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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
SIAM FR26: FEM with B-Splines Introduction 1, page 1
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
SIAM FR26: FEM with B-Splines Introduction 1, page 1
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
SIAM FR26: FEM with B-Splines Introduction 1, page 1
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
SIAM FR26: FEM with B-Splines Introduction 1, page 1
Literature
google: > 10000000 pages
www.web-spline.de (K. Hollig, U.Reif, J. Wipper)→ papers, dissertations, masters theses
K. Hollig: 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=1482pages)
. . .
related method, using b-splines:J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis, JohnWiley & Sons Ltd., 2009.
SIAM FR26: FEM with B-Splines Introduction 1, page 2
Literature
google: > 10000000 pages
www.web-spline.de (K. Hollig, U.Reif, J. Wipper)→ papers, dissertations, masters theses
K. Hollig: 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=1482pages)
. . .
related method, using b-splines:J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis, JohnWiley & Sons Ltd., 2009.
SIAM FR26: FEM with B-Splines Introduction 1, page 2
Literature
google: > 10000000 pages
www.web-spline.de (K. Hollig, U.Reif, J. Wipper)→ papers, dissertations, masters theses
K. Hollig: 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=1482pages)
. . .
related method, using b-splines:J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis, JohnWiley & Sons Ltd., 2009.
SIAM FR26: FEM with B-Splines Introduction 1, page 2
Literature
google: > 10000000 pages
www.web-spline.de (K. Hollig, U.Reif, J. Wipper)→ papers, dissertations, masters theses
K. Hollig: 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=1482pages)
. . .
related method, using b-splines:J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis, JohnWiley & Sons Ltd., 2009.
SIAM FR26: FEM with B-Splines Introduction 1, page 2
Literature
google: > 10000000 pages
www.web-spline.de (K. Hollig, U.Reif, J. Wipper)→ papers, dissertations, masters theses
K. Hollig: 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=1482pages)
. . .
related method, using b-splines:J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis, JohnWiley & Sons Ltd., 2009.
SIAM FR26: FEM with B-Splines Introduction 1, page 2
Literature
google: > 10000000 pages
www.web-spline.de (K. Hollig, U.Reif, J. Wipper)→ papers, dissertations, masters theses
K. Hollig: 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=1482pages)
. . .
related method, using b-splines:J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis, JohnWiley & Sons Ltd., 2009.
SIAM FR26: FEM with B-Splines Introduction 1, page 2
Literature
google: > 10000000 pages
www.web-spline.de (K. Hollig, U.Reif, J. Wipper)→ papers, dissertations, masters theses
K. Hollig: 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=1482pages)
. . .
related method, using b-splines:J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis, JohnWiley & Sons Ltd., 2009.
SIAM FR26: FEM with B-Splines Introduction 1, page 2
History of Finite Elements and Splines
FEM Splines
Engineering
Turner, Clough, Martin,and Topp (1956)
Argyris (1960)Clough (1960)
de Casteljau (1959)Bezier (1966)
Mathematics
Rayleigh (1870)Ritz (1908)Galerkin (1915)Courant (1943)Strang and Fix (1973)
Schoenberg (1946)de Boor (1972)
SIAM FR26: FEM with B-Splines Introduction 1, page 3
Splines as Finite Elements
grid with inner and outer B-splines
SIAM FR26: FEM with B-Splines Introduction 1, page 4
principal difficulties
essential boundary conditions∑k
ukbk = 0 on ∂D =⇒ uk = 0, k ∼ ∂D
poor approximation order
stability
‖ck‖ 6� ‖∑k
ckbk‖ (h→ 0)
ill-conditioned systems, slow convergence of iterative schemes
SIAM FR26: FEM with B-Splines Introduction 1, page 5
principal difficulties
essential boundary conditions∑k
ukbk = 0 on ∂D =⇒ uk = 0, k ∼ ∂D
poor approximation order
stability
‖ck‖ 6� ‖∑k
ckbk‖ (h→ 0)
ill-conditioned systems, slow convergence of iterative schemes
SIAM FR26: FEM with B-Splines Introduction 1, page 5
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
bi → bi +∑j∈J(i)
ei ,jbj , i ∈ I
based on Marsden’s identity weighted extended B-splines (web-splines)
Bi = γiw
bi +∑j∈J(i)
ei ,jbj
with standard properties of finite elements
SIAM FR26: FEM with B-Splines Introduction 1, page 6
Weighted Extended B-Splines
homogeneous boundary conditions, modeled with a weight function
bk → wbk , k ∈ K
suggested by Kantorovich and Krylow, studied by Rvachevstabilization via extension of inner B-splines
bi → bi +∑j∈J(i)
ei ,jbj , i ∈ I
based on Marsden’s identity
weighted extended B-splines (web-splines)
Bi = γiw
bi +∑j∈J(i)
ei ,jbj
with standard properties of finite elements
SIAM FR26: FEM with B-Splines Introduction 1, page 6
Weighted Extended B-Splines
homogeneous boundary conditions, modeled with a weight function
bk → wbk , k ∈ K
suggested by Kantorovich and Krylow, studied by Rvachevstabilization via extension of inner B-splines
bi → bi +∑j∈J(i)
ei ,jbj , i ∈ I
based on Marsden’s identity weighted extended B-splines (web-splines)
Bi = γiw
bi +∑j∈J(i)
ei ,jbj
with standard properties of finite elements
SIAM FR26: FEM with B-Splines Introduction 1, page 6
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
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
compatibility with CAD/CAM systems
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
compatibility with CAD/CAM systems
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
compatibility with CAD/CAM systems
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
compatibility with CAD/CAM systems
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
compatibility with CAD/CAM systems
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
compatibility with CAD/CAM systems
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
compatibility with CAD/CAM systems
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
compatibility with CAD/CAM systems
SIAM FR26: FEM with B-Splines Introduction 1, page 7
Notation
skipping dependencies on parameters
bk = bnk,h, . . .
constants in estimates
≤ const(p1, p2, . . .)
inequalities up to constants
�, �, �
spline approximation with coefficient vector U = {uk}k∈K
u ≈ uh =∑k
ukbk ,
vectors and matricesG = {gk,i}k,i∈I
products UV without transposition
SIAM FR26: FEM with B-Splines Introduction 1, page 8
Notation
skipping dependencies on parameters
bk = bnk,h, . . .
constants in estimates
≤ const(p1, p2, . . .)
inequalities up to constants
�, �, �
spline approximation with coefficient vector U = {uk}k∈K
u ≈ uh =∑k
ukbk ,
vectors and matricesG = {gk,i}k,i∈I
products UV without transposition
SIAM FR26: FEM with B-Splines Introduction 1, page 8
Notation
skipping dependencies on parameters
bk = bnk,h, . . .
constants in estimates
≤ const(p1, p2, . . .)
inequalities up to constants
�, �, �
spline approximation with coefficient vector U = {uk}k∈K
u ≈ uh =∑k
ukbk ,
vectors and matricesG = {gk,i}k,i∈I
products UV without transposition
SIAM FR26: FEM with B-Splines Introduction 1, page 8
Notation
skipping dependencies on parameters
bk = bnk,h, . . .
constants in estimates
≤ const(p1, p2, . . .)
inequalities up to constants
�, �, �
spline approximation with coefficient vector U = {uk}k∈K
u ≈ uh =∑k
ukbk ,
vectors and matricesG = {gk,i}k,i∈I
products UV without transposition
SIAM FR26: FEM with B-Splines Introduction 1, page 8
Notation
skipping dependencies on parameters
bk = bnk,h, . . .
constants in estimates
≤ const(p1, p2, . . .)
inequalities up to constants
�, �, �
spline approximation with coefficient vector U = {uk}k∈K
u ≈ uh =∑k
ukbk ,
vectors and matricesG = {gk,i}k,i∈I
products UV without transposition
SIAM FR26: FEM with B-Splines Introduction 1, page 8
