On Fixed Point (Trial) Methods for Free Boundary Problems

In this note we consider the trial methods for solving steady state free boundary problems. For two test examples (electrochemical machining and continuous casting) we discuss the convergence of a fixed point method. Moreover, using the techniques of shape optimization we introduce a modification of the method, which gives us superlinear convergence rate. This is also confirmed numerically.

Heat transfer in conducting and radiating bodies

Abstract We introduce briefly some nonlocal models for heat transfer in conducting and radiating media. The goal is to give an idea of the general mathematical structure and related existence results for such models.

Stefan-Boltzmann Radiation on Non-convex Surfaces

We consider the stationary heat equation for a non-convex body with Stefan–Boltzmann radiation condition on the surface. The main virtue of the resulting problem is non-locality of the boundary condition. Moreover, the problem is non-linear and in the general case also non-coercive and non-monotone. We show that the boundary value problem has a maximum principle. Hence, we can prove the existence of a weak solution assuming the existence of upper and lower solutions. In the two dimensional case or when a part of the radiation can escape the system we obtain coercivity and stronger existence result. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.

Simplification of Models

In practical applications the “complete” model, i.e., a model that contains all features that the experts in the application domain consider important, is often quite complicated and difficult to analyse mathematically. A straightforward numerical realization is often costly and may give very little qualitative understanding of the situation. It is therefore important to study if the model can be systematically simplified in order to enhance a qualitative analysis/understanding.

Shape Optimization in Contact Problems. 1. Design of an Elastic Body. 2. Design of an Elastic Perfectly Plastic Body

The optimal shape design of a two dimensional body on a rigid foundation is analyzed. The problem is how to find the boundary part of the body where the unilateral boundary conditions are assumed in such a way that a certain energy integral (total potential energy, for example) will be minimized. It is assumed that the material of the body is elastic. Some remarks will be given concerning the design of an elastic perfectly plastic body. Numerical examples will be given.

Rinnakkaislaskenta osittaisdifferentiaaliyhtälöiden numeerisessa ratkaisemisessa

Free surfaces: shape sensitivity analysis and numerical methods

Laskennallinen tiede - tieteen kolmas menetelmä : tilannekatsaus 2011

Professori Osmo Pekonen (1960-2022)

Single-trial-based Temporal Principal Component Analysis on Extracting Event-related Potentials of Interest for an Individual Subject

Abstract Temporal principal component analysis (t-PCA) has been widely used to extract event-related potentials (ERPs) at the group level of multiple subjects’ ERP data. The key assumption of group t-PCA analysis is that desired ERPs of all subjects share the same waveforms (i.e., temporal components), whereas waveforms of different subjects’ ERPs can be variant in phases, peak latencies and so on, to some extent. Additionally, several PCA-extracted components coming from the same ERP dataset failed to be statistically analysed simultaneously because their polarities and amplitudes were indeterminate. To fill these gaps, a novel technique was proposed and employed to extract desired ERP fro…

Hybrid Threats against Industry 4.0 : Adversarial Training of Resilience

Industry 4.0 and Smart Manufacturing are associated with the Cyber-Physical-Social Systems populated and controlled by the Collective Intelligence (human and artificial). They are an important component of Critical Infrastructure and they are essential for the functioning of a society and economy. Hybrid Threats nowadays target critical infrastructure and particularly vulnerabilities associated with both human and artificial intelligence. This article summarizes some latest studies of WARN: “Academic Response to Hybrid Threats” (the Erasmus+ project), which aim for the resilience (regarding hybrid threats) of various Industry 4.0 architectures and, especially, of the human and artificial de…

Yliopistojen tutkintokoulutuksen ja tutkimuksen rahoitus ja tulokset vuosina 2000-2004

A DNA‐Encoded FRET Biosensor for Visualizing the Tension across Paxillin in Living Cells upon Shear Stress

Paxillin is a potential participant in the direct intracellular force transmission which is considered as the foundation of cells sensing and responding to extracellular environment. However, the detection of tension across paxillin has not been achieved due to lacking microsized tools. Herein, a paxillin tension sensor (PaxTs) based on Fluorescence Resonance Energy Transfer (FRET) technique was constructed. PaxTs can be expressed and assembled to FA sites spontaneously to visualize the tension across paxillin with FRET efficiency of ~62.4% in living cells. The tension across paxillin was found to decrease upon shear stress, in which the membrane fluidity and contractility of actin acted as…

Gradient with respect to nodes for non-isoparametric finite elements

We consider the problem of controlling the solution of a finite element model using the nodal co-ordinates as control variables. The main emphasis is on the study of the applicability of the domain deformation method for different element types. The results are applied to a simple problem of finite element grid optimization.

A Boundary Control Approach to an Optimal Shape Design Problem

Abstract We consider the problem of controlling the coincidence set in connection with an obstacle problem. We shall transform the obtained optimal shape design problem into a boundary control problem with Dirichlet boundary conditions.

Time-dependent simulation of Czochralski silicon crystal growth

We have developed a detailed mathematical model and numerical simulation tools based on the streamline upwind/Petrov-Galerkin (SUPG) finite element formulation for the Czochralski silicon crystal growth. In this paper we consider the mathematical modeling and numerical simulation of the time-dependent melt flow and temperature field in a rotationally symmetric crystal growth environment. Heat inside the Czochralski furnace is transferred by conduction, convection and radiation, Radiating surfaces are assumed to be opaque, diffuse and gray. Hence the radiative heat exchange can be modeled with a non-local boundary condition on the radiating part of the surface. The position of the crystal-me…

Single-trial-based temporal principal component analysis on extracting event-related potentials of interest for an individual subject.

Background: Temporal principal component analysis (tPCA) has been widely used to extract event-related potentials (ERPs) at group level of multiple subjects ERP data and it assumes that the underlying factor loading is fixed across participants. However, such assumption may fail to work if latency and phase for one ERP vary considerably across participants. Furthermore, effect of number of trials on tPCA decomposition has not been systematically examined as well, especially for within-subject PCA. New method: We reanalyzed a real ERP data of an emotional experiment using tPCA to extract N2 and P2 from single-trial EEG of an individual. We also explored influence of the number of trials (con…

Thermal deformations of inhomogeneous elastic plates

We consider thermal deformations of transversally inhomogenous elastic plates. Thin plate equations are derived as limits of full three-dimensional models both in the linear was well as in the non-linear case with appropriate convergence proofs. In the non-linear case also the corresponding von Karman equations are formulated. Its is obtained that the inhomogeneity leads to the loss of some symmetry properties at the von Karman equations

Trial Methods for Nonlinear Bernoulli Problem

In this article we consider a free boundary problem which is related to formation of waves on a fluid surface (for example the ship waves). We study the possibility to construct ‘trial’ methods where one solves a sequence of standard flow problems formulated in different geometries that converge to the final free boundary. Furthermore, we use the shape optimization techniques to analyse the convergence of the fixed point iteration near a fixed point. For stream function case we conclude that the fast convergence can be obtained by using non-standard boundary conditions and we present numerical results to confirm the analysis.

Optimal shape design and unilateral boundary value problems: Part II

In the first part we give a general existence theorem and a regularization method for an optimal control problem where the control is a domain in R″ and where the system is governed by a state relation which includes differential equations as well as inequalities. In the second part applications for optimal shape design problems governed by the Dirichlet-Signorini boundary value problem are presented. Several numerical examples are included.

Integro-differential equation modelling heat transfer in conducting, radiating and semitransparent materials

In this work we analyse a model for radiative heat transfer in materials that are conductive, grey and semitransparent. Such materials are for example glass, silicon, water and several gases. The most important feature of the model is the non-local interaction due to exchange of radiation. This, together with non-linearity arising from the well-known Stefan-Boltzmann law, makes the resulting heat equation non-monotone. By analysing the terms related to heat radiation we prove that the operator defining the problem is pseudomonotone. Hence, we can prove the existence of weak solution in the cases where coercivity can be obtained. In the general case, we prove the solvability of the system us…

A nonlocal problem arising from heat radiation on non-convex surfaces

We consider both stationary and time-dependent heat equations for a non-convex body or a collection of disjoint conducting bodies with Stefan-Boltzmann radiation conditions on the surface. The main novelty of the resulting problem is the non-locality of the boundary condition due to self-illuminating radiation on the surface. Moreover, the problem is nonlinear and in the general case also non-coercive. We show that the non-local boundary value problem admits a maximum principle. Hence, we can prove the existence of a weak solution assuming the existence of upper and lower solutions. This result is then applied to prove existence under some hypotheses that guarantee the existence of sub- and…

FINITE ELEMENT APPROXIMATION OF NONLOCAL HEAT RADIATION PROBLEMS

This paper focuses on finite element error analysis for problems involving both conductive and radiative heat transfers. The radiative heat exchange is modeled with a nonlinear and nonlocal term that also makes the problem non-monotone. The continuous problem has a maximum principle which suggests the use of inverse monotone discretizations. We also estimate the error due to the approximation of the boundary by showing continuous dependence on the geometric data for the continuous problem. The final result of this paper is a rigorous justification and error analysis for methods that use the so-called view factors for numerical modeling of the heat radiation.

Yliopistojen tutkintokoulutuksen ja tutkimuksen rahoitus ja tulokset vuosina 2000-2004 ja 2005-2009

Shape optimization in contact problems based on penalization of the state inequality

The paper deals with the approximation of optimal shape of elastic bodies, uni­laterally supported by a rigid, frictionless foundation. Original state inequality, describing the behaviour of such a body is replaced by a family of penalized state problems. The relation between optimal shapes for the original state inequality and those for penalized state equations is established. peerReviewed