Showing papers by "Ewa Majchrzak published in 2009"

Numerical Modeling of Short-Pulse Laser Interactions with Multi-Layered Thin Metal Films

Abstract: Multi-layered thin metal film subjected to a short-pulse laser heating is considered. Mathematical description of the process discussed bases on the equation in which there appear the relaxation time and the thermalization time (dualphase-lag-model). In this study we develop a three level implicit finite difference scheme for numerical modelling of heat transfer in non-homogeneous metal film. At the interfaces an ideal contact between successive layers is assumed. At the stage of computations a solution of only one three-diagonal linear system corresponds to transition from time t to t +∆t. The mathematical model, numerical algorithm and examples of computations are presented in the paper.

Numerical simulation of thermal processes proceeding in a multi-layered film subjected to ultrafast laser heating

Abstract: In the paper, the mathematical model, numerical algorithm and examples of computationsconcerningthermalprocessesproceedinginamulti-layeredthin film subjected to an ultrafast laser pulse are discussed. Theequations describing a course of the analysedprocess correspondto the dual-phase-lag model

Numerical modelling of temperature field in the tissue with a tumor subjected to the action of two external electrodes

Abstract: The domain of tissue with a tumor subjected to the action of electrodes located on the skin surface is considered. External electric field causes the heat generation in the domain analyzed. The distribution of electric potential is described by the system of Laplace’s equations, while the temperature field is described by the system of Pennes’ equations. On the contact surface between healthy tissue and tumor region the ideal electric and ideal thermal contacts are assumed. To assure the optimum conditions of tumor destruction the magnetic nanoparticles are introduced to the tumor region. The aim of investigations is to determine the temperature field in the domain considered for different size and positions of external electrodes, in other words to choose such electrodes which assure the cancer destruction. To solve the coupled problem connected with the biological tissue heating the boundary element method is used. In the final part of the paper the examples of computations are shown. 1. Governing equations The potential ( ) φ , e x y inside the healthy tissue (e = 1) and tumor region (e = 2) (Fig. 1) is described by the system of Laplace’s equations ( ) ( ) 2 , : e φ , 0 e e e x y x y ∈Ω ∇ = (1) where e e [C 2 /(Nm 2 )] is the dielectric permittivity of sub-domain Ωe . At the interface Γc of the tumor and healthy tissue the ideal electric contact is assumed ( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 2 φ , φ , , : φ , φ , e e c x y x y x y x y x y n n  =  ∈Γ  ∂ ∂ − = −  ∂ ∂  (2) On the external surface of tissue being in a contact with the electrodes the following condition is given Please cite this article as: Ewa Majchrzak, Marek Paruch, Numerical modelling of temperature field in the tissue with a tumor subjected to the action of two external electrodes, Scientific Research of the Institute of Mathematics and Computer Science, 2009, Volume 8, Issue 1, pages 137-145. The website: http://www.amcm.pcz.pl/ E. Majchrzak, M. Paruch 138 ( ) ( ) ( ) ( ) 1 1 2 2 , : φ , , : φ , x y x y U x y x y U ∈Γ = ∈Γ = − (3) where U [V] is the electric potential of the electrode relative to the ground. On the remaining external boundary of tissue the ideal electric isolation is assumed: ( ) 1 1 e φ , / 0 x y n − ∂ ∂ = . The electric field inside the tissue is determined by equation ( ) ( ) , φ , e e x y x y = −∇ E (4) Fig. 1. Action of electric field on the tissue with a tumor hyperthermia system The temperature field in the healthy tissue and the tumor region with embedded magnetic nanoparticles is described by the system of Pennes’ equations [1, 2] ( ) ( ) ( ) 2 λ , , , 0 e e e B e met e e T x y k T T x y Q Q x y ∇ +  −  + + =   (5) where e = 1, e = 2 correspond to the healthy tissue and tumor region, respectively, Te denotes temperature, λe [W/(mK)] is the thermal conductivity, ke = GBecB (GBe [1/s] is the perfusion coefficient, cB [J/(m 3 K)] is the volumetric specific heat of blood), TB is the supplying arterial blood temperature, Qmet e [W/m 3 ] is the metabolic heat source, Qe(x, y) [W/m 3 ] is the heat source connected with the electromagnetic field action. It should be pointed out that the thermal conductivity λ2 of tumor region with nanoparticles can be calculated as follows: ( ) 2 2 3 1/ λ 1 / λ / λ ′ = − Θ + Θ , where 2 3 λ , λ ′ are the thermal conductivities of tumor and nanoparticles, respectively and 2 π n r Θ = is the concentration of particles (n is the number of particles, r is the radius of particle). Numerical modelling of temperature field in the tissue with a tumor subjected ... 139 Source function Q1 [W/m 3 ] connected with the electromagnetic dissipated power in healthy tissue depends on the conductivity σ1 [S/m] and the electric field E1 [1] ( ) ( ) 1 1 1 σ , , 2 x y Q x y = E (6) The tumor region with embedded magnetic particles is treated as a composite and due to the assumed homogeneity of Ω2 the mean value of electrical conductivity σ2 of this sub-domain can be approximated as: ( ) 2 2 3 1/ σ 1 / σ / σ ′ = − Θ + Θ , where 2 3 σ , σ ′ are the electrical conductivities of tumor and particles, respectively. Under the assumption that Pt is the tumor area, for ( ) 2 , x y ∈Ω one has ( ) ( ) ( ) 2 2 2 σ , , , 2 t SPM t t x y P Q x y P x y P P − Θ Θ = + E (7) where PSPM is the heat generation connected with the superparamagnetism (SPM) [1]. At the contact surface Γc between the tumor and healthy tissue the ideal contact is assumed ( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 2 , , , : , , λ λ c T x y T x y x y T x y T x y n n  =  ∈Γ  ∂ ∂ − = −  ∂ ∂  (8) On the upper and lower surfaces of healthy tissue domain (skin surface) the Robin condition (convection) is assumed ( ) ( ) 1 1 1 , λ α , w w T x y T x y T n ∂ − =  −    ∂ (9) where αw [W/(m 2 K)] is the heat transfer coefficient between the skin surface and the cooling water, Tw is the cooling water temperature. On the remaining boundaries the adiabatic condition 1 1 λ / 0 T n − ∂ ∂ = can be taken into account. This condition results from the consideration that at the positions far from the center of the domain the temperature field is almost not affected by the external heating [1]. 2. Boundary element method To solve the equations describing the potential of electric field and the temperature field in the domain considered the boundary element method has been applied [3, 4]. E. Majchrzak, M. Paruch 140 The boundary integral equations corresponding to the equations (1) can be expressed as ( ) ( ) ( ) ( ) ( ) ( ) * * ξ, η φ ξ, η ψ , φ ξ, η, , d φ , ψ ξ, η, , d , 1,2 e e e e

Application of the boundary element method coupled with the artificial heat source procedure for numerical modelling of freezing process

Abstract: The freezing process of biological tissue subjected to the action of external cylindrical cryoprobe is analyzed. From the mathematical point of view the problem belongs to the group of moving boundary ones because the shape and dimensions of frozen region are time-dependent. In the paper the mathematical model of the process considered and the method of solution are presented. Next, the example of computations is shown. 1. Mathematical model The freezing process proceeding in the domain of biological tissue subjected to the action of external cylindrical cryoprobe (Fig. 1) can be described by following equation [1-6] ( , , ) ( , ) : ( )

'One-shot' identification of laser intensity in a process of thin metal film heating

Abstract: Thermal interactions between thin metal film and external laser pulse are considered. In particular on a basis of the knowledge of surface temperature distribution the laser intensity is estimated. The problem is described by dual-phase-lag model in which two relaxation parameters (relaxation time and thermalization time) are introduced in order to take into account the microscopic thermal interactions. The laser action is taken into account by additional term (source function) supplemented the basic energy equation. In the paper the mathematical model of the thin film heating process is discussed, the inverse problem is formulated and also the example of computations is presented. 1. Formulation of the problem We will consider a thin film of thickness L and an initial temperature distribution T (x, 0) = T 0 . The constant thermal properties, this means the thermal conductivity λ [W/(mK)] and the volumetric specific heat c [J/(m 3 K)] are assumed. A front surface x = 0 is irradiated by high-intensity and ultra-short duration laser beam as shown in Figure 1. A heat transfer in direction perpendicular to the layer is taken into account (this assumption is entirely acceptable) and then the temperature distribution is described by the following equation [1, 2]

Optimal location of sensors for estimation of cast iron latent heat

Abstract: The thermal processes proceeding in a system casting-mould are considered. The casting is made from cast iron and the austenite and eutectic latent heats should be identified. To estimate these parameters the knowledge of temperature history at the points selected from the domain considered is necessary. The location of sensors should assure the best conditions of identification process. In the paper the method of optimum location of sensors basing on the D-optimality criterion is presented. 1. Formulation of the problem A system casting-mould-environment is considered. Temperature field in casting domain is described by equation [1, 2] ( ) 2 ( , ) : ( ) λ , T x t x C T T x t t ∂ ∈Ω = ∇ ∂ (1) where C(T ) is the substitute thermal capacity of cast iron, λ is the thermal conductivity, T, x, t denote the temperature, geometrical co-ordinates and time. The following approximation of substitute thermal capacity is taken into account

Identification of internal hole parameters on the basis of boundary temperature

Abstract: The Laplace equation describing temperature field in 2D domain with an internal hole of circle shape supplemented by adequate boundary conditions is considered. On the basis of known temperature at the fragment of boundary the position of circle center or its radius is identified. To solve the inverse problem discussed the least square criterion is formulated, and next the gradient method coupled with the boundary element method is applied. To determine sensitivity coefficients the shape sensitivity analysis is used. In the final part of the paper the examples of computations are shown. 1. Direct problem The steady state temperature field T (x, y) in domain limited by boundary Γ = = Γ1 ∪ Γ2 ∪ Γ3 (Fig. 1) is described by the Laplace equation 2 ( , ) : ( , ) 0 x y T x y ∈Ω ∇ = (1) Fig. 1. Domain considered Please cite this article as: Ewa Majchrzak, Damian Tarasek, Identification of internal hole parameters on the basis of boundary temperature, Scientific Research of the Institute of Mathematics and Computer Science, 2009, Volume 8, Issue 1, pages 147-154. The website: http://www.amcm.pcz.pl/

Experiment design for parameters estimation of nonlinear Poisson equation - Part I

Abstract: In this part of the paper the nonlinear Poisson equation is considered, this means the conductivity is a function of the form D(x) = p1x1x2+p2, where p1, p2 are the parameters and x = {x1, x2}, −1≤x1, x2≤1 are the spatial co-ordinates. Sensitivity analysis with respect to parameters p1, p2 using the direct differentiation approach is discussed. The basic problem and additional ones are solved using the finite difference method. In the final part of the paper the results of computations are shown. 1. Formulation of the problem The two-dimensional elliptic equation is considered [ ] 0 ) ( ) ( ) ( : = + ∇ ∇ Ω ∈ x Q x U x D x (1) where D (x) is the coefficient of conductivity, Q (x) is the source function, x = = {x1, x2}, Ω = {x1, x2: −1≤x1≤1, −1≤x2≤1}. The function Q (x) is defined as follows [ ] 2 2 2 1 ) 75 . 0 ( ) 75 . 0 ( exp 10 ) ( − + − = x x x Q (2) while the conductivity coefficient is expressed as 1 2 1 2 ( ) D x = + p p x x (3) where p1, p2 are the parameters (p1 = 1.05, p2 = 4.09). The equation (1) is supplemented by Dirichlet boundary condition ( ) 0 x : U x = ∈Γ (4) It should be pointed out that the mathematical model presented above taken from [1] is connected with computer-assisted tomography. The aim of investigations is to solve the problem formulated and to determine the sensitivity functions ∂U/∂p1, ∂U/∂p2 using the direct differentiation method. Please cite this article as: Ewa Majchrzak, Katarzyna Freus, Sebastian Freus, Experiment design for parameters estimation of nonlinear Poisson equation Part I, Scientific Research of the Institute of Mathematics and Computer Science, 2009, Volume 8, Issue 1, pages 113-118. The website: http://www.amcm.pcz.pl/ E. Majchrzak, K. Freus, S. Freus 114 2. Sensitivity models The equation (1) in Cartesian co-ordinate takes the form 0 ) , ( ) , ( ) ( ) , ( ) ( 2 1 2 2 1 2 2 1 1 2 1 2 1 2 2 1 1 1 = +       ∂ ∂ + ∂ ∂ +       ∂ ∂ + ∂ ∂ x x Q x x x U p x x p x x x x U p x x p x (5)