A new multidimensional integral relationship between heat flux and temperature for direct internal assessment of heat flux

This paper derives a new integral relationship between heat flux and temperature in a transient, two-dimensional heat conducting half space. A unified mathematical treatment is proposed that is extendable to higher-dimensional and finite-region geometries. The analytic expression provides the local heat flux perpendicular to the front surface solely based on an embedded line of temperature sensors parallel to the surface. The relationship does not require apriori knowledge of the surface boundary condition. A new sensor strategy is analytically conceived based on the integral relationship for estimating the local, in-depth heat flux without surface instrumentation. It should further be clarified that the integral relationship requires only knowledge of the local, in-depth temperature and heating/cooling rate (time rate of change of temperature). The resulting formulation is mildly ill-posed and either requires digital filtering of the temperature signal to remove high frequency components of noise or the development of direct heating/cooling rate sensors. This paper (a) develops the new mathematical relationship; (b) demonstrates that the proposed relationship reduces to well-known (i) one-dimensional results under the appropriate assumptions; and, (ii) two-dimensional surface results; and, (c) provides a simple numerical example validating the concept.     

A new multidimensional integral relationship between heat flux and temperature for direct internal assessment of heat flux

This paper derives a new integral relationship between heat flux and temperature in a transient, two-dimensional heat conducting half space. A unified mathematical treatment is proposed that is extendable to higher-dimensional and finite-region geometries. The analytic expression provides the local heat flux perpendicular to the front surface solely based on an embedded line of temperature sensors parallel to the surface. The relationship does not require apriori knowledge of the surface boundary condition. A new sensor strategy is analytically conceived based on the integral relationship for estimating the local, in-depth heat flux without surface instrumentation. It should further be clarified that the integral relationship requires only knowledge of the local, in-depth temperature and heating/cooling rate (time rate of change of temperature). The resulting formulation is mildly ill-posed and either requires digital filtering of the temperature signal to remove high frequency components of noise or the development of direct heating/cooling rate sensors. This paper (a) develops the new mathematical relationship; (b) demonstrates that the proposed relationship reduces to well-known (i) one-dimensional results under the appropriate assumptions; and, (ii) two-dimensional surface results; and, (c) provides a simple numerical example validating the concept.     

Boundary integral operators for the heat equation

We study the integral operators on the lateral boundary of a space-time cylinder that are given by the boundary values and the normal derivatives of the single and double layer potentials defined with the fundamental solution of the heat equation. For Lipschitz cylinders we show that the 2×2 matrix of these operators defines a bounded and positive definite bilinear form on certain anisotropic Sobolev spaces. By restriction, this implies the positivity of the single layer heat potential and of the normal derivative of the double layer heat potential. Continuity and bijectivity of these operators in a certain range of Sobolev spaces are also shown. As an application, we derive error estimates for various Galerkin methods. An example is the numerical approximation of an eddy current problem which is an interface problem with the heat equation in one domain and the Laplace equation in a second domain. Results of numerical computations for this problem are presented.Parts of this work were done while the author had visiting positions at the Carnegie Mellon University, Pittsburgh, USA, and at the Université de Nantes, France, or was supported by the DFG-Forschergruppe KO 634/32-1.     

Coupling of numerical Green''s matrix and boundary integral equations for the elastodynamic analysis of inhomogeneous bodies on an elastic half-space

A coupled system of integral equations (of the domain and boundary types) is formulated for the elastodynamic response analysis of a locally inhomogeneous body on a homogeneous elastic half-space. The method uses the fundamental solution for homogeneous elastostatics in the inhomogeneous domain owing to the lack of a fundamental solution in inhomogeneous elastodynamics.

The integral representation of displacements in the inhomogeneous domain is formulated with the help of this elastostatic fundamental solution by considering the term induced by the inhomogeneity of materials and the acceleration term as the body force term. Then the Green's matrix is obtained numerically from this integral representation and combined with the ordinary boundary integral equations, which are valid in the exterior homogeneous half-space.

Some numerical examples show the efficiency and the versatility of this coupled method.     

A sequential algorithm and error sensitivity analysis for the inverse heat conduction problems with multiple heat sources

This paper proposes a sequential approach to determine the unknown parameters for inverse heat conduction problems which have multiple time-dependent heat sources. There are two main aims in this study, one is to derive an inverse algorithm that can estimate the unknown conditions effectively, and the other is to bring up a theoretical sensitivity analysis to discuss what causes the growth of errors. This paper has three major achievements with regard to the literature on IHCPs, as follows: (1) proposing an efficient sequential inverse algorithm that can simultaneously determine several unknown time-dependent parameters; (2) exploring why the sequential function specification method can provide a stable but inaccurate estimation when tackling problems with larger measurement errors; and (3) discussing the sensitivity problem and analyzing what factors cause the growth in error sensitivity. Three examples are applied to demonstrate the performance of the proposed method, and the numerical results show that the accurate estimations can be obtained by alleviating the error sensitivity when the measurement error is considered.     

Local weak solution of the isentropic compressible Navier-Stokes equations in a half-space

In this paper, we establish the local existence of weak solutions with higher regularity of the three-dimensional half-space compressible isentropic Navier-Stokes equations with the adiabatic exponent γ > 1 in the presence of vacuum. Here we do not need any smallness of the initial data.

     

Local integral manifolds for a nonautonomous parabolic equation

The paper deals with a quasilinear nonautonomous parabolic equation. A finite-dimensional Lipschitz integral manifold is constructed in a neighborhood of an equilibrium point of the limiting autonomous equation. It is proved that the integral manifold attracts exponentially all the solutions of the initial nonautonomous equation, passing through this neighborhood. Bibliography: 13 titles. Dedicated to Olga A. Oleinik This work was supported in part by the Russian Foundation for Basic Research and by the International Science Foundation. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 19, pp. 000-000, 0000.     

Local asymptotic attractivity for nonlinear quadratic functional integral equations

In this paper, an existence result for local asymptotic attractivity of the solutions is proved for a nonlinear quadratic functional integral equation under certain growth conditions which in turn gives the existence as well as asymptotic stability of solutions. A couple of examples are provided for indicating the natural realizations of abstract theory presented in the paper.     

Thermo-elastic stresses in a half-space having an isolated heat source on the boundary

Zusammenfassung Die Arbeit behandelt das stationäre thermoelastische Spannungsfeld, das in einem Halbraum durch eine Wärmequelle an der Grenzebene erzeugt wird, während für den Rest der Grenzebene eine Strahlungsbedingung gilt. Die Lösung wird mit Hilfe von Hankel-Transformationen erhalten, und es stellt sich heraus, dass der Spannungszustand eben sowie parallel zur Oberfläche ist.     

Path integral formulae for heat kernels and their derivatives

Summary The heat kernel and its derivatives of a vector Laplacian on the sections of a bundle over a compact Riemannian manifold are expressed as products of the scalar heat kernel of the manifold and path integrals over the Brownian bridge. The small-time asymptotics of these integrals are computed.     

Boundary integral solution of the two-dimensional heat equation

Here we consider initial boundary value problems for the heat equation by using the heat potential representation for the solution. Depending on the choice of the representation we are led to a solution of the various boundary integral equations. We discuss the solvability of these equations in anisotropic Sobolev spaces. It turns out that the double-layer heat potential D and its spatial adjoint D′ have smoothing properties similar to the single-layer heat operator. This yields compactness of the operators D and D′. In addition, for any constant c ≠ 0, cI + D′ and cI + D′ are isomorphisms. Based on the coercivity of the single-layer heat operator and the above compactness we establish the coerciveness of the hypersingular heat operator. Moreover, we show an equivalence between the weak solution and the various boundary integral solutions. As a further application we describe a coupling procedure for an exterior initial boundary value problem for the non-homogeneous heat equation.     

Local a-posteriori error indicators for the Galerkin discretization of boundary integral equations

In this paper we present local a-posteriori error indicators for the Galerkin discretization of boundary integral equations. These error indicators are introduced and investigated by Babuška-Rheinboldt [3] for finite element methods. We transfer them from finite element methods onto boundary element methods and show that they are reliable and efficient for a wide class of integral operators under relatively weak assumptions. These local error indicators are based on the computable residual and can be used for controlling the adaptive mesh refinement. Received March 4, 1996 / Revised version received September 25, 1996     

On the convergence of the heat balance integral method

Convergence properties are established for the piecewise linear heat balance integral solution of a benchmark moving boundary problem, thus generalising earlier results [Numer. Heat Transfer 8 (1985) 373]. A convergence rate of O(n⁻¹) is identified with minor effects at large values of the Stefan number β (slow interface movement). The correct O(n^−1/2) behaviour for incident heat flux is recovered for β → 0 (pure heat conduction) as previously found [Numer. Heat Transfer 8 (1985) 373–382]. Numerical illustrations support the theoretical findings.     

Determination of the characteristics of concentrated heat sources in a half-space under prescribed conditions for the heat exchange and temperature distribution on its surface

A method is given for solving the inverse problem for determining the intensity and location of concentrated heat sources in a half-space under prescribed conditions on the heat exchange with the external medium and a distribution of surface temperature with error. Using methods of the variational calculus, we obtain a system of nonlinear equations for the characteristics of subsurface heat sources, leading to a temperature function on the surface which is closest to the given distribution. Several individual examples are solved.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 46–51, 1989.     

A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source

A stochastic half-space problem, driven by an additive Gaussian white noise, is considered within the context of the theory of generalized thermoelastic diffusion with one relaxation time. The bounding surface is traction free and subjected to a time dependent thermal shock. A permeating substance is considered in contact with the bounding surface. Laplace transform technique is used to obtain the solution in the transformed domain by using a direct approach. The mean and variance are derived and analyzed for temperature, displacement, stress, strain, concentration and chemical potential. The asymptotic behavior for the solution is discussed. Numerical results are carried out and represented graphically. The second sound effect is observed in the simulation.     

An integral representation of elasticity and sensitivity for stochastic volatility models

This paper presents a generic probabilistic approach to study elasticities and sensitivities of financial quantities under stochastic volatility models. We describe the shock elasticity, the quantile sensitivity and the vega value of cash flows with respect to perturbation of the volatility function of the model. The main contribution is to establish explicit formulae for these elasticities and sensitivities based on a novel application of the exponential measure change technique in Palmowski and Rolski (Bernoulli 8(6):767–785 2002). We carry out explicit calculations for the Heston model and the 3/2 stochastic volatility model, and derive explicit expressions in terms of model parameters.     

A new look at the heat balance integral method

The heat balance integral method is a familiar technique for treating transport problems, particularly phase-change scenarios. Here a number of differences arising in the method's implementation are investigated that result in quantitatively distinct solutions. As a consequence some guidance is provided for selecting the appropriate implementation of the method.     

Asymptotics for a nonlinear integral equation with a generalized heat kernel

This paper is concerned with a nonlinear integral equation $$(P)\qquad u(x, t)=\int_{{\bf R}^N}G(x-y, t)\varphi(y){d}y+\int_0^t \int_{{\bf R}^N}G(x-y, t-s)f(y, s:u){d}y{d}s, \quad $$ where N ≥  1, \({\varphi \in L^\infty({\bf R}^N) \cap L^1({\bf R}^N,(1+|x|^K){d}x)}\) for some K ≥  0. Here, G = G(x,t) is a generalization of the heat kernel. We are interested in the asymptotic expansions of the solution of (P) behaving like a multiple of the integral kernel G as \({t \to \infty}\) .