The article presents a new vision of the process of approximating the solution of differential equations based on the construction of geometric objects of multidimensional space incident to nodal points, called geometric interpolants, which have pre-defined differential characteristics corresponding to the original differential equation. The incidence condition for a geometric interpolant to nodal points is provided by a special way of constructing a tree of a geometric model obtained on the basis of the moving simplex method and using special arcs of algebraic curves obtained on the basis of Bernstein polynomials. A fundamental computational algorithm for solving differential equations based on geometric interpolants of multidimensional space is developed. It includes the choice and analytical description of the geometric interpolant, its coordinate-wise calculation and differentiation, the substitution of the values of the parameters of the nodal points and the solution of the system of linear algebraic equations. The proposed method is used as an example of solving the inhomogeneous heat equation with a linear Laplacian, for approximation of which a 16-point 2-parameter interpolant is used. The accuracy of the approximation was estimated using scientific visualization by superimposing the obtained surface on the surface of the reference solution obtained on the basis of the variable separation method. As a result, an almost complete coincidence of the approximation solution with the reference one was established.
The paper proposes an approach to the comparison of multidimensional geometric objects, which is used to assess the variational geometric models of multifactor processes and phenomena obtained using the geometric theory of multidimensional interpolation. The proposed approach consists of two stages, the first of which consists in the discretization of multidimensional geometric objects in the form of a set of discretely given points, and the second is in comparing the obtained discrete point sets using a criterion that is essentially similar to the coefficient of determination. In this case, one of the discrete point sets is taken as a reference for comparison with another point set. For a correct comparison of multidimensional geometric models in the form of point equations, which are reduced to a system of parametric equations, it is necessary to perform interconnection of parameters. A computational experiment was carried out on the example of comparing geometric models of the physical and mechanical properties of fine-grained concrete. It showed the possibility of using the proposed approach for comparing multidimensional geometric objects and the reliability of the results obtained in comparison with scientific visualization methods. On the same example, it was found that for an accurate comparison of the investigated geometric models of the physical and mechanical properties of fine-grained concrete, it is enough to discretize 100 points. A further increase in the set of discrete points of the compared geometric objects has no significant effect on the criterion for assessing their similarity.
The paper describes an example of modeling an arc of a 2nd order curve using an engineering discriminant and its analytical description based on a graphical algorithm for constructing a curve in point calculus. Examples of modeling the surfaces of engineering structures shells on an elliptical and rectangular plan are given. Research methods include geometric algorithms: modeling of 2nd order curves passing through 3 predetermined points in advance and having tangents at the start and end points, and shell surfaces based on them; analytical definition of curves arcs and sections of surfaces using the mathematical apparatus point calculation in a given parametrization and taking into account all predetermined geometric conditions. This approach can be widely used in the practice of modeling the shells of engineering structures for various technical purposes. It allows the designer to choose the best curvature of the shell surface, which will have the necessary strength characteristics, technical aesthetics and artistic expressiveness. The possibility of dividing the surface of the shell into finite elements of a given amount is also provided for studying the stress-strain state of the shell under the action of various loads in the systems of finite element analysis.
The work is investigated by the influence of variable geometric algorithms in modeling multifactor processes using multidimensional interpolation. Geometric models of multifactorial processes obtained using multidimensional interpolation inherent variability, which is a consequence of the multiplicity of the choice of reference lines during the development of geometric modeling schemes. At the same time, all possible variations of geometric interpolyns are fully satisfying the initial data. It has been established that the number of variations of geometric schemes directly depends on the number of current parameters and the dimension of the space in which the simulated geometrical object is located. Thus, a variable approach to geometrical modeling of multifactor processes generates a number of scientific tasks, the main one is the need to determine the effect of the variability of geometric algorithms on the final results of the computational experiment and, as a result, the choice of the best modeling results. To this end, the article presents the studies of variable geometric algorithms and computational experiments on the example of 2-parametric geometric interpolyns. A classification of 2-parametric geometric interpolytesses, which were conditionally divided into 3 types. Depending on the geometric scheme of constructing interpolynta, the square geometric scheme, a rectangular geometric scheme, a mixed geometric scheme. As a result of computational experiments, it was found that for a square geometric scheme, the variability does not affect the final results, in rectangular geometric schemes, the variability has a slight influence, and mixed geometric schemes may have significant differences and require additional research to select the highest quality geometric process model. Comparison of geometric models were performed by the methods of scientific visualization by overlaying the response surfaces on each other.
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