Automatic Generation of Numerical Codes Jože Korelc University of Ljubljana, Slovenia

Automatic Generation of Numerical Codes

Jože KorelcUniversity of Ljubljana, Slovenia

www.fgg.uni-lj.si/Symech/

2 J. KORELC, UNIVERSITY OF LJUBLJANA

Solution procedure required by modern engineering

M5

AceGen

0.465

Min.

0.1277e 1

Max.

DeformedMeshX

0.40

0.34

0.28

0.22

0.16

0.10

0.47e 1

Multi-field

Multi-scale

Multi-body

Multi-phase

Multi-objective

F F


Hybrid system for FE code generation and FE analysis

AceGencode generator

ComputationalTemplatespackage

FE environmentdependend user subroutine

interface

C FORTRANMathematica

AceGen input

user subroutines• tangent, residual• post-processing

• ....

suplementary subroutines• initialization

• numerical integration

Mathematica

element source file

CDriver• C code

•Math link

FEAPELFENMDriver• full symbolic

cappabilities• Mathematica

ABAQUS

Driver

Optimisation environment

FE environment

Mathematica Inverse


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


Multi-language code generation with AceGen

1. Mathematical description 2. Symbolic description in AceGenSMSInitialize@"test", Language - >"C++"DSMSModule@"Test",8u1$$@3D, x$$, L$$, g$$@3D<D;8x, L<Ł8SMSReal@x$$D, SMSReal@L$$D<;uiŁ Array@SMSReal@u$$@#DD&, 3D;NiŁ9x

L, 1 -

xL,xLI1 - x

LM=;

uŁ Ni.ui;

fŁ u2;gŁ SMSD@f, uiD;SMSExport@g, g$$D;SMSWrite@D;

3. Automatically generated code

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

"Fortran"

2

)}1(,1,{

uf

L

x

L

x

L

x

L

x

uNu ii

N

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];};

"C++"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];]

"Mathematica"


SMT̀ SetElSpec["test",idata$$_,ic_,gd_]:=Block[{q1,q2,q3},q3=SMTNumIntegration[ic];q1={"test",{"SKR" -> SMT̀ SKR, _ -> Null},{"SpecIndex",2,20,0,4,2,5,ic, "NoTimeStorage",0,q3[[1]],0,0,1},"Q1", {"Const 1", "Const 2"},{},{},{1,2,3,4,0} ,{5, 5, 5, 5},{},{},gd,q3[[2]]//Transpose};q1[[3,9]]=0;q1];

(******************* M O D U L E ***************)SetAttributes[SMT̀ SKR,HoldAll];SMT̀ SKR["test",es$$_,ed$$_,nd$$_,rdata$$_,idata$$_ ,p$$_,s$$_]:=Module[{},p$$[[1]]=nd$$[[1,3,1]];];

Multi-environment code generation

SMSInitialize"test", "Language" "C";SMTInitialize"test", "CDriver","SMTTopology" "Q1", "SMTDOFGlobal" 5,

"SMTGroupDataNames" "Const 1", "Const 2";SMTUserSubroutine"Tangent and residual";SMSExportSMSRealnd$$1, "X", 1, p$$1;SMSWrite"Splice" SMTSplice;

SMSInitialize"test", "Language" "Mathematica";SMTInitialize"test", "MDriver","SMTTopology" "Q1", "SMTDOFGlobal" 5,

"SMTGroupDataNames" "Const 1", "Const 2";SMTUserSubroutine"Tangent and residual";SMSExportSMSRealnd$$1, "X", 1, p$$1;SMSWrite"Splice" SMTSplice;

MDriver CDriver

#include "sms.h"void SKR(double v[501],ElementSpec *es,ElementData *ed,NodeData **nd,double *rdata,int *idata,double*p,double **s);__declspec(dllexport) void SMTSetElSpec(ElementSpec *es,int *idata,int *ic,double *gd){ static int pn[5]={1, 2, 3, 4, 0}; static int dof[4]={5, 5, 5, 5}; static char *gdcs[]={"Const 1","Const 2"}; static char *gpcs[]={""};static char *npcs[]={""}; es->Code="test";es->id.NoDimensions=2;es->id.NoDOFGlobal=20; es->id.NoDOFCondense=0;es->id.NoNodes=4; es->id.NoGroupData=2;es->id.NoSegmentPoints=5; es->id.IntCode=*ic;es->id.NoElementData=0; es->Segments=pn;es->DOFGlobal=dof;es->Data=gd; es->id.NoGPostData=0;es->id.NoNPostData=0; es->id.SymmetricTangent=1; es->IntPoints=SMTIntPoints(ic);es->id.NoTimeStorage=0; es->Topology="Q1";es->GroupDataNames=gdcs; es->GPostNames=gpcs;es->NPostNames=npcs;es->user.SKR=SKR;};

void SKR(double v[501],ElementSpec *es,ElementData *ed,NodeData **nd,double*rdata,int *idata,double *p,double **s){p[0]=nd[0]->X[0];};

Suplementaryroutines

usersubroutines


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


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.

InitializationSMSInitialize"test", "Language"

"C", "Mode" "Optimal";Definition of a new subroutineSMSModule"Test",

Realx$$, f$$, dfdx$$;Numericsymbolic interfacex SMSRealx$$;Derivation of the problem Cosx;SMSIfSMSAbs 1010;f Sin2 2;

SMSElse;f

SMSFreezeLimitSin2 2, 0,"Dependency",, LimitDSin2 2,

Evaluate, 0;SMSEndIfTrue, f; Automatic differentiationdfdx SMSDf, x;Symbolicnumeric interafceSMSExportdfdx, dfdx$$;Code generationSMSWrite;

v exp A new auxiliary variable is created ifAceGen finds out that the introductionof the new variable is necessary,otherwise vexp

v exp A new auxiliary variable is created,regardless on the contents of exp.

v exp Create variable which will appearmore than once on a lefthand sideof equationmultivalued variables.

v exp A new valueexpis assigned to thepreviously created auxiliary variable v.


Optimisation of an matrix - common sub expressions

K

EI

L

EI

L

EI

L

EI

LEI

L

EI

L

EI

L

EI

LEI

L

EI

L

EI

L

EI

LEI

L

EI

L

EI

L

EI

L

0

3 2 3 2

2 2

3 2 3 2

2 2

12 6 12 6

6 4 6 2

12 6 12 6

6 2 6 4

Original matrix(input for code generator):

Code generator

v EI

L

v L v L

12

2

2

331 2

Vector of 3 new auxiliary variables:

internal data base

K

v v v v

v v vv

v v v v

vv

v v

S

1 2 1 2

2 3 23

1 2 1 2

23

2 3

2

2

Result is simplified matrix, expressed with the new auxiliary variables

Mathematica


Symbolic system - flow control• "If" construct

• "Do" construct




SMSElse;f SMSFreezeLimit

Sin2 2, 0, "Dependency",, LimitDSin2 2, Evaluate, 0;

SMSEndIfTrue, f; Automatic differentiationdfdx SMSDf, x;Symbolicnumeric interafceSMSExportdfdx, dfdx$$;Code generationSMSWrite;

SMSIfx 0;f x2;

SMSElse;f Sinx;SMSEndIff;

f 1;

SMSDoi, 1, n, 1, f;f f xi;

SMSEndDof;

fictive variable $V[i,3]

$V[i,1]

$V[i,2]


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.





SMSElse;f SMSFreezeLimit

Sin2 2, 0, "Dependency",, LimitDSin2 2, Evaluate, 0;

SMSEndIfTrue, f; Automatic differentiationdfdx SMSDf, x;Symbolicnumeric interafceSMSExportdfdx, dfdx$$;Code generationSMSWrite;

0))2sin(

(

0))2sin(

(:

0)2sin(

0)2sin(

:

2

0

2

2

0

2

Lim

f

Lim

f

AD exceptions

Exceptions handling function:

SMSFreezeexp, Dependency p1,exp

p1,p2,

exp

p2, ...,pn,

exp

pn

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:


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

AceGen0.1225

Min.

0.2000e3

Max.

Misses

0.201e20.400e20.600e20.800e20.100e30.120e30.140e30.160e30.180e3

ultimate loaduq

x)(xh

bb

A

Av

u

v

Responsedirect analysis

optimal solution

optimal shape?


Formulation of the problem• Three finite elemnts are needed to describe the problem:

– 2D elasto plastic element

– line load element

– prescribed displacement constrain element

• Objective function

uu u u

u

u

K K Ψ

K K Ψ

obt. el.el.-plast.

el.-plast.

obt. el.

{ , }

displacements , load factor

standard ideal elasto-plastic formulation equations

. load element equations

prescribed d

u

u u u

u

u

A v p

d

v v

a u

u

ΨΨ

Ψ

Ψ Ψ Ψ

Ψ

Ψ q u

Ψ isplacement constrain element equations

20 0 penaltyconstrain ( ) 0min ; ( ) olumeu h x

k

V

factor loadlimit calculated

factor loadlimit prescribed

u

Newton-Raphson scheme to solve direct problem


Solution• Analytical solution can be obtained by assuming:

– Bernoully beam theory

– in optimal case every material point is in plastic condition

• Optimal finite element solution for given load

AceGen0.2456e2

Min.

0.2000e3

Max.

Misses

0.496e2

0.746e2

0.997e2

0.124e3

0.149e3

0.174e3

2( )( ) ...

( )u

y ypl y

qM xh x x yield stress

W x b

solution numerical2.21)0(

solution analytical41.20)0(

h

h


Typical examples: Multi - field problems• unduction heating • T. Šuštar , C3M - Ljubljana, Slovenia

Magnetic Thermal Mechanical


Typical examples: Multi-scale problems

MACRO - elastic properties of contact surface

MICRO - 3D surface roughness

MACRO - thermal properties of contact surfaceMACRO - lubricant flow in contact surface

S. Stupkiewicz, IPPT, Warsaw, Poland


Typical examples: structural analysis• Sensitivity analysis of single-storey steel building

• four node shell elements and large displacement truss elements

A

0

2

4

6

8

10

12

14

16

18

0 10 20 30 40 50Roof angle (deg)

Sen

siti

vity

Analitic (elastic)

Finite differeces (elastic)

Analitic (plastic)

Finite differeces (plastic)


Typical examples: Simulation of technological proces

Two phase forging

J. Lengiewicz, IPPT, Warsaw, Poland


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/

Documents

Automatic Generation of Numerical Codes Jože Korelc University of Ljubljana, Slovenia