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Additional resources for Physics Reports vol.106
Fig. 4. The temperature distribution in a plate’s centre in time (the problem of the non-stationary heat transfer during a thin plate’s vibrations). inertia forces into account. In order to solve this problem we shall use the upper relaxation method. 108) where Λ denotes a diﬀerence analogue of Laplace’s diﬀerential operator. 45), thus the algorithms realising calculation methods are diﬀerent. , y sM ), M = N1 N2 N3 , k - number of iterations, ai - coeﬃcients at the unknowns. 2. First, initial approximations are set in the entire field of the mesh and boundary conditions are set on its boundaries, where Dirichlet’s problem is considered.
Variational-diﬀerence methods are the most appropriate for analysing the problems discussed in this chapter and they retain the properties of a diﬀerential system. 10). 34) 22 1 Three–Dimensional Problems of Theory of Plates in Temperature Field ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎣ τ1 I= τ0 Ω 3 s=1 3 3 3 sm kαβ s=1 α,β=1 n=1 ∂θ ∂u s ∂v s ∂un ∂v s + −β vs − ρ ∂xβ ∂xα ∂x s ∂τ ∂τ ⎤ ⎥⎥⎥ ⎥⎥⎥ dΩdτ. , M}. For every continuous function f (x, τ) set in field Qτ , functions fi jk (x, x4 ) = f (xi , x j , xk , x4 ) are going to be constructed and defined within ωτ .
Ladyzhenskaya’s work contains derivations of the first initially-boundary problem for a parabolic and a hyperbolic equation in a general form. Works [231, 241, 492] address extended research into hybrid types of problems. Treating those references as basis we are going to prove a theorem that refers to stability of approximate solutions to the coupled thermoelasticity problems for three-dimensional plates. 2 Coupled 3D Thermoelasticity Problem for a Cubicoid 31 To make things simpler let us assume that hi = h.