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FEM Simulation of 3D Deformable Solids: A practitioner’s guide to theory, discretization and model reduction. Part 2: Model reduction
 SIGGRAPH 2012 COURSE NOTES
, 2012
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Subspace fluid resimulation
 ACM Trans. Graph
, 2013
"... Figure 1: An efficient subspace resimulation of novel fluid dynamics. This scene was generated an order of magnitude faster than the original. The solver itself, without velocity reconstruction (§5), runs three orders of magnitude faster. We present a new subspace integration method that is capable ..."
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Cited by 11 (2 self)
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Figure 1: An efficient subspace resimulation of novel fluid dynamics. This scene was generated an order of magnitude faster than the original. The solver itself, without velocity reconstruction (§5), runs three orders of magnitude faster. We present a new subspace integration method that is capable of efficiently adding and subtracting dynamics from an existing highresolution fluid simulation. We show how to analyze the results of an existing highresolution simulation, discover an efficient reduced approximation, and use it to quickly “resimulate ” novel variations of the original dynamics. Prior subspace methods have had difficulty resimulating the original input dynamics because they lack efficient means of handling semiLagrangian advection methods. We show that multidimensional cubature schemes can be applied to this and other advection methods, such as MacCormack advection. The remaining pressure and diffusion stages can be written as a single matrixvector multiply, so as with previous subspace methods, no matrix inversion is needed at runtime. We additionally propose a novel importance samplingbased fitting algorithm that asymptotically accelerates the precomputation stage, and show that the Iterated Orthogonal Projection method can be used to elegantly incorporate moving internal boundaries into a subspace simulation. In addition to efficiently producing variations of the original input, our method can produce novel, abstract fluid motions that we have not seen from any other solver.
Subspace Integration with Local Deformations
"... Subspace techniques greatly reduce the cost of nonlinear simulation by approximating deformations with a small custom basis. In order to represent the deformations well (in terms of a global metric), the basis functions usually have global support, and cannot capture localized deformations. While re ..."
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Cited by 8 (0 self)
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Subspace techniques greatly reduce the cost of nonlinear simulation by approximating deformations with a small custom basis. In order to represent the deformations well (in terms of a global metric), the basis functions usually have global support, and cannot capture localized deformations. While reducedspace basis functions can be localized to some extent, capturing truly local deformations would still require a very large number of precomputed basis functions, significantly degrading both precomputation and online performance. We present an efficient approach to handling local deformations that cannot be predicted, most commonly arising from contact and collisions, by augmenting the subspace basis with custom functions derived from analytic solutions to static loading problems. We also present a new cubature scheme designed to facilitate fast computation of the necessary runtime quantities while undergoing a changing basis. Our examples yield a two order of magnitude speedup over fullcoordinate simulations, striking a desirable balance between runtime speeds and expressive ability.
Smooth Skinning Decomposition with Rigid Bones
"... Figure 1: A set of example poses are decomposed into rigid bone transformations B and a sparse, convex bonevertex weight map W (left hand side) by our block coordinate descent algorithm (right hand side). During the process, the example poses (indicated as blue dots) can be reconstructed more accur ..."
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Cited by 7 (0 self)
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Figure 1: A set of example poses are decomposed into rigid bone transformations B and a sparse, convex bonevertex weight map W (left hand side) by our block coordinate descent algorithm (right hand side). During the process, the example poses (indicated as blue dots) can be reconstructed more accurately by alternatively updating W and B while the other is kept fixed. This paper introduces the Smooth Skinning Decomposition with Rigid Bones (SSDR), an automated algorithm to extract the linear blend skinning (LBS) from a set of example poses. The SSDR model can effectively approximate the skin deformation of nearly articulated models as well as highly deformable models by a low number of rigid bones and a sparse, convex bonevertex weight map. Formulated as a constrained optimization problem where the least squared error of the reconstructed vertices by LBS is minimized, the SSDR model can be solved by a block coordinate descentbased algorithm to iteratively update the weight map and the bone transformations. By employing the sparseness and convex constraints on the weight map, the SSDR model can be used for traditional skinning decomposition tasks such as animation compression and hardwareaccelerated rendering. Moreover, by imposing the orthogonal constraints on the bone rotation matrices (rigid bones), the SSDR model can also be applied in motion editing, skeleton extraction, and collision detection tasks. Through qualitative and quantitative evaluations, we show the SSDR model can measurably outperform the stateoftheart skinning decomposition schemes in terms of accuracy and applicability.
Nonpolynomial galerkin projection on deforming meshes
 ACM Trans. Graph
, 2013
"... Figure 1: Our method enables reduced simulation of fluid flow around this flying bird over 2000 times faster than the corresponding full simulation and reduced radiosity computation in this architectural scene over 113 times faster than the corresponding full radiosity. This paper extends Galerkin p ..."
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Cited by 7 (2 self)
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Figure 1: Our method enables reduced simulation of fluid flow around this flying bird over 2000 times faster than the corresponding full simulation and reduced radiosity computation in this architectural scene over 113 times faster than the corresponding full radiosity. This paper extends Galerkin projection to a large class of nonpolynomial functions typically encountered in graphics. We demonstrate the broad applicability of our approach by applying it to two strikingly different problems: fluid simulation and radiosity rendering, both using deforming meshes. Standard Galerkin projection cannot efficiently approximate these phenomena. Our approach, by contrast, enables the compact representation and approximation of these complex nonpolynomial systems, including quotients and roots of polynomials. We rely on representing each function to be modelreduced as a composition of tensor products, matrix inversions, and matrix roots. Once a function has been represented in this form, it can be easily modelreduced, and its reduced form can be evaluated with time and memory costs dependent only on the dimension of the reduced space.
Geodesic Voxel Binding for Production Character Meshes
"... We propose a fully automatic method for specifying influence weights for closedform skinning methods, such as linear blend skinning. Our method is designed to work with production meshes that may contain nonmanifold geometry, be nonwatertight, have intersecting triangles, or be comprise of multi ..."
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Cited by 4 (0 self)
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We propose a fully automatic method for specifying influence weights for closedform skinning methods, such as linear blend skinning. Our method is designed to work with production meshes that may contain nonmanifold geometry, be nonwatertight, have intersecting triangles, or be comprise of multiple connected components. Starting from a character rest pose mesh and skeleton hierarchy, we first voxelize the input geometry. The resulting voxelization is then used to calculate binding weights, based on the geodesic distance between each voxel lying on a skeleton “bone ” and all nonexterior voxels. This yields smooth weights at interactive rates, without timeconstants, iteration parameters, or costly optimization at bind or pose time. By decoupling weight assignment from distance computation we make it possible to modify weights interactively, at pose time, without additional preprocessing or computation. This allows artists to assess impact of weight selection in the context in which they are used.
BoundaryAware MultiDomain Subspace Deformation
"... In this paper, we propose a novel framework for multidomain subspace deformation using nodewise corotational elasticity. With the proper construction of subspaces based on the knowledge of the boundary deformation, we can use the Lagrange multiplier technique to impose coupling constraints at the ..."
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Cited by 3 (0 self)
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In this paper, we propose a novel framework for multidomain subspace deformation using nodewise corotational elasticity. With the proper construction of subspaces based on the knowledge of the boundary deformation, we can use the Lagrange multiplier technique to impose coupling constraints at the boundary without overconstraining. In our deformation algorithm, the number of constraint equations to couple two neighboring domains is not related to the number of the nodes on the boundary but is the same as the number of the selected boundary deformation modes. The crack artifact is not present in our simulation result and the domain decomposition with loops can be easily handled. Experimental results show that the single core implementation of our algorithm can achieve realtime performance in simulating deformable objects with around quarter million tetrahedral elements.
Simulating Articulated Subspace SelfContact
"... Figure 1: A hand mesh composed of 458K tetrahedra, running at 5.8 FPS (171 ms), including both selfcontact detection and resolution. Our algorithm accelerates the computation of complex selfcontacts by a factor of 5 × to 52 × over other subspace methods and 166 × to 391× over fullrank simulations ..."
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Cited by 3 (1 self)
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Figure 1: A hand mesh composed of 458K tetrahedra, running at 5.8 FPS (171 ms), including both selfcontact detection and resolution. Our algorithm accelerates the computation of complex selfcontacts by a factor of 5 × to 52 × over other subspace methods and 166 × to 391× over fullrank simulations. Our selfcontact computation never dominates the total time, and takes up at most 46 % of a single frame. We present an efficient new subspace method for simulating the selfcontact of articulated deformable bodies, such as characters. Selfcontact is highly structured in this setting, as the limited space of possible articulations produces a predictable set of coherent collisions. Subspace methods can leverage this coherence, and have been used in the past to accelerate the collision detection stage of contact simulation. We show that these methods can be used to accelerate the entire contact computation, and allow selfcontact to be resolved without looking at all of the contact points. Our analysis of the problem yields a broader insight into the types of nonlinearities that subspace methods can efficiently approximate, and leads us to design a posespace cubature scheme. Our algorithm accelerates selfcontact by up to an order of magnitude over other subspace simulations, and accelerates the overall simulation by two orders of magnitude over fullrank simulations. We demonstrate the simulation of high resolution (100K – 400K elements) meshes in selfcontact at interactive rates (5.8 – 50 FPS).
Interactive authoring of simulationready plants
 ACM Trans. Graph
, 2013
"... Figure 1: Simulation of a peach tree with anatomically realistic geometry (Prunus Persica), with fracture. Peaches fall from the tree swaying in the spacetime Perlin wind. 299,707 triangles, 237 branches, 3,556 twigs, 18,536 leaves, 330 fruits, 2,950 reduced DOFs, 7 hierarchy levels, 5 msec of simu ..."
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Cited by 3 (0 self)
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Figure 1: Simulation of a peach tree with anatomically realistic geometry (Prunus Persica), with fracture. Peaches fall from the tree swaying in the spacetime Perlin wind. 299,707 triangles, 237 branches, 3,556 twigs, 18,536 leaves, 330 fruits, 2,950 reduced DOFs, 7 hierarchy levels, 5 msec of simulation per graphical frame. Physically based simulation can produce quality motion of plants, but requires an authoring stage to convert plant “polygon soup” triangle meshes to a format suitable for physically based simulation. We give a system that can author complex simulationready plants in a manner of minutes. Our system decomposes the plant geometry, establishes a hierarchy, builds and connects simulation meshes, and detects instances. It scales to anatomically realistic geometry of adult plants, is robust to nonmanifold input geometry, gaps between branches or leaves, freeflying leaves not connected to any branch, spurious geometry, and plant selfcollisions in the input configuration. We demonstrate the results using a FEM model reduction simulator that can compute largedeformation dynamics of complex plants at interactive rates, subject to user forces, gravity or randomized wind. We also provide plant fracture (with prespecified patterns), inverse kinematics to easily pose plants, as well as interactive design of plant material properties. We authored and simulated over 100 plants from diverse climates and geographic regions, including broadleaf (deciduous) trees and conifers, bushes and flowers. Our largest simulations involve anatomically realistic adult trees with hundreds of branches and over 100,000 leaves.
Subspace Condensation: Full Space Adaptivity for Subspace Deformations
"... Figure 1: (a) The simulation runs at 16 FPS, entirely within the subspace, 67 ⇥ faster than a full space simulation over the entire mesh. (b) Novel wall collisions begin, activating full space tets, shown in red in the inset. The simulation still runs at 2.1 FPS, a 7.7 ⇥ speedup. (c) Collisions prod ..."
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Figure 1: (a) The simulation runs at 16 FPS, entirely within the subspace, 67 ⇥ faster than a full space simulation over the entire mesh. (b) Novel wall collisions begin, activating full space tets, shown in red in the inset. The simulation still runs at 2.1 FPS, a 7.7 ⇥ speedup. (c) Collisions produce a deformation far outside the basis, and 49 % of the tets are simulated in full space. The step runs at 0.5 FPS; still a 1.9⇥ speedup. (d) The collisions are removed, and the 67 ⇥ speedup returns. Subspace deformable body simulations can be very fast, but can behave unrealistically when behaviors outside the prescribed subspace, such as novel external collisions, are encountered. We address this limitation by presenting a fast, flexible new method that allows full space computation to be activated in the neighborhood of novel events while the rest of the body still computes in a subspace. We achieve this using a method we call subspace condensation, a variant on the classic static condensation precomputation. However, instead of a precomputation, we use the speed of subspace methods to perform the condensation at every frame. This approach allows the full space regions to be specified arbitrarily at runtime, and forms a natural twoway coupling with the subspace regions. While condensation is usually only applicable to linear materials, the speed of our technique enables its application to nonlinear materials as well. We show the effectiveness of our approach by applying it to a variety of articulated character scenarios.