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Variability Response Function for Steady-State Heat Conduction

The concept of variability response function based on the weighted integral method is extended to steady-state heat conduction. The conductivity is considered to be homogenous stochastic field. The stochastic element conductivity matrix is represented as linear combination of deterministic element conductivity matrix and random variables, weighted integrals. The choice of different finite element leads to different number of random variables. The variability response function is calculated for one-dimensional and two-dimensional steady-state heat conduction. The concept of the variability response function is used to compute upper bounds of the response variability. The first and second moment of the stochastic conductivity are used as input quantities to describe its randomness. The response variability is calculated using the first-order Taylor expansion approximation of the variability response function. Numerical results are compared with analytical solution in one-dimensional case. The local averaging method is introduced for two-dimensional case to show the influence of the reducing the computational effort on the loss of accuracy. Numerical examples are provided for different boundary conditions, different wave numbers and different number of finite elements. The rate of convergence of calculated sequences of numerical solutions is evaluated to show the quality of its approximation.

variability response function; steady-state heat conduction; stochastic conductivity

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