Past research

Research topics I am working on now

Anisotropic particle on substrate With S. Gavhale (Kyoto University) |
Multiphase dynamics of surfaces with anisotropic energiesA problem important in crystal growth, coating or development of materials with nanostructures such as nanowires. The aim is to mathematically understand motion of multiple interfaces with energy depending on the orientation of the interface, and to develop efficient numerical methods for its simulation. In particular, I focus on the approximation via BMO algorithm (thresholding scheme) which is able to deal with topological changes. Published results: S. Gavhale, K. Svadlenka, Dewetting dynamics of anisotropic particles - a level set numerical approach, Applications of Mathematics, 2021, doi:10.21136/AM.2021.0040-21.Joint work with: Siddharth Gavhale (Kyoto University). |

Hyperbolic mean curvature flow With E. Ginder (Meiji University) |
Hyperbolic interface evolutionEvolution of interfaces of hyperbolic type, such as curvature accelerated motion, have only started being studied, although it applies to many areas from water waves to string theory. We are analyzing such problems from the mathematical and numerical point of view. The main idea is to look at level sets of solutions to wave-type PDEs and thus realize the implicit approach that allows for topology changes. Published results: E. Ginder, K. Svadlenka, Wave-type threshold dynamics and the hyperbolic mean curvature flow, Japan J. Indust. Appl. Math. 33 (2), pp. 501–523, 2016Joint work with: E. Ginder (Meiji University), R. Mohammad (University of the Philippines) |

Cellular pattern formation With R. Mohammad (University of the Philippines) |
Modeling of cellular rearrangement in morphogenesisCells in epithelial tissues of sensory and other organs form regular geometrical patterns. It is known that cell-cell adhesion plays an important role in the formation process but the overall mechanism is still not well understood. We are trying to elucidate the principles behind this mysterious morphogenetic phenomenon by modeling cellular rearrangements as a gradient flow of an interfacial network with nonuniform surface tension. We succeeded in reproducing the observed cell patterns through a mathematical model (see paper cited below) but to capture the process more precisely, we now consider a model coupling interface motion with an evolving chemical field that determines the surface tensions / adhesion strengths of individual interfaces. Published results: R. Z. Mohammad, H. Murakawa, K. Svadlenka, H. Togashi, A numerical algorithm for modeling cellular rearrangements in tissue morphogenesis, Communications Biology 5, 239 (2022), DOI: 10.1038/s42003-022-03174-6Joint work with: R. Mohammad (University of the Philippines), H. Togashi (Kobe University), H. Murakawa (Kyushu University) |

String hitting an flat obstacle With S. Omata (Kanazawa University) et al. |
Hyperbolic free boundary problemsWhen a membrane or a ball or other elastic body bounces from the ground or a bubble moves on water surface, the moving object vibrates while hitting the obstacle, which in mathematical terms leads to a hyperbolic problem with free boundary. Although this type of problems has been studied only sporadically, we obtained a few initial mathematical results by discretizing time and formulating a corresponding minimization problem. However, extension to higher dimensions or to operators more complicated than the wave operator remains still a challenge. Published results: Y. Akagawa, E. Ginder, S. Koide, S. Omata, K. Svadlenka, A Crank-Nicolson type minimization scheme for a hyperbolic free boundary problem, Discrete and Continuous Dynamical Systems Series B 27(5), pp. 2661-2681 (2022), DOI: 10.3934/dcdsb.2021153, etc.Joint work with: S. Omata (Kanazawa University), E. Ginder (Meiji University) |

Simulation of kink formation in compressed elastoplastic material |
Mathematical modeling of kink strengthening in mille-feuille structured materialsWhen certain magnesium alloys featuring layered structure of soft and hard layers are compressed, kinks (wedge-like structures) are formed, which leads to a significant strengthening of the material. If the mechanism of this strengthening is revealed, it can be applied to the design of various materials with outstanding properties. Mathematically it is a very interesting problem spanning geometry and calculus of variations, and we are attempting to discover the strengthening principle using an elastoplasticity model with variational structure. The main tools are Gamma-convergence and homogenization but we look also at the influence of inertia and implement simulations reproducing kink formation. This is joint work within the JSPS Grant-in-Aid for Scientific Research on Innovative Areas. See the official HP for more details. Published results: M. Kruzik, K. Mathis, K. Svadlenka, J. Valdman, Elastoplastic deformations of layered structure, preprint, 2022Joint work with: M. Kruzik (Czech Academy of Sciences), K. Mathis (Charles University), J. Valdman (Czech Academy of Sciences) |

Simulation of moving filaments |
Analysis of co-dimension 2 geometrical motionsWhen we want to understand motion of vortex rings in fluid or motion of dislocation and disclinations in materials, it is necessary to think of motions of 1-dimensional curves in 3-dimensional space. I would like to answer the following questions about motion of such filaments in case when topological changes occur: How to define the motion mathematically in the parabolic and hyperbolic cases? How to design an efficient numerical algorithm for their simulation? |

Simulation of lung undergoing mechanical destruction |
Modeling of lung destruction in COPD diseaseCOPD is a lung disease whose effective treatment is still not discovered because there is a lack of understanding of the mechanism through which lung is destroyed. Collaborators doing research in respiratory medicine hypothesized that mechanical factors are significant during the destruction process and we are trying to support this conjecture through mathematical modeling and numerical simulations of evolving alveolar network, which is based on the minimization of an elastic energy functional. Published results: Y. Hamakawa, K. Svadlenka et al., Spatial patterns of alveolar damage in emphysemous lung originate from alveolar wall stiffening and bronchoconstriction, preprint, 2022This is joint research with professors Y. Hamakawa, A. Sato, S. Sato, N. Tanabe (Kyoto University, Faculty of Medicine), R. Suzuki, M. Tanaka (Kyoto University, iCeMS-CiMPhy) |