One steady-state heat-conduction problem for a polygonal region with mixed boundary conditions

An exact solution is given for one heat-conduction problem involving a polygonal region with mixed boundary conditions. The solution method is based on development of a method presented by the author in previous publications.Translated from Inzhenerno-Fizi-cheskii Zhurnal, Vol. 17, No. 2, pp. 306–312, August, 1969.     

An approximate solution for a stationary heat-conduction problem in a half-space with mixed boundary conditions

A stationary axisymmetric problem with conditions of convective heat transfer is considered. A special method is employed to find an approximate solution in analytic form and its accuracy is evaluated.     

Solution of some mixed heat-conduction boundary-value problems

Some mixed boundary-value problems for steady-state heat conduction in a rectangular domain with a variable heat-transfer coefficient are solved by reduction to infinite systems.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 27, No. 2, pp. 335–340, August, 1974.     

A mixed two-dimensional stationary heat-conduction problem for a cylinder

We investigate the exact solution to the problem of calculating the stationary thermal field external to a cylinder on a portion of the surface of which the thermal flow density is constant and on the remaining portion of which the temperature is constant.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 21, No. 3, pp. 476–479, September, 1971.     

An inverse Laplace transformation for solving heat-conduction problems with discontinuous boundary conditions of the second kind

An inverse Laplace transformation is found for a class of functions encountered in heat-conduction problems with discontinuous boundary conditions.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 50, No. 5, pp. 852–859, May, 1986.     

Numerical treatment of mixed boundary value problems in two-dimensional elastostatics

Finite difference treatment of two-dimensional problems in elastostatics is usually based on the differential equations for the displacement vector or the Airy stress function, depending on whether boundary conditions are on displacement or stress. In either case, determination of stresses requires numerical differentiation and therefore use of a rather fine grid. Moreover, neither method is suited to the treatment of mixed boundary conditions. The alternative method developed in this paper uses the first derivatives of the displacement components at the grid points as basic variables and hence does not require numerical differentiation in the evaluation of stresses. Appropriate finite difference equations are established, and their use is discussed in connection with a specific example with known explicit solution.     

Application of the meshless Galerkin boundary node method to potential problems with mixed boundary conditions

In this paper, the meshless Galerkin boundary node method is developed for boundary-only analysis of two- and three-dimensional potential problems with mixed boundary conditions of Dirichlet and Neumann type. This meshless algorithm leads to a symmetric and positive definite system of linear equations. Additionally, boundary conditions can be implemented directly and easily despite the fact that the employed meshless shape functions lack the delta function property. Theoretical error analysis and numerical results indicate that it is an efficient and accurate numerical method.     

Solution of two-dimensional heat-conduction problems by the Z transform method

The Z-transform (discrete Laplace transform) is used to solve heat-conduction problems in axisymmetric bodies of arbitrary shape for different types of boundary conditions.     

Some heat-conduction problems in the electromagnetic and acoustic interaction with the dielectrics

We solve the inverse problem which consists of the determination of the intensity distribution of a heat source from the temperature at the boundary of a semiinfinite medium, and of the determination of the corresponding temperature field.Bashkirsk State University of the 40th Anniversary of October, UfaTranslated from Inzhenerno-Fizicheskii Zhurnal, Vol. 41, No. 5, pp. 916–921, November, 1981.     

A method of constructing a solution of the heat-conduction equation with complex boundary conditions

It is proposed to solve the heat conduction equation with complicated boundary conditions using the notion of R-functions. A solution which satisfies exactly mixed boundary conditions or boundary conditions of the first, second, or third kind is constructed.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol.21, No. 5, pp.909–913, November, 1971.     

Numerical method for solving heat-conduction problems for two-dimensional bodies of complex shape

A finite-difference scheme is described for a curvilinear orthogonal net which permits the use of a single algorithm for calculating bodies of various shapes.Notation x, y independent variables - u, v orthogonal coordinates - F(w)=F(u + iv) function of a complex variable - g(u,v)= F(w)/w Jacobian of transformation from (u,v) to (x,y) - thermal conductivity - c volumetric heat capacity - Q heat release per unit volume - T temperature - f value of temperature on boundary of region - time - L, L₁, L₂ differential operators - (u,v) solution of differential problem in canonical region - ^j, ₁ ^j , ₂ ^j , t^J, t ₁ ^j , t ₂ ^j network functions in canonical region - ^j, t^*j solutions of difference problems using rectangular and orthogonal nets respectively - {u_i, v_k} rectangular net in canonical region G - {x_i,k, y_i, k} orthogonal net in given region G^* - u_i, v_k dimensions of cell of rectangular net - u_i,v _i,k dimensions of cell of orthogonal net - h, maximum dimension of cell for rectangular and orthogonal nets respectively - ₁, ₂, difference operators for rectangular and orthogonal nets - A, B, C, D, A^*, B^*, C^*, D^* coefficients of difference scheme for rectangular net - D, Ã, B coefficients of difference scheme for orthogonal net Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 40, No. 3, pp. 503–509, March, 1981.     

Algorithms and results of solving the inverse heat-conduction boundary problem in a two-dimensional formulation

The effectiveness of gradient algorithms for solving the inverse problem which are regulated in terms of the number of iterations is investigated.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 48, No. 4, pp. 658–666, April, 1985.     

On modified incomplete cholesky factorization methods for the solution of problems with mixed boundary conditions and problems with discontinuous material conefficients

In Reference 1 a class of first-order factorization methods for the solution of large, sparse systems of equations, of which the coefficient matrix is a symmetric M-matrix, was introduced. Asymptotic results for the computational complexity was proved in the case of systems arising from finite difference approximation of second-order self-adjoint elliptic partial differential equation problems in two dimensions with Dirichlet boundary conditions. In this paper the result is extended to cover even mixed boundary conditions and problems with discontinuous material coefficients. Results from numerical experiments with various finite difference and finite element approximations are presented and comparisons with direct and other interative methods are made.     

Nonstationary nonlinear heat-conduction problems

An exact analytical solution is constructed for the one-dimensional nonlinear nonstationary problem of heat conductivity with boundary conditions of the third kind.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 35, No. 4, pp. 743–746, October, 1978.     

More accurate difference schemes for solving the heat-conduction equation with boundary conditions of the third kind

A difference analog of a boundary condition of the third kind is obtained. By integrating the heat-conduction equation numerically the surface temperature can be calculated with an error of the fourth order of smallness in the size of the space step.     

An approximate method of solving stationary heat conduction problems with mixed boundary conditions of the third kind

We consider an approximate method of determining temperature fields when the parameter in the boundary conditions for convective heat transfer has different values on different portions of the boundary surface. We illustrate the effectiveness of our method with examples.     

Extended-domain method in heat-conduction problems

An effective method for the solution of mathematical physics problems is discussed in the example of heat-conduction problems. Numerical computations are carried out to illustrate the accuracy and convergence of the method.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 35, No. 10, pp. 728–733, October, 1978.     

Macro-elements in the mixed boundary value problems

 The symmetric Galerkin boundary element method (SGBEM), applied to elastostatic problems, is employed in defining a model with BE macro-elements. The model is governed by symmetric operators and it is characterized by a small number of independent variables upon the interface between the macro-elements. The kinematical and mechanical transmission conditions across the interface are imposed in global form, whereas the response of the boundary discretized elastic problem is provided in terms of forces on the interface boundary sides and of displacements at the interface nodes. A variational formulation is presented in which the boundary transmission conditions are derived by Polizzotto's boundary min-max variational principle. Simple numerical applications are shown. Received 8 December 99     

Some steady-state problems in heat conduction theory for wedges with boundary conditions of the third kind

It is shown that a steady-state problem of heat conduction theory for a wedge releasing heat according to Newton's law is reduced, by means of an integral transformation, to solution of a certain functional equation. For a wedge angle of 2 =/m (m=1, 2, 3, ...) an exact solution of the latter equation is found, and formulas for the temperature distribution are obtained.