Modeling of bodies with spherical pores by generalized linear interpolation

Damdinova Tatiana Tsybikovna PhD in Technical Science Associate Professor, East-Siberian State University of Technology and Management 670000, Russia, respublika Buryatiya, g. Ulan-Ude, ul. Klyuchevskaya, 40 V dtatyanac@mail.ru Other publications by this author     Ayusheev Tumen Vladimirovich Doctor of Technical Science Associate Professor, Department of Engineering and Computer Graphics, East Siberian State University of Technology and Management 670013, Russia, respublika Buryatiya, g. Ulan-Ude, ul. Klyuchevskaya, 40V, of. 731 atv62@bk.ru Balzhinimaeva Svetlana Mikhailovna Postgraduate Student, Department of Engineering and Computer Graphics, East Siberian State University of Technology and Management 670013, Russia, respublika Buryatiya, g. Ulan-Ude, ul. Klyuchevskaya, 40V, of. 731 ikg.esstu@bk.ru

Abatnin Aleksandr Andreevich Student, Department of Engineering and Computer Graphics, East Siberian State University of Technology and Management 670013, Russia, respublika Buryatiya, g. Ulan-Ude, ul. Klyuchevskaya, 40V, of. 731 abatnin@mail.ru

Introduction

Increasing the volume of high-resolution image data requires the development of algorithms capable of processing large images with reduced computational costs. Many unambiguous methods of solid-state representation of porous media based on images, such as primitive instantiation, cell decomposition, constructive solid-state geometry, representation using fractal geometry, have the limitation that they do not offer ways to represent internal behavior.

A representation is considered unambiguous when it corresponds to one and only one object in the object space. The developed methods ^[1-4] are suitable for many modeling and design applications, but mostly assume the internal uniformity of the model both in the composition of objects and in the structure of the body itself. More complex physical models, which require scalar, vector and tensor physical fields, increase the need for modeling both the shape and distribution of fields as initial or boundary conditions of modeling.

This article considers an approach to the description of parametric bodies with spherical pores using linear basis functions. The available digital data on the geometry of the sample pores is converted directly into the geometry of bodies consisting of cubic portions. Parametric porous bodies are used to demonstrate internal modeling ^[5-9]. This type of parametric porous bodies is used as initial or boundary conditions in numerical modeling. When forming objects of complex shape, parametric solid-state elements can be connected together.

A lot of research is devoted to the reconstruction of the geometric structure of porous materials based on digital images of objects for a better understanding and representation of physical processes in a porous medium ^[10-15]. A detailed understanding of the microstructure can be used to determine the physical properties, and then to evaluate and improve the characteristics of the simulated objects and processes in them. Also, these studies are due to the development of 3D printing in order to optimize the printed objects, increase their strength or determine and change other properties of the model. In geometric modeling of complex objects, Boolean operations of union, subtraction and intersection are used ^{[16, 17]}.

Problem statement

Let an arbitrary curved frame of a body with spherical pores be given in the three-dimensional space R3, which determines the shape of a solid-state cubic element, or a portion of the object being modeled (Fig. 1). Let's assume that the parameters u, v and w vary from 0 to 1 along the corresponding boundaries of the portion of the body. Then the vector function represents the interior of a portion of the body, representing six known boundary surfaces. The task is set: to construct a vector function that, when or, represents the desired boundary surface.

Figure 1. Portion of a body with spherical pores

A portion of a solid body with Koons boundary surfaces

Let us first consider the problem of constructing the equation of a portion of a solid body without spherical pores. Using the Koons portion method, it is possible to generalize and construct the portion equation of a parametric body: 

                                                  (1)

where

where u, v, w is ^[0,1], is a column vector.

In equation (1) is a trilinear interpolation of eight points. The vector function r1(u,v,w) is a bilinear interpolation between pairs of compatible boundary curves. - this is a linear interpolation between pairs of opposite portions of boundary surfaces.  

When constructing the boundary curve of the portion of the body, we will use the cubic Hermite interpolation function. With its help, a segment of a parametrically defined curve can be represented through the position of its end points, as well as through the values of tangents in them. Then the equations for the boundary curves and transverse gradients will have the form:

      (2)

where ? 0, ? 1, ? 0, ? 1 are Hermite polynomials.

The boundary surfaces of the body will be described by the following equations:

                                                           (3)

where u, v, w is ^[0,1],

Portion of a body with spherical pores

Now we can proceed to solving the problem of constructing a spherical pore inside a portion of the body.

The equation of the surface of a sphere in the Cartesian coordinate system is defined as:

                                                              (4)

where x 0, y 0,z 0 are the coordinates of the center of the sphere, R is the radius of the sphere.

Formula (4) can be used to define a spherical pore if it does not change its position and size during deformation (stretching or compression) of the body relative to the Cartesian coordinate system.

In this case , the portion of a body with spherical pores is determined by

                                                                       (5)

where

            - Cartesian coordinates and radii of spherical pores.

For our purposes, it is necessary to define a spherical pore in the system of curvilinear coordinates of the body

                                                            (6)

where u 0, v 0, w 0, are the curvilinear coordinates of the center of the spherical pore. In this case, the spherical pore will also change when the position and shape of the body change.

In the first case , the portion of a body with spherical pores is determined by

                                                                       (7)

where

            - Cartesian coordinates and radii of spherical pores.

In the second case , the portion of the body is determined by

                                                                      (8)

where

            - curvilinear coordinates and radii of spherical pores.

Results

A computational experiment was carried out with the obtained model of a portion of a porous body in the MathCAD environment. When constructing a portion of the body in the form of a cube, the following coordinates of corner points were set: P 000(0,0,0), P 100(70,0,0), P 010(0,70.0), P 110(70,70,0), P 001(0,0,70), P 101(70,0,70), P 011(0,70,70), P 111(70,70,70). A sphere with a radius of 15 mm was defined inside the cube at the point (35,35,35) (Fig. 3). The surfaces of the inner space of the cube were constructed with the value of the parameter g equal to 0, 18, 35, 53, 70. Figures 3 and 4 show a surface g equal to 35 mm. Then the coordinates of the corner points of the cube were changed: P 000(0,0,0), P 100(70,10,0), P 010(0,65,0), P 110(50,70,0), P 001(0,0,70), P 101(50,0,70), P 011(10,80,60), P 111(50,75,45) (Fig. 4). The surface of a spherical pore was given by the equation in parametric form:

When specifying a portion of the body in the shape of a cube, the hole on the intermediate surface has the shape of a circle (Fig. 2). When changing the position of the corner points of the cube, the intermediate surfaces are transformed into the shape of a hyperbolic paraboloid, and the holes into the shape of an oval (Fig. 3).

Fig. 2. A portion of a porous body in the form of a cube

Fig. 3. Changing the shape of the cube when moving the angular points of a portion of a porous body

To automate 3D modeling of real objects with many spherical pores, images of cheese slices were obtained based on digital image processing as an object of research (Fig.4a). Next, the position and size of the pores were determined by layers (Fig.4b).

Then the data was processed in a program developed in Python and an OpenSCAD CAD script was obtained that automates the creation of a 3D model with Boolean operations (Fig.5).

Fig.5. 3D model of a body with spherical pores.

Conclusion

The proposed method for modeling complex porous media of composite form uses portions of a three-dimensional parametric body. The continuity between the elements is determined similarly to the modeling of cubic parametric splines. The examples given show the determination of the internal structure when changing the shape of a porous body. Its distinctive feature is the possibility of constructing internal intermediate grids in the presence of digital data of spherical pores inside a composite body. 

The software implementation of this method has shown that algorithms based on it are characterized by computational stability and speed. This provides undoubted advantages in visualization tasks, in geometric modeling of various structures with spherical pores under conditions of free deformation.