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