Schedule for: 17w5008 - Geometry & Computation for Interactive Simulation

Visual computing is a wide area that includes computer graphics and image processing, where the 'eye-norm' rules. I will briefly discuss two case studies involving numerical methods and analysis applied to this area. The first involves motion simulation and calibration of soft objects such as cloth, plants and skin. The governing elastodynamics PDE system, discretized in space already at the variational level using FEM,leads to a large, expensive to assemble, dynamical system in time, where the damped motion may mask highly oscillatory stiffness. Geometric integration ideas are making their way into visual computing research these days in search for more quantitative computations. The other case study involves some image processing problems where theremis a premium for local approaches that do not necessarily use underlying PDEs. Here the trend is in the other direction. I will demonstrate and discuss.

Numerical algorithms for simulating continua have long been based on the idea that high resolution and high accuracy go hand-in-hand. While engineering and computational mechanics have pushed resolution limits in pursuit of simulation fidelity, computer graphics has often gone the other way, sacrificing resolution and accuracy for performance. However, what if we want both high-performance and predictive accuracy ? In this talk I will discuss how to avoid this seeming fundamental trade-off using measurement-based and data-driven simulation techniques as well as some examples of these methods applied to both forward and inverse problems in graphics and 3D printing. I will conclude with some thoughts on where these methods might take us in the future.

In this talk, we will discuss model reduction techniques that can be used to construct fast approximation algorithms for shape optimization problems. The goal is to obtain approximate solutions at run times that are independent of the resolution of the discrete shapes to be optimized. As applications we will discuss methods for real-time elasticity-based shape interpolation and the processing of curves in shape spaces in which a shape is a single point. Finally, I will outline the concept of compressed vibration modes of elastic bodies, which in contrast to the natural vibration modes, are localized ("sparse") deformations.

A reconfigurable is an object or collection of objects whose trans- formation between various states defines its functionality or aes- thetic appeal. For example, consider a mechanical assembly com- posed of interlocking pieces, a transforming folding bicycle, or a space-saving arrangement of apartment furniture. Traditional computer-aided design tools are best intended for static objects. I will report on some of our recent attempts to create optimization-based tools for computational design of reconfigurable shapes. I will first discuss a prototype of a general purpose tool. Then I will dive into the curious and delightful special case of computational design of generalized Matryoshka, a.k.a. Russian Nesting Dolls.

personal 1-1 introductions

In this talk we will discuss strategies for general purpose software implementation of isogeometric analysis. More than a decade after the introduction of the latter, industrialization of the technology remains a major challenge, since it involves a new way of thinking (a paradigm shift) with respect to existing computer-aided design and finite element analysis software and requires a re-engineering of current design and simulation practices.

An brief overview of different surface representations and their associated analysis (FEM) functions.

We study the properties of C1-smooth isogeometric function spaces over multi-patch domains. In this context, an isogeometric function is a function defined on a B-spline domain, whose graph surface also has a B-spline representation. Here, the domain of interest is composed of multiple B-spline patches. We analyze the space of C1-smooth functions for the special class of analysis-suitable G1 multi-patch domains and construct a basis for a suitable subspace. The subspace possesses optimal approximation properties. The proposed approach is very flexible, can be implemented easily and can be incorporated directly into an IGA framework. We will also discuss possible extensions and generalizations.

Advances in manufacturing, most prominently in additive manufacturing or 3d printing, are enabling the production of high-performance products with ever increasing functional and geometric complexity. Typically, the geometry features that can be designed reach from the product-scale in the order of tens of centimetres down to submillimetre scale. Novel interactive computational tools are indispensable to representing and exploring the corresponding vast design space. Against this backdrop, for the last several years we have been developing robust and scalable immersed/embedded boundary finite elements, which have clear advantages when applied to highly optimised geometrically complex parts. In contrast to conventional finite elements there is no need to generate and painstakingly maintain a boundary-fitted mesh. It is sufficient to have a non-boundary-fitted hexahedral voxel grid that is combined with auxiliary techniques for enforcing boundary conditions. Moreover, the voxel grid makes it possible to employ a host of multiresolution surface and volume representation techniques already available in computer graphics and computer-aided design.

I will first give a short presentation of the objectives and scope of the CGAL open source project. I will the recently added algorithms that are relevant for geometric modeling and simulation: point set processing, polygon mesh processing and mesh generation. For each software component I will discuss the underlying design principles, show a live demo and explain how users can adapt and extend it to their specific needs. Finally, I will briefly review the existing projects for extending the mesh generation toolbox of CGAL.

The collocation method with weighted extended B-splines (WEB-splines) represents a recently published approach for the spline approximation of the solution of stationary partial differential equations. In contrast to standard finite element methods, WEB-collocation requires no mesh generation and numerical integration, which leads to considerably faster computation times and an easier implementation. In this talk, the basics of WEB-collocation for general boundary value problems with mixed boundary conditions are described and the advantages over finite element methods are illustrated for Poisson's equation as typical model problem. On this basis, current research results from the application to singular and time-dependent problems are presented. The utilization of uniform spline spaces permits a straightforward generalization of the basic concept to hierarchical bases and the development of intuitive refinement strategies. The benefits of these adaptive WEB-collocation algorithms are shown in case of the model problem with a singular solution. Furthermore, considering the problem of simulating a tsunami, the combination of the WEB-collocation concept and a time-step iteration is presented to demonstrate a novel approximation scheme for time-dependent equations.

Since the introduction of Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering for DG approximation of univariate hyperbolic equations by Cockburn et al., many generalizations of SIAC filtering have been proposed. Recently, new advancements in connecting the spline theory and SIAC filtering have paved the way for a more geometric view of this filtering technique. Based on which, various generalizations of the SIAC kernel have been proposed to make the filtering viable for more realistic applications. Examples include the introduction of SIAC line integral with applications for streamlining and flow visualization, hexagonal SIAC using nonseparable splines, and position dependent SIAC with nonuniform knot sequences. In this talk, I will introduce the basic concept of the SIAC filtering, its connection with well-established concepts from approximation theory, and discuss the recent advances in SIAC filtering.

Use of computational topology for pattern analysis in turbulent flows Atmospheric science presents us with the problem of complex turbulent flows. We will present some techniques from computational topology which can be used in quantifying this spatio-temporal complexity, and for detecting minimal flow structures in direct numerical simulations.

OFA is an open-source framework for multi-physics simulation. SOFA aims at interactive and real-time applications, with an emphasis on medical simulation. SOFA benefits today from large, active and international community, including international universities, startups and companies. For more flexibility, SOFA is made up of a stable open-source core and many optional plugins (>100 plugins), providing innovative numerical methods and state-of-the-art algorithms. The SOFA core has a LGPL license (permissive and non-contaminating) fostering development of prototypes and products under any commercial license.

Blender2SOFA is a software bridge that semi-automates the scene-generation cycle, a key bottleneck in authoring, modeling and developing VR units for surgery simulation.

I will present a novel approach for computing a homeomorphic map between two discrete surfaces. We optimize a map by computing a mass transport plan between two surfaces. This non-linear problem, which amounts to minimizing the Dirichlet energy of both the map and its inverse, is solved using two alternating convex optimization problems in a coarse-to-fine fashion. Computational efficiency is achieved through the use of Sinkhorn iterations, modified to handle minimal regularization and unbalanced transport plans.

Usually shapes are modeled through a top down approach. Think of smooth surfaces defined by a few control points. In this talk I will propose an opposite paradigm. This approach creates shapes from a bottom up approach. Think of the "Game of life." Shapes emerge from small scale interactions through local interactions. This approach is inspired by micro-biology. Key to this approach is to use a dynamics solver. More details can be found on the following web page. https://www.autodeskresearch.com/publications/jmi2012

We solve a fractional-time diffusion equation by a collocation-Galerkin method that uses the refinable spaces generated by the fractional B-splines as approximating spaces. The main advantage in using the fractional B-splines is in that their derivatives of both integer and fractional order can be expressed in a closed form that involves just the fractional difference operator. We analyze the performance of the method by solving some test problems.

Six-DoF haptic rendering is useful for interactive applications in virtual assembly and maintenance of complex machinery, such as, for example, car engines and landing gears. There are many technological challenges to overcome before such applications can become commonplace. In simulations involving complex distributed contact, there are typically many simultaneous individual contacts, posing stability issues due to accumulated stiffness. In order for simulations to be useful, they must eliminate (or at least minimize) false-positives, i.e., paths that violate contact due to errors in the contact resolution algorithm. Friction is non-trivial, due to the large number of contacts and stringent time requirements. Even preparing the signed distance fields and point clouds can take an unreasonably amount of time with models of realistic complexity. I will give recent advances on these problems in my group at USC. I will discuss signed distance field generation, continuous collision detection, adaptive stiffness and friction.

A variety of techniques were proposed to model smooth surfaces of arbitrary topology based on tensor product splines (e.g. subdivision surfaces, free-form splines, T-splines). Conversion of an input surface into such a representation is commonly achieved by constructing a global seamless parametrization, possibly aligned to a guiding cross-field and using this parametrization as a domain to construct the spline-based surface. (Informally, seamless parametrizations can be thought of as paramezations of surfaces cut to disks, with isoparametric line directions and spacing on the surface matching perfectly across the cuts). One major fundamental difficulty in designing robust algorithms for this task is the fact that for common types, e.g. subdivision surfaces (requiring a conforming domain mesh) or T-spline surfaces reliably obtaining a suitable parametrization that has the same topological structure (matching singularities and more generally rotations of parametric line directions along loops matching that of the cross-field) as the guiding field poses a major challenge. Even worse, not all fields do admit suitable parametrizations, and no concise conditions are known as to which fields do. I will discuss our recent work that addresses the problem by introducing two new concepts: (1) seamless similarity maps -- a relaxation of the seamless parametrization idea, allowing scale jumps across cuts (2) splines with half-edge knots, that relax the global knot interval consistency requirements on surfaces with nontrivial genus. It turns out, that for any given guiding field structure, a compatible parametrization of this kind exists and can be computed by a relatively simple algorithm; at the same time, for any such parametrization, a smooth piecewise rational surface with exactly the same structure as the input field can be constructed from it. This leads to fully automatic construction of high-order approximations of arbitrary surfaces, even with hiighly complex topology, potentially enabling, e.g., robust automatic conversion of surfaces to isogeometric form.

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ketch and sculpt interfaces have often been touted as “natural” approaches to interactive conceptual modeling. This talk explores the coupling of interactive sketching and sculpting with physical simulation to produce structurally stable designs, as well as design forms interactively aided by physical simulation.

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