arXiv subjects

Yoshiki Jikumaru

Publications and source records attributed to Yoshiki Jikumaru.

7 recordsLinked to original sources

arXiv2026

Michell-Prager type truss structures constructed from integrable discrete power function and discrete logarithmic function

The Michell-Prager type truss structures are constructed from the integrable discrete power and logarithmic functions. It is demonstrated that specific sublattices, subject to suitable boundary conditions, yield approximate Michell trusses, which are theoretically optimal structures achieving force equilibrium with material economy. The result of shape optimization minimizing the Michell functional is provided for numerical evidence of their optimality.

math.DG↗

arXiv2025

On the governing equations of membrane O surfaces

It is known that a shell membrane in equilibrium where a constant purely normal load $q_n$ acts on the membrane, and where the principal curvature lines coincide with the principal stress lines, forms an integrable system called a membrane O surface. This paper formulates the governing equations for membrane O surfaces of the 1st and 2nd kind, which are analogues to Guichard surfaces of the 1st and 2nd kind introduced by Calapso. Furthermore, under this formulation, we show that membrane O surfaces are a subclass of Demoulin's $Ω$ surfaces, and that the Bäcklund transformation for membrane O surfaces preserves membrane O surfaces of the 1st and 2nd kind, respectively.

math.DG↗

arXiv2024

Symmetries in the linear theory of the deformation of thin shells

In this paper, we show that symmetries, which are known in the theory of integrable systems, naturally appeared in the classical linear theory of deformations of thin shells. Our result shows that if the middle surface of a shell becomes `integrable', infinitely many deformations exist that have no shear strains and twisting of the coordinate lines.

math.DG↗

arXiv2024

A derivation of first variation formulas from the strain-displacement relations in thin shell theory

In this paper, we derive the first variation formulas for surfaces in 3-dimensional Euclidean space by using the ``strain-displacement relations'' known in thin shell theory. For applications to architectural surface design, we focus on the objective function which has linear Weingarten surfaces as stationary points. This article aims to provide an elementary and cross-disciplinary exposition for applications without any tensor calculus, and thus, we do not give any new mathematical results.

math.DG↗

arXiv2021

Log-Aesthetic Curves: Similarity Geometry, Integrable Discretization and Variational Principles

In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which are used in computer aided geometric design. We consider these curves in the framework of the similarity geometry and characterize them as invariant curves under the integrable flow on plane curves which is governed by the Burgers equation. We propose a variational principle for these curves, leading to the stationary Burgers equation as the Euler-Lagrange equation. As an application of the formulation developed here, we propose a discretization of these curves and the associated variational principle which preserves the underlying integrable structure. We finally present algorithms for the generation of discrete log-aesthetic curves for given ${\rm G}^1$ data based on the similarity geometry. Our method is able to generate $S$-shaped discrete curves with an inflection as well as $C$-shaped curves according to the boundary condition. The resulting discrete curves are regarded as self-adaptive discretization and thus high-quality even with a small number of points.

nlin.SI↗

arXiv2020

Stability problem of equilibrium discrete planar curves

In this paper, we study planar polygonal curves from the variational methods. We show an unified interpretation of discrete curvatures and the Steiner-type formula by extracting the notion of the discrete curvature vector from the first variation of the length functional. Moreover, we determine the equilibrium curves for the length functional under the area-constraint condition and study their stability.

math.DG↗

arXiv2019

Non-uniqueness of closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature

An anisotropic surface energy is the integral of an energy density that depends on the normal at each point over the considered surface, and it is a generalization of surface area. The minimizer of such an energy among all closed surfaces enclosing the same volume is unique and it is (up to rescaling) so-called the Wulff shape. We prove that, unlike the isotropic case, there exists an anisotropic energy density function such that there exist closed embedded equilibrium surfaces with genus zero in the three-dimensional Euclidean space each of which is not (any homothety and translation of) the Wulff shape. We also give nontrivial self-similar shrinking solutions of anisotropic mean curvature flow. These results are generalized to hypersurfaces in the (n+1)-dimensional Euclidean space.

math.DG↗