### Belyaev V.A. Shapeev V.P.

## Versions of the collocation and least residuals method for solving biharmonic equations in the non-canonical domains

### Reporter: Belyaev V.A.

Versions of the collocation and least residuals method for solving biharmonic equations in the non-canonical domains

Belyaev V.A., Shapeev V.P.

Khristianovich Institute of Theoretical and Applied Mechanics,

Novosibirsk State University

New versions of the collocations and least residuals (CLR) method are proposed and implemented for the numerical solution of boundary value problems for biharmonic equations in non-canonical domains. Differential problems are projected into the space of fourth-degree polynomials by the CLR method. The boundary conditions for the approximate solution are put down exactly on the boundary of the computational domain Ω. The collocation and matching points that are situated outside the domain are used for approximation of solution in the boundary cells. The conditionality of the approximate problems obtained in different versions of the method is investigated. The effect of conditionality on the convergence rate of the iterative processes of problem solving is shown. The comparative advantages of the considered versions of the method are pointed out in this paper.

The proposed versions of the CLR method are used for the calculation of the stress-strain state of isotropic plates of arbitrary forms under the action of the transverse load. The stress-strain state of such plates is described by solution of the biharmonic equation

𝜕^{4}𝜔(𝑥_{1},𝑥_{2})/𝜕^{4}𝑥_{1}+2𝜕^{4}𝜔(𝑥_{1},𝑥_{2})/𝜕^{2}𝑥_{1}𝜕^{2}𝑥_{2}+𝜕^{4}𝜔(𝑥_{1},𝑥_{2})/𝜕^{4}𝑥_{2}=𝑞(𝑥_{1},𝑥_{2})/𝐷, (𝑥_{1},𝑥_{2})∈Ω (1)

with the corresponding different boundary conditions, where 𝜔(𝑥_{1},𝑥_{2}) denotes the bending of the middle surface, 𝑞(𝑥_{1},𝑥_{2}) – the transverse load, 𝐷 – the rigidity of the plate in bending.

Numerical experiments on the convergence of the approximate solution of various problems constructed by the CLR method for equation (1) are performed on a sequence of grids of the following sizes: 10×10, 20×20, 40×40, 80×80. It is shown that the approximate solution obtained by CLR converges with high order and matches with high accuracy the analytical solution of the test problem in the case of known solution.

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