Optimal Random Packing of Spheres and Extremal Effective Conductivity
A close relation between the optimal packing of spheres in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup><...
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MDPI AG
2021-06-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/6/1063 |
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| author | Vladimir Mityushev Zhanat Zhunussova |
| author_facet | Vladimir Mityushev Zhanat Zhunussova |
| author_sort | Vladimir Mityushev |
| collection | DOAJ |
| description | A close relation between the optimal packing of spheres in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></semantics></math></inline-formula> and minimal energy <i>E</i> (effective conductivity) of composites with ideally conducting spherical inclusions is established. The location of inclusions of the optimal-design problem yields the optimal packing of inclusions. The geometrical-packing and physical-conductivity problems are stated in a periodic toroidal <i>d</i>-dimensional space with an arbitrarily fixed number <i>n</i> of nonoverlapping spheres per periodicity cell. Energy <i>E</i> depends on Voronoi tessellation (Delaunay graph) associated with the centers of spheres <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">a</mi><mi>k</mi></msub></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow></semantics></math></inline-formula>). All Delaunay graphs are divided into classes of isomorphic periodic graphs. For any fixed <i>n</i>, the number of such classes is finite. Energy <i>E</i> is estimated in the framework of structural approximations and reduced to the study of an elementary function of <i>n</i> variables. The minimum of <i>E</i> over locations of spheres is attained at the optimal packing within a fixed class of graphs. The optimal-packing location is unique within a fixed class up to translations and can be found from linear algebraic equations. Such an approach is useful for random optimal packing where an initial location of balls is randomly chosen; hence, a class of graphs is fixed and can dynamically change following prescribed packing rules. A finite algorithm for any fixed <i>n</i> is constructed to determine the optimal random packing of spheres in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></semantics></math></inline-formula>. |
| first_indexed | 2024-03-10T10:27:17Z |
| format | Article |
| id | doaj.art-5d7899ec8863462883d11037664c3e97 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T10:27:17Z |
| publishDate | 2021-06-01 |
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| spelling | doaj.art-5d7899ec8863462883d11037664c3e972023-11-21T23:56:40ZengMDPI AGSymmetry2073-89942021-06-01136106310.3390/sym13061063Optimal Random Packing of Spheres and Extremal Effective ConductivityVladimir Mityushev0Zhanat Zhunussova1Department of Computer Science and Computational Methods, Institute of Computer Sciences, Pedagogical University of Krakow, Podchorążych 2, 30-084 Krakow, PolandInstitute of Mathematics and Mathematical Modeling, Almaty 050010, KazakhstanA close relation between the optimal packing of spheres in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></semantics></math></inline-formula> and minimal energy <i>E</i> (effective conductivity) of composites with ideally conducting spherical inclusions is established. The location of inclusions of the optimal-design problem yields the optimal packing of inclusions. The geometrical-packing and physical-conductivity problems are stated in a periodic toroidal <i>d</i>-dimensional space with an arbitrarily fixed number <i>n</i> of nonoverlapping spheres per periodicity cell. Energy <i>E</i> depends on Voronoi tessellation (Delaunay graph) associated with the centers of spheres <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">a</mi><mi>k</mi></msub></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow></semantics></math></inline-formula>). All Delaunay graphs are divided into classes of isomorphic periodic graphs. For any fixed <i>n</i>, the number of such classes is finite. Energy <i>E</i> is estimated in the framework of structural approximations and reduced to the study of an elementary function of <i>n</i> variables. The minimum of <i>E</i> over locations of spheres is attained at the optimal packing within a fixed class of graphs. The optimal-packing location is unique within a fixed class up to translations and can be found from linear algebraic equations. Such an approach is useful for random optimal packing where an initial location of balls is randomly chosen; hence, a class of graphs is fixed and can dynamically change following prescribed packing rules. A finite algorithm for any fixed <i>n</i> is constructed to determine the optimal random packing of spheres in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></semantics></math></inline-formula>.https://www.mdpi.com/2073-8994/13/6/1063optimal packing of sphereseffective conductivityVoronoi tessellationperiodic graphstructural optimization |
| spellingShingle | Vladimir Mityushev Zhanat Zhunussova Optimal Random Packing of Spheres and Extremal Effective Conductivity Symmetry optimal packing of spheres effective conductivity Voronoi tessellation periodic graph structural optimization |
| title | Optimal Random Packing of Spheres and Extremal Effective Conductivity |
| title_full | Optimal Random Packing of Spheres and Extremal Effective Conductivity |
| title_fullStr | Optimal Random Packing of Spheres and Extremal Effective Conductivity |
| title_full_unstemmed | Optimal Random Packing of Spheres and Extremal Effective Conductivity |
| title_short | Optimal Random Packing of Spheres and Extremal Effective Conductivity |
| title_sort | optimal random packing of spheres and extremal effective conductivity |
| topic | optimal packing of spheres effective conductivity Voronoi tessellation periodic graph structural optimization |
| url | https://www.mdpi.com/2073-8994/13/6/1063 |
| work_keys_str_mv | AT vladimirmityushev optimalrandompackingofspheresandextremaleffectiveconductivity AT zhanatzhunussova optimalrandompackingofspheresandextremaleffectiveconductivity |