Origin of the Finite Element Method G. Strang and G. Fix: . . . Surely Argyris in Germany and England, and Martin and Clough in ” America, were among those responsible; we dare not guess who was first. . . .“
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Introduction
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Origin of the Finite Element Method G. Strang and G. Fix: . . . Surely Argyris in Germany and England, and Martin and Clough in ” America, were among those responsible; we dare not guess who was first. . . .“ J.H. Argyris: Energy Theorems and Structural Analysis, Butterworth, London, 1960. M. J. Turner, R. W. Clough, H. C. Martin, and L. C. Topp: Stiffness and deflection analysis of complex structures, J. Aeronaut. Sci. 23 (1956), 805–823, 854.
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Origin of the Finite Element Method G. Strang and G. Fix: . . . Surely Argyris in Germany and England, and Martin and Clough in ” America, were among those responsible; we dare not guess who was first. . . .“ J.H. Argyris: Energy Theorems and Structural Analysis, Butterworth, London, 1960. M. J. Turner, R. W. Clough, H. C. Martin, and L. C. Topp: Stiffness and deflection analysis of complex structures, J. Aeronaut. Sci. 23 (1956), 805–823, 854. earlier theoretical papers: R. Courant 1943, B.G. Galerkin 1915, Ritz 1908, J. W. S. Rayleigh 1870
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Origin of the Finite Element Method G. Strang and G. Fix: . . . Surely Argyris in Germany and England, and Martin and Clough in ” America, were among those responsible; we dare not guess who was first. . . .“ J.H. Argyris: Energy Theorems and Structural Analysis, Butterworth, London, 1960. M. J. Turner, R. W. Clough, H. C. Martin, and L. C. Topp: Stiffness and deflection analysis of complex structures, J. Aeronaut. Sci. 23 (1956), 805–823, 854. earlier theoretical papers: R. Courant 1943, B.G. Galerkin 1915, Ritz 1908, J. W. S. Rayleigh 1870 → (Rayleigh –) Ritz – Galerkin Method
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Literature google: > 10000000 pages
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FEM with B-Splines
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Literature google: > 10000000 pages www.web-spline.de (K. H¨ ollig, U.Reif, J. Wipper) → papers, dissertations, masters theses
SIAM FR26:
FEM with B-Splines
Introduction
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Literature google: > 10000000 pages www.web-spline.de (K. H¨ ollig, U.Reif, J. Wipper) → papers, dissertations, masters theses K. H¨ollig: Finite Element Methods with B-Splines, SIAM, 2003.
SIAM FR26:
FEM with B-Splines
Introduction
1, page 2
Literature google: > 10000000 pages www.web-spline.de (K. H¨ ollig, U.Reif, J. Wipper) → papers, dissertations, masters theses K. H¨ollig: Finite Element Methods with B-Splines, SIAM, 2003. G. Strang and G.J. Fix: An Analysis of the Finite Element Method, Prentice–Hall, Englewood Cliffs, NJ, 1973.
SIAM FR26:
FEM with B-Splines
Introduction
1, page 2
Literature google: > 10000000 pages www.web-spline.de (K. H¨ ollig, U.Reif, J. Wipper) → papers, dissertations, masters theses K. H¨ollig: Finite Element Methods with B-Splines, SIAM, 2003. G. Strang and G.J. Fix: An Analysis of the Finite Element Method, Prentice–Hall, Englewood Cliffs, NJ, 1973. O.C. Zienkiewicz and R.I. Taylor: Finite Element Method, Vol. I–III, Butterworth & Heinemann, London, 2000. (689+459+334=1482 pages)
SIAM FR26:
FEM with B-Splines
Introduction
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Literature google: > 10000000 pages www.web-spline.de (K. H¨ ollig, U.Reif, J. Wipper) → papers, dissertations, masters theses K. H¨ollig: Finite Element Methods with B-Splines, SIAM, 2003. G. Strang and G.J. Fix: An Analysis of the Finite Element Method, Prentice–Hall, Englewood Cliffs, NJ, 1973. O.C. Zienkiewicz and R.I. Taylor: Finite Element Method, Vol. I–III, Butterworth & Heinemann, London, 2000. (689+459+334=1482 pages) ...
SIAM FR26:
FEM with B-Splines
Introduction
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Literature google: > 10000000 pages www.web-spline.de (K. H¨ ollig, U.Reif, J. Wipper) → papers, dissertations, masters theses K. H¨ollig: Finite Element Methods with B-Splines, SIAM, 2003. G. Strang and G.J. Fix: An Analysis of the Finite Element Method, Prentice–Hall, Englewood Cliffs, NJ, 1973. O.C. Zienkiewicz and R.I. Taylor: Finite Element Method, Vol. I–III, Butterworth & Heinemann, London, 2000. (689+459+334=1482 pages) ... related method, using b-splines: J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis, John Wiley & Sons Ltd., 2009.
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History of Finite Elements and Splines Engineering
Turner, Clough, Martin, and Topp (1956) Argyris (1960) Clough (1960)
de Casteljau (1959) Bezier (1966)
FEM
Splines
Rayleigh (1870) Ritz (1908) Galerkin (1915) Courant (1943) Strang and Fix (1973)
Schoenberg (1946) de Boor (1972)
Mathematics
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Splines as Finite Elements
grid with inner and outer B-splines
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principal difficulties essential boundary conditions X uk bk = 0 on ∂D
=⇒
uk = 0, k ∼ ∂D
k
poor approximation order
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principal difficulties essential boundary conditions X uk bk = 0 on ∂D
=⇒
uk = 0, k ∼ ∂D
k
poor approximation order stability kck k 6 k
X
ck bk k (h → 0)
k
ill-conditioned systems, slow convergence of iterative schemes
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Weighted Extended B-Splines homogeneous boundary conditions, modeled with a weight function bk → wbk ,
k ∈K
suggested by Kantorovich and Krylow, studied by Rvachev
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Weighted Extended B-Splines homogeneous boundary conditions, modeled with a weight function bk → wbk ,
k ∈K
suggested by Kantorovich and Krylow, studied by Rvachev stabilization via extension of inner B-splines X bi → bi + ei,j bj , i ∈ I j∈J(i)
based on Marsden’s identity
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Weighted Extended B-Splines homogeneous boundary conditions, modeled with a weight function bk → wbk ,
k ∈K
suggested by Kantorovich and Krylow, studied by Rvachev stabilization via extension of inner B-splines X bi → bi + ei,j bj , i ∈ I j∈J(i)
based on Marsden’s identity weighted extended B-splines (web-splines) X Bi = γi w bi + ei,j bj j∈J(i)
with standard properties of finite elements SIAM FR26:
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Advantages of WEB-Splines flexibility of mesh-based elements and computational efficiency of B-splines
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Advantages of WEB-Splines flexibility of mesh-based elements and computational efficiency of B-splines meshless method
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Advantages of WEB-Splines flexibility of mesh-based elements and computational efficiency of B-splines meshless method uniform grid
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Advantages of WEB-Splines flexibility of mesh-based elements and computational efficiency of B-splines meshless method uniform grid exact fulfilment of boundary conditions
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Advantages of WEB-Splines flexibility of mesh-based elements and computational efficiency of B-splines meshless method uniform grid exact fulfilment of boundary conditions simple parallelization and efficient multigrid techniques
SIAM FR26:
FEM with B-Splines
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Advantages of WEB-Splines flexibility of mesh-based elements and computational efficiency of B-splines meshless method uniform grid exact fulfilment of boundary conditions simple parallelization and efficient multigrid techniques accurate approximations with relatively low-dimensional subspaces
SIAM FR26:
FEM with B-Splines
Introduction
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Advantages of WEB-Splines flexibility of mesh-based elements and computational efficiency of B-splines meshless method uniform grid exact fulfilment of boundary conditions simple parallelization and efficient multigrid techniques accurate approximations with relatively low-dimensional subspaces arbitrary smoothness and approximation order
SIAM FR26:
FEM with B-Splines
Introduction
1, page 7
Advantages of WEB-Splines flexibility of mesh-based elements and computational efficiency of B-splines meshless method uniform grid exact fulfilment of boundary conditions simple parallelization and efficient multigrid techniques accurate approximations with relatively low-dimensional subspaces arbitrary smoothness and approximation order adaptive refinement via hierarchical bases
SIAM FR26:
FEM with B-Splines
Introduction
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Advantages of WEB-Splines flexibility of mesh-based elements and computational efficiency of B-splines meshless method uniform grid exact fulfilment of boundary conditions simple parallelization and efficient multigrid techniques accurate approximations with relatively low-dimensional subspaces arbitrary smoothness and approximation order adaptive refinement via hierarchical bases compatibility with CAD/CAM systems
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Notation skipping dependencies on parameters n bk = bk,h , ...
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Notation skipping dependencies on parameters n bk = bk,h , ...
constants in estimates ≤ const(p1 , p2 , . . .)
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Notation skipping dependencies on parameters n bk = bk,h , ...
constants in estimates ≤ const(p1 , p2 , . . .) inequalities up to constants , ,
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Notation skipping dependencies on parameters n bk = bk,h , ...
constants in estimates ≤ const(p1 , p2 , . . .) inequalities up to constants , , spline approximation with coefficient vector U = {uk }k∈K X uk bk , u ≈ uh = k
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Notation skipping dependencies on parameters n bk = bk,h , ...
constants in estimates ≤ const(p1 , p2 , . . .) inequalities up to constants , , spline approximation with coefficient vector U = {uk }k∈K X uk bk , u ≈ uh = k
vectors and matrices G = {gk,i }k,i∈I products UV without transposition SIAM FR26:
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