions to the problems of velocity and fluid kinematics in this paper calls for the incorporation of courses in numerical analysis and applied mathematics [1,2] into the curriculum of undergraduate and graduate education in computational sciences and engineering.

I encourage all readers without a strong background in numerical mathematics to browse the fundamentals of fluid mechanics [3], Lagrangian fluid dynamics [4], and numerical methods [1], prior to exploring the insightful mathematical ideas of computer graphics presented in this paper.