Optimal Shape Design of Inductor Coils for Surface Hardening

A shape optimization problem is considered related to the design of induction hardening facilities. The mathematical model consists of a vector potential formulation for Maxwell's equations coupled with the energy balance and an ODE to describe the solid-solid phase transition in steel during heating. Depending on the shape of the coil we control the volume fraction of the high temperature phase. The coil is modeled as a tube and is defined by a unit-speed curve. The shape optimiza- tion problem is formulated over the set of admissible curves. The existence of an optimal control is proved. To obtain the form of the shape gradient of the cost functional, the material derivative method is applied. Finally, the first order necessary optimality conditions are estabished for an optimal tube

Optimal Shape Design of Inductor Coils for Surface Hardening

A shape optimization problem is considered related to the design of induction hardening facilities. The mathematical model consists of a vector potential formulation for Maxwell's equations coupled with the energy balance and an ODE to describe the solid-solid phase transition in steel during heating. Depending on the shape of the coil we control the volume fraction of the high temperature phase. The coil is modeled as a tube and is defined by a unit-speed curve. The shape optimiza- tion problem is formulated over the set of admissible curves. The existence of an optimal control is proved. To obtain the form of the shape gradient of the cost functional, the material derivative method is applied. Finally, the first order necessary optimality conditions are estabished for an optimal tube

On the effective properties of suspensions

Einstein's formula for the viscosity of dilute suspensions describes how rigid particles immersed in a Stokes-fluid increase its macroscopic viscosity in terms of the particle volume density ∅. However, up to now, a rigorous justification has only been obtained for dissipation functionals of the flow feld. In this thesis, a cloud of N spherical rigid particles of radius R suspended in a fluid of viscosity μ is considered. It is rigorously shown that the homogenized fluid in the regime NR3 → 0 as N → ∞ has, in accordance with Einstein's formula, the viscosityμ' = μ ( 1 + 5/2∅) to first order in ∅. This is done by establishing L∞ and Lpp estimates for the difference of the solution to the microscopic problem and the solution to the homogenized equation. Regarding the distribution of the particles, it is assumed that the particles are contained in some bounded region and are well separated in the sense that the minimal distance is comparable to the average one. The main tools for the proof are a dipole approximation of the flow feld of the suspension together with the so-called method of reflections and a coarse graining of the volume density. By a very close mathematical analogy to electrostatics a similar result, regarding Maxwell's formula for the conductivity of suspensions, is proven, namely that the conductivity of the homogenized material isη' = η (1 + 3∅) to first order in ∅

Application of Helmholtz/Hodge Decomposition to Finite Element Methods for Two-Dimensional Maxwell\u27s Equations

In this work we apply the two-dimensional Helmholtz/Hodge decomposition to develop new finite element schemes for two-dimensional Maxwell\u27s equations. We begin with the introduction of Maxwell\u27s equations and a brief survey of finite element methods for Maxwell\u27s equations. Then we review the related fundamentals in Chapter 2. In Chapter 3, we discuss the related vector function spaces and the Helmholtz/Hodge decomposition which are used in Chapter 4 and 5. The new results in this dissertation are presented in Chapter 4 and Chapter 5. In Chapter 4, we propose a new numerical approach for two-dimensional Maxwell\u27s equations that is based on the Helmholtz/Hodge decomposition for divergence-free vector fields. In this approach an approximate solution for Maxwell\u27s equations can be obtained by solving standard second order scalar elliptic boundary value problems. This new approach is illustrated by a P1 finite element method. In Chapter 5, we further extend the new approach described in Chapter 4 to the interface problem for Maxwell\u27s equations. We use the extraction formulas and multigrid method to overcome the low regularity of the solution for the Maxwell interface problem. The theoretical results obtained in this dissertation are confirmed by numerical experiments

Mathematical source references

This list of references is intended to be a convenient reference source for those interested in the historical origin of common mathematical ideas, The topics mentioned are mostly those met in a degree course in mathematics. For each entry the list attempts to give an exact source reference with comments about priority. There are now available other historical reference sources for mathematics on the internet but with a different style of presentation.<br/

On the McKean-Vlasov dynamics with or without common noise

McKean-Vlasov stochastic differential equations may arise from a probabilistic interpretation of certain non-linear PDEs or as the limiting behaviour of mean field particle systems (those whose interactions are through the empirical measure) as the population size increases to infinity. Interest in this topic has grown enormously in recent times following the introduction of the related mean field games. These are models derived from the infinite population limit of games with finitely many players and mean field structure, i.e. the dynamics and rewards of one player depend on the other players through the empirical measure. Naturally, it is imperative that the dynamics of the models are well-posed. This question comprises the majority of this text in two stochastic contexts: with or without a common noise. In the more often studied case where the particles are driven by independent Brownian motions, results are provided that pertain to the weak-existence and pathwise continuous dependence on the initial condition. These results adapt a method of Gyöngy and Krylov for Itô's stochastic differential equations to the McKean-Vlasov setting. Should the coefficients and initial distribution satisfy a certain Lyapunov condition, well-posedness of the dynamics may be established along with the existence of an invariant measure for an associated semi-group. These conditions allow for potentially unbounded coefficients, with growth intrinsically linked to the Lyapunov condition. In the second context, particle systems driven by correlated noises are considered. In particular, the particles are each driven by two Brownian motions: one common to all particles and a private Brownian motion independent of all others. The connection between these particle systems and related McKean-Vlasov models through the conditional propagation of chaos is discussed. Existence and uniqueness of weak solutions to the corresponding McKean-Vlasov dynamics is proved in a particular framework that allows for a discontinuous drift coefficient at a price of non-degenerate noise

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal