Towards Real-Time Simulation Of Hyperelastic Materials

We propose a new method for physics-based simulation supporting many different types of hyperelastic materials from mass-spring systems to three-dimensional finite element models, pushing the performance of the simulation towards real-time. Fast simulation methods such as Position Based Dynamics exist, but support only limited selection of materials; even classical materials such as corotated linear elasticity and Neo-Hookean elasticity are not supported. Simulation of these types of materials currently relies on Newton\u27s method, which is slow, even with only one iteration per timestep. In this work, we start from simple material models such as mass-spring systems or as-rigid-as-possible materials. We express the widely used implicit Euler time integration as an energy minimization problem and introduce auxiliary projection variables as extra unknowns. After our reformulation, the minimization problem becomes linear in the node positions, while all the non-linear terms are isolated in individual elements. We then extend this idea to efficiently simulate a more general spatial discretization using finite element method. We show that our reformulation can be interpreted as a quasi-Newton method. This insight enables very efficient simulation of a large class of hyperelastic materials. The quasi-Newton interpretation also allows us to leverage ideas from numerical optimization. In particular, we show that our solver can be further accelerated using L-BFGS updates (Limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm). Our final method is typically more than ten times faster than one iteration of Newton\u27s method without compromising quality. In fact, our result is often more accurate than the result obtained with one iteration of Newton\u27s method. Our method is also easier to implement, implying reduced software development costs

Fast Motion Model of Road Vehicles with Artificial Neural Networks

Nonlinear optimization-based motion planning algorithms have been successfully used for dynamically feasible trajectory planning of road vehicles. However, the main drawback of these methods is their significant computational effort and thus high runtime, which makes real-time application a complex problem. Addressing this field, this paper proposes an algorithm for fast simulation of road vehicle motion based on artificial neural networks that can be used in optimization-based trajectory planners. The neural networks are trained with supervised learning techniques to predict the future state of the vehicle based on its current state and driving inputs. Learning data is provided for a wide variety of randomly generated driving scenarios by simulation of a dynamic vehicle model. The realistic random driving maneuvers are created on the basis of piecewise linear travel velocity and road curvature profiles that are used for the planning of public roads. The trained neural networks are then used in a feedback loop with several variables being calculated by additional numerical integration to provide all the outputs of the original dynamic model. The presented model can be capable of short-term vehicle motion simulation with sufficient precision while having a considerably faster runtime than the original dynamic model

Efficient Deformations Using Custom Coordinate Systems

Physics-based deformable object simulations have been playing an increasingly important role in 3D computer graphics. They have been adopted for humanoid character animations as well as special effects such as fire and explosion. However, simulations of large, complex systems can consume large amounts of computation and mostly remain offline, which prohibits their use for interactive applications.We present several highly efficient schemes for deformable object simulation using custom spatial coordinate systems. Our choices span the spectrum of subspace to full space and both Lagrangian and Eulerian viewpoints.Subspace methods achieve massive speedups over their “full space” counterparts by drastically reducing the degrees of freedom involved in the simulation. A long standing difficulty in subspace simulation is incorporating various non-linearities. They introduce expensive computational bottlenecks and quite often cause novel deformations that are outside the span of the subspace.We address these issues in articulated deformable body simulations from a Lagrangian viewpoint. We remove the computational bottleneck of articulated self-contact handling by deploying a pose-space cubature scheme, a generalization of the standard “cubature” approximation. To handle novel deformations caused by arbitrary external collisions, we introduce a generic approach called subspace condensation, which activates full space simulation on the fly when an out-of-basis event is encountered. Our proposed frameworkefficiently incorporates various non-linearities and allows subspace methods to be used in cases where they previously would not have been considered.Deformable solids can interact not only with each other, but also with fluids. Wedesign a new full space method that achieves a two-way coupling between deformable solids and an incompressible fluid where the underlying geometric representation is entirely Eulerian. No-slip boundary conditions are automatically satisfied by imposing a global divergence-free condition. We are able to simulate multiple solids undergoing complex, frictional contact while simultaneously interacting with a fluid. The complexity of the scenarios we are able to simulate surpasses those that we have seen from any previous method

Efficient Motion Planning for Deformable Objects with High Degrees of Freedom

Many robotics and graphics applications need to be able to plan motions by interacting with complex environmental objects, including solids, sands, plants, and fluids. A key aspect of these deformable objects is that they have high-DOF, which implies that they can move or change shapes in many independent ways subject to physics-based constraints. In these applications, users also impose high-level goals on the movements of high-DOF objects, and planning algorithms need to model their motions and determine the optimal control actions to satisfy the high-level goals. In this thesis, we propose several planning algorithms for high-DOF objects. Our algorithms can improve the scalability considerably and can plan motions for different types of objects, including elastically deformable objects, free-surface flows, and Eulerian fluids. We show that the salient deformations of elastically deformable objects lie in a low-dimensional nonlinear space, i.e., the RS space. By embedding the configuration space in the RS subspace, our optimization-based motion planning algorithm can achieve over two orders of magnitude speedup over prior optimization-based formulations. For free surface flows such as liquids, we utilize features of the planning problems and machine learning techniques to identify low-dimensional latent spaces to accelerate the motion planning computation. For Eulerian fluids without free surfaces, we present a scalable planning algorithm based on novel numerical techniques. We show that the numerical discretization scheme exhibits strong regularity, which allows us to accelerate optimization-based motion planning algorithms using a hierarchical data structure and we can achieve 3-10 times speedup over gradient-based optimization techniques. Finally, for high-DOF objects with many frictional contacts with the environment, we present a contact dynamic model that can handle contacts without expensive combinatorial optimization. We illustrate the benefits of our high-DOF planning algorithms for three applications. First, we can plan contact-rich motion trajectories for general elastically deformable robots. Second, we can achieve real-time performance in terms of planning the motion of a robot arm to transfer the liquids between containers. Finally, our method enables a more intuitive user interface. We allow animation editors to modify animations using an offline motion planner to generate controlled fluid animations.Doctor of Philosoph