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

New Possibilities and Applications of the Least Squares Collocation Method

“…Since it is not possible to describe them in detail here, we only point out their characteristic features which are also used in other methods. For example, it became apparent that using a regular grid within the domain Ω is advisable in versions with rectangular cells implemented in [9,10]. In this case, the transition from cell to cell during calculations is done by simply shifting the index (cell number) by 1.…”

“…A sample grid built with mixed algorithm is displayed in figure 1. Presently, three types of algorithms have been implemented which use 2nd, 4th, 8th degree polynomial spaces for solving the Dirichlet problem for the Poisson equation over arbitrary triangular and convex quadrilateral domains (figure 1), and a domain with curvilinear boundary comprised of smooth curve pieces [9,10]. A curvilinear boundary domain is represented either as a set of joined analytic-form curve pieces, or a discrete set of points located on the boundary (figure 2).…”

“…Let us examine the problem with no known analytical solution. Assume the isotropic triangular plate with vertices (0, 0), (10,0), (3,6) is under the uniform transverse load q = 1 MPa. All plate edges are clamped (figure 3b).…”

“…During this process, coefficients c in neighbor cells can be taken from the previous global iteration (Gauss-Jacobi process) or current iteration (Gauss-Seidel process). Implemented versions of LSC are further detailed in [1,2,[7][8][9][10].…”