Automatic Generation of Numerical Codes
Jože Korelc
University of Ljubljana, Slovenia
www.fgg.uni-lj.si/Symech/
Solution procedure required by modern engineering
Multi-field
Multi-scale
Multi-body
F
F
M5
Multi-phase
Multi-objective

0.47e1

0.10

0.16

0.22

0.28

0.34

0.40
DeformedMeshX
M
a
x
.
0.1277e1
Min.
0.465
AceGen
2 J. KORELC, UNIVERSITY OF LJUBLJANA
Hybrid system for FE code generation and FE analysis
Mathematica
AceGen input
FE environment
dependend user subroutine
interface
AceGen
code generator
user subroutines
• tangent, residual
•post-processing
•....
element source file
C
FE
CDriver
environment • C code
Mathematica
Driver
•Math link
Optimisation
environment
3 J. KORELC, UNIVERSITY OF LJUBLJANA
Mathematica
Computational
Templates
package
suplementary subroutines
• initialization
•numerical integration
FORTRAN
MDriver
• full symbolic
cappabilities
• Mathematica
Inverse
ELFEN
FEAP
ABAQUS
Concepts of AceGen code generator
 simultaneous optimization of expressions and introduction of intermediate
variables,
 expression optimisation based on heuristic procedures,
 generation of characteristic formulae,
 the whole program structure can be generated,
 automatic differentiation technique (forward and backward mode),
 multi-language code generation (FORTRAN, C/C++, Mathematica language),
 multi-environment code generation (automatic interface to numerical environments)
 methods for exploring of generated formulae and run time debugging of the
generated programs
•
•
special procedures are needed for non-local (explicit form of the expressions is lost)
AD inside a single subroutine
4 J. KORELC, UNIVERSITY OF LJUBLJANA
Multi-language code generation with AceGen
1. Mathematical description
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2. Symbolic description in AceGen
SMSInitialize "test", Language - > "C++ "
SMSModule "Test", u1$$ 3 , x$$, L$$, g$$ 3
x, L Ł SMSReal x$$ , SMSReal L$$ ;
u  N i ui
ui Ł Array SMSReal u$$ #
x
x x
x
N  { ,1  , (1  )}
L
L L
L
2
f u
3. Automatically generated code
"Fortran"
SUBROUTINE Test(v,u1,x,L,g)
REAL*8 v(500),u1(3),x,L,g(3)
v(6)=x/L
v(7)=1d0 - v(6)
v(8)=v(6)*v(7)
v(12)=2d0*(u(1)*v(6) +
- u(2)*v(7) + u(3)*v(8))
g(1)=v(12)*v(6)
g(2)=v(12)*v(7)
g(3)=v(12)*v(8)
END
5 J. KORELC, UNIVERSITY OF LJUBLJANA
"C++"
x
x x
,1,
L
L L
u Ł Ni . ui;
Ni Ł
1-
x
L
;
&, 3 ;
;
f Ł u2 ;
g Ł SMSD f, ui ;
SMSExport g, g$$ ;
SMSWrite ;
void Test(double v[501],
double u1[3],double *x,
double *L,double g[3])
{
v[6]=*x/*L;
v[7]=1e0 - v[6];
v[8]=v[6]*v[7];
v[12]=2e0*(u[0]*v[6] +
u[1]*v[7]+ u[2]*v[8]);
g[0]=v[12]*v[6];
g[1]=v[12]*v[7];
g[2]=v[12]*v[8];
};
"Mathematica"
Test[]:=Module[{},
$VV[6]=x$$/L$$;
$VV[7]=1 - $VV[6];
$VV[8]=$VV[6]*$VV[7];
$VV[12]=2*(u$$[1]*$VV[6] +
u$$[2]*$VV[7]+u$$[3]*$VV[8]);
g$$[1]=$VV[6]*$VV[12];
g$$[2]=$VV[7]*$VV[12];
g$$[3]=$VV[8]*$VV[12];
]
MDriver
Multi-environment code generation
CDriver
Suplementary
routines
user
subroutines
6 J. KORELC, UNIVERSITY OF LJUBLJANA
Open problems
 Interactions:
 symbolic system - simultaneous simplification procedure
 symbolic system - automatic differentiation
 symbolic system - simultaneous generation of program structure
 automatic differentiation - simultaneous generation of program
structure
 Verification & validation of generated codes:
 Are the equations solved correctly?
 Are the righ equations solved?
 Interactive debugging of automatically generated numerical
codes
7 J. KORELC, UNIVERSITY OF LJUBLJANA
Symbolic system - simultaneous simplification
• Simultaneous stochastic optimisation procedure (Korelc 1997)
Additional operators
Auxiliary variables have standardized form $V[i, j],
where i is an index of auxiliary
variable and j is an instance of the i-th auxiliary variable.
8 J. KORELC, UNIVERSITY OF LJUBLJANA
Optimisation of an matrix - common sub expressions
Vector of 3 new auxiliary variables:

1
2


v





3
1
2
E
I v
L2
v
L
L 2 3


Original matrix
(input for code generator):
 12 EI
 L3
 6 EI
 2
L
 K0    12 EI
 L3
 6 EI
 2
 L
6 EI
 2
L
4 EI
L
6 EI
L2
2 EI
L
12 EI
L3
6 EI
L2
12 EI
L3
6 EI
L2
6 EI 
 2 
L
2 EI 

L 
6 EI 
L2 
4 EI 

L 
internal
data base
Code
generator
Mathematica
v2 v1 v2 
 v1

v3 
v
v

v
 2

3
2
2
KSv1 v2 v1 v2


v
3
v
v2 v3 
2


2


Result is simplified matrix, expressed with the
new auxiliary variables
9 J. KORELC, UNIVERSITY OF LJUBLJANA
Symbolic system - flow control
• "If" construct
SMSIf x
0 ;
$V[i,1]
2
f x ;
SMSElse
;
f Sin x ;
$V[i,2]
SMSEndIf f ;
fictive variable
$V[i,3]
• "Do" construct
f 1;
SMSDo i, 1, n, 1, f ;
f f
i
x;
SMSEndDo f ;
10 J. KORELC, UNIVERSITY OF LJUBLJANA
Symbolic system - automatic differentiation
• Automatic differentiation technique (AD):
– differentiation of the whole program
– automatic differentiation tool generates a program code for the derivative from a
program code for the basic function
• Reverse mode / Forward mode
• Enhancements with respect to the standard AD technique:
– AD procedure can be initiated at any time and at any point of
derivation of the formulas and as many times as required
– AD as code-to-code translator has to be replaced by the method that
consistently extends current code rather than produce a new one
– the results of all previous uses of AD have to be accounted for when AD is
used several times inside the same subroutine
– The user has to be able to use all the capabilities of the symbolic system on
the final and intermediate results of AD procedure.
11 J. KORELC, UNIVERSITY OF LJUBLJANA
AD exceptions
The basic situations that have to be considered are:
•
there exists implicit dependency between variables
that has to be considered for the differentiation
•
there exists explicit dependency between variables
that has to be neglected for the differentiation
•
the evaluation of the derivative code would lead to
numerical errors
Example:
 sin( 2 2 )
 0


f : 
2
 Lim sin( 2 )   0
   0

  sin( 2 2 )
)  0
 (
f


: 
 
 sin( 2 2 )
Lim
(
)  0
   0 

Exceptions handling function:
SMSFreeze exp, Dependency
12 J. KORELC, UNIVERSITY OF LJUBLJANA
p1 ,
exp
, p2,
p1
exp
, ..., pn ,
p2
exp
pn
Example: limit load optimisation of cantilever beam
• Task: find the shape (h(x)) of cantilever beam that has:
– minimal volume
– given ultimate load
– ideal elasto-plastic material
• Properties of the problem:
– transient coupled problem
– gradient based optimisation

Response
direct analysis
qu ultimate load
u
0.180e3
0.160e3
0.140e3
0.120e3
0.100e3
0.800e2
0.600e2
0.400e2
0.201e2
A
Misses
Max.
h(x)
x
0.2000e3
vA
optimal
solution
Min.
0.1225
AceGen
optimal shape?
13 J. KORELC, UNIVERSITY OF LJUBLJANA
b
b
v
Formulation of the problem
• Three finite elemnts are needed to describe the problem:
– 2D elasto plastic element
– line load element
– prescribed displacement constrain element
a  {u, }
u displacements ,  load factor
Ψ 
Ψ u
Ψ  
obt. el.
el.-plast.
Ψu  Ψu  Ψu
Ψu
el.-plast.
Ψu
obt. el.
standard ideal elasto-plastic formulation equations
    q. u d  load element equations
Newton-Raphson scheme to
solve direct problem
 K uu K u   u   Ψu 
K
     Ψ 
K
  
  
 u

Ψ   v A  v v p prescribed displacement constrain element equations
• Objective function
min  0 ;  0  (  u ) 2  Volume    penaltyconstrain h ( x ) 0
u prescribed limit load factor
 calculated limit load factor
14 J. KORELC, UNIVERSITY OF LJUBLJANA
k
Solution
• Analytical solution can be obtained by assuming:
– Bernoully beam theory
– in optimal case every material point is in plastic condition
2 qu
M ( x)
  y  h( x ) 
x
Wpl ( x)
b y
 y ... yield stress
• Optimal finite element solution for given load
0.174e3
0.149e3
0.124e3
0.997e2
0.746e2
0.496e2
Misses
Max.
0.2000e3
Min.
0.2456e2
AceGen
h(0)  20.41
analytical solution
h(0)  21.2
numerical solution
15 J. KORELC, UNIVERSITY OF LJUBLJANA
Typical examples: Multi - field problems
• unduction heating
•
T. Šuštar , C3M - Ljubljana, Slovenia
Magnetic
16 J. KORELC, UNIVERSITY OF LJUBLJANA
Thermal
Mechanical
Typical examples: Multi-scale problems
S. Stupkiewicz, IPPT, Warsaw, Poland
MICRO - 3D surface roughness
MACRO - elastic properties of contact surface
MACRO - lubricant flow in contact surface
17 J. KORELC, UNIVERSITY OF LJUBLJANA
MACRO - thermal properties of contact surface
Typical examples: structural analysis
• Sensitivity analysis of single-storey steel building
• four node shell elements and large displacement truss elements
A

Sensitivity
18
16
Analitic (elastic)
14
Finite differeces (elastic)
12
Analitic (plastic)
10
Finite differeces (plastic)
8
6
4
2
0
0
18 J. KORELC, UNIVERSITY OF LJUBLJANA
10
20
30
Roof angle (deg)
40
50
Typical examples: Simulation of technological proces
J. Lengiewicz, IPPT, Warsaw, Poland
Two phase forging
19 J. KORELC, UNIVERSITY OF LJUBLJANA
Conclusions
• Hybrid system presented is a powerfull tool for solving
large-scale engineering problems
• System can be dowloaded from www.fgg.uni-
lj.si/Symech/
20 J. KORELC, UNIVERSITY OF LJUBLJANA