Introduction to Automated Computational Modelling · Symbolic ⇧(u) r = programming @u K = @r @u...
2
Introduction to Automated Computational Modelling COURSE May 8 th , 9-11 Aula 8, Polo Nuovo Via Ferrata, 3 – Pavia The seminar will introduce basic notions of Automated Computational Modelling. Applications for the solution of nonlinear problems and the analysis of nonlinear constitutive laws will be shown. The lecture will conclude with a brief overview of the potentialities in the context of Finite Elements Michele Marino, PhD, MSc, Group Leader “Predictive Simulations in Biomechanics” Institute of Continuum Mechanics Leibniz Universität Hannover, Germany Università degli Studi di Pavia Computational Mechanics & Advanced Materials Group - DICAr Symbolic programming ⇧(u) Theory r = @⇧ @u , K = @r @u Numeric
Transcript of Introduction to Automated Computational Modelling · Symbolic ⇧(u) r = programming @u K = @r @u...
Introduction to Automated Computational Modelling
CO
UR
SE
May 8th, 9-11Aula 8, Polo Nuovo
Via Ferrata, 3 – Pavia
The seminar will introduce basic notions of Automated ComputationalModelling. Applications for the solution of nonlinear problems and theanalysis of nonlinear constitutive laws will be shown. The lecture willconclude with a brief overview of the potentialities in the context of FiniteElements
Michele Marino, PhD, MSc,Group Leader “Predictive Simulations in Biomechanics”Institute of Continuum MechanicsLeibniz Universität Hannover, Germany
Università degli Studi di PaviaComputational Mechanics & Advanced Materials Group - DICAr
Symbolicprogramming⇧(u) r =
@⇧
@uK =
@r
@u(1)
1
Theory⇧(u) r =
@⇧
@u, K =
@r
@u(1)
1
Numeric
10 20 30 40X
10
20
30
40
50
60
Y
" = "e + "p f(�, q) < 0 f(�, q) = 0
f(�, q) = �vm(�)� (�yo � q)
("e,↵) =1
2"e · C"e +
1
2H↵2
� =@
@"e, q =
@
@↵
"̇p = �̇@f
@�, ↵̇ = �̇
@f
@q
�̇ � 0 , f(�, q) 0 , �̇f = 0
R(u,h) = 0 hg =
0
@"p↵�
1
A h = [ghg
Qg(u,hg) =
0
@Q✏(u,hg)Q↵(u,hg)Q�(u,hg)
1
A =
0
@"p � "pn � (�� �n)n�
↵� ↵n � (�� �n)nq
f(�, q)
1
A = 0 n� =@f
@�nq =
@f
@q
Qg(u,hg) = 0 tn t un hgn
�hg = A�1g Qg|u,hg hg ! hg +�hg
Ag =@Qg
@hg
�u = K�1totR|u,h u ! u+�u
Ktot =@R
@u+
@R
@h⌦ @h
@u
@h
@u= �
✓@Q
@h
◆�1 @Q
@u
2
6666666666664
K11 K12 0 · · · 0
K21 K22 K23 0...
.... . .
. . .. . .
...0 · · · 0 Kk k�1 Kk k Kk k+1 0 · · · 0...
. . .. . .
. . ....
... Knn�1 Knn Knn+1
0 · · · 0 Kn+1n Kn+1n+1
3
7777777777775
0
BBBBBBBBBBBBBBBB@
u1......
uk�1
ukuk+1......
un+1
1
CCCCCCCCCCCCCCCCA
=
0
BBBBBBBBBBBB@
f1......fk......
fn+1
1
CCCCCCCCCCCCA
4
" = "e + "p f(�, q) < 0 f(�, q) = 0
f(�, q) = �vm(�)� (�yo � q)
("e,↵) =1
2"e · C"e +
1
2H↵2
� =@
@"e, q =
@
@↵
"̇p = �̇@f
@�, ↵̇ = �̇
@f
@q
�̇ � 0 , f(�, q) 0 , �̇f = 0
R(u,h) = 0 hg =
0
@"p↵�
1
A h = [ghg
Qg(u,hg) =
0
@Q✏(u,hg)Q↵(u,hg)Q�(u,hg)
1
A =
0
@"p � "pn � (�� �n)n�
↵� ↵n � (�� �n)nq
f(�, q)
1
A = 0 n� =@f
@�nq =
@f
@q
Qg(u,hg) = 0 un, tn u, t hgn
�hg = A�1g Qg|u,hg hg ! hg +�hg
Ag =@Qg
@hg
�u = K�1totR|u,h u ! u+�u
Ktot =@R
@u+
@R
@h⌦ @h
@u
@h
@u= �
✓@Q
@h
◆�1 @Q
@u
2
6666666666664
K11 K12 0 · · · 0
K21 K22 K23 0...
.... . .
. . .. . .
...0 · · · 0 Kk k�1 Kk k Kk k+1 0 · · · 0...
. . .. . .
. . ....
... Knn�1 Knn Knn+1
0 · · · 0 Kn+1n Kn+1n+1
3
7777777777775
0
BBBBBBBBBBBBBBBB@
u1......
uk�1
ukuk+1......
un+1
1
CCCCCCCCCCCCCCCCA
=
0
BBBBBBBBBBBB@
f1......fk......
fn+1
1
CCCCCCCCCCCCA
4
" = "e + "p f(�, q) < 0 f(�, q) = 0
f(�, q) = �vm(�)� (�yo � q)
("e,↵) =1
2"e · C"e +
1
2H↵2
� =@
@"e, q =
@
@↵
"̇p = �̇@f
@�, ↵̇ = �̇
@f
@q
�̇ � 0 , f(�, q) 0 , �̇f = 0
R(u,h) = 0 hg =
0
@"p↵�
1
A h = [ghg
Qg(u,hg) =
0
@Q✏(u,hg)Q↵(u,hg)Q�(u,hg)
1
A =
0
@"p � "pn � (�� �n)n�
↵� ↵n � (�� �n)nq
f(�, q)
1
A = 0 n� =@f
@�nq =
@f
@q
Qg(u,hg) = 0 tn t un hgn
�hg = A�1g Qg|u,hg hg ! hg +�hg
Ag =@Qg
@hg
�u = K�1totR|u,h u ! u+�u
Ktot =@R
@u+
@R
@h⌦ @h
@u
@h
@u= �
✓@Q
@h
◆�1 @Q
@u
2
6666666666664
K11 K12 0 · · · 0
K21 K22 K23 0...
.... . .
. . .. . .
...0 · · · 0 Kk k�1 Kk k Kk k+1 0 · · · 0...
. . .. . .
. . ....
... Knn�1 Knn Knn+1
0 · · · 0 Kn+1n Kn+1n+1
3
7777777777775
0
BBBBBBBBBBBBBBBB@
u1......
uk�1
ukuk+1......
un+1
1
CCCCCCCCCCCCCCCCA
=
0
BBBBBBBBBBBB@
f1......fk......
fn+1
1
CCCCCCCCCCCCA
4
" = "e + "p f(�, q) < 0 f(�, q) = 0
f(�, q) = �vm(�)� (�yo � q)
("e,↵) =1
2"e · C"e +
1
2H↵2
� =@
@"e, q =
@
@↵
"̇p = �̇@f
@�, ↵̇ = �̇
@f
@q
�̇ � 0 , f(�, q) 0 , �̇f = 0
R(u,h) = 0 hg =
0
@"p↵�
1
A h = [ghg
Qg(u,hg) =
0
@Q✏(u,hg)Q↵(u,hg)Q�(u,hg)
1
A =
0
@"p � "pn � (�� �n)n�
↵� ↵n � (�� �n)nq
f(�, q)
1
A = 0 n� =@f
@�nq =
@f
@q
Qg(u,hg) = 0 un, tn u, t hgn
�hg = A�1g Qg|u,hg hg ! hg +�hg
Ag =@Qg
@hg
�u = K�1totR|u,h u ! u+�u
Ktot =@R
@u+
@R
@h⌦ @h
@u
@h
@u= �
✓@Q
@h
◆�1 @Q
@u
2
6666666666664
K11 K12 0 · · · 0
K21 K22 K23 0...
.... . .
. . .. . .
...0 · · · 0 Kk k�1 Kk k Kk k+1 0 · · · 0...
. . .. . .
. . ....
... Knn�1 Knn Knn+1
0 · · · 0 Kn+1n Kn+1n+1
3
7777777777775
0
BBBBBBBBBBBBBBBB@
u1......
uk�1
ukuk+1......
un+1
1
CCCCCCCCCCCCCCCCA
=
0
BBBBBBBBBBBB@
f1......fk......
fn+1
1
CCCCCCCCCCCCA
4
" = "e + "p f(�, q) < 0 f(�, q) = 0
f(�, q) = �vm(�)� (�yo � q)
("e,↵) =1
2"e · C"e +
1
2H↵2
� =@
@"e, q =
@
@↵
"̇p = �̇@f
@�, ↵̇ = �̇
@f
@q
�̇ � 0 , f(�, q) 0 , �̇f = 0
R(u,h) = 0 hg =
0
@"p↵�
1
A h = [ghg
Qg(u,hg) =
0
@Q✏(u,hg)Q↵(u,hg)Q�(u,hg)
1
A =
0
@"p � "pn � (�� �n)n�
↵� ↵n � (�� �n)nq
f(�, q)
1
A = 0 n� =@f
@�nq =
@f
@q
Qg(u,hg) = 0
�hg = A�1g Qg|u,hg hg ! hg +�hg
Ag =@Qg
@hg
�u = K�1totR|u,h u ! u+�u
Ktot =@R
@u+
@R
@h⌦ @h
@u
@h
@u= �
✓@Q
@h
◆�1 @Q
@u
2
6666666666664
K11 K12 0 · · · 0
K21 K22 K23 0...
.... . .
. . .. . .
...0 · · · 0 Kk k�1 Kk k Kk k+1 0 · · · 0...
. . .. . .
. . ....
... Knn�1 Knn Knn+1
0 · · · 0 Kn+1n Kn+1n+1
3
7777777777775
0
BBBBBBBBBBBBBBBB@
u1......
uk�1
ukuk+1......
un+1
1
CCCCCCCCCCCCCCCCA
=
0
BBBBBBBBBBBB@
f1......fk......
fn+1
1
CCCCCCCCCCCCA
4
" = "e + "p f(�, q) < 0 f(�, q) = 0
f(�, q) = �vm(�)� (�yo � q)
("e,↵) =1
2"e · C"e +
1
2H↵2
� =@
@"e, q =
@
@↵
"̇p = �̇@f
@�, ↵̇ = �̇
@f
@q
�̇ � 0 , f(�, q) 0 , �̇f = 0
R(u,h) = 0 h = [ghg hg =
0
@"p↵�
1
A
Qg(u,hg) =
0
@Q✏(u,hg)Q↵(u,hg)Q�(u,hg)
1
A =
0
@"p � "pn � (�� �n)n�
↵� ↵n � (�� �n)nq
f(�, q)
1
A = 0 n� =@f
@�nq =
@f
@q
�hg = A�1g Qg|u,hg hg ! hg +�hg
Ag =@Qg
@hg
�u = K�1totR|u,h u ! u+�u
Ktot =@R
@u+
@R
@h⌦ @h
@u
@h
@u= �
✓@Q
@h
◆�1 @Q
@u
2
6666666666664
K11 K12 0 · · · 0
K21 K22 K23 0...
.... . .
. . .. . .
...0 · · · 0 Kk k�1 Kk k Kk k+1 0 · · · 0...
. . .. . .
. . ....
... Knn�1 Knn Knn+1
0 · · · 0 Kn+1n Kn+1n+1
3
7777777777775
0
BBBBBBBBBBBBBBBB@
u1......
uk�1
ukuk+1......
un+1
1
CCCCCCCCCCCCCCCCA
=
0
BBBBBBBBBBBB@
f1......fk......
fn+1
1
CCCCCCCCCCCCA
4
Symbolicprogramming⇧(u) r =
@⇧
@uK =
@r
@u(1)
1
Theory⇧(u) r =
@⇧
@u, K =
@r
@u(1)
1
Numeric
10 20 30 40X
10
20
30
40
50
60
Y
" = "e + "p f(�, q) < 0 f(�, q) = 0
f(�, q) = �vm(�)� (�yo � q)
("e,↵) =1
2"e · C"e +
1
2H↵2
� =@
@"e, q =
@
@↵
"̇p = �̇@f
@�, ↵̇ = �̇
@f
@q
�̇ � 0 , f(�, q) 0 , �̇f = 0
R(u,h) = 0 hg =
0
@"p↵�
1
A h = [ghg
Qg(u,hg) =
0
@Q✏(u,hg)Q↵(u,hg)Q�(u,hg)
1
A =
0
@"p � "pn � (�� �n)n�
↵� ↵n � (�� �n)nq
f(�, q)
1
A = 0 n� =@f
@�nq =
@f
@q
Qg(u,hg) = 0 tn t un hgn
�hg = A�1g Qg|u,hg hg ! hg +�hg
Ag =@Qg
@hg
�u = K�1totR|u,h u ! u+�u
Ktot =@R
@u+
@R
@h⌦ @h
@u
@h
@u= �
✓@Q
@h
◆�1 @Q
@u
2
6666666666664
K11 K12 0 · · · 0
K21 K22 K23 0...
.... . .
. . .. . .
...0 · · · 0 Kk k�1 Kk k Kk k+1 0 · · · 0...
. . .. . .
. . ....
... Knn�1 Knn Knn+1
0 · · · 0 Kn+1n Kn+1n+1
3
7777777777775
0
BBBBBBBBBBBBBBBB@
u1......
uk�1
ukuk+1......
un+1
1
CCCCCCCCCCCCCCCCA
=
0
BBBBBBBBBBBB@
f1......fk......
fn+1
1
CCCCCCCCCCCCA
4
" = "e + "p f(�, q) < 0 f(�, q) = 0
f(�, q) = �vm(�)� (�yo � q)
("e,↵) =1
2"e · C"e +
1
2H↵2
� =@
@"e, q =
@
@↵
"̇p = �̇@f
@�, ↵̇ = �̇
@f
@q
�̇ � 0 , f(�, q) 0 , �̇f = 0
R(u,h) = 0 hg =
0
@"p↵�
1
A h = [ghg
Qg(u,hg) =
0
@Q✏(u,hg)Q↵(u,hg)Q�(u,hg)
1
A =
0
@"p � "pn � (�� �n)n�
↵� ↵n � (�� �n)nq
f(�, q)
1
A = 0 n� =@f
@�nq =
@f
@q
Qg(u,hg) = 0 un, tn u, t hgn
�hg = A�1g Qg|u,hg hg ! hg +�hg
Ag =@Qg
@hg
�u = K�1totR|u,h u ! u+�u
Ktot =@R
@u+
@R
@h⌦ @h
@u
@h
@u= �
✓@Q
@h
◆�1 @Q
@u
2
6666666666664
K11 K12 0 · · · 0
K21 K22 K23 0...
.... . .
. . .. . .
...0 · · · 0 Kk k�1 Kk k Kk k+1 0 · · · 0...
. . .. . .
. . ....
... Knn�1 Knn Knn+1
0 · · · 0 Kn+1n Kn+1n+1
3
7777777777775
0
BBBBBBBBBBBBBBBB@
u1......
uk�1
ukuk+1......
un+1
1
CCCCCCCCCCCCCCCCA
=
0
BBBBBBBBBBBB@
f1......fk......
fn+1
1
CCCCCCCCCCCCA
4
" = "e + "p f(�, q) < 0 f(�, q) = 0
f(�, q) = �vm(�)� (�yo � q)
("e,↵) =1
2"e · C"e +
1
2H↵2
� =@
@"e, q =
@
@↵
"̇p = �̇@f
@�, ↵̇ = �̇
@f
@q
�̇ � 0 , f(�, q) 0 , �̇f = 0
R(u,h) = 0 hg =
0
@"p↵�
1
A h = [ghg
Qg(u,hg) =
0
@Q✏(u,hg)Q↵(u,hg)Q�(u,hg)
1
A =
0
@"p � "pn � (�� �n)n�
↵� ↵n � (�� �n)nq
f(�, q)
1
A = 0 n� =@f
@�nq =
@f
@q
Qg(u,hg) = 0 tn t un hgn
�hg = A�1g Qg|u,hg hg ! hg +�hg
Ag =@Qg
@hg
�u = K�1totR|u,h u ! u+�u
Ktot =@R
@u+
@R
@h⌦ @h
@u
@h
@u= �
✓@Q
@h
◆�1 @Q
@u
2
6666666666664
K11 K12 0 · · · 0
K21 K22 K23 0...
.... . .
. . .. . .
...0 · · · 0 Kk k�1 Kk k Kk k+1 0 · · · 0...
. . .. . .
. . ....
... Knn�1 Knn Knn+1
0 · · · 0 Kn+1n Kn+1n+1
3
7777777777775
0
BBBBBBBBBBBBBBBB@
u1......
uk�1
ukuk+1......
un+1
1
CCCCCCCCCCCCCCCCA
=
0
BBBBBBBBBBBB@
f1......fk......
fn+1
1
CCCCCCCCCCCCA
4
" = "e + "p f(�, q) < 0 f(�, q) = 0
f(�, q) = �vm(�)� (�yo � q)
("e,↵) =1
2"e · C"e +
1
2H↵2
� =@
@"e, q =
@
@↵
"̇p = �̇@f
@�, ↵̇ = �̇
@f
@q
�̇ � 0 , f(�, q) 0 , �̇f = 0
R(u,h) = 0 hg =
0
@"p↵�
1
A h = [ghg
Qg(u,hg) =
0
@Q✏(u,hg)Q↵(u,hg)Q�(u,hg)
1
A =
0
@"p � "pn � (�� �n)n�
↵� ↵n � (�� �n)nq
f(�, q)
1
A = 0 n� =@f
@�nq =
@f
@q
Qg(u,hg) = 0 un, tn u, t hgn
�hg = A�1g Qg|u,hg hg ! hg +�hg
Ag =@Qg
@hg
�u = K�1totR|u,h u ! u+�u
Ktot =@R
@u+
@R
@h⌦ @h
@u
@h
@u= �
✓@Q
@h
◆�1 @Q
@u
2
6666666666664
K11 K12 0 · · · 0
K21 K22 K23 0...
.... . .
. . .. . .
...0 · · · 0 Kk k�1 Kk k Kk k+1 0 · · · 0...
. . .. . .
. . ....
... Knn�1 Knn Knn+1
0 · · · 0 Kn+1n Kn+1n+1
3
7777777777775
0
BBBBBBBBBBBBBBBB@
u1......
uk�1
ukuk+1......
un+1
1
CCCCCCCCCCCCCCCCA
=
0
BBBBBBBBBBBB@
f1......fk......
fn+1
1
CCCCCCCCCCCCA
4
" = "e + "p f(�, q) < 0 f(�, q) = 0
f(�, q) = �vm(�)� (�yo � q)
("e,↵) =1
2"e · C"e +
1
2H↵2
� =@
@"e, q =
@
@↵
"̇p = �̇@f
@�, ↵̇ = �̇
@f
@q
�̇ � 0 , f(�, q) 0 , �̇f = 0
R(u,h) = 0 hg =
0
@"p↵�
1
A h = [ghg
Qg(u,hg) =
0
@Q✏(u,hg)Q↵(u,hg)Q�(u,hg)
1
A =
0
@"p � "pn � (�� �n)n�
↵� ↵n � (�� �n)nq
f(�, q)
1
A = 0 n� =@f
@�nq =
@f
@q
Qg(u,hg) = 0
�hg = A�1g Qg|u,hg hg ! hg +�hg
Ag =@Qg
@hg
�u = K�1totR|u,h u ! u+�u
Ktot =@R
@u+
@R
@h⌦ @h
@u
@h
@u= �
✓@Q
@h
◆�1 @Q
@u
2
6666666666664
K11 K12 0 · · · 0
K21 K22 K23 0...
.... . .
. . .. . .
...0 · · · 0 Kk k�1 Kk k Kk k+1 0 · · · 0...
. . .. . .
. . ....
... Knn�1 Knn Knn+1
0 · · · 0 Kn+1n Kn+1n+1
3
7777777777775
0
BBBBBBBBBBBBBBBB@
u1......
uk�1
ukuk+1......
un+1
1
CCCCCCCCCCCCCCCCA
=
0
BBBBBBBBBBBB@
f1......fk......
fn+1
1
CCCCCCCCCCCCA
4
" = "e + "p f(�, q) < 0 f(�, q) = 0
f(�, q) = �vm(�)� (�yo � q)
("e,↵) =1
2"e · C"e +
1
2H↵2
� =@
@"e, q =
@
@↵
"̇p = �̇@f
@�, ↵̇ = �̇
@f
@q
�̇ � 0 , f(�, q) 0 , �̇f = 0
R(u,h) = 0 h = [ghg hg =
0
@"p↵�
1
A
Qg(u,hg) =
0
@Q✏(u,hg)Q↵(u,hg)Q�(u,hg)
1
A =
0
@"p � "pn � (�� �n)n�
↵� ↵n � (�� �n)nq
f(�, q)
1
A = 0 n� =@f
@�nq =
@f
@q
�hg = A�1g Qg|u,hg hg ! hg +�hg
Ag =@Qg
@hg
�u = K�1totR|u,h u ! u+�u
Ktot =@R
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Automated Computational Modelling for the Finite Element Method
CO
UR
SE
May 8th, 11-13 May 9th, 9-11 & 11-13
Aula MS1, DICArVia Ferrata, 3 – Pavia
This seminar will introduce to applications of Automated ComputationalModelling for Finite Element problems. Starting from standard linear elasticproblems, lectures will guide through the generalization to mixedformulations in linear elasticity and to nonlinear problems in finite strainkinematics. The lecture will conclude with a brief overview on approachesfor material nonlinearities, such as elasto-plasticity
Michele Marino, PhD, MSc,Group Leader “Predictive Simulations in Biomechanics”Institute of Continuum MechanicsLeibniz Universität Hannover, Germany
Università degli Studi di PaviaComputational Mechanics & Advanced Materials Group - DICAr
