The research projects have been organized within the SCAT (Scientific
Computing and Applications in Technology) program. The SCAT program
consists of projects in five main research areas. In the year 1994,
these areas were
- foundations of computational methods,
- computational mechanics and physics,
- control and identification,
- computational methods of large-scale systems, and
- pilot projects.
Within each research area a reasonable level of synergy between different
projects is generally provided either by use of similar methods or by study of
similar problems with different types of methods.
The different research areas have a common mathematical background which
makes the communication possible also between projects in different areas. Moreover, much effort has been invested in the common computational environment
which on the one hand reduces the routine work in the projects and on the other
hand makes it easier for the projects to communicate with each other.
Although many of the projects deal with fundamental questions, the main emphasis is in most of them in some special problems. That is why only relatively
few projects can be considered to belong to this category.
Finite Element Methods and Applications (MICHEL KRÍZEK, RAINO A.E. MÄKINEN, PEKKA NEITTAANMÄKI, VALERY RIVKIND and TIMO TIIHONEN)
The project deals with the finite element analysis of some important,
mostly nonlinear, problems. The goal is to give a rigorous analysis
for the theoretical background of these questions. Of course, it is
not possible to cover all technically interesting cases.
Nevertheless, several typical techniques are introduced for solving
nonlinear problems which can then be modified for other problems.
There is no general numerical method capable of solving an arbitrary
nonlinear problem and, therefore, every class of problems has to be
investigated individually.
The investigation of nonlinear problems consists of several steps. The
problem is split into a series of subproblems, including, as a rule,
the proof of the existence (and, if applicable, also uniqueness)
of the true solution, the construction of concrete finite elements,
the treatment of convergence questions together with the numerical
solution of the approximate problem. Among others, Maxwell equa-
tions, contact problems, free boundary problems, Stefan-like problems,
crystal growth problems and problems in elasticity and plasticity have
been handled.
Research collaboration with researchers from Czech
Republic, Germany, USA, France, England, Italy, Poland, Russia and
other universities in Finland.
This project consists of two parts. The first part deals with
nonsmooth analysis and optimization. Subdifferential calculus has
been applied to generalize the optimality conditions of
nondifferentiable functions. This theoretical basis has made it
possible to develop effective bundle-type methods for nonsmooth and
nonconvex optimization problems. The methods have been implemented as
a subroutine library NSOLIB (see chapter 4.3).
The second part deals with multiobjective optimization. A
methodological state- of-the art survey with firm theoretical
foundations has been prepared. The actu- al aim has been to generate
effective and user-friendly methods for complicated real-world
problems. As a concrete result, an interactive method, NIMBUS, for
nondifferentiable multiobjective optimization problems has been
developed. The algorithm is based on the classification of the
objective functions. According to the classification, a new
(multiobjective) optimization problem is formed and solved employing
NSOLIB. Some optimal control problems arising from optimal shape design
and continuous casting processes have been solved as applications. An
implementation of NIMBUS in the MS-Windows environment has been
produced.
The goal of the project is to construct simulation models for some key
parts of a paper machine, as well as to study the properties of
paper. A so-called former section of a paper machine (where fluidlike
suspension of water and fibres gradually solidifies to a porous
fibre net) has been modelled. Mathematical properties of the model
are studied and numerical methods are being developed.
A new project in the modelling of copying behaviour of paper has been
started. The research focuses on modelling the heat and moisture
transfer in paper during hot roll nip actions which are common in
copying and laser printing machines. Experimental tests have also been
carried out in order to obtain verification data for the developed
model.
Free Boundary Problems (JAROSLAV HASLINGER, JARI JÄRVINEN, PEKKA NEITTAANMÄKI, VALERY RIVKIND and TIMO TIIHONEN)
Free boundary problems form one of the main research fields of the
laboratory. In the past, the work on free boundary problems and related
control problems initiated the research in nonsmooth optimization and
in state constrained control problems. Currently, the laboratory
takes part in the ESF programme Mathematical Treatment of Free
Boundary Problems.
The theoretical work on free boundary problems concentrates now on
modelling the evaporation process. The goal is to find adequate free
boundary conditions for a coupled system of Navier-Stokes and Stefan
equations, as well as to develop suitable techniques for mathematical
analysis and numerical solution of such problems.
Another main line in the research is the use of techniques developed
for shape optimization in solving free boundary problems and
analysing the solution methods. Often, the free boundary conditions can
be formulated as optimality conditions of a shape optimization
problem.
There is also work going on in numerical simulation of an industrial
process for producing silicon crystals. A numerical model has been
constructed which takes into account all the major mechanisms of heat
transfer including solidification and heat radiation.
The goal has been to introduce efficient and reliable methods for the
synthesis of magnetic and electric fields. The results have been
documented in a forth-coming monograph "Inverse Problems and Optimal
Design in Electricity and Magnetism" (in printing at the Oxford
University Press).
A special attention has been paid to an algorithmic approach. The book
deals with the solution of real-life synthesis problems in electricity
and magnetism, using the numerical techniques (for solving partial
differential equations, optimization, inverse, and shape design
problems) introduced. Inverse problems have been classified from an
engineering viewpoint. Classical and simple examples have been found
from the following categories: synthesis of sources, synthesis of
boundary conditions, synthesis of material properties, and synthesis of
shapes. A survey of solved problems which have appeared in the
literature has been done.
Some practical topics have been handled. In particular,
implementations of the finite-element method, optimal shape design and
sensitivity analysis, including automatic differentiation of computer
programs, are available. Finally, a survey of subroutine libraries
for the solution of partial differential equations, electromagnetic
problems and optimization problems including nonsmooth and
multiobjective optimization, and shape optimal design has been made.
Timing Simulation of Digital VLSI Circuits (CORNELIU A. MARINOV
PEKKA NEITTAANMÄKI and JUKKA-PEKKA SANTANEN)
The origin of the project is the interest of designers in high-speed
simulators for large scale integrated electrical circuits. There is
also need to model the circuits by distributed parameters. The
numerical simulation methods include waveform relaxation techniques for
linear and nonlinear parameters. Monotone iterative techniques are
applied to differential-algebraic equations to evaluate the circuit
performance. The proposed techniques assure a faster circuit
simulation for design purposes with a high accuracy and good
stability properties. They allow the system to be decomposed into
decoupled subsystems which can be analysed in parallel. A faster timing
simulator for MOS circuits is the main objective.
The research also concentrates on bounding techniques for performance
evaluation. The methods derive a priori closed form bounds for
signal delay and amplitude when networks of distributed parameter lines
are modelled by partial differential equations. These methods give
tighter and computationally simple bounds for voltages and for the
delay in lumped parameter circuits. The studies lead not only to faster
timing simulators, but to accurate ones from the point of view of
the modelling exactness.
The purpose is to discuss optimal control problems, especially,
systems arising from free boundary problems. The main interest has
been in optimal control problems governed by nonlinear parabolic
systems, including, among others, parabolic variational inequalities
and systems with phase transitions. The aim is twofold: firstly, to
give a theoretical approach to the subject and, secondly, to present
detailed algorithms (with convergence proofs) that are necessary in
computerizing the optimal control processes. Several practical
examples are being worked out in detail in order to demonstrate the
usefulness of the proposed methods. The results have been collected in
the monograph [10.1, 1].
Collaboration with researchers from France, Germany, USA, Japan and
Romania.
Optimization of Distributed Systems with Applications (TIMO MÄNNIKKÖ)
The main subject of this project is the theory of sentinels and their
applications. Sentinels are "filters" which are sensitive to the
variations of some specific parameters, and at the same time
insensitive to the variations of all the other unknown parameters
appearing in the system.
As an application, sentinels are used to identify unknown source terms
in a parabolic partial differential equation with incompletely defined
boundary and/or initial conditions.
International co-operation with researchers from Compiegne University
of Technology, France.
In this project, hemivariational inequalities, generalized variational
inequalities involving nonmonotone, multivalued inclusion, are
studied. The existence results have been shown for a constrained
stationary hemivariational inequality and for a parabolic
hemivariational inequality. In addition, a stable and convergent FEM
approximation has been developed for these equations. This
approximation has been used in the numerical solutions of nonmonotone
contact problems of linear elasticity by nonsmooth, nonconvex
optimization methods.
Collaboration with Aristotle University, Greece (prof. P.D.
Panagiotopoulos).
The aim is to develop methods for the identification of a functional
coefficient. It is assumed that one has a distributed observation of
the solution of an elliptic or parabolic partial differential
equation. These observations are used to determine an unknown
coefficient in the equation. In physical systems, this parameter can
represent the heat conductivity or the diffusion coefficient.
Several different discretization schemes based on variants of finite
element, spectral and multigrid methods are considered. In
addition to new efficient numerical algorithms, one of the purposes
is the improvement of techniques for convergence analysis of different
approaches. At the moment, there exist few error estimates for
parameter identification problems, and only for elliptic equations. One
goal is also the treatment of quasilinear and nonlinear problems
with some linearization techniques.
General Regularization Methods for Solutions of Ill-Posed Problems
(TIMO TASKINEN)
A problem is called well-posed if it is uniquely solvable for each
data and if the solution depends continuously on the data. Otherwise,
the problem is called ill-posed. One way to overcome the ill-posedness
is to use regularization methods. In this project, new
regularization schemes are developed and modified to be in their most
suitable form for some particular problems. Error bounds obtainable
and the speed of convergence are also matters of interest.
Co-operation with the University of Tartu, Estonia (prof. G. Vainikko).
Nowadays, simulation models are widely used in process control. In the
dynamic simulation, the computing time and the accuracy are important.
In this project, a parallelized simulation program is developed for
controlling the cooling of a continuous casting machine. The model has
been installed in a casting machine which casts about 500 000 tons
steel in a year. The complicated model which consumes a lot of
computer resources has been parallelized for several transputers
inside a microcomputer.
In addition, a control program for controlling the fibre orientation
of paper in a paper making process is developed. This application is
based on Fourier analysis, optimization and simple analysis of fluid
flows. The control model uses a computer network and the knowledge
base of a paper factory.
The programming tools used are C++ and MS-Windows. Collaboration with
Rautaruukki steel works in Raahe and Kymi paper mills in Kuusankoski.
Parameter Estimation for Continuous Casting Process (ERKKI LAITINEN
and MARKKU MALINEN)
The estimation of the secondary cooling parameters is crucial for the
correct simulation of the heat transfer in continuous casting. Based on
the measurements of the water flow rate and the surface temperature in
a real casting process one can calculate the cooling parameters. In
this project, the aim is to find the optimal heat transfer
coefficients for the simulation and to develop an easy-to-use interface to execute that searching in the MS-Windows environment.
Collaboration with Rautaruukki steel works in Raahe.
The project considers fast direct methods for solving separable linear
systems arising from finite difference or finite element approximations
of partial differential equations on rectangular domains. The
fast direct methods have many important applications, like various
fictitious domain algorithms. The fictitious domain method is a way
to construct an efficient preconditioning matrix for iterative
methods for solving problems in nonrectangular domains. The fictitious
domain methodology has also been used for solving problems arising from
fluid dynamics, acoustics and electromagnetism.
Parallel numerical methods for solving various linear and nonlinear
problems have been developed in the project. The purpose is to
construct efficient imple- mentation algorithms suitable for
distributed memory multiprocessor systems. This project has three main
topics. The first one is to develop a general purpose parallel
finite element package. The second one is to study domain decomposition
methods, especially, applied to free boundary problems. The
parallelization of the fast direct solvers has become a new
important topic within this project. The main goal is to develop
parallel versions of various fictitious domain algorithms.
In shape optimization, the optimization variable is the shape of the
domain on which a partial differential equation (the state equation) is
posed. The research has been oriented towards finding efficient
numerical methods for general optimal shape design problems. In
addition, shape optimization has been applied to solve shape design
problems in fluid dynamics.
The most crucial step in shape optimization is sensitivity analysis,
that is, the calculation of the gradient of the cost function.
There are basically two ways to perform the sensitivity analysis.
The first approach uses the discretized model and carries out design
sensitivity analysis by differentiating the algebraic equa- tions.
The second approach uses the material derivative of continuum mechanics
to analyse the changes in the shape of the domain.
The algebraic approach has been applied to solve shape optimization
problems for a transonic flow. In co-operation with Dr. J. Chleboun
(Czech Republic) a primal hybrid variational formulation has been
applied to solve shape optimization problems. The formulation has
the advantage of giving an accurate approximation of boundary fluxes or
stresses. This is important when boundary formulae are used for the
sensitivity analysis. Shape optimization problems governed by Navier-
Stokes equations with non-standard boundary conditions have also been
studied. In a joint project with Dassault Aviation (France), the
fictitious domain method is applied to multidisciplinary shape
optimization. The objective of this study is to develop an efficient
fictitious domain approach for shape optimization.
Cellular Radio Networks (VEIKKO HARA, ERKKI
LAITINEN, MARKKU MALINEN and
LIINA NENONEN)
The research has been concentrated on two applications which model the
operation of cellular radio networks. The first application is a
model in which optimal base station parameters are estimated by
using a simulation model and optimization methods. The parameters to
be optimized are the handover limits, that is, the values of the signal
strength and the C/I-ratio, where the mobile station has to switch
from one base station to another. The model has been implemented by
using SLAM II and Fortran in the VAX/VMS environment.
The second application handles the transmission power control of
cellular radio networks. This application consists of the
simulation of a cellular network and a transmission power controlling
system. The simulation is highly simplified in comparison with real
mobile networks, and here it works mainly as a testing en- vironment
where different controlling systems can be implemented and compared.
The aim of the application is to control the transmission power in some
n connections systems. In that way, much lower transmission powers
can be used, causing significantly less interference to other
connections. Finally, one can get a considerably better quality of
connections in the whole cellular radio network (specially in
situations or areas which have earlier been problematic).
Collaboration with the Telecommunication Laboratory of the Technical
Research Centre of Finland.
The aim of the project is to adapt some basic deterministic and
stochastic methods of dynamic programming onto neural networks
(feedback-recurrent and feedforward). The actual goal is to get a
powerful tool for handling large scale optimization problems arising
from telecommunication planning. The main topics have been sequential
decision processes (like Markov decision problems and other approaches
which can be finally set as Markovian decision problems) with a special
emphasis on telecommunication network flow control and routing.
The theory of diffusion processes and brownian control are used to be
able to describe more accurately the stochastic nature of the arrival
and flow processes of the customers in small and medium sized
communication networks. The intention is to construct a soft computing
tool for planning.
Collaboration with the Telecommunication Laboratory of the Technical
Research Centre of Finland.
Voice identification and the telematic control of computers are of
great interest nowadays. In this pilot project, the possibility of
sending voice messages with computers and the remote control of a
computer by telephone are studied. The pilot application is to build
up a control program which controls alarm messages in an alarm
centre. The main objective of the program is to translate digital alarm
messages into voice messages and send them forward. On the other hand,
the program must be able to handle incoming telecommunication (voice or
digital) messages.
Collaboration with Hedegren security.
Mathematical hypermedia enables the creation of a mathematical virtual
reality on a computer where mathematics can be studied with the aid of
hypertext, graphics, animation, digital videos, etc. The aim of the
project is to prepare hypermedia teaching applications in
mathematics. They are to support the traditional lecture teaching
and partly replace it at universities. These kinds of applications would
afford the students an opportunity to revise their knowledge in
mathematics and to study mathematics on their own. A future intention
is to add intelligence and adaptiveness to bring individual tutoring
and individual feedback.
Collaboration with Tampere University of Technology, Helsinki University of
Technology and Lappeenranta University of Technology.
The aim of the project is to produce a MS-Windows-based hypermedia
application for quality management. The system contains two modules.
The first module contains the basic targets: standards and criteria of
quality in the production. There is also information about different
quality management techniques and different quality tools to help in
reaching the settled targets. The second module contains the emblems
of the company involved: financing of quality, feedback from
customers, delivery reliability, etc. These emblems come from the
process and show the quality of the production.
With this system, the company can improve the quality of its
production and products by changing the attitudes and improving the
knowledge of employees. The employees can be better educated to realize
how important their individual work is and to see the influences of
their work on the production.
Collaboration with Rautaruukki steel works in Raahe.
Program development has a major role in many of the projects. In this
respect, many projects have shared interests and needs. This has led to
the development of program libraries common to the whole laboratory.
Locally developed software packages are called FEMPAK and NSOLIB.
FEMPAK (RAINO A.E. MäKINEN and JARI TOIVANEN)
FEMPAK 1.0 is a portable software package for the numerical solution
of two- dimensional partial differential equations. It is based on the
finite element method. The package consists of a set of low-level
subroutines for the assembly and solution of the discrete finite
element equations, a set of example driver programs and software to
visualize the results. The graphical devices supported are X-windows
and PostScript.
NSOLIB (MARKO M. MÄKELÄ)
NSOLIB is a Fortran subroutine library for nonsmooth and nonconvex
optimization problems with single or multiple objective functions.
The methods are able to handle either simple bounds for variables,
linear, nonlinear or nonsmooth con- straints, or all of them at the
same time. NSOLIB subroutines are implementations of the proximal
bundle method. They have been tested with various standard test
examples and in several research projects of the laboratory in
different computing environments (microcomputers, workstations,
mainframe and supercomputers). The need of NSOLIB was noticed since the
commercial subroutine packages do not contain efficient codes for
nonsmooth optimization.
Commercial subroutine libraries mainly used are NAG and DISSPLA.
Computer Facilities
The Laboratory of Scientific Computing and the Department of
Mathematics had 30 HP-9000 series workstations and X-terminals and 35
486-level PCs for researchers and students in 1994. In addition, the
facilities at the Finnish National Supercomputer Centre (Cray XMP
EA/432, Convex C3840 and IBM SP2) and the Computer Centre of the
University of Jyvaskyla (VAX 4000-300 and SUN 4/670 computers connected
to terminals and PCs with a local area network) are available.
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