Introduction to the Web-method and its applications
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Springer 2005
Introduction to the Web-method and its applications Klaus Höllig, Christian Apprich and Anja Streit ∗ Universität Stuttgart, Fachbereich Mathematik, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Received 27 March 2003; accepted 2 February 2004 Communicated by Z. Wu and B.Y.C. Hon
The Web-method is a meshless finite element technique which uses weighted extended B-splines (Web-splines) on a tensor product grid as basis functions. It combines the computational advantages of B-splines and standard mesh-based elements. In particular, degree and smoothness can be chosen arbitrarily without substantially increasing the dimension. Hence, accurate approximations are obtained with relatively few parameters. Moreover, the regular grid is well suited for hierarchical refinement and multigrid techniques. This article should serve as an introduction to finite element approximation with B-splines. We first review the construction of Web-bases and discuss their basic properties. Then we illustrate the performance of Ritz–Galerkin schemes for a model problem and applications in linear elasticity. Finally, we discuss several implementation aspects. Keywords: finite element, meshless method, Web-spline, B-spline, weight function, stability AMS subject classification: 65N30, 41A15, 74S05
1.
Introduction
The finite element method has become the method of choice for solving many types of partial differential equations in engineering and physical sciences. Important applications include structural mechanics, fluid flow, thermodynamics, and electromagnetic fields [25]. The basic idea is very elegant and dates back to the classical work of Rayleigh, Ritz, and Galerkin almost a century ago. Guided by physical principles, an elliptic boundary value problem is stated in variational form: 1 Q(u) = a(u, u) − λ(u) → min, u ∈ H. (1) 2 Usually, the quadratic functional Q represents the total energy in the underlying model. The symmetric bilinear form a and the linear form λ correspond to contributions from internal and applied forces, and H is a Hilbert space which incorporates the boundary conditions. A classical example is the analysis of elastic deformations, which has been the starting point of finite element analysis [1,24]. ∗ Present address: Fraunhofer ITWM, 67663 Kaiserslautern, Germany.
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The Ritz–Galerkin method restricts the minimization of (1) to a finite-dimensional subspace Vh = span Bi ⊂ H. i
Under mild assumptions, the resulting approximation ui Bi ≈ u, uh = i
defined by Q(uh ) = min Q(vh ), vh ∈Vh
will converge to u as the discretization parameter h (typically a grid width) tends to 0. A crucial requirement is that the bilinear form is elliptic. In the symmetric case this means that a induces an equivalent norm on H and implies u − uh H const(a) inf u − vh H . vh ∈Vh
(2)
This fundamental inequality, due to Céa, follows directly from the characterization of uh as the best approximation to u in the a-norm [23]. The numerical computation of uh is relatively straight
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