arXiv subjects

Takayuki Yamada

Publications and source records attributed to Takayuki Yamada.

At least 19 recordsLinked to original sources

arXiv2026

Topology optimization of isotropic viscoelastic microstructures based on periodic homogenization

Mitigating low-frequency noise is particularly challenging due to its limited natural attenuation. This study aims to design viscoelastic composite microstructures that achieve both low acoustic reflection and high internal damping by simultaneously enhancing their effective acoustic impedance and attenuation characteristics. Using complex-valued periodic homogenization theory and density-based topology optimization, viscoelastic and impedance-matching materials are designed within a highly symmetric unit cell to manipulate these isotropic properties. Numerical results show that the optimized isotropic design robustly outperforms its constituent materials and simple anisotropic laminate structures, exhibiting performance that is stable across a wide frequency band, independent of orientation, and stable across various structural scales. This demonstrates the potential of microstructural engineering for effective low-frequency noise mitigation.

physics.app-ph↗

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↗

arXiv2026

A variational geometric framework for multi-objective level set topology optimization

This paper proposes a variational framework for multi-objective level set topology optimization. The approach interprets the level set function as a generalized coordinate of a fictitious material and derives its equation of motion from Hamilton's principle, resulting in a damped wave equation governing the optimization process. The objective functionals are combined using a weighted sum formulation. An analysis of the underlying system structure reveals a geometric interpretation of the problem, shifting the perspective beyond conventional approaches based on purely discrete approximations of the Pareto frontier. Under suitable regularity assumptions, the set of stationary solutions forms a structured subset in objective space, in which the Pareto frontier is locally embedded and the weighting factors act as intrinsic coordinates. This perspective motivates the introduction of a dynamic evolution of the weights, leading to a coupled dynamical system for the level set function and the weighting parameters that enables adaptive exploration of the objective landscape. Numerical results demonstrate that the proposed framework provides a stable and uniform approximation of the Pareto frontier and scales to higher-dimensional objective spaces.

math.OC↗

arXiv2026

Topology optimization concerning the mass distribution via filtered gradient flows on the Wasserstein space

In this article, we formulate topology optimization problems concerning the mass distribution as minimization problems for functionals on the Wasserstein space. We relax optimization problems regarding non-convex objective functions on the Wasserstein space by using the Neumann heat semigroup and prove the existence of minimizers of relaxed problems. Furthermore, we introduce the filtered Wasserstein gradient flow and derive the error estimate between the original Wasserstein gradient flow and the filtered one in terms of the Wasserstein distance. We also construct a candidate for the optimal mass distribution for a given fixed total mass and simultaneously obtain the shape of the material by the numerical calculation of filtered Wasserstein gradient flows.

math.OC↗

arXiv2025

Non-Bloch band theory for time-modulated discrete mechanical systems

This study establishes a non-Bloch band theory for time-modulated discrete mechanical systems. We consider simple mass-spring chains whose stiffness is periodically modulated in time. Using the temporal Floquet theory, the system is characterized by linear algebraic equations in terms of Fourier coefficients. This allows us to employ a standard linear eigenvalue analysis. Unlike non-modulated linear systems, the time modulation makes the coefficient matrix non-Hermitian, which gives rise to, for example, parametric resonance, non-reciprocal wave transmission, and non-Hermitian skin effects. In particular, we study finite-length chains consisting of spatially periodic mass-spring units and show that the standard Bloch band theory is not valid for estimating their eigenvalue distribution. To remedy this, we propose a non-Bloch band theory based on a generalized Brillouin zone. The proposed theory is verified by some numerical experiments.

physics.class-ph↗

arXiv2025

On a calculation method of the thickness via partial differential equations

This paper presents a mathematical analysis of an elliptic partial differential equation (PDE) designed to compute the geometric thickness of a given shape. The PDE-based formulation provides a direct and systematic approach to evaluate thickness through the elliptic equation, whose solution yields a vector field from which the thickness is extracted as the divergence. While the convergence of this PDE-based thickness to the geometric thickness had been rigorously justified only for simple geometries such as intervals and straight bands, its validity for more general shapes remained open. In this work, we extend the analysis to annular domains, where curvature effects are nontrivial. We prove that the PDE-based thickness converges to the geometric thickness as the diffusion parameter tends to zero by estimating the difference between two notions of thickness with the square root of the diffusion parameter. Explicit expressions involving modified Bessel functions are obtained for annuli, together with sharp inequalities for their ratios. These results provide a rigorous mathematical foundation for the PDE-based thickness and demonstrate its potential as a reliable tool in shape analysis and topology optimization.

math.AP↗

arXiv2025

On sensitivities regarding shape and topology optimization as derivatives on Wasserstein spaces

In this paper, we apply the framework of optimal transport to the formulation of optimal design problems. By considering the Wasserstein space as a set of design variables, we associate each probability measure with a shape configuration of a material in some ways. In particular, we focus on connections between differentials on the Wasserstein space and sensitivities in the standard setting of shape and topology optimization in order to regard the optimization procedure of those problems as gradient flows on the Wasserstein space.

math.OC↗

arXiv2025

PDE methods for extracting normal vector fields and distance functions of shapes

Partial differential equations can be used for extracting geometric features of shapes. This article summarizes recent methods to extract the normal vector field from an elliptic equation proposed by Yamada and from the heat equation, and also a method to extract the (signed) distance function from an elliptic equation that generalizes Varadhan's in 1967.

math.AP↗

arXiv2025

Exceptional points and defective resonances in an acoustic scattering system with sound-hard obstacles

This paper is concerned with non-Hermitian degeneracy and exceptional points associated with resonances in an acoustic scattering problem with sound-hard obstacles. The aim is to find non-Hermitian degenerate (defective) resonances using numerical methods. To this end, we characterize resonances of the scattering problem as eigenvalues of a holomorphic integral operator-valued function. This allows us to define defective resonances and associated exceptional points based on the geometric and algebraic multiplicities. Based on the theory on holomorphic Fredholm operator-valued functions, we show fractional-order sensitivity of defective resonances with respect to operator perturbation. This property is particularly important in physics and associated with intriguing phenomena, e.g., enhanced sensing and dissipation. A defective resonance is sought based on the perturbation analysis and Nyström discretization of the boundary integral equation. Numerical evidence of the existence of a defective resonance is provided. The numerical results combined with theoretical analysis provide a new insight into novel concepts in non-Hermitian physics.

math-ph↗

arXiv2025

A level set topology optimization theory based on Hamilton's principle

In this paper, we present a novel framework for deriving the evolution equation of the level set function in topology optimization, departing from conventional Hamilton-Jacobi based formulations. The key idea is the introduction of an auxiliary domain, geometrically identical to the physical design domain, occupied by fictitious matter which is dynamically excited by the conditions prevailing in the design domain. By assigning kinetic and potential energy to this matter and interpreting the level set function as the generalized coordinate to describe its deformation, the governing equation of motion is determined via Hamilton's principle, yielding a modified wave equation. Appropriate combinations of model parameters enable the recovery of classical physical behaviors, including the standard and biharmonic wave equations. The evolution problem is formulated in weak form using variational methods and implemented in the software environment FreeFEM++. The influence of the numerical parameters is analyzed on the example of minimum mean compliance. The results demonstrate that topological complexity and strut design can be effectively controlled by the respective parameters. In addition, the method allows for the nucleation of new holes and eliminates the need for re-initializing the level set function. The inclusion of a damping term further enhances numerical stability. To showcase the versatility and robustness of our method, we also apply it to compliant mechanism design and a bi-objective optimization problem involving self-weight and compliance minimization under local stress constraints.

math.OC↗

arXiv2025

Topology Optimization considering Shielding and Penetrating Features based on Fictitious Physical Model

This paper proposes topology optimization for considering shielding and penetrating features. Based on the fictitious physical model, which is a useful approach to control geometric features, the proposed method analyzes fictitious steady-state temperature fields and interprets target geometric features by examining the temperature change. First, the concept of topology optimization based on the level set method is introduced. Next, the basic idea of the fictitious physical model for considering geometric features is explained. Then, the differences between the shielding and penetrating features are clarified, and the fictitious physical model for evaluating these features is proposed. Furthermore, topology optimization for the minimum mean compliance problem with geometric conditions is formulated. Finally, 2D and 3D numerical examples are presented to validate the proposed method.

math.OC↗

arXiv2024

Mathematical analysis of a partial differential equation system on the thickness

This study focuses on linear partial differential equation (PDE) systems that arise in topology optimization where the thickness of a structure is constrained. The thickness derived from the PDE is a fictitious one, and the key challenge of this work is to verify its equivalence to the intuitive, geometrically defined thickness. The main difficulty lies in that while intuitive thickness is determined solely by the shape, the thickness defined by the PDE depends not only on the shape but also on the entire domain and the diffusion coefficients used in solving the PDE. In this paper, we demonstrate that the thickness of an infinite, straight film as a simple shape with constant thickness is equivalent within a general domain. The proof involves constructing a reference solution within a special domain and evaluating the difference using the maximum (modulus) principle and an interior $H^1$ estimate. Additionally, we provide an estimate of the dependence of thickness on the diffusion coefficient.

math.AP↗

arXiv2024

Observation of exceptional points in a spherical open elastic system

Exceptional points (EPs) are spectral singularities in non-Hermitian systems where eigenvalues and their corresponding eigenstates coalesce simultaneously. In this study, we calculate scattering poles in an open spherical solid and propose a depth-first search-based method to identify EPs. Using the proposed method, we numerically identify multiple EPs in a parameter space and confirm the simultaneous degeneracy of scattering poles through numerical experiments. The proposed method and findings enable the exploration of applications in practical three-dimension models.

physics.class-ph↗

arXiv2024

Optimal design of unimorph-type cantilevered piezoelectric energy harvesters using level set-based topology optimization by considering manufacturability

In this study, we propose a design methodology for a piezoelectric energy-harvesting device optimized for maximal power generation at a designated frequency using topology optimization. The proposed methodology is adapted to the design of a unimorph-type piezoelectric energy harvester, wherein a piezoelectric film is affixed to a singular side of a silicon cantilever beam. Both the substrate and the piezoelectric film components undergo concurrent optimization. Constraints are imposed to ensure that the resultant design is amenable to microfabrication, with specific emphasis on the etchability of piezoelectric energy harvesters. Several numerical examples are provided to validate the efficacy of the proposed method. The results show that the proposed method yields optimized substrate and piezoelectric designs with an enhanced electromechanical coupling coefficient, while allowing the eigenfrequency of the device and the minimum output voltage to be set to the desired values. Furthermore, the proposed method can provide solutions that satisfy the cross-sectional shape, substrate-dependent, and minimum output voltage constraints. The solutions obtained by the proposed method are manufacturable in the field of microfabrication.

cs.CE↗

arXiv2023

Orientation Optimization Based on Topological Derivatives in Cooperation with Multi-Material Topology Optimization Based on Extended Level Set Method

This paper provides an orientation angle optimization method for the design of fiber-reinforced composite materials using topology optimization. The orientation angle optimization is based on a topological derivative, which measures the sensitivity of an objective function with respect to a topological change of anisotropic materials. The sensitivity is incorporated into a new gradient-based optimization algorithm. This method allows us to avoid local optima and seek a global optimal solution. We provide some numerical examples and verify the effectiveness of the proposed method.

cs.CE↗

arXiv2023

Level set-based shape optimization of deformable structures for manipulating sound propagation

In this paper, we propose a level set-based shape optimization method for acoustic wave propagation problems with a deformable structure. First, we propose a mathematical model for acoustic wave propagation with a deformed structure based on coordinate transformation and the Eulerian approach. Next, we formulate the shape optimization problem and perform sensitivity analysis based on the shape derivative concept. We then construct an optimization algorithm based on the framework of a level set-based shape and topology optimization method. Finally, we provide two-dimensional optimization examples that demonstrate the effectiveness of our proposed method in providing optimized designs with desired functionality, while considering the structural deformation.

cs.CE↗

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↗