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211
Multilevel Partition of Unity Implicits
 ACM TRANSACTIONS ON GRAPHICS
, 2003
"... We present a shape representation, the multilevel partition of unity implicit surface, that allows us to construct surface models from very large sets of points. There are three key ingredients to our approach: 1) piecewise quadratic functions that capture the local shape of the surface, 2) weighti ..."
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Cited by 218 (7 self)
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We present a shape representation, the multilevel partition of unity implicit surface, that allows us to construct surface models from very large sets of points. There are three key ingredients to our approach: 1) piecewise quadratic functions that capture the local shape of the surface, 2) weighting functions (the partitions of unity) that blend together these local shape functions, and 3) an octree subdivision method that adapts to variations in the complexity of the local shape. Our approach
Multiscale scientific computation: Review 2001
 Multiscale and Multiresolution Methods
, 2001
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The Discontinuous Enrichment Method
, 2000
"... We propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it. The enrichment contains freespace solutions of the homogeneous differential equation that are not represented by the underlyin ..."
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Cited by 50 (6 self)
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We propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it. The enrichment contains freespace solutions of the homogeneous differential equation that are not represented by the underlying polynomial field. Continuity of the enrichment across element interfaces is enforced weakly by Lagrange multipliers. In this manner, we expect to attain high coarsemesh accuracy without significant degradation of conditioning, assuring good performance of the computation at any mesh resolution. Examples of application to acoustics and advectiondiffusion are presented. Key words: Finite elements; discontinuous enrichment; acoustics; advectiondiffusion 1 Introduction The standard finite element method is based on continuous, piecewise polynomial, Galerkin approximation. This approach is optimal for the Laplace operator in the sense that it minimizes the error in the energy normthe H ...
Implementation of a boundary element method for high frequency scattering by convex polygons
 ADVANCES IN BOUNDARY INTEGRAL METHODS (PROCEEDINGS OF THE 5TH UK CONFERENCE ON BOUNDARY INTEGRAL METHODS
"... In this paper we consider the problem of timeharmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. H ..."
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Cited by 43 (20 self)
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In this paper we consider the problem of timeharmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretisation of a wellknown second kind combinedlayerpotential integral equation. We provide a proof that this equation and its adjoint are wellposed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.
The generalized interpolation material point method. Computer Modeling
, 2004
"... Abstract: The Material Point Method (MPM) discrete solution procedure for computational solid mechanics is generalized using a variational form and a Petrov– Galerkin discretization scheme, resulting in a family of methods named the Generalized Interpolation Material Point (GIMP) methods. The genera ..."
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Cited by 42 (0 self)
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Abstract: The Material Point Method (MPM) discrete solution procedure for computational solid mechanics is generalized using a variational form and a Petrov– Galerkin discretization scheme, resulting in a family of methods named the Generalized Interpolation Material Point (GIMP) methods. The generalization permits identification with aspects of other point or node based discrete solution techniques which do not use a body–fixed grid, i.e. the “meshless methods”. Similarities are noted and some practical advantages relative to some of these methods are identified. Examples are used to demonstrate and explain numerical artifact noise which can be expected in MPM calculations. This noise results in nonphysical local variations at the material points, where constitutive response is evaluated. It is shown to destroy the explicit solution in one case, and seriously degrade it in another. History dependent, inelastic constitutive laws can be expected to evolve erroneously and report inaccurate stress states because of noisy input. The noise is due to the lack of smoothness of the interpolation functions, and occurs due to material points crossing computational grid boundaries. The next degree of smoothness available in the GIMP methods is shown to be capable of eliminating cell crossing noise.
Plane wave discontinuous Galerkin methods
, 2007
"... Abstract. We are concerned with a finite element approximation for timeharmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in mediumfrequency ..."
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Cited by 40 (8 self)
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Abstract. We are concerned with a finite element approximation for timeharmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in mediumfrequency regimes. As an alternative, methods that incorporate information about the solution in the form of plane waves have been proposed. Among them the ultra weak variational formulation (UWVF) of Cessenat and Despres [O. Cessenat and B. Despres, Application of an ultra weak variational formulation of elliptic PDEs to the twodimensional Helmholtz equation, SIAM J. Numer. Anal., 35 (1998), pp. 255–299.]. We identify the UWVF as representative of a class of Trefftztype discontinuous Galerkin methods that employs trial and test spaces spanned by local plane waves. In this paper we give a priori convergence estimates for the hversion of these plane wave discontinuous Galerkin methods. To that end, we develop new inverse and approximation estimates for plane waves in two dimensions and use these in the context of duality techniques. Asymptotic optimality of the method in a mesh dependent norm can be established. However, the estimates require a minimal resolution of the mesh beyond what it takes to resolve the wavelength. We give numerical evidence that this requirement cannot be dispensed with. It reflects the presence of numerical dispersion. Key words. Wave propagation, finite element methods, discontinuous Galerkin methods, plane waves, ultra weak variational formulation, duality estimates, numerical dispersion AMS subject classifications. 65N15, 65N30, 35J05
Solving Differential Equations with Radial Basis Functions: Multilevel Methods and Smoothing
 Advances in Comp. Math
"... . Some of the meshless radial basis function methods used for the numerical solution of partial differential equations are reviewed. In particular, the differences between globally and locally supported methods are discussed, and for locally supported methods the important role of smoothing within a ..."
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Cited by 35 (7 self)
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. Some of the meshless radial basis function methods used for the numerical solution of partial differential equations are reviewed. In particular, the differences between globally and locally supported methods are discussed, and for locally supported methods the important role of smoothing within a multilevel framework is demonstrated. A possible connection between multigrid finite elements and multilevel radial basis function methods with smoothing is explored. Various numerical examples are also provided throughout the paper. 1. Introduction During the past few years the idea of using socalled meshless methods for the numerical solution of partial differential equations (PDEs) has received much attention throughout the scientific community. As a few representative examples we mention Belytschko and coworker's results [3] using the socalled elementfree Galerkin method, Duarte and Oden's work [11] using hp clouds, Babuska and Melenk 's work [1] on the partition of unity method, ...
The Meshless Local PetrovGalerkin (MLPG) Method for Solving Incompressible NavierStokes Equations
 CMES
, 2001
"... The truly Meshless Local PetrovGalerkin (MLPG) method is extended to solve the incompressible NavierStokes equations. The local weak form is modified in a very careful way so as to ovecome the socalled BabuskaBrezzi conditions. In addition, The upwinding scheme as developed in Lin and Atluri (20 ..."
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Cited by 35 (7 self)
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The truly Meshless Local PetrovGalerkin (MLPG) method is extended to solve the incompressible NavierStokes equations. The local weak form is modified in a very careful way so as to ovecome the socalled BabuskaBrezzi conditions. In addition, The upwinding scheme as developed in Lin and Atluri (2000a) and Lin and Atluri (2000b) is used to stabilize the convection operator in the streamline direction. Numerical results for benchmark problems show that the MLPG method is very promising to solve the convection dominated fluid mechanics problems.
Generalized Multiscale Finite Element Methods (GMsFEM)
, 2013
"... In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach i ..."
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Cited by 30 (10 self)
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In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space. In the proposed approach, we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage. The enrichment in the online stage is performed based on a spectral decomposition of the offline space. In the online stage, for any input parameter, a multiscale space is constructed to solve the global problem on a coarse grid. The online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues. The computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be reused for solving the forward problem with any forcing and boundary condition. Compared with the other approaches where global snapshots are used, the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarsegrid level. We present various examples in the paper and some numerical results to demonstrate the effectiveness of our method. 1
Computational Aspects of the Ultra Weak Variational Formulation
 Journal of Computational Physics
, 2002
"... this paper we consider computational aspects of the ultra weak variational formulation for the inhomogeneous Helmholtz problem. We introduce a method to improve the UWVF scheme and compare iterative solvers for the resulting linear system. Computations for the acoustic transmission problem in 2D sho ..."
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Cited by 24 (8 self)
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this paper we consider computational aspects of the ultra weak variational formulation for the inhomogeneous Helmholtz problem. We introduce a method to improve the UWVF scheme and compare iterative solvers for the resulting linear system. Computations for the acoustic transmission problem in 2D show that the new approach enables solving Helmholtz problems on a relatively coarse mesh for a wide range of wave numbers. 1.