arXiv subjects

Tomoyuki Oka

Publications and source records attributed to Tomoyuki Oka.

10 recordsLinked to original sources

arXiv2026

Construction of signed distance functions through an elliptic equation

Motivated by recent advances in structural optimization, we propose a novel method for constructing the distance function to the boundary of a given domain. Building on and extending the celebrated Varadhan asymptotic theory, our approach reformulates the governing equation into a more appropriate framework. A central contribution of this work is the derivation of convergence rates within this new setting, which are shown to be optimal in one dimension and offer significant improvements over existing results in higher dimensions.

math.AP↗

arXiv2025

Approximation in an optimal design problem governed by the heat equation

This paper studies a two-material optimal design problem for the time-averaged duality pairing between a (possibly time-dependent) heat source and the weak solution of an initial-boundary value problem for the heat equation with a two-material diffusion coefficient, under a volume constraint. In general, such optimal designs are not guaranteed to exist, and geometric constraints such as the perimeter are required. As an approximation of the problem with an additional perimeter constraint, a material representation based on a level set function, together with a perturbation of the Dirichlet energy, is employed. It is then shown that optimal level set functions exist for the perturbation problem, and the corresponding minimum value converges to that of the elliptic case, thereby elucidating the long-time behavior. Furthermore, two-material domains satisfying this property are also constructed via the nonlinear diffusion-based level set method. In particular, the asymptotic behavior with respect to the perturbation parameter is clarified, and the validity of the approximation is established.

math.AP↗

arXiv2024

Optimal design problem with thermal radiation

This paper is concerned with configurations of two-material thermal conductors that minimize the Dirichlet energy for steady-state diffusion equations with nonlinear boundary conditions described mainly by maximal monotone operators. To find such configurations, a homogenization theorem will be proved and applied to an existence theorem for minimizers of a relaxation problem whose minimum value is equivalent to an original design problem. As a typical example of nonlinear boundary conditions, thermal radiation boundary conditions will be the focus, and then the Fréchet derivative of the Dirichlet energy will be derived, which is used to estimate the minimum value. Since optimal configurations of the relaxation problem involve the so-called grayscale domains that do not make sense in general, a perimeter constraint problem via the positive part of the level set function will be introduced as an approximation problem to avoid such domains, and moreover, the existence theorem for minimizers of the perimeter constraint problem will be proved. In particular, it will also be proved that the limit of minimizers for the approximation problem becomes that of the relaxation problem in a specific case, and then candidates for minimizers of the approximation problem will be constructed by employing time-discrete versions of nonlinear diffusion equations. In this paper, it will be shown that optimized configurations deeply depend on force terms as a characteristic of nonlinear problems and will also be applied to real physical problems.

math.AP↗

arXiv2023

Nesterov's acceleration for level set-based topology optimization using reaction-diffusion equations

This paper discusses level set-based structural optimization. Level set-based structural optimization is a method used to determine an optimal configuration for minimizing an objective functional by updating level set functions characterized as solutions to partial differential equations (PDEs) (e.g., Hamilton-Jacobi and reaction-diffusion equations). In this study, based on Nesterov's accelerated method, a nonlinear (damped) wave equation will be derived as a PDE satisfied by level set functions and applied to a minimum mean compliance problem. Numerically, the method developed in this study will yield convergence to an optimal configuration faster than methods using only a reaction-diffusion equation, and moreover, its FreeFEM++ code will also be described.

math.AP↗

arXiv2023

Topology optimization method with nonlinear diffusion

This paper is concerned with topology optimization based on a level set method using (doubly) nonlinear diffusion equations. Topology optimization using the level set method is called level set-based topology optimization, which is possible to determine optimal configurations that minimize objective functionals by updating level set functions. In this paper, as an update equation for level set functions, (doubly) nonlinear diffusion equations with reaction terms are derived, and then the singularity and degeneracy of the diffusion coefficient are applied to obtain fast convergence of configurations and damping oscillation on boundary structures. In particular, the reaction terms in the proposed method do not depend on the topological derivatives, and therefore, sensitivity analysis to determine a descent direction for objective functionals is relaxed. Furthermore, a numerical algorithm for the proposed method is constructed and applied to typical minimization problems to show numerical validity. This paper is a justification and generalization of the method using reaction-diffusion equations developed by one of the authors in Yamada et al. (2010).

math.OC↗

arXiv2022

Corrector results for space-time homogenization of nonlinear diffusion

The present paper concerns a space-time homogenization problem for nonlinear diffusion equations with periodically oscillating (in space and time) coefficients. Main results consist of corrector results (i.e., strong convergences of solutions with corrector terms) for gradients, diffusion fluxes and time-derivatives without assumptions for smoothness of coefficients. Proofs of the main results are based on the space-time version of the unfolding method, which is deeply concerned with the strong two-scale convergence theory.

math.AP↗

arXiv2021

Space-time homogenization problems for porous medium equations with nonnegative initial data

This paper concerns a space-time homogenization limit of nonnegative weak solutions to porous medium equations. In particular, the so-called homogenized matrix will be characterized in terms of solutions to cell problems, which drastically vary in a scaling parameter $r > 0$. A similar problem has already been studied in [1], where the growth of the power nonlinearity is strictly restricted due to some substantial obstacles. In the present paper, such obstacles will be overcome by developing local uniform estimates for the gradients of nonnegative weak solutions.

math.AP↗

arXiv2021

Space-time arithmetic quasi-periodic homogenization for damped wave equations

This paper is concerned with space-time homogenization problems for damped wave equations with spatially periodic oscillating elliptic coefficients and temporally (arithmetic) quasi-periodic oscillating viscosity coefficients. Main results consist of a homogenization theorem, qualitative properties of homogenized matrices which appear in homogenized equations and a corrector result for gradients of solutions. In particular, homogenized equations and cell problems will turn out to deeply depend on the quasi-periodicity as well as the log ratio of spatial and temporal periods of the coefficients. Even types of equations will change depending on the log ratio and quasi-periodicity. Proofs of the main results are based on a (very weak) space-time two-scale convergence theory.

math.AP↗

arXiv2020

Space-time homogenization for nonlinear diffusion

The present paper is concerned with a space-time homogenization problem for nonlinear diffusion equations with periodically oscillating (in space and time) coefficients. Main results consist of a homogenization theorem (i.e., convergence of solutions as the period of oscillation goes to zero) as well as a characterization of homogenized equations. In particular, homogenized matrices are described in terms of solutions to cell-problems, which have different forms depending on the log-ratio of the spatial and temporal periods of the coefficients. At a critical ratio, the cell problem turns out to be a parabolic equation in microscopic variables (as in linear diffusion) and also involves the limit of solutions, which is a function of macroscopic variables. The latter feature stems from the nonlinearity of the equation, and moreover, some strong interplay between microscopic and macroscopic structures can be explicitly seen for the nonlinear diffusion. As for the other ratios, the cell problems are always elliptic (in micro-variable only) and do not involve any macroscopic variables, and hence, micro- and macrostructures are weakly interacting each other. Proofs of the main results are based on the two-scale convergence theory (for space-time homogenization). Furthermore, finer asymptotics of gradients, diffusion fluxes and time-derivatives with certain corrector terms will be provided, and a qualitative analysis on homogenized matrices will be also performed.

math.AP↗

arXiv2019

The homogenization method for topology optimization of structures: old and new

These are the lecture notes of a short course on the homogenization method for topology optimization of structures, given by Grégoire Allaire, during the "GSIS International Summer School 2018" at Tohoku University (Sendai, Japan). The goal of this course is to review the necessary mathematical tools of homogenization theory and apply them to topology optimization of mechanical structures. The ultimate application, targeted in this course, is the topology optimization of structures built with lattice materials. Practical and numerical exercises are given, based on the finite element free software FreeFem++.

math.AP↗