Influence of the Parameters of Discretization on the Accuracy of Numerical Solution of the Three-Dimensional Problem of
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INFLUENCE OF THE PARAMETERS OF DISCRETIZATION ON THE ACCURACY OF NUMERICAL SOLUTION OF THE THREE-DIMENSIONAL PROBLEM OF HYDROGEN DIFFUSION O. V. Hembara,1 O. Ya. Chepil’,1,2 and N. T. Hembara3
UDC 620.197.5: 669.788
We propose an algorithm and programs for the solution of the three-dimensional problem of hydrogen diffusion. Some numerical solutions of this problem and the problem of heat conduction in the threedimensional statement are verified by analyzing test problems whose analytic solutions are known. The dependence of the relative error on the density of elements of the grid and time intervals is constructed by using the finite-element method. Keywords: heat conduction, hydrogen diffusion, finite-element method, three-dimensional problem, relative error.
In the process of operation of steel structures, hydrogen penetrating into the metal as a result of reactions with ambient or technological media facilitates the initiation and propagation of microcracks in structural steels due to the physicochemical processes and the electrochemical actions. It is practically impossible to experimentally estimate the degree of hydrogenation of a working structural element over the thickness of the metal [1]. Therefore, it is reasonable to apply the numerical methods. In what follows, we develop an algorithm and the corresponding software for the determination of the temperature field and hydrogen concentration in the entire volume of a structural element with regard for the influence of force and thermal factors. Statement of the Problem In the general case of an inhomogeneous material and the presence of inhomogeneous fields, the concentration gradients, mechanical stresses, and temperatures can play the role of driving forces of the diffusion processes. According to the literature data [1], the distribution of the mean value of hydrogen concentration C in a macroisotropic material is described by the Fick equation:
! C 1 Q ⎞ ⎛ + grad(VHσ h ) + gradT ⎟ , J H = − DC ⎜ grad ln ⎠ ⎝ kS RT RT
! ∂C = − div J H . ∂t
(1)
! Here, J H is the density of the diffusion flow of hydrogen, D is the coefficient of hydrogen diffusion in the metal, k s is the coefficient of solubility of hydrogen in the metal, R is the universal gas constant, T is abso1 2 3
Karpenko Physicomechanical Institute, Ukrainian National Academy of Sciences, Lviv, Ukraine. Corresponding author; e-mail: [email protected]. I. Franko Lviv National University, Lviv, Ukraine.
Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 52, No. 2, pp. 119–123, March–April, 2016. Original article submitted October 12, 2015. 280
1068-820X/16/5202–0280
© 2016
Springer Science+Business Media New York
INFLUENCE OF THE PARAMETERS OF DISCRETIZATION ON THE ACCURACY OF NUMERICAL SOLUTION
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lute temperature, VH is the partial molar volume of hydrogen in the metal, σ h is the hydrostatic component of the stress tensor in the metal, Q is the heat of hydrogen transfer in the metal, and t is time.
In the three-dimensional statement, it is difficult to find analytic so
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