Subspace Simulation or Model reduction is a technique to simplify simulations of systems modeled by Ordinary Differential Equations. The essential idea is to project the high dimensional sparse equations on to lower dimensional dense equations and then re-projecting the solution back to the original space. Such a method has a much faster computation and lower memory footprint as compared to a full space simulation. Recalculating the subspace basis of a deformable body is a computationally expensive yet mandatory operation. We show that the subspace of a modified body can be efficiently obtained from the subspace of its original version if mesh changes are small. Our basic idea is to approximate the stiffness matrix by its low-frequency component, so we can calculate new linear deformation modes by solving an incremental eigenvalue decomposition problem. We also show a hybridized approach to calculate the modal derivatives and finally we demonstrate that the cubature samples trained for the original mesh can be reused in fast reduced force and stiffness matrix evaluation.
Next, we handle the problem of inverse simulation in deformable bodies. We present a novel system for interactive elastic shape design in both forward and inverse fashions. Using this system, the user can choose to edit the rest shape or the quasistatic shape of an elastic solid and obtain the other shape that matches under the quasistatic equilibrium condition at the same time. The development of this system is based on the discovery that inverse quasistatic simulation can be immediately solved by Newton’s method with a direct solver. To implement our simulator, we propose a Jacobian matrix evaluation scheme for the inverse elastic problem and we present step length and matrix evaluation
techniques that improve the simulation performance. While our simulator is efficient, it is still not fast enough for the system to generate the result in real time. Our solution is a shape initialization method using the recent projective dynamics technique. Shape initialization not only works as a fast preview function during the user editing process, but also speeds up the convergence of quasistatic or inverse quasistatic simulation afterwards. The use of a heterogeneous algorithm structure allows the system to further reduce its preview cost, by utilizing the power of both the CPU and the GPU. Our experiment demonstrates that the whole system is fast, robust, and convenient for the designer to use in both forward and inverse elastic shape design. It can handle a variety of nonlinear elastic material models, and its run-time performance has space for more improvement.
Finally, we look at the problem of fluid simulation. Small-scale liquid flows on solid surfaces provide convincing details in liquid animation, but they are difficult to be simulated with efficiency and fidelity, mostly due to the complex nature of the surface tension at the contact front where liquid, air, and solid meet. In this section, we propose to simulate the dynamics of new liquid drops from captured real-world liquid flow data, using deep neural networks. To achieve this goal, we develop a data capture system that acquires liquid flow patterns from hundreds of real-world water drop experiments. We then convert raw data into compact data for training neural networks, in which water drops are represented by their contact fronts only. Based on LSTM, our neural networks serve three purposes in our system: predicting the contour of a contact front, predicting the color field gradient of a contact front, and finally predicting whether a contact front is going to break or not. Using these predictions, our simulator recovers the interior shape of a water drop at each time step and handles merging and splitting events by geometric operations. The experiment shows that our trained neural networks are able to perform predictions well. The whole simulator is robust, convenient to use, and capable of generating realistic small-scale liquid effects in animation.