Numerical implementation of the Lame equation with complex boundary conditions

A discrete model is constructed for calculating the Lame equation with complex boundary conditions. The model is tested on an analytical solution. A complex boundary condition arises when a microcrack is specified on one of the boundaries. Calculation of microcracks will enable better assessment of the relevance of the simulation and finding out which mechanisms will occur in the case of plasma flow heating in modern plasma and future thermonuclear installations.


Introduction
One of the most critical problems in the international experimental thermonuclear reactor (ITER project) is design of first wall and divertor plates. Plates material should conduct heat well, spray a small amount of particles into plasma, accumulate little hydrogen and so on. These materials must not be destroyed mechanically, melt and spray out under action of the expected effects in a tokamak. Problems of durability of materials exposed to the high-power plasma streams relevant for fusion reactors on the basis of a different geometry of the magnetic field. Results on the heating of a tungsten plate by a pulsed electron beam were obtained on the experimental setup BETA (Beam of Electrons for materials Test Applications) at BINP SB RAS [1]. The full-scale experiment goes in parallel with computational ones [2]. The calculation of displacements around microcracks perpendicular to the surface is very relevant. The work is devoted to the numerical implementation of the Lame equation with complex boundary conditions. A complex boundary condition arises when a microcrack is specified on one of the boundaries. The aim of the study is to simulate the erosion of the sample surface as a result of evaporation and penetration of heat flux into the material taking into account microcracks [3].

Problem definition
Consider a mathematical two-dimensional model of the linear static theory of elasticity [4]. The model includes the equation of equilibrium, Hooke's law and relations "displacement-deformation": To solve the problem, we will use the statement "in displacements". We obtain the necessary equation for determining the displacements by substituting the expression for ε from the "displacement-deformation" relation in Hooke's law, then we substitute the result into the equation of equilibrium. As a result, we get: uf . The main difficulty in discretization this problem is the construction of an approximation of the boundary conditions specified in terms of stresses. The need to preserve the basic properties of the operator of a differential problem at a discrete level requires cumbersome and far from obvious arguments. The discrete analogue construction scheme described in the next section is quite simple to implement and obviously leads to self-conjugate positive definite approximations of the static problem of the theory of elasticity in the formulation of "displacement". In this case, the boundary conditions specified in terms of stresses are approximated automatically.

The difference scheme
The mathematical model of the problem of the theory of elasticity belongs to the class of models having a conjugate-operator structure [5,6]. To maintain this structure at the difference level, the following procedure is used to construct a discrete analogue [6,7]:  The operator R is selected as the support.  An approximation of h R is constructed.  The approximation of * R conjugate to the selected operator is constructed as the operator conjugate to h R . Consider a rectangular domain We introduce two rectangular grids ( Fig. 1):     : To approximate R , we will use the following operator  ). In this case, the scheme is applicable only if a displacement vector u is specified at the boundary, which imposes significant restrictions on the statement of the problem.
For problems where the stress tensor σ is specified at some boundaries, the operator needs to be expanded. The following is an example for calculating at the boundary of the domain for 0 i  : As a result, a discrete model of the problem is constructed. It has the same structure as the original conjugate-operator:

 
From it, just as at the differential level, we obtain an approximation of the Lame equations: In the case under consideration, it is not difficult to show that the constructed difference scheme approximates the original second-order problem on smooth solutions. Its stability can be established by the method described in [7].
We will solve the resulting problem by the method of minimal residuals (the index h is omitted)

Simulation results
The constructed difference scheme was tested on a static problem with a solution At the three boundaries of the domain, stress tensors were specified as boundary conditions. The fourth boundary (right side of the rectangle) was fixed by the condition  u0 . The vector of mass forces was also calculated using the exact solution.
The calculation results are presented on Fig. 2, 3. The test was performed for  The obtained test results approximate the exact solution well.

Conclusion
A discrete model is constructed for calculating the Lame equation with complex boundary conditions. The model is tested on an analytical solution. A complex boundary condition arises when a microcrack is specified on one of the boundaries. Calculation of microcracks will enable better assessment of the relevance of the simulation and finding out which mechanisms will occur in the case of plasma flow heating in modern plasma and future thermonuclear installations.

Acknowledgements
Problem statement have been obtained under support of the Russian Science Foundation (project No 19-19-00272). The reported study (computational experiment) was funded by the RFBR under the research project 18-31-00303. Lazareva G.G. was partly supported by the "RUDN University Program 5-100".