Introduction to the Web-method and its applications

Höllig, Klaus; Apprich, Christian; Streit, Anja

doi:10.1007/s10444-004-1811-y

Cited by 67 publications

(50 citation statements)

References 18 publications

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“…Tensor product approximants can be used to generalize 1D methods such as B-Spline to multiple dimensions, but require structured grids. Recent examples include the References [6,18]. Subdivision approximations [15] can deal with unstructured data, but so far these approximants are theoretically well founded only in 2D.…”

Section: Sme Approximantsmentioning

confidence: 99%

Smooth, second order, non‐negative meshfree approximants selected by maximum entropy

Cyron

Arroyo

Ortiz

2009

Numerical Meth Engineering

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SUMMARYWe present a family of approximation schemes, which we refer to as second-order maximum-entropy (max-ent) approximation schemes, that extends the first-order local max-ent approximation schemes to second-order consistency. This method retains the fundamental properties of first-order max-ent schemes, namely the shape functions are smooth, non-negative, and satisfy a weak Kronecker-delta property at the boundary. This last property makes the imposition of essential boundary conditions in the numerical solution of partial differential equations trivial. The evaluation of the shape functions is not explicit, but it is very efficient and robust. To our knowledge, the proposed method is the first higher-order scheme for function approximation from unstructured data in arbitrary dimensions with non-negative shape functions. As a consequence, the approximants exhibit variation diminishing properties, as well as an excellent behavior in structural vibrations problems as compared with the Lagrange finite elements, MLS-based meshfree methods and even B-Spline approximations, as shown through numerical experiments. When compared with usual MLS-based second-order meshfree methods, the shape functions presented here are much easier to integrate in a Galerkin approach, as illustrated by the standard benchmark problems. 1 Smooth, second order, non-negative meshfree approximants selected by maximum entropy, Cyron, ;Arroyo, M.; Ortiz, M.; International

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Section: Sme Approximantsmentioning

confidence: 99%

Smooth, second order, non‐negative meshfree approximants selected by maximum entropy

Cyron

Arroyo

Ortiz

2009

Numerical Meth Engineering

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“…The FCM system matrix is defined as 11) and the following is noted: 12) showing that the FCM norm of a function v h is equal to the energy norm of the corresponding vector y with system matrix A. Therefore the · 2 norm of A and its inverse according to (3.10) can be interpreted as the quotient of the FCM norm of a function and the Euclidean norm of the corresponding vector:…”

Section: Condition Numbers In Fcmmentioning

confidence: 99%

“…This can either be done by combining them with geometrically nearby functions [23,24] (as is also done in Web-splines [13,12]) or by simply excluding them from the approximation space, e.g., [7,29,33].…”

Section: Introductionmentioning

confidence: 99%

Condition number analysis and preconditioning of the finite cell method

Prenter

Verhoosel

Zwieten

et al. 2017

Computer Methods in Applied Mechanics and Engineering

146

179

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The (Isogeometric) Finite Cell Method -in which a domain is immersed in a structured background mesh -suffers from conditioning problems when cells with small volume fractions occur. In this contribution, we establish a rigorous scaling relation between the condition number of (I)FCM system matrices and the smallest cell volume fraction. Ill-conditioning stems either from basis functions being small on cells with small volume fractions, or from basis functions being nearly linearly dependent on such cells. Based on these two sources of ill-conditioning, an algebraic preconditioning technique is developed, which is referred to as Symmetric Incomplete Permuted Inverse Cholesky (SIPIC). A detailed numerical investigation of the effectivity of the SIPIC preconditioner in improving (I)FCM condition numbers and in improving the convergence speed and accuracy of iterative solvers is presented for the Poisson problem and for two-and three-dimensional problems in linear elasticity, in which Nitche's method is applied in either the normal or tangential direction. The accuracy of the preconditioned iterative solver enables mesh convergence studies of the finite cell method.

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“…It is shown that the underlying problems related to the large number and mesh-dependence of design variables, the discretization of DBC as well as mesh updating can be completely avoided. According to [22][23][24][25][26], DBC is satisfied by penalizing displacement field with the weighting function defined by the LSF. This means that Dirichlet boundary shapes are described by implicit representations with continuous design variables independent of the mesh discretization.…”

Section: Introductionmentioning

confidence: 99%