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Introduction to the web-method and its applications. (English) Zbl 1070.65118
The proposed finite element method is based on a tensor product grid, hence no mesh generation is needed. On this grid B-splines are introduced and modified using a positive weight function. Using this weight function boundary conditions are implemented, hence there is no need to start with boundary conforming elements. Suggestions on how to construct such a weight function are given, the treatment of splines with part of their support outside of the computational domain is also discussed. Optimal order of approximation is shown under an assumption of smoothness on the solution and the weight function. The behavior of the method is demonstrated on three applications (Laplace equation, linear elasticity, thin plate model).

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
41A15 Spline approximation
74S05 Finite element methods applied to problems in solid mechanics
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
74B05 Classical linear elasticity
74K20 Plates
35J25 Boundary value problems for second-order elliptic equations
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[1] J.H. Argyris, Energy Theorems and Structural Analysis (Butterworths, London, 1960).
[2] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming and P. Krysl, Meshless methods: An overview and recent developments, Comput. Methods Appl. Mech. Engrg. 139(1-4) (1996) 3-47. · Zbl 0891.73075
[3] P. Bochev and M. Gunzburger, Finite element methods of least squares type, SIAM Rev. 40 (1998) 789-837. · Zbl 0914.65108
[4] P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978). · Zbl 0383.65058
[5] P.G. Ciarlet, Mathematical Elasticity, Vol. 3: Theory of Shells (North-Holland, Amsterdam, 2000). · Zbl 0953.74004
[6] F. Cirak, M. Ortiz and P. Schröder, Subdivision surfaces: A new paradigm for thin-shell finite-element analysis, Internat. J. Numer. Methods Engrg. 47 (2000) 2039-2072. · Zbl 0983.74063
[7] E. Cohen, R.F. Riesenfeld and G. Elber, Geometric Modeling with Splines: An Introduction (A.K. Peters, 2001). · Zbl 0980.65016
[8] C. de Boor, A Practical Guide to Splines (Springer, Berlin, 1978). · Zbl 0406.41003
[9] K. Höllig, Finite element approximation with splines, in: Handbook of Computer Aided Geometric Design, eds. G. Farin, J. Hoschek and M.S. Kim (Elsevier, Amsterdam, 2002) pp. 283-308.
[10] K. Höllig, Finite Element Methods with B-Splines, Frontiers in Applied Mathematics, Vol. 26 (SIAM, Philadelphia, PA, 2003). · Zbl 1020.65085
[11] K. Höllig and U. Reif, Nonuniform Web-splines, Computer Aided Geometric Design 20 (2003) 277-294. · Zbl 1069.65502
[12] K. Höllig, U. Reif and J. Wipper, Weighted extended B-spline approximation of Dirichlet problems, SIAM J. Numer. Anal. 39(2) (2001) 442-462. · Zbl 0996.65119
[13] K. Höllig, U. Reif and J. Wipper, Verfahren zur Erhöhung der Leistungsfähigkeit einer Computereinrichtung bei Finite-Elemente-Simulationen und eine solche Computereinrichtung, Deutsche Patentschrift DE 100 233 77 C2 (2003).
[14] K. Höllig, U. Reif and J. Wipper, Multigrid methods with Web-splines, Numer. Math. 91(2) (2002) 237-256. · Zbl 0996.65138
[15] K. Höllig, U. Reif and J. Wipper, Process for increasing the efficiency of a computer in finite element simulations and a computer for performing that process, United States Patent Application, US2002/0029135A1 (2002).
[16] L.W. Kantorowitsch and W.I. Krylow, Näherungsmethoden der Höheren Analysis (VEB Deutscher Verlag der Wissenschaften, Berlin, 1956).
[17] L.D. Landau and E.M. Lifshitz, Theory of Elasticity, 3rd ed. (Pergamon Press, Elmsford, NY, 1986). · Zbl 0178.28704
[18] J.A. Nitsche, On Korn?s second inequality, RAIRO J. Numer. Anal. 15 (1981) 237-248. · Zbl 0467.35019
[19] V.L. Rvachev and T.I. Sheiko, R-functions in boundary value problems in mechanics, Appl. Mech. Rev. 48(4) (1995) 151-188.
[20] L.L. Schumaker, Spline Functions: Basic Theory (Wiley-Interscience, New York, 1980). · Zbl 0449.41004
[21] Special Issue on Meshless Methods, Comput. Mech. Appl. Mech. Engrg. 139 (1996).
[22] E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton Univ. Press, Princeton, 1970). · Zbl 0207.13501
[23] G. Strang and G.J. Fix, An Analysis of the Finite Element Method (Prentice-Hall, Englewood Cliffs, NJ, 1973). · Zbl 0356.65096
[24] M.J. Turner, R.W. Clough, H.C. Martin and L.C. Topp, Stiffness and deflection analysis of complex structures, J. Aeronaut. Sci. 23(9) (1956) 805-823, 854. · Zbl 0072.41003
[25] O.C. Zienkiewicz and R.I. Taylor, Finite Element Method, Vols. I?III (Butterworth & Heinemann, 2000). · Zbl 0991.74003
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