Research interests (old and new)

Timo Tiihonen

Heat radiation

Existence theory

Physical models that involve heat radiation lead either to systems with transport equations or (in simpler situations) to integro-differential equations where the radiative heat exchance shows up as a non-local term. If the surfaces are opaque, diffuse and gray and the surrounding medium is transparent the radiative heat exchange takes place between the surfaces and is modeled with a non-linear integral term on the boundary. The resulting problem is non-monotone and non-coercive. However, it has a maximum principle which leads to existence proofs under quite general conditions. Different heat radiation models (surface, volumetric, (an)isotropic materials) have similar mathematical structure and the corresponding theory can be derived from common abstract framework.

Numerical approximation

From numerical point of view radiation problems combine the difficulties of PDEs (large, badly conditioned matrices) with those of the integral equations (full matrices, unbounded kernels). The problems are non-linear and non-monotone which means that continuous dependence on data is not straight forward but requires inverse monotonicity for example. The geometries are non-convex which makes the use of polyhedral approximate domains non-trivial, not to mention the problem of approximating the boundaries where non-local conditions apply. It turns out to be easier to prove the continuous dependence on the geometry for the original problem before going to discretization and its analysis.

Asymptotic analysis

The question of finding asymptotic models for optically thick materials is important in many practical applications as full volumetric models can be extremely costly to solve numerically. Using asymptotic analysis we have been able to derive rigorous error estimates for several Rosseland type diffusion approximations which allow computationally efficient treatment of nearly opaque materials.

Publications

• Timo Tiihonen: Stefan-Boltzmann radiation on non-convex surfaces, Math. Meth. Appl. Sci., vol 20, pp. 47-57 (1997).
  
• Timo Tiihonen: A nonlocal problem arising from heat radiation on non-convex surfaces , Euro J. Appl. Math., vol 8, pp. 403--416, 1997.
  
• Mika Laitinen, Timo Tiihonen: Heat transfer in conducting, radiating and semitransparent materials Math. Meth. Appl. Sci., Vol 21, pp. 375-392, 1998.
• Mika Laitinen, Timo Tiihonen: Heat transfer in conducting and radiating bodies. Appl. Math. Lett., vol. 10, N. 5, pp. 5--8, 1997.
  
• Tiihonen, T.: Stefan problem with non-local radiation condition, in Free Boundary Problems, Theory and Applications, Ed. by M. Niezgodka, P. Strzelski, Pitman, pp. 52-58, 1996.
  
• T. Tiihonen: Non-local problem arising from heat radiation on non-convex surfaces, in Integral Methods in Science and Engineering, Volume one: Analytic Methods, (Eds. C. Constanda, J. Saranen and S. Seikkala), Pitman, pp. 195--199, 1997.
  
• Timo Tiihonen: Finite element approximation of non-local heat radiation problems, MMMAS, Vol 8, pp. 1071-1089, 1998.
• M. Laitinen, T. Tiihonen: Conductive-Radiative Heat Transfer in Grey Materials, Quart. Appl. Math., vol 59, pp 737-768, 2001.

Free boundary problems

Trial methods for FBPs

We study the use of shape optimization techniques for deriving convergent fixed point schemes (trial methods) for stationary free boundary problems. The method is to formulate the FBP to shape optimization problem and then to analyse the Hessian of the cost functional. If good preconditioner for the Hessian can be found the preconditioned steepest descent method gives a convergent fixed point scheme for the problem.
The research aims at developing simple and reliable methods that can be implemented also with black box PDE solvers.

Publications

• T. Tiihonen, J. Järvinen: On the fixed point (trial) methods for free boundary problems, in "Free boundary problems in continuum mechanics", Ed. by S.N. Antontsev, K.H. Hoffmann and A.M. Khludnev, Free boundary problems in continuum mechanics, Birkhauser Verlag, Basel, 339--350, 1992.
  
• T. Tiihonen: Shape optimization and trial methods for free boundary problems, RAIRO M2AN, vol 31, pp. 805-825, 1997,
  
• K. Kärkkäinen, T. Tiihonen: Trial methods for a non-linear Bernoulli problem , 1996,
  
• K. Kärkkäinen, T. Tiihonen: Trial methods for a non-linear Bernoulli problem, in Progress in Industrial Mathematics (ECMI-96), (Eds. Brons, Bendsoe, Sorensen), Teubner, 1997, pp. 260--267
  
• T. Tiihonen: Fixed point methods for internal free boundary problem, Num. Funct. Anal. Optim., vol. 19, pp. 399-413, 1998.
• K. Kärkkäinen, T. Tiihonen: Free surfaces: shape sensitivity analysis and numerical methods, Intnl. J. Num. Meth. Eng., vol 44, pp. 1079-1098.

Mathematical modelling

Asymptotic analysis

In the past we have worked on some plate models decribing the moisture induced deformations of paper sheets. Now we are interested for example in the asymptotic limits of radiation transport in semi-transparent media.

Phase transition problems

We have been interested in heat and mass transfer in the paper during drying or copying. Some phenomenological models have been derived and compared with experiments. Deeper theoretical work is so far missing but hopefully not for ever.

Crystal growth

Here the focus has been in the numerical simulation of the complete industrial process of silicon crystal growth, including all the major mechanisms related to heat and mass transfer, like radiation, solidification free and forced convection. Using stabilized finite elements quite satisfactory results have been obtained in the cylindrically symmetric case.

Publications

• Tiihonen, T., Pietikäinen, R., Thermal Deformations of Inhomogeneous Elastic Plates, Mathematical Methods in the Applied Sciences 18, 423-436, 1995.
  
• Järvinen, J., Nieminen, R., Tiihonen, T., Mathematical Modelling and Numerical Simulation of Melt Flow in Czochralski Crystal Growth, in "Numerical Methods in Engineering '96", Ed. by Desideri J.-A., Le Tallec P., Onate E., Periaux J. Stein E., Second ECCOMAS Conference on Numerical Methods in Engineering, John Wiley & Sons, 719-724, 1996,
  
• Pietikäinen R., Tiihonen T., Modelling of coupled heat and mass transfer in copying paper, in "Proceedings of ICIAM-95, Issue 4: Applied Sciences, especially Mechanics (Minisymposia)", Ed. by E. Kreuzer, O. Mahrenholtz, Akademie Verlag, 74-76, 1996
  
• Järvinen, J., Nieminen, R., Tiihonen, T., Time-Dependent Simulation of Czochralski Silicon Crystal Growth, Journal of Crystal Growth, vol. 180, pp. 468--476, 1997