Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 9 (204 ) 63 68 XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) (TFOC204) Solving an Unsteady-State and Non-Uniform Heat Conduction Transfer Problem Using Discrete-Analytical Method Vladimir N. Sidorov a, Sergey M. Matskevich a * a Department of Computer Science and Applied Mathematics, Moscow State University of Civil Engineering (26, Yaroslavskoye Shosse, Moscow 29337, Russia) Abstract The paper presents one of the methods to determine heat spread patterns in objects. Mathematical model of the process is a differential equation of the second order with initial and boundary conditions, which can be solved by only one function U(x, y, z, t). In this paper the problem of unsteady-state and non-uniform heat conduction transfer for 2 dimensions, with imposed initial and boundary conditions of the first, second and third kind, is solving using discrete-analytical method. The main idea of this method is to combine discrete and analytical method. In this case, initial problem is divided to 2 stages: in the first stage a discrete technique along ones directions will be applied; in the second stage an analytical method along other directions will be applied. The result will be a discrete set of analytical functions. For discrete stage is used a well-known method of finite differences, and for analytical stage is applied the virtue of the matrix exponent. In the general case, the problem can be submitted in operator form with non-orthogonal quadrangular mesh which is topologically equivalent to square mesh. 204 The Authors. Published by Elsevier Ltd. 204 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil (http://creativecommons.org/licenses/by-nc-nd/3.0/). Engineering (23RSP). Peer-review under responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) Keywords: heat conduction transfer; discrete-analytic method; differential equation; unsteady-state; non-uniform; finite differences; matrix exponent; operator form; quadrangular mesh; Nomenclature c w F specific heat capacity of environment density thermal conductivity coefficient coefficient of thermal diffusivity coefficient of heat transfer ambient temperature power of any possible heat sources * Corresponding author. Tel.: +7-926-888-743. E-mail address: zext@mail.ru 877-7058 204 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) doi:0.06/j.proeng.204.2.03
64 Vladimir N. Sidorov and Sergey M. Matskevich / Procedia Engineering 9 ( 204 ) 63 68. Introduction The paper presents unsteady and non-uniform heat conduction transfer problem statement and algorithm of its solving by discrete-analytical method. Mathematical definition of the problem is a differential equation, based on term of heat flow. Heat transfer equation postulates the fact of thermal balance of the internal energy of the object of Fourier: de Q Q or in tp 2 2 2 U U U U c F 2 2 2 t x y z () Let us consider the discrete-analytical solution of this equation, which combines numerical solutions along directions x, y, z and analytical solution along time direction t. Characters c,, and F in a general case are functions of arguments x, y, z, t and also depend on the current temperature value U. 2. Problem statement In this paper is considered an algorithm for solving unsteady, non-uniform, two-dimensional heat conduction transfer problem: U, U x y x, y U F( x, y, t) t x x y y (2) where x, y - coefficients of thermal diffusivity (=c), corresponding to varying properties of object along x, y, respectively; For the unsteady heat conduction equation must be set the initial condition and boundary conditions of, 2 or 3 kind. The first kind means that the distribution of the temperature is set on boundaries of the area as values of function g. The second kind determines the heat flux density at the boundaries of the area. The third kind corresponds to the case when the ambient temperature w is set and a law of heat transfer between the surface of the body and the environment is set as the Newton-Richman s law. 4. Discrete-analytical method 3.. Approximation First of all, we should approximate the area with finite difference method along directions x, y, but for variable of time - t we will leave analytic relation. We define the space-time area in the form of the 3-coordinate planes, where along the X and Y axes is a space (x, y), and along the T-axis is time (t). First of all, let us consider approximation of the rectangular area. We divide the area with rectangular mesh using increments on the X and Y axes (see Figure ). Mesh nodes: j=0,,,n, N+ coordinates of mesh nodes along, number of nodes is N+2, at that x 0=0 and x N+=l boundary nodes (where we set boundary conditions); i=0,,,n2, N2+ coordinates of mesh nodes along Y, number of nodes is N2+2, at that y 0=0 and y N2+=m boundary nodes (where we set boundary conditions); Along the T axis we leave analytical dependence of temperature on time.
Vladimir N. Sidorov and Sergey M. Matskevich / Procedia Engineering 9 ( 204 ) 63 68 65 Fig.. Finite-difference mesh of the source area. We do finite-difference approximation of the Laplace operator in the spatial coordinates x, y, considering the non-uniformity of the material. In this case, the coefficients of thermal diffusivity will be taken as their half-sums for the adjacent cells in the respective directions: U t U t U t U t i, ji, j i, j i, j i, j i, j i, j i, j 2 x x t x x U i i i i i, j U t U t U t U t i, j i, j i, j i, j i, j i, j i, j 2 y y i i i y y i F t i, j (3) 3.2. Basic variations method To form the matrix of the coefficients of the variables we use the method of basic variations [2]. We write the Laplace operator in the form of a linear operator on the function U: U U LU xy, xy, x x y y (4) Discrete form of this operator can be written in terms of a finite-difference representation, written above. Basic variations method allows by acting discrete linear operator not on the unknown function U, but on the basis function (in this, discrete case - acting on the basis vectors), to write the formula for automatic finding of all the matrix coefficients depending on the indices. km k A a Le (5) m where e m (k) = (k,m), k,m=,2, ; (k,m) = (when m=k) or =0 (when mk) - Kronecker delta symbol.
66 Vladimir N. Sidorov and Sergey M. Matskevich / Procedia Engineering 9 ( 204 ) 63 68 3.3. Normal system of differential equations Let us return to the original differential equation (2), replacing its right-hand side with approximating functions, founded in the previous step. As a result it will be a normal system of differential equations. U AU F t (6) Known solution of such a system [3], according to the theory of ordinary differential equations is a vector: t At 0 At U e e F d (7) Matrix exponent e At, according to the theory of matrix functions for symmetric matrices can be determine in the form of the Jordan decomposition as: 2 n e At Te S T, e S diag{ e t, e t,..., e t } (8) where T - a matrix of eigenvectors, n eigen values 5. Operator formulation of the problem with the characteristic functions of area An important advantage of the operator statement it enables writing the common formulas, describing of both relations within the area and boundary conditions. That is the union of the differential equation and boundary conditions in one equation with the relevant weight characteristics. Let us write a normal form of the second boundary unsteady heat transfer problem: U U U xy, xy, F, xy,, t0 t x x y y lu q, x, y U x, y,0, t 0 (9) Now we write relevant characteristic functions of area. Those are the characteristic functions of area of coordinates and Heaviside function of time, xyt,,, xy,, t 0 xyt,, or xy xy, t t; xy ; t 0, xyt,, 0, xy, 0, t 0 (0) Then, problem can be described as a single operator expression: N N N N cu aij U aij U F t q t xyc () t x x y y i j i j where a ij - coefficients of thermal conductivity.
Vladimir N. Sidorov and Sergey M. Matskevich / Procedia Engineering 9 ( 204 ) 63 68 67 6. Non-orthogonal mesh of initial area Let us take the considered area and divide it into finite quadrangular elements (see Figure 2 - left). Then approximate it to the rectangular area with the unit cells (cells with unit sizes) by a non-degenerate deformation of its elements (see Figure 2 - left). This can be done by passing from global coordinates of nodes of the original mesh in the chosen coordinate system to local coordinates (see Figure 2 - right). Fig. 2. Approximation of the source area to topologically equivalent square mesh (left), transition from global coordinates to local (right). Coordinate transformation can be performed using the following algorithm []: Let (x,x 2) be the original coordinate system. We introduce a new coordinate system, acting locally within each element. And also let us write operators of simple difference:,,,,, D i j i j D i j i j (2) S S 2 S S Then we can write the coordinates of the middle node within the element, made up by a linear shape function on the angular nodes.,,,, xt EtD x i j Et2D2 x i j xt2 EtD x2 i j Et2D2 x2 i j (3) After the change of variables, difference operators will be carried out according to the formulas for differentiating a composite function. xt xt 2 x xt x xt2 x x 2 xt or xt xt 2 2 22 x2 xt x2 xt2 x2 x xt (4) where Jacobi matrix, which can be found as = - and final Jacobean of system will be J: xt ; x D x ; D D Et D ; J det (5) S P PS pq q P P P 3P 3P 22 2 2 xp xtq
68 Vladimir N. Sidorov and Sergey M. Matskevich / Procedia Engineering 9 ( 204 ) 63 68 Then, the total operator statement for non-orthogonal mesh will be: * * JcU J aij U J aij U F t q t xyc t x x y y (6) 7. Example In the shown example is given a numerically simulated picture of the heat distribution in a non-uniform twodimensional plate consisting of several materials (see Figure 3, left). On two edges temperature was maintained at 00 C on two other 50 C. At the initial time the temperature across the plate is zero. The result is shown in the following time points: 0, 00, 000, 00 000 sec. (see Figure 3 (right) Figure 4 (left)). Figure 4, right also shows the calculated steady-state problem for the same area. Fig. 3. Square mesh partition (left); Temperature at t=0 s, t=00 s (right). Fig. 4. Temperature at t=000 s, t=000 s (left), solving steady-state problem (right). References [] Akimov P.A., Correct Discrete-Continual Finite Element Method of Structural Analysis Based on Precise Analytical Solutions of Resulting Multipoint Boundary Problems for Systems of Ordinary Differential Equations. // Applied Mechanics and Materials Vols. 204-208 (202), pp. 4502-4505. [2] Akimov P.A., Sidorov V.N., Mozgaleva M.L., Construction of stiffness matrices of three-dimensional discrete-continual finite element with quadrangular cross-section by method of basic variation, Moscow, 2008. [3] Akimov P.A., Sidorov V.N., Correct Method of Analytical Solution of Multipoint Boundary Problems of Structural Analysis for Systems of Ordinary Differential Equations with Piecewise Constant Coefficients. // Advanced Materials Research Vols. 250-253, 20, pp. 3652-3655.