Sparse Surface Constraints for Combining Physics-based Elasticity Simulation and Correspondence-Free Object Reconstruction

We address the problem to infer physical material parameters and boundary conditions from the observed motion of a homogeneous deformable object via the solution of an inverse problem. Parameters are estimated from potentially unreliable real-world data sources such as sparse observations without correspondences. We introduce a novel Lagrangian-Eulerian optimization formulation, including a cost function that penalizes differences to observations during an optimization run. This formulation matches correspondence-free, sparse observations from a single-view depth sequence with a finite element simulation of deformable bodies. In conjunction with an efficient hexahedral discretization and a stable, implicit formulation of collisions, our method can be used in demanding situation to recover a variety of material parameters, ranging from Young's modulus and Poisson ratio to gravity and stiffness damping, and even external boundaries. In a number of tests using synthetic datasets and real-world measurements, we analyse the robustness of our approach and the convergence behavior of the numerical optimization scheme

CECM: A continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element models

We present the Continuous Empirical Cubature Method (CECM), a novel algorithm for empirically devising efficient integration rules. The CECM aims to improve existing cubature methods by producing rules that are close to the optimal, featuring far less points than the number of functions to integrate. The CECM consists on a two-stage strategy. First, a point selection strategy is applied for obtaining an initial approximation to the cubature rule, featuring as many points as functions to integrate. The second stage consists in a sparsification strategy in which, alongside the indexes and corresponding weights, the spatial coordinates of the points are also considered as design variables. The positions of the initially selected points are changed to render their associated weights to zero, and in this way, the minimum number of points is achieved. Although originally conceived within the framework of hyper-reduced order models (HROMs), we present the method's formulation in terms of generic vector-valued functions, thereby accentuating its versatility across various problem domains. To demonstrate the extensive applicability of the method, we conduct numerical validations using univariate and multivariate Lagrange polynomials. In these cases, we show the method's capacity to retrieve the optimal Gaussian rule. We also asses the method for an arbitrary exponential-sinusoidal function in a 3D domain, and finally consider an example of the application of the method to the hyperreduction of a multiscale finite element model, showcasing notable computational performance gains. A secondary contribution of the current paper is the Sequential Randomized SVD (SRSVD) approach for computing the Singular Value Decomposition (SVD) in a column-partitioned format. The SRSVD is particularly advantageous when matrix sizes approach memory limitations

Seeing and Hearing Fluid Subspaces

Fluids have inspired generations of artists and scientists throughout history. Aesthetically, the wide variety of abstract shapes they form is both surprising and pleasing. Besides visual art, which until the digital age mostly captured frozen moments in time, late 19th-century composers such as Debussy and Ravel wrote works of music inspired by the movement of fluids over time. With the framework of several basic conservation laws of physics, earlier 19th-century scientific work discovered a set of differential equations called the Navier-Stokes equations that described the time evolution of fluid velocity fields. In recent years, the advent of higher computing power and the birth of computer graphics as a discipline has given rise to computational methods for approximating and visualizing solutions to the Navier-Stokes equations, which had previously remained intractably complex. Many artists and musicians have also embraced digital technologies, allowing for the development of algorithmically generated music as well as multimodal representations of large, complex data sets. With this new technology, it is natural to consider the following question: is it possible to systematically generate sounds from fluid dynamics while retaining an underlying musicality? In this dissertation, we present a framework for generating correlated correlated fluid motions and musical sounds using the empirical eigenvalues of a subspace fluid simulation. Our method is multimodal in nature, allowing for the generation of musical sound as well as novel visual forms. The specific mapping from fluid velocity to sound chosen allows for control and modulation of both the visuals and the audio in an integrated, unifying fashion.The method of subspace simulation, which our mapping framework relies on, has a known drawback of high memory consumption. As a means of overcoming this technical obstacle, we also present a data compression framework for fluid subspaces. Our proposed algorithm can achieve an order of magnitude data compression without any noticeable visual artifacts. Using this compression algorithm allows the potential for simulating greater variety of complex scenes on powerful computers as well as the ability to run previously too-complex scenes on a laptop

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