International audienceThis paper is about derivatives techniques and their composition for semi-numerical models. Techniques such as symbolic derivation and automatic differentiation are addressed. All techniques are illustrated for the gradient based optimization of a magnetic nano switch

Coulomb, Jean-Louis

Delinchant, Benoît

Du Peloux, Bertrand

Llie, C.

Pham Quang, Phuong

Slusanschi, E.

English

Hal - Université Grenoble Alpes

Mixing Techniques to Compute Derivatives of semi-numerical models: Application to Magnetic Nano Switch Optimization

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Mixing Techniques to Compute Derivatives of
semi-numerical models: Application to Magnetic Nano
Switch Optimization
Phuong Pham Quang, Benoît Delinchant, C. Llie, E. Slusanschi, Jean-Louis
Coulomb, Bertrand Du Peloux
To cite this version:
Phuong Pham Quang, Benoît Delinchant, C. Llie, E. Slusanschi, Jean-Louis Coulomb, et al.. Mixing
Techniques to Compute Derivatives of semi-numerical models: Application to Magnetic Nano Switch
Optimization. COMPUMAG 2011, Jul 2011, Sydney, Australia. ￿hal-00610867￿
10. Optimization and Design, 4. Nano-Electromagnetic Computation and Applications 
Abstract — This paper is about derivatives techniques and 
their composition for semi-numerical models. Techniques such 
as symbolic derivation and automatic differentiation are 
addressed. All techniques are illustrated for the gradient based 
optimization of a magnetic nano switch. 
I. INTRODUCTION 
Sizing by optimization is nowadays of major interest of 
the design process of electromagnetic device. Our goal is to 
use optimization algorithms in conjunction with semi- 
analytical model in order to solve the sizing problem. 
Among the deterministic algorithms, the constrained 
gradient based optimization using Sequential Quadratic 
Programming (SQP) algorithms is very efficient. Such 
algorithms require accurate values of the objective function 
and constraint derivatives. Unfortunately, it is often difficult 
to obtain the symbolic expression of the derivatives of the 
sizing model. But several techniques as Automatic 
Differentiation (AD) can be used to compute these 
derivatives. 
The paper highlights a generic framework of model 
composition using different derivation techniques. This 
approach is shown through the example of a magnetic nano 
switch optimization. 
II. DERIVATION TECHNIQUES 
The finite difference approximation leads to an 
approximation suffering from both truncation and 
cancellation errors. It is very easy to set-up, but it is very 
difficult to settle the adjustment parameter. 
The derivatives can be computed symbolically. This 
method is reliable and the faster that can be found, not only 
with symbolic expression but also with functions resulting 
from numerical integration or from implicit solver (thanks 
to implicit theorem). The main drawback of this method is 
the impossibility to differentiate the computing code (with 
conditional instructions, loops, etc.). 
Automatic differentiation (AD) exploits the natural 
process of source code compilation and makes use of the 
intermediate representation of implemented functions in 
different programming languages. The derivatives of each 
elementary operation can be obtained in a straightforward 
way and can be combined according to the chain rule of 
differential calculus in order to obtain the derivative of the 
initial function. Thus, depending on the chosen 
accumulation strategy, functions of arbitrary complexity can 
be differentiated in two ways, namely forward and reverse. 
In the forward mode one propagates derivatives of 
intermediate variables with respect to the independent 
variables, while in reverse, derivatives of the function with 
respect to intermediate variables – also called adjoints – are 
propagated. 
Typically, AD can be implemented using either the 
operator overloading (ADOL-C [4], CppAD, etc) or the 
source transformation technique (ADIFOR, ADiJaC [6], 
etc). In operator overloading one overloads the operators 
which are applied on new variable types, with the routine 
call performing the actual derivative computation. The 
source transformation approach examines the source code 
of the original function and generates new code that 
computes the desired derivatives in the same time that the 
original function.  
III. SENSITIVITY PROPAGATION OF COMPOUND MODEL 
As studied in [2] , and as shown in Fig. 1 and Fig. 2, two 
kinds of sensitivity propagation mechanism can be 
considered to build global sensitivities. 
 
 
Fig. 1. A compound model based on partial derivative propagation 
 
 
Fig. 2. A compound model based on differential propagation 
Each method owns its advantages and drawbacks. The 
choice of partial derivative propagation has been made with 
interface standard in CADES framework (developed in 
Grenoble laboratory) to ensure interoperability between 
models from different derivation techniques. 
IV. DERIVATIVES OF MAGNETIC NANO SWITCH MODEL  
The semi-numerical model of magnetic nano switch was 
presented in [3]. It involves a coupled magneto-mechanical 
deformation of a cantilever beam, and allows evaluating the 
contact length and the contact force. The following part 
details how the derivatives of each model blocks are 
computed. 
Mixing Techniques to Compute Derivatives of semi-numerical models:  
Application to Magnetic Nano Switch Optimization 
P. Pham-Quang1,2, B. Delinchant1, C. Ilie3,  
E. Slusanschi3, J.L. Coulomb1, and B. du Peloux2 
1Grenoble Electrical Engineering Laboratory, G2ELab, F-38402 St Martin d’Hères Cedex, France 
2Cedrat SA, 15, Chemin de Malacher, F-38246 Meylan Cedex, France 
3Computer Science and Engineering, University Politehnica of Bucharest, 060042 Bucharest, Romania 
Email: phuong.pham-quang@g2elab.grenoble-inp.fr 
10. Optimization and Design, 4. Nano-Electromagnetic Computation and Applications 
A. Magnetic model 
For specific configuration of MEMS/NEMS, the 
Coulombian approach is used for the modeling of magnet, 
leading to a full analytical magnetic field model. Forces and 
magnetic torques applied on the beam (in magnetic parts) 
are computed by adaptive numerical integrations. The 
derivatives of this magnetic model can be computed 
symbolically with accuracy and rapidity [5]. 
B. Mechanical deformation model 
The model used to compute deformation in the presence 
of contact of a cantilever beam has inputs and outputs as 
shown in Fig. 3 
 
Fig. 3. Inputs and outputs of mechanical deformation model 
ADiJaC [6] was used in order to manage the derivation 
of this complex algorithm. ADiJaC is a source 
transformation AD tool that implements both the forward 
and reverse modes on Java codes. To date it is the first 
useable AD tool for the Java programming language. 
Differentiating the NEMS model with ADiJaC was 
improved to deal with special cases of matrix operations, 
nested method calls, special functions (e.g. “length”), and 
special array initializations – all intra-procedural 
transformations. During this work, ADiJaC’s inter-
procedural analysis was also enhanced, so as to allow the 
tool to differentiate functions returning arrays, and receiving 
any number of vector and scalar parameters. 
C. Magneto-Mechanic coupling 
 
Fig. 4. Sequential magnetic-mechanical coupling 
The sequential magneto-mechanical coupling described 
in Fig. 4. has been solved by an implicit solver with Gauss 
method. Its derivatives are computed symbolically thanks to 
a classical implicit theorem. 
D. Composition in the CADES framework 
 
 
Fig. 5. Sensitivity propagation of magnetic nano switch in CADES 
All these models are given with their own Jacobian 
using different techniques (symbolic or AD). CADES 
framework defines a standard to make the composition 
possible. Each model implementing this standard can be 
integrated into the framework. Then the overall is treated 
automatically by another AD tool (ADOL-C) to build the 
global model (Fig. 5).   
V. OPTIMIZATION RESULTS 
Working principle of magnetic nano switch was 
presented in [3]. Geometry and parameters to be optimized 
are given in Fig. 6.  The optimization aims to determine 
dimensions of both fixed and mobile magnets and their 
positions on the beam. The objective is to minimize the 
volume of magnets, while respecting constraints such as 
length of contact and contact force, in order to ensure the 
quality of contact or the contact resistances are below a 
desired value.  
 
 
Fig. 6. Geometry and parameters to be optimized 
A genetic algorithm (without gradients) and the SQP 
algorithm (using gradients) were used. Both gave the same 
results but the first algorithm needed 3 hours computation 
time while the second only 92 seconds.  
VI. CONCLUSIONS  
Deterministic optimization using gradient based 
algorithms is very efficient for sizing problems with 
constraints. However, such algorithms meet a major lack of 
usage because of the difficulty to obtain exactly and rapidly 
the derivatives values. This paper highlights the 
composition of different derivation techniques in a 
compound model through the example of magnetic nano 
switch. 
The full paper will detail the propagation mechanism of 
derivatives. Each derivation techniques will be detailed, 
highlighting advantages and drawbacks. More details will 
be given on the optimization procedure and results. 
VII. REFERENCES 
[1] J. Sobieszczanski-Sobieski, “Sensitivity of complex, internally 
coupled systems,” AIAA J., vol. 28, no. 1, pp. 153–160, Jan. 1990. 
[2] B. Delinchant, F. Wurtz and E. Atienza, “Reducing sensitivity 
analysis times-cost of compound model”, IEEE Transactions on 
Magn., vol. 40 no. 2, pp. 1216-1219, March 2004. 
[3] P. Pham-Quang, B. Delinchant, J.L. Coulomb and B. du Peloux, 
“Semi-analytical magneto-mechanic coupling with contact analysis 
for MEMS/NEMS”, to be published IEEE Transactions on Magn., 
[4] P. Enciu, F. Wurtz, L. Gerbaud, B. Delinchant “Automatic 
differentiation for electromagnetic models used in optimization”, 
COMPEL; Vol 28, Issue: 5, pp 1313 - 1326, 2009, ISSN: 0332-1649 
[5] B. Delinchant, et al. “Gradient based optimization of semi-numerical 
models with symbolic sensitivity”, IJAEM, vol.30, pp.189-200, 2009. 
[6] E. Slusanschi, Algorithmic differentiation of Java programs, Ph.D. 
thesis, RWTH Aachen University,  2008. 
Magnetic 
Mechanic 
Forces, Torques Positions 
Mechanical  
deformation 
model 
Deformations 
Contact’s length 
Contact’s force 
 
 
Substrate 
Zf 
Zm z 
x 
y 
Ym 
Magnetic  
(Symbolic) 
Mechanic  
(ADiJaC) 
Coupling  
(Symbolic) 
Supplementary equations  
(ADOL-C) 
ADOL-C for composition 
Geometry 
Young’s modulus 
Punctual forces 
Punctual torques 
Y_offset 


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Mixing Techniques to Compute Derivatives of semi-numerical models: Application to Magnetic Nano Switch Optimization

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