International Scientific Journals
of Scientific Technical Union of Mechanical Engineering "Industry 4.0"

This paper defines constrained functional similarity between 2-D trajectories via minimizing the H¹ semi-norm of the difference between the trajectories. An exact general solution is obtained for the case wherein the components of the trajectories are mesh- functions defined on a uniform mesh and the imposed constraints are linear. Various examples are presented, one of which features application to mechanics and two-point boundary value problems. A MATLAB code is given for the solution of one of the examples. The code could easily be adjusted to other cases.

This paper shows that the constrained similarity transformation of surfaces under boundary constraints is a Laplacian preserving transformation. First, a general proof is presented and then the result is verified for mesh-functions through particular examples. The fact that the constrained similarity transformation, subject to boundary constraints, is a Laplacian preserving transformation is used to construct a method for solving boundary value problems for the Laplace’s and the Poisson’s equations. Given any solution to these equations we apply the constrained similarity transformation to get the particular solution that satisfies the given boundary conditions.

This paper defines constrained similarity between surfaces via minimizing the L² norm of the gradient of the difference between the surfaces. An exact general solution is obtained for the case wherein the surfaces are given as mesh-functions defined on a uniform mesh and the imposed constraints are linear. Various examples are presented as well as a MATLAB code for the solution of one of the examples. The code could be adjusted to other cases.