## 1 Introduction  
Covering a certain region by simple shapes has various applications. Our interest in covering problems is motivated by packing particles arising in science and engineering applications. Packing problems are widely used in modeling liquid and glass structures [1; 2], representing granular materials [3], packing beds, and cermet [4], as well as in many other applications (see, e.g., [5; 6; 7; 8; 9; 10]). Many algorithms have been proposed for packing spherical particles (see, e.g., [11; 12; 13; 14; 15; 16] and the references therein). However, to have a better and more adequate representation of particulate microstructure, packing non-spherical particles must be considered. The approaches proposed to handle non-spherical shapes can be divided in three large groups [17]. Techniques in the first group use analytical equations for the shapes of the particles, e.g., ellipsoids, and the main modeling problem is to state analytically non-overlapping and containment conditions [18; 19; 20; 21; 22; 23]. The second approach is based on tessellating the container/particles shapes with a grid and then approximating them by corresponding collections of grid nodes (see, e.g., [15; 24] and the references therein). This way, detecting the overlap is reduced to verify if two shapes share the same node. However, to get a reasonable approximation, fine grids must be used resulting in large-scale and memory-consuming problems. In the third approach, the shape of the particle is represented approximately by a collection of spheres having different sizes and positions (see, e.g., [25; 26; 27]). Then, detecting the overlap is reduced to verifying the overlapping for two spheres from different particle collections. This problem is much simpler than detecting overlapping in the first approach. The third approach can be considered as a compromise between the simplicity of the tessellating techniques and the rigor of the methods based on analytical shapes presentation. The efficiency of the third approach depends on the quality of the multi-spherical approximation of the shape of the particle. The number of the spheres representing the shape of the particle must be small enough to simplify the non-overlapping (pairwise) test. At the same time, the union of the spheres in the collection must represent the original shape of the particle closely enough.  
Among the different shapes used to represent the microstructure of non-spherical particles, the spheroid (ellipsoid of revolution) is one of the most frequently used (see, e.g., [5; 6; 7; 8; 10]). One of the reasons to use the spheroid is its relative simplicity comparing to the general ellipsoid. Moreover, in many cases, analyses of 3D spheroids are reduced to the examination of the 2D ellipses used to generate the spheroid.  
In this paper, the problem of covering the spheroid by different spheres is studied. An \(\varepsilon\)-cover is introduced as an outer multi-spherical approximation of the spheroid within the error \(\varepsilon\). A fast heuristic algorithm is proposed to construct the optimized \(\varepsilon\)-cover giving a reasonable balance between the value of the proximity parameter \(\varepsilon\) and the number of spheres used. Computational results are provided to demonstrate efficiency of the approach.  
The main contributions of the paper are as follows:  
* The concept of the \(\varepsilon\)-cover is introduced for the outer multi-spherical approximation of the spheroid.
* A fast two-stage approach is proposed to get a reasonable (optimized) \(\varepsilon\)-cover.
* Numerical results are provided to illustrate the main constructions.  
The paper is organized as follows. Section 2 presents the basic definitions and formulations used in the paper as well as the two-stage conceptual approach. Solution techniques are presented in detail in Section 3. Numerical results are presented in Section 4, while Section 5 presents the conclusions. Expressions for the parameters used in Section 3 are derived in the Appendix A.