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80
Interactive SkeletonDriven Dynamic Deformations
 ACM Transactions on Graphics
, 2002
"... This paper presents a framework for the skeletondriven animation of elastically deformable characters. A character is embedded in a coarse volumetric control lattice, which provides the structure needed to apply the finite element method. To incorporate skeletal controls, we introduce line constrai ..."
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Cited by 95 (1 self)
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This paper presents a framework for the skeletondriven animation of elastically deformable characters. A character is embedded in a coarse volumetric control lattice, which provides the structure needed to apply the finite element method. To incorporate skeletal controls, we introduce line constraints along the bones of simple skeletons. The bones are made to coincide with edges of the control lattice, which enables us to apply the constraints efficiently using algebraic methods. To accelerate computation, we associate regions of the volumetric mesh with particular bones and perform locally linearized simulations, which are blended at each time step. We define a hierarchical basis on the control lattice, so for detailed interactions the simulation can adapt the level of detail. We demonstrate the ability to animate complex models using simple skeletons and coarse volumetric meshes in a manner that simulates secondary motions at interactive rates.
DimensionAdaptive TensorProduct Quadrature
 Computing
, 2003
"... We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the highdimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approxi ..."
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Cited by 74 (12 self)
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We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the highdimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lowerdimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself.
A Multiresolution Framework for Dynamic Deformations
, 2002
"... We present a novel framework for dynamic simulation of elastically deformable solids. Our approach combines classical finite element methodology with subdivision wavelets to meet the needs of computer graphics applications. We represent deformations using a wavelet basis constructed from volumetric ..."
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Cited by 72 (2 self)
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We present a novel framework for dynamic simulation of elastically deformable solids. Our approach combines classical finite element methodology with subdivision wavelets to meet the needs of computer graphics applications. We represent deformations using a wavelet basis constructed from volumetric CatmullClark subdivision. CatmullClark subdivision solids allow the domain of deformation to be tailored to objects of arbitrary topology. The domain of deformation can correspond to the interior of a subdivision surface or can enclose an arbitrary surface mesh. Within the wavelet framework we develop the equations of motion for elastic deformations in the presence of external forces and constraints. We solve the resulting differential equations using an implicit method, which lends stability. Our framework allows tradeoff between speed and accuracy. For interactive applications, we accelerate the simulation by adaptively refining the wavelet basis while avoiding visual "popping" artifacts. Offline simulations can employ a fine basis for higher accuracy at the cost of more computation time. By exploiting the properties of smooth subdivision we can compute less expensive solutions using a trilinear basis yet produce a smooth result that meets the constraints.
Corotational Simulation of Deformable Solids
, 2004
"... The classical formulation of large displacement viscoelasticity requires the geometrically nonlinear Green tensor. Keeping track of the rotational part of strain permits alternative formulations, that allow the tensor to stay linear, and at the same time maintaining rotational invariance. We replac ..."
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Cited by 51 (2 self)
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The classical formulation of large displacement viscoelasticity requires the geometrically nonlinear Green tensor. Keeping track of the rotational part of strain permits alternative formulations, that allow the tensor to stay linear, and at the same time maintaining rotational invariance. We replace a recently proposed heuristical warping technique by the application of the polar decomposition. The polar decomposition exactly extracts rotations, thus enhances stability and accuracy. We combine it with a hierarchical finite element basis, which allows us to compute accurate rotations from a coarse level nonlinear simulation and use them with corotated tensors for finer detail.
The Multilevel Finite Element Method for Adaptive Mesh Optimization and Visualization of Volume Data
 In Proceedings Visualization
, 1997
"... Multilevel representations and mesh reduction techniques have been used for accelerating the processing and the rendering of large datasets representing scalar or vector valued functions defined on complex 2 or 3 dimensional meshes. We present a method based on finite element approximations which co ..."
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Cited by 46 (5 self)
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Multilevel representations and mesh reduction techniques have been used for accelerating the processing and the rendering of large datasets representing scalar or vector valued functions defined on complex 2 or 3 dimensional meshes. We present a method based on finite element approximations which combines these two approaches in a new and unique way that is conceptually simple and theoretically sound. The main idea is to consider mesh reduction as an approximation problem in appropriate finite element spaces. Starting with a very coarse triangulation of the functional domain a hierarchy of highly nonuniform tetrahedral (or triangular in 2D) meshes is generated adaptively by local refinement. This process is driven by controlling the local error of the piecewise linear finite element approximation of the function on each mesh element. A reliable and efficient computation of the global approximation error combined with a multilevel preconditioned conjugate gradient solver are the key co...
Local and parallel finite element algorithms based on twogrid discretizations
 Math. Comput
"... Abstract. A number of new local and parallel discretization and adaptive finite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components ..."
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Cited by 38 (13 self)
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Abstract. A number of new local and parallel discretization and adaptive finite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. The theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for finite element solutions on general shaperegular grids. Some numerical experiments are also presented to support the theory. 1.
On twogrid convergence estimates
 Numer. Linear Algebra Appl
"... Abstract. We derive a new representation for the exact convergence factor of the classical twolevel and twogrid preconditioners. Based on this result, we establish necessary and sufficient conditions for constructing the components of efficient AMG (algebraic multigrid) methods. The relation of th ..."
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Cited by 26 (8 self)
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Abstract. We derive a new representation for the exact convergence factor of the classical twolevel and twogrid preconditioners. Based on this result, we establish necessary and sufficient conditions for constructing the components of efficient AMG (algebraic multigrid) methods. The relation of the sharp estimate to the classical twolevel hierarchical basis methods is discussed as well. Lastly, as an application, we give an optimal twogrid convergence proof of a purely algebraic “window”AMG method. twogrid, two–level methods, convergence, sharp estimates, algebraic multigrid 1.
Reproducing Kernel Hierarchical Partition of Unity
 Internat. J. Numer. Methods Engineering
, 1998
"... This work is concerned with developing the hierarchical basis for meshless methods. A reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart. To form such hierarchical partition, a class of basic wavelet fun ..."
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Cited by 25 (5 self)
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This work is concerned with developing the hierarchical basis for meshless methods. A reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart. To form such hierarchical partition, a class of basic wavelet functions are introduced. Based upon the builtin consistency conditions, the differential consistency conditions for the hierarchical kernel functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, which is theoretically proven and numerically validated. For a special interpolant with different combinations of the hierarchical kernels, a synchronized convergence effect may be observed. Being different from the conventional Legendre function based p type hierarchical basis, the new hierarchical basis is an intrinsic pseudospectral basis, which can remain as a partition of unity in a local region, because the discrete wavelet kernels fo...
Algebraic multilevel preconditioning of finite element matrices using local Schur complements, Numerical Linear Algebra with Applications
"... Abstract. We consider an algebraic multilevel preconditioning method for SPD matrices resulting from finite element discretization of elliptic PDEs. In particular, we focus on nonM matrices. The method is based on element agglomeration and assumes access to the individual element matrices. The coar ..."
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Cited by 22 (4 self)
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Abstract. We consider an algebraic multilevel preconditioning method for SPD matrices resulting from finite element discretization of elliptic PDEs. In particular, we focus on nonM matrices. The method is based on element agglomeration and assumes access to the individual element matrices. The coarsegrid element matrices are simply Schur complements computed from local neighborhood matrices (agglomerate matrices), i.e., small collections of element matrices. Assembling these local Schur complements results in a global Schur complement approximation. In addition, performing the elimination of finedegrees of freedom locally, but then without neglecting any fillin, offers the opportunity to construct a new kind of incomplete LU factorization of the pivot matrix at every level. Based on these components an algebraic multilevel preconditioner is defined. The method can also be applied to systems of PDEs. A numerical analysis shows its efficiency and robustness.
A posteriori error estimates based on the polynomial preserving recovery
 SIAM J. Numer. Anal
"... Abstract Superconvergence of order O(h 1+ρ), for some ρ> 0, is established for the gradient recovered with the Polynomial Preserving Recovery (PPR) when the mesh is mildly structured. Consequently, the PPRrecovered gradient can be used in building an asymptotically exact a posteriori error estim ..."
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Cited by 21 (11 self)
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Abstract Superconvergence of order O(h 1+ρ), for some ρ> 0, is established for the gradient recovered with the Polynomial Preserving Recovery (PPR) when the mesh is mildly structured. Consequently, the PPRrecovered gradient can be used in building an asymptotically exact a posteriori error estimator.